# Considering light-matter interactions in Friedmann equations

V. Vavryčuk  The Czech Academy of Sciences  Boční II 1401, 141 00 Praha 4
<EMAIL_ADDRESS>
(Received May 22, 2020)

###### Abstract

The Friedmann equations valid for the transparent universe are modified for
the universe with opacity caused by absorption of light by ambient cosmic dust
in intergalactic space. The modified equations lead to a cosmological model,
in which cosmic opacity produces radiation pressure that counterbalances
gravitational forces. The proposed model predicts a cyclic
expansion/contraction evolution of the Universe within a limited range of
scale factors with no initial singularity. The maximum redshift, at which the
contraction of the Universe stops, is $z\approx$ 14-15. The model avoids dark
energy and removes some other tensions of the standard cosmological model.

early universe – cosmic background radiation – dust, extinction – universe
opacity – dark energy

##  1 Introduction

Dust is an important component of the interstellar and intergalactic medium,
which interacts with the stellar radiation. Dust grains absorb and scatter the
starlight and reemit the absorbed energy at infrared, far-infrared and
microwave wavelengths (Mathis, 1990; Schlegel et al., 1998; Calzetti et al.,
2000; Draine, 2003, 2011; Vavryčuk, 2018). Since galaxies contain interstellar
dust, they lose their transparency and become opaque (Calzetti, 2001; Holwerda
et al., 2005, 2007; Finkelman et al., 2008; Lisenfeld et al., 2008).
Similarly, the Universe is not transparent but partially opaque due to ambient
cosmic dust. The cosmic opacity is very low in the local Universe (Chelouche
et al., 2007; Muller et al., 2008), but it might steeply increase with
redshift (Ménard et al., 2010b; Xie et al., 2015; Vavryčuk, 2017b).

The fact that the Universe is not transparent but partially opaque might have
fundamental cosmological consequences, because the commonly accepted
cosmological model was developed for the transparent universe. Neglecting
cosmic opacity produced by intergalactic dust may lead to distorting the
observed evolution of the luminosity density and the global stellar mass
density with redshift (Vavryčuk, 2017b). Non-zero cosmic opacity may
invalidate the interpretation of the Type Ia supernova (SNe Ia) dimming as a
result of dark energy and the accelerating expansion of the Universe (Aguirre,
1999a, b; Aguirre & Haiman, 2000; Ménard et al., 2010a; Vavryčuk, 2019).
Intergalactic dust can partly or fully produce the cosmic microwave background
(CMB) (Wright, 1982; Bond et al., 1991; Narlikar et al., 2003). For example,
Vavryčuk (2018) showed that thermal radiation of dust is capable to explain
the spectrum, intensity and temperature of the CMB including the CMB
temperature/polarization anisotropies.

If cosmic opacity and light-matter interactions are considered, the Friedmann
equations must be modified and the radiation pressure caused by absorption of
photons by dust grains must be incorporated. Based on numerical modeling and
observations of basic cosmological parameters, I show that the modified
Friedmann equations avoid the initial singularity and lead to a cyclic model
of the Universe with expansion/contraction epochs within a limited range of
scale factors.

##  2 Theory

###  2.1 Friedmann equations for the transparent universe

The standard Friedmann equations read (Peacock, 1999, p. 665)

${\left({\frac{\dot{a}}{a}}\right)}^{2}=\frac{8\pi
G}{3}\rho-\frac{kc^{2}}{a^{2}}\,,$ (1) $\frac{\ddot{a}}{a}=-\frac{4\pi
G}{3}\left(\rho+\frac{3p}{c^{2}}\right)\,,$ (2)

where $a=R/R_{0}=\left(1+z\right)^{-1}$ is the relative scale factor, $G$ is
the gravitational constant, $\rho$ is the mass density, $k/a^{2}$ is the
spatial curvature of the universe, $p$ is the pressure, and $c$ is the speed
of light. Considering the mass density $\rho$ as a sum of matter and radiation
contributions and including the vacuum contribution, we get

$\frac{8\pi
G}{3}\rho=H^{2}_{0}\left[{\Omega_{m}a^{-3}+\Omega_{r}a^{-4}+\Omega_{\Lambda}}\right]\,.$
(3)

