# Thermal SZ effect in a magnetized IGM dominated by interacting DM
decay/annihilation during dark ages

Arun Kumar Pandey *  |1 , Sunil Malik *  | 2,3
1Department of Physics and Astrophysics, University of Delhi, New Delhi-110
007, India,
2 Institute fur Physik und Astronomie Universitat Potsdam, Haus 28, Karl-
Liebknecht-Str 24/25, D-14476 Potsdam, Germany,
3Deutsches Elektronen-Synchrotron DESY, Platanenallee 6, 15738 Zeuthen,
Germany E-mail<EMAIL_ADDRESS>(arun_pandey@prl.iitgn.ac.in)E-mail:
<EMAIL_ADDRESS>
###### Abstract

During cosmic dawn, the thermal history of the universe is well studied, and a
study of this era can give us some of the most useful insight into the
universe before the recombination epoch. Its precise modeling and future high-
precision measurements will be a valuable tool for determining the thermal
history of the universe. In the present work, we study the thermal and
ionization history of IGM in the presence of decaying magnetic fields via
ambipolar and turbulent decay, Baryon-Dark matter (BDM) interaction, including
the DM decay/annihilation. The BDM interaction cross-sections considered are
of the form $\sigma=\sigma_{0}v^{n}$, where $n=-2$ and $n=-4$. In this work,
we show that in the current scenario, the decay/annihilation of the DM
particles have a considerable impact on the temperature and ionization
histories at low redshift. With the addition of the concept of fractional
interaction, which states that if a fraction of the DM particles interacts
with the baryons, the temperature and ionization fraction of the baryons show
a strong dependence on the percentage of DM particles interacting with the
baryons. We have also studied the interesting consequences of the present
scenario on the thermal Sunyaev-Zeldovich (tSZ) effect. We show that the
highest value of the absolute value of the mean $y-$parameter in the current
DM decay/annihilation scenario is well within the values derived from
experimental data such as PLANCK, FIRAS, and PIXIE. Later we calculate the
bound on the ordinary magnetic fields originating from the Dark photons.

###### keywords:

SZ effect, CMB, dark matter, IGM, magnetic fields, ambipolar diffusion,
turbulent decay, dark matter decay/annihilation, baryon-DM interaction, Dark
photons

##  1 Introduction

In the standard $\Lambda$CDM model of cosmology, thermal and the ionization
history of the universe, between the recombination epoch and the first-star
formation, is well studied Peebles (1968); Zeldovich et al. (1969); Ali-
Haimoud & Hirata (2011); Chluba & Thomas (2011). After the recombination
epoch, due to Compton scattering between the residual free electrons and the
CMB photons, temperature of the matter $T_{m}$ and CMB temperature $T_{\rm
cmb}$ remains same, i.e. $T_{m}\approx T_{\rm cmb}$ till redshift $z\sim 100$.
Subsequently, the interaction rate becomes so small that matter and CMB
photons decouple, and the temperature of the matter evolves with redshift as
$T_{m}(z)\propto(1+z)^{2}$. This behavior of matter temperature and hence the
ionization fraction can change significantly and can evolve differently if
there is some other factor involved. Some important factors are magnetic
fields, dark matter (DM)-baryon interactions, and DM decay or annihilation in
the IGM. Any deviation from the standard history is thus an indication of a
new mechanism of heating and cooling. The ‘Experiment to Detect the Global
Epoch of reionization Signature’ (EDGES) collaboration has discovered an
absorption trough signal at $78$ MHz in the $14<z<20$ shift region in a 21-cm
signal Bowman et al. (2018). The measured trough in the 21-cm absorption at
$z\sim 17.2$ is $T_{21}=-500^{+200}_{-500}$ mK with 90% confidence level. This
observation of signals by the collaborations opens a new window into the dark
ages, shedding new light on the thermal and ionization history. Since the
publication of the EDGES results, there were several models developed,
considering colder than expected gas temperature or additional sources of
photons. There were few models given where baryons and dark matter
(represented by $\chi$ in the upcoming sections) interactions are considered
to lower the gas temperature by cooling it by absorption of heat from the gas
to DM (assuming DM to be colder than gas) Muñoz et al. (2015); Muñoz & Loeb
(2017); Barkana et al. (2018); Pandey et al. (2020). The most commonly studied
portal through which the interaction happens is via some mediator, for
example, a vector mediator. In this case, interaction takes place because of
the kinetic mixing between one dark and one ordinary Abelian gauge boson. The
ordinary gauge bosons are considered to be from the $U(1)$ gauge group of the
standard model or hypercharge fields above the electroweak symmetry breaking
scale. However, the photon of the dark sector comes to be identified as the
boson of $U(1)^{\prime}$ group of the dark sector. The coupling between the
standard model particles (SMP) and Dark matter particles (DMP) may occur
through following channels:
$\text{SMP}~{}\xrightarrow{e}~{}A_{\mu}\xrightarrow{\epsilon
e^{\prime}}~{}\text{DMP}\xrightarrow{e^{\prime}}A_{\mu}^{\prime}$ or
$A_{\mu}\xrightarrow{e}~{}\text{SMP}\xrightarrow{\epsilon
e}~{}A_{\mu}^{\prime}~{}\xrightarrow{e^{\prime}}~{}\text{DMP}$ (where $e$ and
$e^{\prime}$ are the coupling constant s of the ordinary and dark photons
respectively sectors and $\epsilon=\text{sin}\theta$, $\theta$ being the
rotation angle). The low energy Lagrangian of this interaction could be
written as Chen et al. (2021)

