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Cosmology and Cosmic Rays Propagation in the Relativity with a Preferred Frame

By Georgy I. Burde

Reviewed: October 1st 2021Published: December 8th 2021

DOI: 10.5772/intechopen.101032

Downloaded: 35

Abstract

In this chapter, cosmological models and the processes accompanying the propagation of the cosmic rays on cosmological scales are considered based on particle dynamics, electrodynamics and general relativity (GR) developed from the basic concepts of the ‘relativity with a preferred frame’. The ‘relativity with a preferred frame’, designed to reconcile the relativity principle with the existence of the cosmological preferred frame, incorporates the preferred frame at the fundamental level of special relativity (SR) while retaining the fundamental space-time symmetry which, in the standard SR, manifests itself as Lorentz invariance. The cosmological models based on the modified GR of the ‘relativity with a preferred frame’ allow us to explain the SNIa observational data without introducing the dark energy and also fit other observational data, in particular, the BAO data. Applying the theory to the photo pion-production and pair-production processes, accompanying the propagation of the Ultra-High Energy Cosmic Rays (UHECR) and gamma rays through the universal diffuse background radiation, shows that the modified particle dynamics, electrodynamics and GR lead to measurable signatures in the observed cosmic rays spectra which can provide an interpretation of some puzzling features found in the observational data. Other possible observational consequences of the theory, such as the birefringence of light propagating in vacuo and dispersion, are discussed.

Keywords

  • general relativity
  • FRW models
  • late-time cosmic acceleration
  • dark energy
  • UHECR
  • gamma rays
  • photo pion-production
  • pair-production

1. Introduction

Lorentz symmetry is arguably the most fundamental symmetry of physics, at least in its modern conception. Physical laws are Lorentz-covariant among inertial frames; namely, the form of a physical law is invariant under the Lorentz group of space-time transformations. Therefore, the Lorentz symmetry sets a fundamental constraint for physical theories. Nevertheless, modifications of special relativity (SR) and possible violations of Lorentz invariance have recently obtained increased attention. Although, the success of general relativity (GR) to describe all observed gravitational phenomena proves the fundamental importance of Lorentz invariance in our current understanding of gravitation, some of the modern theories (unification theories, extensions of the standard model and so on) suggest a violation of special relativity. The aim of most of the Lorentz violating theories is to modify a Lorentz invariant theory by introducing small phenomenological Lorentz-violating terms into the basic relations of the theory (Lagrangian density, dispersion relation and so on) and predict what can be expected from it. Reviews of the most popular approaches [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26] to parameterizing Lorentz violating physics in the context of their relation to the ‘relativity with a preferred frame’ can be found in [27, 28]. Some of those studies are discussed in the following sections about the results obtained in the present paper.

The theory termed ‘relativity with a preferred frame’ developed in [27, 28, 29] represents a very special type of a Lorentz violating theory that is conceptually different from others found in the literature. It is not even a preferred frame that makes a difference—all violations of Lorentz invariance, made by distorting Lorentz-invariant relations of the theory, imply the existence of a preferred frame for the formulation of the physical laws, the one in which all the calculations need to be carried out, since breaking relativistic invariance also invalidates the transformations that allow us to change reference frame. The first major difference of the present analysis from the above-mentioned studies is that the Lorentz violation is not introducedinto the theory but it is a result of using freedom in formulation one of two basic principles of special relativity, the principle of universality of the speed of light. In other terms, Lorentz’s violation is ingrained into the framework of the theory at some fundamental level. The second major difference is that the relativistic invariance, in the sense that the form of a physical law is invariant under the space-time transformations between inertial frames, is not violated—it is a Lorentz violation without violation of relativistic invariance.

To outline the framework of the theory named ‘relativity with a preferred frame’ one has to start from the definition of the preferred frame. In the ‘relativity with a preferred frame’, the preferred frame is defined as the only frame where propagation of light is isotropic, while it is anisotropic in all other frames moving relative to the preferred one (it is a common definition in the studies investigating the fundamentals of special relativity and its potential breaking).1 Discussing the anisotropy of propagation of light one has to distinguish between the two-wayspeed of light, i.e. the average speed from source to observer and back, and the one-wayspeed which is a speed of light in one direction—eitherfrom source to observer orback. In the ‘relativity with a preferred frame’, it is the one-wayspeed of light that is assumed to be anisotropic in all the frames except the preferred frame, while the two-wayspeed of light is isotropic and equal to cin all inertial frames.2 The analysis is based on the invariance of the equation of (anisotropic) light propagation for the space-time transformations between inertial frames and the group structure of the transformations plays a central role in the analysis. Although, the existence of the preferred frame seems to be in contradiction both with the basic principles of special relativity and with the group property of the transformations, in the framework of the ‘relativity with a preferred frame’, those principles are retained. The crucial element, which allows retaining the relativistic invariance and the group property of the space-time transformations, is that the anisotropy parameter k, figuring in the equation of the anisotropic light propagation, is treated as a variable that takes part in the group transformations (for more details, see Section 2). Then the preferred frame, in which k=0, enters the analysis on equal footing with other frames since nothing distinguishes the transformations to/from that frame from the transformations between two frames with k0. The space-time symmetry underlying the group of transformations between inertial frames, which in the standard SR is expressed by the existence of the combination invariant under the transformations (interval), in the ‘relativity with a preferred frame’, reveals itself also in the form of the invariant combination, a counterpart of the interval of the standard SR. Such a ‘modified space-time symmetry’ paves the way to extensions of the kinematics of the ‘relativity with a preferred frame’ to free-particle dynamics, general relativity and electromagnetic field theory.

The above-described generalization of special relativity cannot be validated by experiments measuring the speed of light since only the two-way speed of light, the same in all the frames, can be measured. For creating a physical theory, predictions of which can be compared with observational data, it is needed to identify the preferred frame of the present analysis, which is defined by the property of isotropy of the one-way speed of light, with a frame possessing the property that velocity of any other frame relative to it can be measured using some physical phenomena. In the present analysis, that preferred frame is a comoving frame of cosmology or the CMB frame (note that identifying the preferred frame with the CMB frame is a common feature of practically all Lorentz-violating theories). It is the only frame possessing the property, that motion of any other frame relative to it is distinguishable, and, in addition, this frame, like the preferred frame of the present analysis, is defined based on the isotropy property. As a result of specifying the preferred frame, all the relations of the ‘relativity with a preferred frame’, as well as of its extensions, contain only one universal constant bwhich is a parameter to be adjusted for fitting the results of the theory to observational data.

Identifying the preferred frame with the cosmological comoving frame implies that the theory should be applied to phenomena on cosmological scales. Studying different phenomena requires extensions of the modified SR kinematics to different areas of physics. The purpose of this chapter is to present a unified view of the extensions and their applications based on the concept of the modified space-time symmetry. This includes extension to general relativity (Section 4.1) and constructing cosmological models based on the modified general relativity (Section 4.2); extension to the dynamics of the free particles (Section 3.1) and its application to the processes accompanying the Ultra High Energy Cosmic Rays (UHECR) and the gamma-rays propagation (Sections 5.1 and 5.2); extension to electromagnetic field (Section 3.2) and studying electromagnetic waves based on the modified electrodynamics (Section 3.3) with application to the gamma-rays propagation (Section 5.3).

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2. Special relativity kinematics

Kinematics of the ‘relativity with a preferred frame’ will be only outlined in this section, for a detailed presentation see [27, 28, 29].

