# Universe opacity and CMB

Václav Vavryčuk,
Institute of Geophysics, The Czech Academy of Sciences, Boční II, Praha 4,
14100, Czech Republic
E-mail<EMAIL_ADDRESS>
(Accepted 2018 April 16. Received 2018 April 16; in original form 2017 August
15)

###### Abstract

A cosmological model, in which the cosmic microwave background (CMB) is a
thermal radiation of intergalactic dust instead of a relic radiation of the
Big Bang, is revived and revisited. The model suggests that a virtually
transparent local Universe becomes considerably opaque at redshifts $z>2-3$.
Such opacity is hardly to be detected in the Type Ia supernova data, but
confirmed using quasar data. The opacity steeply increases with redshift
because of a high proper density of intergalactic dust in the previous epochs.
The temperature of intergalactic dust increases as $(1+z)$ and exactly
compensates the change of wavelengths due to redshift, so that the dust
radiation looks apparently like the radiation of the blackbody with a single
temperature. The predicted dust temperature is $T^{D}=2.776\,\mathrm{K}$,
which differs from the CMB temperature by 1.9% only, and the predicted ratio
between the total CMB and EBL intensities is 13.4 which is close to 12.5
obtained from observations. The CMB temperature fluctuations are caused by EBL
fluctuations produced by galaxy clusters and voids in the Universe. The
polarization anomalies of the CMB correlated with temperature anisotropies are
caused by the polarized thermal emission of needle-shaped conducting dust
grains aligned by large-scale magnetic fields around clusters and voids. A
strong decline of the luminosity density for $z>4$ is interpreted as the
result of high opacity of the Universe rather than of a decline of the global
stellar mass density at high redshifts.

###### keywords:

cosmic background radiation – dust, extinction – early Universe – galaxies:
high redshift – galaxies: ISM – intergalactic medium

††pubyear: 2017††pagerange: Universe opacity and CMB–LABEL:lastpage

##  1 Introduction

The cosmic microwave background (CMB) radiation was discovered by Penzias &
Wilson (1965) who reported an isotropic and unpolarized signal in the
microwave band characterized by a temperature of about 3.5 K. After this
discovery, many experiments have been conducted to provide more accurate CMB
observations. The rocket measurements of Gush et al. (1990) and FIRAS on the
COBE satellite (Mather et al., 1990; Fixsen et al., 1996) proved that the CMB
has almost a perfect thermal blackbody spectrum with an average temperature of
$T=2.728\pm 0.004\,\mathrm{K}$ (Fixsen et al., 1996), which was further
improved using the WMAP data to $T=2.72548\pm 0.00057\,\mathrm{K}$ (Fixsen,
2009).

The CMB temperature consists of small directionally dependent large- and
small-scale fluctuations analysed, for example, by the WMAP (Bennett et al.,
2003) or Planck (Planck Collaboration et al., 2014a, b, 2016a) data. The
large-scale fluctuation of $\pm 0.00335\,\mathrm{K}$ with one hot pole and one
cold pole is called the ’dipole anisotropy’ being caused by motion of the
Milky Way relative to the Universe (Kogut et al., 1993). The small-scale
fluctuations are of about $\pm 70\,\mu\mathrm{K}$ being studied, for example,
by the WMAP (Bennett et al., 2003; Hinshaw et al., 2009; Bennett et al.,
2013), ACBAR (Reichardt et al., 2009), BOOMERANG (MacTavish et al., 2006), CBI
(Readhead et al., 2004), and VSA (Dickinson et al., 2004) instruments using
angular multipole moments. The measurements of the CMB temperature were
supplemented by detection of the CMB polarization anomalies by the DASI
telescope at a subdegree angular scale by Kovac et al. (2002) and Leitch et
al. (2002). The DASI polarization measurements were confirmed and extended by
the CBI (Readhead et al., 2004), CAPMAP (Barkats et al., 2005), BOOMERANG
(Montroy et al., 2006) and WMAP (Page et al., 2007) observations. The
measurements indicate that the polarization anomalies and the temperature
anisotropies are well correlated.

Immediately, after the discovery of the CMB by Penzias & Wilson (1965), Dicke
et al. (1965) proposed to interpret the CMB as a blackbody radiation
originated in the hot Big Bang. Since the blackbody radiation has been
predicted for the expanding universe by several physicists and cosmologists
before the CMB discovery (Alpher & Herman, 1948; Gamow, 1952, 1956), the
detection of the CMB by Penzias & Wilson (1965) was a strong impulse for a
further development of the Big Bang theory. Over years, the CMB radiation
became one of the most important evidences supporting this theory. The
temperature fluctuations have several peaks attributed to some cosmological
parameters such as the curvature of the universe or the dark-matter density
(Hu & Dodelson, 2002; Spergel et al., 2007; Komatsu et al., 2011). The
polarization anomalies are interpreted as a signal generated by Thomson
scattering of a local quadrupolar radiation pattern by free electrons during
the last scattering epoch (Hu & White, 1997).

The CMB as a relic radiation of the hot Big Bang is now commonly accepted even
though it is not the only theory offering an explanation of the CMB origin.
Another option, discussed mostly in the cold Big Bang theory (Hoyle &
Wickramasinghe, 1967; Layzer & Hively, 1973; Rees, 1978; Hawkins & Wright,
1988; Aguirre, 2000) and in steady-state and quasi-steady-state cosmology
(Bondi & Gold, 1948; Hoyle, 1948; Arp et al., 1990; Hoyle et al., 1993, 1994)
is to assume that the CMB does not originate in the Big Bang but is a
radiation of intergalactic dust thermalized by the light of stars (Wright,
1982; Pan, 1988; Bond et al., 1991; Peebles, 1993; Narlikar et al., 2003). The
’dust theory’ assumes the CMB to be produced by dust thermalization at high
redshifts. It needs the high-redshift Universe to be significantly opaque at
optical wavelengths which is now supported by observations of the
intergalactic opacity (Ménard et al., 2010; Xie et al., 2015) and by the
presence of dust in damped Lyman $\alpha$ absorbers in intergalactic space at
high redshift (Vladilo et al., 2006; Noterdaeme et al., 2017; Ma et al.,
2017).

Both the hot Big Bang theory and the dust theory are faced with difficulties
when modelling properties of the CMB. The hot Big Bang works well under the
assumption of a transparent Universe but it cannot satisfactorily explain how
the CMB could survive the opaque epochs of the Universe without being
significantly distorted by dust. The distortion should be well above the
sensitivity of the COBE/FIRAS, WMAP or Planck flux measurements and should
include a decline of the spectral as well as total CMB intensity due to
absorption (Vavryčuk, 2017b). Detailed analyses of the CMB anisotropies by
WMAP and Planck also revealed several unexpected features at large angular
scales such as non-Gaussianity of the CMB (Vielva et al., 2004; Cruz et al.,
2005; Planck Collaboration et al., 2014c) and a violation of statistical
isotropy and scale invariance following from the Big Bang theory (Schwarz et
al., 2016).

By contrast, the dust theory has troubles with explaining why intergalactic
dust radiates as a blackbody, why the CMB temperature is unaffected by a
variety of redshifts of radiating dust grains (Peacock, 1999, p. 289), and why
the CMB is almost isotropic despite the variations of the dust density and of
the starlight in the Universe. The theory should also satisfactorily explain
correlated observations of the temperature and polarization fluctuations in
the CMB.

The assumption of the blackbody radiation of intergalactic dust was
questioned, for example, by Purcell (1969) who analysed the Kramer-Kronig
relations applied to space sparsely populated by spheroidal grains. He argued
that intergalactic dust grains whose size is less than 1 $\mu$m are very poor
radiators of millimetre waves and thus cannot be black. On the other hand,
Wright (1982) demonstrated that needle-shaped conducting grains could provide
a sufficient long-wavelength opacity. The long-wavelength absorption is also
strengthened by complex fractal or fluffy dust aggregates (Wright, 1987;
Henning et al., 1995). Hence, it now seems that the opacity of intergalactic
dust is almost unconstrained and the assumption of the blackbody radiation of
intergalactic dust is reasonable (Aguirre, 2000).

In this paper, I address the other objections raised to the dust theory. I
show that under some assumptions about the stellar and dust mass evolution in
the Universe the idea of the CMB produced by dust thermalization can be
reconciled with observations and that the controversies of the dust theory
might be apparent. I present formulas for the redshift-dependent extragalactic
background light (EBL), which is the main source of the intergalactic dust
radiation. Subsequently, I determine the redshift-dependent temperature of
dust and establish a relation between the intensity of the EBL and CMB. Based
on observations of the opacity of the Universe, the maximum redshift of dust
contributing to the observed CMB is estimated. Finally, I discuss why the CMB
temperature is so stable and how the small-scale temperature and polarization
anisotropies in the CMB can be explained in the dust theory.

##  2 Extragalactic background light (EBL)

###  2.1 Observations of EBL

The EBL covers the near-ultraviolet, visible and infrared wavelengths from 0.1
to 1000 $\mu$m and has been measured by direct as well as indirect techniques.
The direct measurements were provided, for example, by the IRAS, DIRBE on
COBE, FIRAS, ISO, and SCUBA instruments, for reviews see Hauser & Dwek (2001);
Lagache et al. (2005); Cooray (2016). The direct measurements are supplemented
by analysing integrated light from extragalactic source counts (Madau &
Pozzetti, 2000; Hauser & Dwek, 2001) and attenuation of gamma rays from
distant blazars due to scattering on the EBL (Kneiske et al., 2004; Dwek &
Krennrich, 2005; Primack et al., 2011; Gilmore et al., 2012).

The spectrum of the EBL has two distinct maxima: at visible-to-near-infrared
wavelengths (0.7 - 2 $\mu$m) and at far-infrared wavelengths (100-200 $\mu$m)
associated with the radiation of stars and with the thermal radiation of dust
in galaxies (Schlegel et al., 1998; Calzetti et al., 2000). Despite the
extensive measurements of the EBL, the uncertainties are still large (see
Figure 1). The best constraints at the near- and mid-infrared wavelengths come
from the lower limits based on the integrated counts (Fazio et al., 2004; Dole
et al., 2006; Thompson et al., 2007) and from the upper limits based on the
attenuation of gamma-ray photons from distant extragalactic sources (Dwek &
Krennrich, 2005; Aharonian et al., 2006; Stecker et al., 2006; Abdo et al.,
2010). Integrating the lower and upper limits of the spectral energy
distributions shown in Figure 1, the total EBL should fall between 40 and
$200\,\mathrm{nWm}^{-2}\mathrm{sr}^{-1}$. The most likely value of the total
EBL from 0.1 to 1000 $\mu$m is about
$80-100\,\mathrm{nWm}^{-2}\mathrm{sr}^{-1}$ (Hauser & Dwek, 2001).

Figure 1:  Spectral energy distribution (SED) of the extragalactic background
light (EBL) with estimates of its minimum and maximum limits (black lines).
The observations reported by various authors are marked by colour symbols
(modified after Domínguez et al. (2011)).

###  2.2 Redshift dependence of EBL

Let us assume that the comoving galaxy and dust number densities, galaxy
luminosity and galactic and intergalactic opacities are conserved with cosmic
time. The EBL in a transparent expanding universe with no light sources is
calculated by the equation of radiative transfer (Peebles, 1993, his equation
5.158 with luminosity density $j=0$)

$\frac{d}{dt}I_{\nu}^{\mathrm{EBL}}+3HI_{\nu}^{\mathrm{EBL}}=0\,,$ (1)

where $I_{\nu}^{\mathrm{EBL}}(t)$ is the specific intensity of the EBL (in
$\mathrm{Wm}^{-2}\mathrm{Hz}^{-1}\mathrm{sr}^{-1}$), $H(t)=\dot{R}/R$ is the
Hubble parameter and $R$ is the scale factor. Solving equation (1)

$\int\frac{dI_{\nu}^{\mathrm{EBL}}}{I_{\nu}^{\mathrm{EBL}}}=\int-3Hdt\,,$ (2)

and taking into account the time-redshift relation (Peebles, 1993, his
equation 13.40)

$dt=\frac{1}{H}\frac{dz}{1+z}\,,$ (3)

the specific intensity $I_{\nu}^{\mathrm{EBL}}$ at redshift $z$ is

$I_{\nu}^{\mathrm{EBL}}=\left(1+z\right)^{3}I_{\nu 0}^{\mathrm{EBL}}\,,$ (4)

and subsequently the bolometric intensity $I^{\mathrm{EBL}}$ at redshift $z$

$I^{\mathrm{EBL}}\left(z\right)=\left(1+z\right)^{4}I_{0}^{\mathrm{EBL}}\,,$
(5)

where $I_{\nu 0}^{\mathrm{EBL}}$ and $I_{0}^{\mathrm{EBL}}$ are the specific
and bolometric EBL intensities at $z=0$. Equation (5) expresses the fact that
the bolometric EBL scales with the expansion as $(1+z)^{-4}$. Since light
sources and absorption are not considered, the EBL declines with the expansion
of the Universe. The decline is $(1+z)^{-3}$ due to the volume expansion and
$(1+z)^{-1}$ due to the redshift.

If light sources and absorption are considered, the equation of radiative
transfer must be modified (Peacock, 1999, his equation 12.13)

$\frac{d}{dt}I_{\nu}^{\mathrm{EBL}}+3HI_{\nu}^{\mathrm{EBL}}=\frac{c}{4\pi}j_{\nu}-c\kappa_{\nu}I_{\nu}^{\mathrm{EBL}}\,,$
(6)

where $j_{\nu}(t)$ is the luminosity density at frequency $\nu$ (in
$\mathrm{Wm}^{-3}\mathrm{Hz}^{-1}$) and $\kappa_{\nu}$ is the opacity at
frequency $\nu$.

If we assume that the comoving stellar and dust mass densities are constant,
the comoving specific intensity of the EBL is also constant. Consequently, the
radiation from light sources is exactly compensated by light absorption and
the right-hand side of equation (6) is zero. It physically means that the
light produced by stars is absorbed by galactic and intergalactic dust. The
process is stationary because the energy absorbed by intergalactic dust
produces its thermal radiation which keeps the dust temperature constant.
Since dust grains are very small, any imbalance between the radiated stellar
energy and energy absorbed by dust would produce fast observable changes in
the dust temperature. Hence,

$I_{\nu}^{\mathrm{EBL}}=\frac{1}{4\pi}\frac{j_{\nu}}{\kappa_{\nu}}\,,$ (7)

which should be valid for all redshifts $z$. The specific luminosity density
$j_{\nu}(z)$ in equation (7) increases with $z$ as $(1+z)^{3}$ and the opacity
$\kappa_{\nu}$ is redshift independent, because the number of absorbers in the
comoving volume is constant (the proper attenuation coefficient per unit ray
path increases with $z$ but the proper length of a ray decreases with $z$).
Hence equation (7) predicts $I_{\nu}^{\mathrm{EBL}}$ to increase with $z$ as
$(1+z)^{3}$ similarly as in the case of no light sources and no absorption,
see equation (4). Consequently, the bolometric EBL intensity increases with
$z$ in an expanding dusty universe with galaxies according to equation (5)
derived originally for the transparent universe with no light sources.

