The Planck Power – A Numerical Coincidence or a Fundamental Number in Cosmology? The Planck Power – A Numerical Coincidence or a Fundamental Number in Cosmology?

The Planck system of units has been recognized as the most fundamental such system in physics ever since Dr. Max Planck ﬁrst derived it in 1899. The Planck system of units in general, and especially the Planck power in particular, suggest a simple and interesting cosmological model. Perhaps this model may at least to some degree represent the real Universe; even if it does not, it seems interesting conceptually. The Planck power equals the Planck energy divided by the Planck time, or equivalently the Planck mass times c 2 divided by the Planck time. We show that the nongravitational mass-energy of our local region (L-region) of the Universe is, at least approximately, to within a numerical factor on the order of 2, equal to the Planck power times the elapsed cosmic time since the Big Bang. This result is shown to be consistent, to within a numerical factor on the order of 2, with results obtained via alternative derivations. We justify employing primarily L-regions within an observer’s cosmological event horizon, rather than O-regions (observable regions) within an observer’s cosmological particle horizon. Perhaps this might imply that as nongravitational mass-energy leaves the cosmological event horizon of our L-region via the Hubble ﬂow, it is replaced at the rate of the Planck power and at the expense of negative gravitational energy. Thus the total mass-energy of our L-region, and likewise of all L-regions, is conserved at the value zero. Some questions concerning the Second Law of Thermodynamics and possible thwarting of the heat death of the Universe predicted thereby, whether via Planck-power input or via some other agency, are discussed. Abstract The Planck system of units has been recognized as the most fundamental such system in physics ever since Dr. Max Planck ﬁrst derived it in 1899. The Planck system of units in general, and especially the Planck power in particular, suggest a simple and interesting cosmological model. Perhaps this model may at least to some degree represent the real Universe; even if it does not, it seems interesting conceptually. The Planck power equals the Planck energy divided by the Planck time, or equivalently the Planck mass times c 2 divided by the Planck time. We show that the nongravitational mass-energy of our local region (L-region) of the Universe is, at least approximately, to within a numerical factor on the order of 2, equal to the Planck power times the elapsed cosmic time since the Big Bang. This result is shown to be consistent, to within a numerical factor on the order of 2, with results obtained via alternative derivations. [We justify employing primarily L-regions within an observer’s cosmological event horizon, rather than O-regions (observable regions) within an observer’s cosmological particle horizon.] Perhaps this might imply that as nongravitational mass-energy leaves the cosmological event horizon of our L-region via the Hubble ﬂow, it is replaced at the rate of the Planck power and at the expense of negative gravitational energy. Thus the total mass-energy of our L-region, and likewise of all L-regions, is conserved at the value zero. Some questions concerning the Second Law of Thermodynamics and possible thwarting of the heat death of the Universe predicted thereby, whether via Planck-power input or via some other agency, are discussed. We then give a brief review of the Multiverse, and of some alternative viewpoints. regions), O-regions (observable regions), comoving frame, Second Law of Thermodynamics, heat death, Planck power versus heat death, low-entropy boundary conditions versus heat death, kinetic versus thermodynamic control, kinetic control versus heat death, minimal Boltzmann brains, extraordinary observers. favor ever-expanding alternative model of is in in alternative model of an oscillating Universe investigated in this a Big Rip is to be consistent with indeed part life is closer of one. A oscillating Λ Second Law of Thermodynamics. This kinetic as opposed to thermodynamic control could similarly obtain at the initial creation in accordance with Eq. (8) of an oscillating Universe with two-time low-entropy boundary conditions at the


Introduction
In Sect. 2 we define and distinguish between local regions (L-regions) within an observer's cosmological event horizon and observable regions (O-regions) within an observer's cosmological particle horizon, of the Universe, and justify primarily employing L-regions. In Sect. 3 we discuss the importance of the Planck system of units, which has been recognized as the most fundamental such system in physics ever since Dr. Max Planck first derived it in 1899. We then consider a possibly important role of the Planck system of units, especially of the Planck power, in cosmology. Perhaps the ensuing cosmological model may at least to some degree represent the real Universe; even if it does not, it seems interesting conceptually. The Planck power equals the Planck energy divided by the Planck time, or equivalently the Planck mass times c 2 divided by the Planck time. In Sect. 3 we show that the nongravitational mass-energy of our local region (L-region) of the Universe is, at least approximately, to within a numerical factor on the order of 2, equal to the Planck power times the elapsed cosmic time since the Big Bang. This result is shown to be consistent, to within a numerical factor on the order of 2, with results obtained via alternative derivations. We consider the possible inference that as nongravitational mass-energy leaves the cosmological event horizon of our L-region via the Hubble flow, it is replaced at the rate of the Planck power and at the expense of negative gravitational energy. The problem of consistency with astronomical and astrophysical observations is discussed in Sect. 4. In Sects. 3 and 4 we consider only nonoscillating cosmologies (except for brief parenthetical mentions of oscillating ones in the second-to-last paragraph of Sect. 4). In Sects. 5-8 we consider both nonoscillating and oscillating cosmologies. Some questions concerning the Second Law of Thermodynamics and possible thwarting of the heat death predicted thereby are discussed with respect to the Planck power in Sect. 4, with respect to cosmology in general and minimal Boltzmann brains in particular in Sect. 5, with respect inflation to in Sect. 6, and with respect to kinetic versus thermodynamic control in Sects. 4 and 7. (We discuss possible thwarting of the heat death with respect to kinetic versus thermodynamic control mainly as regards the Planck power in particular in Sect. 4 but more generally in Sect. 7.) A brief review concerning the Multiverse, and some alternative viewpoints, are given in Sect. 8.

L-regions and O-regions
In this chapter we will consider primarily local regions or L-regions of the Universe rather than observable regions or O-regions [1] thereof, although we will also consider O-regions as necessary [1]. 1 We now define and distinguish between L-regions and O-regions, and justify primarily employing L-regions, as opposed to O-regions only occasionally, as necessary [1]. Let R be the radial ruler distance or proper distance [2] to the boundary of our L-region, that is to our cosmological event horizon [3], where the Hubble flow is at c, the speed of light in vacuum; beyond this horizon it exceeds c. Thus if the Hubble constant H (τ) does not vary with cosmic time [4,5] τ and is always equal to its present value H 0 , then light emitted at the present cosmic time [4,5] τ 0 by sources beyond our cosmological event horizon [2,3] and hence beyond our L-region can never reach us. Likewise, light emitted at the present cosmic time [4,5] τ 0 by us can never reach them. Also, if the Hubble constant H (τ) does not vary with cosmic time [4,5] τ and is always equal to its present value H 0 , then our cosmological event horizon [2,3] is always at fixed ruler distance R 0 = c/H 0 away and hence our L-region

Introduction
In Sect. 2 we define and distinguish between local regions (L-regions) within an observer's cosmological event horizon and observable regions (O-regions) within an observer's cosmological particle horizon, of the Universe, and justify primarily employing L-regions. In Sect. 3 we discuss the importance of the Planck system of units, which has been recognized as the most fundamental such system in physics ever since Dr. Max Planck first derived it in 1899. We then consider a possibly important role of the Planck system of units, especially of the Planck power, in cosmology. Perhaps the ensuing cosmological model may at least to some degree represent the real Universe; even if it does not, it seems interesting conceptually. The Planck power equals the Planck energy divided by the Planck time, or equivalently the Planck mass times c 2 divided by the Planck time. In Sect. 3 we show that the nongravitational mass-energy of our local region (L-region) of the Universe is, at least approximately, to within a numerical factor on the order of 2, equal to the Planck power times the elapsed cosmic time since the Big Bang. This result is shown to be consistent, to within a numerical factor on the order of 2, with results obtained via alternative derivations. We consider the possible inference that as nongravitational mass-energy leaves the cosmological event horizon of our L-region via the Hubble flow, it is replaced at the rate of the Planck power and at the expense of negative gravitational energy. The problem of consistency with astronomical and astrophysical observations is discussed in Sect. 4. In Sects. 3 and 4 we consider only nonoscillating cosmologies (except for brief parenthetical mentions of oscillating ones in the second-to-last paragraph of Sect. 4). In Sects. 5-8 we consider both nonoscillating and oscillating cosmologies. Some questions concerning the Second Law of Thermodynamics and possible thwarting of the heat death predicted thereby are discussed with respect to the Planck power in Sect. 4, with respect to cosmology in general and minimal Boltzmann brains in particular in Sect. 5, with respect inflation to in Sect. 6, and with respect to kinetic versus thermodynamic control in Sects. 4 and 7. (We discuss possible thwarting of the heat death with respect to kinetic versus thermodynamic control mainly as regards the Planck power in particular in Sect. 4 but more generally in Sect. 7.) A brief review concerning the Multiverse, and some alternative viewpoints, are given in Sect. 8.

L-regions and O-regions
In this chapter we will consider primarily local regions or L-regions of the Universe rather than observable regions or O-regions [1] thereof, although we will also consider O-regions as necessary [1]. 1 We now define and distinguish between L-regions and O-regions, and justify primarily employing L-regions, as opposed to O-regions only occasionally, as necessary [1]. Let R be the radial ruler distance or proper distance [2] to the boundary of our L-region, that is to our cosmological event horizon [3], where the Hubble flow is at c, the speed of light in vacuum; beyond this horizon it exceeds c. Thus if the Hubble constant H (τ) does not vary with cosmic time [4,5] τ and is always equal to its present value H 0 , then light emitted at the present cosmic time [4,5] τ 0 by sources beyond our cosmological event horizon [2,3] and hence beyond our L-region can never reach us. Likewise, light emitted at the present cosmic time [4,5] τ 0 by us can never reach them. Also, if the Hubble constant H (τ) does not vary with cosmic time [4,5] τ and is always equal to its present value H 0 , then our cosmological event horizon [2,3] is always at fixed ruler distance R 0 = c/H 0 away and hence our L-region of the Universe is always of fixed size. [We denote the value of a given quantity Q today (at the present cosmic time τ 0 ) by Q 0 and its value at general cosmic time τ by Q (τ).] Light emitted at past cosmic times τ < τ 0 (but not too far in the past) by sources now beyond (but not too far beyond) our cosmological event horizon [R 0 = c/H 0 always if H (τ) = H 0 always] and hence beyond our L-region but still within our O-region [1][2][3] can reach us, because when this light was emitted these sources were still within our L-region. Likewise, light emitted in the past τ < τ 0 (but not too far in the past) by us can reach them. The boundary of our O-region of the Universe is our cosmological particle horizon [1][2][3]. The boundary of our O-region (our cosmological particle horizon) is further away than the boundary of our L-region (our cosmological particle horizon) [1][2][3]. If H (τ) = H 0 always, not only is the boundary of our O-region currently at ruler distance R 0 > R 0 = c/H 0 but R (τ) gets further away with increasing cosmic time τ [4,5], while the boundary of our L-region R (τ) always remains fixed at R 0 = c/H 0 . The fixed size of our (or any) L-region given constant H (τ) = H 0 simplifies our discussions. More importantly, all parts of our (or any) L-region are always in casual contact, while outer parts of our (or any) O-region beyond the limit of the corresponding L-region were but no longer are in causal contact. Hence we will primarily employ L-regions rather than O-regions.
Hubble flow exceeding c may seem to violate Special Relativity. But General Relativitynot Special Relativity -is applicable in cosmology [6]. Special Relativity is applicable only within local inertial frames, and any given observer is not -indeed cannot be -in the same local inertial frame as this observer's cosmological event horizon [1][2][3][4][5][6] (and even less so as this observer's cosmological particle horizon [1][2][3][4][5][6]). Thus Hubble flow exceeding c does not violate General Relativity [6]. It should also be noted that the Hubble flow is motion with space rather than through space -every object in the Hubble flow is at rest in the comoving frame [7]. An object's motion, if any, relative to the comoving frame [7] is its peculiar motion. 2 At the 27th Texas Symposium on Relativistic Astrophysics [8], values of the Hubble constant today H 0 from the upper 60s to the low 70s (km / s) / Mpc were given [8], so H 0 ≈ 70 (km / s) / Mpc splits the difference [8]. These values were essentially unchanged from those obtained shortly preceding this Symposium [9,10]. The Planck 2015 results [11] state a value of H 0 = 68 (km / s) / Mpc [11], but this Planck 2015 work [11] also cites other recent results that range from the low 60s (km / s) / Mpc to the low 70s (km / s) / Mpc. Thus the value H 0 = 68 (km / s) / Mpc [11] not only is the most reliable and most recent one as of this writing, but it also splits the difference of the range of other recent results cited in this Planck 2015 work [11]. Hence we take the Hubble constant today to be H 0 .
= 68 (km / s) / Mpc ≈ 2.2 × 10 −18 (km / s) / km = 2.2 × 10 −18 s −1 [11]. 3 2 (Re: Entry [7], Ref. [2]) An observer in the comoving frame (ideally in intergalactic space as far removed as possible from local gravitational fields such as those of galaxies, stars, etc.) sees the 2.7 K cosmic background radiation as isotropic (apart from fluctuations of fractional magnitude F ≈ 10 −5 , which can be "smoothed out" via, say, computer processing to yield a uniform background). But even Earth is a fairly good approximation to the comoving frame: Earth's peculiar motion ≈ 380 km / s c (see p. 352 of Ref. [2]) with respect to the cosmic background radiation is fairly slow, and local gravitational fields are fairly weak (vescape c).

The Planck power in cosmology
The Planck system of units has been recognized as the most fundamental such system in physics ever since Dr. Max Planck first derived it in 1899 [12][13][14][15]. It is based on Planck's reduced constanth ≡ h/2π (or Planck's original constant h), the speed of light in vacuum c, and the universal gravitational constant G, with Boltzmann's constant k usually also included. These four fundamental physical constants are seen by everything, corresponding to the Planck system of units encompassing universal domain. By contrast, for example, the fundamental electric charge is seen only by electrically-charged particles. 4 The Planck system of units in general, and especially the Planck power in particular, suggest a simple and interesting cosmological model. Perhaps this model may at least to some degree represent the real Universe; even if it does not, it seems interesting conceptually.

