The relation expressing the *Luminosity Distance dL* of a cosmological source in terms of its redshift *z* is 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

*Cosmology and Cosmic Rays Propagation in the Relativity with a Preferred Frame DOI: http://dx.doi.org/10.5772/intechopen.101032*

$$d\_L = H\_0^{-1} \left( z + \frac{1}{2} \left( \mathbf{1} - q\_0^{(D)} \right) z^2 + \cdots \right) \tag{89}$$

where the deceleration parameter *q* ð Þ *D* 0 is positive for all three possible values of the curvature parameter *Kc* which 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 *q* ð Þ *D* 0 <0 which 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 *dL* is obtained in the form [27]

$$d\_L = a(\eta\_0)(\mathbf{1} + z)\mathbb{S}(\chi\_1) \tag{90}$$

which coincides with a common form of the relation for *dL* [37, 42]. Nevertheless, even though it does not contain the factor *B β*�1 ð Þ, the dependence of *dL* on *z* obtained by eliminating *χ*1 from Eqs. (90) and (88) will differ from the common one since the relation (88) for *z* does contain the factor *B β*�1 ð Þ. To derive the dependence *dL*ð Þ*z* in a closed-form using Eqs. (90) and (88), the function *a*ð Þ*η* determining 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 *ρ* and the form of equation of state giving the pressure *p* as a function of the energy density. The energy density *ρ*ð Þ*t* is usually assumed to be a mixture of non-relativistic matter with equation of state *p* ¼ 0 and dark energy with equation of state *p* ¼ *wρ* while ignoring the relativistic matter (radiation). In the commonly accepted *Λ*CDM model, the dark energy obeys the equation of state with *w* ¼ �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])

$$\left(\left(\mathbf{x}'\right)^2 = H\_0^2 \mathbf{x}^2 \left(\boldsymbol{\Omega}\_{\Lambda} + \boldsymbol{\Omega}\_{\mathbf{M}} \boldsymbol{\infty}^{-3} + \boldsymbol{\Omega}\_{\mathbf{K}} \boldsymbol{\infty}^{-2}\right) \tag{91}$$

where

$$\alpha(t) = \frac{a(t)}{a\_0}, \quad a\_0 = a(t\_0) \tag{92}$$

and the parameters *ΩΛ*, *ΩM* and *ΩK* are defined by

$$\mathcal{Q}\_{\Lambda} = \frac{\rho\_{\rm V0}}{\rho\_{\rm c}}, \quad \mathcal{Q}\_{\rm M} = \frac{\rho\_{\rm M0}}{\rho\_{\rm c}}; \qquad \rho\_{\rm c} = \frac{\mathfrak{H}\_{\rm 0}^{2}}{8\pi G}, \quad \mathfrak{Q}\_{\rm K} = -\frac{K\_{\rm c}}{d\_{0}^{2}H\_{0}^{2}} \tag{93}$$

where *G* is Newton's gravitational constant, *ρV*0 and *ρM*0 are the present energy densities in the vacuum and non-relativistic matter and *ρc* is the critical energy density. Being evaluated at *t* ¼ *t*0 Eq. (91) becomes

$$
\Omega\_{\Lambda} + \Omega\_{\Lambda} + \Omega\_{K} = \mathbf{1} \tag{94}
$$

The Friedmann Eq. (91) allows us to calculate the radial coordinate *χ*1 of an object of a given redshift *z*. Eq. (86) defining *χ*1 can be represented in the form

$$\chi\_1 = \eta\_0 - \eta\_1 = \int\_{\eta\_1}^{\eta\_0} d\eta = \int\_{t\_1}^{t\_0} \frac{dt}{a(t)} = \frac{1}{a\_0} \int\_{\infty}^{1} \frac{d\infty}{\mathbf{x}' \infty} \tag{95}$$

where *x*0 is a function of *x* defined by the Friedmann Eq. (91) and *x*1 ¼ *a t*ð Þ1 *=a*0. Then using Eq. (91) in (95) yields

$$\chi\_1 = \int\_{\infty\_1}^1 \frac{d\mathbf{x}}{a\_0 H\_0 \mathbf{x}^2 \sqrt{\mathcal{Q}\_\Lambda + \mathcal{Q}\_\Lambda \mathbf{x}^{-\beta} + \mathcal{Q}\_\mathcal{K} \mathbf{x}^{-2}}} \tag{96}$$

In the standard cosmology, Eq. (88) (with *B β*�1 ð Þ¼ 1) provides a simple relation

$$x\_1 = \frac{1}{1+z} \tag{97}$$

so that (96) becomes a closed-form relation for *χ*1ð Þ*z* . For a 'concordance' model, which is the flat space *Λ*CDM model, *ΩK* ¼ 0 and *ΩΛ* ¼ 1 � *ΩM*, Eq. (96) can be represented in the form

$$\chi\_{1m}^{\epsilon}(\mathbf{z}\_1) = \int\_0^{\mathbf{z}\_1} \frac{d\mathbf{z}}{\sqrt{\mathbf{1} - \mathfrak{Q}\_M + \mathfrak{Q}\_M(\mathbf{1} + \mathbf{z})^3}}, \quad \chi\_{1m}^{\epsilon} = \chi\_1^{\epsilon} a\_0 H\_0 \tag{98}$$

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

$$d\_L^{\varepsilon}(\mathbf{z}\_1) = \frac{1}{H\_0} (\mathbf{1} + \mathbf{z}\_1) \chi\_{1m}^{\varepsilon}(\mathbf{z}\_1) \tag{99}$$

In the framework of the present analysis, expressing *χ*1 as a function of *z*1 by combining Eqs. (96) and (88) becomes more complicated in view of the fact that *β*�1, and so the factor *B β*�1 ð Þ, depend on *χ*1. We will outline the calculations for the case of a flat universe, *ΩK* ¼ 0, which is also the assumption of the concordance model.¶ 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,

¶ 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.