The transformations between two arbitrary inertial reference frames Sand S, with the coordinate systems XYZTand xyztin the standard configuration (with the y- and z-axes parallel to the Y- and Z-axes and Smoving relative to Swith the velocity vin the positive direction of the common x-axis), are considered. In the subsequent analysis, the group property of the space-time transformations is used as a primary tool. Groups of transformations are sought using the condition of invariance of the equation of anisotropic light propagation [30]

ds2=c2dt22kcdtdx1k2dx2dy2dz2=0E1

where kis the anisotropy parameter such that speeds of light in the positive and negative x-directions are

c+=c1+k,c=c1kE2

Eq. (1) incorporates both the anisotropy of the one-wayspeed of light as equation (2) shows and the universality of the two-wayspeed of light in the sense that it is equal to cin all inertial frames (see, e.g., [33, 34]).3 The transformations involve both the space and time coordinates xyztand the anisotropy parameter kso that the equations of light propagation in the frames Sand Sare

c2dT22KcdTdX1K2dX2dY2dZ2=0,E3
c2dt22kcdtdx1k2dx2dy2dz2=0E4

where Kand kare the values of the anisotropy parameter in the frames Sand Srespectively. The one-parameter (a) group of transformations of variables from XYZTKto xyztk, which converts (3) into (4), is sought in the form

x=fXTKa,t=qXTKa;y=gYZKa,z=hYZKa;k=pKaE5

where, based on the symmetry arguments, it is assumed that the transformations of the variables xand tdo not involve the variables yand zand vice versa. According to the Lie group method (see, e.g., [35, 36])., the infinitesimal transformations corresponding to (5) are introduced, as follows

xX+ξXTKa,tT+τXTKa,yY+ηYZKa,zZ+ζYZKa,kK+κKaE6

Proceeding by the usual Lie group technique (see [27, 28, 29] for details) one can define the form of the transformations in xyztkvariables. Calculating invariants of the group one can define a combination (a counterpart of the interval of the standard relativity) that is invariant under the transformations, namely

ds˜2=1λk2c2dt22kcdtdx1k2dx2dy2dz2E7

where

λk=exp0kpκpdpE8

with κkbeing the group generator for the variable ka, see equation (6).

Furthermore, introducing the new variables

t˜=1kctkx,x˜=1λkx,y˜=1λky,z˜=1λkzE9

converts the invariant combination (7) into the Minkowski interval

ds˜2=c2dt˜2dx˜2dy˜2dz˜2E10

while the transformations take the form of rotations in the x˜t˜space (Lorentz transformations). However, in the calculation of physical effects, the ‘true’ time and space intervals in the ‘physical’ variables txyz, obtained from t˜x˜y˜z˜by the transformation inverse to (9), are to be used.

The expression (7) for the modified interval and the transformations (9) contain the function λkwhich depends on the unspecified function κk, the infinitesimal group generator for the variable k. This uncertainty reflects the fact that, within the above-developed framework, there is no possibility to determine the value of the anisotropy parameter kor, in other terms, to determine which frame is the preferred one, since only the two-way speed of light, equal to cin all the frames, can be measured. To specify the theory, such that its predictions could be compared with observations, there should exist a possibility to measure the frame velocity relative to a preferred frame using some other physical phenomena. Under the assumption that it is possible, the argument, that anisotropy of the one-way speed of light in an arbitrary inertial frame is due to its motion for a preferred frame, combined with group properties of the transformations, leads to the conclusion that the anisotropy parameter kin a frame moving relative to a preferred frame with velocity β¯=v¯/cshould be given by some universal function of that velocity, as follows

k=Fβ¯orβ¯=fkE11

where β¯=fkis a function inverse to Fβ¯. Then the group generator κkis calculated by (see [27, 28, 29] for details)

κk=1f2kfkE12

which allows to calculate the factor λkfrom (8). Next, with the expression (11) for kintroduced into (8), the factor λkbecomes a function Bβ¯of the frame velocity β¯relative to a preferred frame, as follows

λ(kβ¯Bβ¯=exp0β¯Fm1m2dmE13

In the subsequent analysis, those general relations are specified using an approximation for Fβ¯based on the following argument. An expansion of the function Fβ¯in series in β¯should not contain even powers of β¯since it is expected that a direction of the anisotropy vector changes to the opposite if a direction of motion for a preferred frame is reversed: Fβ¯=Fβ¯. Thus, with accuracy up to the third order in β¯, the dependence of the anisotropy parameter on the velocity for a preferred frame can be approximated by

k=Fβ¯bβ¯,β¯=fkk/bE14

With this approximation, the group generator κkcalculated using (12) takes the form

κk=bk2bE15

and, correspondingly, the factors λkand Bβ¯calculated from equations (8) and (13) become

λk=1k2b2b/2E16
Bβ¯=1β¯2b/2E17

Thus, after the specification, all the equations contain only one undefined parameter, a universal constant b. It is worth reminding that, even though the specified law (14) is linear in β, it does include the second-order term which is identically zero. Therefore describing the anisotropy effects, which are of the order of β2, by the above equations, is legitimate. In particular, the expression (17) for Bβ¯is valid up to the second-order in βand, with the same order of approximation, it can be represented as

Bβ¯=1b2β¯2E18
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3. Extensions to other areas of physics

3.1 Free particle dynamics

In this section, the free particle dynamics of the ‘relativity with a preferred frame’ developed in [28] is presented in a shortened form. The modified dynamics is developed based on the existence of the invariant combination ds˜(a counterpart of the interval of the standard SR) defined by equation (7). Then the action integral for a free material particle is [37]

S=mcabds=tatbLdtE19

where the integral is along the world line between two given world points and Lrepresents the Lagrange function. The invariant ds˜defined by (7) can be represented in the form

ds˜=cdtQkβxβλkE20

where

Qkβx.β=1kβx2β2;βx=vxc,β2=vx2+vy2+vz2c2E21

and

vx=dxdt,vy=dydt,vz=dzdtE22

are components of the velocity vector. Then the Lagrange function is defined by

L=mc2Qkβx.βλkE23

which is used to obtain expressions for the momentum Pand energy Eof a particle, as follows

Px=1cLβx=mck+βx1k2λkQkβx.β,Py=1cLβy=mcβyλkQkβx.β,Pz=1cLβz=mcβzλkQkβx.βE24

and

E==Pxvx+Pyvy+PzvzL=mc21kβxλkQkβx.βE25

Proceeding with the four-dimensional formulation, we will use the variables t˜x˜y˜z˜defined by (9) which allows converting the invariant combination (7) into the form (10) of the Minkowski interval. Introducing the four-dimensional contrainvariant radius vector by

x0x1x2x3=ctxyz=1λkctkxxyzE26

we define the contrainvariant four-velocity vector as

ui=dxidsE27

where the superscript iruns from 0 to 3. Using (26) and (20) in (27) yields

u0u1u2u3=1Qkβx.β1kβxβxβyβzE28

where Qkβx.βis defined by (21). Correspondingly, covariant four-dimensional radius-vector and velocity vector are defined by

x0x1x2x3=ctxyz,E29
u0u1u2u3=1Qkβx.β1kβxβxβyβzE30

and the following relations hold

dxidxi=ds2E31
uiui=1E32

where a common rule of summation over repeated indexes is assumed.

Next, recalling that the momentum four-vector is defined by

pi=SxiE33

and using the principle of the least action [37] we find (see [28] for details) that

pi=mcuiE34

while the contravariant components of the four-momentum vector are

pi=mcuiE35

Then from the identity (32) we get

pipi=m2c2E36

Recalling that

Px=Sx,Py=Sy,Pz=Sz,E=StE37

with allowance for (26) and (33), we have

Px=1λkSx1kSx0=kp0p1λk,Py=1λkSx2=p2λk,Pz=1λkSx3=p3λk,E=cλkSx0=cp0λkE38

which, upon using (34) and (30), yields the relations (24) and (25) for the three-momentum and energy. Solving equations (38) for the components of the four-momentum vector we get

p0=kc,p1=λkkEcPx,p2=λkPy,p3=λkPzE39

Then using (39) in (36) yields a dispersion relation for a free particle which can be represented in the form

Ec+PxEc+Px=Py2+Pz2+m2c2λk2E40

where the speeds of light c+and cin the positive and negative x-directions are defined by equation (2). It follows from (40) that for massless particles moving along the x-axis in the positive xdirection

Px=Ec+=E1+kcE41

while for massless particles moving in the negative xdirection

Px=Ec=E1kcE42

3.2 Electromagnetic field equations

The invariant action integral for a charged material particle in the electromagnetic field is made up of two parts: the action for the free particle defined by (19) and a term describing the interaction of the particle with the field. The invariance is provided by using the combinations that are invariant in the Minkowskian variables (26) so that the action integral takes the form [37]

S=abmcdsecAidxiE43

where the coordinates xiare related to physical coordinates txyzby (26) and Aiare components of the (covariant) four-potential vector expressed through the contravariant components Aiby

A0A1A2A3=A0A1A2A3E44

Upon representing the four-potential as

A0A1A2A3=ϕA˜=ϕ˜A˜xA˜yA˜zE45

where A0=ϕ˜is a scalar potential and the three-dimensional vector A˜is the vector potential of the field, the electromagnetic part of the action integral can be written in the form

S=t1t2ecAveϕdtE46

Here and in what follows, ‘tilde’ indicates that variables and operations are in Minkowskian space-time variables (26). Note that, while scalars and components of three-dimensional vectors in the Minkowskian formulation appear with ‘tilde’, four-dimensional Minkowskian variables are not supplied with ‘tilde’. It does not lead to any confusion since the four-dimensional notation does not applicable to the formulation in physical variables.