The Planck power in cosmology
The Planck system of units has been recognized as the most fundamental such system in physics ever since Dr. Max Planck first derived it in 1899 [12][13][14][15]. It is based on Planck's reduced constanth ≡ h/2π (or Planck's original constant h), the speed of light in vacuum c, and the universal gravitational constant G, with Boltzmann's constant k usually also included. These four fundamental physical constants are seen by everything, corresponding to the Planck system of units encompassing universal domain. By contrast, for example, the fundamental electric charge is seen only by electrically-charged particles. 4 The Planck system of units in general, and especially the Planck power in particular, suggest a simple and interesting cosmological model. Perhaps this model may at least to some degree represent the real Universe; even if it does not, it seems interesting conceptually.
Multiply the Planck mass m Planck = (hc/G) 1/2 by c 2 to obtain the Planck energy E Planck = hc 5 /G 1/2 [12][13][14][15]. Divide the Planck energy by the Planck time t Planck = hG/c 5 1/2 to obtain the Planck power P Planck = c 5 /G . = 3.64 × 10 52 W ⇐⇒ P Planck /c 2 = c 3 /G . = 4.05 × 10 35 kg / s [12][13][14][15]. [The dot-equal sign ( . =) means "very nearly equal to."] Note that -unlike the Planck length, mass, energy, time, and temperature T Planck = E Planck /k = hc 5 for the mass of our L-region (not considering the negative gravitational energy). But M 0 ≈ 1.8 × 10 53 kg is of order-of-magnitude agreement with an estimate of M 0 assuming that the mass-energy density of of our L-region of the Universe [1,3] equals the critical density ρ crit [16], as seems to be the case if not exactly then at least to within a very close approximation. The density critical density ρ crit corresponds to the borderline between ever-expanding and oscillating Universes given vanishing cosmological constant, i.e., Λ = 0, and to spacetime being flat, and hence space Euclidean, on the largest scales, i.e., to the spatial curvature index being 0 rather than +1 or −1, given any value of Λ [16-20]. 5 The critical density is Applying the most recent and best result for H 0 , namely H 0 ≈ 68 (km / s) / Mpc ≈ 2.2 × 10 −18 (km / s) / km = 2.2 × 10 −18 s −1 [11], yields as an estimate of M 0 In Eq.
(3) we assume that the volume of our L-region is given by the Euclidean value 4πR 3 0 /3. But since astronomical observations indicate that spacetime is flat, and hence space is Euclidean, on the largest scales, i.e., that the spatial curvature index is 0 rather than +1 or −1, this assumption seems justified [11,[16][17][18][19][20]. Is the order-of-magnitude agreement between Eqs. (1) and (3) merely a numerical coincidence? Or does it suggest that the Planck power plays a fundamental role in cosmology -entailing a link between the smallest (Planck-length and Planck-time) and largest (cosmological) scales?
While there is order-of magnitude agreement between Eqs. (1) and (3), there is a discrepancy between them by a factor of ≈ 2. That is, Planck-power input as per Eq. (1) seems to imply ρ ≈ 2ρ crit . Since in this era of precision cosmology all quantities in Eqs. (1)-(3) are known far more accurately than to within a factor of 2, it seems that this factor of ≈ 2 cannot simply be dismissed. But we admit that we have no explanation for this factor of ≈ 2. Furthermore, we will see that Eqs. (5)-(7) seem to imply a discrepancy with Eq. (1) by a factor of ≈ 3/2 in the opposite direction, i.e., that Planck-power input as per Eq. (1) seems to imply ρ ≈ 2ρ crit /3. Such discrepancies by numerical factors on the order of 2 may prove our Planck-power hypothesis to be wrong. At the very least they prove that even if it is right in general it is only an introductory hypothesis whose details still need to be understood. Then again, perhaps because there is consistency to within a small numerical factors of O ∼ 2, our Planck-power hypothesis may be correct in general as an introductory hypothesis, even though, even if correct in general, its details still need to be understood. Do our considerations so far in this Sect. 3 suggest that, even though the Universe certainly began with the Big Bang, there has been since the Big Bang mass-energy input, at least on the average, at the Planck power, into our L-region of the Universe? We list several alternative proposals for such input (this list probably is not exhaustive): (a) steady-state-theory mass-energy input ex nihilo [21-23], (b) mass-energy input ex nihilo via other means [24,25], (c) mass-energy input at the expense of negative gravitational energy [26][27][28][29][30][31][32] rather than ex nihilo, or (d) mass-energy input at the expense of nongravitational negative energy, for example, at the expense of the negative-energy C field in some versions of the steady-state theory [33][34][35]. If at the expense of negative gravitational energy as per proposal (c), then forever the total (mass plus gravitational) energy of our L-region, and likewise of any L-region, of the Universe, and hence of the Universe as a whole, is conserved at the value and to spacetime being flat, and hence space Euclidean, on the largest scales, i.e., to the spatial curvature index being 0 rather than +1 or −1, given any value of Λ [16-20]. 5 The critical density is Applying the most recent and best result for H 0 , namely H 0 ≈ 68 (km / s) / Mpc ≈ 2.2 × 10 −18 (km / s) / km = 2.2 × 10 −18 s −1 [11], yields as an estimate of M 0 In Eq.
(3) we assume that the volume of our L-region is given by the Euclidean value 4πR 3 0 /3. But since astronomical observations indicate that spacetime is flat, and hence space is Euclidean, on the largest scales, i.e., that the spatial curvature index is 0 rather than +1 or −1, this assumption seems justified [11,[16][17][18][19][20]. Is the order-of-magnitude agreement between Eqs. (1) and (3) merely a numerical coincidence? Or does it suggest that the Planck power plays a fundamental role in cosmology -entailing a link between the smallest (Planck-length and Planck-time) and largest (cosmological) scales?
While there is order-of magnitude agreement between Eqs. (1) and (3), there is a discrepancy between them by a factor of ≈ 2. That is, Planck-power input as per Eq. (1) seems to imply ρ ≈ 2ρ crit . Since in this era of precision cosmology all quantities in Eqs. (1)-(3) are known far more accurately than to within a factor of 2, it seems that this factor of ≈ 2 cannot simply be dismissed. But we admit that we have no explanation for this factor of ≈ 2. Furthermore, we will see that Eqs. (5)-(7) seem to imply a discrepancy with Eq. (1) by a factor of ≈ 3/2 in the opposite direction, i.e., that Planck-power input as per Eq. (1) seems to imply ρ ≈ 2ρ crit /3. Such discrepancies by numerical factors on the order of 2 may prove our Planck-power hypothesis to be wrong. At the very least they prove that even if it is right in general it is only an introductory hypothesis whose details still need to be understood. Then again, perhaps because there is consistency to within a small numerical factors of O ∼ 2, our Planck-power hypothesis may be correct in general as an introductory hypothesis, even though, even if correct in general, its details still need to be understood. Do our considerations so far in this Sect. 3 suggest that, even though the Universe certainly began with the Big Bang, there has been since the Big Bang mass-energy input, at least on the average, at the Planck power, into our L-region of the Universe? We list several alternative proposals for such input (this list probably is not exhaustive): (a) steady-state-theory mass-energy input ex nihilo [21-23], (b) mass-energy input ex nihilo via other means [24,25], (c) mass-energy input at the expense of negative gravitational energy [26][27][28][29][30][31][32] rather than ex nihilo, or (d) mass-energy input at the expense of nongravitational negative energy, for example, at the expense of the negative-energy C field in some versions of the steady-state theory [33][34][35]. If at the expense of negative gravitational energy as per proposal (c), then forever the total (mass plus gravitational) energy of our L-region, and likewise of any L-region, of the Universe, and hence of the Universe as a whole, is conserved at the value zero [26][27][28][29][30][31][32]. (There are "certain 'positivity' theorems ... which tell us that the total energy of a system, including the 'negative gravitational potential energy contributions' ..., cannot be negative [32]." But positivity theorems do seem to allow the total energy of a system, including the negative gravitational energy, to be strictly zero. Also, perhaps positivity theorems need necessarily apply only for isolated sources in asymptotically-flat spacetime.) In this chapter we will mainly presume proposal (c) from the immediately preceding list, for the following reasons: (i) Unlike proposals (a) and (b), proposal (c) entails no violation of the First Law of Thermodynamics (conservation of mass-energy). (ii) Negative gravitational energy is known to exist, unlike the negative-energy C field of proposal (d), which was perhaps introduced at least partially ad hoc to render the steady-state theory consistent with the First Law of Thermodynamics (conservation of mass-energy). Moreover, unlike gravity, the C field not only has never been observed, but also entails difficulties of its own [34,35]. (iii) We will show that proposal (c) need not be inconsistent with the observed features of the Universe.
The Universe clearly shows evolutionary rather than steady-state [21-23, [33][34][35] behavior since the Big Bang. But it could stabilize to a steady state in the future. It could already now be thus stabilizing or even thus stabilized in the very recent past with as yet no or at most very limited observational evidence that might be suggestive of such stabilization. Thus even if there is steady-state-type creation of mass-energy since the Big Bang at the rate of the Planck power (we presume, in light of the immediately preceding paragraph, most likely at the expense of the Universe's negative gravitational energy), perhaps this might be compatible with the observed evolutionary behavior of the Universe since the Big Bang. (This point and related ones will be discussed in more detail in Sect. 4.) Although General Relativity is required for an accurate consideration of the Universe's gravity, the following Newtonian approximation may be valid as an order-of-magnitude estimate [26][27][28][29][30][31][32]. Such an estimate is suggestive in favor of Planck-power input at the expense of negative gravitational energy [26][27][28][29][30][31][32], which does not require a violation of the First Law of Thermodynamics (conservation of mass-energy) [26][27][28][29][30][31][32], as opposed to Planck-power input ex nihilo [21][22][23][24][25], which would require such a violation, or via C-field input, the C field never having been observed and also entailing its own difficulties [34,35]. In accordance with the last paragraph of Sect. 2, we take the Hubble constant today to be H 0 .
= 68 (km / s) / Mpc ≈ 2.2 × 10 −18 (km / s) / km = 2.2 × 10 −18 s −1 [8][9][10][11]. Thus neglecting any variation of H (τ) with τ, τ 0 = 1/H 0 ≈ 4.5 × 10 17 s consistently with the previously given value, and the ruler radius of our L-region of the Universe is R 0 = cτ 0 = c/H 0 ≈ 1.4 × 10 23 km = 1.4 × 10 26 m. The positive mass-energy of our L-region of the Universe within our cosmological event horizon is M 0 c 2 and the negative Newtonian gravitational energy of our L-region is ≈ −GM 2 0 /R 0 . Hence in the Newtonian approximation setting the total energy equal to zero yields [26][27][28][29][30][31][32] zero [26][27][28][29][30][31][32]. (There are "certain 'positivity' theorems ... which tell us that the total energy of a system, including the 'negative gravitational potential energy contributions' ..., cannot be negative [32]." But positivity theorems do seem to allow the total energy of a system, including the negative gravitational energy, to be strictly zero. Also, perhaps positivity theorems need necessarily apply only for isolated sources in asymptotically-flat spacetime.) In this chapter we will mainly presume proposal (c) from the immediately preceding list, for the following reasons: (i) Unlike proposals (a) and (b), proposal (c) entails no violation of the First Law of Thermodynamics (conservation of mass-energy). (ii) Negative gravitational energy is known to exist, unlike the negative-energy C field of proposal (d), which was perhaps introduced at least partially ad hoc to render the steady-state theory consistent with the First Law of Thermodynamics (conservation of mass-energy). Moreover, unlike gravity, the C field not only has never been observed, but also entails difficulties of its own [34,35].
In accordance with the last paragraph of Sect. 2, we take the Hubble constant today to be H 0 ≈ 68 (km / s) / Mpc ≈ 2.2 × 10 −18 (km / s) / km = 2.2 × 10 −18 s −1 [8][9][10][11]. Thus neglecting any variation of H (τ) with τ, τ 0 = 1/H 0 ≈ 4.5 × 10 17 s consistently with the previously given value, and the ruler radius of our L-region of the Universe is R 0 = cτ 0 = c/H 0 ≈ 1.4 × 10 23 km = 1.4 × 10 26 m. The positive mass-energy of our L-region of the Universe within our cosmological event horizon is M 0 c 2 and the negative Newtonian gravitational energy of our L-region is ≈ −GM 2 0 /R 0 . Hence in the Newtonian approximation setting the total energy equal to zero yields [26][27][28][29][30][31][32]  (4) is fulfilled as closely as we can expect, especially given that our Newtonian approximation should be expected to provide only order-of-magnitude estimates, and also perhaps because (even after an initial fast inflationary stage) H (τ) may not be strictly constant.
There is yet another order-of-magnitude result that is consistent with our Planck-power hypothesis. Applying Eq. (1), rate of Planck-power mass input into our L-region is Letting ρ be the average density of our L-region, the rate of Hubble-flow mass-export from our L-region is In Eq. (6) we assume that the surface area bounding our L-region is given by the Euclidean value 4πR 2 0 . But since astronomical observations indicate that spacetime is flat, and hence space is Euclidean, on the largest scales, i.e., that the spatial curvature index is 0 rather than +1 or −1, this assumption seems justified [11,[16][17][18][19][20]. For steady-state to obtain we must have The numerical value for ρ obtained in the last line of Eq. (7) is in order-of-magnitude agreement with ρ crit as per Eq. (2), as well as in order-of-magnitude agreement with observations.
Recalling the second paragraph following that containing Eqs. (1)-(3), note that Eqs. (5)-(7) seem to imply a discrepancy with Eq. (1) by a factor of ≈ 3/2 in the opposite direction from the discrepancy with Eq. (1) by a factor of ≈ 2 implied by Eq. (3). Planck-power input as per Eq. (1) seems to imply ρ ≈ 2ρ crit , while Eqs. (5)-(7) seem to imply ρ ≈ 2ρ crit /3. Since in this era of precision cosmology all quantities in Eqs. (1)-(3) and (5)-(7) are known far more accurately than to within a factor of 2, such discrepancies by numerical factors of O ∼ 2 may prove our Planck-power hypothesis to be wrong. At the very least they prove that even if it is (4) is fulfilled as closely as we can expect, especially given that our Newtonian approximation should be expected to provide only order-of-magnitude estimates, and also perhaps because (even after an initial fast inflationary stage) H (τ) may not be strictly constant.
There is yet another order-of-magnitude result that is consistent with our Planck-power hypothesis. Applying Eq. (1), rate of Planck-power mass input into our L-region is Letting ρ be the average density of our L-region, the rate of Hubble-flow mass-export from our L-region is In Eq. (6) we assume that the surface area bounding our L-region is given by the Euclidean value 4πR 2 0 . But since astronomical observations indicate that spacetime is flat, and hence space is Euclidean, on the largest scales, i.e., that the spatial curvature index is 0 rather than +1 or −1, this assumption seems justified [11,[16][17][18][19][20]. For steady-state to obtain we must have The numerical value for ρ obtained in the last line of Eq. (7) is in order-of-magnitude agreement with ρ crit as per Eq. (2), as well as in order-of-magnitude agreement with observations.
Recalling the second paragraph following that containing Eqs. (1)-(3), note that Eqs. (5)-(7) seem to imply a discrepancy with Eq. (1) by a factor of ≈ 3/2 in the opposite direction from the discrepancy with Eq. (1) by a factor of ≈ 2 implied by Eq. (3). Planck-power input as per Eq. (1) seems to imply ρ ≈ 2ρ crit , while Eqs. (5)-(7) seem to imply ρ ≈ 2ρ crit /3. Since in this era of precision cosmology all quantities in Eqs. (1)-(3) and (5)-(7) are known far more accurately than to within a factor of 2, such discrepancies by numerical factors of O ∼ 2 may prove our Planck-power hypothesis to be wrong. At the very least they prove that even if it is right in general it is only an introductory hypothesis whose details still need to be understood. Then again, perhaps because there is consistency to within small numerical factors of O ∼ 2, our Planck-power hypothesis may be correct in general as an introductory hypothesis, even though, even if correct in general, its details still need to be understood.
While, even accepting discrepancies by a factor of O ∼ 2, the fulfillment of Eqs. (1)- (7) does not constitute proof of Planck-power input, it at least seems suggestive. Could Planck-power input, if it exists, be a classical process independent of quantum effects, if not absolutely then at least via opposing quantum effects canceling out, ash cancels out in the division P Planck = E Planck /t Planck ? Note that perhaps similar canceling out obtains with respect to the Planck speed l Planck /t Planck = c: c is the fundamental speed in the classical (nonquantum) theories of Special and General Relativity.
(The difficulty in squaring inflation with the Second Law of Thermodynamics, and a possible resolution of this difficulty, will be discussed in Sect. 6.) Observational evidence that in early 2014 initially seemed convincing for inflation in general [41,42], albeit possibly ruling out a few specific types of inflation [41,42,47], has been questioned [43][44][45][46][47][48][49][50], but not disproved [43][44][45][46][47][48][49][50]. 6 Moreover, even if an inflationary model is correct, recent observational findings disfavor simple models of inflation, such as quadratic and natural inflation [47]. Even if such one-time initial mass-energy inputs [36-40] occurred, could sustained mass-energy input then continue indefinitely such that the Planck power is at least a floor below which the average rate of mass-energy input into our L-region of the Universe cannot fall? It at least appears not to have fallen below this floor [16]. By the cosmological principle [51], if this is true of our L-region of the Universe then it must be true of any L-region thereof.
Thus our Planck-power hypothesis at least appears to entail a link between the smallest (Planck mass and Planck time) and largest (cosmological) scales, rather than being merely a numerical coincidence.