*Cosmology and Cosmic Rays Propagation in the Relativity with a Preferred Frame DOI: http://dx.doi.org/10.5772/intechopen.101032*

*ΩΛ* ¼ 0, the relation *χ*1ð Þ *x*1 is obtained from (96) in an analytical form, which allows finding the dependence *x*1 *χ*1 ð Þ by inverting the result, as follows

$$\chi\_{1m} = 2(1 - \sqrt{\chi\_1}) \quad \Rightarrow \quad \varkappa \mathbf{1} = \frac{1}{4}(\chi\_{1m} - 2)^2, \quad \chi\_{1m} = a\_0 H\_0 \chi\_1 \tag{100}$$

The dependence *B χ*1 ð Þ, with the accuracy up to third order in *χ*1, is given by [27]

$$B(\bar{\beta}\_1(\chi\_1)) = \mathbf{1} - \frac{b}{2} \left( \chi\_{1m}^2 + \mathcal{Q}\_M \chi\_{1m}^3 \right) \tag{101}$$

Substituting (101) and (100) into (88) reduces the problem to a transcendental equation for *χ*1*m*ð Þ *z*1 , as follows

$$\frac{1}{4}(\chi\_{1m}(\mathbf{z}\_1) - \mathbf{2})^2 (\mathbf{z}\_1 + \mathbf{1}) \left(\mathbf{1} - \frac{\mathbf{b}}{2} \left(\chi\_{1m}(\mathbf{z}\_1)^2 + \chi\_{1m}(\mathbf{z}\_1)^3\right)\right) = \mathbf{1} \tag{102}$$

Representing the solution of (102) as a series in *z*1 yields

$$\chi\_{1m}(z\_1) = z\_1 + \frac{1}{4}(-3 - 2b)z\_1^2 + \frac{1}{8}(5 + 4b + 4b^2)z\_1^3 \tag{103}$$

Then the relation *dL*ð Þ *z*1 , calculated from (90) with *S χ*1 ð Þ¼ *χ*1, is

$$d\_L(x\_1) = \frac{1}{H\_0} \left( x\_1 + \frac{1}{4}(1 - 2b)z\_1^2 + \frac{1}{8}(4b^2 - 1)z\_1^3 \right) \tag{104}$$

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 *dL*ð Þ*z* derived 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 *Λ*CDM 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 *dL*ð Þ*z* produced by the present model with *dc L*ð Þ*z* for the concordance model while using constraints on the parameter *Ωc M* from the SNIa data analysis. It is found that, for every value of *Ωc M* from the interval, defined by fitting the SNIa data to the concordance model, the parameter *b* can be chosen such that the dependence *dL*ð Þ*z* coincided with *dc L*ð Þ*z* with a quite high accuracy (were graphically undistinguishable). An example is given in **Figure 1** where the dependence *dL*ð Þ*z* for *ΩM* ¼ 1 (flat universe), defined by Eq. (104), is plotted for three different values of *b* together with *dc L*ð Þ*z* of the concordance model. It demonstrates that there exists a value of *b* (in the present case it is *b* ¼ 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* ¼ 1 � *ΩM*) show that for every value of *ΩM* >0 there exists the value of *b*, for which the deviation *dL*ð Þ*z* from *dc L*ð Þ*z* is negligible. It is worth clarifying again that the above is intended to be a comparison of the dependence *dL*ð Þ*z* yielded by the present model with that derived from the SNIa observations so that the dependence *dc L*ð Þ*z* for the 'concordance' model plays a role of a fitting formula for the SNIa data.

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

**Figure 1.**

*Dependence of the luminosity distance dL on the red-shift z: thin solid line for the concordance model with Ωc M* ¼ 0*:*31*; short-dashed for the present model with ΩM* ¼ 1*, b* ¼ �1*:*2*; long-dashed for the present model with ΩM* ¼ 1*, b* ¼ �0*:*672*; thick solid for the present model with ΩM* ¼ 1*, b* ¼ �0*:*2*.*

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 *DM*ð Þ*z* and the Hubble parameter *H z*ð Þ respectively. 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 *DM*ð Þ*z* and *H z*ð Þ that 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 *DM*ð Þ*z* and *H z*ð Þ. The two regions in the plane ð Þ *ΩM*, *b* defined 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 *H z*ð Þ and *DM*ð Þ*z* are 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 ð Þ *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.

#### **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 [45–47]) is considered. The interactions of the UHECR with the CMB photons are characterized by a *Cosmology and Cosmic Rays Propagation in the Relativity with a Preferred Frame DOI: http://dx.doi.org/10.5772/intechopen.101032*

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

$$\frac{Eth}{Est} = \left(1 - z^2\right)^{-b}; \quad Est = \frac{\varepsilon\_\pi \left(2\varepsilon\_p + \varepsilon\_\pi\right)}{4E^{(r)}}\tag{105}$$

where *Eth* is the threshold value of the UHECR protons energy calculated using equations of the relativity with a preferred frame, *Est* is the standard value of the GZK threshold calculated using equations of the standard relativity, *εp* ¼ *mpc*2 and *επ* ¼ *mπc*2 are the proton and pion rest energies and *E*ð Þ*γ* is 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 1 � *z*2 ð Þ�*b* . The universal constant *b* is 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).

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 1018 eV and then the fraction of protons is progressively decreasing up to

#### **Figure 2.**

*Left panel: dependence of the correction factor to the GZK threshold on the source redshift z for different values of the parameter b: shot-dashed for b* ¼ �0*:*4*; long-dashed for b* ¼ �0*:*7*; solid for b* ¼ �1*. Right panel: Number of sources ns (in arbitrary units), that may contribute to the observed flux of protons at the energy Ep, versus Ep Est, where Est is the standard GZK threshold value, for different values of b: shot-dashed for b* ¼ �0*:*4*; long-dashed for b* ¼ �0*:*7*; solid for b* ¼ �1*.*

energies of 1019*:*6 eV. It seemed to be not consistent with the general consensus, that UHECRs are mostly protons and that sources should accelerate them to > 1020 eV. 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 *Ep* of the proton energy, there exists a value *zth* of the redshift (distance *Dth*) such that, for the UHECR sources with *D* > *Dth*, the GZK threshold *Eth* is less than *Ep* and so the protons with the energy *Ep* injected by the sources at the distances *D* > *Dth* cannot 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 *Dth* decreasing when the proton energy *Ep* is 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*þ*e*� through the process of pair production

*Cosmology and Cosmic Rays Propagation in the Relativity with a Preferred Frame DOI: http://dx.doi.org/10.5772/intechopen.101032*

*γ* þ *γb* ! *e* þ þ *e* � (106)

which has the effect of a significant energy attenuation in the flux of highenergy 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 (*Eth γb* ) that a *γb* needs to produce a *e*þ*e*�.