Eq. (1) is then rewritten as

$H^{2}\left(a\right)=H^{2}_{0}\left[{\Omega_{m}a^{-3}+\Omega_{r}a^{-4}+\Omega_{\Lambda}+\Omega_{k}a^{-2}}\right]\,,$
(4)

with the condition

$\Omega_{m}+\Omega_{r}+\Omega_{\Lambda}+\Omega_{k}=1\,,$ (5)

where $H(a)=\dot{a}/a$ is the Hubble parameter, $H_{0}$ is the Hubble
constant, and $\Omega_{m}$, $\Omega_{r}$ , $\Omega_{\Lambda}$ and $\Omega_{k}$
are the normalized matter, radiation, vacuum and curvature terms. Assuming
$\Omega_{r}=0$ and $\Omega_{k}=0$ in Eq. (4), we get the $\Lambda$CDM model

$H^{2}\left(a\right)=H^{2}_{0}\left[{\Omega_{m}a^{-3}+\Omega_{\Lambda}}\right]\,,$
(6)

which describes a flat, matter-dominated universe. The universe is
transparent, because any interaction of radiation with matter is neglected.
The vacuum term $\Omega_{\Lambda}$ is called dark energy and it is responsible
for the accelerating expansion of the Universe. The dark energy is introduced
into Eqs (3-5) to fit the $\Lambda$CDM model with observations of the Type Ia
supernova dimming.

###  2.2 Friedmann equations for the opaque universe

The basic drawback of the $\Lambda$CDM model is its assumption of transparency
of the Universe and neglect of the universe opacity caused by interaction of
light with intergalactic dust. Absorption of light by cosmic dust produces
radiation pressure acting against the gravity, but this pressure is ignored in
the $\Lambda$CDM model.

Let us assume a space filled by light and cosmic dust formed by uniformly
distributed spherical dust grains. The dust grains absorb photons and reemit
them in the form of thermal radiation. The total force produced by absorption
of photons, which acts on dust in a unit volume of the Universe, is

$M_{D}\ddot{R}=S_{D}p_{D}\,,$ (7)

where $M_{D}$ and $S_{D}$ are the mass and surface of all dust grains in the
spherical volume of radius $R$, and $p_{D}$ is the radiation pressure caused
by dust absorption of the extragalactic background light (EBL) present in the
cosmic space

$p_{D}=\frac{\lambda}{c}I^{\mathrm{EBL}}\,,$ (8)

where $\lambda$ is the bolometric cosmic opacity (defined as attenuation per
unit raypath), and $I^{\mathrm{EBL}}$ is the bolometric intensity of the EBL,
which depends on redshift as (Vavryčuk, 2018, his eq. 5)

$I^{\mathrm{EBL}}=I^{\mathrm{EBL}}_{0}\left(1+z\right)^{4}\,,$ (9)

where subscript ’0’ means the quantity at $z=0$. Since the production and
absorption of photons should be in balance, the EBL intensity
$I_{0}^{\mathrm{EBL}}$ is related to the luminosity density $j$ at $z=0$ as
(Vavryčuk, 2018, his eq. 7)

$\lambda_{0}I^{\mathrm{EBL}}_{0}=\frac{j_{0}}{4\pi}\,.$ (10)

If the comoving number density of dust grains is constant, the opacity
$\lambda$ in Eq. (8) is redshift independent, $\lambda=\lambda_{0}$ (the
proper attenuation coefficient per unit ray path increases with $z$, but the
proper length of a ray decreases with $z$). Hence, the pressure $p_{D}$ in Eq.
(8) reads

$p_{D}=\frac{j_{0}}{4\pi c}\left(1+z\right)^{4}\,.$ (11)

Inserting Eq. (11) into Eq. (7) and substituting $R$ by the relative scale
factor $a=R/R_{0}$, we obtain

$\ddot{a}=\frac{S_{D}}{M_{D}}\frac{j_{0}}{4\pi c}\frac{1}{a^{4}}\,,$ (12)

where $R_{0}=1$. Integrating Eq. (12) in time

${\left({\frac{\dot{a}}{a}}\right)}^{2}=\frac{S_{D}}{M_{D}}\frac{j_{0}}{6\pi
c}\frac{1}{a^{5}}\,,$ (13)

and including absorption terms defined in Eqs (12-13) into Eqs (1-2), we get a
new form of the Friedmann equations valid for a model of the opaque universe

