Missing dust signature in the cosmic microwave background

I examine a possible spectral distortion of the Cosmic Microwave Background (CMB) due to its absorption by galactic and intergalactic dust. I show that even subtle intergalactic opacity of $1 \times 10^{-7}\, \mathrm{mag}\, h\, \mathrm{Gpc}^{-1}$ at the CMB wavelengths in the local Universe causes non-negligible CMB absorption and decline of the CMB intensity because the opacity steeply increases with redshift. The CMB should be distorted even during the epoch of the Universe defined by redshifts $z<10$. For this epoch, the maximum spectral distortion of the CMB is at least $20 \times 10^{-22} \,\mathrm{Wm}^{-2}\, \mathrm{Hz}^{-1}\, \mathrm{sr}^{-1}$ at 300 GHz being well above the sensitivity of the COBE/FIRAS, WMAP or Planck flux measurements. If dust mass is considered to be redshift dependent with noticeable dust abundance at redshifts 2-4, the predicted CMB distortion is even higher. The CMB would be distorted also in a perfectly transparent universe due to dust in galaxies but this effect is lower by one order than that due to intergalactic opacity. The fact that the distortion of the CMB by dust is not observed is intriguing and questions either opacity and extinction law measurements or validity of the current model of the Universe.


I N T RO D U C T I O N
Observations of the cosmic microwave background (CMB) based on rocket measurements of Gush, Halpern & Wishnow (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 ± 0.004 K (Fixsen et al. 1996). The accuracy was improved using the WMAP data, which yielded an average temperature of T = 2.72548 ± 0.00057 K (Fixsen 2009). Observed tiny large-scale variations of the CMB temperature of ±0.00335 K are attributed to the motion (including rotation) of the Milky Way relative to the Universe (Kogut et al. 1993). The small-scale variations of ±300 μK traced, for example, by the WMAP (Bennett et al. 2003;Hinshaw et al. 2009;Bennett et al. 2013), ACBAR (Reichardt et al. 2009) and BOOMERanG (MacTavish et al. 2006) instruments using angular multipole moments are attributed to basic properties of the Universe as its curvature or the dark-matter density (Spergel et al. 2007;Komatsu et al. 2011).
Since the CMB as a relic radiation of the big bang experienced different epochs of the Universe, it interacted with matter of varying physical and chemical properties. Distortions of the CMB due to this E-mail: vv@ig.cas.cz interaction comprise the μ-type (at z 10 5 ) and y-type (at z 10 4 ) distortions related to the photon-electron interactions, distortions produced by the reionized IGM and the presence of galactic and extragalactic foregrounds (Wright 1981;Chluba & Sunyaev 2012;De Zotti et al. 2016). The foreground contamination of the CMB due to diffuse emission of intergalactic dust thermalized by the absorption of starlight was estimated, for example by Imara & Loeb (2016b). They found that the predicted contamination is under the detection of the COBE/FIRAS experiments (Mather et al. 1994;Fixsen et al. 1996) but it should be recognized in observations of the Primordial Inflation Explorer (Kogut et al. 2014) and the Polarized Radiation Imaging and Spectroscopy Mission (André et al. 2014) that would exceed the spectral sensitivity limits of COBE/FIRAS by three to four orders of magnitude.
Another possible origin of distortion of the CMB related to galactic and intergalactic dust is absorption of the CMB by dust. Absorbing properties of dust grains have been discussed by Wright (1987), Wright (1991), Henning, Michel & Stognienko (1995), Stognienko, Henning & Ossenkopf (1995) and others, who pointed out that the long-wavelength absorption of needle-shaped conducting grains or complex fractal or fluffy dust aggregates might provide a sufficient opacity for the CMB. Hence, it is worth to model the CMB attenuation by dust and to check if it is detectable or not. In this paper, I study the spectral and total distortions of the CMB due to absorption by dust. I find that the imprint of cosmic dust in the CMB predicted by theory is not negligible; however, it is missing in observations even though it is above their current detection level.