In the electrodynamics of the standard special relativity (which, in our case, is electrodynamics in Minkowskian variables), the electric and magnetic field intensities are defined based on equations of motion of a charged particle obtained from the Lagrange equations.

ddtLv=LrE47

where, in the Lagrange function L, a part related to the electromagnetic field is given by the integrand of (46). Then the electric and magnetic field intensities E˜and H˜are introduced by separating the right-hand side of the vector equation of motion (the force) into two parts, one of which does not depend on the velocity of the particle and the second depends on the velocity, being proportional to the velocity and perpendicular to it, as follows.

dpdt˜=eE˜+ecv×H˜E48

where p˜is the momentum vector. The electric and magnetic field intensities are related to the potentials by

E˜=1cA˜tgrad˜ϕ;H˜=curl˜A˜E49

The same line of arguments is used to derive equations describing the electromagnetic field in physical variables txyz. The action integral is represented in the form

S=tatbLdtE50

where tis the ‘physical’ time related to the Minkowskian variables via (26) and Lis the Lagrangian in physical variables. The free particle part of Lis defined by Eqs. (21)(23). To obtain the electromagnetic field part of the Lagrangian, the right-hand side of (46) is transformed to physical space-time variables and then the new variables ϕAxAyAz(modified potentials) are introduced by the relations

A0=ϕ˜=λkϕ,A1=A˜x=λkAx,A2=A˜y=λkAy,A3=A˜z=λkAzE51

As the result, the Lagrangian function Lin the action integral (50) takes the form

L=Lp+ecvxAx+vyAy+vzAzE52

where Lpis the free particle part of Ldefined by Eqs. (21)(23). Substituting (52) into the Lagrange equations

ddtLv=LrE53

yields

dpdt=ecAtegradϕ+ecv×curlAE54

Thus, upon using the modified potentials, equations of motion in physical variables have the same form as in the standard relativity and the physical electric and magnetic field intensities are expressed through the modified potentials by the relations

E=1cAtgradϕ;H=curlAE55

of the same form (49) as in the standard relativity.

It is evident that the first pair of the Maxwell equations in physical variables, which is derived from Eq. (55), have the same form as in the standard relativity

curlE=1cHt;divH=0E56

To obtain the second pair of Maxwell equations in physical variables let us calculate the components of the electromagnetic field tensor Fikdefined by

Fik=AkxiAixkE57

Expressing Aiin the right-hand side of (57) through the modified potentials by (51) and then transforming the result to physical space-time variables using (26), with subsequent use of Eq. (55), yields the expressions for the components Fikof the electromagnetic field tensor in terms of physical electric and magnetic field intensities. The result can be written as a matrix in which the index i=0,1,2,3labels the rows, and the index kthe columns, as follows

Fik=λk20ExEyEzEx0Hz+kEyHy+kEzEyHzkEy0HxEzHykEzHx0E58

while

Fik=λk20ExEyEzEx0Hz+kEyHy+kEzEyHzkEy0HxEzHykEzHx0E59

Note that the terms with kin the expressions (58) and (59) spoil the property, that FikFikwhen EE, of the standard relativity electrodynamics.

The electromagnetic field equations are obtained with the aid of the principle of least action [37] in the form

Fikxk=0E60

(only fields in a vacuum, that are relevant to the subject of this paper, are considered). Substituting (59) into (60) and transforming the equations to physical space-time variables, upon combining equations with different ‘i’ and using the first pair of the Maxwell Eq. (56), yields the second pair of the Maxwell equations in the three-dimensional form

divE=kcExt;curlH=1k21cEt2kEx+kgradExE61

An important feature of Eq (61) is their linearity in Eand Hand hence in Ai. The Lorentz-violating terms thereby avoid the complications of nonlinear modifications to the Maxwell equations, which are known to occur in some physical situations such as nonlinear optics or when vacuum polarization effects are included. Another feature is that the extra Lorentz-violating terms involve only the electric field, as well as its derivatives.

Note the existence of an alternative way of the derivation of the modified Maxwell Eqs. (56) and (61). Based on Eqs. (49), (51), and (55), the electric and magnetic field intensities E˜and H˜in Minkowskian formulation can be related to the physical electric and magnetic field intensities Eand H, as follows

Ex˜=λk2Ex,Ey˜=λk2Ey,Ez˜=λk2Ez,Hx˜=λk2Hx,Hy˜=λk2Hy+kEz,Hz˜=λk2HzkEyE62

The same relations are seen in the expressions (58) for the components of the electromagnetic field tensor. It is readily verified that substituting the relations (62) into the Maxwell equations of the standard relativity

curl˜E˜=1cH˜t˜,div˜H˜=0,curl˜H˜=1cE˜t˜,div˜E˜=0E63

as

Ex˜txyz=λk2Exttxxxyyzz,E64

where

ttx=λkt+kcx,xx=λkx.,yy=λky,zz=λkzE65

yields the modified Maxwell Eqs. (56) and (61).

3.3 Electromagnetic waves

Like the electromagnetic wave equation of the standard relativity electrodynamics, the equation describing electromagnetic waves in the electrodynamics of the relativity with a preferred frame can be derived straight from the modified Maxwell equations (reproduced below for convenience)

divH=0,curlE=1cHtE66
divE=kcExt;curlH=1k21cEt2kEx+kgradExE67

Eliminating Hby taking ‘curl’ from the second equation of (66) and substituting curlHfrom the second equation of (67), with the subsequent use of differential consequences of the first equation of (67) for eliminating mixed space derivatives, yields

2fx2+2fy2+2fz21k21c22ft2+2k1c2ftx=0E68

where ftxyzstands for any component of E. It is readily verified that the wave equation for Hobtained from the modified Maxwell equations in a similar way has the same form (68).

Alternatively, the wave Eq. (68) can be derived from (60) expressed in terms of the potentials using (57) while imposing the Lorentz gauge condition

Akxk=0E69

Converting the derivatives in the resulting equation

2Akxkxk=0E70

into derivatives in physical space-time variables yields equations of the form (68) with fbeing any component of Ak. Given the fact, that equations (51) and (55) relating Akto the modified potentials ϕAand then to Eand Hare linear, it is evident that any of those variables obeys Eq. (68).

Much of the propagation behavior of the electromagnetic wave is encoded in its dispersion relation, which provides spectral information for the modes. To find the dispersion relation the ansatz in the form of monochromatic plane waves is used, as follows

ftxyz=faωq)expiqxx+qyy+qzzωtE71

where ωand q=qxqyqzcan be regarded as the frequency and wave vector of the mode or as the associated energy and momentum (taking the real part is understood, as usual). Substituting (71) into (68) yields the dispersion relation

c2q22ckqxω1k2ω2=0whereq2=qx2+qy2+qz2E72

The dispersion relation (72) can be also represented in the form

ωc+qxωc+qx=qy2+qz2E73

where c+and care defined by (2). The form (73) adheres to the dispersion relation (40) for free massless particles with Eand Preplaced by ωand q. In the standard relativity, the polynomial (72) determining ωreduces to one with two quadruply degenerate roots ω=±cqwhich correspond to the opposite directions of the group velocity. In the modified electrodynamics, the polynomial also has two roots

ω=ckqx+1k2q2+k2qx21k2,ω=ckqx1k2q2+k2qx21k2E74

Like as in the standard relativity case, the two roots (74) are obtained from each other by changing the sign of ωbut, in the case of k0, it is accompanied by a change of sign of the anisotropy parameter k.

More insight about the wave motion implied by Eq. (68) can be gained from the modified Maxwell Eqs. (66) and (67). Eq. (66), which are unaffected by the modifications, reduce with the ansatz (71) to

qH=0,ωH=q×EE75

The first of these equations shows that the magnetic field remains transverse to qdespite the Lorentz violation. The second equation shows that the magnetic field His perpendicular to the electric field E. The first equation of (67) reduces to

qE=ωkcExE76

Eq. (76) implies the existence of two modes.

