# On the Galactic Halos Rotation by Planck Data

Noraiz Tahir 1,2,∗, Francesco De Paolis 1,2,3, Asghar Qadir 4,
Achille A. Nucita 1,2,3
1 Department of Mathematics and Physics “Ennio De Giorgi”, University of
Salento,
Via per Arnesano, I-73100 Lecce, Italy;
<EMAIL_ADDRESS>(F.D.P<EMAIL_ADDRESS>(A.A.N.)
2 INFN, Sezione di Lecce, Via per Arnesano, I-73100 Lecce, Italy
3 INAF, Sezione di Lecce, Via per Arnesano, I-73100 Lecce, Italy
4 Abdul Salam School of Mathematics, G. C. University, Lahore 54600, Pakistan;
<EMAIL_ADDRESS>(A.Q.)

###### Abstract

As galactic halos are not directly visible, there are many ambiguities
regarding their composition and rotational velocity. Though most of the dark
matter is non-baryonic, some fraction is, and it can be used to trace the halo
rotation. Asymmetries in the CMB towards M31 had been seen in the Planck data
and ascribed to the rotational Doppler shift of the M31 halo. Subsequently,
the same methods were used in the direction of five other galaxies belonging
to the Local Group, namely M33, M81, M82, NGC 5128, and NGC 4594. It had been
proved that there could be stable clouds of gas and dust in thermal
equilibrium with the CMB at 2.7 K, which had been called “virial clouds”. In
this paper, adopting this scenerio, an attempt is made to constrain the
fraction of dust grains and gas molecules in the clouds.

Keywords: cosmic microwave background; spiral galaxies; molecular clouds; dark
matter; baryons

##  1 Introduction

According to the standard cosmological model ($\Lambda$CDM) baryons contribute
$\sim$5% of the Universe, of which we have only seen about half (Bertone,
2010; Burdyuzha, 2020; Pérez de los Heros, 2020). Where the rest of them are
and how they can be detected are still open questions. This is the so-called
“missing baryon problem” (see e.g., Shull et al. (2012); Driver (2021)). It
has been suggested that the missing baryons might have been ejected from the
galaxies into the intergalactic medium (IGM), and a fraction of that component
may be present in the warm-hot medium around galaxies, at temperatures of
about $10^{5}$–$10^{7}$ K (for details see Refs. (Cen et al., 1999, 2006;
Fraser et al., 2011; Gupta et al., 2012)). On the other hand, it is also
suggested that a non-negligible fraction of these baryons are in the form of
“cold gas clouds” spread out in the galactic halos (De Paolis et al., 1995a;
Gerhard et al., 1996; Pifenniger et al., 1994; Palla et al., 1983).

In 1995, it was proposed that there may be molecular hydrogen (${\rm H_{2}}$)
clouds in galactic halos at exactly the CMB temperature, so that they do not
stand out against it (De Palois et al., 1995). It had been assumed, without
suggesting any model or mechanism for it, that the clouds remain stable and
avoid further gravitational collapse. It has been argued that cold clouds
could not be in equilibrium with the CMB, as they would not be able to absorb
the long wavelengths in it (Alcock et al., 1993; De Rújula et al., 1991), and
so would have to collapse. To see such clouds merged in the CMB, in galactic
halos, it was suggested that one look for $\gamma$-ray scintillation due to
cosmic rays striking the ${\rm H_{2}}$ molecules in the clouds. Observations
by EGRET did display an excess of diffuse $\gamma$-ray emission in the halo of
the Milky Way, but it could also have been due to unresolved sources in the
Galaxy or from high-latitude inverse Compton scattering (see (Dixon et al.,
1998; De Palois et al., 1998; Kalberla et al., 1999)). Another suggestion for
detecting the claimed gas clouds is to look for an asymmetric Doppler shift
due to the rotation of the galaxies. The suggestion had been to consider our
sister galaxy Andromeda (M31), and assume that the clouds rotate with the
whole M31 halo. It is clear that the rotation would induce a red-shift or a
blue-shift of the emitted electromagnetic signal depending on the rotation
direction with respect to the observer (De Paolis et al., 1995b).

In a series of papers, first with the analysis of WMAP and then of Planck
data, CMB data was used to trace the halos of some nearby galaxies (De Paolis
et al., 2011, 2014; De Palois et al., 2015; Gurzadyan et al., 2015; De Paolis
et al., 2016; Gurzadyan et al., 2018; De Paolis et al., 2019) to determine
whether if the data supports the Doppler shift explanation of the temperature
asymmetry. The data uncovered it, not only for the galactic disks but also for
some galactic halos. Further support for the Doppler shift came from the
observation that the temperature asymmetry was almost frequency-independent.

In summary, there is no doubt that this phenomenon is seen in several nearby
galaxies. It remains to investigate whether it is related to cold halo gas
clouds, with or without dust contamination, or is due to other effects. As a
working hypothesis here, we assume that it is caused, at least in part, by
virial clouds populating the galactic halos. For this purpose, we model these
clouds and estimate their physical parameters.

This paper aims to present a model to see the change in the physical
parameters of virial clouds when they are contaminated by different
percentages of dust grains. Then we will try to estimate the rotational
velocity of the halos of several nearby edge-on spiral galaxies, which show a
consistent temperature asymmetry in the CMB data as observed by the Planck
satellite and already analyzed. The gas and dust clouds had been modeled
previously in Refs. Tahir et al. (2019, 2019) by using ad-hoc boundary
conditions and then the halo rotational velocities of the above-mentioned
spiral galaxies estimated. We now use a more reliable model described in
detail in Ref. Qadir et al. (2019) to estimate the physical parameters of
virial clouds with contamination of halo dust grains in them.

The plan of the paper is as follows: in Section 2, the physical parameters of
virial clouds are estimated, as a function of the fraction of molecular
hydrogen (${\rm H_{2}}$), helium (He) and interstellar dust. In Section 3, the
adopted models describing the distribution of the gas clouds in the halo of
the considered galaxies (M31, M33, M81, M82, NGC 5128, and NGC 4594) are
briefly described. These models are those generally adopted for describing the
halo dark matter distribution and include the Navarro–Frenk–White (NFW),
Moore, and Burkert models. Using these models, we then derive the distribution
of the virial clouds and the cloud-filling factor. Then, in Section 4, we give
an expression to estimate the galactic halo rotational velocity. In Section 5,
the halo rotational velocity is estimated, assuming that each virial cloud is
optically thick and the whole halo is composed of such clouds. However, the
results obtained can be easily generalized to the case in which the virial
clouds constitute only a fraction of the galactic halo dark matter and, even
if less straightforwardly, to the not-optically thick clouds. Finally, in
Section 6, the obtained results are discussed.

##  2  Virial Cloud Model

Virial clouds populating the galactic halos are assumed to be spherically
symmetric and at a temperature equal to that of the CMB, i.e., $T\simeq 2.7$
K. Each cloud is assumed to be composed of molecular hydrogen, helium, and
interstellar dust grains with different fractions. It has been suggested that
galactic halos contain interstellar dust grain fraction of about 1–3$\%$ in
mass, and the rest is ${\rm H_{2}}$, He and other heavier molecules (Imara et
al., 2018; Draine, 2003). Halo dust grains are expected to be at a slightly
higher temperature with respect to that of the gas, about 5 K or so, depending
on the cloud position within the galactic halo (Yershov et al., 2020), but as
a first approximation, we assume here that dust and gas are at the same
temperature, i.e., $2.7$ K.

There is no doubt that interstellar dust grains are found in galactic halos.
It can be easily expected that dust grains form in the disk, and then the
radiation pressure from disk stars, exerting a force on the dust grains, can
push them towards the external regions of galaxies Ferrara et al. (1991). This
force can move the dust grains to high galactic latitudes via dust-gas
coupling. This process is similar to the driving agent of cool winds in giant
stars (Salpeter, 1974; Kwok, 1975). Once these grains are above the disk,
i.e., in the galactic halo, the the radiation pressure can expel the grains
out of the galaxy environment (Chiao et al., 1972; Greenberg et al., 1987;
Barsella et al., 1989). The important question that needs to be answered is
then to quantitatively estimate how much these dust grains contaminate the
galactic halos.