Planck power and kinetic control versus heat death: Big-Bang-initiated evolution merging into steady state?
In the simplest ever-expanding cosmologies, the Universe begins with a Big Bang and expands forever, with flat geometry on the largest scales, and with the Hubble constant H (τ) not varying with cosmic time [4,5] τ and always equal to its present value H 0 . As the Universe expands the Hubble flow carries mass-energy past the cosmological event horizon of our L-region of the Universe. But this "loss" is replaced by new positive mass-energy continually created within our L-region of the Universe forever at the rate of the Planck power and at the expense of our L-region's negative gravitational energy. This compensates right in general it is only an introductory hypothesis whose details still need to be understood. Then again, perhaps because there is consistency to within small numerical factors of O ∼ 2, our Planck-power hypothesis may be correct in general as an introductory hypothesis, even though, even if correct in general, its details still need to be understood.
While, even accepting discrepancies by a factor of O ∼ 2, the fulfillment of Eqs. (1)- (7) does not constitute proof of Planck-power input, it at least seems suggestive. Could Planck-power input, if it exists, be a classical process independent of quantum effects, if not absolutely then at least via opposing quantum effects canceling out, ash cancels out in the division P Planck = E Planck /t Planck ? Note that perhaps similar canceling out obtains with respect to the Planck speed l Planck /t Planck = c: c is the fundamental speed in the classical (nonquantum) theories of Special and General Relativity.
(The difficulty in squaring inflation with the Second Law of Thermodynamics, and a possible resolution of this difficulty, will be discussed in Sect. 6.) Observational evidence that in early 2014 initially seemed convincing for inflation in general [41,42], albeit possibly ruling out a few specific types of inflation [41,42,47], has been questioned [43][44][45][46][47][48][49][50], but not disproved [43][44][45][46][47][48][49][50]. 6 Moreover, even if an inflationary model is correct, recent observational findings disfavor simple models of inflation, such as quadratic and natural inflation [47]. Even if such one-time initial mass-energy inputs [36-40] occurred, could sustained mass-energy input then continue indefinitely such that the Planck power is at least a floor below which the average rate of mass-energy input into our L-region of the Universe cannot fall? It at least appears not to have fallen below this floor [16]. By the cosmological principle [51], if this is true of our L-region of the Universe then it must be true of any L-region thereof.
Thus our Planck-power hypothesis at least appears to entail a link between the smallest (Planck mass and Planck time) and largest (cosmological) scales, rather than being merely a numerical coincidence.