Optical depth
Effective optical depth τ (z) for light emitted at redshift z is expressed as (Peebles 1993, his equation 13.42) where n D is the comoving dust number density, σ is the attenuation cross-section, E(z) is the dimensionless Hubble parameter c is the speed of light, H 0 is the Hubble constant, m is the total matter density and is the dimensionless cosmological constant. Equation (1) can be rewritten using galactic and intergalactic attenuation coefficients ε G and ε IG as with where κ is the mean galactic opacity and γ is the mean free path of a light ray between galaxies in the comoving space where a is the mean galaxy radius and n is the galaxy number density in the comoving space. Equation (3) is valid for frequencyindependent attenuation. Considering the 'λ −β extinction law', where λ is the wavelength of light (Mathis 1990;Calzetti, Kinney & Storchi-Bergmann 1994;Charlot & Fall 2000;Draine 2003), we can express the galactic and intergalactic attenuations at frequency ν using the reference quantities related to observed frequency ν 0 , Equation (3) is then modified to expressing the fact that light is more attenuated at high z because of its shift to high frequencies.

Extinction of the CMB
Assuming the CMB to be a perfect blackbody radiation, its spectral intensity (i.e. energy flux received per unit area from a unit solid angle in the frequency interval ν to ν + dν, in Wm −2 Hz −1 sr −1 ) is described by the Planck's law where ν is the frequency, T CMB is the CMB temperature, h is the Planck constant, c is the speed of light and k B is the Boltzmann constant. Since the CMB is attenuated by galactic and intergalactic opacities, we can evaluate the distortion of the spectral CMB intensity at frequency ν along light ray coming from redshift z as where τ ν and I ν are defined in equations (7) and (8). Consequently, the reduction of the total CMB intensity (in Wm −2 sr −1 ) is Evaluating equations (9) and (10) for different redshifts z, we can predict the distortion of the CMB intensity by the opacity of the Universe when going back in cosmic time up to redshift z. Such approach is advantageous because it suppresses uncertainties in observed parameters needed in calculations. We start at present time, when the galactic and intergalactic opacities are best constrained from observations, and gradually extrapolate the prediction to higher redshifts.

O PAC I T Y O B S E RVAT I O N S
In order to evaluate the CMB distortion due to absorption by dust, we need estimates of the dust mass in the Universe and its history. The most straightforward way is to use observations of the galactic and intergalactic opacities at visual wavelengths mapping the distribution of dust in galaxies and intergalactic space and relate the visual and CMB opacities using the extinction law describing the dependence of attenuation of light on wavelength.

Galactic and intergalactic opacities
The opacity of galaxies basically depends on their type and age (for a review, see Calzetti 2001). The most transparent galaxies are elliptical with an effective extinction A V of 0.04 − 0.08 mag. The light extinction by dust in spiral and irregular galaxies is higher (González et al. 1998;Holwerda et al. 2005a. Typical values for the inclination-averaged extinction are as follows: 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). Considering the relative frequency of galaxy types in the Universe, we can average the visual extinctions of individual galaxy types and calculate the mean visual extinction and the mean visual galactic opacity. According to Vavryčuk (2017), the average value of visual opacity κ V is about 0.22 ± 0.08 at z = 0.
The intergalactic opacity is lower by several orders than the galactic opacity being observed, particularly, in galaxy haloes and in cluster centres (Ménard et al. 2010a). The opacity in the galaxy clusters has been measured by reddening of background objects behind the clusters (Chelouche, Koester & Bowen 2007;Bovy, Hogg & Moustakas 2008;Muller et al. 2008). The intergalactic opacity can also be measured by correlations between the positions of lowredshift galaxies and high-redshift quasi-stellar objects. Ménard et al. (2010a) correlated the brightness of ∼85 000 quasars at z > 1 with the position of 24 million galaxies at z ∼ 0.3 derived from the Sloan Digital Sky Survey (SDSS). The estimated value of A V is about 0.03 mag at z = 0.5 and about 0.05-0.09 mag at z = 1. A consistent opacity is reported by Xie et al. (2015) who investigated the redshifts and luminosity of the quasar continuum of ∼90 000 objects. The authors estimated the visual opacity to be ∼0.02 h Gpc −1 at z < 1.5. As mentioned by Ménard, Kilbinger & Scranton (2010b), such opacity is not negligible and can lead to bias in determining cosmological parameters if ignored.