###  2.3 EBL and the Tolman relation

Equation (5) can alternatively be derived using the Tolman relation, which
expresses the redshift dependence of the surface brightness of galaxies in the
expanding universe (Peacock, 1999, his equation 3.90)

$B^{\mathrm{bol}}\left(z\right)=\left(1+z\right)^{-4}B_{0}^{\mathrm{bol}}\,,$
(8)

where $B^{\mathrm{bol}}\left(z\right)$ and $B_{0}^{\mathrm{bol}}$ is the
bolometric surface brightness of a galaxy (in
$\mathrm{Wm}^{-2}\mathrm{sr}^{-1}$) at redshift $z$ and at $z=0$,
respectively. The Tolman relation says that the bolometric surface brightness
of galaxies decreases with the redshift in the expanding universe in contrast
to the static universe, where the surface brightness of galaxies is
independent of their distance.

In the Tolman relation, the observer is at $z=0$ and the redshift dependence
is studied for distant galaxies at high redshift. However, the relation can be
reformulated for an observer at redshift $z$. Obviously, if we go back in
time, the surface brightness of galaxies was higher in the past than at
present by factor $(1+z)^{4}$. If the number density of galaxies is assumed to
be constant in the comoving volume, the EBL for an observer at redshift $z$
should also be higher by the same factor, see equation (8). Hence, the
intensity of the EBL was significantly higher at redshift $z$ than at present.

Strictly speaking, the Tolman relation was originally derived for a
transparent universe with no dust. The presence of dust reduces the intensity
of the EBL and dust must be incorporated into the model. In analogy to the
surface brightness of galaxies, we can introduce a surface absorptivity of
dust as a surface brightness of negative value. Thus instead of radiating
energy, the energy is absorbed. Since the dust density is conserved in the
comoving volume similarly as the number density of galaxies, the intensity of
the EBL will be lower in the partially opaque universe than in the transparent
universe, but the redshift dependence described in equation (8) is conserved.

##  3 Opacity observations

###  3.1 Galactic and intergalactic opacity

The methods for measuring the galactic opacity usually perform multi-
wavelength comparisons and a statistical analysis of the colour and number
count variations induced by a foreground galaxy onto background sources (for a
review, see Calzetti (2001)). The most transparent galaxies are ellipticals
with an effective extinction $A_{V}$ of $0.04-0.08$ mag. The dust extinction
in spiral and irregular galaxies is higher. Holwerda et al. (2005) found that
the dust opacity of the disk in the face-on view apparently arises from two
distinct components: an optically thicker component ($A_{I}=0.5-4$ mag)
associated with the spiral arms and a relatively constant optically thinner
disk ($A_{I}=0.5$ mag). Typical values for the inclination-averaged extinction
are: $0.5-0.75$ mag for Sa-Sab galaxies, $0.65-0.95$ mag for the Sb-Scd
galaxies, and $0.3-0.4$ mag for the irregular galaxies at the B-band
(Calzetti, 2001).

Adopting estimates of the relative frequency of specific galaxy types in the
Universe and their mean visual extinctions (Vavryčuk, 2017a, his table 2), we
can estimate their mean visual opacities and finally the overall mean galactic
opacity. According to Vavryčuk (2017a), the average value of the visual
opacity $\kappa$ is about $0.22\pm 0.08$. A more accurate approach should take
into account statistical distributions of galaxy sizes and of the mean galaxy
surface brightness for individual types of galaxies.

Ménard et al. (2010) estimated the visual intergalactic attenuation to be
$A_{V}=(1.3\pm 0.1)\times 10^{-2}$ mag at a distance from a galaxy of up to
170 kpc and $A_{V}=(1.3\pm 0.3)\times 10^{-3}$ mag on a large scale at a
distance of up to 1.7 Mpc. Similar values are reported by Muller et al. (2008)
and Chelouche et al. (2007) for the visual attenuation produced by
intracluster dust. However, the intergalactic attenuation is redshift
dependent. It increases with redshift, and a transparent universe becomes
significantly opaque (optically thick) at redshifts of $z=1-3$ (Davies et al.,
1997). The increase of intergalactic extinction with redshift is confirmed by
Ménard et al. (2010) by correlating the brightness of $\sim$85.000 quasars at
$z>1$ with the position of 24 million galaxies at $z\sim 0.3$ derived from the
Sloan Digital Sky Survey. The authors estimated $A_{V}$ to about 0.03 mag at
$z=0.5$ but to about $0.05-0.09$ mag at $z=1$. In addition, a consistent
opacity was reported by Xie et al. (2015) who studied the luminosity and
redshifts of the quasar continuum of $\sim 90.000$ objects. The authors
estimated the effective dust density $n\sigma_{V}\sim
0.02\,h\,\mathrm{Gpc}^{-1}$ at $z<1.5$.

Dust extinction can also be estimated from the hydrogen column densities
studied by the Lyman $\alpha$ (Ly$\alpha$) absorption lines of damped Lyman
absorbers (DLAs). Bohlin et al. (1978) determined the column densities of the
interstellar H I towards 100 stars and found a linear relationship between the
total hydrogen column density,
$N_{\mathrm{H}}=2N_{\mathrm{H2}}+N_{\mathrm{HI}}$ , and the colour excess from
the Copernicus data

$N_{\mathrm{H}}/\left(A_{B}-A_{V}\right)=5.8\times
10^{21}\,\mathrm{cm}^{-2}\,\mathrm{mag}^{-1}\,,$ (9)

and

$N_{\mathrm{H}}/A_{V}\approx 1.87\times
10^{21}\,\mathrm{cm}^{-2}\,\mathrm{mag}^{-1}\,\,\mathrm{for}\,\,R_{V}=3.1\,.$
(10)

Rachford et al. (2002) confirmed this relation using the FUSE data and
adjusted slightly the slope in equation (9) to $5.6\times
10^{21}\,\mathrm{cm}^{-2}\,\mathrm{mag}^{-1}$. Taking into account
observations of the mean cross-section density of DLAs reported by Zwaan et
al. (2005)

$\langle n\sigma\rangle=\left(1.13\pm 0.15\right)\times
10^{-5}\,h\,\mathrm{Mpc}^{-1}\,,$ (11)

the dominating column density of DLAs, $N_{\mathrm{HI}}\sim
10^{21}\,\mathrm{cm}^{-2}$ (Zwaan et al., 2005), and the mean molecular
hydrogen fraction in DLAs of about $0.4-0.6$ (Rachford et al., 2002, their
Table 8), equation (10) yields for the intergalactic attenuation $A_{V}$ at
$z=0$: $A_{V}\sim 1-2\times 10^{-5}\,h\,\mathrm{Mpc}^{-1}$. Considering also a
contribution of less massive LA systems, we get basically the result of Xie et
al. (2015): $A_{V}\sim 2\times 10^{-5}\,h\,\mathrm{Mpc}^{-1}$.

###  3.2 Wavelength-dependent opacity

The dust opacity is frequency dependent (see Figure 2). In general, it
decreases with increasing wavelength but displays irregularities. The
extinction law for dust in the Milky Way is well reproduced for infrared
wavelengths between $\sim 0.9\,\mu$m and $\sim 5\,\mu$m by the power-law
$A_{\lambda}\sim\lambda^{-\beta}$ with $\beta$ varying from 1.3 to 1.8
(Draine, 2003). At wavelengths of 9.7 and 18 $\mu$m, the dust absorption
displays two maxima associated with silicates (Mathis, 1990; Li & Draine,
2001; Draine, 2003). At longer wavelengths, the extinction decays according to
a power-law with $\beta=2$. This decay is predicted by numerical modelling of
graphite or silicate dust grains as spheroids with sizes up to 1 $\mu$m
(Draine & Lee, 1984). However, the long-wavelength opacity also depends on the
shape of the dust grains. For example, Wright (1982); Henning et al. (1995);
Stognienko et al. (1995) and others report that needle-shaped conducting
grains or complex fractal or fluffy dust aggregates can produce much higher
long-wavelength opacity than spheroidal grains with the power-law described by
$0.6<\beta<1.4$ (Wright, 1987).

Figure 2:  Normalized frequency-dependent attenuation due to absorption by
dust in the Milky Way (Draine, 2003, his Tables 4-6).

###  3.3 Opacity ratio

Extinction of the EBL is caused by two effects: (1) the galactic opacity
causing obscuration of background galaxies by partially opaque foreground
galaxies, and (2) the intergalactic opacity produced by light absorption due
to intergalactic dust. The distribution of the absorbed EBL energy between
galaxies and intergalactic dust can be quantified by the opacity ratio
(Vavryčuk, 2017a, his equation 16)

$R_{\kappa}=\frac{\lambda_{0}\gamma_{0}}{\kappa}\,,$ (12)

where $\kappa$ is the mean bolometric galactic opacity, $\lambda_{0}$ is the
mean bolometric intergalactic absorption coefficient along a ray path at
$z=0$, and $\gamma_{0}$ is the mean free path of a light ray between galaxies
at $z=0$,

$\gamma_{0}=\frac{1}{n_{0}\pi a^{2}}\,,$ (13)

where $a$ is the mean galaxy radius, and $n_{0}$ is the galaxy number density
at $z=0$.

The opacity ratio is a fundamental cosmological quantity controlled by the
relative distribution of dust masses between galaxies and the intergalactic
space. Since opacity is a relative quantity, it is invariant to the extinction
law and redshift.

Considering observations of the galactic and intergalactic opacity, and
estimates of the mean free path of light between galaxies (see Table 1), the
opacity ratio is in the range of 6-35 with an optimum value of 13.4 (see
Figure 3). This indicates that the EBL is predominantly absorbed by
intergalactic dust. The EBL energy absorbed by galaxies is much smaller being
only a fraction of the EBL energy absorbed by intergalactic dust.

Table 1: Opacity ratio

| $n$ | $\gamma$ | $\kappa$ | $A_{V}$ | $\lambda_{V}$ | $I^{\mathrm{EBL}}$ | $R_{\kappa}^{A}$ | $R_{\kappa}^{B}$
---|---|---|---|---|---|---|---|---
| $(h^{3}\,\mathrm{Mpc}^{-3})$ | $(h^{-1}\,\mathrm{Gpc})$ |  | $(\mathrm{mag}\,h\,\mathrm{Gpc}^{-1})$ | $(h\,\mathrm{Gpc}^{-1})$ | $(\mathrm{nWm}^{-2}\,\mathrm{sr}^{-1})$ |  |
Minimum ratio | 0.025 | 130 | 0.30 | 0.015 | 0.0138 | 200 | 6.0 | 5.0
Maximum ratio | 0.015 | 210 | 0.14 | 0.025 | 0.0230 | 40 | 34.5 | 24.9
Optimum ratio | 0.020 | 160 | 0.22 | 0.020 | 0.0184 | 80 | 13.4 | 12.5

  * •

$n$ is the number density of galaxies, $\gamma$ is the mean free path between
galaxies defined in equation (13), $A_{V}$ is the visual intergalactic
extinction, $\lambda_{V}$ is the visual intergalactic extinction coefficient,
$\kappa$ is the mean visual opacity of galaxies, $I^{\mathrm{EBL}}$ is the
total EBL intensity, $R_{\kappa}^{A}$ is the opacity ratio calculated using
equation (12), and $R_{\kappa}^{B}$ is the opacity ratio calculated using
equation (22). The mean effective radius of galaxies $a$ is considered to be
10 kpc in equation (13). All quantities are taken at $z=0$.

Figure 3:  The opacity ratio $R_{\kappa}$ as a function of the intergalactic
visual attenuation $A_{V}$ and the mean free path between galaxies $\gamma$.

##  4 Thermal radiation of intergalactic dust

The energy of light absorbed by galactic or intergalactic dust heats up the
dust and produces its thermal radiation. The temperature of dust depends on
the intensity of light absorbed by dust grains. Within galaxies, the light
intensity is high, the galactic dust being heated up to 20-40 K and emitting a
thermal radiation at infrared (IR) and far-infrared (FIR) wavelengths
(Schlegel et al., 1998; Draine & Li, 2007). Since the intensity of the EBL is
lower than the intensity of light within galaxies, the intergalactic dust is
colder and emits radiation at microwave wavelengths. At these wavelengths, the
only dominant radiation is the CMB, see Figure 4.

Figure 4:  Spectral energy distribution of the EBL limits (blue lines) and of
the CMB (red line). The EBL limits are taken from Figure 1. The numbers
indicate the total intensities of the EBL and CMB.

###  4.1 Absorption of EBL by intergalactic dust

Assuming intergalactic dust to be the ideal blackbody, its temperature $T^{D}$
is calculated using the Stefan-Boltzmann law

$T^{D}=\left(\frac{\,\,I^{D}}{\pi\sigma}\right)^{\frac{1}{4}}\,,$ (14)

where $\sigma=5.67\times 10^{−8}\,\mathrm{Wm}^{-2}\,\mathrm{K}^{-4}$ is the
Stefan-Boltzmann constant, and $I^{D}$ is the total dust intensity (radiance)
in $\mathrm{Wm}^{-2}\,\mathrm{sr}^{-1}$.

If we consider the thermal energy radiated by dust equal to the EBL absorbed
by dust, and insert the lower and upper limits of the EBL, 40 and 200
$\mathrm{nWm}^{-2}\,\mathrm{sr}^{-1}$, obtained from observations (Figure 1)
into equation (14), the temperature of the intergalactic dust ranges from 1.22
to 1.82 K. These values are much lower than the observed temperature of 2.725
K of the CMB. In order to heat up intergalactic dust to the CMB temperature,
the energy flux absorbed by dust should be 996
$\mathrm{nWm}^{-2}\,\mathrm{sr}^{-1}$. This value is $5-25$ times higher than
the total intensity of the EBL. Hence, if the CMB is related to the thermal
radiation of intergalactic dust, the EBL forms just a fraction of the energy
absorbed by dust.

Obviously, assuming that the dust radiation is only a reprocessed EBL is not
correct. A more appropriate approach should consider the absorption of the
thermal radiation of dust itself and a balance between the energy radiated and
absorbed by dust and by galaxies.

###  4.2 Energy balance of thermal dust radiation

Both galaxies and intergalactic dust radiate and absorb energy. Galaxies
radiate light in optical, IR and FIR spectra; intergalactic dust radiates
energy in the microwave spectrum. Radiation of galaxies produces the EBL with
the total intensity $I^{\mathrm{EBL}}$, which is partly absorbed by galaxies
($I^{\mathrm{EBL}}_{AG}$) and partly by dust ($I^{\mathrm{EBL}}_{AD}$),

$I^{\mathrm{EBL}}=I^{\mathrm{EBL}}_{AG}+I^{\mathrm{EBL}}_{AD}\,.$ (15)

The same applies to dust radiation with the total intensity $I^{D}$

$I^{D}=I^{D}_{AG}+I^{D}_{AD}\,.$ (16)

If the energy radiated by dust is completely absorbed by dust (no dust
radiation is absorbed by galaxies, $I^{D}_{AG}=0$) and no other sources of
light are present ($I^{\mathrm{EBL}}_{AD}=0$), the dust temperature is
constant. If dust additionally absorbs some light emitted by galaxies
($I^{\mathrm{EBL}}_{AD}\neq 0$), it is being heated up and the dust
temperature increases continuously with no limit (see Figure 5a). The process
of heating can be terminated only if some energy emitted by dust is absorbed
back by galaxies ($I^{D}_{AG}\neq 0$). In this case, dust warming continues
until the intergalactic dust reaches energy equilibrium. Since the dust grains
are small, the process of dust thermalization by the EBL is fast and the
effect of universe expansion can be neglected.