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

Let us equate the square of the total 4—momentum *p*ð Þ *L* ¼ *p*ð Þ*γ* þ *p γb* ð Þ in the lab frame before the collision with the square of the total 4—the momentum of the outgoing particles *p*ð Þ *CM* ¼ *p*ð Þ þ þ *p*ð Þ � in their center of mass frame after the collision

$$\left(p^{(\gamma)} + p^{(\gamma\_b)}\right)^2 = \left(p^{(+)} + p^{(-)}\right)^2\tag{107}$$

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

$$\left(\mathbf{p}^{(+)} + \mathbf{p}^{()}\right)^2 = \left(\mathbf{p}\_0^{(+)} + \mathbf{p}\_0^{(-)}\right)^2 - \left(\mathbf{p}\_1^{(+)} + \mathbf{p}\_1^{(-)}\right)^2 - \left(\mathbf{p}\_2^{(+)} + \mathbf{p}\_2^{(\cdot)}\right)^2 - \left(\mathbf{p}\_3^{(+)} + \mathbf{p}\_3^{(-)}\right)^2 \tag{108}$$

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* ¼ *βy* ¼ *βz* ¼ 0 for both particles. As the result, we obtain the following expression for the square of the total 4—momentum of outgoing particles

$$\left(p^{(+)} + p^{(-)}\right)^2 = c^2(m\_\epsilon + m\_\epsilon)^2\tag{109}$$

Note that, although *Px* does not vanish for *βx* ¼ *βy* ¼ *βz* ¼ 0, the component *p*1 of the four-momentum does vanish since, in the expression (39) for *p*1, 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 *x* direction while the background photon moves, according to the threshold assumption (ii), in the positive *x* direction. Thus, the momenta of the photons are related to their energies using Eq. (41), as follows

$$P\_\mathbf{x}^{(\mathbf{y})} = -\frac{E\_\mathbf{y}(\mathbf{1} - k)}{c}, \quad P\_\mathbf{x}^{(\mathbf{y}\_b)} = \frac{E\_{\mathbf{y}\_b}(\mathbf{1} + k)}{c} \tag{110}$$

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

$$\begin{aligned} \left(p^{(\gamma)} + p^{(\gamma\_b)}\right)^2 &= \left(p\_0^{(\gamma)} + p\_0^{(\gamma\_b)}\right)^2 - \left(p\_1^{(\gamma)} + p\_1^{(\gamma\_b)}\right)^2\\ &= \left(\frac{E\_\gamma \lambda(k)}{c} + \frac{E\_{\gamma\_b} \lambda(k)}{c}\right)^2 - \left(\lambda(k)\left(\frac{kE\_\gamma}{c} - P\_x^{(\gamma)}\right) + \lambda(k)\left(\frac{kE\_{\gamma\_b}}{c} - P\_x^{(\gamma\_b)}\right)\right)^2 \end{aligned} \tag{111}$$

Substituting (110) for *P*ð Þ*γ x* and *P γb* ð Þ *x* into (111) yields

$$\left(p^{(\gamma)} + p^{(\gamma\_b)}\right)^2 = 4\lambda(k)^2 \frac{E\_\gamma E\_{\gamma\_b}}{c^2} \tag{112}$$

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

$$E\_{\gamma}^{th} = \frac{m\_e^2 c^4}{\lambda(k)^2 E\_{\gamma\_b}} \tag{113}$$

or the expression for the threshold energy of the background photon (minimum energy to produce *e*þ*e*�)

$$E\_{\gamma\_b}^{th} = \frac{m\_e^2 c^4}{\lambda(k)^2 E\_\gamma} \tag{114}$$

The factor *λ*ð Þ*k* can be represented as a function *B*ð Þ *β*� of the frame velocity *β*� relative to a preferred frame which, with an accuracy up to ð Þ *β*� 3 , is given by the expression (see (17))

$$B(\bar{\beta}) = \left(\mathbf{1} - \bar{\beta}^2\right)^{b/2} \tag{115}$$

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*�ð Þ*z* in series, besides the leading term *z*, includes terms of the order *z*2 and higher, they do not contribute to the expression for *β*�2 up to the terms of the order *z*3 and so, with the accuracy of the expression (115), *β*�2 can be replaced by *z*2. Then the threshold equation takes the form

$$\frac{E\_{\gamma\_b}^{th}}{E\_{\gamma\_b}^{\rm Sth}} = \left(\mathbf{1} - z^2\right)^{-b}; \quad E\_{\gamma\_b}^{\rm Sth} = \frac{m\_e^2 c^4}{E\_\gamma} \tag{116}$$

where *Eth γb* is the modified value of the threshold and *ESth γb* is 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 1 � *z*2 ð Þ�*b* . The universal constant *b* is 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*).

*Cosmology and Cosmic Rays Propagation in the Relativity with a Preferred Frame DOI: http://dx.doi.org/10.5772/intechopen.101032*

Attenuation of gamma rays with the energy *Eγ* from the source at redshift *zs* due to the pair production process is characterized by the optical depth *τγ Eγ* , *zs* � �. For *zs* not too large one typically has *τγ* ð Þ *E*0, *zs* <1 so that the Universe is optically thin along the line of sight of the source and if it happens that *τγ* ð Þ *E*0, *zs* >1 the Universe becomes optically thick at some point along the line of sight. The value *zh* such that *τγ* ð Þ¼ *E*0, *zs* 1 defines the *γ*-ray horizon for a given *E*0, and sources beyond the horizon tend to become progressively invisible as *zs* further increases. The optical depth is evaluated by

$$\pi\_{\mathcal{I}}(E\_{\mathcal{I}}, z\_{\circ}) = \int\_{0}^{l\_{\circ}(x\_{\circ})} dl \, K\_{\mathcal{I}\mathcal{I}b} \left( E\_{\mathcal{I}}, l(z) \right) \tag{117}$$

where *Kγγb Eγ* , *l z*ð Þ � � is the *γ*-ray absorption coefficient, which represents the probability per unit path length, *l*, that a *γ*-ray will be destroyed by the pairproduction process. The absorption coefficient is calculated by convolving the spectral number density *nb Eγb* , *z* � � of background photons at a redshift *z* with the cross section of the pair production process *σ Eγ* , *Eγb* , *θ*, *z* �� � (*θ* is the angle between the direction of propagation of both photons) for fixed values of *Eγb* and *θ* and next integrating over these variables [55], as follows

$$K\_{\mathcal{I}\gamma\_b}(E\_{\mathcal{I}},z) = \int\_{-1}^{1} d(\cos\theta) \, \frac{1-\cos\theta}{2} \int\_{E\_{\gamma\_b}^{\text{id}}}^{\infty} dE\_{\gamma\_b} \, n\_b(E\_{\gamma\_b},z) \, \sigma\left((E\_{\mathcal{I}},E\_{\gamma\_b},\theta,z)\right) \tag{118}$$

