## Abstract

When supernova observations in the end of the 1990s showed the cosmic expansion to be accelerating, it became necessary to reintroduce the cosmological constant Λ as a fitting parameter. Although its physical origin has remained a mystery, it has generally been interpreted as some kind of energy field referred to as “dark energy.” This interpretation however implies a cosmic coincidence problem because we happen to live at a time when dark energy becomes the dominant driver of the expansion. Here we present an alternative explanation: The Λ term is induced by a global boundary constraint that ties its value to the conformal age of the universe. The cosmic coincidence problem then goes away. We illustrate how the cosmological evolution that is implied by this constraint differs from standard cosmology. Without the use of any free parameters, the theory predicts a present value of Λ that is within 2σ from the value derived from CMB observations with the Planck satellite. The universe is found to be mildly inflationary throughout the entire radiation-dominated era. This obviates the need to postulate a hypothetical, violent grand unification theory (GUT) era inflation to explain the observed large-scale homogeneity and isotropy of the universe.

### Keywords

- dark energy
- cosmology
- theory
- inflation
- gravitation
- early universe

## 1. Introduction

The term “dark energy” refers to the cosmological constant

Dark energy is widely regarded as one of the biggest problems in contemporary physics (for a review, cf. [4]). All conceivable ways to modify gravity have been tried. Different approaches to model the observational data have been explored, e.g. [5]. Elaborate laboratory experiments have been performed in the search for new scalar fields that would modify gravity [6]. On top of this, evidence against the earlier interpretation of the supernova observations in terms of dark energy has been discovered [7].

Recently [8] it was shown that there is an alternative way to explain the need for a cosmological constant, namely, as the result of a global cosmic boundary constraint instead of through the introduction of some new physical field. This approach leads to a new cosmological framework that brings a resolution to several outstanding enigmas, including the cosmic coincidence problem. Without the use of any free parameters,

In Section 2 we review the arguments that have been presented in [8] for the origin of the global constraint that governs the value of

## 2. Resonant origin of the Λ term

In standard cosmological models, the universe is assumed to be homogeneous and isotropic on the largest scales, because this is what observations tell us. The cosmological evolution can then conveniently be described in terms of a scale factor

where

Besides “proper time”

In terms of the temporal coordinates

The conformal metric of the first of these two equations shows that the metric coefficients are proportional to

The word “Euclidian” as the term for the second metric in Eq. (3) does not refer to the flatness assumption but to the signature of the metric: (

The transformation to Euclidian spacetime leads to remarkable advantages and insights, which have found important applications in various areas in the form of Euclidian field theory, e.g., in solid-state physics [10]. The Hamiltonian in ordinary spacetime becomes the Lagrangian in Euclidian spacetime. Quantum field theory QFT in Euclidian spacetime has the structure of statistical mechanics in ordinary spacetime. The oscillating phase factors in QFT become the Boltzmann factors, while the path integral becomes the partition function. Euclidian spacetime has long been known to provide a direct and elegant route to the derivation of the Hawking temperature of black holes, cf. [8, 11].

In the following we will show how the oscillating phase factors of the Euclidian metric field contain a resonance that fixes the value of the cosmological constant

This is the appropriate form to be used with the weak-field approximation, because the right-hand side of Eq. (4) represents the source term for gravitational waves when making a Fourier expansion, while the left-hand side describes the evolution of the vacuum fields, cf. [12]. We have here adopted the standard sign convention with (

In the weak-field approximation and the harmonic gauge,

While the vacuum fields without physical sources (the

With the period of the oscillation given by

In standard cosmology

where

Inserting the value of

### 2.1 Link between Λ and the age of the universe

In standard cosmology

The existence of such a physical link means that we need to single out, among all the solutions of the oscillator equation in Eq. (5), the Fourier component with a wavelength that corresponds to the conformal age of the universe. This only makes sense if time is bounded between the Big Bang and the Now, which seems to contradict the Einsteinian view that all future times somehow “already preexist” and that the experienced split between past, present, and future is just a stubborn illusion. Here we will argue (for details, see Section 3) that the Einsteinian view only makes sense in a universe devoid of observers and that this is not the universe that we inhabit. Like in quantum physics the observer plays a fundamental role in defining the nature of reality. The split between past, present, and future is not some illusion that we need to come to terms with, but is deeply physical. As soon as we introduce an observer (which can be a test particle, without brains or consciousness!) in Einstein’s universe, the split occurs. In any observable universe the future does not exist, even in principle. The only accessible region is between the Big Bang and the Now, and this region is bounded. The theory has to be applied to the observable universe, not to some idealized universe without observers. This is not just some alternative philosophical viewpoint but has profound physical consequences. It leads to a very different cosmological framework, as will be made clear in the following sections.