Figure 5:  Scheme of the energy balance between the EBL (black arrows) and CMB
(grey arrows) radiated and absorbed by galaxies (ellipses) and intergalactic
dust (grey area). (a) Dust is warmed by a part of the EBL which is absorbed by
dust. (b) Dust is in thermal equilibrium, when the EBL flux absorbed by
intergalactic dust is compensated by the CMB flux absorbed by galaxies.

Under the thermal equilibrium of dust, the energy interchanged between
galaxies and dust is mutually compensated

$I^{\mathrm{EBL}}_{AD}=I^{D}_{AG}\,,$ (17)

and the total energy of dust is conserved (see Figure 5b).

Since the proportion between the energy absorbed by intergalactic dust and by
galaxies is controlled by the opacity ratio

$I^{\mathrm{EBL}}_{AD}=R_{\kappa}I^{\mathrm{EBL}}_{AG}\,,\,I^{D}_{AD}=R_{\kappa}I^{D}_{AG}\,,$
(18)

we can rewrite equations (15) and (16) to read

$I^{\mathrm{EBL}}=\frac{1+R_{\kappa}}{R_{\kappa}}I^{\mathrm{EBL}}_{AD}\,,\,I^{D}=\left(1+R_{\kappa}\right)I^{D}_{AG}\,\,,$
(19)

and the relation between the intensity of dust radiation and the EBL is
finally expressed using equation (17) as

$I^{D}=R_{\kappa}I^{\mathrm{EBL}}\,,$ (20)

where $R_{\kappa}$ is defined in equation (12) and estimated in Table 1.

Equation (20) is invariant to the cosmological model considered and its
validity can be verified by observations. The EBL intensity estimated using
current measurements ranges from 40 to
$200\,\mathrm{nWm}^{-2}\,\mathrm{sr}^{-1}$ (see Figure 4) with an optimum
value of about $80\,\mathrm{nWm}^{-2}\,\mathrm{sr}^{-1}$. The optimum dust
temperature predicted from equation (20) when inserting
$80\,\mathrm{nWm}^{-2}\,\mathrm{sr}^{-1}$ for the $I^{\mathrm{EBL}}$ and 13.4
for the $R_{\kappa}$ is

$T^{D}_{\mathrm{theor}}=2.776\,\mathrm{K}\,,$ (21)

which is effectively the CMB temperature. The difference between the predicted
temperature of dust and the observed CMB temperature is about 1.9% being
caused by inaccuracies in the estimates of the EBL intensity and of the
opacity ratio.

If we substitute the predicted dust intensity $I^{D}$ corresponding to
temperature 2.776 K by the CMB intensity $I^{\mathrm{CMB}}$ corresponding to
temperature 2.725 K in equation (20), we can calculate the opacity ratio
$R_{\kappa}$ defined in equation (12) in the following alternative way:

$R_{\kappa}=\frac{I^{\mathrm{CMB}}}{I^{\mathrm{EBL}}}\,.$ (22)

This ratio lies in the range of $5-25$ with the optimum value of 12.5 which is
quite close to the value of 13.4 obtained from measurements of the galactic
and intergalactic opacity using equation (12), see Table 1.

###  4.3 Redshift dependence of the dust temperature and dust radiation

The thermal radiation of the intergalactic dust must depend on redshift
similarly as any radiation in the expanding universe. The redshift dependence
of the intensity of dust radiation is derived from equation (20). Since the
opacity ratio $R_{\kappa}$ does not depend on redshift and the redshift
dependence of $I^{\mathrm{EBL}}$ is described by equation (5), we get

$I^{D}\left(z\right)=\left(1+z\right)^{4}I_{0}^{D}\,,$ (23)

where $I_{0}^{D}$ is the intensity of dust radiation at redshift $z=0$.

Inserting equation (23) into equation (14) the dust temperature at redshift
$z$ comes out

$T^{D}\left(z\right)=\left(1+z\right)T_{0}^{D}\,,$ (24)

where $T_{0}^{D}$ is the temperature of dust at $z=0$. Hence, the dust
temperature linearly increases with redshift $z$. Similarly as the Tolman
relation, equation (24) is invariant to the cosmological model applied, being
based only on the assumptions of conservation of the galaxy number density,
dust density and constant galaxy luminosity in the comoving volume.

##  5 Spectral and total intensity of dust radiation

###  5.1 Spectral intensity of dust radiation

If we assume dust to be the blackbody, its thermal radiation (i.e., energy
emitted per unit projected area into a unit solid angle in the frequency
interval $\nu$ to $\nu+d\nu$) is described by the Planck’s law

$I_{\nu}\left(\nu,T^{D}\right)=\frac{2h\nu^{3}}{c^{2}}\frac{1}{e^{h\nu/k_{B}T^{D}}-1}\,\,\,(\mathrm{in\,Wm}^{-2}\,\mathrm{sr}^{-1}\mathrm{Hz}^{-1})\,,$
(25)

where $\nu$ is the frequency, $T^{D}$ is the dust temperature, $h$ is the
Planck constant, $c$ is the speed of light, and $k_{B}$ is the Boltzmann
constant. The dust temperature is uniform for a given time instant, but
increases with redshift $z$, see equation (24). Since the received wavelengths
also increase with redshift $z$, we arrive at

$\frac{h\nu}{k_{B}T^{D}}=\frac{h\nu_{0}\left(1+z\right)}{k_{B}T_{0}^{D}\left(1+z\right)}=\frac{h\nu_{0}}{k_{B}T_{0}^{D}}\,.$
(26)

Hence, the temperature increase with $z$ exactly eliminates the frequency
redshift of the thermal radiation in equation (25). Consequently, the
radiation of dust observed at all distances looks apparently as the radiation
of the blackbody with a single temperature.

###  5.2 Total intensity of dust radiation

Assuming the temperature of dust particles at redshift $z$ to be

$T^{D}\left(z\right)=\left(1+z\right)T_{0}^{\mathrm{CMB}}\,,$ (27)

we can calculate the total intensity of thermal dust radiation.

The total (bolometric) intensity $I^{D}$ of the dust radiation (in
$\mathrm{Wm}^{-2}\,\mathrm{sr}^{-1}$) is expressed as an integral over
redshift $z$

$I^{D}=\frac{1}{4\pi}\int_{0}^{\infty}j^{D}\left(z\right)\,e^{-\tau^{D}\left(z\right)}\,\frac{c}{H_{0}}\frac{dz}{E\left(z\right)}\,,$
(28)

where $j^{D}\left(z\right)$ is the luminosity density of dust radiation, and
$\tau^{D}\left(z\right)$ is the effective optical depth of the Universe at the
CMB wavelengths produced by intergalactic dust. The term
${1/\left(1+z\right)^{2}}$ expressing the reduction of the received energy
caused by redshift $z$ and present, for example, in a similar formula for the
EBL (Vavryčuk, 2017a, his equation 1) is missing in equation (28), because the
energy reduction is eliminated by the redshift dependence of dust temperature.
Since the temperature increases linearly with $z$, the dust luminosity density
$j^{D}\left(z\right)$ reads

$j^{D}\left(z\right)=\left(1+z\right)^{4}j_{0}^{D}\,,$ (29)

where $j_{0}^{D}$ is the dust luminosity density in the local Universe
($z=0$). Consequently,

$I^{D}=\frac{j_{0}^{D}}{4\pi}\int_{0}^{\infty}\left(1+z\right)^{4}\,e^{-\tau^{D}\left(z\right)}\,\frac{c}{H_{0}}\frac{dz}{E\left(z\right)}\,.$
(30)

The effective optical depth $\tau^{D}\left(z\right)$ reads

$\tau^{D}\left(z\right)=\frac{c}{H_{0}}\int_{0}^{z}\lambda_{0}^{D}\left(1+z^{\prime}\right)^{4}\,\,\frac{dz^{\prime}}{E\left(z^{\prime}\right)}\,,$
(31)

where $\lambda_{0}^{D}$ is the mean intergalactic absorption coefficient of
dust radiation along the ray path. The term describing the absorption of the
CMB by galaxies is missing in equation (31) because it is exactly compensated
by the EBL radiated by galaxies and absorbed by intergalactic dust, see
equation (17).

Taking into account the following identity

$\int_{0}^{\infty}f\left(z\right)\,\mathrm{exp}\left(-\int_{0}^{z}f\left(z^{\prime}\right)dz^{\prime}\right)dz=1\,,$
(32)

assuming $f\left(z\right)\rightarrow\infty$ for $z\rightarrow\infty$, the
intensity of dust radiation comes out as

$I^{D}=\frac{1}{4\pi}\frac{j_{0}^{D}}{\lambda_{0}^{D}}\,.$ (33)

Since the luminosity density of dust radiation $j_{0}^{D}$ reads

$j_{0}^{D}=n_{0}^{D}E^{D}=4n_{0}^{D}\sigma^{D}L^{D}=4\pi
n_{0}^{D}\sigma^{D}I^{\mathrm{CMB}}\,,$ (34)

where $n_{0}^{D}$ is the number density of dust particles at $z=0$, $E^{D}$ is
the total luminosity of one dust particle (in W), $\sigma^{D}$ is the mean
cross-section of dust particles, $L^{D}$ is the energy flux radiated per unit
surface of dust particles (in Wm-2), and $I^{\mathrm{CMB}}$ is the intensity
radiated by a blackbody with the CMB temperature (in
$\mathrm{Wm}^{-2}\,\mathrm{sr}^{-1}$). Since

$\lambda_{0}^{D}=n_{0}^{D}\sigma_{D}\,,$ (35)

equations (33) and (34) yield

$I^{D}=I^{\mathrm{CMB}}\,.$ (36)

Equation (36) is valid independently of the cosmological model considered and
states that the energy flux received by the unit area of the intergalactic
space is equal to the energy flux emitted by the unit area of intergalactic
dust particles. This statement is basically a formulation of the Olbers’
paradox (Vavryčuk, 2016, his equation 9) applied to dust particles instead of
to stars. Since the sky is fully covered by dust particles and distant
background particles are obscured by foreground particles, the energy fluxes
emitted and received by dust are equal. This is valid irrespective of the
actual dust density in the local Universe.

##  6 Saturation redshift of CMB

The total intensity of the CMB is calculated by summing the intensity over all
redshifts $z$, see equation (30). Since the intensity is attenuated due to the
exponential term with the optical depth in equation (30), the contribution of
the dust radiation to the energy flux decreases with redshift $z$. Since most
of the CMB energy is absorbed by intergalactic dust but not by galaxies, the
optical depth depends basically on the attenuation of intergalactic dust at
the CMB wavelengths $\lambda^{\mathrm{CMB}}$, which can be expressed as

$\lambda^{\mathrm{CMB}}=k^{\mathrm{CMB}}\lambda_{V}\,,$ (37)

where $k^{\mathrm{CMB}}$ is the ratio between the attenuation of intergalactic
dust at the CMB and visual wavelengths. The lower the CMB attenuation, the
slower the decrease of dust radiation with redshift. Consequently, we can
define the so-called saturation redshift $z^{*}$ as the redshift for which the
CMB intensity reaches 98% of its total value:

$I^{D}\left(z^{*}\right)=\frac{j_{0}^{D}}{4\pi}\int_{0}^{z^{*}}\left(1+z\right)^{4}\,e^{-\tau^{D}\left(z\right)}\,\frac{c}{H_{0}}\frac{dz}{E\left(z\right)}=0.98\,I^{D}\,.$
(38)

Assuming that the expanding history of the Universe is correctly described by
equation (38) for redshifts up to 50-60 and inserting
$0.02\,\mathrm{mag}\,h\,\mathrm{Gpc}^{-1}$ for the visual intergalactic
opacity and $1\times 10^{-4}$ for the ratio between the CMB and visual
extinctions (see Figure 2), we get the CMB to be saturated at redshifts of
about $z^{*}=55$ (see Figure 6). If the ratio is lower by one order, the
saturation redshift is about 100. A rather high value of the CMB saturation
redshift indicates that the observed CMB intensity is a result of dust
radiation summed over vast distances of the Universe. As a consequence, the
CMB intensity must be quite stable with only very small variations with
direction in the sky.

Figure 6:  The intensity of the CMB radiated by dust at redshift $z$ (a-b),
and the cumulative intensity of the CMB emitted by dust up to redshift $z$
(c-d). (a) and (c) - intergalactic attenuation $A_{V}$ is $1.5\times 10^{-2}$
(blue dashed line), $2.0\times 10^{-2}$ (blue solid line) and $2.5\times
10^{-2}\,\mathrm{mag}\,h\,\mathrm{Gpc}^{-1}$ (blue dotted line). The ratio
between the CMB and visual attenuations is $1.0\times 10^{-4}$. (b) and (d) -
the ratio between the CMB and visual attenuations is $0.5\times 10^{-4}$ (blue
dashed line), $1.0\times 10^{-4}$ (blue solid line) and $2.0\times 10^{-4}$
(blue dotted line). The intergalactic attenuation $A_{V}$ is $2.0\times
10^{-2}\,\mathrm{mag}\,h\,\mathrm{Gpc}^{-1}$. Cosmological parameters:
$H_{0}=67.7\,\mathrm{km}\,\mathrm{s}^{-1}\mathrm{Mpc}^{-1}$, $\Omega_{m}=0.3$,
and $\Omega_{\Lambda}=0.7$.

##  7 CMB temperature anisotropies

So far, we have assumed the EBL to be perfectly isotropic with no directional
variation, being a function of redshift only. Obviously, this assumption is
not correct because of clustering of galaxies and presence of voids in the
Universe (Bahcall, 1999; Hoyle & Vogeley, 2004; Jones et al., 2004; von Benda-
Beckmann & Müller, 2008; Szapudi et al., 2015). Consequently, the EBL displays
fluctuations (Figure 7a) manifested as small-scale EBL anisotropies reported
mostly at IR wavelengths (Kashlinsky et al., 2002; Cooray et al., 2004;
Matsumoto et al., 2005; Kashlinsky et al., 2007; Matsumoto et al., 2011;
Cooray et al., 2012; Pyo et al., 2012; Zemcov et al., 2014). Since the EBL
forms about 7% of the total energy absorbed by dust and reradiated as the CMB,
the EBL fluctuations should affect the intensity of the CMB (Figure 7b). The
remaining 93% of the total energy absorbed by dust is quite stable because it
comes from the CMB itself which is averaged over large distances.