Now, an objection possibly arising is whether magnetic fields and the particle
destruction rate in the hot halos can play a major role in the dust grain
evolution so that, eventually, these grains might not move outside the hot
halo environment. In this respect, Ferrara et al. (1991) have shown that
different dust grain compounds respond differently to ambient conditions. For
example, graphites move faster than silicates, since graphite is much lighter
than silicates grains, but both types of grains can reach values in excess of
about 100 km s-1; thus, they can move further outside the halos depending upon
the halo gas temperature, galaxy mass, and radiation pressure coefficients. It
is also found in the literature that most of the halo dust grains are in the
form of amorphous carbon, and silicates, all the other kinds of molecules and
atoms that are thought to be present might be dominated by the effects of the
hot gaseous halos, i.e., the high radiation pressure and magnetic field.
Hence, the molecules and atoms other than graphite and silicates might not be
able to reach the outer parts of the halos. So, these outer regions of the
halos might contain traces of amorphous carbon and silicate dust grains, whose
masses are quite similar. Therefore, the average mass of a single halo dust
grain can be assumed to be $\simeq 5.65\times 10^{-23}$ g (Hirashita et al.,
2020).

The cloud mass within the galactocentric distance $r$ can be written as

$M_{cl}(r)=\alpha M_{H_{2}}(r)+\beta M_{He}(r)+\gamma M_{d}(r),$ (1)

where $\alpha$, $\beta$, and $\gamma$ are the fractions of $H_{2}$, He, and
dust, with the condition that $\alpha+\beta+\gamma=1$. Of course,
$M_{H_{2}}(r)$, $M_{He}(r)$, and $M_{d}(r)$ are the total mass of ${\rm
H_{2}}$, He and dust within the distance $r$. If the cloud is at the CMB
temperature; hence, the CMB can be considered a heat bath for the virial
clouds. One can use the canonical ensemble distribution to obtain the density
profile of these clouds, which can be written as

$\displaystyle\rho_{cl}^{3}(r)=\left(\frac{512\pi^{9/2}}{3^{9/2}}\right)\left(\frac{G}{kT}\right)^{9/2}(\rho_{c_{H_{2}}}\rho_{c_{He}}\rho_{c_{d}})^{3/2}\times(m_{H_{2}}m_{He}m_{d})^{5/2}\delta,$
(2)

where

$\displaystyle\delta=exp\left[-\left(A+B+C\right)\right].$ (3)

Here,

$A=\frac{\alpha GM_{H_{2}}(r)m_{H_{2}}}{rkT},$ (4) $B=\frac{\beta
GM_{He}(r)m_{He}}{rkT},$ (5)

and

$C=\frac{\gamma GM_{d}(r)m_{d}}{rkT}.$ (6)

Here, $G$ is Newton’s gravitational constant, $k$ is the Boltzmann constant,
$T$ is the cloud temperature, $\rho_{c_{H_{2}}}$, $\rho_{c_{He}}$, and
$\rho_{c_{d}}$ are the central density of ${\rm H_{2}}$, He and dust,
$m_{H_{2}}$, $m_{He}$, and $m_{d}$ are the masses of the single ${\rm H_{2}}$
molecule, He atom, and halo dust grain, respectively. Taking the natural
logarithm of Equation (2), and then substituting the obtained expression in
Equation (1), we obtain

$\displaystyle\alpha M_{H_{2}}(r)m_{H_{2}}$ $\displaystyle+\beta
M_{He}(r)m_{He}+\gamma M_{d}(r)m_{d}=-\frac{3rkT}{4\pi
G}\ln\left(\frac{\rho_{cl}(r)}{\lambda}\right),$ (7)

where
$\lambda=(8\pi^{3/2}/3^{3/2})[G/(kT)]^{3/2}(\rho_{c_{H_{2}}}\rho_{c_{He}}\rho_{c_{d}})^{1/2}[m_{H_{2}}m_{He}m_{d}]^{5/6}$.
The boundary conditions are at $r=0$, where $\rho_{cl}\rightarrow\rho_{c}$,
and $(d\rho(r)/dr)|_{r\rightarrow 0}=0$. Taking the derivative of Equation (2)
with respect to $r$ and using Equation (7), we obtain the following
differential equation

$\displaystyle r\frac{d\rho_{cl}(r)}{dr}-r^{2}\left(\frac{2\pi
G}{kT}\right)[\rho_{cl}(r)($
$\displaystyle\alpha\rho_{c_{H_{2}}}m_{H_{2}}+\beta\rho_{c_{He}}m_{He}+\gamma\rho_{c_{d}}m_{d})]$
$\displaystyle-\rho_{cl}(r)\ln\left(\frac{\rho_{cl}(r)}{\lambda}\right)=0,$
(8)

In Figure 1, we give the density profile of virial clouds with different
fractions of dust, i.e., $\gamma=0.01$ (red curve), $0.02$ (black curve), and
$0.03$ (blue curve). As one can see, by increasing the contamination of dust
grains, the cloud’s central density remains approximately the same, but the
size of the cloud decreases, and so as its mass. This occurs because heavier
molecules are pulled in more with gravity, so the cloud shrinks in size. In
Table 1, we give the physical parameters (cloud central density, Jeans mass,
and radius) of the three fluids virial cloud with different values of the
parameters $\alpha$, $\beta$, and $\gamma$.

Figure 1: The density profile of three fluids virial cloud with different fractions of ${\rm H_{2}}$, He, and dust. The red curve is for parameter values $\alpha=0.74$, $\beta=0.25$, and $\gamma=0.01$; the black curve corresponds to the profile for $\alpha=0.74$, $\beta=0.24$, and $\gamma=0.02$; the blue curve is obtained in the case of $\alpha=0.73$, $\beta=0.24$, and $\gamma=0.03$. $\alpha$ | $\beta$ | $\gamma$ | $\rho_{c}$ | $M_{J}$ | $R_{J}$
---|---|---|---|---|---
|  |  | $(10^{-21}\,{\rm g\,cm^{-3}})$ | $M_{\odot}$ | pc
0.74 | 0.01 | 0.25 | 1.44 | 0.020 | 0.025
0.74 | 0.02 | 0.24 | 1.79 | 0.019 | 0.018
0.73 | 0.03 | 0.24 | 2.04 | 0.017 | 0.011
Table 1: Physical parameters of three fluids virial cloud. The virial cloud
central density, Jeans mass, and radius are given for three different choices
of the parameters $\alpha$, $\beta$, and $\gamma$, which define the cloud
composition (see text for details).

##  3  Virial clouds distribution in the galactic halos

As next, we assume that a fraction $f$ of the galactic halo dark matter is in
the form of virial clouds. If these clouds are contributing to the halo dark
matter, we can say that their distribution follows the dark matter
distribution profile, which is generally modeled by using the
Navarro–Frenk–White (NFW), the Moore, and the Burkert density profiles Navarro
et al. (1996); Moore et al. (1999); Burkert (1995). These radial density
profiles are, respectively,

${\displaystyle\rho_{{}_{N}}(r)=\frac{\rho_{\circ}^{N}}{\left(\frac{r}{r_{\circ}^{N}}\right)\left(1+\frac{r}{r_{\circ}^{N}}\right)^{2}}},$
(9)

(NFW model)

${\displaystyle\rho_{{}_{M}}(r)=\frac{\rho_{\circ}^{M}}{\left(\frac{r}{r_{\circ}^{M}}\right)^{1.5}\left(1+\left(\frac{r}{r_{\circ}^{M}}\right)^{1.5}\right)}},$
(10)

(Moore model)

and

${\displaystyle\rho_{{}_{B}}(r)=\frac{\rho_{\circ}^{B}}{\left(1+\frac{r}{r_{\circ}^{B}}\right)\left(1+\left(\frac{r}{r_{\circ}^{B}}\right)^{2}\right)}}.$
(11)

(Burkert model).