The existence of a metric resonance with respect to Euclidian time

Note that the Euclidian metric and scale factor have here been treated like a quantum field by allowing them to have an analytical continuation into the complex plane. When we however convert back to ordinary conformal time

### 2.2 Resonant amplitude and the validity of the weak-field approximation

According to Euclidian field theory, the oscillating QFT phase factors in Euclidian spacetime become Boltzmann factors in ordinary spacetime, if the field has periodic boundary conditions. When interpreted as due to a cosmic resonance, our finding that

It gives us the temperature

The identical result can be obtained with the help of Heisenberg’s uncertainty principle. For a system in thermal equilibrium at temperature

Inserting the value for

Alternatively we could have started from Eqs. (13) and (14) to obtain Eq. (12).

Replacing

it follows from Eq. (12) that

This comparison serves to demonstrate that the temperature

### 2.3 Nature of the global constraint for Λ

We have shown how

The nature of the boundary condition is however fundamentally different in our

The value of

The choice of observer defines the observable universe and its age. The observer is by definition always located at redshift

## 3. The participatory role of observers

Although Einstein’s opinion on the split between past, present, and future seems to have been somewhat ambivalent, his most quoted statement on the subject is that this split is an illusion, “but a very stubborn one.” He tended to regard all temporal instants along the infinite timeline as somehow already preexisting as part of a 4D map. This map contains both past and future, in spite of the fact that no observer is able to directly experience any other time than what we refer to as “Now.” Nevertheless the physical meaning of the concept of “Now” remained elusive to him.

The Einsteinian view of a 4D spacetime that maps all times is meaningful only in a universe devoid of observers. As soon as one introduces an observer, the timeline automatically splits up, because the presence of an observer implies a “Here” and “Now.” This split is profoundly physical, because we know from experience that the future is not part of the observable universe. It is not accessible to any observer, even in principle. This is the only universe in which our cosmological theories can be tested, not in some idealized universe devoid of observers, to which nobody can belong.

We are not merely dealing with an alternative philosophical viewpoint, because the introduction of observers leads to a different physical theory with different testable consequences. In the observable universe, time is always bounded, between the Big Bang as one edge and the Now as the other edge. In contrast, in the Einsteinian universe, time is unbounded in the future. The finite temporal dimension allows a global boundary constraint that leads to the emergence of a

A fundamental difference between classical and quantum physics concerns the role of observers. We can introduce test observers in classical physics, but they are not participatory in the way that they are in quantum physics. The classical world represents an objective reality that exists in a form that is independent of the presence of observers. It is the Einsteinian universe. In contrast, the quantum reality comes into existence through the participation of observers. It is the reason for the fundamental quantum fuzziness or uncertainty, the probabilistic causality, and the irreversibility through the collapse of the wave function. While the evolution of the wave function is time symmetric and deterministic, the act of “observation” or “measurement” leads to the profoundly different nature of quantum reality.

Although the role of our cosmological observers is very different from that of quantum theory, the comparison with quantum physics serves to indicate ways in which observer participation profoundly affects the nature of the theory. While abandoning the traditional classical view by allowing observer participation, we transform the theory into something that in at least this respect is closer to the nature of quantum physics. The consequence in our case is that the value of the cosmological constant gets uniquely determined in a way that leads to a very different cosmological framework.

The presence of participating observers also changes our interpretation of spacetime in a profound way by introducing a distinction between local and nonlocal time, a distinction that is absent in a universe without observers. With nonlocal time we here mean the same thing as look-back time. In contrast, dynamical time is the same as local time, because it is the only time that an observer can experience directly. The observables are redshifts, apparent brightnesses, structuring of celestial objects, etc. The observer is by definition always at redshift

In both standard cosmology and our alternative theory, the value of

## 4. Derivation of the cosmological evolution

The choice of observer defines the age

The dynamical time scale is the local time scale that is experienced by a comoving observer and which characterizes the age

In both standard cosmology and AC theory, the expansion rate of the universe, as represented by the Hubble constant, is governed by the equation

if we assume zero spatial curvature (see Section 4.2 for a justification of this assumption). Since

as follows from Eqs. (7) and (8). In standard cosmology

The scale factor normalized to epoch

The redshifts

which satisfies the requirement of Eq. (17) that

### 4.1 Key difference between standard cosmology and AC theory

The values of

according to Eq. (6). It is the fundamental equation that sets AC theory apart from standard theory.

Because the conformal age

Here the dimensionless parameter

### 4.2 Theoretical justification for the flatness assumption

While observations support our assumption of vanishing spatial curvature, AC theory requires it on theoretical grounds, in contrast to standard theory. Since curvature is induced by the presence of matter-energy sources, which may include the vacuum energy

### 4.3 Iterative solution of the basic equations

Because the value of the conformal age

From the relation

To find the

The

First of all, the value of

where

cf. [13].