${\left({\frac{\dot{a}}{a}}\right)}^{2}=\frac{8\pi
G}{3}\rho-\frac{S_{D}}{M_{D}}\frac{j_{0}}{6\pi
c}\frac{1}{a^{5}}-\frac{kc^{2}}{a^{2}}\,,$ (14)
$\frac{\ddot{a}}{a}=-\frac{4\pi
G}{3}\left(\rho+\frac{3p_{D}}{c^{2}}\right)+\frac{S_{D}}{M_{D}}\frac{j_{0}}{4\pi
c}\frac{1}{a^{5}}\,,$ (15)

which read for dust formed by spherical grains as

${\left({\frac{\dot{a}}{a}}\right)}^{2}=\frac{8\pi G}{3}\rho-\frac{1}{2\pi
c}\frac{j_{0}}{\rho_{D}R_{D}}\frac{1}{a^{5}}-\frac{kc^{2}}{a^{2}}\,,$ (16)
$\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}\rho+\frac{3}{4\pi
c}\frac{j_{0}}{\rho_{D}R_{D}}\frac{1}{a^{5}}\,,$ (17)

where $R_{D}$ and $\rho_{D}$ are the radius and the specific density of dust
grains. In Eq. (17), we omit gravity forces produced by pressure $p_{D}$,
because they are negligible with respect to the other terms. Consequently, the
Hubble parameter reads

$H^{2}\left(a\right)=H^{2}_{0}\left[{\Omega_{m}a^{-3}+\Omega_{r}a^{-4}+\Omega_{a}a^{-5}+\Omega_{k}a^{-2}}\right]\,,$
(18)

which simplifies for a matter-dominated opaque universe ($\Omega_{r}=0$) as

$H^{2}\left(a\right)=H^{2}_{0}\left[{\Omega_{m}a^{-3}+\Omega_{a}a^{-5}+\Omega_{k}a^{-2}}\right]\,,$
(19)

with the condition

$\Omega_{m}+\Omega_{a}+\Omega_{k}=1\,,$ (20)

where $\Omega_{m}$, $\Omega_{a}$ and $\Omega_{k}$ are the normalized gravity,
absorption and curvature terms, respectively,

$\Omega_{m}=\frac{1}{H^{2}_{0}}\left({\frac{8}{3}\pi G\rho_{0}}\right)\,,$
(21) $\Omega_{a}=-\frac{1}{H^{2}_{0}}\left({\frac{1}{2\pi
c}\frac{j_{0}}{\rho_{D}R_{D}}}\right)\,,$ (22)
$\Omega_{k}=-\frac{kc^{2}}{H^{2}_{0}}\,.$ (23)

The minus sign in Eq. (22) means that the radiation pressure due to absorption
acts against the gravity. The dark energy is missing in Eqs (18-20), because
the Type Ia supernova dimming can successfully be explained by cosmic opacity,
as discussed in Vavryčuk (2019).

Eq. (19) shows that the increase of the absorption term $\Omega_{a}$ with
redshift is enormously high. The reasons for such a steep rise of $\Omega_{a}$
with $z$ are, however, straightforward. The steep rise combines the three
following effects: (1) the increase of photon density with $(1+z)^{3}$ due to
the space contraction, (2) the increase of absorption of photons with $(1+z)$
due to the shorter distance between dust grains, and (3) the increase of rate
of absorbed photons by dust grains with $(1+z)$ due to time dilation.

###  2.3 Distance-redshift relation

The scale factor $a$ of the Universe with the zero expansion rate is defined
by the zero Hubble parameter in Eq. (19), which yields a cubic equation in $a$

$\Omega_{k}a^{3}+\Omega_{m}a^{2}+\Omega_{a}=0\,.$ (24)

Taking into account that $\Omega_{m}>0$ and $\Omega_{a}<0$, Eq. (24) has two
distinct real positive roots for

$\left({\frac{\Omega_{m}}{3}}\right)^{2}>\left({\frac{\Omega_{k}}{2}}\right)^{2}\left|\Omega_{a}\right|\,\,\,\mathrm{and}\,\,\,\Omega_{k}<0\,.$
(25)

Negative $\Omega_{a}$ and $\Omega_{k}$ imply that

$\Omega_{m}>1\,\,\,\mathrm{and}\,\,\,\rho_{0}>\rho_{c}=\frac{8\pi
G}{3H^{2}_{0}}\,.$ (26)