Here, $r_{\circ}^{N,M,B}$ are the halo core radius of the adopted models, and
$\rho_{\circ}^{N,M,B}$ are the relative core densities. It can be easily seen
that the core density corresponds to the central density of the considered
galactic model. Of course, the values of both the core radius and core density
depend on the considered halo model and are given in columns 3–8 of Table 2.
These values are taken from Tamm et al. (2012); Tahir et al. (2019) and are
reproduced in Table 2 for the advantage of the reader.

Galaxy | $i$ | $r_{\circ}^{N}$ | $r_{\circ}^{M}$ | $r_{\circ}^{B}$ | $\rho_{\circ}^{N}$ | $\rho_{\circ}^{M}$ | $\rho_{\circ}^{B}$
---|---|---|---|---|---|---|---
| (deg) | kpc | kpc | kpc | $({\rm g\leavevmode\nobreak\ cm^{-3}})$ | $({\rm g\leavevmode\nobreak\ cm^{-3}})$ | $({\rm g\leavevmode\nobreak\ cm^{-3}})$
M31 | 77 | 16.5 | 31 | 9.1 | $3.06\times 10^{-24}$ | $5.05\times 10^{-24}$ | $1.25\times 10^{-23}$
M33 | 59 | 35 | 18 | 12 | $7.00\times 10^{-25}$ | $7.60\times 10^{-26}$ | $5.00\times 10^{-25}$
M81 | 14 | 10 | 18 | 8.9 | $5.10\times 10^{-24}$ | $7.65\times 10^{-25}$ | $3.68\times 10^{-24}$
M82 | 26 | 11 | 13 | 9 | $4.00\times 10^{-27}$ | $1.79\times 10^{-27}$ | $6.29\times 10^{-27}$
NGC 5128 | 14.6 | 21 | 12 | 9.8 | $1.67\times 10^{-25}$ | $4.40\times 10^{-25}$ | $1.00\times 10^{-24}$
NGC 4594 | 83 | 4.5 | 7.4 | 3.2 | $1.96\times 10^{-22}$ | $3.18\times 10^{-23}$ | $3.93\times 10^{-22}$
Table 2: Adopted physical parameters for the NFW, Moore, and Burkert models
for each considered galaxy. Note: For each considered galaxy, indicated in
column 1, the inclination angle $i$ is given in the second column, the core
radius for each adopted model is also given (columns 3–5), together with the
respective core density (columns 6–8).

The surface density of the virial clouds obtained in a galactic halo is given
by (Lokas et al., 2001)

$\Sigma(a)=2\int_{a}^{\infty}\frac{r\rho_{{}_{N,M,B}}(r)}{(r^{2}-a^{2})^{1/2}}dr.$
(12)

and, therefore, the total number of virial clouds per unit area can be
estimated as

${\displaystyle N(a)=f\left(\frac{\Sigma(a)}{M_{J}}\right)},$ (13)

where $f$ is the fraction of halo dark matter present in the form of virial
clouds, $M_{J}$ is the Jeans mass of a single virial cloud, and $a$ is a given
value of the halo projected radius.

Once we have estimated the total number of virial clouds in the three
considered galactic dark matter halo models using Equation (13), we are in the
position to estimate the cloud filling factor $S$, which is actually the ratio
of the area covered by all the clouds within the halo radius $R$ and the total
projected area and is given by

${\displaystyle S(\leq R)=\frac{\int_{0}^{R}2\pi
aN(a)r_{cl}^{2}da}{\int_{0}^{R}2\pi ada}},$ (14)

where $r_{cl}$ is the virial cloud radius, and $M_{cl}$ its mass.

##  4  Estimating the galaxy halo rotational velocity

Starting from the cumulative radial distribution of the virial clouds and
considering the temperature asymmetry in the CMB data towards a given galaxy,
we can then estimate the halo rotational velocity, which is given by

${\rm v_{rot}}=\left(\frac{\Delta
T}{T}\right)\left(\frac{c}{2\sin(i)Sf<\tau>}\right),$ (15)

where $c$ is the light speed in a vacuum, $i$ is the inclination angle of the
considered galaxy, $S$ is the virial cloud filling factor as calculated from
Equation (14), $<\tau>$ is the average optical depth of the virial clouds, and
$\Delta T/T$ is the temperature asymmetry detected in the CMB towards the two
opposite quadrants of the galactic halo (for details see (De Paolis et al.,
2014; De Palois et al., 2015; Gurzadyan et al., 2015; De Paolis et al., 2016;
Gurzadyan et al., 2018; De Paolis et al., 2019)). The values of $\Delta T/T$
are obtained from the Planck data analyzed for the SMICA band and previously
analyzed in Refs. De Paolis et al. (2016); Gurzadyan et al. (2018); Tahir et
al. (2019). For estimating ${\rm v_{rot}}$, we assume, as a first
approximation, that the virial clouds are characterized by $<\tau>\simeq 1$,
which means that the virial clouds are assumed to emit as black bodies at the
CMB temperature. However, one needs to estimate the average optical depth of
the virial clouds, $<\tau>$, over a detector frequency range,
$\nu_{1}-\nu_{2}$, which is, therefore, given as
$<\tau>=\displaystyle{\frac{1}{\nu_{1}-\nu_{2}}\int_{\nu_{1}}^{\nu_{2}}\tau_{\nu}\leavevmode\nobreak\
d\nu}$. The evaluation of this relation allows for determining whether the
considered virial clouds are optically thick or thin. An alternative way to
check the stability of the virial clouds is by estimating the “stability time”
of these clouds, which is related to the probability of interaction of the
molecules contaminating the virial clouds with the CMB photons. In Qadir et
al. (2019), it was estimated the stability time for the virial clouds composed
of purely ${\rm H_{2}}$ molecules. The estimated stability time results are
much less than the collapsing time of the cloud, which showed that these
clouds become stable before they could collapse. The considered situation was
very simple, but when dust and heavier molecules are assumed to contaminate
the clouds, the stability time and the collapsing time could vary because
there could be quantum effects coming in, i.e., additional translational,
rotational, and vibrational modes due to the interaction of the incoming
radiation with heavier and complex molecules.

##  5 Halo Model

The aim of this section is to implement the generalized model we give in the
previous sections, to the considered galaxy halos. As a first approximation,
we assume that all the dark matter in the considered galactic halos is in the
form of virial clouds. We also assume, for definiteness, that a dust grain
mass fraction contamination of the order of 1–3% is present in the considered
galaxy halos.

We start by modeling the M31 galactic halo, and then we extend our analysis
considering the other galaxies. Following Tamm et al. (2012), we assume that
the M31 halo size is $R\simeq 200$ kpc. However, the reader should keep in
mind that there is not a definite estimate for the size of the M31 halo. For
example, Lehner et al. (2020) used ultraviolet absorption measurements of Si
II, Si III, Si IV, C II, and C IV from the Hubble Space Telescope/Cosmic
Origins Spectrograph and O IV from the far Ultraviolet Spectrographic Explorer
to estimate how metals are distributed in the circumgalactic medium (CGM) of
M31 and found that the halo of M31 might extend up to $\approx$300 kpc.

In Figure 2, we give the cloud filling factor obtained through Equation (14).
It is seen that for all the considered virial cloud models, the filling factor
turns out to be less than 1. It is also seen that the obtained values of the
filling factor for the three dark matter models when we consider three fluids
virial clouds are more or less the same.

The virial cloud cumulative radial distribution towards the M31 halo, for a
dust concentration fraction in the range 1–3%, is shown in Figure 3, assuming
$f=1$. From these plots, it is clear that the number of virial clouds in the
M31 halo does not depend strongly on the considered halo model, but it depends
more heavily on the cloud chemical composition and dust abundance.

We are now in a position to estimate the rotational velocity of the M31 halo,
within various galactocentric distances using Equation (15). We use the SMICA
processed data to obtain the values of $\Delta T$ for M31 galaxy, which is
already analyzed in (Tahir et al., 2019). The SMICA data were chosen because
they are less contaminated than the other available Planck bands. We estimated
the rotational velocity values within 41.5 kpc to 103.8 kpc, and the obtained
results are shown in Table 3.