$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle\mathcal{L}_{\rm
SM}+\mathcal{L}_{\rm
DM}-\frac{1}{4}X_{\mu\nu}\,X^{\mu\nu}+\frac{1}{2}m^{2}_{\chi}X_{\mu}X^{\mu}$
(1) $\displaystyle-$
$\displaystyle\frac{\epsilon}{2}F_{\mu\nu}X^{\mu\nu}+\epsilon
eJ_{\mu}X^{\mu}+\epsilon e^{\prime}J^{\prime}_{\mu}A^{\mu}$

where $\mathcal{L}_{\rm SM}$ and $\mathcal{L}_{\rm DM}$ are the Lagrangian
belonging to the SM particles and the DM particles respectively. The third and
fourth terms are the kinetic and mass terms for the dark photons,
respectively. The fifth term is nothing but a kinetic mixing of dark photon
field $A_{\mu}^{\prime}$ and the ordinary photon field $A_{\mu}$ Masaki & Soda
(2018); Berezhiani et al. (2013); Kandus et al. (2011). The last two terms are
the coupling between SM and dark photons and vice versa. The product of two
gauge groups $U(1)\times U(1)^{\prime}$ is eventually broken down to some
diagonal $U(1)_{\rm em}$ group which represents the ordinary massless photon
eigenstate $\gamma$ while another mass eigenstate $\gamma^{\prime}$ becomes
heavy. This ordinary photon may interact primarily with SM particles with the
coupling constant $e$, but it will perceive DM particles as having electric
charges $e^{\prime}=\epsilon e$ Holdom (1986).

To cool the gas, a Rutherford-like interaction cross-section between the DM
and the gas particles is considered Prinz et al. (1998); Spergel & Steinhardt
(2000); Davidson et al. (2000); Dubovsky et al. (2004); Tulin et al. (2013a);
Barkana (2018). A generalized form of the interaction cross-section is given
by $\sigma(v)=\sigma_{0}(v/c)^{n}$ (where $v$ is the relative velocity of the
DM and gas particles), $n=-1$ for the Yukawa potential (massive Boson
exchange) Aviles & Cervantes-Cota (2011), $n=-2$ is for an electric dipole
moment Sigurdson et al. (2004) and $n=-4$ corresponds to millicharge DM
particles McDermott et al. (2011); Dolgov et al. (2013). Above velocity-
dependent DM and gas particles interaction seems a best way to evade present
day, astrophysical and cosmological constraints Jaeckel & Ringwald (2010),
including the constraint on relativistic degree of freedom from the big bang
nucleosynthesis (BBN) and CMB Davidson et al. (2000); Vogel & Redondo (2014),
anisotropy in CMB Dubovsky et al. (2004); Dolgov et al. (2013); Dvorkin et al.
(2014), fifth force experiments Adelberger et al. (2003); Williams et al.
(2004); Kapner et al. (2007); Schlamminger et al. (2008); Xu et al. (2018),
the stellar cooling Hardy & Lasenby (2017) and bounds on the self-interactions
of DM Tulin et al. (2013b, a). Based on a possible spectral distortion of CMB
photons, due to the interaction of the dark matter with the baryons, bounds on
the spectral distortion are given for the interaction cross-section McDonald
et al. (2000). Self-interactions and the interaction with the standard model
(SM) particles are the deciding factors for the velocity distribution of the
DM. In the $\Lambda$CDM universe, the cold DM and baryon fluids have a
relative velocity, resulting in supersonic coherent baryon flows after
recombination Tseliakhovich & Hirata (2010). If the momentum transfer rate
between DM and the baryons while interacting is low, the drag between the two
fluids may not efficiently dissipate their relative bulk velocity. The
relative motion dominates over the thermal motions when the universe is
sufficiently cooled down. Furthermore, if the relative bulk velocity is
considerably large before recombination, the Boltzmann equations used to
understand the fluid velocity fluctuations become nonlinear due to mixing the
individual Fourier modes. To avoid the mixing problem for our convenience, we
have used the root mean square velocity of the relative bulk velocity as a
correction to the thermal velocity dispersion Dvorkin et al. (2014); Xu et al.
(2018); Slatyer & Wu (2018). The frequent self-interaction causes DM
velocities to reshuffle towards the thermal Maxwell-Boltzmann (MB)
distribution, with the highest entropy. The DM velocity distribution does not
have to be MB when the DM thermally decouples from SM particles if the
timescale for velocity redistribution through self-interactions is longer than
the expansion period Ali-Haïmoud (2021). For a limiting case when the self-
interactions are negligible, the heat exchange between the DM particles with
baryons can differ by up to a factor of $\sim 2-3$ only Ali-Haïmoud (2019).