Under these conditions, Eq. (19) describes a universe with a cyclic
expansion/contraction history and the two real positive roots
$a_{\mathrm{min}}$ and $a_{\mathrm{max}}$ define the minimum and maximum scale
factors of the Universe. For $\Omega_{a}\ll 1$, the scale factors
$a_{\mathrm{min}}$ and $a_{\mathrm{max}}$ read approximately

$a_{\mathrm{min}}\cong\sqrt{\left|{\frac{\Omega_{a}}{\Omega_{m}}}\right|}\,\,\,\mathrm{and}\,\,\,a_{\mathrm{max}}\cong\left|{\frac{\Omega_{m}}{\Omega_{k}}}\right|\,,$
(27)

and the maximum redshift is

$z_{\mathrm{max}}=\frac{1}{a_{\mathrm{min}}}-1\,.$ (28)

The scale factors $a$ of the Universe with the maximum expansion/contraction
rates are defined by

$\frac{d}{da}H^{2}\left(a\right)=0\,,$ (29)

which yields a cubic equation in $a$

$2\Omega_{k}a^{3}+3\Omega_{m}a^{2}+5\Omega_{a}=0\,.$ (30)

Taking into account Eq. (17) and Eqs (21-23), the deceleration of the
expansion reads

$\ddot{a}=-\frac{1}{2}H^{2}_{0}\left[\Omega_{m}a^{-2}+3\Omega_{a}a^{-4}\right]\,.$
(31)

Hence, the zero deceleration is for the scale factor

$a=\sqrt{\left|{\frac{3\Omega_{a}}{\Omega_{m}}}\right|}\,.$ (32)

Finally, the comoving distance as a function of redshift is expressed from Eq.
(19) as follows

$\begin{split}dr=\frac{c}{H_{0}}\frac{dz}{\sqrt{\Omega_{m}\left(1+z\right)^{3}+\Omega_{a}\left(1+z\right)^{5}+\Omega_{k}\left(1+z\right)^{2}}}\,.\end{split}$
(33)

##  3 Modeling

Table 1: Maximum redshift and scale factor in the cyclic model of the opaque
universe

Model | $\varepsilon$ | $c_{\mathrm{S/M}}$ | $j_{0}$ | $\Omega_{m}$ | $\Omega_{a}$ | $\Omega_{k}$ | $a_{\mathrm{max}}$ | $z_{\mathrm{max}}$
---|---|---|---|---|---|---|---|---
|  |  | $(10^{8}\,h\,L_{\odot}\,\mathrm{Mpc}^{-3})$ |  |  |  |  |
A | 20 | 4.0 | 3.8 | 1.2 | $\,\,-7.2\times 10^{-3}$ | $\,\,-0.192$ | 6.2 | 11.8
B | 5 | 1.9 | 2.5 | 1.2 | $\,\,-2.3\times 10^{-3}$ | $\,\,-0.198$ | 6.1 | 21.8
C | 15 | 3.3 | 3.1 | 1.2 | $\,\,-4.8\times 10^{-3}$ | $\,\,-0.195$ | 6.1 | 14.7
D | 15 | 3.3 | 3.1 | 1.1 | $\,\,-4.8\times 10^{-3}$ | $\,\,-0.095$ | 11.6 | 14.1
E | 15 | 3.3 | 3.1 | 1.3 | $\,\,-4.8\times 10^{-3}$ | $\,\,-0.295$ | 4.4 | 15.3

  * •

Parameter $\varepsilon$ is the ratio of the major to minor axis of the prolate
spheroidal dust grains, $c_{\mathrm{S/M}}$ is the correction for the S/M ratio
of the spheroidal to spherical dust grains, $j_{0}$ is the bolometric
luminosity density at $z=0$, $\Omega_{m}$, $\Omega_{a}$, and $\Omega_{k}$ are
the matter, absorption and curvature terms, and $a_{\mathrm{max}}$ and
$z_{\mathrm{max}}$ are the maximum scale factor and redshift, respectively.
Models A, B and C predict low, high and optimum values of $z_{\mathrm{max}}$.
Models E, D and C predict low, high and optimum values of $a_{\mathrm{max}}$.

###  3.1 Parameters for modeling

For calculating the expansion history and cosmic dynamics of the Universe, we
need observations of intergalactic dust grains, the galaxy luminosity density,
the mean mass density, and the expansion rate and curvature of the Universe at
the present time.