Then the integral over distance *l* in (117) is represented as an integral over *z* to arrive at the expression for the optical depth in the form

$$\pi\_{\gamma}(E\_{\gamma}, z\_{\circ}) = \int\_{0}^{\varepsilon\_{\gamma}} dz \, \frac{dl(z)}{dz} \, K\_{\gamma \eta\_{b}}(E\_{\gamma}, z) \tag{119}$$

The threshold energy of background photons *Eth γb* taking part in the expressions (118) and (119) is corrected according to (116) such that *Eth γb* decreases 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 *dl* is defined as *dl* ¼ *a t*ð Þ*dχ* where *a t*ð Þ is the scale factor and *χ* is 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 *χ* where the parameters are to be specified according to the cosmological model accepted. Commonly the quantity *dl z*ð Þ *dz* is calculated within the standard 'concordance' *Λ*CDM cosmological model, where the expression (96) is specified to *ΩK* ¼ 0, *ΩΛ* ¼ 1 � *ΩM* and *x*1 given by (97), which yields

$$\frac{dl(z)}{dz} = \frac{1}{H\_0} \frac{1}{\left(z+1\right)\sqrt{1-\Omega\_M+\Omega\_M(1+z)^3}}\tag{120}$$

In the cosmology of the relativity with a preferred frame, *ΩΛ* ¼ 0 and *ΩK* ¼ 1 � *ΩM* and, upon using these values in (96), one has for *dl z*ð Þ *dz* the following

*Dark Matter - Recent Observations and Theoretical Advances*

$$\frac{dl(z)}{dz} = \frac{1}{H\_0} \frac{a(t)}{a(t\_0)} \frac{1}{\sqrt{1 - \mathcal{Q}\_M + \mathcal{Q}\_M(1+z)}}\tag{121}$$

where the quantity *a t*ð Þ *a t*ð Þ 0 is to be calculated using several other equations as it is done (for the particular case *ΩM* ¼ 1) in equations from (100) to (104). Similar calculations for the general case *ΩM* 6¼ 1 lead to the expression for *dl z*ð Þ *dz* represented as series in *z*, as follows

$$\begin{split} \frac{dl(z)}{dz} &= \frac{1}{H\_0} \left( 1 + \left( -2 - b - \frac{\Omega\_M}{2} \right) z + \left( 3 + 3b + \frac{3b^2}{2} + \Omega\_M + \frac{3\Omega\_M^2}{8} \right) z^2 \\ &+ \left( -3 - 4b - \frac{5b^2}{2} - \Omega\_M - \frac{3\Omega\_M^2}{8} \right) z^3 \right) \end{split} \tag{122}$$

In the concordance model relation (120), the value *ΩM* ¼ 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 *ΩM* and the corresponding values of *b*, within which the results fit both the SNIa and BAO data [27]. The curvature *Kc* in the present model is not obligatory zero but the value of *ΩM* ¼ 1 corresponding to the flat universe is within the interval of allowed values of *ΩM*. Although Eqs. (120) and (122) defining dependence *dl z*ð Þ *dz* on *z* in 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 *τγ Eγ* , *zs* � � only via the threshold value *Eth γb* in (118), like in other Lorentz-violating theories (see, e.g., [56–58]).

#### **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* ¼ *qz* ¼ 0 and *qx* ¼ *q*, the two routes (74) become

$$
\omega = \pm \frac{c}{1 \pm k} q \tag{123}
$$

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

*Cosmology and Cosmic Rays Propagation in the Relativity with a Preferred Frame DOI: http://dx.doi.org/10.5772/intechopen.101032*

#### **Figure 3.**

*The dependence of dl z*ð Þ *dz (multiplied by H*0*) on z for the concordance model with ΩM* ¼ 0*:*31 *(solid) and for the cosmological model, based on 'relativity with a preferred frame' [27], with ΩM* ¼ 1*, b* ¼ �0*:*672 *(dashed) and ΩM* ¼ 0*:*5*, b* ¼ �0*:*495 *(dotted) where the values of the parameters ΩM and b are chosen from those consistent with both the SNIa and BAO data (see [27]).*

$$\frac{\partial \phi}{\partial q} = -\frac{c}{1 - k} \tag{124}$$

It does not depend on *q* and 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, 59–62]). 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 *Fik* defined by (57) and the potentials *Ak* , as follows

$$L = -\frac{1}{4}F\_{ik}F^{ik} - \frac{1}{4}(k\_F)\_{nmik}F^{nm}F^{ik} + \frac{1}{2}(k\_{AF})^n e\_{nmik}A^mF^{ik} \tag{125}$$

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* ¼ � 1 4 *FikFik* using equations (58) and (59) yields

*Dark Matter - Recent Observations and Theoretical Advances*

$$L = \lambda(k)^4 \left(\frac{\mathbf{1}}{2} \left(\mathbf{E}^2 - \mathbf{H}^2\right) + k \left(E\_\gamma H\_\mathbf{z} - E\_\mathbf{z} H\_\mathbf{y}\right) - k^2 \frac{\mathbf{1}}{2} \left(E\_\gamma^2 + E\_\mathbf{z}^2\right)\right) \tag{126}$$

It is seen that the form (126) is in a sense more general than (125) because of the Lorentz violating multiplier *λ*ð Þ*k* 4 . 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

$$L\_{add} = k \left( F^{02} F^{12} + F^{13} F^{03} \right) - \frac{1}{2} k^2 \left( F^{02} F^{02} + F^{03} F^{03} \right) \tag{127}$$

which fits the form (125) with the coefficients

$$(k\_F)\_{0212} = -4k,\ (k\_F)\_{0313} = -4k,\ (k\_F)\_{0202} = 2k^2,\ (k\_F)\_{0303} = 2k^2\tag{128}$$

while other ð Þ *kF nmik* as well as all *kAF* are zeros. The second term on the righthand 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.