Planck power and kinetic control versus heat death: Big-Bang-initiated evolution merging into steady state?
In the simplest ever-expanding cosmologies, the Universe begins with a Big Bang and expands forever, with flat geometry on the largest scales, and with the Hubble constant H (τ) not varying with cosmic time [4,5] τ and always equal to its present value H 0 . As the Universe expands the Hubble flow carries mass-energy past the cosmological event horizon of our L-region of the Universe. But this "loss" is replaced by new positive mass-energy continually created within our L-region of the Universe forever at the rate of the Planck power and at the expense of our L-region's negative gravitational energy. This compensates for "losses" streaming past the cosmological event horizon of our L-region of the Universe via the Hubble flow -and does so consistently with the First Law of Thermodynamics (conservation of mass-energy) [26][27][28][29][30][31][32]. Because by the cosmological principle [51] our L-region is nothing special, the same is true of any L-region [3] of the Universe. Thus forever the total (mass plus gravitational) energy of our L-region, and likewise of any L-region, of the Universe, and hence of the Universe as a whole, is conserved at the value zero [26][27][28][29][30][31][32].
There is needed a mechanism whereby a sufficiently large fraction f of the Planck-power mass-energy input within the cosmological event horizon of our L-region, and of any L-region, of the Universe is produced in the form of hydrogen  [12], and in conventional units in Appendix A in the inside front cover of Ref. [14] and in Table 8.1 on p. 219 of Ref. [10].) This small value f ∼ 10 −5 is sufficient to sustain star formation forever not only at the periphery but even at the center [58] of our island Universe [1] and likewise of every other island Universe [1] in the Multiverse [52 -58]. The remainder of the Planck-power input would be in forms other than hydrogen (perhaps traces of heavier elements, elementary particles of normal and/or dark matter, dark energy, etc.?).
But perhaps the simplest mode of Planck-power input is initially in the form of the simplest possible type of dark energy, corresponding to positive constant Λ -a positive cosmological constant. Constancy of Λ is required for constancy of Planck-power input initially in the form of Λ. Positivity of Λ seems to be required for positivity of Planck-power input being initially in the form of Λ, because negative Λ corresponds to contraction of space and hence to diminution of Λ-mass-energy. Thus the simplest possible type of dark energy, corresponding to positive constant Λ -a positive cosmological constant -is perhaps the type of dark energy that is most easily reconcilable with Planck-power input, in particular with for "losses" streaming past the cosmological event horizon of our L-region of the Universe via the Hubble flow -and does so consistently with the First Law of Thermodynamics (conservation of mass-energy) [26][27][28][29][30][31][32]. Because by the cosmological principle [51] our L-region is nothing special, the same is true of any L-region [3] of the Universe. Thus forever the total (mass plus gravitational) energy of our L-region, and likewise of any L-region, of the Universe, and hence of the Universe as a whole, is conserved at the value zero [26][27][28][29][30][31][32].
There is needed a mechanism whereby a sufficiently large fraction f of the Planck-power mass-energy input within the cosmological event horizon of our L-region, and of any L-region, of the Universe is produced in the form of hydrogen  [12], and in conventional units in Appendix A in the inside front cover of Ref. [14] and in Table 8.1 on p. 219 of Ref. [10].) This small value f ∼ 10 −5 is sufficient to sustain star formation forever not only at the periphery but even at the center [58] of our island Universe [1] and likewise of every other island Universe [1] in the Multiverse [52 -58]. The remainder of the Planck-power input would be in forms other than hydrogen (perhaps traces of heavier elements, elementary particles of normal and/or dark matter, dark energy, etc.?).
But perhaps the simplest mode of Planck-power input is initially in the form of the simplest possible type of dark energy, corresponding to positive constant Λ -a positive cosmological constant. Constancy of Λ is required for constancy of Planck-power input initially in the form of Λ. Positivity of Λ seems to be required for positivity of Planck-power input being initially in the form of Λ, because negative Λ corresponds to contraction of space and hence to diminution of Λ-mass-energy. Thus the simplest possible type of dark energy, corresponding to positive constant Λ -a positive cosmological constant -is perhaps the type of dark energy that is most easily reconcilable with Planck-power input, in particular with constancy of Planck-power input. Moreover, constant Λ -a cosmological constant -is the only, unique, choice for Λ that can be put on the left-hand (geometry) side of Einstein's field equations without altering their symmetric and divergence-free form [65][66][67], "belonging to the field equations much as an additive constant belongs to an indefinite integral [65 -67]. 8 Nevertheless the current trend is to put Λ on the right-hand (mass-energy-stress) side of Einstein's field equations, which allows more freedom [66]. But if Λ is put on the right-hand side "the rationale for its uniqueness then disappears: it no longer needs to be a divergence-free 'geometric' tensor, built solely from the g µν ... the geometric view of Λ ... is undoubtedly the simplest [66]". Thus we might speculate about a link between constancy of Λ as a (positive) cosmological constant [65-67] and constancy of (positive) Planck-power input: Perhaps Planck-power input occurs initially as (positive) cosmological-constant Λ, with f ∼ 10 −5 thereof, then hopefully, somehow, via an as-yet-unknown mechanism, being transformed into hydrogen. It is important to note that -unlike equilibrium blackbody radiation -(positive) cosmological-constant-Λ dark energy seems to be at less than, indeed at far less than, maximum entropy. Thus there seems to be more than enough entropic "room" for f ∼ 10 −5 of positive-cosmological-constant-Λ dark energy to decay into hydrogen, without requiring decay all the way to iron. Positive-cosmological-constant-Λ Planck-power input thus seems to offer the benefits but not the liabilities of the steady-state theory [21-23, [33][34][35] [violation of mass-energy conservation without the C field (which has never been observed) and which also entails other difficulties with it [34,35] -recall the paragraph immediately following that containing Eq. (1)]. Positive cosmological-constant Λ also implies, or at least is consistent with, constant H (τ) = H 0 at all cosmic times τ, and hence a fixed size of our L-region, with its boundary (event horizon [2,3]) R (τ) always fixed at R 0 = c/H 0 . Thus to sum up this paragraph, the simplest model overall seems to entail (a) positive-cosmological-constant Λ, (b) Planck-power input initially as positive-cosmological-constant Λ at the expense of negative gravitational energy, with (c) f ∼ 10 −5 of Planck-power input, then hopefully, somehow, via an as-yet-unknown mechanism, being transformed into hydrogen. We note that the most reliable and most recent astronomical and astrophysical observations and measurements as of this writing are consistent with positive cosmological-constant-Λ dark energy [68,69], indeed possibly or even probably more consistent with positive cosmological-constant-Λ dark energy than with any other alternative [68,69]. But, of course, this issue is far from being definitely decided [68,69]. Even though our main point in this chapter most naturally based on positive constant Λ, in Sects. 5-7 some other possibilities for Λ will be qualitatively considered.
We cannot help but notice that temperature fluctuations in the cosmic background radiation have a typical fractional magnitude of F ≈ 10 −5 [70,71]. The observed and measured value F ≈ 10 −5 [70,71] is obviously far more certain than the speculated value f ∼ 10 −5 ; hence the distinction between the ≈ symbol as opposed to the ∼ symbol. Although it is unlikely that there is a connection between F ≈ 10 −5 [70,71] and f ∼ 10 −5 , it doesn't seem to hurt if we at least mention this numerical concurrence -just in case there might be a connection.
But the following question arises: Even if there is Planck-power input, why is not all of it in a thermodynamically-most-probable maximum-entropy form such as (iron + equilibrium constancy of Planck-power input. Moreover, constant Λ -a cosmological constant -is the only, unique, choice for Λ that can be put on the left-hand (geometry) side of Einstein's field equations without altering their symmetric and divergence-free form [65][66][67], "belonging to the field equations much as an additive constant belongs to an indefinite integral [65 -67]. 8 Nevertheless the current trend is to put Λ on the right-hand (mass-energy-stress) side of Einstein's field equations, which allows more freedom [66]. But if Λ is put on the right-hand side "the rationale for its uniqueness then disappears: it no longer needs to be a divergence-free 'geometric' tensor, built solely from the g µν ... the geometric view of Λ ... is undoubtedly the simplest [66]". Thus we might speculate about a link between constancy of Λ as a (positive) cosmological constant [65-67] and constancy of (positive) Planck-power input: Perhaps Planck-power input occurs initially as (positive) cosmological-constant Λ, with f ∼ 10 −5 thereof, then hopefully, somehow, via an as-yet-unknown mechanism, being transformed into hydrogen. It is important to note that -unlike equilibrium blackbody radiation -(positive) cosmological-constant-Λ dark energy seems to be at less than, indeed at far less than, maximum entropy. Thus there seems to be more than enough entropic "room" for f ∼ 10 −5 of positive-cosmological-constant-Λ dark energy to decay into hydrogen, without requiring decay all the way to iron. Positive-cosmological-constant-Λ Planck-power input thus seems to offer the benefits but not the liabilities of the steady-state theory [21-23, [33][34][35] [violation of mass-energy conservation without the C field (which has never been observed and which also entails other difficulties [34,35]) -recall the second paragraph following that containing Eqs.  blackbody radiation) and none of it as hydrogen -why is not f = 0 [52-57]? If this were the case then the heat death predicted by the Second Law of Thermodynamics [59-63] would not be thwarted even with Planck-power input. While we are not sure of an answer to this question, we can venture what prima facie at least seems to be a reasonable guess: (a) Planck-power input (if it exists) generates equal nonzero quantities of both positive mass-energy and negative gravitational energy starting from (zero positive energy + zero negative energy = zero total energy), and the entropy of (zero positive energy + zero negative energy = zero total energy) is perforce zero. (b) Planck-power input is a steady-state but nonequilibrium process that does not allow enough time for complete thermalization of the input from the initial value of zero entropy of (zero positive energy + zero negative energy = zero total energy) to the maximum possible positive entropy of (nonzero positive energy + nonzero negative energy = zero total energy) in a form such as (iron + equilibrium blackbody radiation). That is, Planck-power input is kinetically rather than thermodynamically controlled [72][73][74][75][76][77]. 9 Thus even though, thermodynamically, Planck-power input should be in a maximum-entropy form such as (iron + equilibrium blackbody radiation), kinetically the reaction zero positive energy + zero negative energy = zero total energy −→ nonzero positive energy + nonzero negative energy = zero total energy. (8) occurs too quickly to allow thermodynamic equilibrium = maximum entropy to be attained. Yet even Planck-power input initially as positive-cosmological-constant Λ, with a fraction f ∼ 10 −5 of Planck-power input hopefully, somehow, via an as-yet-unknown mechanism, being transformed into hydrogen, entails some entropy increase. The entropy increase ∆S that it does entail is sufficient to render the probability of its reversal as per Boltzmann's relation between entropy and probability, Prob (∆S) = exp (−∆S/k), equal to zero for all practical purposes. Thus we are justified in placing only a forward arrow (no reverse arrow) at the beginning of the second line of Eq. (8). Thus Planck-power input entails enough entropy increase to stabilize it and prevent its reversal. But it occurs quickly enough to allow kinetic control [72][73][74][75][76][77] to prevent it from entailing maximal entropy increase.
To recapitulate out considerations thus far in Sect. 4: Perhaps the simplest possible Planck-power input is initially as positive-cosmological-constant Λ. Positivity of Λ is required for positivity of Planck-power input, and constancy of Λ is required for constancy of Planck-power input.  [75]. Reference [73] does not render Ref. [72] obsolete, because Ref. [72] discusses aspects not discussed in Ref. [73], and vice versa. Likewise, Reference [77] does not render Ref. [76] obsolete, because Ref. [76] discusses aspects not discussed in Ref. [77], and vice versa.  63] would not be thwarted even with Planck-power input. While we are not sure of an answer to this question, we can venture what prima facie at least seems to be a reasonable guess: (a) Planck-power input (if it exists) generates equal nonzero quantities of both positive mass-energy and negative gravitational energy starting from (zero positive energy + zero negative energy = zero total energy), and the entropy of (zero positive energy + zero negative energy = zero total energy) is perforce zero: There is only one way for there to be nothing (Ω = 1), and hence by Boltzmann's relation between entropy and probability S = k ln Ω = k ln 1 = 0. (b) Planck-power input is a steady-state but nonequilibrium process that does not allow enough time for complete thermalization of the input from the initial value of zero entropy of (zero positive energy + zero negative energy = zero total energy) to the maximum possible positive entropy of (nonzero positive energy + nonzero negative energy = zero total energy) in a form such as (iron + equilibrium blackbody radiation). That is, Planck-power input is kinetically rather than thermodynamically controlled [72][73][74][75][76][77]. 9 Thus even though, thermodynamically, Planck-power input should be in a maximum-entropy form such as (iron + equilibrium blackbody radiation), kinetically the reaction zero positive energy + zero negative energy = zero total energy −→ nonzero positive energy + nonzero negative energy = zero total energy. (8) occurs too quickly to allow thermodynamic equilibrium = maximum entropy to be attained. Yet even Planck-power input initially as positive-cosmological-constant Λ, with a fraction f ∼ 10 −5 of Planck-power input hopefully, somehow, via an as-yet-unknown mechanism, being transformed into hydrogen, entails some entropy increase. The entropy increase ∆S that it does entail is sufficient to render the probability of its reversal as per Boltzmann's relation between entropy and probability, expressed in the form Prob (∆S) = exp (−∆S/k), equal to zero for all practical purposes. Thus we are justified in placing only a forward arrow (no reverse arrow) at the beginning of the second line of Eq. (8). Thus Planck-power input entails enough entropy increase to stabilize it and prevent its reversal. But it occurs quickly enough to allow kinetic control [72][73][74][75][76][77] to prevent it from entailing maximal entropy increase.
To recapitulate out considerations thus far in Sect. 4: Perhaps the simplest possible Planck-power input is initially as positive-cosmological-constant Λ. Positivity of Λ is required for positivity of Planck-power input, and constancy of Λ is required for constancy of Planck-power input. Constancy of Λ is requisite for Λ to be most simply encompassed within Einstein's field equations [65-67], besides correlating with constancy of Planck-power input. Positive-cosmological-constant Λ also implies, or at least is consistent with, constant (event horizon [2,3]) R (τ) always fixed at R 0 = c/H 0 . But we wish for a fraction f ∼ 10 −5 of Planck-power input hopefully, somehow, via an as-yet-unknown mechanism, being transformed into hydrogen. Hydrogen, so that stars can have fuel. But why hydrogen? Why not a thermodynamically dead form such as (iron + equilibrium blackbody radiation)? Because kinetically, it would be much more difficult for positive-cosmological-constant Λ to be transformed into a complex atom such as iron than into the simplest one -hydrogen. Thus while thermodynamic control would favor iron, if kinetic control wins then hydrogen is favored [72][73][74][75][76][77]. Note that kinetic control is vital not only in initial creation of hydrogen, but also in then preserving hydrogen long enough for it to be of use. It is owing to kinetic control that the Sun and all other main-sequence stars fuse hydrogen only to helium, not to iron, and are restrained to doing so slowly enough to give them usefully-long lifetimes. Main-sequence fusion of hydrogen to iron is thermodynamically favored, but kinetically its rate of occurrence is for all practical purposes zero. Thus kinetic control wins, limiting main-sequence fusion to helium and at a slow enough rate to give stars usefully-long lifetimes [72][73][74][75][76][77]. Indeed it is owing to kinetic control that not only hydrogen, but also all other elements except iron, do not instantaneously decay to iron. Kinetic control may also argue against positive-cosmological-constant Λ being completely transformed into equilibrium blackbody radiation (without iron). A single hydrogen atom can be created at rest with respect to the comoving frame [7]. By contrast, to conserve momentum, at least two photons must be created simultaneously, which may impose a bottleneck that diminishes the rate of such a process kinetically. Hence f ∼ 10 −5 of the Planck-power input in the positive-but-much-less-than-maximal-entropy form of hydrogen may at least prima facie seem plausible. Again it doesn't seem to hurt to at least mention the numerical concurrence between F ≈ 10 −5 [70,71] and f ∼ 10 −5 , even if any connection is unlikely.
Note that a zero value for the initial entropy for would also obtain if Planck-power input were ex nihilo [21][22][23][24][25] or at the expense of a negative-energy C field (despite its never having been observed and its other difficulties [34,35]) or other negative-energy field rather than at the expense of negative gravitational potential energy: the entropy of (zero positive energy + zero negative energy = zero total energy) would still perforce be zero. Thus our considerations of this Sect. 4, including that of dominance of kinetic over thermodynamic control [72][73][74][75][76][77], would still be applicable.
Our L-region and O-region clearly manifest evolutionary behavior, for example increasing metallicity [52-55] and a decreasing rate of star formation [52][53][54][55]. But our Planck-power hypothesis seems to suggest that its evolutionary behavior could gradually merge towards steady-state behavior. Early in the history of our L-region and O-region, star formation occurred at a much faster rate than now, and stars were on the average much more massive and hence very much faster-burning [hydrogen-burning rate ∼ (mass of star) 3 ]. Thus hydrogen was consumed faster than a conversion of f ∼ 10 −5 of Planck-power input could replace it: Stars were burning capital in addition to (Planck-power) incomeindeed more capital than income. But with decreasing rate of star formation and decreasing average stellar mass, perhaps a steady-state balance between hydrogen consumption and its replacement via f ∼ 10 −5 of Planck-power input could be approached, with stars living solely on (Planck-power) income. Merging of evolutionary towards steady-state behavior could already be beginning or could have even begun in the very recent past with as yet no or at most very limited observational evidence that might be suggestive of it. If such merging exists then both metallicity and star formation rate could stabilize in the future.
H (τ) = H 0 at all cosmic times τ, and hence a fixed size of our L-region, with its boundary (event horizon [2,3]) R (τ) always fixed at R 0 = c/H 0 . But we wish for a fraction f ∼ 10 −5 of Planck-power input hopefully, somehow, via an as-yet-unknown mechanism, being transformed into hydrogen. Hydrogen, so that stars can have fuel. But why hydrogen? Why not a thermodynamically dead form such as (iron + equilibrium blackbody radiation)? Because kinetically, it would be much more difficult for positive-cosmological-constant Λ to be transformed into a complex atom such as iron than into the simplest one -hydrogen. Thus while thermodynamic control would favor iron, if kinetic control wins then hydrogen is favored [72][73][74][75][76][77]. Note that kinetic control is vital not only in initial creation of hydrogen, but also in then preserving hydrogen long enough for it to be of use. It is owing to kinetic control that the Sun and all other main-sequence stars fuse hydrogen only to helium, not to iron, and are restrained to doing so slowly enough to give them usefully-long lifetimes. Main-sequence fusion of hydrogen to iron is thermodynamically favored, but kinetically its rate of occurrence is for all practical purposes zero. Thus kinetic control wins, limiting main-sequence fusion to helium and at a slow enough rate to give stars usefully-long lifetimes [72][73][74][75][76][77]. Indeed it is owing to kinetic control that not only hydrogen, but also all other elements except iron, do not instantaneously decay to iron. Kinetic control may also argue against positive-cosmological-constant Λ being completely transformed into equilibrium blackbody radiation (without iron). A single hydrogen atom can be created at rest with respect to the comoving frame [7]. By contrast, to conserve momentum, at least two photons must be created simultaneously, which may impose a bottleneck that diminishes the rate of such a process kinetically. Hence f ∼ 10 −5 of the Planck-power input in the positive-but-much-less-than-maximal-entropy form of hydrogen may at least prima facie seem plausible. Again it doesn't seem to hurt to at least mention the numerical concurrence between F ≈ 10 −5 [70,71] and f ∼ 10 −5 , even if any connection is unlikely.
Note that a zero value for the initial entropy as per the paragraph containing Eq. (8) would also obtain if Planck-power input were ex nihilo [21][22][23][24][25] or at the expense of a negative-energy C field (despite its never having been observed and its other difficulties [34,35]) or other negative-energy field rather than at the expense of negative gravitational potential energy: the entropy of (zero positive energy + zero negative energy = zero total energy) would still perforce be zero. Thus our considerations of this Sect. 4, including that of dominance of kinetic over thermodynamic control [72][73][74][75][76][77], would still be applicable.
Our L-region and O-region clearly manifest evolutionary behavior, for example increasing metallicity [52-55] and a decreasing rate of star formation [52][53][54][55]. But our Planck-power hypothesis seems to suggest that its evolutionary behavior could gradually merge towards steady-state behavior. Early in the history of our L-region and O-region, star formation occurred at a much faster rate than now, and stars were on the average much more massive and hence very much faster-burning [main-sequence hydrogen-burning rate ∼ (mass of star) 3 ]. Thus hydrogen was consumed faster than a conversion of f ∼ 10 −5 of Planck-power input could replace it: Stars were burning capital in addition to (Planck-power) incomeindeed more capital than income. But with decreasing rate of star formation and decreasing average stellar mass, perhaps a steady-state balance between hydrogen consumption and its replacement via f ∼ 10 −5 of Planck-power input could be approached, with stars living solely on (Planck-power) income. Merging of evolutionary towards steady-state behavior could already be beginning or could have even begun in the very recent past with as yet no or at most very limited observational evidence that might be suggestive of it. If such Perhaps they even could already now be stabilizing or have even already begun stabilizing in the very recent past with as yet no or at most very limited observational evidence that might be suggestive of such stabilization in particular, or of such merging in general. This stabilization, if it exists, would require only a small fraction f ∼ 10 −5 of Planck-power as hydrogen to maintain the current status quo in our L-region and O-region. This would allow star formation to continue forever not merely at the peripheries of island Universes, but even in their central regions [58]. We note that there is observational evidence that might at least be suggestive of "unexplained" hydrogen [78], which perhaps might qualify as such very limited suggestive observational evidence of merging towards steady-state behavior [78].
It should perhaps be re-emphasized that even Planck-power input as hydrogen entails some entropy increase and therefore is thermodynamically irreversible, consistently with the Second Law of Thermodynamics while still thwarting the heat death. The heat death is thus thwarted via dilution of entropy as an island Universe [1]  Thus with only f ∼ 10 −5 of the Planck-power input as hydrogen, the heat death predicted by the Second Law of Thermodynamics [59-63] of our L-region of our island Universe [1], and likewise of any L-region of any island Universe [1], is thwarted forever. The heat death is thwarted forever not only at the periphery but even at the center [58] of our and every other island Universe [1]. The heat death is thwarted consistently with, not in violation of, the Second Law of Thermodynamics [59][60][61][62][63]. Hubble flow export of entropy (along with mass-energy) out of our L-region of our island Universe [1], and likewise out of any L-region of any island Universe [1], as its expansion creates more volume forever, is compensated forever by creation of thermodynamically fresh but still positive-entropy mass-energy -most importantly, hopefully, the fraction f ∼ 10 −5 thereof as hydrogen -via Planck-power input.
Steady-state balance between Planck-power input and Hubble-flow expansion of space can allow both the entropy density and the nongravitational mass-energy density in our L-region of our island Universe [1], and likewise in any L-region of any island Universe [1], to remain constant, even as the total entropy and nongravitational mass-energy of the entire island Universe increase indefinitely. As mass-energy creation at the rate of the Planck power and at the expense of negative gravitational energy is matched by mass-energy dilution via an island Universe's expanding space, so is entropy production matched by entropy dilution. Thus the negative-energy gravitational field of an island Universe is an inexhaustible fuel (positive mass-energy and negative-entropy = negentropy = less-than-maximum-entropy) source. Gravity is a bank that provides an infinite line of credit and never requires repayment [79]. Planck-power input draws on this infinite line of credit [79], which never runs out -indeed which cannot run out. [Of course, if (positive) nongravitational mass-energy density remains constant, then so must (negative) gravitational energy density, if the balance of zero total energy is to be maintained. Thus Planck-power input if it exists is really equally of positive nongravitational mass-energy and negative gravitational energy simultaneously.] Additional questions bearing on the Second Law of Thermodynamics will be discussed in Sects. 5-7. In this chapter, whether concerning Planck-power input or otherwise, we limit ourselves to considerations of thwarting the heat death within the restrictions of merging exists then both metallicity and star formation rate could stabilize in the future. Perhaps they even could already now be stabilizing or have even already begun stabilizing in the very recent past with as yet no or at most very limited observational evidence that might be suggestive of such stabilization in particular, or of such merging in general. This stabilization, if it exists, would require only a small fraction f ∼ 10 −5 of Planck-power as hydrogen to maintain the current status quo in our L-region and O-region. This would allow star formation to continue forever not merely at the peripheries of island Universes, but even in their central regions [58]. We note that there is observational evidence that might at least be suggestive of "unexplained" hydrogen [78], which perhaps might qualify as such very limited suggestive observational evidence of merging towards steady-state behavior [78].
It should perhaps be re-emphasized that even Planck-power input as hydrogen entails some entropy increase and therefore is thermodynamically irreversible, consistently with the Second Law of Thermodynamics, while still thwarting the heat death. The heat death is thus thwarted via dilution of entropy as an island Universe [1]  Thus with only f ∼ 10 −5 of the Planck-power input as hydrogen, the heat death predicted by the Second Law of Thermodynamics [59-63] of our L-region of our island Universe [1], and likewise of any L-region of any island Universe [1], is thwarted forever. The heat death is thwarted forever not only at the periphery but even at the center [58] of our and every other island Universe [1]. The heat death is thwarted consistently with, not in violation of, the Second Law of Thermodynamics [59][60][61][62][63]. Hubble flow export of entropy (along with mass-energy) out of our L-region of our island Universe [1], and likewise out of any L-region of any island Universe [1], as its expansion creates more volume forever, is compensated forever by creation of thermodynamically fresh but still positive-entropy mass-energy -most importantly, hopefully, the fraction f ∼ 10 −5 thereof as hydrogen -via Planck-power input.
Steady-state balance between Planck-power input and Hubble-flow expansion of space can allow both the entropy density and the nongravitational mass-energy density in our L-region of our island Universe [1], and likewise in any L-region of any island Universe [1], to remain constant, even as the total entropy and nongravitational mass-energy of the entire island Universe increase indefinitely. As mass-energy creation at the rate of the Planck power and at the expense of negative gravitational energy is matched by mass-energy dilution via an island Universe's expanding space, so is entropy production matched by entropy dilution. Thus the negative-energy gravitational field of an island Universe is an inexhaustible fuel (positive mass-energy and negative-entropy = negentropy = less-than-maximum-entropy) source. Gravity is a bank that provides an infinite line of credit and never requires repayment [79]. Planck-power input draws on this infinite line of credit [79], which never runs out -indeed which cannot ever run out. [Of course, if (positive) nongravitational mass-energy density remains constant, then so must (negative) gravitational energy density, if the balance of zero total energy is to be maintained. Thus Planck-power input if it exists is really equally of positive nongravitational mass-energy and negative gravitational energy simultaneously.] Additional questions bearing on the Second Law of Thermodynamics will be discussed in Sects. 5-7. In this chapter, whether concerning Planck-power input or otherwise, we the Second Law. Nonetheless we note that the universal validity of the Second Law of Thermodynamics has been seriously questioned [80][81][82][83][84], albeit with the understanding that even if not universally valid at the very least it has a very wide range of validity [80][81][82][83][84].
There are two difficulties that should at least be briefly mentioned and, even if only briefly and only incompletely, also addressed.  67], and thence of hydrogen atoms (and/or other entities) newly created therefrom via Planck-power input can be instantaneously "rubber-stamped" onto our entire L-region at once, rather than being transmitted as a "signal" from one place to another within our L-region. (ii) Even if an interaction, or any other process such as "rubber-stamping," can be infinitely fast or instantaneous -and hence universally simultaneous -it can be so in only one reference frame [86]. A superluminal phenomenon, even be it only the motion of a geometric point that possesses no mass-energy and carries no information (for example the intersection point of scissors blades) [86], can be infinitely fast and hence instantaneous -universally simultaneous -in only one reference frame [86] (as a subluminal phenomenon can be infinitely slow -at rest -in only one reference frame [86]). 10 But there is a natural choice for this frame: The comoving frame [7], in which the cosmic background radiation and Hubble flow are isotropic [7], even if not an absolute rest frame, is at least a preferred rest frame [87], indeed the preferred rest frame [87], of our L-region of the Universe. If any one reference frame can claim to be preferred, it is the comoving frame [7,87]. Since by the cosmological principle [51] there is nothing special about our L-region of the Universe, the same likewise obtains in any other L-region thereof. The existence of this universal preferred frame [7,87] implies the existence of a preferred, perhaps even absolute, cosmic time τ [4, 5,87] 11 A clock in the comoving frame also measures the longest possible elapsed time ∆τ corresponding to a given decrease in the temperature of the cosmic limit ourselves to considerations of thwarting the heat death within the restrictions of the Second Law. Nonetheless we note that the universal validity of the Second Law of Thermodynamics has been seriously questioned [80][81][82][83][84], albeit with the understanding that even if not universally valid at the very least it has a very wide range of validity [80][81][82][83][84].
There are two difficulties that should at least be briefly mentioned and, even if only briefly and only incompletely, also addressed. (i) In order for negative gravitational energy to balance positive mass-energy of a hydrogen atom (or of any other entity), a hydrogen atom (or other entity) newly created via Planck-power input would have to interact gravitationally infinitely fast or instantaneously [85,86] -and hence universally simultaneously [85,86] -with our entire L-region of the Universe within our cosmological event horizon [3]. But if a signal of mass-energy and/or information is not transmitted, no violation of relativity is required [85,86].
Perhaps this may be possible if, as suggested in the third paragraph of this Sect. 4, Planck-power input occurs initially as positive-cosmological-constant Λ [65-67], with f ∼ 10 −5 thereof, then hopefully, somehow, via an as-yet-unknown mechanism, being transformed into hydrogen. Perhaps the gravitational interaction of positive-cosmological-constant Λ [65-67], and thence of hydrogen atoms (and/or other entities) newly created therefrom via Planck-power input can be instantaneously "rubber-stamped" onto our entire L-region at once, rather than being transmitted as a "signal" from one place to another within our L-region. (ii) Even if an interaction, or any other process such as "rubber-stamping," can be infinitely fast or instantaneous -and hence universally simultaneous -it can be so in only one reference frame [86]. A superluminal phenomenon, even be it only the motion of a geometric point that possesses no mass-energy and carries no information (for example the intersection point of scissors blades) [86], can be infinitely fast and hence instantaneous -universally simultaneous -in only one reference frame [86] (as a subluminal phenomenon can be infinitely slow -at rest -in only one reference frame [86]). 10 But there is a natural choice for this frame: The comoving frame [7], in which the cosmic background radiation and Hubble flow are isotropic [7], even if not an absolute rest frame, is at least a preferred rest frame [87], indeed the preferred rest frame [87], of our L-region of the Universe. If any one reference frame can claim to be preferred, it is the comoving frame [7,87]. Since by the cosmological principle [51] there is nothing special about our L-region of the Universe, the same likewise obtains in any other L-region thereof. The existence of this universal preferred frame [7,87] implies the existence of a preferred, perhaps even absolute, cosmic time τ [4, 5,87] 5,87]. A clock moving at velocity v relative to the comoving frame [7,87] measures times shorter by a ratio of 1 − v 2 /c 2 1/2 [7,87]. Thus the existence of this universal preferred frame and hence of cosmic time [4,5] weakens [87] the concept of relativity of simultaneity [85] as obtains within "the featureless vacuum of Special Relativity" [4,5,85-87]: Events, even if spatially separated, can be considered absolutely simultaneous if they occur when -with "when" having an absolute meaning -the cosmic background radiation as observed in the comoving frame has the same temperature, this temperature currently decreasing monotonically with increasing cosmic time τ since the Big Bang [4, 5,87]. 12 (Simultaneity of non-spatially-separated events is absolute even in Special Relativity [85].) Also, the contribution to the total nongravitational mass of our L-region of the Universe of a body of rest-mass [95] m is equal to m only if it is at rest in the comoving frame; if it moves at velocity v relative to the comoving frame [7,87] then its contribution is m 1 − v 2 /c 2 −1/2 [7,87]. For a zero-rest-mass particle the contribution is m = E/c 2 where E is its energy as measured in the comoving frame (for example m = E/c 2 = hν/c 2 for a photon of frequency ν as measured in the comoving frame). Thus the total nongravitational mass-energy M 0 of our L-region as per Eq. (1) is that measured with respect to the comoving frame.
It should be noted that these two difficulties (i) and (ii) discussed in the immediately preceding paragraph [85,86] 98] with no higher entropy (or entropy density if they are infinite) than that of their precursor Universe could perhaps obtain for reasons similar to Planck-power input into our L-region being at positive but less-than-maximum entropy as per our considerations in this Sect. 4 -perhaps most importantly kinetic control winning over thermodynamic control [72][73][74][75][76][77]. False-vacuum high-energy-density-scalar-field regions -the inflaton field -of the Multiverse separating island Universes [1] inflate much faster than they decay to non-inflating true-vacuum regions. Hence while inflation had a beginning once begun it is eternal [99]. Within island Universes high-cosmological-"constant" regions play essentially the same role that inflationary regions play between island Universes: they double in size much faster than their half-life against decay, so each island Universe expands forever, albeit more slowly than inflationary regions separating island Universes [1,100]. Yet the 12 (Re: Entry [87]) The phrase "the featureless vacuum of Special Relativity" is a quote from a very thoughtful and insightful letter from Dr. Wolfgang Rindler, most probably in the 1990s, in reply to a question that I raised concerning relativity of simultaneity.  [7,87]. For a zero-rest-mass particle the contribution is m = E/c 2 where E is its energy as measured in the comoving frame (for example m = E/c 2 = hν/c 2 for a photon of frequency ν as measured in the comoving frame). Thus the total nongravitational mass-energy M 0 of our L-region as per Eq. (1) is that measured with respect to the comoving frame.