The size $a$ of dust grains is in the range of $0.01-0.2\,\mu$m with a power-
law distribution $a^{-q}$ with $q=3.5$ (Mathis et al., 1977; Jones et al.,
1996), but silicate and carbonaceous grains dominating the scattering are
typically with $a\approx 0.1\mu$m (Draine & Fraisse, 2009; Draine, 2011). The
grains of size $0.07\,\mu\mathrm{m}\leq a\leq 0.2\,\mu\mathrm{m}$ are also
ejected to the IGM most effectively (Davies et al., 1998; Bianchi & Ferrara,
2005). The grains form complicate fluffy aggregates, which are often elongated
or needle-shaped (Wright, 1982, 1987). Considering that the density of
carbonaceous material is $\rho\approx 2.2\,\mathrm{g\,cm}^{-3}$, and the
silicate density is $\rho\approx 3.8\,\mathrm{g\,cm}^{-3}$ (Draine, 2011), the
average density of porous dust grains is $\approx 2\,\mathrm{g\,cm}^{-3}$ or
less (Flynn, 1994; Kocifaj et al., 1999; Kohout et al., 2014).

The galaxy luminosity density is determined from the Schechter function
(Schechter, 1976). It has been measured by large surveys 2dFGRS (Cross et al.,
2001), SDSS (Blanton et al., 2001, 2003) or CS (Brown et al., 2001). The
luminosity function in the R-band was estimated at $z=0$ to be $\left(1.84\pm
0.04\right)\times 10^{8}\,\,h\,L_{\odot}\,\,\mathrm{Mpc^{-3}}$ for the SDSS
data (Blanton et al., 2003) and $\left(1.9\pm 0.6\right)\times
10^{8}\,\,h\,L_{\odot}\,\,\mathrm{Mpc^{-3}}$ for the CS data (Brown et al.,
2001). The bolometric luminosity density is estimated by considering the
spectral energy distribution (SED) of galaxies averaged over different galaxy
types, being thus $1.4-2.0$ times larger than that in the R-band (Vavryčuk,
2017b, his table 2): $j=2.5-3.8\times
10^{8}\,\,h\,L_{\odot}\,\mathrm{Mpc^{-3}}$.

The Hubble constant $H_{0}$ is measured by methods based on the Sunyaev-
Zel’dovich effect (Birkinshaw, 1999; Bonamente et al., 2006) or gravitational
lensing (Suyu et al., 2013; Bonvin et al., 2017), gravitational waves (Vitale
& Chen, 2018; Howlett & Davis, 2020) or acoustic peaks in the CMB spectrum
provided by Planck Collaboration et al. (2016), and they yield values mostly
ranging between 66 and 74 $\mathrm{km\,s^{-1}\,Mpc^{-1}}$. Here I use an
estimate $H0=69.8\pm 2.5\,\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$ of $H_{0}$ obtained
by Freedman et al. (2019) using the SNe Ia with a red giant branch
calibration.

Assuming the $\Lambda$CDM model, the CMB and BAO observations indicate a
nearly flat Universe (Planck Collaboration et al., 2016). This method is not,
however, model independent and ignores an impact of cosmic dust on the CMB. A
model-independent method proposed by Clarkson et al. (2007) is based on
reconstructing the comoving distances by Hubble parameter data and comparing
with the luminosity distances (Li et al., 2016; Wei & Wu, 2017) or the angular
diameter distances (Yu & Wang, 2016). The cosmic curvature can also be
constrained using strongly gravitational lensed SNe Ia (Qi et al., 2019) and
using lensing time delays and gravitational waves (Liao, 2019). The authors
report the curvature term $\Omega_{k}$ ranging between -0.3 to 0 indicating a
closed universe, not significantly departing from flat geometry.

Figure 1:  Maximum redshift as a function of $\Omega_{m}$ and $\Omega_{a}$.