The first one corresponds to the electric field with Ex=0. Then it follows from (76) that the electric field is perpendicular to q. Further, the condition Ex=0implies that the vector Elies in the plane yzand so the vector qis directed along the x-axis (the direction of the anisotropy vector k). Therefore Hx=0and also, based on rotational symmetry in the plane yz, it can be set Hz=0which implies Ey=0. In such a case, the first equation of (66) shows that qy=0and the first equation of (66) shows that qz=0. Then the second equation of (66) and the second equation of (67) reduce to the system of equations for the two nonzero components of the electric and magnetic field intensities Ezand Hywhile the requirement of vanishing the determinant of the system yields the dispersion relation (72). Thus, the mode with Ex=0represents a usual electromagnetic plane wave with the magnetic and electric fields transverse to the direction of propagation of the wave qand perpendicular to each other, which propagates along the direction of the anisotropy vector (but with the modified dispersion relation).

The second mode corresponds to the case Ex0. Then it follows from (76) that the electric field vector is not normal to q. Since, according to the second equation of (75), His normal to the plane of Eand q, one can choose, without loosing generality, the direction of Hto be along the y-axis and the plane of the vectors qand Eto be the xz-plane. Then the first equation of (75) gives qy=0and it is readily verified that the remaining equations of (66) and (67) can be satisfied only if qz0with ω, qxand qzobeying the dispersion relation (72) where it is set qy=0. Note the particular case, when Eis directed along the x-axis (Ez=0), in which the dispersion relation degenerates to

ω=cqxk,qz=±qxkE77

Thus, the second mode represents electromagnetic wave, in which the magnetic field His transverse to direction of propagation qand perpendicular to the electric field E, like as in the regular wave, but, as distinct from the regular wave, the electric field is not normal to q. Another characteristic feature of such a wave, that distinguishes it from the first mode, is that the direction of propagation is not along the anisotropy vector kand so not along with the velocity of relative motion of the source and the observer. It implies that in the case when the relative motion velocity is only the cosmological recession velocity, such a wave propagates not along a line of sight.

It is worthwhile to note a distinguishing feature of the above analysis as compared with other studies of electromagnetic waves in the presence of the Lorentz violation. Typically, different modes arising due to the Lorentz violation correspond to different roots of the modified dispersion relation (see, e.g., [6, 38, 39, 40]). The present analysis provides an unusual example when two different modes correspond to the same root of the dispersion relation (for the waves propagating to the observer. it is the second root of (74)). The existence of two modes is revealed only when one studies the corresponding solutions of the modified Maxwell equations. It is worth also noting that the present analysis is performed solely in terms of field intensities Eand Hwhile most studies of electromagnetic waves in the presence of the Lorentz violation involve also the electromagnetic field potentials Akwhich are accompanied by extensive discussions of different gauge choices and their influence on the results.

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4. Cosmology

4.1 General relativity

The basic principle of general relativity, the Equivalence Principle (see, e.g. [41]), which asserts that at each point of spacetime it is possible to choose a ‘locally inertial’ coordinate system where objects obey Newton’s first law, is valid independently of the law of propagation of light assumed. In other terms, it can be applied when the processes in the locally inertial frame are governed by the laws of ‘relativity with a preferred frame’. Based on that there exists the invariant combination (7), which by the change of variables (9) is converted into the Minkowski interval, one can state that the general relativity equations in arbitrary coordinates x0x1x2x3are valid if the locally inertial coordinates ξ0ξ1ξ2ξ3are

ξ0=ct,ξ1=x,ξ2=y,ξ3=zE78

where t, x, yand zare defined by (9). In these variables, the invariant spacetime distance squared ds2=gikdxidxkis equal to ds˜2=ηikdξidξk(the notation ηikis used for the Minkowski metric and the rule of summation over repeated indices is implied). Thus, the apparatus of general relativity is applied in the coordinates x0x1x2x3while, in the calculation of the ‘true’ time and space intervals, the ‘physical’ variables txyz(it is the new notation for what was before txyz) are to be used. Eq. (9) relating the physical coordinates to the ‘locally inertial’ coordinates, rewritten with allowance for (78) and (9), are

t=1cλkξ0+kξ1,x=λkξ1,y=λkξ2,z=λkξ3E79

The ‘true’ time and space intervals can be determined using a procedure similar to that described in [37]. Applying that procedure (see [27] for details) yields the following relations for the ‘true’ proper time interval dtand the element dlof ‘the true’ spatial distance:

dt=1cλkg00dx0E80
dl=λkγαβdxαdxβ,γαβ=gαβ+g0αg0βg00E81

where gik(i,k=0,1,2,3) are components of the space-time metrical tensor and γαβ(α,β=1,2,3) are components of the space metrical tensor. It is important to note, that the expression for the proper velocity of a particle v=dl/dtis not modified, since the time and the distance intervals are modified by the same factor λk.

4.2 Cosmological models

Modern cosmological models assume that, at each point of the universe, the ‘typical’ (freely falling) observer can define the (preferred) Lorentzian frame in which the universe appears isotropic. The metric derived based on isotropy and homogeneity (the Robertson-Walker metric) has the form [41, 42]

ds2=dt2a2tdr21Kcr2+r2dΩ,dΩ=dθ2+sin2θdϕ2E82

where a comoving reference system, moving at each point of space along with the matter located at that point, is used. This implies that the coordinates rθϕare unchanged for each typical observer. In (82), and further throughout this section, the system of units in which the speed of light is equal to unity, is used. The time coordinate x0=tis the synchronous proper time at each point of space. The constant Kc(this notation is used, instead of common kor K, to avoid confusion with the symbols for the anisotropy parameter) by a suitable choice of units for rcan be chosen to have the value +1, 0, or 1.

Introducing, instead of r, the radial coordinate χby the relation r=Sχwith

Sχ=sinχforKc=1sinhχforKc=1χforKc=0E83

and replacing the time tby the conformal timeηdefined by

dt=atE84

converts (82) into the form

ds2=a2ηdη2dχ2S2χE85

The information about the scale factor atin the Robertson-Walker metric can be obtained from observations of shifts in the frequency of light emitted by distant sources. The frequency shift can be calculated by considering the propagation of a light ray in isotropic space with the metric (85) adopting a coordinate system in which we are at the center of coordinates χ=0and the source is at the point with a coordinate χ=χ1. A light ray propagating along the radial direction obeys the equation dη2dχ2=0. For a light ray coming toward the origin from the source, that equation gives

χ1=η1+η0E86

where η1corresponds to the moment of emission t1and η0corresponds to the moment of observation t0. The red-shift parameterzis defined by

z=ν1ν01E87

where ν0is the observed frequency and ν1is the frequency of the emitted light which coincides with the frequency of a spectral line observed in terrestrial laboratories. Calculations within the framework of the relativity with a preferred frame (see details in [27]) lead to the relation

z=aη0aη0χ1Bβ¯11E88

The relation expressing the Luminosity DistancedLof a cosmological source in terms of its redshift zis one of the fundamental relations in cosmology. It has been exploited to get information about the time evolution of the expansion rate. In a matter-dominated cosmological model of the universe (Friedman-Robertson-Walker model) based on the standard GR, solving the gravitational field equations yields the luminosity distance-redshift relation of the form

dL=H01z+121q0Dz2+E89

where the deceleration parameter q0Dis positive for all three possible values of the curvature parameter Kcwhich means that, in that model, the expansion of the universe is decelerating. However, recent observations of Type Ia supernovae (SNIa), fitted into the luminosity distance versus redshift relation of the form (89), corresponding to the deceleration parameter q0D<0which indicates that the expansion of the universe is accelerating. This result is interpreted as that the time evolution of the expansion rate cannot be described by a matter-dominated cosmological model. To explain the discrepancy within the context of general relativity and fit the theory to the SNIa data, the dark energy, a new component of the energy density with strongly negative pressure that makes the universe accelerate, is introduced (see, e.g., [42]).

In the relativity with a preferred frame, solving the modified GR equations for a matter-dominated model lead to the luminosity distance-redshift relation of the form, which allows fitting the results of observations with supernovae so that the acceleration problem can be naturally resolved—there is no acceleration and so no need in introducing the dark energy. Below, the calculations leading to the modified luminosity distance-redshift relation are outlined (for more details see [27]).