The Planck power: One-time and two-time low-entropy boundary conditions, and minimal Boltzmann brains
It should be noted that these two difficulties (i) and (ii) discussed in the immediately preceding paragraph [85,86] 98] with no higher entropy (or entropy density if they are infinite) than that of their precursor Universe could perhaps obtain for reasons similar to Planck-power input into our L-region being at positive but less-than-maximum entropy as per our considerations in this Sect. 4 -perhaps most importantly kinetic control winning over thermodynamic control [72][73][74][75][76][77]. False-vacuum high-energy-density-scalar-field regions -the inflaton field -of the Multiverse separating island Universes [1] inflate much faster than they decay to non-inflating true-vacuum regions. Hence while inflation had a beginning once begun it is eternal [99]. Within island Universes high-cosmological-"constant" regions play essentially the same role that inflationary regions play between island Universes: they double in size much faster than their half-life against decay, so each island Universe expands forever, cosmological "constant" is not high everywhere in an island Universe [1]; in L-regions and O-regions such as ours regions it is sedate. As decay of the inflaton field gives birth to island Universes, within each island Universe decay of high-cosmological-constant-field regions gives birth to new sedate L-regions and O-regions such as ours. In these sedate L-regions and O-regions, the cosmological "constant" may eventually decay to negative values, resulting in a Big Crunch -and perhaps oscillatory behavior, even as entire island Universes expand forever and the spaces between them expand forever even faster. For simplicity, as noted in the first paragraph of this Sect. 5, we thus far in this chapter (except for brief parenthetical remarks in the second-to-last paragraph of Sect.  13 Shortly we will discuss Dr. Roger Penrose's central point [61,62] concerning entropy in the context of both ever-expanding and oscillatory behavior.

The Planck power: One-time and two-time low-entropy boundary conditions, and minimal Boltzmann brains
If Planck-power input is positive when our L-region expands, could it be negative if and when it contracts? Could this reduce or at least help to reduce the (nongravitational) mass-energy, and hence also entropy, during contraction, possibly to zero, by the time of the Big Crunch? If so, could a singularity at the Big Crunch thereby be evaded, thus ensuring a new thermodynamically fresh Big Bang to begin a new cycle? Moreover, since the Planck power (whether or not divided by c 2 ) does not containh, but only G and c, would or at least might this evading of a Big Crunch singularity be a classical process independent of 13  , but nevertheless seemed to favor an ever-expanding Universe. An alternative model of an oscillating Universe is discussed in the work in Ref. [63] cited in Entry [92]. In the alternative model of an oscillating Universe investigated in this work, even a Big Rip is shown to be consistent with and indeed part of an oscillating Universe's life cycle. The work in Ref. [63] cited in Entry [93] considers related issues. Also, we should mention that an oscillatory Universe is closer to Einstein's conception of cosmology than a nonoscillatory one. A closed oscillating Universe with Λ = 0, similar to that considered by Dr. Albert Einstein in the early 1930s, is discussed in the material from Ref. [12] cited in Entry [94]. albeit more slowly than inflationary regions separating island Universes [1,100]. Yet the cosmological "constant" is not high everywhere in an island Universe [1]; in L-regions and O-regions such as ours regions it is sedate. As decay of the inflaton field gives birth to island Universes, within each island Universe decay of high-cosmological-constant-field regions gives birth to new sedate L-regions and O-regions such as ours. In these sedate L-regions and O-regions, the cosmological "constant" may eventually decay to negative values, resulting in a Big Crunch -and perhaps oscillatory behavior, even as entire island Universes expand forever and the spaces between them expand forever even faster. For simplicity, as noted in the first paragraph of this Sect. 5 , but nevertheless seemed to favor an ever-expanding Universe. An alternative model of an oscillating Universe is discussed in the work in Ref. [63] cited in Entry [92]. In the alternative model of an oscillating Universe investigated in this work, even a Big Rip is shown to be consistent with and indeed part of an oscillating Universe's life cycle. The work in Ref. [63] cited in Entry [93] considers related issues. Also, we should mention that an oscillatory Universe is closer to Einstein's conception of cosmology than a nonoscillatory one. A closed oscillating Universe with Λ = 0, similar to that considered by Dr. Albert Einstein in the early 1930s, is discussed in the material from Ref. [12] cited in Entry [94]. quantum effects, if not absolutely then at least via opposing quantum effects canceling out, ash cancels out in the division P Planck = E Planck /t Planck [12][13][14][15]? Note again that perhaps similar canceling out obtains with respect to the Planck speed l Planck /t Planck = c: c is the fundamental speed in the classical (nonquantum) theories of Special and General Relativity.
But negative Planck-power input requires entropy reduction. Hence it seems to require two-time low-entropy boundary conditions [101][102][103][104][105] at the Big Bang and at the Big Crunch -although two-time, or one-time, low-entropy boundary conditions can also obtain in a "traditional" oscillating Universe without any (positive or negative) Planck In nonoscillating, ever-expanding, cosmologies, only one-time low-entropy boundary conditions can occur.) Two-time low-entropy boundary conditions require that not only the Big Bang but also the Big Crunch must be special [61,62,[101][102][103][104][105]. But even one-time low-entropy boundary conditions at the Big Bang that are required for our L-region and O-region to exist as it is currently observed are equally special [61,62]. We will not address the question of whether or not the decrease in entropy during the contracting phase of an oscillating universal cycle imposed by two-time low-entropy boundary conditions [101][102][103][104][105] should be construed as contravening the Second Law of Thermodynamics. It could perhaps be argued that, within the restrictions of the Second Law, given two-time low-entropy boundary conditions [101][102][103][104][105] there is no net decrease in entropy for an entire cycle, or that two-time low-entropy boundary conditions [101][102][103][104][105] impose such a tight constraint on an oscillating Universe's journey through phase space that there is no change in entropy from the initial and final low value during a cycle. In accordance with the third-to-last paragraph of Sect. 4, in ideas developed in this chapter per se (as opposed to brief descriptions of ideas developed in cited references) we limit ourselves to considerations of thwarting the heat death within the restrictions of the Second Law of Thermodynamics. Nonetheless we again note that the universal validity of the second law has been seriously questioned [80][81][82][83][84], albeit with the understanding that even if not universally valid at the very least it has a very wide range of validity [80][81][82][83][84].
The reduction of the (nongravitational) mass-energy of a contracting Universe to zero or at least close to zero at the Big-Crunch/Big-Bang = Big Bounce event might thus be a way, although not necessarily the only way [101][102][103][104][105], to ensure zero entropy -the entropy of nothing is perforce zero -or at least low entropy at the Big Bounce. It should be noted that a zero-or at least low-entropy state at the Big Bounce is imposed in models with two-time low-entropy boundary conditions [101][102][103][104][105].  [5], in ever-expanding cosmological models Poincaré fluctuations on the scale of galactic -or smaller, indeed, even minimal-Boltzmann-brain -dimensions in spite of the dissipation due to expansion would not be expected, because the energy of starlight and ultimately all energy would be irrevocably lost from each and every galaxy into infinitely-expanding space and (without compensating input via a Planck-power or other mechanism, which is not considered on p. 192 of Ref. [5]) never replaced. In nonoscillating, ever-expanding, cosmologies, only one-time low-entropy boundary conditions can occur.) Two-time low-entropy boundary conditions require that not only the Big Bang but also the Big Crunch must be special [61,62,[101][102][103][104][105]. But even one-time low-entropy boundary conditions at the Big Bang that are required for our L-region and O-region to exist as it is currently observed are equally special [61,62]. We will not address the question of whether or not the decrease in entropy during the contracting phase of an oscillating universal cycle imposed by two-time low-entropy boundary conditions [101][102][103][104][105] should be construed as contravening the Second Law of Thermodynamics. It could perhaps be argued that, within the restrictions of the Second Law, given two-time low-entropy boundary conditions [101][102][103][104][105] there is no net decrease in entropy for an entire cycle, or that two-time low-entropy boundary conditions [101][102][103][104][105] impose such a tight constraint on an oscillating Universe's journey through phase space that there is no change in entropy from the initial and final low value during a cycle. In accordance with the third-to-last paragraph of Sect. 4, in ideas developed in this chapter per se (as opposed to brief descriptions of ideas developed in cited references) we limit ourselves to considerations of thwarting the heat death within the restrictions of the Second Law of Thermodynamics. Nonetheless we again note that the universal validity of the Second Law has been seriously questioned [80][81][82][83][84], albeit with the understanding that even if not universally valid at the very least it has a very wide range of validity [80][81][82][83][84].
The reduction of the (nongravitational) mass-energy of a contracting Universe to zero or at least close to zero at the Big-Crunch/Big-Bang = Big Bounce event might thus be a way, although not necessarily the only way [101][102][103][104][105], to ensure zero entropy -the entropy of nothing is perforce zero [recall the paragraph containing Eq. (8) in Sect. 4] -or at least low entropy at the Big Bounce. It should be noted that a zero-or at least low-entropy state at the Big Bounce is imposed in models with two-time low-entropy boundary conditions [101][102][103][104][105]. Thus the cosmic time [4,5] interval from the Big Bang to the Big Crunch can be incomparably shorter than and is totally unrelated to the Poincaré recurrence time [108]. 14 point [61,62] concerning entropy survives unscathed. This point had been brought out previously [108][109][110], but Dr. Roger Penrose's more modern analysis [61,62] takes into consideration inflation, which was not generally recognized prior to the late 1970s [108][109][110]. (See Sect. 6 concerning the connection with inflation.) This point begins with but does not end with recognizing that the L-region and O-region of our Universe are not merely special. They are much more special than they have to be -their negentropy is much greater than is required for conscious observers to exist. By far the minimum negentropy consistent with conscious observation would be that required for the minimal existence of a single minimally-conscious observer -one and only one minimal Boltzmann brain [111][112][113][114][115][116][117][118] with no body or sense organs, and with zero information including zero sensory input even if fictitious [112] and zero memory even if fictitious [113], save only the minimal information that one exists and is conscious and even this minimal information only for most minimal fleeting split-second of conscious existence consistent with recognition that one exists and is conscious, in an otherwise maximum-entropy and therefore dead L-region and O-region of our Universe -no other observers, no Sun or other stars, no Earth or other planets, no Darwinian evolution, no nothing (at any rate no nothing worthwhile). Input of any sensory information even if fictitious [112], and/or any memory even if fictitious [113], is incompatible with the minimalness of a Boltzmann brain required by Boltzmann's exponential relation between negentropy σ ≡ S max − S and its associated probability Prob (σ) = exp (−σ/k).
[Note: Negentropy σ ≡ S max − S should not be confused with the entropy change ∆S associated with a given reaction or process introduced in the paragraph containing Eq. (8).] Even fictitious sensory input [112] or fictitious memory [113], as in a dream or in a simulated Universe, requires larger σ than none at all and hence is exponentially forbidden. Thus Boltzmann's exponential relation Prob (σ) = exp(−σ/k) allows not any Boltzmann brain but only a minimal Boltzmann brain -and only one of them. Based solely on Boltzmann's exponential relation Prob (σ) = exp(−σ/k) a lone minimal Boltzmann brain is not merely by far but exponentially by far the most probable type of observer to be and exponentially by far the most probable type of L-region and O-region of our Universe -or of any Universe in the Multiverse -to find oneself in: One should then expect not even fictitious sensory input [112], not even fictitious memory [113], but only the most fleeting split-second of conscious existence consistent with recognition that one is conscious.
But a basis solely on Boltzmann's relation Prob (σ) = exp(−σ/k) is incorrect, or at the very least incomplete. Boltzmann's relation Prob (σ) = exp(−σ/k) is valid only assuming thermodynamic equilibrium -that the ensemble of L-regions and O-regions corresponds to that at thermodynamic equilibrium. Probably the most powerful argument against this being the case is the vast disparity between our L-region and O-region that we actually observe and what one would observe as per the immediately preceding paragraph based solely on Boltzmann's relation Prob (σ) = exp(−σ/k). This disparity, the minimal-Boltzmann-brain disparity, by a factor of O ∼ 10 10 123 [61,62] [61,62] takes into consideration inflation, which was not generally recognized prior to the late 1970s [108][109][110]. (See Sect. 6 concerning the connection with inflation.) This point begins with but does not end with recognizing that the L-region and O-region of our Universe are not merely special. They are much more special than they have to be -their negentropy is much greater than is required for conscious observers to exist. By far the minimum negentropy consistent with conscious observation would be that required for the minimal existence of a single minimally-conscious observer -one and only one minimal Boltzmann brain [111][112][113][114][115][116][117][118] with no body or sense organs, and with zero information including zero sensory input even if fictitious [112] and zero memory even if fictitious [113], save only the minimal information that one exists and is conscious and even this minimal information only for most minimal fleeting split-second of conscious existence consistent with recognition that one exists and is conscious, in an otherwise maximum-entropy and therefore dead L-region and O-region of our Universe -no other observers, no Sun or other stars, no Earth or other planets, no Darwinian evolution, no nothing (at any rate no nothing worthwhile). Input of any sensory information even if fictitious [112], and/or any memory even if fictitious [113], is incompatible with the minimalness of a Boltzmann brain required by Boltzmann's exponential relation between negentropy σ ≡ S max − S and its associated probability Prob (σ) = exp (−σ/k).
[Note: Negentropy σ ≡ S max − S should not be confused with the entropy change ∆S associated with a given reaction or process introduced in the paragraph containing Eq. (8), even though Boltzmann's relation has the same exponential form for both.] Even fictitious sensory input [112] or fictitious memory [113], as in a dream or in a simulated Universe, requires larger σ than none at all and hence is exponentially forbidden. Thus Boltzmann's exponential relation Prob (σ) = exp(−σ/k) allows not any Boltzmann brain but only a minimal Boltzmann brain -and only one of them. Based solely on Boltzmann's exponential relation Prob (σ) = exp(−σ/k) a lone minimal Boltzmann brain is not merely by far but exponentially by far the most probable type of observer to be and exponentially by far the most probable type of L-region and O-region of our Universe -or of any Universe in the Multiverse -to find oneself in: One should then expect not even fictitious sensory input [112], not even fictitious memory [113], but only the most fleeting split-second of conscious existence consistent with recognition that one exists and is conscious.
But a basis solely on Boltzmann's relation Prob (σ) = exp(−σ/k) is incorrect, or at the very least incomplete. Boltzmann's relation Prob (σ) = exp(−σ/k) is valid only assuming thermodynamic equilibrium -that the ensemble of L-regions and O-regions corresponds to that at thermodynamic equilibrium. Probably the most powerful argument against this being the case is the vast disparity between our L-region and O-region that we actually observe and what one would observe as per the immediately preceding paragraph based solely on Boltzmann's relation Prob (σ) = exp(−σ/k). This disparity, the minimal-Boltzmann-brain disparity, by a factor of O ∼ 10 10 123 [61,62] may be significant, they are utterly dwarfed by the minimal-Boltzmann-brain disparity that constants. 15 In contrast to minimal Boltzmann brains, we are sometimes dubbed "ordinary observers" [115-117] -but based solely on Boltzmann's relation Prob (σ) = exp(−σ/k) dubbing us even as extraordinary observers would be a vast understatement. Indeed the same reasoning can be extended to extraordinary observers. For, based solely on Boltzmann's Prob (σ) = exp(−σ/k), exponentially by far the most probable extraordinary observer (say, a human with a typical life span) is a minimal extraordinary observer, and only one of these per L-region or O-region. While σ required for a lone minimal extraordinary observer greatly exceeds that required for a lone minimal Boltzmann brain, it is still utterly dwarfed by the actual σ of our L-region and O-region: The disparity of Prob (σ) = exp(−σ/k) between that corresponding to a lone minimal extraordinary observer and that corresponding to our observed L-region and O-region is still by a factor of O ∼ 10 10 123 [61,62]. We are privileged to be not merely minimal extraordinary observers but super-extraordinary observers -more correctly hyper-extraordinary observers -with an entire Universe to explore and enjoy [61,62].
There are many arguments against Boltzmann-brain hypotheses [111][112][113][114][115][116][117][118]. Indeed, if there exist (a) imposed one-time low-entropy boundary conditions, (b) imposed two-time low-entropy boundary conditions [16,[88][89][90][91][92][93][94][101][102][103][104][105] in an oscillating L-region and O-region, or (c) [33][34][35]) imposed low-entropy mass-energy input such as hydrogen in a nonoscillating one [78], then such imposition would preclude thermodynamic equilibrium. Indeed, given (b) or (c), thermodynamic equilibrium would not only be precluded but be precluded forever. Given (b) or (c), there would be no need to assume a decaying or finite-lived Universe [117] to help explain consistency with our observations. But even given (a) the heat death σ = 0 need not be the most probable current state of the L-region or O-region of our Universe and hence a minimal Boltzmann brain [111][112][113][114][115][116][117][118] need not be the most probable current observer therein, because at the current cosmic time decay to maximum entropy has not yet occurred. Since by the cosmological principle [51] our L-region and O-region are nothing special, this must likewise be true with respect to any L-region or O-region in our island Universe -and likewise with respect to those in any other island Universe in the Multiverse. Moreover, it has even been argued that low-entropy boundary conditions are not required to avoid minimal Boltzmann brains being exponentially by far the most probable type of observer, or even the most probable type of observer at all [116]. Also, it has been argued that special, i.e., low-entropy, conditions are not required at Big Bangs or Big Bounces [123,124]. [Clustering of matter at t = 0, which might typically be expected to increase entropy in the presence of gravity [125,126], does not do so because in this model [123,124] it is prevented owing to positive kinetic energy equaling negative gravitational energy in magnitude, so that the total energy (which in a Newtonian model excludes mass-energy) equals zero. But on pp. 3-4 of Ref. [124], friction, which generates entropy, is invoked during the time evolution of the system. Frictional damping, by degrading part of the macroscopic kinetic energy of any given pair of objects into microscopic kinetic energy (heat), facilitates their settling into a bound Keplerian-orbit state. But because friction thus generates entropy, this may correspond to a hidden, overlooked, pre-friction low-entropy assumption concerning the initial t = 0 state of this model [123,124] in either of its two directions of time [123,124]. But a Kepler pair can be formed without friction, for example via a three-body collision wherein a third body removes enough macroscopic obtains even given these requisite fundamental and effective laws of physics and physical constants. 15 In contrast to minimal Boltzmann brains, we are sometimes dubbed "ordinary observers" [115-117] -but based solely on Boltzmann's relation Prob (σ) = exp(−σ/k) dubbing us even as extraordinary observers would be a vast understatement. Indeed the same reasoning can be extended to extraordinary observers. For, based solely on Boltzmann's relation Prob (σ) = exp(−σ/k), exponentially by far the most probable extraordinary observer (say, a human with a typical life span) is a minimal extraordinary observer, and only one of these per L-region or O-region. While σ required for a lone minimal extraordinary observer greatly exceeds that required for a lone minimal Boltzmann brain, it is still utterly dwarfed by the actual σ of our L-region and O-region: The disparity of Prob (σ) = exp(−σ/k) between that corresponding to a lone minimal extraordinary observer and that corresponding to our observed L-region and O-region is still by a factor of O ∼ 10 10 123 [61,62]. We are privileged to be not merely minimal extraordinary observers but super-extraordinary observers -more correctly hyper-extraordinary observers -with an entire Universe to explore and enjoy [61,62].
There are many arguments against Boltzmann-brain hypotheses [111][112][113][114][115][116][117][118]. Indeed, if there exist (a) imposed one-time low-entropy boundary conditions, (b) imposed two-time low-entropy boundary conditions [16,[88][89][90][91][92][93][94][101][102][103][104][105] in an oscillating L-region and O-region, or (c) [33][34][35]) imposed low-entropy mass-energy input such as hydrogen in a nonoscillating one [78], then such imposition would preclude thermodynamic equilibrium. Indeed, given (b) or (c), thermodynamic equilibrium would not only be precluded but be precluded forever. Given (b) or (c), there would be no need to assume a decaying or finite-lived Universe [117] to help explain consistency with our observations. But even given (a) the heat death σ = 0 need not be the most probable current state of the L-region or O-region of our Universe and hence a minimal Boltzmann brain [111][112][113][114][115][116][117][118] need not be the most probable current observer therein, because at the current cosmic time decay to maximum entropy has not yet occurred. Since by the cosmological principle [51] our L-region and O-region are nothing special, this must likewise be true with respect to any L-region or O-region in our island Universe -and likewise with respect to those in any other island Universe in the Multiverse. Moreover, it has even been argued that low-entropy boundary conditions are not required to avoid minimal Boltzmann brains being exponentially by far the most probable type of observer, or even the most probable type of observer at all [116]. Also, it has been argued that special, i.e., low-entropy, conditions are not required at Big Bangs or Big Bounces [123,124]. [Clustering of matter at t = 0, which might typically be expected to increase entropy in the presence of gravity [125,126], does not do so because in this model [123,124] it is prevented owing to positive kinetic energy equaling negative gravitational energy in magnitude, so that the total energy (which in a Newtonian model excludes mass-energy) equals zero. But on pp. 3-4 of Ref. [124], friction, which generates entropy, is invoked during the time evolution of the system. Frictional damping, by degrading part of the macroscopic kinetic energy of any given pair of objects into microscopic kinetic energy (heat), facilitates their settling into a bound Keplerian-orbit state. But because friction thus generates entropy, this may correspond to a hidden, overlooked, pre-friction low-entropy assumption concerning the initial t = 0 state of this model [123,124] in either of its two directions of time [123,124]. But a Kepler pair can be formed without friction, for example via a three-body collision wherein a third body removes enough macroscopic kinetic energy from the other two (without degrading any into heat) that they can settle into a bound Keplerian-orbit state.] Perhaps we should also note that the fraction f ∼ 10 −5 of Planck-power input as hydrogen mentioned in Sect. 4 would maintain our L-region and O-region much farther from thermodynamic equilibrium than is required for existence of one and only one minimal-Boltzmann-brain. Thus if Planck-power input exists then f ∼ 10 −5 rather than f = 0 cannot be explained owing to our L-region being lucky: Boltzmann's exponential relation Prob (σ) = exp(−σ/k) on the one hand, and σ being a monotonically increasing function of f on the other, rules out any values of σ and f larger than the absolute minima that allow the existence of one and only one minimal-Boltzmann-brain obtaining by dumb luck. Thus if Planck-power input exists then perhaps there is an underlying principle or law of physics requiring f ∼ 10 −5 not only in our L-region but in accordance with the cosmological principle [51] in every L-region of our, and also every other, island Universe [1] in the Multiverse [52-58].