###  3.2 Results

Figure 2:  The evolution of the Hubble parameter with redshift in the past and
with the scale factor in the future (in $\mathrm{km\,s^{-1}\,Mpc^{-1}}$). (a)
The blue dashed, dotted and solid lines show Models A, B and C in Tab. 2. (b)
The blue solid, dashed, and dotted lines show Models C, D and E in Tab. 2. The
black dotted lines mark the predicted maximum redshifts (a) and maximum scale
factors (b) for the models considered. The black dot denotes the state in C
when the deceleration of the expansion is zero. The dot is not at the maximum
of $H(z)$ because the zero deceleration is with respect to time but not with
respect to $z$. The red solid line shows the flat $\Lambda$CDM model with
$H_{0}=69.8\,\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$, taken from Freedman et al.
(2019), and with $\Omega_{m}=0.3$ and $\Omega_{\Lambda}=0.7$.  Figure 3:
Comoving distance as a function of redshift $z$. The blue dashed, dotted and
solid lines show Models A, B and C in Tab. 2. The black dotted lines mark the
predicted maximum redshifts for the models considered. The red solid line
shows the flat $\Lambda$CDM model with
$H_{0}=69.8\,\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$, taken from Freedman et al.
(2019), and with $\Omega_{m}=0.3$ and $\Omega_{\Lambda}=0.7$.

Estimating the required cosmological parameters from observations (see Table
1), I calculate the upper and lower limits of the volume of the Universe and
the evolution of the Hubble parameter with time. The mass density of the
Universe higher than the critical density is considered, and subsequently
$\Omega_{m}$ is higher than 1. The Hubble constant is
$H_{0}=69.8\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$, taken from Freedman et al.
(2019). The dust grains are assumed to be prolate spheroids with a varying
shape ratio. The specific dust density is $2000\,\mathrm{kg\,m^{-3}}$.
Parameter $\Omega_{a}$ varies from $-7.6\times 10^{-3}$ to $-2.5\times
10^{-3}$ depending on the luminosity density $j_{0}$ and the spheroidal shape
of the dust grains (see Eq. (22) and Table 1).

As seen in Fig. 1, the maximum redshift of the Universe depends mostly on
$\Omega_{a}$, and ranges from 11.5 to 21.3. The maximum redshift
$z_{\mathrm{max}}$ calculated approximately by Eqs (26-27) has an accuracy
higher than 1% compared to the exact solution of Eq. (24). In contrast to
$a_{\mathrm{min}}$ depending mostly on $\Omega_{a}$, the maximum scale factor
$a_{\mathrm{max}}$ of the Universe depends primarily on $\Omega_{m}$. The
limiting value is $\Omega_{m}=1$, when $a_{\mathrm{max}}$ is infinite. For
$\Omega_{m}=1.1$, 1.2, 13 and 1.5, the scale factor $a_{\mathrm{max}}$ is
11.6, 6.5, 4.4 and 3.0, respectively.

The history of the Hubble parameter $H(z)$ and its evolution in the future
$H(a)$ calculated by Eq. (19) is shown in Fig. 2 for five scenarios summarized
in Table 1. As mentioned, the form of $H(z)$ is controlled by $\Omega_{a}$
(Fig. 2a), while the form of $H(a)$ is controlled by $\Omega_{m}$ (Fig. 2b).
The Hubble parameter $H(z)$ increases with redshift up to its maximum. After
that the function rapidly decreases to zero. The drop of $H(z)$ is due to a
fast increase of light attenuation producing strong repulsive forces at high
redshift. For future epochs, function $H(a)$ is predicted to monotonously
decrease to zero. The rate of decrease is controlled just by gravitational
forces; the repulsive forces originating in light attenuation are negligible.
For a comparison, Fig. 2 (red line) shows the Hubble parameter $H(a)$ for the
standard $\Lambda$CDM model (Planck Collaboration et al., 2016), which is
described by Eq. (6) with $\Omega_{m}=0.3$ and $\Omega_{\Lambda}=0.7$. The
deceleration of the expansion becomes zero before $H(z)$ attains its maximum
(see the black dot in Fig. 2a). The redshift of the zero deceleration is about
2/3 of the maximum achievable redshift.

The distance-redshift relation for the proposed cyclic model of the Universe
is quite different from the standard $\Lambda$CDM model (see Fig. 3). In both
models, the comoving distance monotonously increases with redshift, but the
redshift can go possibly to 1000 or more in the standard model, while the
maximum redshift is likely 14-15 in the cyclic model. The increase of distance
with redshift is remarkably steeper for the $\Lambda$CDM model than for the
cyclic model. The ratio between distances in the cyclic and $\Lambda$CDM
models is about 0.54.