When going to a different epoch with a different

During the iteration we enforce the correct

Besides

The scale factor

We further note that the scale factor uniquely determines the temperature of the cosmic radiation background through

which is valid back to a temperature

Because

In Figure 1 the parameters

### 4.4 Solution for the time scale

The next step of the calculation is to use the solution for

The proper age

It can be solved by integration to obtain the proper age

The present age

which readily follows from the definition of

The left panel of Figure 2 shows

Note also that the evolutionary time scales are quite different in the two theories. While both curves coincide at the present epoch, simply because they share the same normalization

Since the AC evolution is so close to linear in the log–log diagram, it is meaningful to represent it in the form of a power law:

For clarity we have explicitly written the exponent

Comparison with Eq. (30) shows that

Since

Overall the temporal variations of

Note that the level

### 4.5 Solution for the expansion rate

Similar to Eq. (32) we obtain from the definition of

Alternatively we may replace time

Knowing both

In the right panel of Figure 2, the Hubble time *T* = 2.725 K), the epoch of equipartition between matter and radiation, and the BBN epoch when the radiation temperature is

While the solid and dashed curves for

This huge difference has major implications for our understanding of BBN physics. At a first glance, it might seem that it would make AC theory incompatible with the constraints imposed by the observed abundances of the light chemical elements, because the BBN predictions depend on the value of the expansion rate. However, a closer look at the BBN problem shows that the situation is much more complex, because we are in a totally different regime. AC theory may still be compatible with the observational constraints, but this remains an open question. At the time of writing, the required BBN modeling with AC theory is still work in progress.

Similarly the significantly slower expansion rate in AC theory around the epochs of equipartition and recombination will require a reevaluation of the processes that govern the formation of the CMB spectrum. This is needed to allow AC theory to be confronted with the constraints that are imposed by the observed CMB signatures.

## 5. Natural inflation without new fields

Let us next determine the cosmic acceleration parameter

Its derivative with respect to

In contrast, in standard cosmology

Figure 3 shows how standard theory (represented by the dashed curve) has three distinct levels for

In AC theory there is only a gentle transition around equipartition from a level of

It may seem confusing that the universe is currently accelerating according to standard cosmology, while both Figure 3 and the right panel of Figure 1 show it to be decelerating according to AC theory. The reason is that

When we throughout this chapter have referred to the “observed acceleration” of the cosmic expansion, we have implicitly meant the acceleration that is inferred when the observational data are interpreted with the Friedmann-Lemaître models, because no other framework has been available for describing the observations in terms of an evolving scale factor. The discovery with the supernova observations was that a positive cosmological constant

The inference of an acceleration from redshift data depends on the way in which redshift

In standard cosmology an inflationary period in the early universe has been postulated to provide a solution to two fundamental cosmological problems: the horizon and the flatness problem [14]. The remarkable smoothness of the observed CMB tells us that the universe was homogeneous and isotropic on large scales to an extremely high degree (of order

In a decelerating universe, the radius of the cosmic horizon (e.g., the Hubble radius

The described properties are illustrated in Figure 4, where we have plotted the comoving Hubble radius

Causal contact is only possible over distances that are smaller than the comoving horizon size. As seen by the dashed curve in Figure 4, the largest scales that we observe today (of order 10 Gly, the approximate present Hubble radius) only came into causal contact very recently, well after the time of recombination (

To solve this problem, an early inflationary phase without known physical origin was postulated [14]. With its negative slope for

After the inflation idea was introduced, there have been a plethora of theoretical papers on the subject, which now has a prominent place in all modern cosmology textbooks. Still, four decades after its invention, the hypothetical inflaton field that is assumed to be responsible for the phenomenon has not been identified, in spite of an abundance of searches with string theory, supersymmetric grand unified theories, or other exotic alternatives. The existence of a violently inflationary phase around the GUT era, when the universe was a tiny fraction of a second old, is often treated as a fact, while fundamental arguments against it, like in [15, 16], are largely ignored.

In contrast, the solid curve of the AC theory in Figure 4 shows that the comoving Hubble radius

In the right panel of Figure 4, we have let AC theory be represented by two curves. The solid curve is based on the use of a value 73.5 km s

The linear representation of the right panel of Figure 4 again highlights how the present epoch is singled out by the standard model as something extraordinarily special. The comoving Hubble radius has one single narrow peak throughout all of cosmic history, and this peak is located where we happen to live in cosmic time.

## 6. Conclusions

The cosmological constant

Forty years ago another inflationary phase was postulated to occur in the GUT era of the very early universe, in order to answer the question why the universe is observed to be so homogeneous and isotropic on large scales [14]. The scalar inflaton field needed to drive the inflation has however not been identified in spite of a profusion of papers on this topic.

In the present work, we show that both these problems are connected and can be solved, if the

We have derived and solved the mathematical equations that follow from this approach. It leads to a very different cosmological framework, which we refer to as the “AC theory” (AC for alternative cosmology). Some implications of this theory have been highlighted: The cosmic coincidence problem disappears, our epoch is not special in any way, and we are not privileged observers. The boundary constraint leads to an evolving scale factor that describes an accelerating, inflating phase from the beginning of the Big Bang throughout the entire radiation-dominated era. There is no need to postulate some early violent inflation driven by some hypothetical inflaton field, because the boundary constraint automatically causes the universe to inflate. The theory reproduces the observed value of

As the cosmic expansion rate is found to have been much slower in the past than it was according to standard cosmology, the various observational data need to be reinterpreted with the new framework, in particular the BBN predictions of the abundances of the light chemical elements, and the observed signatures in the cosmic microwave background. The confrontation of the theory with such observational constraints represents work in progress that may ultimately determine the viability of the theory in its present form.