#### **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* ¼ *F*ð Þ *β*� , defining dependence of the anisotropy parameter on the frame velocity relative to the preferred frame, by the expression *F*ð Þ¼ *β*� *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 *b* is 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

#### *Cosmology and Cosmic Rays Propagation in the Relativity with a Preferred Frame DOI: http://dx.doi.org/10.5772/intechopen.101032*

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 *b* such 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 *DM*ð Þ*z* which is the comoving angular diameter distance. Measurements of the redshift distribution of galaxies yield the value of the Hubble parameter *H z*ð Þ. The fact that the two regions in the plane ð Þ *ΩM*, *b* , within which the predictions of the present theory fit the *DM* data and the *H z*ð Þ data, are overlapped, both provides a support for the theory and places quite tight constraints on the values of the parameters *ΩM* and *b* since 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 *b* is 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 *b* confined within a quite narrow interval (approximately from *b* ¼ �0*:*4 to *b* ¼ �0*:*8).

Next, it might be expected that some constraints on allowed values of *b* could 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 1018 eV and then the fraction of protons is progressively decreasing up to energies of 1019*:*6 eV, contradict the previous consensus that UHECRs are mostly protons accelerated in the sources to >1020 eV. 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 *b* defined 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 *b* or 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, 59–62]) 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 Lorentzviolating 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.

### **Author details**

Georgy I. Burde

Alexandre Yersin Department of Solar Energy and Environmental Physics, Swiss Institute for Dryland Environmental and Energy Research, Jacob Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Israel

\*Address all correspondence to<EMAIL_ADDRESS>
© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Cosmology and Cosmic Rays Propagation in the Relativity with a Preferred Frame DOI: http://dx.doi.org/10.5772/intechopen.101032*

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#### **Chapter 8**

## Dark Energy as an Information Field and Attribute of the Universe

*Alexander Bogomolov*

#### **Abstract**

The hypothesis that information is the same attribute of the Universe as space, time and matter, which appeared after the Big Bang and its carrier is the dark energy, is proposed. We assume that along with the development of matter and space–time continuum the development of information took place, i.e. its accumulation and complication of forms of its realization in the Universe, and the formation of information field. The proposed concept is close to or coincides with the bio-centrism concept. They substantiate the connection between information and dark energy from philosophical and physical positions. Including properties of "black holes" of the Universe. The rate of increase of knowledge and volumes of created information in the process of human evolution is very similar to the rate of increase of "mass" of dark energy of the Universe. They concluded that structured information in the information field, which forms the noosphere or thinking shell, plays a decisive role both in life of an individual and humanity. Comparison of quantum properties of information field of the Universe (noosphere) and dark energy may lead to new discoveries of the essence of life and the Universe. Studying and mastering of noosphere will mean the transition of humanity to a cosmic stage of its development.

 Introduction**

We live in a universe that is expanding and developing. But once upon a time, the Universe, as most scientists and religious thinkers believe, did not exist, and it arose from something or was created. There are many scientific theories of the origin of the universe, but most scientists have abandoned the idea of an infinite and timeless universe since the 20th century. The most developed, widespread and confirmed by indirect evidence recently got because of new astronomical and cosmological discoveries is the Big Bang theory. According to this theory, the Universe was born because of the Big Bang of some singularity (point) having structure and content about which we know nothing, except that the laws of physics known to us did not operate in it. We will probably never know the cause of the explosion, which occurred about 13.8 billion years ago. After the Big Bang came time, space, which expanded, and matter as elementary particles that later formed various elements (matter). The scattered tiny elements gave rise to the world we know today. After the Big Bang, the universe expanded and is still expanding today (**Figure 1**).

**Figure 1.** *The birth of the universe. Source: Yandex.ru.*

The main disadvantage of the Big Bang theory is the inability to explain the cause of the explosion and the mechanisms of the origin of space–time-matter [1]. Also, the Big Bang theory has difficulties when trying to describe the behavior of the Universe from the position of quantum mechanics. I believe that since the Big Bang emerged such attributes of the universe as space, time, matter. In modern science, several scientists believe that information may also be one attribute of the Universe [2–5]. It is logical to assume that along with the development of matter and space–time continuum, there was the development of information, i.e. its accumulation and complication of forms of its realization.

The modern level of scientific cognition increasingly allows us to connect the progressive development of matter with the processes of reflection and with the accumulation of structural information. The accumulation of information in the process of life not only in an individual person but also in the evolutionary process of the development of society, the transformation of information into knowledge, is characterized in the last 100 years by an explosive character. By 2020 we estimate the accumulated

**Figure 2.** *Explosive increase in information (source IDC).*

#### *Dark Energy as an Information Field and Attribute of the Universe DOI: http://dx.doi.org/10.5772/intechopen.99044*

volume of information at about 40 zettabytes, which is 57 times more than the number of grains of sand on all the beaches of the planet (**Figure 2**).

The growth rate of knowledge and volumes of information created in the process of the evolution of mankind is very similar to the growth rate of the "mass" of the dark energy of the Universe. Its special properties allow it to claim the role of a carrier of information field of the Universe.

#### **2. Dark energy—the main riddle of the fundamental physics of the XXI century**

#### **2.1 Religion about the creation of the world**

The theories of the origin of the universe in various religious movements agree that it was created as a result of a conscious and spiritual action by God or the Creator. The major difference between scientific and religious theories of the origin of the universe is primarily in terminology. Scientific theories use the term "origin" instead of "creation," the term "Nature" instead of "Creator," etc. One can find many similarities between scientific and religious theories of the origin of the universe.

Religious teachings, in fact, have not strayed that far from the model of the creation of the world by the Big Bang. For example, the Bible says that the Word (information) was at the beginning of creation. The question of the reason for the appearance (creation) of the World has concerned thinkers since the fourth century A.D. 1600 years ago, the theologian Augustine the Blessed tried to understand the nature of God before the creation of the universe. He concluded that time was part of God's creation, and there simply was no "before" the creation of the world. The belief in God's creation of the world out of nothing is the basic doctrine of creation in traditional Christian theology.