##  4 Other supporting evidence

The cyclic cosmological model of the opaque universe successfully removes some
tensions of the standard $\Lambda$CDM model:

  * •

The model does not limit the age of stars in the Universe. For example,
observations of a nearby star HD 140283 (Bond et al., 2013) with age of
$14.46\pm 0.31\,\mathrm{Gyr}$ are in conflict with the age of the Universe,
$13.80\pm 0.02\,\mathrm{Gyr}$, determined from the interpretation of the CMB
as relic radiation of the Big Bang (Planck Collaboration et al., 2016).

  * •

The model predicts the existence of very old mature galaxies at high
redshifts. The existence of mature galaxies in the early Universe was
confirmed, for example, by Watson et al. (2015) who analyzed observations of
the Atacama Large Millimetre Array (ALMA) and revealed a galaxy at $z>7$
highly evolved with a large stellar mass and heavily enriched in dust.
Similarly, Laporte et al. (2017) analyzed a galaxy at $z\approx 8$ with a
stellar mass of $\approx 2\times 10^{9}\,\,M_{\odot}$ and a dust mass of
$\approx 6\times 10^{6}\,\,M_{\odot}$. A large amount of dust is reported by
Venemans et al. (2017) for a quasar at $z=7.5$ in the interstellar medium of
its host galaxy. In addition, a remarkably bright galaxy at $z\approx 11$ was
found by Oesch et al. (2016) and a significant increase in the number of
galaxies for $8.5<z<12$ was reported by Ellis et al. (2013). Note that the
number of papers reporting discoveries of galaxies at $z\approx 10$ or higher
is growing rapidly (Hashimoto et al., 2018; Hoag et al., 2018; Oesch et al.,
2018; Salmon et al., 2018).

  * •

The model is capable to explain the SNe Ia dimming discovered by Riess et al.
(1998) and Perlmutter et al. (1999) without introducing dark energy as the
hypothetical energy of vacuum (Vavryčuk, 2019), which is difficult to explain
under the quantum field theory (Weinberg et al., 2013). Moreover, the speed of
gravitational waves and the speed of light differ for most of dark energy
models (Sakstein & Jain, 2017; Ezquiaga & Zumalacárregui, 2017), but
observations of the binary neutron star merger GW170817 and its
electromagnetic counterparts proved that both speeds coincide with a high
accuracy.

  * •

The model avoids a puzzle, how the CMB as relic radiation could survive the
whole history of the Universe without any distortion (Vavryčuk, 2017a) and why
several unexpected features at large angular scales such as non-Gaussianity
(Vielva et al., 2004; Cruz et al., 2005; Planck Collaboration et al., 2014)
and a violation of statistical isotropy and scale invariance are observed in
the CMB.

##  5 Discussion and conclusions

The radiation pressure as a cosmological force acting against the gravity has
not been proposed yet, even though its role is well known in the stellar
dynamics (Kippenhahn et al., 2012). The radiation pressure is important in the
evolution of massive stars (Zinnecker & Yorke, 2007), in supernovae stellar
winds and in galactic wind dynamics (Aguirre, 1999b; Martin, 2005; Hopkins et
al., 2012; Hirashita & Inoue, 2019). Apparently, the radiation pressure in the
evolution of the Universe was overlooked, because the Universe was assumed to
be transparent. By contrast, the role of radiation pressure is essential in
the opaque universe model, because it is produced by absorption of photons by
cosmic dust. Since the cosmic opacity and the intensity of the EBL steeply
rise with redshift, the radiation pressure, negligible at present, becomes
significant at high redshifts and can fully eliminate gravity and stop the
universe contraction.

Hence, the expansion/contraction evolution of the Universe might be a result
of imbalance of gravitational forces and radiation pressure. Since the
comoving global stellar and dust masses are basically independent of time with
minor fluctuations only, the evolution of the Universe is stationary.
Obviously, the recycling processes of stars and galaxies (Segers et al., 2016;
Anglés-Alcázar et al., 2017) play a more important role in this model than in
the standard cosmology.

The age of the Universe in the cyclic model is unconstrained and galaxies can
be observed at any redshift less than the maximum redshift $z_{\mathrm{max}}$.
The only limitation is high cosmic opacity, which can prevent observations of
the most distant galaxies. Hypothetically, it is possible to observe galaxies
from the previous cycle/cycles, if their distance is higher than that
corresponding to $z_{\mathrm{max}}\approx 14-15$. The identification of
galaxies from the previous cycles will be, however, difficult, because their
redshift will be a periodic function with increasing distance.

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