It is known that magnetic fields of different amplitudes are present at all
length scales Bhatt & Pandey (2016); Subramanian (2016); Anand et al. (2017);
Pandey & Anand (2021). The origin of the seed magnetic field remains an open-
ended subject in current cosmology, despite having so many fundamental
impacts. The pervasive magnetic fields are anticipated to have a significant
impact on a variety of processes, including baryogenesis Giovannini &
Shaposhnikov (1998); Simone et al. (2011); Kushwaha & Shankaranarayanan
(2021), primordial nucleosynthesis Grasso & Rubinstein (1996), structure
formation Sethi & Subramanian (2005); Kahniashvili et al. (2013); Katz et al.
(2021), and cosmic microwave background physics Barrow et al. (1997); Yamazaki
et al. (2010); Kunze & Komatsu (2015) (for more details, see the review
article Grasso & Rubinstein (2001)). In reference Berezhiani et al. (2013),
authors have discussed the origin of these fields during galaxy formation,
when a finite interaction between the baryons and DM particles is considered.
The mechanism discussed here has some implications for our present work in the
context of DM particles. The amplitude and the properties of these magnetic
fields depend on the origin mechanism, and it is characterized by present-day
field strength $B_{\lambda}$ at a given length scale $\lambda$. If these
magnetic fields are present at the time of decoupling, due to the Lorentz
force in the ionized plasma, significant density fluctuations can be generated
by Magnetohydrodynamic motions and heat the gas. Heating of the gas occurs
from the ambipolar diffusion and turbulent decay of the magnetic fields Sethi
& Subramanian (2005). The heating of gas affects the structure formation and
hence can affect the reionization and other further processes after the
decoupling epoch. Cowling (1956); Wasserman (1978); Kim et al. (1996); Sethi &
Subramanian (2005). The effect of these magnetic fields on the neutral
hydrogen in the IGM by hydrodynamic effects is used to constrain the strength
of the magnetic fields Tashiro & Sugiyama (2006).

In addition to energy deposition from the interaction of the DM and gas
particles, DM decays or annihilation may also inject energy Mapelli et al.
(2006); Galli et al. (2009); Giesen et al. (2012); Lopez-Honorez et al.
(2013); Ade et al. (2016); Liu et al. (2016); Liu & Slatyer (2018); Fraser et
al. (2018); Mitridate & Podo (2018); Aghanim et al. (2020). Annihilation/decay
of the DM could inject electromagnetically interacting particles into the
universe, with a potentially wide range of observable consequences. These
ionizing particles during the cosmic dark ages may change the ionization and
the thermal history along with the last scattering surface width and hence can
modify the anisotropy of the cosmic microwave background (CMB) Adams et al.
(1998); Chen & Kamionkowski (2004); Padmanabhan & Finkbeiner (2005). One of
the proposed origin mechanisms of magnetic fields, as stated above, is when DM
particles interact with baryons in some way. The extra energy induced due to
the DM annihilation or decay will leave a noticeable impact on the 21-cm
signal and its fluctuations Short et al. (2020). Recently, it has been shown
that energy injection by the DM annihilation impacted CMB, and a novel way of
constraining DM annihilation, based on PLANCK 2018, is discussed Huang et al.
(2021). The discussion thus far has been focused mainly on the single species
of DM, weakly interacting with the baryons at all times. However, it is known
that the bounds on interaction cross-section from the CMB suggest weak
coupling at all times only for certain species of the DM particle (i.e.,
specific values of ‘n’). It is known that for redshift $z\gtrsim
10^{4}-10^{5}$, coupling is stronger for $n\geq 0$ Gluscevic & Boddy (2018);
Boddy & Gluscevic (2018). In this era, the baryon and DM remain strongly
coupled with each other and the DM behaves as an additional baryonic
component, except that they do not participate in the recombination process.
For $n=-2$ and $n=-4$, the CMB bounds on $\sigma_{0}$ does not violate, if we
assume a fraction $f_{\chi_{,\rm in}}=\rho_{d}/\rho_{\rm DM}$ of the total DM
density $\rho_{\rm DM}$ only interact with the baryons. In this case, the
remaining fraction of the DM is cold collisionless DM Boddy et al. (2018). In
the present work, we have studied thermal and ionization history for various
values of $f_{\chi_{,\rm in}}$. A value of $f_{\chi_{,\rm in}}=1$ means all
the DM particles are interacting with the baryons. The baryon-DM interaction
is considered under the following assumption: (i). the DM particle has no
interaction with the neutral component of baryons, implying that it has little
interaction with atomic dipole moments, (ii). the interactions between the
charged components and the rest of the matter (and within the remaing matter)
are large enough to maintain a constant velocity and temperature for all
baryonic components (iii). A nonzero and tiny electric charge exists in a
portion of the DM (i.e., $f_{\chi,{\rm int}}<1$) that does not interact with
the neutral fraction (iv). the charged component of the DM particles is a
Fermion with mass $m_{\chi}>1$ MeV and has a fractional charge, and the
positively and negatively charged components have particle-antiparticle
asymmetry.