#### **2.2 String theory**

The theory explaining the origin of the universe because of the Big Bang is not the only one. For example, string theory assumes a cyclical birth and death of the universe2. According to string theory, they do not build the entire world of particles, as we believe it today, but of infinitely thin objects with the ability to make vibrations, which allows us to draw an analogy with strings. With the help of complex mathematical apparatus it is possible to connect these vibrations with energy and hence with mass any particle arises because of one or another type of quantum string vibrations.

Like any unproven theory, string theory has several difficulties, which suggests that it requires enormous refinement.

#### **2.3 Dark matter and dark energy**

Scientists today believe that matter (that we see around us and in the Universe) makes up only less than 5% of the known Universe. The rest is a dark matter at 25% and about 70% is dark energy.

Dark matter is akin to ordinary matter in the sense that it can clump together (say, the size of a galaxy or galaxy cluster) and takes part in gravitational interactions in the same way as ordinary matter. Most likely, it comprises new particles, not yet discovered in terrestrial conditions. This itself testifies that there is a new, not yet discovered law of conservation in nature that forbids these particles to

disintegrate. In our Galaxy, near the Sun, the mass of dark matter is approximately equal to the mass of ordinary matter. These particles must accumulate in the centers of the planets and in the center of the Sun (the matter is practically transparent for them, and they can fall inside the Earth or the Sun). There they annihilate with each other, and other particles, including neutrinos, are produced.

Dark energy is a much stranger substance than dark matter. The mystery of dark energy originated much earlier than the now common term. The starting point of this story was astronomers' estimates of the masses of various galaxies. The estimated mass of the galaxies was about 10 times the total mass of all their stars. This means that some unknown substance handles this mass [6–9]. This substance is called "dark energy". It is the one that dominates the Universe.

They do not collect dark energy in clumps, but is uniformly "spilled" in the Universe. In galaxies and clusters of galaxies there is just as much of it as outside of them. The most unusual thing is that dark energy generates antigravity. This distinguishes it dramatically from ordinary forms of matter. The accumulation and increase of dark energy leads to the expansion of the universe.

#### **2.4 Dark energy and the expansion of the universe**

References astronomical observations show that today (and in the not too distant past) the Universe is expanding with acceleration: the rate of expansion increases with time [10]. In this sense we can talk about antigravity: the usual gravitational attraction would slow the expansion of galaxies, but in our Universe, it turns out that it is the opposite. Dark energy handles the expansion of the Universe with modern astronomical methods it is possible not only to measure the current rate of expansion of the Universe, but also to determine how it has changed. What can be the reason for the increase of an antigravity mass of dark energy, what processes in the Universe also occur at an increasing rate? One of such processes is the process of generation, accumulation and transformation of information in the Universe, in particular on the Earth. What is information?

#### **2.5 Information and the information field**

"Information is information, not matter and not energy" -N. Wiener.

Despite its widespread occurrence, the concept of information remains one of the most debatable in science, and the term may have different meanings in different branches of human activity. The modern level of scientific cognition increasingly allows us to link the progressive development of matter with the processes of reflection and with the accumulation of structural information (knowledge). Information, as a measure of ordering of material structures and their interaction, is an objective participant at all stages of organization of matter. Information participated in the processes of self-organization of matter, contributing to the emergence of life. The most important epistemological problem is understanding the place and role of information in the universe and human life.

One of the most common concepts in physics is the concept of a field. It also used this concept in many other fields of knowledge: cosmogony, astrophysics, quantum physics, biology, etc. The concept of a field also applies to information. The mathematical theory of the information field (IFT) is based on a serious mathematical apparatus [11].

The IFT theory uses computational methods developed for quantum field theory and statistical field theory, allows mathematical derivation of optimal visualization algorithms, data analysis methods and even computer simulation schemes.

Application of IFT algorithms to astronomical datasets provides high accuracy images of the Universe and facilitates the search for weak statistical signals from the Big Bang.

#### **2.6 Noosphere—the information field of the universe**

They commonly referred the information field of the Universe to as the noosphere. В. Vernadsky introduced the concept of the noosphere as the highest form of development of the biosphere [12]. The central theme of the doctrine of the noosphere is the unity of the biosphere and mankind. Vernadsky in his works reveals the roots of this unity, the importance of the biosphere's organization in the development of mankind. This allows us to understand the place and role of historical development of humanity in the biosphere's evolution, the regularities of its transition to the noosphere.

One of the key ideas underlying Vernadsky's theory of the noosphere is that man is not a creature living separately according to its own laws, he exists within nature and is part of it. This unity is primarily because of the functional inseparability of the environment and man, which Vernadsky tried to show. Humanity itself is a natural phenomenon, and it is natural that the influence of the biosphere affects not only the environment of life but also its thinking sphere.

But it is not only nature that affects man; there is also a feedback. And it is not superficial, reflecting the physical influence of man on the environment, it is much deeper. This is proved because recently the planetary geological forces have noticeably intensified. " … we are seeing more and more vividly the geological forces around us in action. This has coincided, hardly coincidentally, with the penetration into the scientific consciousness of the belief about the geological significance of *Homo sapiens*, with the identification of a new state of the biosphere - the noosphere - and is one form of its expression. It is connected, of course, first with the clarification of natural scientific work and thought within the biosphere, where living matter plays a major role".

Developing V. Vernadsky's ideas, P. Teilhard de Chardin [13] called the noosphere a certain "shell of thought" over the Earth. He imagined mind as a flame in which the globe warms up and which gradually covers the planet, forming its new cover: "The Earth is not only covered by myriads of thought grains, but is enveloped by a single thinking shell that forms functionally one vast thought grains on a cosmic scale. The multitude of individual thoughts is grouped and amplified in the act of one unanimous thought.

The "thought shell" over the Earth, or the Noosphere, includes a multitude of individual thought-images (the author's definition) of individual living beings of the Earth, including humans (**Figure 3**).

One can consider that the biofield of an individual human being is his individual noosphere. **Figure 4** shows a symbolic image of man and his noosphere.