Let the luminosity density of dust radiation $j^{D}$ display small-scale
variations with distance $\Delta\left(r\right)$ reflecting the EBL
fluctuations in the Universe. Transforming distance to redshift and taking
into account the redshift dependence of $j^{D}$, we get

$j^{D}\left(z\right)=\left(1+\Delta\left(z\right)\right)\left(1+z\right)^{4}j_{0}^{D}\,,$
(39)

where $\Delta(z)$ is much smaller than 1 and has a zero mean value. Inserting
equation (39) into equation (28), the variation of the total intensity of the
dust radiation $\Delta I^{D}$ (in $\mathrm{Wm}^{-2}\,\mathrm{sr}^{-1}$) reads

$\Delta
I^{D}=\frac{j_{0}^{D}}{4\pi}\int_{0}^{\infty}\Delta\left(z\right)\left(1+z\right)^{4}\,e^{-\tau^{D}\left(z\right)}\,\frac{c}{H_{0}}\frac{dz}{E\left(z\right)}\,,$
(40)

where the optical depth $\tau^{D}\left(z\right)$ is defined in equation (31).
The variation of the CMB temperature corresponding to $\Delta I^{D}$ is
obtained using equation (14).

The sensitivity of the intensity of the dust radiation and the dust
temperature to the EBL fluctuations can be tested numerically. Figures 8 and 9
show synthetically generated fluctuations of the luminosity density
$\Delta\left(r\right)$. The fluctuations were generated by bandpass filtering
of white noise. To mimic real observations, we kept predominantly fluctuations
of size between 20-100 Mpc, corresponding to typical cluster, supercluster and
void dimensions (Bahcall, 1999; Hoyle & Vogeley, 2004; von Benda-Beckmann &
Müller, 2008). The other sizes of fluctuations were suppressed. The
probability density function of $\Delta\left(r\right)$ is very narrow with a
standard deviation of 0.02 (Figure 8). The noise level of 2% expresses the
fact that the variations of the EBL should contribute to the dust radiation by
less than 10% only.

Considering the luminosity density fluctuations shown in Figure 9 and the
intergalactic opacity at the CMB wavelengths $1\times 10^{-4}$ lower than that
at visual wavelengths (see Fig. 2) in modelling of the intensity variation
$\Delta I^{D}$ using equation (40), we find that $\Delta I^{D}$ attains values
up to $\pm 0.25\,\mathrm{nWm}^{-2}\,\mathrm{sr}^{-1}$ and the standard
deviation of dust temperature is $\pm 60\,\mu\mathrm{K}$. The maximum
variation of the dust temperature is up to $\pm 170\,\mu\mathrm{K}$. The
standard deviations and the maximum limits were obtained from 1000 noise
realizations. Compared to observations, the retrieved variations are
reasonable. Taking into account very rough estimates of input parameters, the
predicted temperature variation fits well the observed small-scale
anisotropies of the CMB attaining values up to $\pm 70\,\mu\mathrm{K}$
(Bennett et al., 2003; Hinshaw et al., 2009; Bennett et al., 2013; Planck
Collaboration et al., 2014a, b). More accurate predictions are conditioned by
a detailed mapping of EBL fluctuations by planned cosmological missions such
as Euclid, LSST or WFIRST (van Daalen et al., 2011; Masters et al., 2015). For
example, the NASA explorer SPHEREx will be able to conduct a three-dimensional
intensity mapping of spectral lines such as H$\alpha$ at $z\sim 2$ and
Ly$\alpha$ at $z>6$ over large areas in the sky (Cooray, 2016; Fonseca et al.,
2017; Gong et al., 2017).

The relation between the presence of voids, clusters and the CMB anisotropies
predicted in this paper has been recently supported by observations of several
authors. The studies were initiated by detecting an extreme cold spot (CS) of
$-70\,\mu$K in the WMAP images (Vielva et al., 2004; Cruz et al., 2005) which
violated a condition of Gaussianity of the CMB required in the Big Bang
theory. Later, the origin of the CS was attributed to the presence of a large
void detected by the NRAO VLA Sky Survey (Rudnick et al., 2007). The presence
of the supervoid was later confirmed by Gurzadyan et al. (2014) using the
Kolmogorov map of Planck’s 100 GHz data and by Szapudi et al. (2015) using the
WISE-2MASS infrared galaxy catalogue matched with Pan-STARRS1 galaxies. The
radius of the supervoid was estimated by Szapudi et al. (2015) to be
$R_{\mathrm{void}}=220\pm 50\,h^{-1}\,\mathrm{Mpc}$.

A physical relation between large-scale structures and the CMB anisotropies
has been confirmed also for other spots. For example, a large low-density
anomaly in the projected WISE-2MASS galaxy map called the Draco supervoid was
aligned with a CMB decline by Finelli et al. (2016). The imprint of
superstructures on the CMB was also statistically evidenced by stacking CMB
temperatures around the positions of voids from the SDSS DR7 spectroscopic
galaxy catalogue (Cai et al., 2014). In addition, Kovács et al. (2017) probed
the correlation between small temperature anomalies and density perturbations
using the data of the Dark Energy Survey (DES). They identified 52 large voids
and 102 superclusters at redshifts $0.2<z<0.65$ and performed a stacking
measurement of the CMB temperature field based on the DES data. They detected
a cumulative cold imprint of voids with $\Delta T=−5.0\pm 3.7\,\mu K$ and a
hot imprint of superclusters $\Delta T=5.1\pm 3.2\,\mu K$.

Figure 7:  Examples of the angular fluctuations of (a) the specific intensity
of the EBL, and (b) the CMB temperature. The EBL plot is taken from Zemcov et
al. (2014, their Figure S8) and represents a differenced 1.1 $\mu$m image of
two fields (NEP - ELAIS-N1) smoothed with the Gaussian function with FWHM of
7.2’ to highlight a large-scale structure. The differenced image is used by
Zemcov et al. (2014) to reduce the effect of flat-fielding errors. The CMB
plot was produced using tools of the NASA/IPAC Infrared Science Archive and
represents a smoothed image of the Planck public intensity data (Release 2) at
353 GHz for the ELAIS-N1 field at 16h11m.5, 54d37m.9. The image is rescaled to
temperature. Since the EBL plot does not show an actual field but a
differenced field, the patterns of the EBL and CMB plots indicate a rough
similarity but they cannot mutually correlate.  Figure 8:  The probability
density function of the modelled luminosity density fluctuations of dust
radiation.  Figure 9:  The luminosity density fluctuations of dust radiation
(a) and its spectrum (b) as a function of distance.  Figure 10:  Optical depth
and colour excess of intergalactic space as a function of redshift. The
extinction coefficient $R_{V}$ is assumed to be 5. $A_{V}$ \- extinction at
the visual band, $A_{B}$ \- extinction at the B band.

##  8 CMB polarization

Small-scale CMB anisotropies are expressed not only in temperature but also in
polarization being mutually correlated. In the dust theory, the CMB
polarization can be explained by interaction of dust with a cosmic magnetic
field produced by large-scale structures in the Universe. Magnetic fields are
present in all types of galaxies, clusters and superclusters. The Milky Way
has a typical interstellar magnetic field strength of 2 $\mu$G in a regular
ordered component on kpc scales (Kulsrud, 1999). Other spiral galaxies have
magnetic field strengths of 5 to 10 $\mu$G, with field strengths up to 50
$\mu$G in starburst galaxy nuclei (Beck et al., 1996). The magnetic fields in
galaxy clusters and in the intergalactic medium (IGM) have strength of
$10^{-5}-10^{-7}$ G (Widrow, 2002; Vallée, 2004; Giovannini, 2004). They are
present even in voids with a minimum strength of about $10^{-15}$ G on Mpc
scales (Beck et al., 2013). The intercluster magnetic fields have been
measured using synchrotron relic and halo radio sources within clusters,
inverse Compton X-ray emission from clusters, surveys of Faraday rotation
measures of polarized radio sources both within and behind clusters, and
studies of cluster cold fronts in X-ray images (Carilli & Taylor, 2002). The
measurements suggest substantially magnetized atmospheres of most clusters.
Models of the magnetic field of the IGM typically involve an ejection of the
fields from normal or active galaxies (Heckman, 2001). Kronberg et al. (1999)
considered this mechanism and showed that a population of dwarf starburst
galaxies at $z\geq 6$ could magnetize almost 50% of the universe.

The magnetic fields are easily traced via polarization of radiation resulting
from extinction or/and emission by aligned dust grains in the interstellar or
intergalactic medium (Lazarian, 2007; Andersson, 2015). The grain alignment by
magnetic fields proved to be an efficient and rapid process which causes a
linear polarization of starlight when passing through the dust and a polarized
thermal emission of dust (Lazarian, 2007). Hence, optical, IR and FIR
polarimetry can reveal the presence and the detailed structure of magnetic
fields in our Galaxy (Planck Collaboration et al., 2015, 2016b) as well as in
large-scale formations in the Universe (Feretti et al., 2012). The polarized
light can be used not only for tracing magnetic fields but also for detecting
dust. For example, the observation of submillimetre polarization proves a
needle origin for the cold dust emission and presence of metallic needles in
the ejecta from core-collapse supernovae (Dwek, 2004; Dunne et al., 2009).

Tracing magnetic fields in our Galaxy is particularly important for the
analysis of the CMB because the polarized galactic dust forms a foreground
which should be eliminated from the CMB polarization maps (Lazarian & Prunet,
2002; Gold et al., 2011; Ichiki, 2014; Planck Collaboration et al., 2015,
2016b). Some authors also point to a possible interaction of the CMB with
intergalactic magnetic fields (Ohno et al., 2003) and admit that these fields
may modify the pattern of the CMB anisotropies and, eventually, induce
additional anisotropies in the polarization (Giovannini, 2004). However, since
the CMB is believed to be a relic radiation originating in the Big Bang, the
possibility that the CMB is actually a dust radiation with polarization
tracing the large-scale magnetic fields has not been investigated or proposed.

Assuming that the CMB is produced by thermal radiation of intergalactic dust,
the small-scale polarization anisotropies of the CMB are readily explained by
the polarized thermal radiation of needle-shaped conducting dust grains
present in the IGM and aligned by cosmic magnetic fields produced by large-
scale structures in the Universe. The phenomenon is fully analogous to the
polarized interstellar dust emission in the Milky Way, which is observed at
shorter wavelengths because the temperature of the interstellar dust is higher
than that of the intergalactic dust. Since both the temperature and
polarization anisotropies of the CMB are caused by clusters and voids in the
Universe, they are spatially correlated.

##  9 Dust in the high-redshift Universe

The presence of a significant amount of dust is unexpected at high redshifts
in the Big Bang theory but reported in recent years by many authors. For
example, a submillimitre radiation coming from warm dust in the quasar host
galaxy is detected for a large number of high-redshift quasars ($z>5-6$)
observed by the IRAM 30-m telescope or SCUBA (Priddey et al., 2003; Fan, 2006;
Priddey et al., 2008) or mm and radio radiation of quasars observed with the
Max Planck Millimetre Bolometer Array (MAMBO) at 250 GHz (Wang et al., 2008).
Similarly, the existence of mature galaxies in the early Universe indicates
that this epoch was probably not as dark and young as so far assumed. Based on
observations of the Atacama Large Millimetre Array (ALMA), Watson et al.
(2015) investigated a galaxy at $z>7$ highly evolved with a large stellar mass
and heavily enriched in dust. Similarly, Laporte et al. (2017) analysed a
galaxy at a photometric redshift of $z\sim 8$ with a stellar mass of $\sim
2\times 10^{9}\,M_{\sun}$, a SFR of $\sim 20M_{\sun}/\mathrm{yr}$ and a dust
mass of $\sim 6\times 10^{6}M_{\sun}$. Also a significant increase in the
number of galaxies for $8.5<z<12$ reported by Ellis et al. (2013) and the
presence of a remarkably bright galaxy at $z\sim 11$ found by Oesch et al.
(2016) questions the assumption of the age and darkness of the high-redshift
Universe.

Although numerous recent observations confirm significant reddening of
galaxies and quasars caused by the presence of dust at high redshifts, it is
unclear which portion of the reddening is produced by local dust in a galaxy
and by intergalactic dust along the line of sight. Xie et al. (2015, 2016)
tried to distinguish between both sources of extinction by studying spectra of
$\sim 90.000$ quasars from the SDSS DR7 quasar catalogue (Schneider et al.,
2010). They calculated composite spectra in the redshift intervals
$0.71<z<1.19$ and $1.80<z<3.15$ in four bolometric luminosity bins and
revealed that quasars at higher redshifts have systematically redder UV
continuum slopes indicating intergalactic extinction $A_{V}$ of about $2\times
10^{-5}\,h\,\mathrm{Mpc}^{-1}$, see Figure 10 for its redshift dependence.

The dust content in the IGM can also be probed by studying absorption lines in
spectra of high-redshift quasars caused by intervening intergalactic gaseous
clouds. The massive clouds reach neutral hydrogen column densities
$N_{\mathrm{HI}}>10^{19}\,\mathrm{cm}^{-2}$ (Lyman-limit systems, LLS) or even
$N_{\mathrm{HI}}>2\times 10^{20}\,\mathrm{cm}^{-2}$ (damped Lyman systems,
DLA) and they are self-shielded against ionizing radiation from outside (Wolfe
et al., 2005; Meiksin, 2009). They have higher metallicities than any other
class of Lyman absorbers ([M/H] $\sim$ -1.1 dex; Pettini et al. (1994) and
they are expected to contain dust. The dust content is usually estimated from
the abundance ratio [Cr/Zn] assuming that this stable ratio is changed in
dusty environment because Cr is depleted on dust grains but Zn is undepleted.
For example, Pettini et al. (1994) analysed the [Cr/Zn] ratio of 17 DLAs at
$z_{\mathrm{abs}}\sim 2$ and reported a typical dust-to-gas ratio of 1/10 of
the value in the interstellar medium (ISM) in our Galaxy. Another analysis of
18 DLAs at $0.7<z_{\mathrm{abs}}<3.4$ performed by Pettini et al. (1997)
yielded a dust-to-gas ratio of about 1/30 of the Galaxy value. Other dust
indicators such as depletion of Fe and Si relative to S and Zn were used by
Petitjean et al. (2002) who studied a dust pattern for a DLA at
$z_{\mathrm{abs}}=1.97$ and provided evidence for dust grains with an
abundance similar to that in the cold gas in the Galaxy.

Similarly as for the interstellar medium, dust also well correlates with the
molecular hydrogen $\mathrm{H}_{2}$ in clouds (Levshakov et al., 2002;
Petitjean et al., 2008), because $\mathrm{H}_{2}$ is formed on the surfaces of
dust grains (Wolfe et al., 2005). Since the molecular hydrogen fraction
sharply increases above a H I column density of $5\times
10^{20}\,\mathrm{cm}^{-2}$ (Noterdaeme et al., 2017), DLAs can be expected to
form reservoirs of dust. For example, Noterdaeme et al. (2017) discovered a
molecular cloud in the early Universe at $z_{\mathrm{abs}}=2.52$ with a
supersolar metallicity and an overall molecular hydrogen fraction of about
50%, which contained also carbon monoxide molecules. The authors suggest the
presence of small dust grains to explain the observed atomic and molecular
abundances.