In summary, to study the thermal and ionization history of IGM, we have
considered the following effects in addition to the cooling effects from
Bremsstrahlung, collisional excitation, recombination, and collisional
ionization cooling:

  * •

extra energy injection from the annihilation/decay of the DM particles,

  * •

decaying magnetic fields in an early universe plasma via ambipolar diffusion
and turbulent processes,

  * •

energy exchange between the baryons and DM particle due to their non-standard
velocity-dependent interactions.

Since the thermal Sunyaev–Zeldovich (tSZ) effect directly probes the thermal
energy of the universe, precise modeling and future high-accuracy measurements
will provide a powerful way to constrain the thermal history of the universe.
The tSZ effect occurs when ionized electrons with thermal motion scatter CMB
photons, resulting in secondary CMB temperature oscillations. The evolution of
the $y-$parameter, which quantifies the distortion in the CMB, with respect to
redshift $z$ is presented in the last section (see subsection (3.2)). We also
compared our findings to previous research on the temperature and ionization
history of the universe during the dark ages. In the current study, we mainly
focus on the redshift $10<z<1100$ to avoid the non-linear evolution of the
density perturbations. This work is divided in following sections: section
(2), contains the thermal and ionization history, where we have address
baryon-DM interaction (see subsection (2.5)) along with the DM
decay/annihilation (2.6) and heating of IGM due to the ambipolar and turbulent
decay of magnetic fields (2.3) in addition to the standard scenario (2.1).
Finally, we have discussed the obtained results in section (3) and concluded
the work in section (4). We want to mention here that we are working with the
MKS unit. In the present work, we have considered the flat FLRW universe and
used following present-day parameters: $H_{0}=h\times 100$ km s-1 (where
$h=0.6766$), $h^{2}\Omega_{b}=0.022$, $h^{2}\Omega_{m}=0.142$, and
$h^{2}\Omega_{\Lambda}=0.318$, Boltzmann constant $k_{B}=1.381\times 10^{-23}$
J K-1, and speed of light $c=3\times 10^{8}$ m s-1 Aghanim et al. (2020).

##  2 Ionization and thermal history of the universe

There is still a major gap in our understanding of the IGM’s thermal and
ionization histories after the recombination epoch. Various theoretical
studies are given to understand the evolution of gas in the IGM. Within the
$\Lambda$CDM model of cosmology, at the end of the recombination epoch, matter
decouples from the CMB photons and the universe temperature reaches around
$3000$ K. After this epoch, the Universe becomes for most of the time,
homogeneous until the epoch of reionization. This period is known as the dark
ages. During this period of time, residual electrons leftover from the
recombination scatter with the baryons and maintain the thermal equilibrium
until $z\approx 200$. As a result of the Universe’s thermal expansion, the gas
cools adiabatically, reaching a temperature of 10 K at redshift $z\approx 17$.
Later, during the epoch of reionization, over-density in the matter
perturbations grows and eventually collapses to form the first star and
galaxy. In the present work, we will study the implications of the DM decay or
annihilation on the thermal and ionization history, in conjunction with the
three other mechanisms that could deepen the 21-cm absorption signal (detected
by EDGES collaboration): i). nonstandard recombination histories (with
magnetic fields); ii). baryon-DM interaction; and iii). an additional source
of 21-cm photons during the dark ages. These mechanisms were given to explain
the EDGES measurements Bowman et al. (2018). In our present work, we include
all these effects to study their role in the thermal and ionization history of
the IGM.