Human noosphere is a structural unit of noosphere of the Earth, and the latter in its turn is a structural unit of noosphere of the Cosmos (**Figure 5**).

Such a hierarchical network structure of the Universe noosphere testifies to the universal connection of events occurring in it. Astrophysicist V.A. Ambartsumyan once said: "I sneeze and the whole cosmos will shake" [14]. By the way, the modern quantum theory (Einstein-Podolsky-Rosen paradox) [15] does not deny it. The EPR paradox speaks of the presence of a single wave function of correlated (bound) quantum particles, i.e., of the universal connection of phenomena in nature.

The eminent scientist and naturalist N. Tesla believed that he received his scientific revelations by drawing information from the single information field of the Earth (noosphere) [16].

**Figure 3.** *Thought-images of a person. Source: Yandex.ru.*

**Figure 4.** *Thought-images of a man form a noosphere around him. Source: Yandex.ru.*

Being convinced that the Universe is alive and humans are to a certain extent "automatons" behaving according to the Creator's plans, Tesla proposed an original theory of memory. He believed that the human brain does not possess the ability

*Dark Energy as an Information Field and Attribute of the Universe DOI: http://dx.doi.org/10.5772/intechopen.99044*

**Figure 5.** *Noosphere of the earth. Source: Yandex.ru.*

to remember in the sense that it is commonly believed (biochemically, or rather biophysically), and that memory is merely the ability of the human brain to retrieve this or that information (thought-image) from the noosphere.

Another eminent scientist, Albert Einstein, also believed that our universe is rational. "I want to know how God created this world. I want to know his thoughts, the rest is particular," Einstein said [17].

#### **2.7 Biocentrism**

We should mention the doctrine of biocentrism [18]. Proponents of biocentrism argue that current theories of the physical world do not work. While physics is foundational to the study of the universe, chemistry is foundational to the study of life, scientists need to place biology before the other sciences to get a theory of everything. The theory of bio-centrism is based on the idea of the primacy of cosmic consciousness, cites in its proof the achievements of quantum physics.

Something specially created the world for the existence of life, not only on a microscopic subatomic level but also on the scale of the entire universe. Wherever life appears, the world grows around it.

Scientists have already found many indications that all matter in the universe, from atoms to stars, seems to have been tailor-made just for us. Other researchers speak of the principle of intelligent design and believe that the cosmos is such a fitting place for us for a reason. True, the second concept is a veritable Pandora's Box, for it is a fertile field for all kinds of biblical interpretations and references to other topics, which in this case are completely unimportant or worse. Whatever these new discoveries are called, they have already sparked some serious discussions both within and outside the professional astrophysics community.

In the late 1960s, it had already become clear that if the Big Bang had been a millionth stronger, cosmic matter would have been blown around too quickly, and stars and planets would not have had time to form. As a result, we would not have existed. Even more incredible coincidence seems that all four basic interactions

existing in the Universe and all constants are perfect for the flow of interatomic interactions, the existence of atoms and chemical elements, planets, liquid water and life. It is enough to change any of these constants slightly and we would not exist.

Such life-affirming physical values are woven into the fabric of reality like cotton and linen fibers into good currency. The best known of these constants is probably the gravitational constant, but the fine structure constant plays an equally key role in the existence of life. If this quantity, called "alpha, "were only 1.1 times or more above its present value, nuclear fusion could not take place in stars. The fine structure constant is causing so much scrutiny because the Big Bang formed only hydrogen and helium - and virtually nothing else. Oxygen and carbon are necessary for the existence of life (suffice it to say that oxygen is necessary for the appearance of water), but there is no shortage of oxygen precisely because it is formed in stars as one product of nuclear fusion.

The concept of biocentrism is based on the following principles.

#### **2.8 The noosphere from the physical point of view**

The manifestation of noosphere as information field, connected with the events in the nature and socio-economic sphere, allows us to speak about their universal connection. Nevertheless, the key question is still the nature and essence of noosphere.

The attempt to find an explanation of the noosphere as an information field from the physical point of view or from the point of view of a scientific materialist is of great interest. For example, the existence of the noosphere agrees also with the notion of black holes in modern physics. Nothing, even the smallest particle of light, can avoid the fate of being absorbed into a black hole if it is nearby. The existence of black holes is proven and we observe them in the Universe.

 But then where does the information go? In the late 1990s, Hawking rejected the idea that black holes extinguish information. Instead, the scientist suggested that information may indeed still exist, but in a very different form [19].

The expansion of the Universe testifies to a constant increase in the "mass" of dark energy since the beginning of the Big Bang, and this increase occurs at an increasing rate. Since the Big Bang the structure of the Universe, i.e. the increase of its information or non-entropy, has also been increasing at an increasing rate. In April 1982 (!) Academician M.A. Markov reported at the Presidium of the USSR Academy of Sciences: " … the information field of the Universe is layered and structurally resembles a "matreshka", and each layer is linked hierarchically with higher layers, up to the Absolute, and besides an information bank it is also a regulator of the beginning in the fate of people and humanity" [20]. And if the information field of the Universe really exists, there is no better candidate for its material embodiment than dark energy.

#### **2.9 The fractal principle of the universe**

The development of the Universe and complication of its structure can be linked with increase of its non-entropy (information) but its entropy is increasing which eventually will lead to its thermal death. These two processes resemble by analogy the birth, development and death of a human, which is also a kind of the Universe.

It manifested fractal principle of the Universe construction because structure and form of some objects are similar or repeated in other objects, including those of quite original nature. For example, some scientists find similarity in the structure of galactic networks and neural networks of human brain. Fractal principle also takes place during realization of evolutionary processes of development of absolutely different observable systems. It is possible to find a deep analogy between birth, life and death of a human, and between birth, development and death of the Big Universe [21].

#### **2.10 Similarity between the development of man and the universe**

Human birth also resembles the Big Bang of some singularity - the size of an egg is about 0.1–0.15 microns. Then man grows, his mass increases, and is complexity as a system grows, with accumulation of his non-entropy (information). Eventually man is destroyed (dies), his entropy reaches a maximum, and his non-entropy reaches a minimum or zero.