##  10 Corrections of luminosity and stellar mass density for intergalactic
dust

The presence of dust in the high-redshift Universe has consequences for
determining the star formation rate (SFR) and the global stellar mass history
(SMH). So far observations of the SFR and SMH are based on measurements of the
luminosity density evolution at UV and NIR wavelengths assuming a transparent
universe. The most complete measurements of the luminosity density evolution
are for the UV luminosity based on the Lyman break galaxy selections covering
redshifts up to 10-12 (Bouwens et al., 2011, 2015; Oesch et al., 2014). The
measurements of the global stellar mass density require surveys covering a
large fraction of the sky such as the SDSS and 2dFGRS. The local stellar mass
function was determined, for example, by Cole et al. (2001) using the 2dFGRS
redshifts and the NIR photometry from the 2MASS, and by Bell et al. (2003)
from the SDSS and 2MASS data. The observed luminosity density averaged over
different types of galaxies steeply increases with redshift as $(1+z)^{4}$ for
$z$ less than 1 (Franceschini et al., 2008; Hopkins, 2004). The luminosity
density culminates at redshifts of about 2-3 and then decreases. The stellar
mass density displays no significant evolution at redshifts $z<1$ (Brinchmann
& Ellis, 2000; Cohen, 2002). However, a strong evolution of the stellar mass
density is found at higher redshifts, $1<z<4$, characterized by a monotonous
decline (Hopkins & Beacom, 2006; Marchesini et al., 2009). This decline
continues even for redshifts $z>4$ (González et al., 2011; Lee et al., 2012).

Obviously, if a non-zero intergalactic opacity is considered, the observations
of the SFR and SMH must be corrected. The apparent stellar mass density
$\rho\left(z\right)$ determined under the assumption of a transparent universe
and the true stellar mass density $\rho_{\mathrm{true}}\left(z\right)$
determined for a dusty universe are related as

$\rho_{\mathrm{true}}\left(z\right)=\rho\left(z\right)e^{\tau\left(z\right)}\,,$
(41)

where $\tau\left(z\right)$ is the optical depth of intergalactic space
(Vavryčuk, 2017a, his equation 19)

$\tau\left(z\right)=\frac{c}{H_{0}}\int_{0}^{z}\lambda_{0}\left(1+z^{\prime}\right)^{2}\,\,\frac{dz^{\prime}}{E\left(z^{\prime}\right)}\,,$
(42)

and $\lambda_{0}$ is the UV intergalactic attenuation at zero redshift.

Analogously, the apparent UV luminosity density $j$ at redshift $z$ is
corrected for dust attenuation by multiplying it with the exponential factor
$e^{\tau\left(z\right)}$

$j_{\mathrm{true}}\left(z\right)=j\left(z\right)\,\left(1+z\right)^{-3}e^{\tau\left(z\right)}\,.$
(43)

The additional term $(1+z)^{-3}$ in equation (43) originates in the
transformation from the comoving to proper volumes (for a detailed discussion,
see Appendix A).

As shown in Vavryčuk (2017a), if the UV luminosity density is corrected for
intergalactic opacity and transformed from the proper to comoving volumes
according to equation (43), it becomes redshift independent, see Figure 11.
The abundance of the apparent luminosity density at redshifts $2<z<4$ is
commonly interpreted as the result of an enormously high SFR in this epoch
(Madau et al., 1998; Kochanek et al., 2001; Franceschini et al., 2008).
However, as indicated in Figure 11a, the luminosity density abundance at
$2<z<4$ is actually caused by the expansion of the Universe. When going back
in time, the Universe occupied a smaller volume and the proper number density
of galaxies and the proper luminosity density produced by galaxies were higher
(see Appendix A). The steep increase of the luminosity density is almost
unaffected by intergalactic opacity for $z<2$ because the Universe is
effectively transparent at this epoch. Since the opacity of the Universe
steeply increases with redshift and light extinction becomes significant for
$z>2-3$, the luminosity density does not increase further and starts to
decline with $z$. This decline continues to very high redshifts. After
correcting the luminosity density for intergalactic extinction, no decline at
high redshifts is observed (Figure 11b).

The theoretical predictions of the corrected SMH calculated for an
intergalactic opacity of $0.075\,h\,\mathrm{mag}\,\mathrm{Gpc}^{-1}$ at UV
wavelengths according to equation (41) are shown in Figure 12. If the
transparent universe characterized by zero attenuation is assumed,
observations suggest a steep decline of the stellar mass with redshift (Figure
12a). However, if intergalactic opacity is taken into account, the true SMH is
constant and independent of redshift (Figure 12b). Hence the decline of the
stellar mass density reported under the assumption of a transparent universe
might be fully an artefact of neglecting the opacity of the intergalactic
space. The redshift-independent comoving SMH (Figure 12b) looks apparently in
contradiction with permanent star-formation processes in the Universe.
However, it is physically conceivable provided the cosmic star formation rate
is balanced by the stellar mass-loss rate due to, for example, core-collapse
supernova explosions and stellar winds or superwinds (Heckman et al., 2000;
Woosley et al., 2002; Heger et al., 2003; Yoon & Cantiello, 2010).

Figure 11:  (a) The apparent comoving UV luminosity density as a function of
redshift assuming a transparent universe. (b) The corrected UV luminosity
density as a function of redshift in a dusty universe. Observations are taken
from Schiminovich et al. (2005, black circles), Reddy & Steidel (2009, green
circles), Bouwens et al. (2014a, blue circles), McLure et al. (2013, red
circles), Ellis et al. (2013, magenta circles), Oesch et al. (2014, cyan
circles), and Bouwens et al. (2014b, yellow circles). The solid line in (a)
and the dotted line in (b) show the predicted proper luminosity density for
the opaque universe with the UV intergalactic extinction of
$0.075\,\mathrm{mag}\,h\,\mathrm{Gpc}^{-1}$. The solid line in (b) shows the
predicted comoving luminosity density corrected for the opaque universe with
the UV intergalactic extinction of
$0.075\,\mathrm{mag}\,h\,\mathrm{Gpc}^{-1}$.  Figure 12:  (a) The apparent
global stellar mass history (Apparent SMH). The colour squares show
observations reported by Pérez-González et al. (2008, grey), Pozzetti et al.
(2010, green), Kajisawa et al. (2009, blue), Marchesini et al. (2009, red),
Reddy et al. (2012, cyan), González et al. (2011, black), Lee et al. (2012,
magenta), and Yabe et al. (2009, yellow). The values are summarized in Table 2
of Madau & Dickinson (2014). The black line shows the apparent stellar mass
history calculated using equation (26) of Vavryčuk (2017a) with the UV
intergalactic opacity of $0.08\,\mathrm{mag}\,h\,\mathrm{Gpc}^{-1}$ at $z=0$.
(b) The corrected global stellar mass history (Corrected SMH). The black
dotted line shows the stellar mass history for a transparent universe, the
black solid line shows the stellar mass history after eliminating the effect
of the intergalactic opacity assuming $A_{UV}$ of
$0.075\,\mathrm{mag}\,h\,\mathrm{Gpc}^{-1}$ at $z=0$.

##  11 Observations of submillimetre galaxies

A promising tool for probing the evolution of the Universe at high redshifts
are observations at submillimetre (submm) wavelengths using instruments such
as SCUBA (Holland et al., 1999), MAMBO (Kreysa et al., 1999), SPIRE (Griffin
et al., 2010) and SCUBA-2 (Holland et al., 2013). The key points that make the
submm observations attractive are: (1) their minimum distortion due to
intergalactic attenuation, and (2) their ability to sample the spectral energy
density (SED) of galaxies at wavelengths close to its rest-frame maximum of
$\sim 100\mu\mathrm{m}$. As a consequence, large negative K-corrections should
enable detecting galaxies at redshifts of up to $z\sim 20$ (Blain et al.,
2002; Casey et al., 2014). However, the submm observations also have
limitations. First, an accurate determination of redshifts is conditioned by
following up the submm sources at other wavelengths which is often difficult.
For example, matching submm sources to radio counterparts proved to be
successful in identifying them at other wavelengths (Chapman et al., 2005) but
sources with no or very weak radio counterparts are lost in this approach.
Second, the detection limit of $z\sim 20$ for submm galaxies is too optimistic
because it neglects the frequency- and redshift-dependent intergalactic
opacity. As shown in Figure 13, only a flux density at wavelengths greater
than 1 mm is essentially undistorted for $z$ up to 15. The flux density at 350
or 500 $\mu\mathrm{m}$ starts to be markedly attenuated at redshifts higher
than 10. Finally, the galaxy radiation targeted by the submm observations is
the thermal radiation of galactic dust with a temperature of $\sim$30-40 K.
Since the temperature of the intergalactic dust increases with redshift as
$(1+z)\,T^{\mathrm{CMB}}$, the galactic and intergalactic dust have similar
temperatures at redshifts of 10-15. Because of no or weak temperature contrast
between the galaxies and intergalactic dust at these redshifts, the thermal
radiation of galactic dust is lost in the background intergalactic radiation.
Consequently, dusty star-forming galaxies with dust temperatures of 30-40 K
cannot be observed at submm wavelengths at $z>10$.

As a result, so far only high-redshift galaxy samples of a limited size and
restricted to redshifts of less than 5-6 are available (Chapman et al., 2005;
Coppin et al., 2009; Cox et al., 2011; Strandet et al., 2016; Ikarashi et al.,
2017) which are not decisive enough for statistically relevant conclusions
about the galaxy number density and properties of galaxies at redshifts $z>5$.

Figure 13:  The predicted flux density of a dusty galaxy as a function of
redshift at several observed submm wavelengths. Full line - predictions of
Blain et al. (2002, their Figure 4) for a transparent Universe for wavelengths
of 1100 $\mu$m (red), 850 $\mu$m (green), 500 $\mu$m (blue) and 350 $\mu$m
(magenta). Dashed line - a correction for a non-zero intergalactic opacity.
Note the high K correction at wavelengths 850 $\mu$m and 1100 $\mu$m, which
yields a flux density almost independent of redshift. For higher frequencies,
this property is lost. The template spectrum is chosen to reproduce typical
properties of dusty galaxies (Blain et al., 2002, their Figure 2). The
corrected curves were calculated according to Vavryčuk (2017b, his equation
7).

##  12 Reionization and the Gunn-Peterson trough

The high-redshift Universe can be studied by an evolution of the neutral
hydrogen fraction in the IGM traced by observations of the Lyman $\alpha$
(Ly$\alpha$) forest of absorption lines in quasar optical spectra. These
absorption lines are produced by neutral hydrogen (H I) in intergalactic
gaseous clouds ionized by ultraviolet radiation at a wavelength of 1216
Å(Wolfe et al., 2005; Meiksin, 2009). The incidence rate of the absorption
lines per unit redshift carries information about the number density of the
Ly$\alpha$ clouds and its evolution in time and the width of the absorption
lines depends on the column density of the clouds.

According to the Big Bang theory, the neutral hydrogen in the IGM is reionized
at redshifts between 6 and 20 by the first quasars and galaxies (Gnedin &
Ostriker, 1997; Gnedin, 2000), so that the IGM becomes transparent for the UV
radiation at lower redshifts. Based on the interpretation of the CMB
polarization as a product of the Thomson scattering (Hu & White, 1997; Hu &
Dodelson, 2002), the reionization as a sudden event is predicted at $z\sim 11$
(Dunkley et al., 2009; Jarosik et al., 2011). Observations of the Ly$\alpha$
forest and the Gunn-Peterson trough in quasar spectra date the end of the
reionization at $z\sim 6-7$ (Becker et al., 2001; Fan et al., 2002, 2004,
2006). Obviously, if the Big Bang theory is not valid and the CMB has another
origin, the idea of dark ages with mostly neutral hydrogen in the IGM and no
sources of ionizing radiation is disputed. Similarly, the hypothesis about the
reionization as an epoch of a transition from neutral to ionized hydrogen due
to high-redshift galaxies and quasars is questioned.

Although, a change in the neutral hydrogen fraction by several tens of per
cent at $z\sim 6-7$, supporting the idea of reionization, has been suggested
by Pentericci et al. (2011), Ono et al. (2012) and Schenker et al. (2012), a
rapid increase of neutral hydrogen is in conflict with a modelling of the
evolution of the IGM, which favours the reionization as a rather gradual
process with a continuous rise of the Ly$\alpha$ photons mean free path
(Miralda-Escudé et al., 2000; Becker et al., 2007; Bolton & Haehnelt, 2013).
Moreover, it is unclear how sources with a declining comoving luminosity
density at $z>6$ (Figure 11a) could reionize the neutral hydrogen in the IGM
(Bunker et al., 2010; McLure et al., 2010; Grazian et al., 2011). Instead, the
Ly$\alpha$ optical depth measurements are more consistent with an essentially
constant and redshift-independent photoionization rate (Faucher-Giguère et
al., 2008, their Figure 14) predicted in Figure 11b and with no strong
evolution in the neutral hydrogen fraction of the IGM (Krug et al., 2012;
Bolton & Haehnelt, 2013).

The redshift-independent ionization background radiation might also explain a
puzzling ubiquity of the Ly$\alpha$ emission of very high-redshift galaxies if
the IGM is considered to be significantly neutral over $7<z<9$ (Stark et al.,
2017; Bagley et al., 2017). Hence, the evolution of the Ly$\alpha$ forest with
redshift and the Gunn-Peterson trough at $z\sim 6-7$ might not be produced by
increasing the abundance of neutral hydrogen in the Universe at $z>6$ but by
the Ly$\alpha$ clouds with a constant comoving neutral hydrogen fraction in a
smaller proper volume of the Universe at high redshift. Since overdense
Ly$\alpha$ regions with non-evolving neutral hydrogen fractions are close to
each other, they start to touch and prevent escaping the Ly$\alpha$ photons.

##  13 Evolution of metallicity

Another tool for probing the history of the Universe is tracing heavy elements
(metals) of the IGM with redshift. Models based on the Big Bang theory predict
a persistent increase of the mean metallicity of the Universe with cosmic time
driven mainly by star formation history (Pei & Fall, 1995; Madau et al.,
1998). The mean metallicity should rise from zero to 0.001 solar by $z=6$ (1
Gyr after Big Bang), reaching about 0.01 solar at $z=2.5$ and 0.09 solar at
$z=0$ (Madau & Dickinson, 2014, their Figure 14). Similarly to the mean
metallicity of the Universe, the abundance of metals dispersed into the IGM by
supernovae ejecta and winds is also continuously rising in the standard
cosmological models (Gnedin & Ostriker, 1997, their Figure 5). This can be
tested by comparing the column density of a singly ionized metal to that of
neutral hydrogen in Ly$\alpha$ systems. Frequently, DLA systems are selected
because the ionization corrections are assumed to be negligible due to the
high column density N(H I) producing the self-shielding of the DLA absorbers
(Prochaska & Wolfe, 2000; Vladilo et al., 2001).