This section is divided into the following subsections. In subsection (2.1),
we have presented the well-known standard thermal history of the universe. In
the next subsection (2.3), we have briefly discussed baryon-DM interaction and
heating of the gas due to a finite difference in the temperature of baryons
and the DM particles; and a non-zero relative velocity. Subsection (2.5)
contains the heating of the gas due to magnetic heating from the ambipolar and
turbulent decay. The last subsection (2.6) of this section briefly summarizes
the energy injection in the system by deposition due to the annihilation or
decay of the DM particles. At the end of this section, we have written a
complete set of equations required for our study in subsection (2.7).

###  2.1 Standard scenario

During and after the recombination epoch, the conventional perception of the
thermal and the ionization history of the universe is well approximated by the
three-level atomic model Peebles (1968); Zeldovich et al. (1969); Ali-Haimoud
& Hirata (2011). In this standard theory, matter and radiation are in thermal
equilibrium before the recombination epoch ($z\gg 1100$). After this epoch,
the interaction between the matter and CMB photons becomes so weak that matter
decouples from the CMB photons and reaches $\sim 3000$ K. After this epoch,
dark ages start, and few residue electrons and CMB photons are the main
ingredients. The residual free electrons allow Compton scattering to maintain
a thermal coupling of the gas to the CMB photons and set kinetic temperature
of the gas $T_{b}$ equals to CMB temperature $T_{\rm CMB}$ between redshift
$200\lesssim z\lesssim 1100$ Pritchard & Loeb (2012). The gas cools
adiabatically as $T_{b}\propto(1+z)^{2}$ for $z_{*}\lesssim z\lesssim 200$
($z_{*}$ is the redshift at which first star formation takes place or
additional energy injected to reionize the system) Hirata & Sigurdson (2007);
Loeb & Zaldarriaga (2004). With the expansion of the universe, the number
density of the gas and collisional coupling becomes weak, and hence
$T_{b}<T_{\rm CMB}$. Below redshift $z\lesssim z_{*}$, for energy injected
from the first-star formation or any other source, kinetic temperature $T_{b}$
becomes larger than the CMB temperature. The evolution equations of the
baryonic matter and the ionization fraction is given by Ali-Haimoud & Hirata
(2011)

$\displaystyle\frac{dT_{b}^{\rm sta}}{dz}$ $\displaystyle=$
$\displaystyle\frac{2T_{b}}{(1+z)}+\frac{\Gamma_{C}}{(1+z)H}(T_{b}-T_{\rm
CMB}),$ (2) $\displaystyle\frac{dx_{e}^{\rm sta}}{dz}$ $\displaystyle=$
$\displaystyle\frac{1}{H(1+z)}\Bigg{[}\left(n_{\small
B}x_{e}^{2}\alpha_{B}-(1-x_{e})\beta_{B}e^{-E_{21}/T_{\rm CMB}}\right)D$ (3)
$\displaystyle-$ $\displaystyle\gamma_{e}\,n_{\small
B}(1-x_{e})\,x_{e}\Bigg{]}.$

In equation (2), $\Gamma_{C}$ is Compton scattering rate and it is defined as
Ali-Haïmoud & Hirata (2010)

$\Gamma_{C}=\frac{8\sigma_{T}x_{e}\rho_{\rm CMB}}{3(1+x_{e}+f_{\rm
He})m_{e}}.$ (4)

The electron fraction and the helium fraction are defined by
$x_{e}=n_{e}/n_{H}$ and $f_{\rm He}=\frac{y_{p}}{N_{\rm tot}(1-y_{p})}$
respectively. Here $N_{\rm tot}=3.971$ and $y_{p}=0.21$, $\rho_{\rm
CMB}=a_{r}T_{\rm CMB}^{4}$ (the radiation constant $a_{r}=7.565\times
10^{-16}$ J-m-3-K-4) and the Thomson scattering cross-section
$\sigma_{T}=6.652\times 10^{-29}$ m2. Coefficients $\alpha_{B}$, $\beta_{B}$
and $\gamma_{e}$ in equation (3) are defined as Peebles (1968)