The growth of non-entropy occurs in an open system because of inflow of energy and information from outside. Similar processes take place, apparently, in the cosmic Universe. Accumulation of information and knowledge on Earth, as well as in other intelligent worlds of the Universe, takes place with increasing pace, as well as the growth of dark energy mass, which gives reason to hypothesize about their strong interconnection or identity.

#### **3. Conclusion**

If dark energy is the information field of the universe.

Most scientists are of the opinion that, besides matter, there is in our world some as yet unknown to science substance, which creates antigravity, almost an order of magnitude greater than the gravity of the material component of the Universe. Before this discovery, astronomy and astrophysics were mainly concerned with the study of the material Universe, comprising atomic matter and radiation. After it will be finally revealed that it is not this part of the Universe that dominates, but a substance of immaterial nature, as yet unknown to science, it will be necessary to make a major revision of current cosmological ideas about our world, which will

lead to a change in the cosmological paradigm, established based on discoveries of the first quarter of the twentieth century.

Someone practically accepted that humanity is at the stage of transition from post-industrial to information society. Information society is such a stage of society development, when the use of information and communication technologies (ICT) has a decisive influence on the main social institutions and spheres of life Information society is a special society, unknown to history. It is difficult or impossible to say what humanity will look like in, say, 100 or 200 years. However, we can probably agree with K.E. Tsiolkovsky that humanity humanity will pass to the "ray form" of its existence and will be spread in such form throughout the Universe [22].

If to suppose that already now informational essence of each person separately, humanity and informational essence of other inhabited worlds are already in the informational field of the Universe - noosphere, then "mastering" of noosphere will mean transition of humanity to the space stage of its development. The transition to the "ray form" or noosphere will occur at some stage of human development. By the way, this may explain why we have not yet met our brothers in mind. Because of the exponential growth of knowledge, intelligent civilizations do not reach the stage of building interstellar ships and interstellar travel. Even earlier, they master knowledge from the cosmic noosphere and go to the ray form, becoming practical gods. The same is waiting for humanity, unless it destroys itself earlier. In the radiant form it is possible to control matter, transforming it into various forms. Perhaps, the results by intelligent entities are displayed in the phenomena that we call UFOs. Otherwise, from the point of view of today's science, it is impossible to explain the appearance, appearance and behavior of UFOs.

Philosophical understanding of the noosphere concept leads to a conclusion formulated by Georg Hegel "Everything real is reasonable, everything reasonable is real". The hypothesis about a close connection between dark energy and the informational field of the Universe, if it finds confirmation, will lead to revolutionary changes in the basic provisions of science, the appearance of new fabulous technologies and radical revision of the tasks and lifestyle of society, putting not material but spiritual values at the top of the list.

#### **Author details**

Alexander Bogomolov Financial Institute under the Government of the Russian Federation, Moscow, Russia

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© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Dark Energy as an Information Field and Attribute of the Universe DOI: http://dx.doi.org/10.5772/intechopen.99044*

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### *Edited by Michael L. Smith*

This book presents several new, important explanations for dark matter, all dissimilar to the discredited subatomic particle-like but invisible matter. One chapter presents evidence that abundant cold hydrogen, baryonic matter, is the source of the missing gravity. Another chapter suggests that dark matter is better explained by stars in spiral galaxies that follow non-Keplerian orbits. A third chapter proposes that gravity attributed to dark matter is due to the sprinkling of black holes throughout galaxies, which is supported by LIGO/Virgo observations. Another chapter questions the assumptions of the Friedmann (FLRW) model, proposing a better method for handling astrophysical data. Additional chapters discuss cosmic ray propagation, axion decay, the cosmological scale factor, and the philosophical outlook of cosmologists when dealing with the questions of dark matter and dark energy.

Published in London, UK © 2022 IntechOpen © Elen11 / iStock

Dark Matter - Recent Observations and Theoretical Advances

Dark Matter

Recent Observations and Theoretical Advances

*Edited by Michael L. Smith*


**2. Dysphagia**

Swallowing is a complex sensoriomotor function that depends on the integrity of the mecha‐ noreceptors and chemoreceptors for the sequential stimulation and inhibition of the upper aerodigestive tract; this coordinated process transports foods and liquids through the mouth and pharynx to the esophagus [10] and simultaneously protect the upper airway [1].

Biomechanical events that contribute to secure bolus transport and airway protective mecha‐ nisms include: closure of the introitus to the trachea by vocal cord adduction, approximation of the adducted arytenoids to close the laryngeal aditus, epiglottal descent, antero-superior displacement of the larynx away from the path of the bolus, and opening of the upper oesophageal sphincter [11]. During swallowing, the closure of the larynx and the respiratory pause during swallowing are vital protective mechanisms that prevent aspiration [12]. This phenomena is describes schematically in Figure 2.

**Figure 2.** Simplified description of the usual swallowing pattern in healthy adults. Swallowing initiates during the ex‐ piratory phase of the breathing cycle. Swallowing interrupts exhalation (swallow apnea), and once the swallow has been completed, breathing resumes with exhalation.

Disordered swallowing, or dysphagia, can develop from lesions in certain areas of the cortex and brainstem that control the swallowing function, or damage to the associated cranial nerves. It is a common problem observed in patients with stroke and head injury [13]. Dysphagia affects at least 12% of patients in acute care hospitals and more than 50% of those in chronic care settings [14]. The presence of dysphagia is associated with aspiration induced chest infections and increases the risk of serious respiratory consequences such as pneumonia. Identification of the patient at risk of aspiration is important from a clinical view point. Due to the risk of aspiration, a significant number of dysphagic patients are fed with a nasogastric tube and /or intravenous fluids [13]. Dysphagia can significantly impact a person's quality of life as well as their health status [15].


**3. Conclusions**

Elastography is a noninvasive, US-based, real-time imaging modality that can be a valuable tool in orienting the diagnosis and can be integrated into imaging protocols already involving the standard US to obtain a multiparametric approach. Although SWE has excellent potential in characterizing FLLs, further research is needed to evaluate the accuracy of these methods and set specific cut-off values.

Published studies have conflicting results, and no consensus has been so far established. This is the main reason that the WFUMB [13] and EFSUMB [21] guidelines do not recommend the use of elastography for differentiation between malignant and benign FLLs. Further multicentric studies with larger and homogenous cohorts are required for these techniques to be appropriately used routinely in a clinical setting.