However, observations do not provide convincing evidence of the predicted
metallicity evolution. The observations indicate (Rauch, 1998; Pettini, 2004;
Meiksin, 2009): (1) a puzzling widespread metal pollution of the IGM and a
failure to detect a pristine material with no metals even at high redshifts,
and (2) an unclear evolution of the metallicity. Prochaska & Wolfe (2000)
found no evolution in the N(H I)-weighted mean [Fe/H] metallicity for
redshifts $z$ from 1.5 to 4.5, but later studies of larger datasets of DLAs
have indicated a decrease of metallicity with increasing redshift (Prochaska
et al., 2003, their Figure 1). Rafelski et al. (2012) combined 241 abundances
of various metals (O I, S II, Si II, Zn II, Fe II and others) obtained from
their data and from the literature, and found a metallicity decrease of -0.22
dex per unit redshift for $z$ from 0.09 to 5.06. The decrease is, however,
significantly slower than the prediction and the scatter of measurements is
quite large (up to 2 dex) making the result unconvincing (Rafelski et al.,
2012, their Figures 5 and 6). Furthermore, observations of the C IV absorbers
do not show any visible redshift evolution over cosmic times from 1 to 4.5 Gyr
after the Big Bang suggesting that a large fraction of intergalactic metals
may already have been in place at $z>6$ (Songaila, 2001; Pettini et al., 2003;
Ryan-Weber et al., 2006).

##  14 Primordial deuterium, helium and lithium abundances

If the CMB is a thermal radiation of dust but not a relic radiation of the Big
Bang, the concept of the Big Bang is seriously disputed. Firstly, except for
the CMB, no direct observations indicate the Big Bang and no measurements
provide information on the actual expanding/contracting history of the
Universe at $z>8-10$. Secondly, predictions of some cosmological constants and
quantities based on the interpretation of the CMB anisotropies such as the
baryonic density, helium abundance and dark matter density in the Universe, or
timing of the reionization epoch at $z\sim 11$ (Spergel et al., 2003, 2007;
Dunkley et al., 2009; Planck Collaboration et al., 2015) are invalidated.

The only remaining argument for the Big Bang theory is its prediction of
primordial abundances of deuterium, helium and lithium in the Universe (Olive
et al., 2000; Cyburt et al., 2016). The Big Bang nucleosynthesis (BBN)
parameterizes the D, 4He and 7Li abundances by the baryon-to-photon ratio
$\eta$ or equivalently the baryon density $\Omega_{b}\,h^{2}$. The baryon
density is usually determined from deuterium abundance observations, so that
the theory is capable of predicting only the other two values: the helium
abundance ${}^{4}\mathrm{He/H}=0.2470$ and the lithium abundance
${}^{7}\mathrm{Li/H}=4.648\times 10^{-10}$. Initially, observations did not
match the predicted 4He/H abundance well (Pagel et al., 1992; Peimbert et al.,
2000) but after two decades of efforts (Peimbert et al., 2007; Izotov et al.,
2014; Aver et al., 2015) when adopting a large number of systematic and
statistical corrections (Peimbert et al., 2007, their Table 7), a satisfactory
fit has finally been achieved (Figure 14). By contrast, the fit of the lithium
abundance is much worse; the predicted 7Li/H abundance is 2-3 times larger
than observations (Cyburt et al., 2008; Fields, 2011). As stated by Cyburt et
al. (2016), to date, there is no solution of the discrepancy of the 7Li
abundance without substantial departures of the BBN theory. Hence, the BBN
theory may not provide us with fully-established firm evidence of the Big
Bang.

Figure 14:  Measurements of the primordial helium abundance $Yp$ derived from
H II regions with a small fraction of heavy metals. The measurements are taken
from Pagel et al. (1992), Peimbert et al. (2000), Luridiana et al. (2003),
Izotov & Thuan (2004), Izotov et al. (2007) and Peimbert et al. (2007). The
values are summarized in Peimbert (2008, his Table 1). The dashed line shows
the theoretical prediction. Modified after Peimbert (2008).

##  15 Dust theory and cyclic cosmology

The model of a dusty universe is based on completely different postulates than
the Big Bang theory. It is assumed that that global stellar mass density and
the overall dust masses within galaxies and in intergalactic space are
essentially constant with cosmic time. Consequently, the cosmic star formation
rate is balanced by the stellar mass-loss rate due to, for example, core-
collapse supernova explosions and stellar winds or superwinds. These
constraints are needed for the EBL to rise as $(1+z)^{4}$ and the dust
temperature to increase exactly as $(1+z)$ with redshift. These assumptions
seem apparently unphysical and contradicting observations but most of these
arguments are not actually well established and can be disproved as shown in
the previous sections.

If the number density of galaxies and the overall dust masses within galaxies
and in the intergalactic space are basically constant with cosmic time, then
this might happen within an oscillating model of the universe with repeating
expansion and contraction periods. The cyclic cosmological model was
originally proposed by Friedmann in 1922 (Friedmann, 1999) and developed later
in many modifications (Steinhardt & Turok, 2002; Novello & Bergliaffa, 2008;
Battefeld & Peter, 2015) including the quasi-steady-state cosmological model
(Narlikar et al., 2007). Obviously, the idea of the universe oscillating
within a given range of redshifts is a mere hypothesis full of open questions.
The primary question is: which forces drive the oscillations. Without
proposing any solution, we can just speculate that formations/destructions of
galaxies and complex recycling processes in galaxies and in the IGM might play
a central role in this model (Segers et al., 2016; Anglés-Alcázar et al.,
2017). Importantly, the upper limit of the redshift should not be very high
$(z\lesssim 20-30)$ to allow the existence of galaxies as independent units
even in the epoch of the minimum proper volume of the Universe. If so, the
oscillations around a stationary state reflect only some imbalance in the
Universe and the CMB comes partly from the previous cosmic cycle or cycles.

##  16 Conclusions

The analysis of the EBL and its extinction caused by opacity of the Universe
indicates that the CMB might be thermal radiation of intergalactic dust. Even
though the local Universe is virtually transparent with an opacity of $\sim
0.01\,\mathrm{mag}\,h\,\mathrm{Gpc}^{-1}$ at visual wavelengths, it might be
considerably opaque at high redshifts. For example, the visual opacity
predicted by the proposed model reaches values of about 0.08, 0.19, 0.34 and
0.69 mag at $z$ = 1, 2, 3 and 5, respectively. Such opacity is hardly to be
detected in the Type Ia supernova data (Jones et al., 2013; Rodney et al.,
2015; de Jaeger et al., 2017), but it can be studied using quasar data (Ménard
et al., 2010; Xie et al., 2015). Since the energy of light is absorbed by
dust, it heats up the dust and produces its thermal radiation. The temperature
of dust depends on the intensity of light surrounding dust particles. Within
galaxies, the light intensity is high the dust being heated up to
$20-40\,\mathrm{K}$ and emitting thermal radiation at IR and FIR wavelengths.
The intensity of light in intergalactic space is much lower than within
galaxies, hence the intergalactic dust is colder and emits radiation at
microwave wavelengths.

The actual intergalactic dust temperature depends on the balance of the energy
radiated and absorbed by galaxies and dust. The EBL energy radiated by
galaxies and absorbed by intergalactic dust is re-radiated in the form of the
CMB. The CMB radiation is mostly absorbed back by intergalactic dust but also
partly by the dust in galaxies. The intergalactic dust is warmed by the EBL
and the warming process continues until intergalactic dust reaches energy
equilibrium. This happens when the energy interchanged between galaxies and
intergalactic dust is mutually compensated. Hence the EBL energy absorbed by
dust equals the CMB energy absorbed by galaxies.

The distribution of the CMB energy between intergalactic dust and galaxies is
controlled by the opacity ratio calculated from the galactic and intergalactic
opacities. The opacity ratio is frequency and redshift independent and
controls the temperature of intergalactic dust. A high opacity ratio means
that a high amount of the CMB is absorbed back by dust. Consequently, dust is
warmed up by the EBL to high temperatures. A low opacity ratio means that a
significant part of the CMB energy is absorbed by galaxies, hence the dust is
warmed up by the EBL to rather low temperatures. The opacity ratio can also be
estimated from the EBL and CMB intensities. The optimum value of the opacity
ratio calculated from observations of the galactic and intergalactic opacities
is 13.4 while that obtained from the EBL and CMB intensities is 12.5. The fit
is excellent considering rather high uncertainties in observations of the EBL
and in the galactic and intergalactic opacities.

The thermal radiation of dust is redshift dependent similarly as the radiation
of any other objects in the Universe. Since its intensity basically depends on
the EBL, which increases with redshift as $\left(1+z\right)^{4}$, the CMB
temperature increases with redshift as $\left(1+z\right)$. The temperature
increase with $z$ exactly eliminates the frequency redshift of the dust
thermal radiation. Consequently, the radiation of dust observed at all
distances looks apparently as the radiation of the blackbody with a single
temperature. This eliminates the common argument against the CMB as the
thermal radiation of dust that the spectrum of dust radiation cannot be
characterized by a single temperature because of redshift (Peacock, 1999, p.
289). The redshift dependence of the EBL intensity, CMB temperature and CMB
intensity are invariant to the cosmology and can be applied to models of the
Universe with a complicated expanding/contracting history.

The CMB is radiated at a broad range of redshifts with the maximum CMB
intensity coming from redshifts of 25-40 provided that the hitherto assumed
expansion history of the Universe is correct. If the expansion history is
different, e.g., if the Universe is oscillating, then part of the CMB might
come from previous cosmic cycles. This indicates that the observed CMB stems
from an enormous space and a long epoch of the Universe. As a consequence, the
observed CMB temperature and intensity must be quite stable with only very
small variations with direction in the sky. These variations reflect the EBL
fluctuations due to the presence of large-scale structures as clusters,
superclusters and voids in the Universe. The predicted CMB variation
calculated from estimates of the EBL fluctuations attains values of tens of
$\mu$K well consistent with observations. The CMB polarization is produced by
a polarized emission of needle-shaped conducting dust grains present in the
IGM aligned by cosmic magnetic fields around large-scale structures in the
Universe. The phenomenon is fully analogous to a polarized interstellar dust
emission in the Galaxy which is observed at shorter wavelengths because the
temperature of the interstellar dust is higher than that of the intergalactic
dust. Since the temperature and polarization anisotropies of the CMB have a
common origin - existence of clusters, superclusters and voids - both
anisotropies are spatially correlated.

The intensity of the CMB exactly corresponds to the intensity radiated by the
blackbody with the CMB temperature. This implies that the energy flux received
at a unit area of intergalactic space is equal to the energy flux emitted by a
unit area of intergalactic dust particles. This statement is basically a
formulation of the Olbers’ paradox applied to dust particles instead of to
stars. Since the sky is fully covered by dust particles and distant background
particles are obscured by foreground particles, the energy fluxes emitted and
received by dust are equal. Consequently, the intensity of the CMB does not
depend on the actual dust density in the local Universe and on the
expanding/contracting history of the Universe.

Further development of the dust theory depends on more accurate measurements
of the EBL, distribution of the galactic and intergalactic dust, and the
opacity of galaxies and of the intergalactic space at high redshifts. More
definitive evidence of the properties of the high-redshift Universe can be
provided by the James Webb Space Telescope (Gardner et al., 2006; Zackrisson
et al., 2011). This telescope can probe the galaxy populations and properties
of the IGM at high redshift and check which cosmological model suits the
observations better.

##  Appendix A EBL and luminosity density

The bolometric intensity of the EBL (in $\mathrm{Wm}^{-2}\mathrm{sr}^{-1}$) is
calculated as an integral of the redshift-dependent bolometric luminosity
density of galaxies reduced by the attenuation-obscuration effect (Vavryčuk,
2017a, his equation 1)

$I_{0}^{\mathrm{EBL}}=\frac{1}{4\pi}\int_{0}^{z_{\mathrm{max}}}\frac{j\left(z\right)}{\left(1+z\right)^{2}}\,e^{-\tau\left(z\right)}\,\frac{c}{H_{0}}\frac{dz}{E\left(z\right)}\,,$
(44)

where $I_{0}^{\mathrm{EBL}}$ is the EBL intensity at present $(z=0)$, $j(z)$
is the proper bolometric luminosity density, $\tau(z)$ is the bolometric
optical depth, $c$ is the speed of light, $H_{0}$ is the Hubble constant, and
$E\left(z\right)$ is the dimensionless Hubble parameter

$E\left(z\right)=\sqrt{\left(1+z\right)^{2}\left(1+\Omega_{m}z\right)-z\left(2+z\right)\Omega_{\Lambda}}\,,$
(45)

where $\Omega_{m}$ is the total matter density, and $\Omega_{\Lambda}$ is the
dimensionless cosmological constant. Equation (A1) is approximate because the
optical depth is averaged over the EBL spectrum; a more accurate formula
should consider the optical depth $\tau\left(z\right)$ as a function of
frequency.

Taking into account that the proper luminosity density $j(z)$ in Equation (A1)
depends on redshift as (Vavryčuk, 2017a, his equation 5)

$j\left(z\right)=j_{0}\left(1+z\right)^{4}\,\,,$ (46)

where $j_{0}$ is the comoving luminosity density, we get

$I_{0}^{\mathrm{EBL}}=\frac{1}{4\pi}\int_{0}^{z_{\mathrm{max}}}j_{0}\left(1+z\right)^{2}\,e^{-\tau\left(z\right)}\,\frac{c}{H_{0}}\frac{dz}{E\left(z\right)}\,.$
(47)

Bear in mind that $j(z)$ in Equation (A1) is the proper luminosity density but
not the comoving luminosity density as commonly assumed, see Dwek et al.
(1998, their equation 9) or Hauser & Dwek (2001, their equation 5).
Integrating the comoving luminosity density in Equation (A1) would lead to
incorrect results because it ignores the fact that we observe the luminosity
density from different epochs of the Universe. When considering the proper
luminosity density in Equation (A1) we actually follow observations
(Franceschini et al., 2001; Lagache et al., 2005; Franceschini et al., 2008)
and sum the individual contributions to the EBL at various redshifts. The
proper luminosity density in the EBL integral is used also by Peacock (1999).
His formula is, however, different from Equation (A4) because he uses the
reference luminosity density $j_{0}$ at early cosmic times. Obviously, fixing
$j_{0}$ to the early cosmic times is possible and mathematically correct
(Peacock, 1999, his equation 3.95) but not applicable to calculating the EBL
using the luminosity density measured at $z=0$.

Equation (A4) can also be derived from equation (A1) in an alternative
straightforward way. The quantity $j\left(z\right)$ in equation (A1) means the
bolometric luminosity density at $z$ and the factor $(1+z)^{-2}$ reflects
reducing this luminosity density due to the expansion of the Universe.
However, if we calculate the EBL from observations, we do not fix $j$ at
redshift $z$ but at the present epoch ($z=0$). Hence, we go back in time and
correct the luminosity density $j_{0}$ at $z=0$ not for the expansion but for
the contraction of the Universe in its early times. As a consequence, instead
of the factor $(1+z)^{-2}$ in equation (A1) we use the factor $(1+z)^{2}$ in
equation (A4).

## Acknowledgements

I thank an anonymous reviewer for helpful comments and Alberto Domínguez for
providing me kindly with Figure 1. This research has made use of the NASA/IPAC
Infrared Science Archive, which is operated by the Jet Propulsion Laboratory,
California Institute of Technology, under contract with the National
Aeronautics and Space Administration.