In the relativity with a preferred frame, the expression for dLis obtained in the form [27]

dL=aη01+zSχ1E90

which coincides with a common form of the relation for dL[37, 42]. Nevertheless, even though it does not contain the factor Bβ¯1, the dependence of dLon zobtained by eliminating χ1from Eqs. (90) and (88) will differ from the common one since the relation (88) for zdoes contain the factor Bβ¯1. To derive the dependence dLzin a closed-form using Eqs. (90) and (88), the function aηdetermining the dynamics of the cosmological expansion it to be defined by solving the gravitational field equations of Einstein which requires to make some tentative assumptions about the cosmic energy density ρand the form of equation of state giving the pressure pas a function of the energy density. The energy density ρtis usually assumed to be a mixture of non-relativistic matter with equation of state p=0and dark energy with equation of state p=while ignoring the relativistic matter (radiation). In the commonly accepted ΛCDM model, the dark energy obeys the equation of state with w=1(vacuum) which is equivalent to introducing into Einstein’s equation a cosmological constant Λ. Then the fundamental Friedmann equation, which is obtained as a consequence of the Einstein field equations, can be written in the form (see, e.g., [42])

x2=H02x2ΩΛ+ΩMx3+ΩKx2E91

where

xt=ata0,a0=at0E92

and the parameters ΩΛ, ΩMand ΩKare defined by

ΩΛ=ρV0ρc,ΩM=ρM0ρc;ρc=3H028πG,ΩK=Kca02H02E93

where Gis Newton’s gravitational constant, ρV0and ρM0are the present energy densities in the vacuum and non-relativistic matter and ρcis the critical energy density. Being evaluated at t=t0Eq. (91) becomes

ΩΛ+ΩM+ΩK=1E94

The Friedmann Eq. (91) allows us to calculate the radial coordinate χ1of an object of a given redshift z. Eq. (86) defining χ1can be represented in the form

χ1=η0η1=η1η0=t1t0dtat=1a0x11dxxxE95

where xis a function of xdefined by the Friedmann Eq. (91) and x1=at1/a0. Then using Eq. (91) in (95) yields

χ1=x11dxa0H0x2ΩΛ+ΩMx3+ΩKx2E96

In the standard cosmology, Eq. (88) (with Bβ¯1=1) provides a simple relation

x1=11+zE97

so that (96) becomes a closed-form relation for χ1z. For a ‘concordance’ model, which is the flat space ΛCDM model, ΩK=0and ΩΛ=1ΩM, Eq. (96) can be represented in the form

χ1mcz1=0z1dz1ΩM+ΩM1+z3,χ1mc=χ1ca0H0E98

Here and in what follows, quantities with a superscript “c” refer to the concordance model, with the original notation secured for the corresponding quantities of the present model. Then the luminosity distance is calculated as

dLcz1=1H01+z1χ1mcz1E99

In the framework of the present analysis, expressing χ1as a function of z1by combining Eqs. (96) and (88) becomes more complicated in view of the fact that β¯1, and so the factor Bβ¯1, depend on χ1. We will outline the calculations for the case of a flat universe, ΩK=0, which is also the assumption of the concordance model.4 With that assumption and the presumption, that in the cosmology based on the relativity with a preferred frame there is no need in introducing dark energy, ΩΛ=0, the relation χ1x1is obtained from (96) in an analytical form, which allows finding the dependence x1χ1by inverting the result, as follows

χ1m=21x1x1=14χ1m22,χ1m=a0H0χ1E100

The dependence Bχ1, with the accuracy up to third order in χ1, is given by [27]

Bβ¯1χ1=1b2χ1m2+ΩMχ1m3E101

Substituting (101) and (100) into (88) reduces the problem to a transcendental equation for χ1mz1, as follows

14χ1mz122z1+11b2χ1mz12+χ1mz13=1E102

Representing the solution of (102) as a series in z1yields

χ1mz1=z1+1432bz12+185+4b+4b2z13E103

Then the relation dLz1, calculated from (90) with Sχ1=χ1, is

dLz1=1H0z1+1412bz12+184b21z13E104

To compare the results produced by the model with those, obtained from an analysis of type Ia supernova (SNIa) observations, one needs some fitting formulas for the dependence dLzderived from the observational data. It is now common, in an analysis of the SNIa data, to fit the Hubble diagram of supernovae measurements to the ΛCDM model (mostly, to the concordance model) and represent the results as constraints on the model parameters (see, e.g. [43]). Therefore, in what follows, a comparison of the results with the SNIa data is made by comparing the dependence dLzproduced by the present model with dLczfor the concordance model while using constraints on the parameter ΩMcfrom the SNIa data analysis. It is found that, for every value of ΩMcfrom the interval, defined by fitting the SNIa data to the concordance model, the parameter bcan be chosen such that the dependence dLzcoincided with dLczwith a quite high accuracy (were graphically undistinguishable). An example is given in Figure 1 where the dependence dLzfor ΩM=1(flat universe), defined by Eq. (104), is plotted for three different values of btogether with dLczof the concordance model. It demonstrates that there exists a value of b(in the present case it is b=0.672) for which the deviation is negligible. As it was mentioned above, in the present model the assumption of the flat universe is not obligatory. Calculations for other values of ΩM(remind that ΩK=1ΩM) show that for every value of ΩM>0there exists the value of b, for which the deviation dLzfrom dLczis negligible. It is worth clarifying again that the above is intended to be a comparison of the dependence dLzyielded by the present model with that derived from the SNIa observations so that the dependence dLczfor the ‘concordance’ model plays a role of a fitting formula for the SNIa data.

Figure 1.

Dependence of the luminosity distancedLon the red-shiftz: thin solid line for the concordance model withΩMc=0.31; short-dashed for the present model withΩM=1,b=1.2; long-dashed for the present model withΩM=1,b=0.672; thick solid for the present model withΩM=1,b=0.2.

The Baryon Acoustic Oscillations (BAO) data are commonly considered as confirming the accelerated expansion and imposing constraints on the dark energy parameters. Applying the cosmological models based on the ‘relativity with a preferred frame’ to the interpretation of the BAO data provides an alternative view on the role of the BAO observations in cosmology. Comparing the predictions of the present model with the recently released galaxy clustering data set of the Baryon Oscillation Spectroscopic Survey (BOSS), part of the Sloan Digital Sky Survey III (SDSS III), shows that the BAO data can be well fit to the present cosmological model. The BAO data include two independent sets of data: the BAO scales in transverse and line-of-sight directions which can be interpreted to yield the comoving angular diameter distance DMzand the Hubble parameter Hzrespectively. In [44], the results of several studies studying the sample provided by the BOSS data with a variety of methods are combined into a set of the final consensus constraints on DMzand Hzthat optimally capture all of the information. It is found (see [27] for details) that the results yielded by the present model are consistent with the consensus constraints of [44] on both DMzand Hz. The two regions in the plane ΩMbdefined by constraints on these two sets are overlapped such that the overlapping area corresponds to the values of the model parameters for which the results on Hzand DMzare consistent both with the BAO data and with each other. And what can be considered as a very convincing proof of the robustness of the present model is that a line in the plane ΩM.b, on which the results produced by the present model fit also the SNIa observational data, passes inside that quite narrow overlapping region defined by the BAO data. Thus, the results produced by the present model fit three different sets of data by adjusting (together with the matter density parameter ΩM) only one universal parameter b. It is worth noting again that, as distinct from the concordance model to which the SNIa and BAO data are commonly fitted by adjusting the dark energy parameters, the present model fits well all the data without introducing dark energy.

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5. Propagation of cosmic rays

5.1 Attenuation of the UHECR due to the pion photoproduction process

In this section, the application of the theory to the description of the effects due to the interactions of the Ultra-High Energy Cosmic Rays (UHECR) with universal diffuse background radiation in the course of the propagation of cosmic rays from their sources to Earth over long distances (see, e.g., review articles [45, 46, 47]) is considered. The interactions of the UHECR with the CMB photons are characterized by a well-defined energy threshold for the energy suppression due to pion photoproduction by UHECR protons—the Greisen-Zatsepin-Kuzmin (GZK) cutoff [48, 49]. The fluxes of cosmic ray protons with energies above this threshold would be strongly attenuated over distances of a few tens of Mpc so that the cosmic ray protons from the sources at a larger distance, even if they were accelerated to energies higher than the threshold, would not be able to survive the propagation. The energy position of the GZK cutoff can be predicted based on special relativity as a theoretical upper limit (‘GZK limit’) on the energy of UHECR set by pion photoproduction in the interactions of cosmic ray particles with the microwave background radiation. Calculating the GZK limit based on the particle dynamics of the special relativity with a preferred frame developed in Section 3.1 (see [28] for details) yields

EthEst=1z2b;Est=επ2εp+επ4EγE105

where Ethis the threshold value of the UHECR protons energy calculated using equations of the relativity with a preferred frame, Estis the standard value of the GZK threshold calculated using equations of the standard relativity, εp=mpc2and επ=mπc2are the proton and pion rest energies and Eγis the CMB photon energy.