Evolution of opacity with redshift
The galactic and intergalactic opacities depend on redshift. First, they increase with redshift due to the expansion of the Universe. This geometrical effect has already been taken into account in equation (1) by considering an increasing dust density with redshift because the Universe occupied a smaller volume in its early epoch. Second, a redshift-dependent formation and evolution of global dust mass in galaxies and in intergalactic space must be taken into account.
Observations indicate that interstellar dust mass M d is strongly linked to the star formation rate (SFR) of galaxies. da Cunha et al.
(2010) analysed 3258 low-redshift SDSS galaxies with z < 0.2 and reported the relation M d ∼ SFR 1.1 . Calura et al. (2017) extended the data set with high-redshift galaxies from Santini et al. (2010) and found a similar relation with a slightly lower slope of ∼0.9. The same slope is also reported by Hjorth, Gall & Michałowski (2014). Adopting the M d -SFR relation, we deduce from the SFR history (see Fig. 1) that the global dust mass steeply increases for z < 2-2.5, it culminates at z = 3-4 and then it starts to decline (Madau et al. 1996;Hopkins & Beacom 2006;Madau & Dickinson 2014;Popping, Somerville & Galametz 2016). The decline is not, however, substantially steep because dust is reported even in starforming galaxies at redshifts of z > 5 (Casey, Narayanan & Cooray 2014). Based on observations of the Atacama Large Millimeter Array, 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 ∼ 8 with a stellar mass of ∼2 × 10 9 M , a SFR of ∼20 M yr −1 and a dust mass of ∼6 × 10 6 M .

Extinction law
The light extinction due to absorption by dust is frequency dependent (see Fig. 2). In general, it decreases with increasing wavelength but displays irregularities. The extinction curve for dust in the Milky Way can be approximated for infrared wavelengths between ∼0.9 and ∼5 μm by a power law A λ ∼ λ −β with β ranging between 1.61 and 1.81 (Draine 2003(Draine , 2011. At wavelengths of 9.7 and 18 μm, the absorption displays two distinct maxima attributed to silicates (Mathis 1990;Li & Draine 2001;Draine 2003). At longer wavelengths, the extinction curve is smooth obeying a power law with β = 2. This decay is also predicted by the Mie theory modelling graphite or silicate dust grains as small spheres or spheroids with sizes up to 1 μm (Draine & Lee 1984). However, Wright (1982), Figure 2. Normalized frequency-dependent attenuation (Draine 2003, tables 4-6). The black and red dashed lines show the long-wavelength asymptotic behaviour predicted by the power law with β = 2 and β = 1.5. Henning et al. (1995) and Stognienko et al. (1995) and others point out that the long-wavelength absorption also depends on the shape of the dust grains and that needle-shaped conducting grains or complex fractal or fluffy dust aggregates can provide higher long-wavelength opacity with the power law described by 0.6 < β < 1.4 (Wright 1987).