We remark here that HI observations towards the Andromeda galaxy obtained with
the Effelsberg and Green Bank 100 m telescopes showed that the rotation curve
in the M31 disk up to a galactocentric distance $\approx 35$ kpc, is
approximately flat with velocity $\approx 226$ km/s. Moreover, M31 rotation
curves up to a galactocentric distance $\approx 100$ kpc were built by
combining disk rotation velocities and radial velocities of satellite galaxies
and globular clusters. Ref. Carigana et al. (2006) adopted the NFW model to
estimate these curves obtaining an approximately flat behavior with
$v_{rot}\approx$ 110–200 km/s. However, as already noted, the M31 halo
rotation velocity is much less constrained by observations with respect to the
disk rotational velocity. In particular, from Table 3, we see that for $f=1$,
the rotational velocity of the M31 halo follows a trend similar to that
obtained for the disk, for the virial cloud model with 1% , 2%, and 3%
contamination of dust grains. The estimated M31 halo rotational velocity turns
out to be ${\rm v_{rot}}\approx$ 50–230 km/s, at large galactocentric
distances (that is, in the region between about 20 kpc and 108 kpc from the
M31 center).

$f_{dust}$ | R | $\Delta T/T$ |  | $N$ |  |  | $S$ |  |  | ${\rm v_{rot}}$ |
---|---|---|---|---|---|---|---|---|---|---|---
| (kpc) | $(10^{-6})$ |  | $10^{8}({\rm kpc^{-2}})$ |  |  | $(10^{-2})$ |  |  | (km/s) |
|  |  | $N^{N}$ | $N^{M}$ | $N^{B}$ | $S_{N}$ | $S_{M}$ | $S_{B}$ | ${\rm v_{rot}^{N}}$ | ${\rm v_{rot}^{M}}$ | ${\rm v_{rot}^{B}}$
1% | 21.4 |  $-$0.98 | 4.36 | 4.52 | 16.25 | 27.20 | 28.28 | 100 |  $-$0.55 |  $-$0.53 |  $-$0.99
| 31.1 | 2.36 | 2.60 | 2.73 | 5.67 | 16.38 | 17.10 | 35.4 | 2.23 | 2.12 | 3.27
| 41.8 | 6.40 | 1.69 | 1.77 | 7.90 | 10.57 | 11.11 | 49.41 | 9.41 | 8.96 | 12.28
| 51.8 | 14.3 | 1.18 | 1.24 | 6.16 | 7.42 | 7.75 | 38.50 | 29.86 | 28.57 | 28.27
| 77.8 | 7.90 | 0.67 | 0.61 | 3.93 | 3.76 | 3.86 | 24.59 | 32.55 | 31.71 | 34.09
| 103.8 | 7.80 | 0.36 | 0.36 | 2.87 | 2.22 | 2.29 | 17.95 | 53.34 | 52.87 | 37.28
2% | 21.4 |  $-$0.98 | 4.59 | 4.76 | 17.11 | 14.89 | 15.43 | 55.45 |  $-$1.01 |  $-$0.98 |  $-$1.82
| 31.1 | 2.36 | 2.74 | 2.88 | 5.97 | 8.99 | 9.33 | 19.34 | 4.08 | 3.89 | 6.00
| 41.8 | 6.40 | 1.78 | 1.87 | 8.32 | 5.77 | 6.05 | 26.96 | 17.25 | 16.43 | 22.59
| 51.8 | 14.3 | 1.24 | 1.30 | 6.48 | 4.04 | 4.23 | 21.01 | 54.72 | 52.35 | 51.82
| 77.8 | 7.90 | 0.63 | 0.65 | 4.14 | 2.20 | 2.11 | 13.42 | 59.65 | 58.12 | 62.48
| 103.8 | 7.80 | 0.38 | 0.31 | 3.02 | 1.24 | 1.25 | 9.79 | 97.76 | 96.90 | 68.32
3% | 21.4 |  $-$0.98 | 5.13 | 5.32 | 19.12 | 6.21 | 6.44 | 23.14 |  $-$2.43 |  $-$2.35 |  $-$4.36
| 31.1 | 2.36 | 3.06 | 3.21 | 6.67 | 3.71 | 3.89 | 8.08 | 9.79 | 9.33 | 14.39
| 41.8 | 6.40 | 1.99 | 2.90 | 9.30 | 2.40 | 2.52 | 11.25 | 41.34 | 39.37 | 53.92
| 51.8 | 14.3 | 1.39 | 1.45 | 7.24 | 1.69 | 1.76 | 8.77 | 131.10 | 125.42 | 119.69
| 77.8 | 7.90 | 0.70 | 0.72 | 4.62 | 0.88 | 0.82 | 5.60 | 142.93 | 139.24 | 124.15
| 103.8 | 7.80 | 0.42 | 0.43 | 3.37 | 0.65 | 0.52 | 4.08 | 234.22 | 232.66 | 163.69
Table 3: Estimated physical parameters of the M31 halo, populated by virial
clouds with 1%, 2%, and 3% dust contamination. Note: The temperature asymmetry
divided by the CMB temperature (column 3), the total number of virial clouds
per unit area within the observed radius (columns 4–6), the filling factor
(columns 7–9) and the estimated rotational velocity of the M31 halo (columns
10–12) are given, with respect to the considered halo radii (column 2), for
the three considered dark matter models.

a

b

c

Figure 2: The filling factor for the three fluids virial cloud model with 1%
(a), 2% (b), and 3% (c) dust contamination is given in the case of the M31
halo. The bold, dotted, and dashed curves correspond to the NFW, Moore, and
Burkert models, respectively.

a

b

c

Figure 3: The cumulative radial distribution of the number of clouds per unit
area for the three fluids virial cloud models with 1% (a), 2% (b), and 3% (c)
dust contamination is given in the case of the M31 halo. The bold curve is for
the NFW model, the dotted curve corresponds to the Moore model, while the
dashed one is for the Burkert model.

Similarly, in the case of other spiral galaxies, we can apply the same
procedure like M31 for estimating the halo rotational velocity of the other
considered galaxies, namely M33, M81, M82, NGC 5128, and NGC 4594,
respectively. We also assumed that 1–3% of dust contaminate virial clouds
present in the halos of these galaxies, and $f=1$ is used as a reference value
in Tables 3–8.

The total number of virial clouds per unit area, the filling factor of the
clouds in each considered galaxy halo, and the halo rotational velocity are
shown in Tables 3–8. We have used the temperature asymmetry data published in
the literature (see in particular the De Paolis et al. (2014); De Palois et
al. (2015); Gurzadyan et al. (2015); De Paolis et al. (2016); Gurzadyan et al.
(2018); De Paolis et al. (2019)) in the 70, 100, and 143 GHz bands in the case
of M33, M81, M82, and NGC 5128, and SMICA band for NGC 5494 at a resolution
corresponding to $N_{side}=2048$, in HEALPix scheme Gorski et al. (2005). The
temperature excesses are shown in Tables 4 and 5 (see column 3 for NGC 5494
and column 4 for M33, M81, M82, and NGC 5128, respectively). Tables 3–8 report
the obtained values of the halo rotational velocity ${\rm v_{rot}}$ for the
various adopted models and for $f_{dust}$ ranging from 1% to 3%. As one can
see, the obtained ${\rm v_{rot}}$ values are in substantial agreement with the
expectations. However, in the case of some galaxies, there are regions in
which apparently inconsistent values of ${\rm v_{rot}}$ are present.
Considering in detail each analyzed galaxy, in the case of the M31 halo (see
Table 3), the rotational velocity is acceptable for all three adopted models
and for all dust fractions. In the case of NGC 5494, M33, and M81 halo,
respectively (see Tables 4–6), the ${\rm v_{rot}}$ values appear acceptable
for $f_{dust}$ of 1% and 2% and only marginally acceptable for $f_{dust}=3\%$,
as a result, we believe that it is not likely to say that $f_{dust}<3\%$ for
these galaxies. In the case of the M82 halo (see Table 7) at a large radius
for all the dust concentrations, the ${\rm v_{rot}}$ values are not acceptable
hence we can say that at larger radius $f_{dust}<1\%$. In the case of NGC 5128
halo (see Table 8), the rotational velocity values appear acceptable for all
the cases.