Dr. Roger Penrose's concerns: Both sides of the inflation issue
We still must consider Dr. Roger Penrose's difficulty with inflation per se, the evidence for inflation not yet being totally beyond doubt [36-50]. Dr. Penrose has shown that, as per Boltzmann's relation between entropy and probability Prob (σ) = exp(−σ/k), the probability Prob 1, per "attempt," of creation of a Universe as far from thermodynamic equilibrium as ours without inflation, while extremely small, is nevertheless enormously larger than the probability Prob 2 with inflation. That is Prob 2 ≪ Prob 1 ≪ 1. At the 27th Texas Symposium on Relativistic Astrophysics [8], I asked Dr. Penrose the following question (I have streamlined the wording for this chapter): No matter how much smaller Prob 2 is than Prob 1 (so long as Prob 2, however miniscule even compared to the already miniscule Prob 1, is finitely greater than zero), inflation has to initiate only once -after initiating once it will then overwhelm all noninflationary regions. Dr. Penrose provided a concise and insightful reply [128], and also suggested that I re-read the relevant sections of his book, "The Road to Reality [15,61,62]" I did so. Dr. Penrose's key argument seems to be centered on squaring inflation with the Second Law of Thermodynamics. Dr. Penrose's central point, already briefly discussed in Sect. 5, begins with but does not end with recognizing that our L-region and O-region are much more thermodynamically atypical -with much lower entropy -than is required for us to exist even as hyper-extraordinary observers, as opposed to only one of us as a minimal extraordinary observer, let alone only one of us as a minimal Boltzmann brain. Our L-region and O-region are thermodynamically extremely atypical not merely with respect to all possible L-regions and O-regions. They kinetic energy from the other two -without degrading any into heat -that they can settle into a bound Keplerian-orbit state.] Perhaps we should also note that the fraction f ∼ 10 −5 of Planck-power input as hydrogen mentioned in Sect. 4 would maintain our L-region and O-region much farther from thermodynamic equilibrium than is required for existence of one and only one minimal Boltzmann brain. Thus if Planck-power input exists then f ∼ 10 −5 rather than f = 0 cannot be explained owing to our L-region being lucky: Boltzmann's exponential relation Prob (σ) = exp(−σ/k) on the one hand, and σ being a monotonically increasing function of f on the other, rules out any values of σ and f larger than the absolute minima that allow the existence of one and only one minimal Boltzmann brain obtaining by dumb luck.
Thus if Planck-power input exists then perhaps there is an underlying principle or law of physics requiring f ∼ 10 −5 not only in our L-region but in accordance with the cosmological principle [51] in every L-region of our, and also every other, island Universe [1] in the Multiverse [52-58].

Dr. Roger Penrose's concerns: Both sides of the inflation issue
We still must consider Dr. Roger Penrose's difficulty with inflation per se, the evidence for inflation not yet being totally beyond doubt [36-50]. Dr. Penrose has shown that, as per Boltzmann's exponential relation between negentropy and probability Prob (σ) = exp(−σ/k), the probability Prob 1, per "attempt," of creation of a Universe as far from thermodynamic equilibrium as ours without inflation, while extremely small, is nevertheless enormously larger than the probability Prob 2 with inflation. That is Prob 2 ≪ Prob 1 ≪ 1.
At the 27th Texas Symposium on Relativistic Astrophysics [8], I asked Dr. Penrose the following question (I have streamlined the wording for this chapter): No matter how much smaller Prob 2 is than Prob 1 (so long as Prob 2, however miniscule even compared to the already miniscule Prob 1, is finitely greater than zero), inflation has to initiate only onceafter initiating once it will then overwhelm all noninflationary regions. Dr. Penrose provided a concise and insightful reply [128], and also suggested that I re-read the relevant sections of his book, "The Road to Reality [15,61,62]" I did so. Dr. Penrose's key argument seems to be centered on squaring inflation with the Second Law of Thermodynamics. Dr. Penrose's central point, already briefly discussed in Sect. 5, begins with but does not end with recognizing that our L-region and O-region are much more thermodynamically atypical -with much lower entropy -than is required for us to exist even as hyper-extraordinary observers, as opposed to only one of us as a minimal extraordinary observer, let alone only one of us as a minimal Boltzmann brain. Our L-region and O-region are thermodynamically extremely atypical not merely with respect to all possible L-regions and O-regions. They are thermodynamically extremely atypical even with respect to the extremely tiny subset of already thermodynamically extremely atypical L-regions and O-regions that allow us to exist as hyper-extraordinary observers, as opposed to only one of us as a minimal extraordinary observer, let alone only one of us as a minimal Boltzmann brain. But now the link to inflation per se: As thermodynamically untypical as our L-region and O-region are today, they become as per Boltzmann's Prob (σ) = exp(−σ/k) exponentially ever more thermodynamically untypical as one considers them backwards in time [61,62]. Thus the disparity today by a factor of O ∼ 10 10 123 between the minimal-Boltzmann-brain or even minimal-extraordinary-observer hypothesis and observation becomes exponentially ever more severe as one considers our L-region and O-region backwards in time [61,62]. Thus the connection with inflation: Since inflation smooths out temperature differences and other nonuniformities, the very existence of temperature differences and other nonuniformities prior to inflation implies lower entropy than without such nonuniformities and hence renders the thermodynamic problem of origins worse not better [61,62]. In fact exponentially worse as per Boltzmann's exponential diminution Prob (σ) = exp(−σ/k) of probability with increasing negentropy ∆S [61,62]. As thermodynamically atypical and hence exponentially improbable as our Big Bang was, it must have been thermodynamically more atypical and hence exponentially more improbable if it was inflation-mediated than if it was not. This is the basic reason for Dr. Penrose's extremely strong inequality Prob 2 ≪ Prob 1.
(We should, however, cite the remark that prior to inflation there may have been little mass-energy to thermalize [129].) Nevertheless my question still persists: In infinite time, or even in a sufficiently long finite time, even the most improbable event (so long as its probability, however miniscule, is finitely greater than zero) not merely can occur but must occur. It has been noted "that whatever physics permitted one Big Bang to occur might well permit many repetitions [130]." But suppose that Universe creations can occur via both noninflationary and inflationary physics. Even if because Prob 2 ≪ Prob 1 there first occurred an enormous but finite number N 1 of noninflationary Big Bangs yielding Universes as far from thermodynamic equilibrium as ours, so long as Prob 2, however miniscule even compared to the already miniscule Prob 1, is finitely greater than zero, after a sufficiently enormous but finite number N 1 of such noninflationary Universe creations inflation must initiate. And it need initiate only once to kick-start the inflationary Multiverse. Thereafter the inflationary Multiverse rapidly attains overwhelming dominance over the noninflationary one -with the number N 2 of inflation-mediated Big Bangs yielding Universes as far from thermodynamic equilibrium as ours henceforth overwhelming the number N 1 of noninflationary ones by an ever-increasing margin. To reiterate, no matter how much smaller Prob 2 is than Prob 1 (so long as Prob 2, however miniscule even compared to the already miniscule Prob 1, is finitely greater than zero), in infinite time, or even in a sufficiently long finite time, inflation must eventually initiate once, kick-starting the inflationary Multiverse, which henceforth becomes ever-increasingly overwhelmingly dominant over the noninflationary one. But even if inflation is eternal, it did have a beginning [99], and hence so did the inflationary Multiverse [99].
While in this Sect. 6 the focus is on thermodynamic issues concerning inflation, we note that Dr. Penrose also considers nonthermodynamic issues, specifically the flatness problem [131]. are thermodynamically extremely atypical even with respect to the extremely tiny subset of already thermodynamically extremely atypical L-regions and O-regions that allow us to exist as hyper-extraordinary observers, as opposed to only one of us as a minimal extraordinary observer, let alone only one of us as a minimal Boltzmann brain. But now the link to inflation per se: As thermodynamically untypical as our L-region and O-region are today, they become as per Boltzmann's relation Prob (σ) = exp(−σ/k) exponentially ever more thermodynamically untypical as one considers them backwards in time [61,62]. Thus the disparity today by a factor of O ∼ 10 10 123 between the minimal-Boltzmann-brain or even minimal-extraordinary-observer hypothesis and observation becomes exponentially ever more severe as one considers our L-region and O-region backwards in time [61,62]. Thus the connection with inflation: Since inflation smooths out temperature differences and other nonuniformities, the very existence of temperature differences and other nonuniformities prior to inflation implies lower entropy than without such nonuniformities and hence renders the thermodynamic problem of origins worse not better [61,62]. In fact exponentially worse as per Boltzmann's exponential diminution Prob (σ) = exp(−σ/k) of probability with increasing negentropy σ [61,62]. As thermodynamically atypical and hence exponentially improbable as our Big Bang was, it must have been thermodynamically more atypical and hence exponentially more improbable if it was inflation-mediated than if it was not. This is the basic reason for Dr. Penrose's extremely strong inequality Prob 2 ≪ Prob 1.
(We should, however, cite the remark that prior to inflation there may have been little mass-energy to thermalize [129].) Nevertheless my question still persists: In infinite time, or even in a sufficiently long finite time, even the most improbable event (so long as its probability, however miniscule, is finitely greater than zero) not merely can occur but must occur. It has been noted "that whatever physics permitted one Big Bang to occur might well permit many repetitions [130]." But suppose that Universe creations can occur via both noninflationary and inflationary physics. Even if because Prob 2 ≪ Prob 1 there first occurred an enormous but finite number N 1 of noninflationary Big Bangs yielding Universes as far from thermodynamic equilibrium as ours, so long as Prob 2, however miniscule even compared to the already miniscule Prob 1, is finitely greater than zero, after a sufficiently enormous but finite number N 1 of such noninflationary Universe creations inflation must initiate. And it need initiate only once to kick-start the inflationary Multiverse. Thereafter the inflationary Multiverse rapidly attains overwhelming dominance over the noninflationary one -with the number N 2 of inflation-mediated Big Bangs yielding Universes as far from thermodynamic equilibrium as ours henceforth overwhelming the number N 1 of noninflationary ones by an ever-increasing margin. To reiterate, no matter how much smaller Prob 2 is than Prob 1 (so long as Prob 2, however miniscule even compared to the already miniscule Prob 1, is finitely greater than zero), in infinite time, or even in a sufficiently long finite time, inflation must eventually initiate once, kick-starting the inflationary Multiverse, which henceforth becomes ever-increasingly overwhelmingly dominant over the noninflationary one. But even if inflation is eternal, it did have a beginning [99], and hence so did the inflationary Multiverse [99].
While in this Sect. 6 the focus is on thermodynamic issues concerning inflation, we note that Dr. Penrose also considers nonthermodynamic issues, specifically the smoothness and flatness problems [131].