$\displaystyle\alpha_{B}$ $\displaystyle=$ $\displaystyle F\times
10^{-19}\times\frac{4.309\times(T_{b}/10^{4})^{-0.616}}{1+0.670\times(T_{b}/10^{4})^{0.530}}\,\,[{\rm
m}^{3}\,{\rm s}^{-1}],$ (5) $\displaystyle\beta_{B}$ $\displaystyle=$
$\displaystyle\alpha_{B}\left(\frac{2\pi m_{e}k_{B}T_{\rm CMB}}{h_{\rm
pl}^{2}}\right)^{3/2}\exp\left(\frac{E_{21}}{k_{B}T_{\rm CMB}}\right)\,[{\rm
s}^{-1}]\,,$ (6) $\displaystyle\gamma_{e}$ $\displaystyle=$ $\displaystyle
0.291\times
10^{-7}\times\left(\left|\frac{E_{1s}}{k_{B}T_{b}}\right|\right)^{0.39}\,\frac{\exp(-|E_{1s}/k_{B}T_{b}|)}{0.232+|E_{1s}/k_{B}T_{b}|}$
(7) $\displaystyle\times[{\rm cm}^{3}{\rm s}^{-1}].$

Here $\alpha_{B}$ is the Case-B recombination coefficient, $\beta_{B}$ is the
photoionization rate and $\gamma_{e}$ is the collisional coefficient. Other
parameters used are: $E_{21}=E_{1s}-E_{2s}=-10.2$ eV is energy of Ly$\alpha$
wavelength photon, $E_{1s}=-13.6$ eV, and $E_{2s}=-3.4$ eV. Coefficient $D$
represents the probability of a hydrogen atom in the state $n=2$ decaying to
the ground state before photoionization can occur. It is defined as Peebles
(1968)

$\displaystyle D=\frac{1+K_{H}\Lambda_{2s,1s}n_{\small
B}(1-x_{e})}{1+K_{H}\,n_{\small B}(1-x_{e})+K_{H}n_{B}\beta_{B}(1-x_{e})}$ (8)

where $K_{H}=\lambda_{H_{2p}^{3}}/8\pi H$, $\lambda_{H_{2p}}=121.568$ nm and
$\Lambda_{2s,1s}=8.22~{}{\rm s}^{-1}$ is the hydrogen two photon decay rate.

###  2.2 Cooling effects

We have also included cooling effects from the bremsstrahlung, collisional
excitation, recombination and collisional ionization cooling. With these
cooling effects, the evolution equation for the baryon temperature has
following form

$\displaystyle\frac{dT_{b}^{\rm cool}}{dz}=\frac{\Gamma_{\rm
cool}}{1.5k_{B}\,n_{\small B}\,(1+z)H},$ (9)

where

$\displaystyle\Gamma_{\rm cool}=\frac{x_{e}n_{H}}{1.5k_{B}}\left[\Theta
x_{e}+\Psi(1-x_{e})+\eta x_{e}+\zeta(1-x_{e})\right],$ (10)

Unit of $\Gamma_{\rm cool}$ is $[\text{J}\,\text{m}^{3}\,\text{s}^{-1}]$. Here
$\Theta$, $\Psi$, $\eta$ and $\zeta$ are defined as (Fukugita & Kawasaki
(1994))

$\displaystyle\centering\Theta\@add@centering$ $\displaystyle=$ $\displaystyle
5.84\times 10^{-37}\,\left(\frac{T_{b}}{10^{5}}\right)^{0.5}\,,$
$\displaystyle\Psi$ $\displaystyle=$ $\displaystyle 7.50\times
10^{-31}\left(1+\left(\frac{T_{b}}{10^{5}}\right)\right)^{-1}\exp\left(-\frac{1.18}{T_{b}/10^{5}}\right),$
$\displaystyle\eta$ $\displaystyle=$ $\displaystyle 2.06\times
10^{-36}\left(\frac{T_{b}}{10^{5}}\right)^{0.5}\left(\frac{T_{b}}{10^{5}}\right)^{-0.2}\left(1+\left(\frac{T_{b}}{10^{5}}\right)^{0.7}\right)^{-1},$
$\displaystyle\zeta$ $\displaystyle=$ $\displaystyle 4.02\times
10^{-32}\left(\frac{T_{b}}{10^{5}}\right)^{0.5}\left(1+\left(\frac{T_{b}}{10^{5}}\right)^{-1}\right)\exp\left(-\frac{1.58}{T_{b}/10^{5}}\right)\,.$

It is important to note here that all these functions are dependent on baryon
temperature; hence they have a redshift dependence. The term on the right-hand
side of the equation (10) has a positive sign; hence redshift lowers the
baryon temperature.