It is seen that the expression (105) for the threshold energy of the proton differs from the common one by the factor 1z2b. The universal constant bis negative, both as it is expected from intuitive arguments and as it is found by fitting the cosmological model developed in the framework of the ‘relativity with a preferred frame’ to the observational data (Section 4.2). Therefore the threshold energy decreases as the distance to the source of the particles (the redshift z) increases (Figure 2, left panel).

Figure 2.

Left panel: dependence of the correction factor to the GZK threshold on the source redshiftzfor different values of the parameterb: shot-dashed forb=0.4; long-dashed forb=0.7; solid forb=1. Right panel: Number of sourcesns(in arbitrary units), that may contribute to the observed flux of protons at the energyEp, versusEpEst, whereEstis the standard GZK threshold value, for different values ofb: shot-dashed forb=0.4; long-dashed forb=0.7; solid forb=1.

This effect may contribute to the interpretation of the data on the mass composition of UHECR which is a key observable in the context of the physics of UHECR as it fixes few fundamental characteristics of the sources. The mass composition of UHECR became a matter of active debate after that the Pierre Auger Collaboration (Auger) reported on its recent observations [50, 51]. The observations of Auger, far the largest experiment set-up devoted to the detection of UHECR, have shown that the UHECR mass composition is dominated by protons only at energies around and below 1018eV and then the fraction of protons is progressively decreasing up to energies of 1019.6eV. It seemed to be not consistent with the general consensus, that UHECRs are mostly protons and that sources should accelerate them to >1020eV. At the same time, the Telescope Array (TA) experiment, even if with 1/10 of the Auger statistics, collected data seemed to confirm the pre-Auger scenario [52]. A common effort of the Auger and TA collaborations allowed to reconcile the interpretations of the Auger and TA observations so that the evidence for a composition becoming gradually heavier towards higher energies is now considered to be well established. It implies that the primary UHECR flux at the sources includes both protons and heavy nuclei which are to be accelerated with very high maximum injection energies. This imposes severe constraints on the parameters of the acceleration models and has served as a stimulus to build new acceleration models or reanimate the previously developed models that can potentially explain the phenomenology of the UHECR mass composition data. The models are characterized by a complex scenario and/or include some exotic assumptions.

The complexity of the scenario and the severe constraints on the model parameters, required in the case of a composition with heavy nuclei, are not present in the case if the UHECR mass composition is dominated by protons. In the latter case, the scenario is much simpler, only protons are accelerated with very high maximum injection energies. The view that the UHECR are mostly protons is, theoretically, a natural possibility. Proton is the most abundant element in the universe and several different astrophysical objects, at present and past cosmological epochs could provide efficient acceleration even if it requires very high luminosities and maximum acceleration energies. The models of interaction of UHECR with the astrophysical background are also much simpler if the UHECR are mostly protons. In this case, the only relevant astrophysical background is the CMB [53, 54]. This fact makes the propagation of UHE protons free from the uncertainties related to the background, being the CMB exactly known as a pure black body spectrum that evolves with red-shift through its temperature.

The results of the present study allow reconciling (at least, partially) the view, that, the primary UHECR flux at the sources is dominated by protons accelerated with very high maximum injection energies, with the observational evidence that the fraction of protons in the UHECR is decreasing towards higher energies. The apparent contradiction can be resolved by taking into account the effect, predicted by the present analysis, that the number of sources, which may contribute to the observed flux of protons at a given energy, is progressively decreasing with the energy increases. This effect is a consequence of the threshold condition (105) which implies that, among protons produced by a source at some z, only those having the energies lower than the threshold energy for that z, can reach the Earth. In other terms, for a given value Epof the proton energy, there exists a value zthof the redshift (distance Dth) such that, for the UHECR sources with D>Dth, the GZK threshold Ethis less than Epand so the protons with the energy Epinjected by the sources at the distances D>Dthcannot reach the Earth. Thus, the sources, that may contribute to the observed flux at the energy Ep, are confined within the sphere of the radius Dth, with Dthdecreasing when the proton energy Epis increasing. If the distribution of the sources in space is more or less uniform, the number of sources Ns, that may contribute to the observed flux at the energy Ep, decreases with Ep(Figure 2, right panel). Thus, reducing the fraction of protons in the observed UHECR flux towards the higher energies can be considered as the result of reducing the number of sources contributing to the flux.

5.2 Attenuation due to the pair-production process

Gamma rays (γ) propagating from distant sources to Earth interact with the photons of the extragalactic background light (γb) being able to produce e+ethrough the process of pair production

γ+γbe++eE106

which has the effect of a significant energy attenuation in the flux of high-energy gamma rays. Such interaction takes place for gamma rays with energies (Eγ) above the threshold of pair production. The existence of a threshold can be also expressed as the minimum energy (Eγbth) that a γbneeds to produce a e+e.

The following assumptions should be made if we intend to calculate the threshold value of the energy of the gamma-rays photons:

  1. It is needed to take the lowest energy the high-energy photon can have to react with the background photon to yield the two particles which correspond to the situation when they both are produced at rest in their center of mass frame after the collision.

  2. To maximize the energy available from the collision, the initial momenta of the two particles in the lab frame should be pointing in opposite directions.

Let us equate the square of the total 4—momentum pL=pγ+pγbin the lab frame before the collision with the square of the total 4—the momentum of the outgoing particles pCM=p++pin their center of mass frame after the collision

pγ+pγb2=p++p2E107

The right-hand side of (107) is calculated, as follows

p++p()2=p0++p02p1++p12p2++p2()2p3++p32E108

where Eq. (39) are to be substituted into (108), with the three-momentum and energy defined by equations (24), (25) and (21) in which it is set βx=βy=βz=0for both particles. As the result, we obtain the following expression for the square of the total 4—momentum of outgoing particles

p++p2=c2me+me2E109

Note that, although Pxdoes not vanish for βx=βy=βz=0, the component p1of the four-momentum does vanish since, in the expression (39) for p1, the first term compensates the non-vanishing part of Px.

The left-hand side of Eq. (107) is to be expressed in terms of the high-energy and background photons energies using the relations between the particle’s momentum and energies obtained from the dispersion relation (40). The high-energy photons move to the observer, in the direction opposite to the direction the velocity of the lab frame relative to the observer (relative to the preferred frame) which is chosen to be a positive direction of the x-axis. So, the high-energy photon moves along the x-axis in the negative xdirection while the background photon moves, according to the threshold assumption (ii), in the positive xdirection. Thus, the momenta of the photons are related to their energies using Eq. (41), as follows

Pxγ=Eγ1kc,Pxγb=Eγb1+kcE110

where kis the anisotropy parameter in the lab frame. Then the left-hand side of (107) is calculated as follows (head-on collision)

pγ+pγb2=p0γ+p0γb2p1γ+p1γb2=Eγλkc+Eγbλkc2λkkEγcPxγ+λkkEγbcPxγb2E111

Substituting (110) for Pxγand Pxγbinto (111) yields

pγ+pγb2=4λk2EγEγbc2E112

Then using Eqs. (112) and (109) in (107) and solving the resulting equation for Eγone obtains the expression for the threshold energy of the high-energy photon

Eγth=me2c4λk2EγbE113

or the expression for the threshold energy of the background photon (minimum energy to produce e+e)

Eγbth=me2c4λk2EγE114

The factor λkcan be represented as a function Bβ¯of the frame velocity β¯relative to a preferred frame which, with an accuracy up to β¯3, is given by the expression (see (17))

Bβ¯=1β¯2b/2E115

In a cosmological context, where β¯is a recession velocity of a source, β¯depends on the cosmological redshift of an object z. Although the expansion of β¯zin series, besides the leading term z, includes terms of the order z2and higher, they do not contribute to the expression for β¯2up to the terms of the order z3and so, with the accuracy of the expression (115), β¯2can be replaced by z2. Then the threshold equation takes the form

EγbthEγbSth=1z2b;EγbSth=me2c4EγE116

where Eγbthis the modified value of the threshold and EγbSthis the standard value of the threshold. It is seen that the expression (116) for the threshold energy of the background photon differs from the standard one by the factor 1z2b. The universal constant bis negative, both as it is expected from intuitive arguments and as it is found by fitting the cosmological model developed in the framework of the ‘relativity with a preferred frame’ to the observational data [27]. Thus, the threshold energy of the background photon decreases with the distance to the source (the redshift z).