P R E D I C T E D C M B D I S TO RT I O N
I consider the intergalactic opacity at visual wavelengths of 0.01 mag h Gpc −1 that is two times lower than that reported by Xie et al. (2015). The ratio of the CMB and visual attenuation ε CMB /ε V of 1 × 10 −5 is taken from Mathis (1990) and Draine (2003). Actually, this ratio is very low being obtained for a steep decrease of attenuation at long wavelengths (β = 2). Realistic values for dust particles with complex shapes might be higher by one order (Wright 1987, β = 1.5). I intentionally use the low value of ε CMB in order to be sure that the predicted level of the CMB distortion is the lower threshold of expected values.
The CMB distortion is calculated for two models. Model A is based on an assumption that the comoving dust density is independent of redshift. Model B adopts an interstellar and an intergalactic dust density evolving with redshift in accordance with the SFR (see Fig. 1). The spectral and total CMB distortions are calculated using equations (9) and (10) with parameters summarized in Table 1. In calculations, both of the galactic and intergalactic opacities (G+IG) or the galactic opacity only (G) is considered. Fig. 3 shows the spectral CMB intensity and its corresponding distortion produced by dust in the epoch of 0 < z < z max with z max of 6 and 10. As expected, the distortion increases with increasing z max , but the effect of dust absorption is visible even for z max of 6. The distortion is more pronounced for Model B than for Model A. This is caused by abundance of dust for z ∼ 2-4 considered in Model B but neglected in Model A. The maximum distortion is observed at a frequency of 300 GHz and reaches a value of 5.1 × 10 −22 Wm −2 Hz −1 sr −1 for Model A and 51.0 × 10 −22 Wm −2 Hz −1 sr −1 for Model B. These values exceed the detection level of the COBE/FIRAS (absolute sensitivity of ∼1-2 × 10 −22 Wm −2 Hz −1 sr −1 ; Fixsen et al. 1996) or WMAP and Planck flux measurements (absolute sensitivity of ∼7 × 10 −23 Wm −2 Hz −1 sr −1 ; Hinshaw et al. 2009;Planck Collaboration VIII 2014). The total CMB distortion is about 0.2 and 1.7 nWm −2 sr −1 for z max = 6 for Models A and B, respectively 10 0.02 160 0.22 2.0 1.4 × 10 −3 9.2 × 10 −3 1.0 × 10 −5 1.4 × 10 −8 9.2 × 10 −8 Note. Quantity a is the mean effective radius of galaxies, n is the comoving number density of galaxies, γ is the mean free path between galaxies, κ V is the mean visual opacity of galaxies, β is the slope in the extinction law, ε G V is the visual galactic attenuation coefficient defined in equation (4), ε IG V is the visual intergalactic attenuation coefficient and ε G CMB and ε IG CMB are the galactic and intergalactic attenuation coefficients at the CMB wavelengths, respectively. . The full black line shows the spectral CMB intensity. Full blue/red lines: z max = 10; dashed blue/red lines: z max = 6. Blue lines: distortions due to galactic and intergalactic dust (G+IG); red lines: distortions due to galactic dust (G). The grey area marks intensities that are under the sensitivity of the COBE/FIRAS measurements at 300 GHz (Fixsen et al. 1996)  ( Fig. 4). Model B predicts a faster increase of the total CMB distortion with z max than Model A. The maximum distortion increases up to z max ∼ 7. At higher z, the CMB is not distorted because the model is effectively free of dust. Note that the reported values are the lower thresholds; the realistic distortions should be higher.

D I S C U S S I O N
It is commonly considered that the CMB is distorted by foreground diffuse far-infrared and submillimetre emission of dust in the Milky Way, other galaxies and intergalactic space (Draine & Fraisse 2009; Imara & Loeb 2016b). However, the CMB can also be distorted due to absorption by dust producing a decline of the CMB intensity at all frequencies. This distortion should be high enough to be observable in the CMB measurements. The maximum spectral distortion of the CMB light coming from z = 10 is predicted at 300 GHz, which is at least 20 times higher than the detection level of the COBE/FIRAS measurements (Fixsen et al. 1996) and at least 35 times higher than the detection level of the WMAP or Planck measurements (Hinshaw et al. 2009;Planck Collaboration VIII 2014). The CMB should also be distorted in a perfectly transparent universe just due to absorption by dust in galaxies. This effect is about one order lower than that for the intergalactic opacity, but still above the detection level of the current CMB measurements. Finally, let us shortly discuss why the imprint of dust is missing on the CMB. First, we can speculate that the parameters used in modelling are seriously biased. However, it contradicts observations of the intergalactic opacity (Ménard et al. 2010a;Xie et al. 2015;Imara & Loeb 2016a), opacity of galaxies (González et al. 1998;Calzetti 2001;Holwerda et al. 2005a and the extinction law data in the Milky Way (Draine & Lee 1984;Mathis 1990;Li & Draine 2001;Draine 2003). Secondly, we can question the big bang as the origin of the CMB and revive theory of the CMB as the thermal radiation of dust itself being produced at much later times than big bang (Layzer & Hively 1973;Wright 1982Wright , 1987Wright , 1991Aguirre 2000). In such theory, the CMB should not be distorted because the CMB would concurrently be absorbed and reradiated by dust. In any case, it is clear that the missing dust imprint on the CMB is an intriguing puzzle that should be further studied and confronted with current measurements and models of the Universe.