Notice that the obtained ${\rm v_{rot}}$ values depend on the following
assumptions: (1) We assume a constant dust fraction in each galaxy,
independent of the galactocentric distance. However, as is explained
previously that the dust grain temperature can easily be higher. It should be
borne in mind that dust grains in the halo follow a distribution profile that
depends upon the galactocentric distance, and to treat this model more
physically, one needs to see how these dust grains are distributed in the
halos; (2) the temperature of the halo dust grains is assumed to be exactly
$2.7$ K (see Yershov et al. (2020)), and hence one needs to see how the
temperature profile of dust changes with the galactocentric distance; (3) the
distribution profile of the clouds in the halos of each considered galaxy: it
can be seen that the NFW model gives more consistent results than the other
two models, and (5) our assumption that the observed asymmetry in the CMB,
i.e., the values of $\Delta T/T$ are solely due to virial clouds in the halo
of each galaxy. However, there are other effects from a theoretical point of
view that give can give rise to this asymmetry in the CMB. To treat the
problem of the halo rotation more physically, one has to see how the halo dust
is distributed and the chemical composition of the virial clouds, for each of
the above galaxies separately, and consider other possible contributing
effects.

$f_{dust}$ | R | $\Delta T/T$ |  | $N$ |  |  | $S$ |  |  | ${\rm v_{rot}}$ |
---|---|---|---|---|---|---|---|---|---|---|---
| (kpc) | $(10^{-6})$ |  | $10^{8}({\rm kpc^{-2}})$ |  |  | $(10^{-2})$ |  |  | (km/s) |
|  |  | $N^{N}$ | $N^{M}$ | $N^{B}$ | $S_{N}$ | $S_{M}$ | $S_{B}$ | ${\rm v_{rot}^{N}}$ | ${\rm v_{rot}^{M}}$ | ${\rm v_{rot}^{B}}$
1% | 33.4 |  $-$8.44 | 20.26 | 15.73 | 12.43 | 12.66 | 19.86 | 17.74 |  $-$10.37 |  $-$12.92 |  $-$17.25
| 66.8 |  $-$6.60 | 5.58 | 4.13 | 6.21 | 3.49 | 2.58 | 4.43 |  $-$28.53 |  $-$38.65 |  $-$42.44
| 100.2 | 0.36 | 2.56 | 1.85 | 4.87 | 1.60 | 1.16 | 3.13 | 34.54 | 47.75 | 49.14
| 167 | 3.66 | 0.95 | 0.67 | 0.57 | 0.59 | 0.42 | 0.18 | 39.32 | 33.67 | 49.58
2% | 33.4 |  $-$8.44 | 21.32 | 16.62 | 13.82 | 6.91 | 5.93 | 4.83 |  $-$18.45 |  $-$23.68 |  $-$23.71
| 66.8 |  $-$6.60 | 5.88 | 4.34 | 6.97 | 1.90 | 1.40 | 1.22 |  $-$52.33 |  $-$70.84 |  $-$60.91
| 100.2 | 0.36 | 2.70 | 1.95 | 5.13 | 0.87 | 0.63 | 0.64 | 63.30 | 87.51 | 92.91
| 167 | 3.66 | 1.00 | 0.70 | 0.30 | 0.32 | 0.22 | 0.09 | 71.00 | 84.25 | 88.93
3% | 33.4 |  $-$8.44 | 23.83 | 18.57 | 14.62 | 2.88 | 2.22 | 1.72 |  $-$44.21 |  $-$56.74 |  $-$31.42
| 66.8 |  $-$6.60 | 6.57 | 4.86 | 7.79 | 0.79 | 0.58 | 0.92 |  $-$125.50 |  $-$169.72 |  $-$143.44
| 100.2 | 0.36 | 3.02 | 2.18 | 5.74 | 0.36 | 0.26 | 0.68 | 151.65 | 200.96 | 160.91
| 167 | 3.66 | 1.11 | 0.79 | 0.32 | 0.13 | 0.09 | 0.04 | 140.74 | 158.01 | 164.51
Table 4: Same as Table 3, but for NGC 5494 halo. Note: All columns have the same meaning as Table 3, but for NGC 5494 halo.  $\nu$ | $f_{dust}$ | R | $\Delta T/T$ |  | $N$ |  |  | $S$ |  |  | ${\rm v_{rot}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---
(GHz) |  | (kpc) | $(10^{-6})$ |  | $10^{8}({\rm kpc^{-2}})$ |  |  | $(10^{-2})$ |  |  | (km/s) |
|  |  |  | $N^{N}$ | $N^{M}$ | $N^{B}$ | $S_{N}$ | $S_{M}$ | $S_{B}$ | ${\rm v_{rot}^{N}}$ | ${\rm v_{rot}^{M}}$ | ${\rm v_{rot}^{B}}$
70 | 1% | 92 | 25.68 | 3.13 | 7.26 | 1.35 | 19.20 | 4.50 | 8.46 | 22.92 | 98.95 | 53.17
|  | 171 | 11.0 | 1.12 | 2.18 | 5.28 | 7.02 | 1.74 | 3.34 | 27.36 | 141.36 | 31.49
|  | 245 | 5.13 | 0.59 | 1.07 | 3.60 | 3.57 | 6.71 | 2.21 | 24.12 | 134.08 | 21.10
| 2% | 92 | 25.68 | 3.30 | 7.65 | 1.42 | 17.89 | 2.43 | 4.65 | 42.04 | 181.33 | 97.45
|  | 171 | 11.0 | 1.18 | 2.29 | 5.55 | 3.99 | 11.33 | 8.14 | 50.14 | 259.06 | 57.81
|  | 245 | 5.13 | 6.27 | 1.12 | 3.79 | 20.77 | 3.05 | 1.96 | 44.20 | 245.71 | 38.67
| 3% | 92 | 21.4 | 3.69 | 8.55 | 1.59 | 4.21 | 0.13 | 0.19 | 100.63 | 434.35 | 233.47
|  | 171 | 11.0 | 1.32 | 2.56 | 6.21 | 1.71 | 0.38 | 0.075 | 120.14 | 620.68 | 138.51
|  | 245 | 5.13 | 7.01 | 1.26 | 4.27 | 0.89 | 0.015 | 0.0052 | 105.91 | 588.67 | 92.64
100 | 1% | 92 | 22.01 | 3.13 | 7.26 | 1.35 | 19.20 | 4.50 | 8.46 | 19.64 | 84.81 | 45.58
|  | 171 | 15.41 | 1.12 | 2.18 | 5.28 | 7.02 | 1.74 | 3.34 | 38.30 | 197.91 | 44.16
|  | 245 | 6.60 | 0.59 | 1.07 | 3.60 | 3.57 | 6.71 | 2.21 | 31.01 | 172.38 | 27.13
| 2% | 92 | 22.01 | 3.30 | 7.65 | 1.42 | 17.89 | 2.43 | 4.65 | 36.01 | 135.42 | 83.52
|  | 171 | 15.41 | 1.18 | 2.29 | 5.55 | 3.99 | 11.33 | 8.14 | 70.20 | 362.69 | 80.94
|  | 245 | 6.60 | 6.27 | 1.12 | 3.79 | 20.77 | 3.05 | 1.96 | 56.83 | 315.91 | 49.71
| 3% | 92 | 22.01 | 3.69 | 8.55 | 1.59 | 4.21 | 0.13 | 0.19 | 86.25 | 372.37 | 200.17
|  | 171 | 15.41 | 1.32 | 2.56 | 6.21 | 1.71 | 0.38 | 0.075 | 168.19 | 868.95 | 193.92
|  | 245 | 6.60 | 7.01 | 1.26 | 4.27 | 0.89 | 0.015 | 0.0052 | 136.17 | 756.87 | 119.11