Attenuation of gamma rays with the energy Eγfrom the source at redshift zsdue to the pair production process is characterized by the optical depth τγEγzs. For zsnot too large one typically has τγE0zs<1so that the Universe is optically thin along the line of sight of the source and if it happens that τγE0zs>1the Universe becomes optically thick at some point along the line of sight. The value zhsuch that τγE0zs=1defines the γ-ray horizon for a given E0, and sources beyond the horizon tend to become progressively invisible as zsfurther increases. The optical depth is evaluated by

τγEγzs=0lszsdlKγγbEγlzE117

where KγγbEγlzis the γ-ray absorption coefficient, which represents the probability per unit path length, l, that a γ-ray will be destroyed by the pair-production process. The absorption coefficient is calculated by convolving the spectral number density nbEγbzof background photons at a redshift zwith the cross section of the pair production process σ(EγEγbθz(θis the angle between the direction of propagation of both photons) for fixed values of Eγband θand next integrating over these variables [55], as follows

KγγbEγz=11dcosθ1cosθ2EγbthdEγbnbEγbzσ(EγEγbθzE118

Then the integral over distance lin (117) is represented as an integral over zto arrive at the expression for the optical depth in the form

τγEγzs=0zsdzdlzdzKγγbEγzE119

The threshold energy of background photons Eγbthtaking part in the expressions (118) and (119) is corrected according to (116) such that Eγbthdecreases with the distance to the source (the redshift z). The cumulative outcome of this phenomenon may result in measurable variations in the expected attenuation of the gamma rays flux reducing the expected flux.

The preferred frame effects may influence the optical depth also via the cosmological part of the expression (119). In the Robertson-Walker metric (82) (or (85)), the distance element dlis defined as dl=atwhere atis the scale factor and χis the radial distance element defined by (83). These quantities are calculated based on the GR equations (more specifically, Friedman equations) which leads to the expression (96) for the radial distance χwhere the parameters are to be specified according to the cosmological model accepted. Commonly the quantity dlzdzis calculated within the standard ‘concordance’ ΛCDM cosmological model, where the expression (96) is specified to ΩK=0, ΩΛ=1ΩMand x1given by (97), which yields

dlzdz=1H01z+11ΩM+ΩM1+z3E120

In the cosmology of the relativity with a preferred frame, ΩΛ=0and ΩK=1ΩMand, upon using these values in (96), one has for dlzdzthe following

dlzdz=1H0atat011ΩM+ΩM1+zE121

where the quantity atat0is to be calculated using several other equations as it is done (for the particular case ΩM=1) in equations from (100) to (104). Similar calculations for the general case ΩM1lead to the expression for dlzdzrepresented as series in z, as follows

dlzdz=1H01+2bΩM2z+3+3b+3b22+ΩM+3ΩM28z2+34b5b22ΩM3ΩM28z3E122

In the concordance model relation (120), the value ΩM=0.31. obtained from the observational data (see [27] for references), is used. In the present model, there is an interval of allowed values of ΩMand the corresponding values of b, within which the results fit both the SNIa and BAO data [27]. The curvature Kcin the present model is not obligatory zero but the value of ΩM=1corresponding to the flat universe is within the interval of allowed values of ΩM. Although Eqs. (120) and (122) defining dependence dlzdzon zin the concordance model and in the present model look completely different, the corresponding dependencies practically coincide as it is seen from Figure 3. Thus, the preferred frame effects influence τγEγzsonly via the threshold value Eγbthin (118), like in other Lorentz-violating theories (see, e.g., [56, 57, 58]).

Figure 3.

The dependence ofdlzdz(multiplied byH0) onzfor the concordance model withΩM=0.31(solid) and for the cosmological model, based on ‘relativity with a preferred frame’ [27], withΩM=1,b=0.672(dashed) andΩM=0.5,b=0.495(dotted) where the values of the parametersΩMandbare chosen from those consistent with both the SNIa and BAO data (see [27]).

5.3 Astrophysical tests for vacuum dispersion and vacuum birefringence

In the literature on Lorentz violation, as major features of the behavior of electromagnetic waves in vacuum in the presence of Lorentz violation, vacuum dispersion and vacuum birefringence are considered. Astrophysical tests for vacuum dispersion of light from astrophysical sources seek differences in the velocity of light at different wavelengths due to Lorentz violation which should result in observed arrival-time differences. For differences in the arrival times of different wavelengths to be interpreted as caused by differences in the light velocities, explosive or pulsed sources of radiation that produce light over a wide range of wavelengths in a short period, such as gamma-ray bursts, pulsars, or blazars, are to be used. All those are point sources, which have the disadvantage (to impose constraints on Lorentz violation) that a single line of sight is involved, which provides sensitivity to only a restricted portion of space for free coefficients of the Lorentz violating models.

The same is valid for the present theory leading to the dispersion relation (72). In the case of the waves propagating along the x-axis (aligned with the anisotropy vector k), when qy=qz=0and qx=q, the two routes (74) become

ω=±c1±kqE123

which corresponds to the waves propagating in the opposite directions. For a wave propagating to the observer from a cosmological source, with the x-axis directed from the observer to the source, the group velocity is

ωq=c1kE124

It does not depend on qand so there is no place for vacuum dispersion.

Another test, that is commonly used for setting constraints on the parameters of the Lorentz-violating theories in electrodynamics, is the vacuum birefringence test. In birefringent scenarios, the two eigenmodes propagate at slightly different velocities. This implies that the superposition of the modes is altered as light propagates in free space. Since the two modes differ in polarization, the change in superposition causes a change in the net polarization of the radiation. However, it does not apply to the present theory leading to the dispersion relation (72). The two roots of the dispersion relation correspond to the waves propagating in different directions. Thus, no two eigenmodes are propagating in the same direction and so there is no possibility for vacuum birefringence. Thus, neither tests for vacuum dispersion nor tests for vacuum birefringence can impose restrictions, additional to those imposed by cosmological data, on the values of the only parameter of the theory b.

The vacuum birefringence and vacuum dispersion are widely discussed in the literature as astrophysical tests of Lorentz violation in the pure photon sector of the standard-model extension (e.g., [6, 38, 59, 60, 61, 62]). Therefore it is of interest, in that context, to compare the Lorentz-violating terms, appearing in the Lagrangian due to the preferred frame effects in the present study, with those introduced as a formal SME extension. Extracted from the SME, the Lorentz-violating electrodynamics can be written in terms of the usual field strength Fikdefined by (57) and the potentials Ak, as follows

L=14FikFik14kFnmikFnmFik+12kAFnεnmikAmFikE125

In what follows, we calculate the Lagrangian of the electrodynamics with a preferred frame and compare the Lorentz violating terms in that Lagrangian with those in (125). Calculating L=14FikFikusing equations (58) and (59) yields

L=λk412E2H2+kEyHzEzHyk212Ey2+Ez2E126

It is seen that the form (126) is in a sense more general than (125) because of the Lorentz violating multiplier λk4. However, since the multiplier does not depend on the field variables and so does not influence the form of the field equations, it can be disregarded. Then the Lorentz-violating terms in (126) can be written based on (59) in terms of the field strength, as follows

Ladd=kF02F12+F13F0312k2F02F02+F03F03E127

which fits the form (125) with the coefficients

kF0212=4k,kF0313=4k,kF0202=2k2,kF0303=2k2E128

while other kFnmikas well as all kAFare zeros. The second term on the right-hand side of (125), not contributing to the Lagrangian of the present theory, could be disregarded from the beginning because it has theoretical difficulties associated with negative contributions to the energy [6, 38]. The Lagrangian defined by (127) (or (128)) provides an example of the Lorentz-violating SME (in a pure photon sector) which leads to equations of the electromagnetic wave propagation not exhibiting the vacuum birefringence and vacuum dispersion effects.