Kinetic control versus both heat death and Boltzmann brains?
A tentative solution to the thermodynamic problem of origins, namely dominance of kinetic over thermodynamic control [72][73][74][75][76][77] has already been proposed, as a reasonable guess, for the special cases of Planck-power input throughout Sect. 4 and Everett-Universe creation in the last paragraph of Sect. 4. We would now like to consider this issue somewhat more generally.
A generalized form of this prima facie perhaps reasonable guess might include: (a) Creation in general, by whatever method, both initial via Big Bang with or without inflation, etc. [26][27][28][29][30][31], via Everett [96][97][98], and sustained via Planck-power (or other [33][34][35]) input of equal nonzero quantities of both positive mass-energy and negative gravitational (or other negative [33][34][35]) energy starting from (zero positive energy + zero negative energy = zero total energy) entails an initial entropy of zero -the entropy of (zero positive energy + zero negative energy = zero total energy) is perforce zero. (b) Creation in general, by whatever method, both initial via Big Bang with or without inflation, etc. [26][27][28][29][30][31], via Everett [96][97][98], and sustained via Planck-power (or other [33][34][35]) input of equal nonzero quantities of both positive mass-energy and negative gravitational (or other negative [33][34][35]) energy starting from (zero positive energy + zero negative energy = zero total energy) is a nonequilibrium process. These processes do not allow enough time for complete thermalization of the input from the initial value of zero entropy of (zero positive energy + zero negative energy = zero total energy) to the maximum possible positive entropy of (nonzero positive energy + nonzero negative energy = zero total energy). Thus even though, thermodynamically, exponentially the most probable creation, initial or sustained, by any method, would yield a maximum-entropy Universe with exponentially the most probable observer a minimal Boltzmann brain, kinetically the reaction zero positive energy + zero negative energy = zero total energy −→ nonzero positive energy + nonzero negative energy = zero total energy (8 (restated)) occurs too quickly to allow thermodynamic equilibrium = maximum entropy to be attained. Thus creation, initial or sustained, by whatever method, yields (nonzero positive energy + nonzero negative energy = zero total energy) at positive but far lass than maximum entropy, consistently with the Second Law of Thermodynamics but not with the heat death. Thus the basis of our proposed tentative solution to the thermodynamic problem of both initial and sustained-input origins: the reaction (rx) of Eq. (8) is kinetically rather than thermodynamically controlled [72][73][74][75][76][77]. This kinetic control does not defeat thermodynamics (specifically the Second Law of Thermodynamics) but it does defeat the heat death. Thus if the reaction of Eq. (8) is kinetically rather than thermodynamically controlled then the heat death is thwarted, but within the restrictions of the Second Law of Thermodynamics. This kinetic as opposed to thermodynamic control could similarly obtain at the initial creation in accordance with Eq. (8) of an oscillating Universe with two-time low-entropy boundary conditions at

Kinetic control versus both heat death and Boltzmann brains?
A tentative solution to the thermodynamic problem of origins, namely dominance of kinetic over thermodynamic control [72][73][74][75][76][77] has already been proposed, as a reasonable guess, for the special case of Planck-power input discussed in association with Eq. (8) in Sect. 4 and Everett-Universe creation in the last paragraph of Sect. 4. We would now like to consider this issue somewhat more generally.
A generalized form of this prima facie perhaps reasonable guess might include: (a) Creation in general, by whatever method, both initial via Big Bang with or without inflation, etc. [26][27][28][29][30][31], via Everett [96][97][98], and sustained via Planck-power (or other [33][34][35]) input of equal nonzero quantities of both positive mass-energy and negative gravitational (or other negative [33][34][35]) energy starting from (zero positive energy + zero negative energy = zero total energy) entails an initial entropy of zero -the entropy of (zero positive energy + zero negative energy = zero total energy) is perforce zero: recall the paragraph containing Eq. (8) in Sect. 4. (b) Creation in general, by whatever method, both initial via Big Bang with or without inflation, etc. [26][27][28][29][30][31], via Everett [96][97][98], and sustained via Planck-power (or other [33][34][35]) input of equal nonzero quantities of both positive mass-energy and negative gravitational (or other negative [33][34][35]) energy starting from (zero positive energy + zero negative energy = zero total energy) is a nonequilibrium process. These processes do not allow enough time for complete thermalization of the input from the initial value of zero entropy of (zero positive energy + zero negative energy = zero total energy) to the maximum possible positive entropy of (nonzero positive energy + nonzero negative energy = zero total energy). Thus even though, thermodynamically, exponentially the most probable creation, initial or sustained, by any method, would yield a maximum-entropy Universe with exponentially the most probable observer a minimal Boltzmann brain, kinetically the reaction zero positive energy + zero negative energy = zero total energy −→ nonzero positive energy + nonzero negative energy = zero total energy (8 (restated)) occurs too quickly to allow thermodynamic equilibrium = maximum entropy to be attained. Thus creation, initial or sustained, by whatever method, yields (nonzero positive energy + nonzero negative energy = zero total energy) at positive but far lass than maximum entropy, consistently with the Second Law of Thermodynamics but not with the heat death. Thus the basis of our proposed tentative solution to the thermodynamic problem of both initial and sustained-input origins: the reaction (rx) of Eq. (8) is kinetically rather than thermodynamically controlled [72][73][74][75][76][77]. This kinetic control does not defeat thermodynamics (specifically the Second Law of Thermodynamics) but it does defeat the heat death. Thus if the reaction of Eq. (8) is kinetically rather than thermodynamically controlled then the heat death is thwarted, but within the restrictions of the Second Law of Thermodynamics. This kinetic as opposed to thermodynamic control could similarly obtain at the initial creation in accordance with Eq. Let ∆S rx be the increase in entropy associated with the reaction (rx) of Eq. (8), with respect to our L-region. If 0 ∆S rx S max ∼ 10 123 k, then, on the one hand, the strong inequality 0 ∆S rx ensures an equilibrium constant K eq = exp(∆S rx /k) sufficiently large that the reverse reaction is forbidden for all practical purposes, thus stabilizing creation [72][73][74][75][76][77]. Thus the strong inequality 0 ∆S rx justifies the placement of only a forward arrow (no reverse arrow) at the beginning of the second line of Eq. (8) [72][73][74][75][76][77]. On the other hand, the strong inequality ∆S rx S max ∼ 10 123 k ensures against the doom and gloom that one would dread based solely on Boltzmann's relation Prob (∆S) = exp(−∆S/k). Note for example that even if ∆S rx = 10 120 k and hence for the reaction (rx) of Eq. (8) K eq = e 10 120 , the entropy of our L-region is still only O ∼ 10 −3 of that corresponding to thermodynamic equilibrium and hence still σ ∼ 10 123 k. [References [73][74][75][76][77] express the equilibrium constant as K eq = exp(−∆G rx /kT), where ∆G rx is the Gibbs free energy change associated with a reaction in the special case of a system maintained at constant temperature T and constant ambient pressure.
(To be precise, the ambient pressure must be maintained strictly constant during a reaction, but the temperature of the reactive system can vary in intermediate states so long as at the very least the initial and final states are at the same temperature, for this definition of ∆G rx to be valid [132][133][134][135]. 16 In this special case, |∆G rx | is the maximum work that a reaction can yield if ∆G rx < 0 and the minimum work required to enable it if ∆G rx > 0. But in this special case ∆G rx = −T∆S rx where ∆S rx is the total entropy change of the (system + surroundings). Hence K eq = exp(−∆G rx /kT) is the corresponding special case of K eq = exp(∆S rx /k). In this chapter ∆S and ∆S rx are always taken to be total entropy changes of the entire Universe or at least of our L-region thereof.] 16 (Re: Entry [132], Ref. [132]) One point: On p. 479 of Ref. [132], it is stated that in an adiabatic process all of the energy lost by a system can be converted to work, but that in a nonadiabatic process less than all of the energy lost by a system can be converted to work. But if the entropy of a system undergoing a nonadiabatic process increases, then more than all of the energy lost by this system can be converted to work, because energy extracted from the surroundings can then also contribute to the work output. In some such cases positive work output can be obtained at the expense of the surroundings even if the change in a system's energy is zero, indeed even if a system gains energy. Examples: (a) Isothermal expansion of an ideal gas is a thermodynamically spontaneous process, yielding work even though the energy change of the ideal gas is zero. (b) Evaporation of water into an unsaturated atmosphere (relative humidity less than 100%) is a thermodynamically spontaneous process, yielding work even though it costs heat, i.e., yielding work even though liquid water gains energy in becoming water vapor: see Refs. [133][134][135] concerning this point.  Let ∆S rx be the increase in entropy associated with the reaction (rx) of Eq. (8), with respect to our L-region. If 0 ∆S rx S max ∼ 10 123 k, then, on the one hand, the strong inequality 0 ∆S rx ensures an equilibrium constant K eq = exp(∆S rx /k) sufficiently large that the reverse reaction is forbidden for all practical purposes, thus stabilizing creation [72][73][74][75][76][77]. Thus the strong inequality 0 ∆S rx justifies the placement of only a forward arrow (no reverse arrow) at the beginning of the second line of Eq. (8) [72][73][74][75][76][77]. On the other hand, the strong inequality ∆S rx S max ∼ 10 123 k ensures against the doom and gloom that one would dread based solely on Boltzmann's relations Prob (∆S) = exp(−∆S/k) and Prob (σ) = exp(−σ/k). Note for example that even if ∆S rx = 10 120 k and hence for the reaction (rx) of Eq. (8) K eq = e 10 120 , the entropy of our L-region is still only O ∼ 10 −3 of that corresponding to thermodynamic equilibrium and hence still σ ∼ 10 123 k − 10 120 k , which is for all practical purposes still σ ∼ 10 123 k. [References [73][74][75][76][77] express the equilibrium constant as K eq = exp(−∆G rx /kT), where ∆G rx is the Gibbs free energy change per molecular reaction in the special case of a system maintained at constant temperature T and constant ambient pressure. (To be precise, the ambient pressure must be maintained strictly constant during a reaction, but the temperature of the reactive system can vary in intermediate states so long as at the very least the initial and final states are at the same temperature, for this definition of ∆G rx to be valid [132][133][134][135]. 16 ) In this special case, |∆G rx | is the maximum work obtainable per molecular reaction if ∆G rx < 0 and the minimum work required to enable it if ∆G rx > 0. But in this special case ∆G rx = −T∆S rx where ∆S rx is the total entropy change of the (system + surroundings) per molecular reaction. Hence K eq = exp(−∆G rx /kT) is the corresponding special case of K eq = exp(∆S rx /k). In this chapter ∆S and ∆S rx are always taken to be total entropy changes of the entire Universe or at least of our L-region thereof.] 16 (Re: Entry [132], Ref. [132]) One point: On p. 479 of Ref. [132], it is stated that in an adiabatic process all of the energy lost by a system can be converted to work, but that in a nonadiabatic process less than all of the energy lost by a system can be converted to work. But if the entropy of a system undergoing a nonadiabatic process increases, then more than all of the energy lost by this system can be converted to work, because energy extracted from the surroundings can then also contribute to the work output. In some such cases positive work output can be obtained at the expense of the surroundings even if the change in a system's energy is zero, indeed even if a system gains energy. Examples: (a) Isothermal expansion of an ideal gas is a thermodynamically spontaneous process, yielding work even though the energy change of the ideal gas is zero. (b) Evaporation of water into an unsaturated atmosphere (relative humidity less than 100%) is a thermodynamically spontaneous process, yielding work even though it costs heat, i.e., yielding work even though liquid water gains energy in becoming water vapor: see Refs. [133][134][135] concerning this point.
As a brief aside, we note that many chemical reactions are similarly kinetically rather than thermodynamically controlled [72][73][74][75][76][77], in like manner as Eq. (8). While only chemical reactions are discussed in Refs. [72][73][74][75][76][77], the same principle likewise applies with respect to all kinetically rather than thermodynamically controlled processes, for example kinetically rather than thermodynamically controlled physical and nuclear reactions. As we discussed in Sect. 4 if nuclear reactions were thermodynamically rather than kinetically controlled then there would be nothing but (iron + equilibrium blackbody radiation) -an iron-dead Universe.  [72][73][74][75][76][77] seems to be at least a reasonable tentative explanation of why we are privileged to be not merely minimal extraordinary observers but super-extraordinary observers -more correctly hyper-extraordinary observers -with an entire Universe to explore and enjoy [61,62]. By the cosmological principle [51] we may hope that this is true everywhere in the Multiverse.