## References

  * Abdo et al. (2010) Abdo A. A., et al., 2010, ApJ, 723, 1082
  * Aguirre (2000) Aguirre A. N., 2000, ApJ, 533, 1
  * Aharonian et al. (2006) Aharonian F., et al., 2006, Nature, 440, 1018
  * Alpher & Herman (1948) Alpher R. A., Herman R., 1948, Nature, 162, 774
  * Andersson (2015) Andersson B.-G., 2015, in Lazarian A., de Gouveia Dal Pino E. M., Melioli C., eds, Astrophysics and Space Science Library Vol. 407, Magnetic Fields in Diffuse Media. p. 59, doi:10.1007/978-3-662-44625-6_4
  * Anglés-Alcázar et al. (2017) Anglés-Alcázar D., Faucher-Giguère C.-A., Kereš D., Hopkins P. F., Quataert E., Murray N., 2017, MNRAS, 470, 4698
  * Arp et al. (1990) Arp H. C., Burbidge G., Hoyle F., Wickramasing N. C., Narlikar J. V., 1990, Nature, 346, 807
  * Aver et al. (2015) Aver E., Olive K. A., Skillman E. D., 2015, J. Cosmology Astropart. Phys., 7, 011
  * Bagley et al. (2017) Bagley M. B., et al., 2017, ApJ, 837, 11
  * Bahcall (1999) Bahcall N. A., 1999, in Dekel A., Ostriker J. P., eds, Formation of Structure in the Universe. p. 135
  * Barkats et al. (2005) Barkats D., et al., 2005, ApJ, 619, L127
  * Battefeld & Peter (2015) Battefeld D., Peter P., 2015, Phys. Rep., 571, 1
  * Beck et al. (1996) Beck R., Brandenburg A., Moss D., Shukurov A., Sokoloff D., 1996, ARA&A, 34, 155
  * Beck et al. (2013) Beck A. M., Hanasz M., Lesch H., Remus R.-S., Stasyszyn F. A., 2013, MNRAS, 429, L60
  * Becker et al. (2001) Becker R. H., et al., 2001, AJ, 122, 2850
  * Becker et al. (2007) Becker G. D., Rauch M., Sargent W. L. W., 2007, ApJ, 662, 72
  * Bell et al. (2003) Bell E. F., McIntosh D. H., Katz N., Weinberg M. D., 2003, ApJS, 149, 289
  * Bennett et al. (2003) Bennett C. L., et al., 2003, ApJS, 148, 1
  * Bennett et al. (2013) Bennett C. L., et al., 2013, ApJS, 208, 20
  * Blain et al. (2002) Blain A. W., Smail I., Ivison R. J., Kneib J.-P., Frayer D. T., 2002, Phys. Rep., 369, 111
  * Bohlin et al. (1978) Bohlin R. C., Savage B. D., Drake J. F., 1978, ApJ, 224, 132
  * Bolton & Haehnelt (2013) Bolton J. S., Haehnelt M. G., 2013, MNRAS, 429, 1695
  * Bond et al. (1991) Bond J. R., Carr B. J., Hogan C. J., 1991, ApJ, 367, 420
  * Bondi & Gold (1948) Bondi H., Gold T., 1948, MNRAS, 108, 252
  * Bouwens et al. (2011) Bouwens R. J., et al., 2011, Nature, 469, 504
  * Bouwens et al. (2014a) Bouwens R. J., et al., 2014a, ApJ, 793, 115
  * Bouwens et al. (2014b) Bouwens R. J., et al., 2014b, ApJ, 795, 126
  * Bouwens et al. (2015) Bouwens R. J., et al., 2015, ApJ, 803, 34
  * Brinchmann & Ellis (2000) Brinchmann J., Ellis R. S., 2000, ApJ, 536, L77
  * Bunker et al. (2010) Bunker A. J., et al., 2010, MNRAS, 409, 855
  * Cai et al. (2014) Cai Y.-C., Neyrinck M. C., Szapudi I., Cole S., Frenk C. S., 2014, ApJ, 786, 110
  * Calzetti (2001) Calzetti D., 2001, PASP, 113, 1449
  * Calzetti et al. (2000) Calzetti D., Armus L., Bohlin R. C., Kinney A. L., Koornneef J., Storchi-Bergmann T., 2000, ApJ, 533, 682
  * Carilli & Taylor (2002) Carilli C. L., Taylor G. B., 2002, ARA&A, 40, 319
  * Casey et al. (2014) Casey C. M., Narayanan D., Cooray A., 2014, Phys. Rep., 541, 45
  * Chapman et al. (2005) Chapman S. C., Blain A. W., Smail I., Ivison R. J., 2005, ApJ, 622, 772
  * Chelouche et al. (2007) Chelouche D., Koester B. P., Bowen D. V., 2007, ApJ, 671, L97
  * Cohen (2002) Cohen J. G., 2002, ApJ, 567, 672
  * Cole et al. (2001) Cole S., et al., 2001, MNRAS, 326, 255
  * Cooray (2016) Cooray A., 2016, Royal Society Open Science, 3, 150555
  * Cooray et al. (2004) Cooray A., Bock J. J., Keatin B., Lange A. E., Matsumoto T., 2004, ApJ, 606, 611
  * Cooray et al. (2012) Cooray A., Gong Y., Smidt J., Santos M. G., 2012, ApJ, 756, 92
  * Coppin et al. (2009) Coppin K. E. K., et al., 2009, MNRAS, 395, 1905
  * Cox et al. (2011) Cox P., et al., 2011, ApJ, 740, 63
  * Cruz et al. (2005) Cruz M., Martínez-González E., Vielva P., Cayón L., 2005, MNRAS, 356, 29
  * Cyburt et al. (2008) Cyburt R. H., Fields B. D., Olive K. A., 2008, J. Cosmology Astropart. Phys., 11, 012
  * Cyburt et al. (2016) Cyburt R. H., Fields B. D., Olive K. A., Yeh T.-H., 2016, Reviews of Modern Physics, 88, 015004
  * Davies et al. (1997) Davies J. I., Phillipps S., Trewhella M., Alton P., 1997, MNRAS, 291, 59
  * Dicke et al. (1965) Dicke R. H., Peebles P. J. E., Roll P. G., Wilkinson D. T., 1965, ApJ, 142, 414
  * Dickinson et al. (2004) Dickinson C., et al., 2004, MNRAS, 353, 732
  * Dole et al. (2006) Dole H., et al., 2006, A&A, 451, 417
  * Domínguez et al. (2011) Domínguez A., et al., 2011, MNRAS, 410, 2556
  * Draine (2003) Draine B. T., 2003, ARA&A, 41, 241
  * Draine & Lee (1984) Draine B. T., Lee H. M., 1984, ApJ, 285, 89
  * Draine & Li (2007) Draine B. T., Li A., 2007, ApJ, 657, 810
  * Dunkley et al. (2009) Dunkley J., et al., 2009, ApJS, 180, 306
  * Dunne et al. (2009) Dunne L., et al., 2009, MNRAS, 394, 1307
  * Dwek (2004) Dwek E., 2004, ApJ, 607, 848
  * Dwek & Krennrich (2005) Dwek E., Krennrich F., 2005, ApJ, 618, 657
  * Dwek et al. (1998) Dwek E., et al., 1998, ApJ, 508, 106
  * Ellis et al. (2013) Ellis R. S., et al., 2013, ApJ, 763, L7
  * Fan (2006) Fan X., 2006, New Astron. Rev., 50, 665
  * Fan et al. (2002) Fan X., Narayanan V. K., Strauss M. A., White R. L., Becker R. H., Pentericci L., Rix H.-W., 2002, AJ, 123, 1247
  * Fan et al. (2004) Fan X., et al., 2004, AJ, 128, 515
  * Fan et al. (2006) Fan X., et al., 2006, AJ, 132, 117
  * Faucher-Giguère et al. (2008) Faucher-Giguère C.-A., Lidz A., Hernquist L., Zaldarriaga M., 2008, ApJ, 688, 85
  * Fazio et al. (2004) Fazio G. G., et al., 2004, ApJS, 154, 39
  * Feretti et al. (2012) Feretti L., Giovannini G., Govoni F., Murgia M., 2012, A&ARv, 20, 54
  * Fields (2011) Fields B. D., 2011, Annual Review of Nuclear and Particle Science, 61, 47
  * Finelli et al. (2016) Finelli F., García-Bellido J., Kovács A., Paci F., Szapudi I., 2016, MNRAS, 455, 1246
  * Fixsen (2009) Fixsen D. J., 2009, ApJ, 707, 916
  * Fixsen et al. (1996) Fixsen D. J., Cheng E. S., Gales J. M., Mather J. C., Shafer R. A., Wright E. L., 1996, ApJ, 473, 576
  * Fonseca et al. (2017) Fonseca J., Silva M. B., Santos M. G., Cooray A., 2017, MNRAS, 464, 1948
  * Franceschini et al. (2001) Franceschini A., Aussel H., Cesarsky C. J., Elbaz D., Fadda D., 2001, A&A, 378, 1
  * Franceschini et al. (2008) Franceschini A., Rodighiero G., Vaccari M., 2008, A&A, 487, 837
  * Friedmann (1999) Friedmann A., 1999, General Relativity and Gravitation, 31, 1991
  * Gamow (1952) Gamow G., 1952, The creation of the universe.
  * Gamow (1956) Gamow G., 1956, Vistas in Astronomy, 2, 1726
  * Gardner et al. (2006) Gardner J. P., et al., 2006, Space Sci. Rev., 123, 485
  * Gilmore et al. (2012) Gilmore R. C., Somerville R. S., Primack J. R., Domínguez A., 2012, MNRAS, 422, 3189
  * Giovannini (2004) Giovannini M., 2004, International Journal of Modern Physics D, 13, 391
  * Gnedin (2000) Gnedin N. Y., 2000, ApJ, 535, 530
  * Gnedin & Ostriker (1997) Gnedin N. Y., Ostriker J. P., 1997, ApJ, 486, 581
  * Gold et al. (2011) Gold B., et al., 2011, ApJS, 192, 15
  * Gong et al. (2017) Gong Y., Cooray A., Silva M. B., Zemcov M., Feng C., Santos M. G., Dore O., Chen X., 2017, ApJ, 835, 273
  * González et al. (2011) González V., Labbé I., Bouwens R. J., Illingworth G., Franx M., Kriek M., 2011, ApJ, 735, L34
  * Grazian et al. (2011) Grazian A., et al., 2011, A&A, 532, A33
  * Griffin et al. (2010) Griffin M. J., et al., 2010, A&A, 518, L3
  * Gurzadyan et al. (2014) Gurzadyan V. G., Kashin A. L., Khachatryan H., Poghosian E., Sargsyan S., Yegorian G., 2014, A&A, 566, A135
  * Gush et al. (1990) Gush H. P., Halpern M., Wishnow E. H., 1990, Physical Review Letters, 65, 537
  * Hauser & Dwek (2001) Hauser M. G., Dwek E., 2001, ARA&A, 39, 249
  * Hawkins & Wright (1988) Hawkins I., Wright E. L., 1988, ApJ, 324, 46
  * Heckman (2001) Heckman T. M., 2001, in Hibbard J. E., Rupen M., van Gorkom J. H., eds, Astronomical Society of the Pacific Conference Series Vol. 240, Gas and Galaxy Evolution. p. 345 (arXiv:astro-ph/0009075)
  * Heckman et al. (2000) Heckman T. M., Lehnert M. D., Strickland D. K., Armus L., 2000, ApJS, 129, 493
  * Heger et al. (2003) Heger A., Fryer C. L., Woosley S. E., Langer N., Hartmann D. H., 2003, ApJ, 591, 288
  * Henning et al. (1995) Henning T., Michel B., Stognienko R., 1995, Planet. Space Sci., 43, 1333
  * Hinshaw et al. (2009) Hinshaw G., et al., 2009, ApJS, 180, 225
  * Holland et al. (1999) Holland W. S., et al., 1999, MNRAS, 303, 659
  * Holland et al. (2013) Holland W. S., et al., 2013, MNRAS, 430, 2513
  * Holwerda et al. (2005) Holwerda B. W., Gonzalez R. A., Allen R. J., van der Kruit P. C., 2005, AJ, 129, 1396
  * Hopkins (2004) Hopkins A. M., 2004, ApJ, 615, 209
  * Hopkins & Beacom (2006) Hopkins A. M., Beacom J. F., 2006, ApJ, 651, 142
  * Hoyle (1948) Hoyle F., 1948, MNRAS, 108, 372
  * Hoyle & Vogeley (2004) Hoyle F., Vogeley M. S., 2004, ApJ, 607, 751
  * Hoyle & Wickramasinghe (1967) Hoyle F., Wickramasinghe N. C., 1967, Nature, 214, 969
  * Hoyle et al. (1993) Hoyle F., Burbidge G., Narlikar J. V., 1993, ApJ, 410, 437
  * Hoyle et al. (1994) Hoyle F., Burbidge G., Narlikar J. V., 1994, MNRAS, 267, 1007
  * Hu & Dodelson (2002) Hu W., Dodelson S., 2002, ARA&A, 40, 171
  * Hu & White (1997) Hu W., White M., 1997, New Astron., 2, 323
  * Ichiki (2014) Ichiki K., 2014, Progress of Theoretical and Experimental Physics, 2014, 06B109
  * Ikarashi et al. (2017) Ikarashi S., et al., 2017, ApJ, 835, 286
  * Izotov & Thuan (2004) Izotov Y. I., Thuan T. X., 2004, ApJ, 602, 200
  * Izotov et al. (2007) Izotov Y. I., Thuan T. X., Stasińska G., 2007, ApJ, 662, 15
  * Izotov et al. (2014) Izotov Y. I., Thuan T. X., Guseva N. G., 2014, MNRAS, 445, 778
  * Jarosik et al. (2011) Jarosik N., et al., 2011, ApJS, 192, 14
  * Jones et al. (2004) Jones B. J., Martínez V. J., Saar E., Trimble V., 2004, Reviews of Modern Physics, 76, 1211
  * Jones et al. (2013) Jones D. O., et al., 2013, ApJ, 768, 166
  * Kajisawa et al. (2009) Kajisawa M., et al., 2009, ApJ, 702, 1393
  * Kashlinsky et al. (2002) Kashlinsky A., Odenwald S., Mather J., Skrutskie M. F., Cutri R. M., 2002, ApJ, 579, L53
  * Kashlinsky et al. (2007) Kashlinsky A., Arendt R. G., Mather J., Moseley S. H., 2007, ApJ, 654, L5
  * Kneiske et al. (2004) Kneiske T. M., Bretz T., Mannheim K., Hartmann D. H., 2004, A&A, 413, 807
  * Kochanek et al. (2001) Kochanek C. S., et al., 2001, ApJ, 560, 566
  * Kogut et al. (1993) Kogut A., et al., 1993, ApJ, 419, 1
  * Komatsu et al. (2011) Komatsu E., et al., 2011, ApJS, 192, 18
  * Kovac et al. (2002) Kovac J. M., Leitch E. M., Pryke C., Carlstrom J. E., Halverson N. W., Holzapfel W. L., 2002, Nature, 420, 772
  * Kovács et al. (2017) Kovács A., et al., 2017, MNRAS, 465, 4166
  * Kovalev et al. (2016) Kovalev Y. Y., et al., 2016, ApJ, 820, L9
  * Kreysa et al. (1999) Kreysa E., et al., 1999, Infrared Physics and Technology, 40, 191