###  2.3 Magnetic heating

In the present work, we consider a statistically homogeneous and isotropic
random Gaussian, non-helical magnetic field background, which means we neglect
any back reaction of the local matter evolution. During this period, we also
assumes that universe is infinitely conducting. In this period $\vec{{\bf
B}}(t,\vec{x})=\tilde{{\bf B}}(\vec{x})/a^{2}(t)$ on length scale larger than
the magnetic Jeans length scale
$\lambda_{J}^{-1}=k_{J}=\sqrt{8\pi\rho_{m}G}/{\rm v}_{A}$, where ${\rm
v}_{A}=B_{0}/\sqrt{4\pi\rho_{b}}$ ($B_{0}$ is the comoving strength of
magnetic fields and $\rho_{b}$ is the baryon density) Sethi & Subramanian
(2005). For the assumptions made above, power spectrum $\mathcal{P}_{B}(k)$ of
the magnetic field is given by the relations Landau L. D. (1987); Brandenburg
et al. (2018)

$\displaystyle\langle\tilde{{\bf B}}^{*}_{i}({\bf k}){\bf B}_{j}({\bf
k}^{\prime})\rangle=\frac{(2\pi)^{3}}{2}\,\delta_{D}({\bf k}-{\bf
k}^{\prime})(\delta_{ij}-\hat{k}_{i}\hat{k}_{j})\mathcal{P}_{B}(k)$ (12)

where ${\bf B}^{*}_{i}({\bf k})$ is the Fourier component of ${\bf B}_{i}({\bf
k})$ with a mode ${\bf k}$. Here power spectrum $\mathcal{P}_{B}(k)$ is
defined as

$\mathcal{P}_{B}(k)=\begin{cases}\frac{(2\pi)^{n_{B}+5}}{\Gamma\left(\frac{n_{B}+3}{2}\right)}B_{\lambda}^{2}\frac{k^{n_{B}}}{k_{\lambda}^{n_{B}+3}}&,~{}~{}~{}\text{for}~{}k<k_{D}\\\
0&,~{}~{}~{}\text{for}~{}k>k_{D}\\\ \end{cases}$ (13)

In the above equation, $B_{\lambda}$ is the strength of magnetic field
smoothed at a length scale of $\lambda=k_{\lambda}^{-1}$ and $n_{B}$ is the
magnetic field spectral index. Here $\lambda_{D}=k_{D}^{-1}$ is given by the
following relation

$\frac{k_{D}}{2\pi{\rm Mpc}^{-1}}=\left[1.32\times
10^{-3}\left(\frac{B_{0}(t)}{1{\rm
nG}}\right)^{2}\left(\frac{\Omega_{b}h^{2}}{0.02}\right)^{-1}\left(\frac{\Omega_{m}h^{2}}{0.15}\right)\right]^{\frac{-1}{(n_{\small
B}+5)}}.$ (14)

Here $B_{0}(t)=B_{0}\,(1+z)^{2}$ is the comoving magnetic field strength. The
spectral index $n_{B}=-2.9$ corresponds to the scale-invariant magnetic
fields. In the present work, we have used values of $n_{B}>-2.9$ to avoid the
infrared divergence of the magnetic fields Chluba et al. (2015).

In the post recombination era, the radiative viscosity decreases quickly and
at length scale $\lambda<\lambda_{J}$ (Jeans length scale), a decaying
Magnetohydrodynamic (MHD) turbulence is generated due to the non-linear
effects Jedamzik et al. (1998); Subramanian & Barrow (1998). At these length
scales, magnetic fields decay by two processes, (i) ambipolar diffusion and
(ii) turbulent decay. The evolution equation of the magnetic energy density is
given by

$\displaystyle\frac{d}{dt}\left(\frac{|{\bf
B}|^{2}}{8\pi}\right)=-4H(t)\left(\frac{|{\bf
B}|^{2}}{8\pi}\right)-\Gamma_{\rm ambi}-\Gamma_{\rm tur}.$ (15)

We have followed the calculations done in Sethi (2003); Sethi & Subramanian
(2005) for the ambipolar and turbulent processes. In our work, ambipolar decay
is given by

$\displaystyle\Gamma_{\rm
ambi}=\frac{\rho_{n}}{16\pi^{2}\gamma\rho_{b}^{2}\rho_{i}}|(\nabla\times{\bf
B})\times{\bf B}|^{2},$ (16)

where $\rho_{n}$, $\rho_{b}$ and $\rho_{i}$ are the mass density of the
neutral, ionized atoms and the baryon densities, respectively. In equation
(16), $\gamma$ is given by (Schleicher et al. (2009))