A brief review concerning the Multiverse, and some alternative viewpoints
As a brief aside, we note that many chemical reactions are similarly kinetically rather than thermodynamically controlled [72][73][74][75][76][77], in like manner as Eq. (8). While only chemical reactions are discussed in Refs. [72][73][74][75][76][77], the same principle likewise applies with respect to all kinetically rather than thermodynamically controlled processes, for example kinetically rather than thermodynamically controlled physical and nuclear reactions. As we discussed in Sect. 4 if nuclear reactions were thermodynamically rather than kinetically controlled then there would be nothing but (iron + equilibrium blackbody radiation) -an iron-dead Universe. We should note that if conscious observers, also referred to as self-aware substructures (SASs) [143][144][145], are not merely self-aware but also have free will, then they have at least some degree of choice concerning creation of Level III Universes: They then have at least some freedom to choose whether or not to make a given observation or measurement, which observations and measurements to make, and when to make them. Even if the Everett interpretation [96-98] of quantum mechanics is incorrect [146] and Level III Universes exist only in potentiality until one and only one of them is actualized [146], say via wave-function collapse [147], then an SAS with free will still has this degree of choice.

A brief review concerning the Multiverse, and some alternative viewpoints
Even if the probabilities of the possible outcomes of any given observation or measurement cannot be altered, the set of possible outcomes on offer to Nature depends on which observations and measurements are chosen by an SAS with free will, and when they are on offer depends on when an SAS with free will chooses to observe or measure. Thus irrespective of the character of Level III Universes, if free will exists then there is this qualitative difference between unchosen observations and measurements made by Nature herself, say via decoherence [148,149], and chosen ones made by an SAS with free will. Moreover, "decoherence" is perhaps too strong a term; "delocalization of coherence" seems more correct. Since quantum-mechanical information in general cannot be destroyed, quantum-mechanical coherence in particular is never really destroyed, merely delocalized. As with any delocalization process there is an accompanying increase in entropy. But within a system of finite volume this increase in entropy is limited to a finite maximum value, implying recoherence, or more correctly relocalization of coherence, after a Poincaré recurrence time [108,150,151]. Of course, typical Poincaré recurrence times [108,150,151] of all but very small systems are inconceivably long, but in a very small system at least partial recoherence, or more correctly relocalization of coherence, may occur in a reasonable time. We should note that even before the term "decoherence" had been coined, some aspects of decoherence, or more correctly delocalization of coherence, had been partially anticipated [152,153]. For general reviews concerning the quantum-mechanical measurement problem see, for example, Refs. [149] and [152][153][154][155] [156]. Alex Vilenkin writes: "Beautiful mathematics combines simplicity with depth [156]." (But also that "simplicity" and "depth" are almost as difficult to define as "beauty [156].") But Dr. Alex Vilenkin also writes: "Mathematical beauty may be useful as a guide, but it is hard to imagine that it would suffice to select a unique theory out of We should note that if conscious observers, also referred to as self-aware substructures (SASs) [143][144][145], are not merely self-aware but also have free will, then they have at least some degree of choice concerning creation of Level III Universes: They then have at least some freedom to choose whether or not to make a given observation or measurement, which observations and measurements to make, and when to make them. Even if the Everett interpretation [96-98] of quantum mechanics is incorrect [146] and Level III Universes exist only in potentiality until one and only one of them is actualized [146], say via wave-function collapse [147], then an SAS with free will still has this degree of choice.
Even if the probabilities of the possible outcomes of any given observation or measurement cannot be altered, the set of possible outcomes on offer to Nature depends on which observations and measurements are chosen by an SAS with free will, and when they are on offer depends on when an SAS with free will chooses to observe or measure. Thus irrespective of the character of Level III Universes, if free will exists then there is this qualitative difference between unchosen observations and measurements made by Nature herself, say via decoherence [148,149], and chosen ones made by an SAS with free will. Furthermore, a choice made by an SAS with free will seems to be an initial condition on the future history of the Universe, or on the future history of the Level III Multiverse of quantum branches given the Everett scenario [96][97][98]. The question then arises of compatibility with the Mathematical Universe Hypothesis (MUH), according to which initial conditions cannot exist [150,151]. But both the very notion of choice [152] and exhortations to "Let's make a difference!" [153] seem incompatible with denial of free will. Moreover, "decoherence" is perhaps too strong a term; "delocalization of coherence" seems more correct. Since quantum-mechanical information in general cannot be destroyed, quantum-mechanical coherence in particular is never really destroyed, merely delocalized. As with any delocalization process there is an accompanying increase in entropy. But within a system of finite volume this increase in entropy is limited to a finite maximum value, implying recoherence, or more correctly relocalization of coherence, after a Poincaré recurrence time [108,154,155]. Of course, typical Poincaré recurrence times [108,150,151] of all but very small systems are inconceivably long, but in a very small system at least partial recoherence, or more correctly relocalization of coherence, may occur in a reasonable time. We should mention that even before the term "decoherence" had been coined, some aspects of decoherence, or more correctly delocalization of coherence, had been partially anticipated [156,157]. For general reviews concerning the quantum-mechanical measurement problem see, for example, Refs. [149] and [156][157][158][159] [99]. The laws of quantum mechanics -our laws of quantum mechanicsgoverned the initial tunneling event that created not merely our Universe but the Multiverse, at least through Level II [99,136]. Thus these laws, on whatever tablets they are written, must have existed before, and must exist independently of, the Multiverse at least through Level II [99,136] -not merely of our island Universe [99]. Concerning Level III, it seems that Levels I+II, or at the very least Level I, must exist first, because Levels I+II, or at the very least Level I, seems prerequisite for the existence of entities capable of executing Dr. Hugh Everett's program [96][97][98]. But might the prerequisites for a beginning and for the pre-existence of our laws of quantum mechanics be general, operative even at Level IV [136]? But if so then might Level IV -but not Levels I, II, and III -be more restricted than has been suggested [136]? For then might our laws of quantum mechanics be part of the one maximally-beautiful mathematical structure that maximally entails both simplicity and depth our fundamental (not merely effective) laws of physics [136] -after all? Then perhaps this one maximally beautiful mathematical structure, this maximal possible entailment of both simplicity and depth, is the only one realized via physically-existing Universes. But if this is the case then the question arises: Why does this one maximally beautiful mathematical structure permit life [159] (at the very least, carbon-based life as we know it on Earth)?
We must admit that in this chapter we have not even scratched the surface, as per this paragraph and the two immediately following ones. There are many alternative viewpoints concerning the Multiverse and related issues. We should at least mention a few of them that we have not mentioned until now. According to at least one of these viewpoints, inflation is eternal into the past as well as into the future, and hence has no beginning as well as no end [160][161][162]. But perhaps this is compatible with inflation having a beginning if regions of inflation in the forward and backward time directions are disjoint and incapable of any interaction with each other [163]. Then perhaps observers in both types of regions would consider their home region to be evolving forward, not backward, in time. According to other viewpoints, inflation not only has a beginning but also has an end -eternal inflation is impossible [164,165]. According to one of these viewpoints, the end of inflation is imposed by the increasingly fractal nature of spacetime [164,165]. We also note that Dr. Roger Penrose considered another difficulty associated with possible fractal nature of spacetime: inflation does not solve the smoothness and flatness problems if the structure of spacetime is fractal, , are equal: some are more beautiful and hence more equal than others [160]. Alex Vilenkin writes: "Beautiful mathematics combines simplicity with depth [160]." (But also that "simplicity" and "depth" are almost as difficult to define as "beauty [160].") But Dr. Alex Vilenkin also writes: "Mathematical beauty may be useful as a guide, but it is hard to imagine that it would suffice to select a unique theory out of the infinite number of possibilities [161]." These points are also considered by Dr. Roger Penrose [162]. Yet even so mathematical beauty should have at least some selective power. A case in point: Newton's laws have both simplicity and depth, and hence are beautiful. But Einstein's laws have both greater simplicity and greater depth, and hence are more beautiful.
The laws of motion have the same form in all reference frames in General Relativity but not in Newton's theory (for example, Newton's theory requires extra terms for centrifugal and Coriolis forces in rotating reference frames), thus General Relativity has greater simplicity; additionally, Newton's theory is a limiting case of Einstein's but not vice versa, thus General Relativity also has greater depth. Hence might a Universe wherein Newton's laws are the fundamental laws, not merely a limiting case of relativity and quantum mechanics, be denied physical existence in a Level IV Multiverse -because even though it is a beautiful mathematical structure, it is not the maximally-beautiful one that maximally entails both simplicity and depth? While (even neglecting quantum mechanics) we cannot be sure if General Relativity is the maximally-beautiful mathematical structure, we can be sure that Newtonian theory, while beautiful, is not maximally beautiful. Moreover, while the Multiverse is eternal, it nonetheless, at least below Level IV [136], did have a beginning [99]. The laws of quantum mechanics -our laws of quantum mechanics -governed the initial tunneling event that created not merely our Universe but the Multiverse, at least through Level II [99,136]. Thus these laws, on whatever tablets they are written, must have existed before, and must exist independently of, the Multiverse at least through Level II [99,136] not merely of our island Universe [99]. Concerning Level III, it seems that Levels I+II, or at the very least Level I, must exist first, because Levels I+II, or at the very least Level I, seems prerequisite for the existence of entities capable of executing Dr. Hugh Everett's program [96][97][98]. But might the prerequisites for a beginning and for the pre-existence of our laws of quantum mechanics be general, operative even at Level IV [136]? But if so then might Level IV -but not Levels I, II, and III -be more restricted than has been suggested [136]? For then might our laws of quantum mechanics be part of the one maximally-beautiful mathematical structure that maximally entails both simplicity and depth -our fundamental (not merely effective) laws of physics [136] -after all? Then perhaps this one maximally beautiful mathematical structure, this maximal possible entailment of both simplicity and depth, is the only one realized via physically-existing Universes. But if this is the case then the question arises: Why does this one maximally beautiful mathematical structure permit life [163] (at the very least, carbon-based life as we know it on Earth)?
We must admit that in this chapter we have not even scratched the surface, as per this paragraph and the two immediately following ones. There are many alternative viewpoints concerning the Multiverse and related issues. We should at least mention a few of them that we have not mentioned until now. According to at least one of these viewpoints, inflation is eternal into the past as well as into the future, and hence has no beginning as well as no end [164][165][166]. But perhaps this is compatible with inflation having a beginning if regions of inflation in the forward and backward time directions are disjoint and incapable of any interaction with each other [167]. Then perhaps observers in both types of regions would consider their home region to be evolving forward, not backward, in time. According to or worse than fractal [166]. According to another of these viewpoints, the end of inflation is imposed by the Big Snap, according to which expansion of space will eventually dilute the number of degrees of freedom per any unit volume, and specifically per Hubble volume, to less than one, although the Universe will probably be in trouble well before the number of degrees of freedom per Hubble volume is reduced to one [167,168]. But perhaps new degrees of freedom can be created to compensate [167,168]. Perhaps Planck-power input, if it exists, can, because it replenishes mass-energy, also replenish degrees of freedom -thereby precluding the Big Snap. In Dr. Max Tegmark's rubber-band analogy, this corresponds to new molecules of rubber being created as the rubber band stretches, thereby keeping the density of rubber constant [167,168]. But, with or without a Big Snap [167,168], if inflation does have an end for any reason whatsoever, then my question to Dr. Roger Penrose in Sect. 6 is answered negatively.
There are also many proposed solutions to the entropy problem (why there is so very much more than one minimal Boltzmann brain in our L-region and O-region), some of which we have already discussed and/or cited in Sects. 5-8, other than Planck-power input. But there are still other proposed solutions to the entropy problem. One other proposed solution that we have not yet cited entails quantum fluctuations ensuring that every baby Universe starts out with an unstable large cosmological-constant, which corresponds to low total entropy because it is thermodynamically favorable for the consequent high-energy false vacuum to decay spontaneously [169,170]. Yet another proposed solution that we have not yet cited entails observer-assisted low entropy [168].
There are also many alternative viewpoints concerning fine-tuning and life in the Universe. It has been noted that since physical parameters such as constants of nature, strengths of forces, masses of elementary particles, etc., all have real-number values, and the range of real numbers is infinite, then if the probability of occurrence of a given real-number value of a given parameter is uniform, or at least non-convergent, then there is only an infinitessimal probability of this value being within any finite range [171]. But if thee are an infinite number of L-regions and O-regions, this infinity may be as large or even larger. We should also note that while some scientists are favorable towards the idea of fine tuning [172], others are sceptical to the point of not requiring a Multiverse to explain it away, but stating that it is an invalid concept even if our O-region constituted the entire Universe [173][174][175]. Even this sceptical viewpoint admits that only a very small range of parameter space is consistent with carbon-based life as we know it on Earth [], but assumes that a much larger range of parameter space is consistent with life in general [174]. But life, at least chemically-based life, probably must be based on carbon, because no other element even comes close to matching carbon's ability to form highly complex, information-rich molecules. Even carbon's closest competitor, silicon, falls woefully short. Also, nucleosynthesis in stars forms carbon more easily that silicon [176], so carbon is more abundant [176].
other viewpoints, inflation not only has a beginning but also has an end -eternal inflation is impossible [168,169]. According to one of these viewpoints, the end of inflation is imposed by the increasingly fractal nature of spacetime [168,169]. We also note that Dr. Roger Penrose considered another difficulty associated with possible fractal nature of spacetime: inflation does not solve the smoothness and flatness problems if the structure of spacetime is fractal, let alone worse than fractal [170]. According to another of these viewpoints, the end of inflation is imposed by the Big Snap, according to which expansion of space will eventually dilute the number of degrees of freedom per any unit volume, and specifically per Hubble volume, to less than one, although the Universe will probably be in trouble well before the number of degrees of freedom per Hubble volume is reduced to one [171,172]. But perhaps new degrees of freedom can be created to compensate [171,172]. Perhaps Planck-power input, if it exists, can, because it replenishes mass-energy, also replenish degrees of freedom -thereby precluding the Big Snap. In Dr. Max Tegmark's rubber-band analogy, this corresponds to new molecules of rubber being created (at the expense of negative gravitational energy of a "rubber-band Universe") the rubber band stretches, thereby keeping the density of rubber constant [171,172]. But, with or without a Big Snap [171,172], if inflation does have an end for any reason whatsoever, then my question to Dr. Roger Penrose in Sect. 6 is answered negatively: then the vast majority of Big Bangs will be noninflationary.
There are also many proposed solutions to the entropy problem (why there is so very much more than one minimal Boltzmann brain in our L-region and O-region), some of which we have already discussed and/or cited in Sects. 5-8, other than Planck-power input. But there are still other proposed solutions to the entropy problem. One other proposed solution that we have not yet cited entails quantum fluctuations ensuring that every baby Universe starts out with an unstable large cosmological-constant, which corresponds to low total entropy because it is thermodynamically favorable for the consequent high-energy false vacuum to decay spontaneously [173,174]. Yet another proposed solution that we have not yet cited entails observer-assisted low entropy [172].
There are also many alternative viewpoints concerning fine-tuning and life in the Universe. It has been noted that since physical parameters such as constants of nature, strengths of forces, masses of elementary particles, etc., all have real-number, or perhaps rational-number, values. The range of real numbers, or even of rational numbers, is infinite. (The countable infinity of rational numbers is smaller than the uncountable infinity of real numbers, but even a countable infinity is still infinite.) Hence, if the probability of occurrence of a given real-number, or even rational-number, value of a given parameter is uniform, or at least non-convergent, then there is only an infinitessimal probability of this value being within any finite range [175,176]. But if thee are an infinite number of L-regions and O-regions, this infinity may be as large or even larger. We should also note that while some scientists are favorable towards the idea of fine tuning [177], others are skeptical to the point of not requiring a Multiverse to explain it away, but stating that it is an invalid concept even if our O-region constituted the entire Universe [178][179][180]. Even this skeptical viewpoint admits that only a very small range of parameter space is consistent with carbon-based life as we know it on Earth [180], but assumes that a much larger range of parameter space is consistent with life in general [180]. But life, at least chemically-based life, probably must be based on carbon, because no other element even comes close to matching carbon's ability to form highly complex, information-rich molecules. Even carbon's closest competitor, silicon, falls woefully short. Also, nucleosynthesis in stars forms carbon more easily than silicon [181], so the relation between thermodynamics on the one hand, and inflation and cosmology on the other, and to Dr carbon is more abundant in the Universe [181]. We conclude by citing a paper that, while favoring a purely materialistic viewpoint, discusses would be required to seriously question it [182], and a book that explores many topics and viewpoints [183]. Reference [183] considers not only many, probably most, of the topics and viewpoints also considered in references that we have previously cited, but also many additional topics and viewpoints.