143 | 1% | 92 | 21.46 | 3.13 | 7.26 | 1.35 | 19.20 | 4.50 | 8.46 | 19.15 | 82.69 | 44.44
|  | 171 | 16.87 | 1.12 | 2.18 | 5.28 | 7.02 | 1.74 | 3.34 | 41.95 | 216.76 | 48.37
|  | 245 | 7.15 | 0.59 | 1.07 | 3.60 | 3.57 | 6.71 | 2.21 | 33.60 | 186.75 | 29.39
| 2% | 92 | 21.46 | 3.30 | 7.65 | 1.42 | 17.89 | 2.43 | 4.65 | 35.10 | 151.54 | 81.44
|  | 171 | 16.87 | 1.18 | 2.29 | 5.55 | 3.99 | 11.33 | 8.14 | 76.89 | 397.23 | 88.65
|  | 245 | 7.15 | 6.27 | 1.12 | 3.79 | 20.77 | 3.05 | 1.96 | 61.57 | 342.23 | 53.86
| 3% | 92 | 21.46 | 3.69 | 8.55 | 1.59 | 4.21 | 0.13 | 0.19 | 84.10 | 363.06 | 195.11
|  | 171 | 16.87 | 1.32 | 2.56 | 6.21 | 1.71 | 0.38 | 0.075 | 184.21 | 951.71 | 212.39
|  | 245 | 7.15 | 7.01 | 1.26 | 4.27 | 0.89 | 0.015 | 0.0052 | 147.52 | 819.94 | 129.04
Table 5: Estimated physical parameters of the clouds in the M33 halo and its rotational velocity. Note: We give, for the considered Planck frequency band (column 1), and for each considered halo radii (column 3), the temperature asymmetry divided by the CMB temperature (column 4), the total number of virial clouds per unit area within the observed radius (column 5–7), the filling factor (column 8–10) and the estimated rotational velocity of the M31 halo (column 11–13), for the three considered dark matter models.  $\nu$ | $f_{dust}$ | R | $\Delta T/T$ |  | $N$ |  |  | $S$ |  |  | ${\rm v_{rot}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---
(GHz) |  | (kpc) | $(10^{-6})$ |  | $10^{8}({\rm kpc^{-2}})$ |  |  | $(10^{-2})$ |  |  | (km/s) |
|  |  |  | $N^{N}$ | $N^{M}$ | $N^{B}$ | $S_{N}$ | $S_{M}$ | $S_{B}$ | ${\rm v_{rot}^{N}}$ | ${\rm v_{rot}^{M}}$ | ${\rm v_{rot}^{B}}$
70 | 1% | 15 | 20.36 | 14.80 | 15.56 | 18.85 | 0.92 | 0.97 | 0.98 | 13.12 | 12.90 | 48.44
|  | 30 | 23.85 | 5.28 | 5.49 | 13.39 | 0.33 | 0.34 | 0.87 | 44.51 | 43.08 | 106.45
|  | 60 | 29.35 | 1.63 | 1.62 | 4.60 | 0.10 | 0.10 | 0.28 | 17.84 | 17.91 | 25.8
| 2% | 15 | 20.36 | 15.60 | 16.37 | 13.37 | 0.50 | 0.53 | 0.98 | 24.44 | 23.79 | 88.76
|  | 30 | 23.85 | 5.56 | 5.78 | 14.47 | 0.18 | 0.18 | 0.47 | 82.09 | 78.94 | 195.08
|  | 60 | 29.35 | 1.71 | 1.71 | 4.84 | 0.055 | 0.055 | 0.15 | 32.69 | 32.83 | 47.30
| 3% | 15 | 20.36 | 17.40 | 18.30 | 13.39 | 0.21 | 0.22 | 0.41 | 54.60 | 57.01 | 212.67
|  | 30 | 23.85 | 6.21 | 6.46 | 16.64 | 0.072 | 0.078 | 0.019 | 194.64 | 189.14 | 467.38
|  | 60 | 29.35 | 1.92 | 1.91 | 5.41 | 0.023 | 0.023 | 0.063 | 78.33 | 78.67 | 113.97
100 | 1% | 15 | 14.67 | 14.80 | 15.56 | 18.85 | 0.92 | 0.97 | 0.98 | 19.81 | 19.31 | 34.91
|  | 30 | 20.18 | 5.28 | 5.49 | 13.39 | 0.33 | 0.34 | 0.87 | 37.86 | 36.45 | 90.07
|  | 60 | 27.52 | 1.63 | 1.62 | 4.60 | 0.10 | 0.10 | 0.28 | 36.26 | 36.79 | 24.23
| 2% | 15 | 14.67 | 15.60 | 16.37 | 13.37 | 0.50 | 0.53 | 0.98 | 17.97 | 17.15 | 63.97
|  | 30 | 20.18 | 5.56 | 5.78 | 14.47 | 0.18 | 0.18 | 0.47 | 69.39 | 66.80 | 165.06
|  | 60 | 27.52 | 1.71 | 1.71 | 4.84 | 0.055 | 0.055 | 0.15 | 30.52 | 30.78 | 44.11
| 3% | 15 | 14.67 | 17.40 | 18.30 | 13.39 | 0.21 | 0.22 | 0.41 | 43.07 | 41.08 | 153.28
|  | 30 | 20.18 | 6.21 | 6.46 | 16.64 | 0.072 | 0.078 | 0.019 | 166.25 | 160.04 | 395.47
|  | 60 | 27.52 | 1.92 | 1.91 | 5.41 | 0.023 | 0.023 | 0.063 | 73.43 | 73.75 | 106.40
143 | 1% | 15 | 16.51 | 14.80 | 15.56 | 18.85 | 0.92 | 0.97 | 0.98 | 11.03 | 10.52 | 39.27
|  | 30 | 18.34 | 5.28 | 5.49 | 13.39 | 0.33 | 0.34 | 0.87 | 34.42 | 33.13 | 81.88
|  | 60 | 27.52 | 1.63 | 1.62 | 4.60 | 0.10 | 0.10 | 0.28 | 16.72 | 16.79 | 24.23
| 2% | 15 | 16.51 | 15.60 | 16.37 | 13.37 | 0.50 | 0.53 | 0.98 | 20.22 | 19.29 | 71.97
|  | 30 | 18.34 | 5.56 | 5.78 | 14.47 | 0.18 | 0.18 | 0.47 | 63.08 | 60.72 | 150.06
|  | 60 | 27.52 | 1.71 | 1.71 | 4.84 | 0.055 | 0.055 | 0.15 | 30.65 | 30.78 | 44.41
| 3% | 15 | 16.51 | 17.40 | 18.30 | 13.39 | 0.21 | 0.22 | 0.41 | 48.45 | 46.22 | 172.44
|  | 30 | 18.34 | 6.21 | 6.46 | 16.64 | 0.072 | 0.078 | 0.019 | 151.13 | 145.49 | 359.54
|  | 60 | 27.52 | 1.92 | 1.91 | 5.41 | 0.023 | 0.023 | 0.063 | 73.43 | 73.75 | 106.40
Table 6: Same as Table 5 but for M81 halo. Note: All columns have the same meaning as Table 5 but for M81 galaxy halo.  $\nu$ | $f_{dust}$ | R | $\Delta T/T$ |  | $N$ |  |  | $S$ |  |  | ${\rm v_{rot}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---
(GHz) |  | (kpc) | $(10^{-6})$ |  | $10^{8}({\rm kpc^{-2}})$ |  |  | $(10^{-2})$ |  |  | (km/s) |
|  |  |  | $N^{N}$ | $N^{M}$ | $N^{B}$ | $S_{N}$ | $S_{M}$ | $S_{B}$ | ${\rm v_{rot}^{N}}$ | ${\rm v_{rot}^{M}}$ | ${\rm v_{rot}^{B}}$
70 | 1% | 15 | 18.16 | 14.57 | 16.61 | 16.74 | 9.10 | 10.03 | 10.46 | 68.25 | 59.85 | 118.15
|  | 30 | 15.59 | 5.31 | 5.35 | 13.02 | 3.32 | 3.34 | 8.13 | 160.69 | 159.58 | 213.09
|  | 60 |  $-$5.87 | 1.66 | 1.49 | 8.03 | 1.04 | 0.93 | 5.04 |  $-$192.72 |  $-$214.66 |  $-$163.03
| 2% | 15 | 18.16 | 15.33 | 17.49 | 17.63 | 4.96 | 5.66 | 5.71 | 125.07 | 109.68 | 216.53
|  | 30 | 15.59 | 5.59 | 5.63 | 13.70 | 1.82 | 1.82 | 4.44 | 294.49 | 294.47 | 390.51