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6. Discussion

The ‘relativity with a preferred frame’ incorporates the existence of the cosmological preferred frame into the framework of the theory while preserving fundamental principles of the SR: the principle of relativity and the principle of universality of the light propagation. The relativistic invariance is preserved in the sense, that the physical laws are covariant (their form does not change) under the group of transformations between inertial frames, and the relativistic symmetry is preserved (although modified) in the sense that there exists a combination, a counterpart of the interval of the standard relativity theory, which is invariant under the transformations. The existence of the modified symmetry provides an extension of the theory to general relativity such that the general covariance is also preserved. Thus, the ‘relativity with a preferred frame’ is a relativity theory, both in the special relativity and in the general relativity parts. Except for identifying the preferred frame with a comoving frame of cosmology, the theory does not include any assumptions. No approximations are involved besides approximating the universal function k=Fβ¯, defining dependence of the anisotropy parameter on the frame velocity relative to the preferred frame, by the expression Fβ¯=bβ¯valid up to the third order in β¯. As the result, all the relations of the theory include only one universal parameter b.

The problem of defining allowed values of bis to be considered in the context of verification of the theory by observations since nothing in the theory itself imposes constraints on the values of b. Discussing the results of the application of the theory to natural phenomena, one can separate the conceptual and quantitative aspects. In the conceptual aspect, the cosmological models, developed using the modified general relativity, are of the most importance. First, it is related to the interpretation of the luminosity distance versus redshift relation deduced from the SNIa data, which has played a revolutionary role in the development of modern cosmology concepts. That relation, corresponding to the negative deceleration parameter, cannot be explained using cosmological matter-dominated models (Friedman-Robertson-Walker models) based on the standard general relativity. To explain the data, in modern cosmology, dark energy, a new type of energy with a peculiar equation of state corresponding to negative pressure, is introduced. In the cosmology of the ‘relativity with a preferred frame’, the luminosity distance versus redshift relation for the matter-dominated cosmological model contains corrections, such that the effective deceleration parameter can be negative. As the result, neither the acceleration of the universe expansion nor the dark energy providing the acceleration is needed. The consistency of the cosmological models, based on the ‘relativity with a preferred frame’, is supported by that, for any reasonable value of the parameter ΩM, there exists a value of bsuch that the luminosity distance versus redshift relation fits with high accuracy the SNIa data.

In the applications of the theory to the BAO data, the conceptual and quantitative aspects go together. The BAO observations provide two different sets of data: BAO scales in transverse and line-of-sight directions. Measurements of the angular distribution of galaxies yield the quantity DMzwhich is the comoving angular diameter distance. Measurements of the redshift distribution of galaxies yield the value of the Hubble parameter Hz. The fact that the two regions in the plane ΩMb, within which the predictions of the present theory fit the DMdata and the Hzdata, are overlapped, both provides a support for the theory and places quite tight constraints on the values of the parameters ΩMand bsince they should be confined within a quite narrow overlapping region. An additional (and quite strong) argument in favor of both consistency of the theory and estimates for the parameter bis that the line in the plane (ΩM,b), on which the results of the present model fit the SNIa data, lies within that narrow region. Thus, the results fit well three different sets of observational data with the values of the theory parameter bconfined within a quite narrow interval (approximately from b=0.4to b=0.8).

Next, it might be expected that some constraints on allowed values of bcould arise as the result of applying the theory to the cosmic rays data. In the propagation of the Ultra-High Energy Cosmic Rays from distant sources to Earth, the most remarkable effect is the attenuation due to pion photoproduction by UHECR protons which is characterized by the GZK threshold. Applying the ‘relativity with a preferred frame’ to the calculation of the energy threshold for the attenuation process results in the correction factor to the GZK limit. Although a comparison of that prediction of the theory with the data on the UHECR flux does not straightforwardly lead to constraints on the values of b, another issue, namely the data on the mass composition of UHECR, provides indirect confirmation of the theory. Those data, showing that the UHECR mass composition is dominated by protons only at energies around and below 1018eV and then the fraction of protons is progressively decreasing up to energies of 1019.6eV, contradict the previous consensus that UHECRs are mostly protons accelerated in the sources to >1020eV. The prediction of the ‘relativity with a preferred frame’, that the GZK threshold energy decreases with the distance to the source of the particles (with the values of the parameter bdefined by the cosmological data) allows to resolve, at least, partially, the contradiction between the view, that the primary UHECR flux is mostly protons accelerated to very high energies, and the observational data showing that the fraction of protons in the UHECR is decreasing towards higher energies. The explanation lies in that, because of decreasing the energy threshold with the distance to the source, the number of sources, contributing to the observed flux of protons at a given energy, should be progressively decreasing with the energy increasing.

Applying the modified particle dynamics to the pair-production process, which is responsible for attenuation of the gamma-rays flux, does not provide quantitative constraints on the values of the parameter bor indirect confirmations of the theory. At the same time, the results of applying the modified electromagnetic field dynamics to the behavior of electromagnetic waves in a vacuum maybe counted as a kind of indirect confirmation of the theory. The vacuum birefringence and vacuum dispersions are the features present in the popular Lorentz-violating theories (e.g., [6, 38, 59, 60, 61, 62]) and the fact, that no indications of the existence of those phenomena are found in observations, imposes constraints on the values of numerous parameters of those theories. On the contrary, the electromagnetic field equations and based on them the electromagnetic wave equation of the present theory, although modified such that the Lorentz invariance is violated, does not predict such features as the vacuum birefringence and vacuum dispersion. Thus the absence of observational evidence for the existence of those phenomena may be considered as an argument in favor of the theory.

In general, the fact, that applying the theory containing only one universal parameter to several different phenomena does not lead to any contradictions, proves a consistency of its basic principles. The presence of only one parameter in the theory is a consequence of the fact that, as distinct from the popular Lorentz-violating theories, where Lorenz violation is introduced phenomenologically by adding Lorentz-violating terms to the Lorentz invariant relations, the ‘relativity with a preferred frame’ starts from the physically reasonable modification of the basic postulates of the SR. The generalized relativistic invariance, and so the Lorentz invariance violation, are ingrained in the theory at the most fundamental level being imbedded into the metric. It is also worth to emphasize that the conceptual basis of the theory has been developed without having in mind possible applications. It is aimed at designing the framework which would allow to incorporate the preferred frame into special relativity while retaining the relativity principle and the fundamental space-time symmetry. Nevertheless, the theory provides explanations of some observational data, that were regarded as puzzling after their discovery (like the SNIa luminosity distance-redshift relation indicating the acceleration of the universe and the absence of high energy protons in the UHECR flux). As the result, the concepts (among which dark energy is the most striking one), introduced to explain those puzzling features, become redundant. All the above justifies treating the ‘relativity with a preferred frame’ as an alternative to some currently accepted theories.

Notes

  • It is worth noting that, although the anisotropy of speed of light is one of the central features of the present analysis, this theory stands apart from the ample literature on the conventionality of simultaneity and clock synchronization. A discussion of those issues in the context of the ‘relativity with a preferred frame’ can be found in [29, 30].
  • In the modern versions of the experiments designed to test special relativity and the so-named ‘test theories’ (e.g., [31, 32], see a discussion in [27, 29, 30]), the tests are meant to detect the anisotropy of the two-way speed of light.
  • Although the form (1) seems to be attributed to the one-dimensional formulation, in the three-dimensional case, the equation has the same form if the anisotropy vector k is directed along the x-axis [30]. In the present analysis, the x-axis defines also the line of relative motion of the two frames but it does not lead to any ambiguity. The assumption, that the anisotropy vector k is along the direction of relative motion of the frames S′ and S, is justified by that one of the frames in a set of frames with different values of k is a preferred frame. Since the anisotropy is attributed to the motion with respect to the preferred frame, it is expected that the axis of anisotropy is either in the direction of motion or opposite to it.
  • In the present model, this assumption is not obligatory. It is worthwhile to note that, despite what is frequently claimed, a flatness of the universe is not stated in modern cosmology. Given the fact, that there is no direct measurement procedure of the curvature of space independent of the cosmological model assumed, the flatness of the space is the result valid only within the framework of the ΛCDM model.

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Georgy I. Burde (December 8th 2021). Cosmology and Cosmic Rays Propagation in the Relativity with a Preferred Frame [Online First], IntechOpen, DOI: 10.5772/intechopen.101032. Available from:

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