  * Kronberg et al. (1999) Kronberg P. P., Lesch H., Hopp U., 1999, ApJ, 511, 56
  * Krug et al. (2012) Krug H. B., et al., 2012, ApJ, 745, 122
  * Kulsrud (1999) Kulsrud R. M., 1999, ARA&A, 37, 37
  * Lagache et al. (2005) Lagache G., Puget J.-L., Dole H., 2005, ARA&A, 43, 727
  * Laporte et al. (2017) Laporte N., et al., 2017, ApJ, 837, L21
  * Layzer & Hively (1973) Layzer D., Hively R., 1973, ApJ, 179, 361
  * Lazarian (2007) Lazarian A., 2007, J. Quant. Spectrosc. Radiative Transfer, 106, 225
  * Lazarian & Prunet (2002) Lazarian A., Prunet S., 2002, in Cecchini S., Cortiglioni S., Sault R., Sbarra C., eds, American Institute of Physics Conference Series Vol. 609, Astrophysical Polarized Backgrounds. pp 32–43 (arXiv:astro-ph/0111214), doi:10.1063/1.1471820
  * Lee et al. (2012) Lee K.-S., et al., 2012, ApJ, 752, 66
  * Leitch et al. (2002) Leitch E. M., et al., 2002, Nature, 420, 763
  * Levshakov et al. (2002) Levshakov S. A., Dessauges-Zavadsky M., D’Odorico S., Molaro P., 2002, ApJ, 565, 696
  * Li & Draine (2001) Li A., Draine B. T., 2001, ApJ, 554, 778
  * Luridiana et al. (2003) Luridiana V., Peimbert A., Peimbert M., Cerviño M., 2003, ApJ, 592, 846
  * Ma et al. (2017) Ma J., Ge J., Zhao Y., Prochaska J. X., Zhang S., Ji T., Schneider D. P., 2017, MNRAS, 472, 2196
  * MacTavish et al. (2006) MacTavish C. J., et al., 2006, ApJ, 647, 799
  * Madau & Dickinson (2014) Madau P., Dickinson M., 2014, ARA&A, 52, 415
  * Madau & Pozzetti (2000) Madau P., Pozzetti L., 2000, MNRAS, 312, L9
  * Madau et al. (1998) Madau P., Pozzetti L., Dickinson M., 1998, ApJ, 498, 106
  * Marchesini et al. (2009) Marchesini D., van Dokkum P. G., Förster Schreiber N. M., Franx M., Labbé I., Wuyts S., 2009, ApJ, 701, 1765
  * Masters et al. (2015) Masters D., et al., 2015, ApJ, 813, 53
  * Mather et al. (1990) Mather J. C., et al., 1990, ApJ, 354, L37
  * Mathis (1990) Mathis J. S., 1990, ARA&A, 28, 37
  * Matsumoto et al. (2005) Matsumoto T., et al., 2005, ApJ, 626, 31
  * Matsumoto et al. (2011) Matsumoto T., et al., 2011, ApJ, 742, 124
  * McLure et al. (2010) McLure R. J., Dunlop J. S., Cirasuolo M., Koekemoer A. M., Sabbi E., Stark D. P., Targett T. A., Ellis R. S., 2010, MNRAS, 403, 960
  * McLure et al. (2013) McLure R. J., et al., 2013, MNRAS, 432, 2696
  * Meiksin (2009) Meiksin A. A., 2009, Reviews of Modern Physics, 81, 1405
  * Ménard et al. (2010) Ménard B., Scranton R., Fukugita M., Richards G., 2010, MNRAS, 405, 1025
  * Miralda-Escudé et al. (2000) Miralda-Escudé J., Haehnelt M., Rees M. J., 2000, ApJ, 530, 1
  * Montroy et al. (2006) Montroy T. E., et al., 2006, ApJ, 647, 813
  * Muller et al. (2008) Muller S., Wu S.-Y., Hsieh B.-C., González R. A., Loinard L., Yee H. K. C., Gladders M. D., 2008, ApJ, 680, 975
  * Narlikar et al. (2003) Narlikar J. V., Vishwakarma R. G., Hajian A., Souradeep T., Burbidge G., Hoyle F., 2003, ApJ, 585, 1
  * Narlikar et al. (2007) Narlikar J. V., Burbidge G., Vishwakarma R. G., 2007, Journal of Astrophysics and Astronomy, 28, 67
  * Noterdaeme et al. (2017) Noterdaeme P., et al., 2017, A&A, 597, A82
  * Novello & Bergliaffa (2008) Novello M., Bergliaffa S. E. P., 2008, Phys. Rep., 463, 127
  * Oesch et al. (2014) Oesch P. A., et al., 2014, ApJ, 786, 108
  * Oesch et al. (2016) Oesch P. A., et al., 2016, ApJ, 819, 129
  * Ohno et al. (2003) Ohno H., Takada M., Dolag K., Bartelmann M., Sugiyama N., 2003, ApJ, 584, 599
  * Olive et al. (2000) Olive K. A., Steigman G., Walker T. P., 2000, Phys. Rep., 333, 389
  * Ono et al. (2012) Ono Y., et al., 2012, ApJ, 744, 83
  * Page et al. (2007) Page L., et al., 2007, ApJS, 170, 335
  * Pagel et al. (1992) Pagel B. E. J., Simonson E. A., Terlevich R. J., Edmunds M. G., 1992, MNRAS, 255, 325
  * Pan (1988) Pan G., 1988, Chinese Astron. Astrophys., 12, 191
  * Peacock (1999) Peacock J. A., 1999, Cosmological Physics
  * Peebles (1993) Peebles P. J. E., 1993, Principles of Physical Cosmology
  * Pei & Fall (1995) Pei Y. C., Fall S. M., 1995, ApJ, 454, 69
  * Peimbert (2008) Peimbert M., 2008, preprint,  (arXiv:0811.2980)
  * Peimbert et al. (2000) Peimbert M., Peimbert A., Ruiz M. T., 2000, ApJ, 541, 688
  * Peimbert et al. (2007) Peimbert M., Luridiana V., Peimbert A., 2007, ApJ, 666, 636
  * Pentericci et al. (2011) Pentericci L., et al., 2011, ApJ, 743, 132
  * Penzias & Wilson (1965) Penzias A. A., Wilson R. W., 1965, ApJ, 142, 419
  * Pérez-González et al. (2008) Pérez-González P. G., et al., 2008, ApJ, 675, 234
  * Petitjean et al. (2002) Petitjean P., Srianand R., Ledoux C., 2002, MNRAS, 332, 383
  * Petitjean et al. (2008) Petitjean P., Ledoux C., Srianand R., 2008, A&A, 480, 349
  * Pettini (2004) Pettini M., 2004, in Esteban C., García López R., Herrero A., Sánchez F., eds, Cosmochemistry. The melting pot of the elements. pp 257–298 (arXiv:astro-ph/0303272)
  * Pettini et al. (1994) Pettini M., Smith L. J., Hunstead R. W., King D. L., 1994, ApJ, 426, 79
  * Pettini et al. (1997) Pettini M., King D. L., Smith L. J., Hunstead R. W., 1997, ApJ, 478, 536
  * Pettini et al. (2003) Pettini M., Madau P., Bolte M., Prochaska J. X., Ellison S. L., Fan X., 2003, ApJ, 594, 695
  * Planck Collaboration et al. (2014a) Planck Collaboration et al., 2014a, A&A, 571, A15
  * Planck Collaboration et al. (2014b) Planck Collaboration et al., 2014b, A&A, 571, A16
  * Planck Collaboration et al. (2014c) Planck Collaboration et al., 2014c, A&A, 571, A24
  * Planck Collaboration et al. (2015) Planck Collaboration et al., 2015, A&A, 576, A104
  * Planck Collaboration et al. (2016a) Planck Collaboration et al., 2016a, A&A, 594, A11
  * Planck Collaboration et al. (2016b) Planck Collaboration et al., 2016b, A&A, 596, A105
  * Pozzetti et al. (2010) Pozzetti L., et al., 2010, A&A, 523, A13
  * Priddey et al. (2003) Priddey R. S., Isaak K. G., McMahon R. G., Robson E. I., Pearson C. P., 2003, MNRAS, 344, L74
  * Priddey et al. (2008) Priddey R. S., Ivison R. J., Isaak K. G., 2008, MNRAS, 383, 289
  * Primack et al. (2011) Primack J. R., Domínguez A., Gilmore R. C., Somerville R. S., 2011, in Aharonian F. A., Hofmann W., Rieger F. M., eds, American Institute of Physics Conference Series Vol. 1381, American Institute of Physics Conference Series. pp 72–83 (arXiv:1107.2566), doi:10.1063/1.3635825
  * Prochaska & Wolfe (2000) Prochaska J. X., Wolfe A. M., 2000, ApJ, 533, L5
  * Prochaska et al. (2003) Prochaska J. X., Gawiser E., Wolfe A. M., Castro S., Djorgovski S. G., 2003, ApJ, 595, L9
  * Purcell (1969) Purcell E. M., 1969, ApJ, 158, 433
  * Pyo et al. (2012) Pyo J., Matsumoto T., Jeong W.-S., Matsuura S., 2012, ApJ, 760, 102
  * Rachford et al. (2002) Rachford B. L., et al., 2002, ApJ, 577, 221
  * Rafelski et al. (2012) Rafelski M., Wolfe A. M., Prochaska J. X., Neeleman M., Mendez A. J., 2012, ApJ, 755, 89
  * Rauch (1998) Rauch M., 1998, ARA&A, 36, 267
  * Readhead et al. (2004) Readhead A. C. S., et al., 2004, ApJ, 609, 498
  * Reddy & Steidel (2009) Reddy N. A., Steidel C. C., 2009, ApJ, 692, 778
  * Reddy et al. (2012) Reddy N., et al., 2012, ApJ, 744, 154
  * Rees (1978) Rees M. J., 1978, Nature, 275, 35
  * Reichardt et al. (2009) Reichardt C. L., et al., 2009, ApJ, 694, 1200
  * Rodney et al. (2015) Rodney S. A., et al., 2015, AJ, 150, 156
  * Rudnick et al. (2007) Rudnick L., Brown S., Williams L. R., 2007, ApJ, 671, 40
  * Ryan-Weber et al. (2006) Ryan-Weber E. V., Pettini M., Madau P., 2006, MNRAS, 371, L78
  * Schenker et al. (2012) Schenker M. A., Stark D. P., Ellis R. S., Robertson B. E., Dunlop J. S., McLure R. J., Kneib J.-P., Richard J., 2012, ApJ, 744, 179
  * Schiminovich et al. (2005) Schiminovich D., et al., 2005, ApJ, 619, L47
  * Schlegel et al. (1998) Schlegel D. J., Finkbeiner D. P., Davis M., 1998, ApJ, 500, 525
  * Schneider et al. (2010) Schneider D. P., et al., 2010, AJ, 139, 2360
  * Schwarz et al. (2016) Schwarz D. J., Copi C. J., Huterer D., Starkman G. D., 2016, Classical and Quantum Gravity, 33, 184001
  * Segers et al. (2016) Segers M. C., Crain R. A., Schaye J., Bower R. G., Furlong M., Schaller M., Theuns T., 2016, MNRAS, 456, 1235
  * Songaila (2001) Songaila A., 2001, ApJ, 561, L153
  * Spergel et al. (2003) Spergel D. N., et al., 2003, ApJS, 148, 175
  * Spergel et al. (2007) Spergel D. N., et al., 2007, ApJS, 170, 377
  * Stark et al. (2017) Stark D. P., et al., 2017, MNRAS, 464, 469
  * Stecker et al. (2006) Stecker F. W., Malkan M. A., Scully S. T., 2006, ApJ, 648, 774
  * Steinhardt & Turok (2002) Steinhardt P. J., Turok N., 2002, Science, 296, 1436
  * Stognienko et al. (1995) Stognienko R., Henning T., Ossenkopf V., 1995, A&A, 296, 797
  * Strandet et al. (2016) Strandet M. L., et al., 2016, ApJ, 822, 80
  * Szapudi et al. (2015) Szapudi I., et al., 2015, MNRAS, 450, 288
  * Thompson et al. (2007) Thompson T. A., Quataert E., Waxman E., 2007, ApJ, 654, 219
  * Vallée (2004) Vallée J. P., 2004, New Astron. Rev., 48, 763
  * Vavryčuk (2016) Vavryčuk V., 2016, Ap&SS, 361, 198
  * Vavryčuk (2017a) Vavryčuk V., 2017a, MNRAS, 465, 1532
  * Vavryčuk (2017b) Vavryčuk V., 2017b, MNRAS, 470, L44
  * Vielva et al. (2004) Vielva P., Martínez-González E., Barreiro R. B., Sanz J. L., Cayón L., 2004, ApJ, 609, 22
  * Vladilo et al. (2001) Vladilo G., Centurión M., Bonifacio P., Howk J. C., 2001, ApJ, 557, 1007
  * Vladilo et al. (2006) Vladilo G., Centurión M., Levshakov S. A., Péroux C., Khare P., Kulkarni V. P., York D. G., 2006, A&A, 454, 151
  * Wang et al. (2008) Wang R., et al., 2008, ApJ, 687, 848
  * Watson et al. (2015) Watson D., Christensen L., Knudsen K. K., Richard J., Gallazzi A., Michałowski M. J., 2015, Nature, 519, 327
  * Widrow (2002) Widrow L. M., 2002, Reviews of Modern Physics, 74, 775
  * Wolfe et al. (2005) Wolfe A. M., Gawiser E., Prochaska J. X., 2005, ARA&A, 43, 861
  * Woosley et al. (2002) Woosley S. E., Heger A., Weaver T. A., 2002, Reviews of Modern Physics, 74, 1015
  * Wright (1982) Wright E. L., 1982, ApJ, 255, 401
  * Wright (1987) Wright E. L., 1987, ApJ, 320, 818
  * Xie et al. (2015) Xie X., Shen S., Shao Z., Yin J., 2015, ApJ, 802, L16
  * Xie et al. (2016) Xie X., Shao Z., Shen S., Liu H., Li L., 2016, ApJ, 824, 38
  * Yabe et al. (2009) Yabe K., Ohta K., Iwata I., Sawicki M., Tamura N., Akiyama M., Aoki K., 2009, ApJ, 693, 507
  * Yoon & Cantiello (2010) Yoon S.-C., Cantiello M., 2010, ApJ, 717, L62
  * Zackrisson et al. (2011) Zackrisson E., Rydberg C.-E., Schaerer D., Östlin G., Tuli M., 2011, ApJ, 740, 13
  * Zemcov et al. (2014) Zemcov M., et al., 2014, Science, 346, 732
  * Zwaan et al. (2005) Zwaan M. A., van der Hulst J. M., Briggs F. H., Verheijen M. A. W., Ryan-Weber E. V., 2005, MNRAS, 364, 1467
  * de Jaeger et al. (2017) de Jaeger T., et al., 2017, ApJ, 835, 166
  * van Daalen et al. (2011) van Daalen M. P., Schaye J., Booth C. M., Dalla Vecchia C., 2011, MNRAS, 415, 3649
  * von Benda-Beckmann & Müller (2008) von Benda-Beckmann A. M., Müller V., 2008, MNRAS, 384, 1189