|  | 60 |  $-$5.87 | 1.75 | 1.57 | 8.46 | 0.56 | 0.51 | 2.74 |  $-$353.18 |  $-$393.38 |  $-$298.78
| 3% | 15 | 18.16 | 17.14 | 19.54 | 19.70 | 2.07 | 2.36 | 2.38 | 299.66 | 262.78 | 518.76
|  | 30 | 15.59 | 6.25 | 6.29 | 15.32 | 0.76 | 0.76 | 1.85 | 705.54 | 700.65 | 935.60
|  | 60 |  $-$5.87 | 1.96 | 1.76 | 9.45 | 0.23 | 0.21 | 1.14 |  $-$846.17 |  $-$942.46 |  $-$715.82
100 | 1% | 15 | 16.88 | 14.57 | 16.61 | 16.74 | 9.10 | 10.03 | 10.46 | 55.62 | 109.80 | 109.80
|  | 30 | 17.61 | 5.31 | 5.35 | 13.02 | 3.32 | 3.34 | 8.13 | 181.49 | 180.23 | 240.67
|  | 60 |  $-$6.60 | 1.66 | 1.49 | 8.03 | 1.04 | 0.93 | 5.04 |  $-$216.87 |  $-$241.49 |  $-$183.34
| 2% | 15 | 16.88 | 15.33 | 17.49 | 17.63 | 4.96 | 5.66 | 5.71 | 116.23 | 101.92 | 201.22
|  | 30 | 17.61 | 5.59 | 5.63 | 13.70 | 1.82 | 1.82 | 4.44 | 332.60 | 330.29 | 441.05
|  | 60 |  $-$6.60 | 1.75 | 1.57 | 8.46 | 0.56 | 0.51 | 2.74 |  $-$397.35 |  $-$442.52 |  $-$336.12
| 3% | 15 | 16.88 | 17.14 | 19.54 | 19.70 | 2.07 | 2.36 | 2.38 | 278.47 | 244.20 | 482.20
|  | 30 | 17.61 | 6.25 | 6.29 | 15.32 | 0.76 | 0.76 | 1.85 | 796.85 | 791.32 | 566.68
|  | 60 |  $-$6.60 | 1.96 | 1.76 | 9.45 | 0.23 | 0.21 | 1.14 |  $-$951.94 |  $-$900.60 |  $-$805.30
143 | 1% | 15 | 22.01 | 14.57 | 16.61 | 16.74 | 9.10 | 10.03 | 10.46 | 82.73 | 72.54 | 143.22
|  | 30 | 22.01 | 5.31 | 5.35 | 13.02 | 3.32 | 3.34 | 8.13 | 226.86 | 225.29 | 300.84
|  | 60 |  $-$4.40 | 1.66 | 1.49 | 8.03 | 1.04 | 0.93 | 5.04 |  $-$144.54 |  $-$160.99 |  $-$122.28
| 2% | 15 | 22.01 | 15.33 | 17.49 | 17.63 | 4.96 | 5.66 | 5.71 | 151.60 | 132.95 | 262.46
|  | 30 | 22.01 | 5.59 | 5.63 | 13.70 | 1.82 | 1.82 | 4.44 | 415.75 | 412.86 | 551.31
|  | 60 |  $-$4.40 | 1.75 | 1.57 | 8.46 | 0.56 | 0.51 | 2.74 |  $-$264.89 |  $-$295.03 |  $-$224.08
| 3% | 15 | 22.01 | 17.14 | 19.54 | 19.70 | 2.07 | 2.36 | 2.38 | 363.22 | 318.52 | 268.81
|  | 30 | 22.01 | 6.25 | 6.29 | 15.32 | 0.76 | 0.76 | 1.85 | 996.06 | 989.15 | 932.08
|  | 60 |  $-$4.40 | 1.96 | 1.76 | 9.45 | 0.23 | 0.21 | 1.14 |  $-$634.63 |  $-$706.85 |  $-$736.68
Table 7: Same as Table 5 but for the case of M82 halo. Note: All columns have the same meaning as Table 5 but for M82 galaxy halo.  $\nu$ | $f_{dust}$ | R | $\Delta T/T$ |  | $N$ |  |  | $S$ |  |  | ${\rm v_{rot}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---
(GHz) |  | (kpc) | $(10^{-6})$ |  | $10^{8}({\rm kpc^{-2}})$ |  |  | $(10^{-2})$ |  |  | (km/s) |
|  |  |  | $N^{N}$ | $N^{M}$ | $N^{B}$ | $S_{N}$ | $S_{M}$ | $S_{B}$ | ${\rm v_{rot}^{N}}$ | ${\rm v_{rot}^{M}}$ | ${\rm v_{rot}^{B}}$
70 | 1% | 92 | 25.68 | 18.68 | 12.75 | 13.48 | 11.16 | 7.97 | 8.42 | 13.09 | 19.16 | 11.22
|  | 171 | 11.74 | 6.22 | 3.76 | 6.02 | 3.89 | 2.35 | 3.76 | 17.94 | 29.67 | 38.63
|  | 245 | 4.40 | 3.19 | 1.84 | 4.74 | 1.99 | 1.15 | 2.96 | 13.14 | 22.72 | 46.30
| 2% | 92 | 25.68 | 19.66 | 13.43 | 14.19 | 6.37 | 4.35 | 4.59 | 23.99 | 35.13 | 18.57
|  | 171 | 11.74 | 6.55 | 3.96 | 6.33 | 2.12 | 1.28 | 2.05 | 32.89 | 54.38 | 57.84
|  | 245 | 4.40 | 3.36 | 1.94 | 4.99 | 1.09 | 0.62 | 1.62 | 24.01 | 41.63 | 84.85
| 3% | 92 | 25.68 | 21.97 | 15.01 | 15.86 | 2.67 | 1.83 | 1.91 | 57.48 | 84.16 | 44.49
|  | 171 | 11.74 | 7.32 | 4.43 | 7.08 | 0.88 | 0.53 | 0.85 | 78.80 | 130.30 | 78.81
|  | 245 | 4.40 | 3.76 | 2.17 | 5.58 | 0.45 | 0.26 | 0.67 | 57.53 | 99.75 | 120.33
100 | 1% | 92 | 22.01 | 18.68 | 12.75 | 13.48 | 11.16 | 7.97 | 8.42 | 11.22 | 16.43 | 18.68
|  | 171 | 15.41 | 6.22 | 3.76 | 6.02 | 3.89 | 2.35 | 3.76 | 23.55 | 38.95 | 30.46
|  | 245 | 6.60 | 3.19 | 1.84 | 4.74 | 1.99 | 1.15 | 2.96 | 19.65 | 34.08 | 69.45
| 2% | 92 | 22.01 | 19.66 | 13.43 | 14.19 | 6.37 | 4.35 | 4.59 | 20.56 | 30.11 | 15.94
|  | 171 | 15.41 | 6.55 | 3.96 | 6.33 | 2.12 | 1.28 | 2.05 | 71.38 | 120.71 | 47.28
|  | 245 | 6.60 | 3.36 | 1.94 | 4.99 | 1.09 | 0.62 | 1.62 | 62.45 | 127.85 | 139.02
| 3% | 92 | 22.01 | 21.97 | 15.01 | 15.86 | 2.67 | 1.83 | 1.91 | 72.14 | 68.13 | 77.63
|  | 171 | 15.41 | 7.32 | 4.43 | 7.08 | 0.88 | 0.53 | 0.85 | 171.02 | 149.63 | 113.27
|  | 245 | 6.60 | 3.76 | 2.17 | 5.58 | 0.45 | 0.26 | 0.67 | 130.49 | 93.49 | 97.36
143 | 1% | 92 | 21.28 | 18.68 | 12.75 | 13.48 | 11.16 | 7.97 | 8.42 | 10.84 | 15.83 | 18.39
|  | 171 | 16.88 | 6.22 | 3.76 | 6.02 | 3.89 | 2.35 | 3.76 | 35.80 | 42.66 | 23.83
|  | 245 | 7.15 | 3.19 | 1.84 | 4.74 | 1.99 | 1.15 | 2.96 | 31.29 | 36.92 | 75.24
| 2% | 92 | 21.28 | 19.66 | 13.43 | 14.19 | 6.37 | 4.35 | 4.59 | 19.81 | 29.17 | 15.83
|  | 171 | 16.88 | 6.55 | 3.96 | 6.33 | 2.12 | 1.28 | 2.05 | 47.28 | 78.18 | 122.38
|  | 245 | 7.15 | 3.36 | 1.94 | 4.99 | 1.09 | 0.62 | 1.62 | 139.02 | 67.66 | 137.89
| 3% | 92 | 21.28 | 21.97 | 15.01 | 15.86 | 2.67 | 1.83 | 1.91 | 47.63 | 69.73 | 36.86
|  | 171 | 16.88 | 7.32 | 4.43 | 7.08 | 0.88 | 0.53 | 0.85 | 113.27 | 187.31 | 154.35
|  | 245 | 7.15 | 3.76 | 2.17 | 5.58 | 0.45 | 0.26 | 0.67 | 93.49 | 162.10 | 130.64
Table 8: Same as Table 5 but for NGC 5128 halo. Note: All columns have the
same meaning as Table 5 but for NGC 5128 galaxy halo.

##  6 Results and Discussions

The observed Doppler shift De Paolis et al. (2011, 2014); De Palois et al.
(2015); Gurzadyan et al. (2015); De Paolis et al. (2016); Gurzadyan et al.
(2018); De Paolis et al. (2019) had been predicted earlier to search for cold
gas clouds in Ref. De Paolis et al. (1995b), which could contain some of the
missing baryons. It has since been used to try to investigate the halo
rotation (see Refs. Tahir et al. (2019, 2019)). Here, the virial model in Ref.
Qadir et al. (2019) is used to obtain more reliable and precise answers
regarding the halo rotation and the composition and distribution of the clouds
in the halos. For this purpose, we used the following assumptions as a first
approximation: (1) all the considered halos contain 1–3% $f_{dust}$ which
remains constant throughout the halo; (2) the entire temperature asymmetry in
the CMB $\Delta T/T$ we observed towards the galaxies is due to virial clouds
in the halos; (3) the temperature of halo dust grains is also exactly $2.7$ K;
and (4) the virial clouds follow the same dark matter profile NFW, Moore or
Burkert models are widely used to map the distribution of dark matter in the
galactic halos.

The radial distribution of virial clouds within the M31 halo, taking $f=1$ for
definiteness, and the cumulative radial profiles were obtained by estimating
the virial cloud surface density $\Sigma(a)$, by using Equation (12), and then
dividing it by the mass of a single cloud (see Figure 3). It was seen that the
total number of clouds per unit area for each model at large galactocentric
distances was approximately the same. We then estimated the cloud-filling
factor by using Equation (14). Our end goal was to estimate the rotational
velocities and the viable chemical composition of the virial clouds of the
galaxy halos under consideration for each of the three dark matter models,
which are given in Tables 3–8. In most cases, the obtained values of the
rotational velocity appear consistent with the expectations, but there
appeared some inconsistent values in the estimated rotational velocity of the
considered halos, which can merely be due to the values chosen for the model
parameters, in particular, $f_{dust}$ and dust temperature at 2.7 K. For
example, at first, we need to see how dust grains are distributed in the halos
of these galaxies and how the halo dust temperature profile can be seen in the
halos. The other point concerns the assumption that the entire temperature
asymmetry in the CMB is due to virial clouds. However, four other
possibilities may be considered to explain the detected temperature towards
the considered galactic halo: (i) the Sunyaev–Zeldovich (SZ) effect, which is
the distortion of the CMB through inverse Compton scattering by high energy
electrons in galaxy clusters, in which the low energy CMB photons receive an
average energy boost during a collision with the high energy cluster
electrons, in particular, the kinetic energy (Matilla et al., 2020); (ii)
synchrotron emission by fast-moving electrons which is a type of non-thermal
radiation generated by charged particles spiraling around magnetic field
lines. Since these electrons are subject to acceleration due to the change of
their direction, they emit photons (Dolag et al., 2000); (iii) free-free
emission (or bremsstrahlung emission) by free electrons passing close to
atomic nuclei (Sun et al., 2010); and (iv) the anomalous microwave emission
(AME) from dust grains, which is the electric dipole radiation from small
spinning dust grains. The anomalous component at 53 GHz is 2.5 times as bright
as the free-free emission traced by $H\alpha$, providing an approximate
normalization for models with significant spinning dust emission (Leitch,
2013). The spinning dust model is thought to be the simplest explanation for
the observed dust correlations, in the microwave data, but the combination of
increasingly sophisticated models and observations with better frequency
resolution will ultimately decide whether spinning dust can explain the
detected anomalous emission. So, to analyze the rotational dynamics of
galactic halos in more detail by using Planck data, it would be necessary to
constrain the contribution of each of the effects (i)–(iv) above in the
temperature asymmetry towards considered halos. In particular, the
contribution of the rotational kinetic Sunyaev–Zeldovich (rkSZ) has been
considered in the case of the M31 halo, and it was found that at the present
level of accuracy, it gives a negligible contribution to the observed
asymmetry towards M31 as seen by the Planck data Tahir et al. (2022).
Consideration of the contribution of the other effects and the halo dust
profile will be attempted in a separate paper.

We stress that we do not claim that the virial clouds should contain all the
baryonic dark matter present in the galactic halos and should solely explain
the observed rotational asymmetry seen towards the considered galaxies in the
CMB by Planck data. Many models that explain the existence of this kind of
small dense clouds have been proposed in the literature.

To take the matter of the true chemical composition and nature of virial
clouds more physically, one would need to trace the cloud evolution from the
time of their formation to the present, allowing for all the major qualitative
changes that occurred during that time. Their formation would have been after
the recombination time (that is at $z<1000$), and they must have been in
equilibrium with the CMB since then. The first step of the evolution of these
clouds from the surface of the last scattering (LSS) up to the formation of
population-III stars has been studied in detail (Tahir et al., 2021). In Ref.
Bromm et al. (1999) hydrodynamic simulations were used to explain the
formation of first-generation stars and investigate the fragmentation process
of a metal-free gas. It was seen that this gas collapsed at $z=30$ in clouds
with a mass of about $10^{6}\leavevmode\nobreak\ M_{\odot}$. Fragmentation
then led to the formation of dense clumps of mass about
$10^{3}\leavevmode\nobreak\ M_{\odot}$. It was seen in Ref. (Tahir et al.,
2021) that at $z=50$ virial clouds which are made of primordial gas have a
mass of about $10^{5}\leavevmode\nobreak\ M_{\odot}$ which does not violate
the results of Ref. Bromm et al. (1999). One could take the next step forward
and move on towards evolution II of these clouds to understand their true
nature, but the next step of evolution is more problematic and ambiguities can
arise. At this stage, we will have to incorporate the quantum calculations
that were found to be negligible in the Evolution I paper. All these things
should be taken into account to understand if the virial cloud model used in
this paper is not in contradiction with the models already present in
literature for such kinds of clouds in the halos. This will be studied later
in more detail.

## Acknowledgments

We acknowledge the use of Planck’s data in the Legacy Archive for Microwave
Background Data Analysis (LAMBDA) and HEALPix Gorski et al. (2005) packages.
The TAsP and Euclid INFN projects are also acknowledged as well as the
anonymous referees for the useful comments.

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