$\displaystyle\gamma=\frac{\frac{1}{2}n_{\rm H}\langle\sigma v\rangle_{{\rm
H}^{+},{\rm H}}+\frac{4}{5}n_{\rm He}\,\langle\sigma v\rangle_{{\rm
H}^{+},{\rm He}}}{m_{\rm H}\,(n_{\rm H}+4n_{\rm He})}\,.$ (17)

Here $\langle\sigma v\rangle_{{\rm H}^{+},{\rm H}}$ and $\langle\sigma
v\rangle_{{\rm H}^{+},{\rm He}}$ is defined as (Schleicher et al. (2008))

$\displaystyle\langle\sigma v\rangle_{{\rm H}^{+},{\rm H}}$ $\displaystyle=$
$\displaystyle 0.649\,T_{b}^{0.375}\times 10^{-15}\,{\rm m}^{3}\,s^{-1},$
$\displaystyle\langle\sigma v\rangle_{{\rm H}^{+},{\rm He}}$ $\displaystyle=$
$\displaystyle[1.424+7.438\times 10^{-6}T_{b}-6.734\times 10^{-9}T_{b}^{2}]$
(18) $\displaystyle\times$ $\displaystyle 10^{-15}\,{\rm m}^{3}\,s^{-1}.$

Magnetic energy decay by MHD turbulence is given by Sethi & Subramanian (2005)

$\displaystyle\Gamma_{{\rm
tur}}=\frac{B_{0}^{2}(t)}{8\pi}\frac{3m}{2}\frac{\left[\ln\left(1+t_{\rm
eddy}/t_{i}\right)\right]^{m}\,H(t)}{\left[\ln\left(1+t_{\rm
eddy}/t_{i}\right)+\ln(t/t_{i})\right]^{m+1}}\,,$ (19)

where $t_{\rm decay}=k_{D}^{-1}/{\rm v}_{\rm A}$ is the dynamical timescale
and $t_{i}=1.248\times 10^{13}$ s. $m$ is given by $m=2(n_{\small
B}+3)/(n_{\small B}+5)$ (Olesen (1997); Shiromizu (1998); Jedamzik et al.
(1998); Christensson et al. (2001); Sethi & Subramanian (2005)). The decay of
magnetic fields via ambipolar and turbulent process heats the baryons and
hence the contribution from the magnetic heating to the baryon temperature is
given by

$\displaystyle\frac{dT_{b}^{\rm
mag}}{dz}=-\frac{2}{3k_{B}n_{H}(1+z)H}(\Gamma_{\rm ambi}+\Gamma_{\rm tur})$
(20)

The heating of the baryons as we move from high to low redshift is represented
by the negative sign in the preceding equation.

(a)

(b)

Figure 1: BDM+PMF: Evolution of baryon temperature and ionization fraction
with respect to redshift, $z$. Magnetic fields contribute via ambipolar
diffusion and magneohydrodynamic turbulent decay. The color description is
given in the plots. We note here that we have considered one on one Dark
matter and baryon interactions (means all Dark matter particles are
interacting with the baryons), and $f_{\chi_{,\rm in}}=1$.

(a)

(b)

Figure 2: PMF+BDM: Evolution of baryon temperature and ionization fraction and
DM with respect to redshift $z$. Color description is given in the plots. In
the left panel, we consider two values of $f_{\chi_{,\rm in}}=0.1$ and $1.0$
at given values of $B_{0}$, $\sigma_{0}$ and $m_{\chi}$. On right panel,
baryon temperature at a fix redshift, with respect to $f_{\chi_{,\rm in}}$. We
use $\lambda_{B}=0.25$ Mpc.

###  2.4 Magnetic fields and the density evolution

In the previous subsection, we have considered the effect of dissipation by
ambipolar and turbulent decay of magnetic fields on the baryon temperature. It
is known that these magnetic fields can generate density perturbations in the
post-recombination epoch and later can collapse to form structures Wasserman
(1978); Kim et al. (1996); Subramanian & Barrow (1998); Gopal & Sethi (2003);
Sethi & Subramanian (2005). The density evolution equations of the baryons and
DM, in presence of the magnetic fields at a length scale larger than the
magnetic Jeans scale, are given by

$\displaystyle\frac{\partial^{2}\delta_{b}}{\partial
t^{2}}+2H(t)\frac{\partial\delta_{b}}{\partial t}-4\pi
G(\rho_{d}\delta_{d}+\rho_{b}\delta_{b})=S(t),$ (21)
$\displaystyle\frac{\partial^{2}\delta_{d}}{\partial
t^{2}}+2H(t)\frac{\partial\delta_{d}}{\partial t}-4\pi
G(\rho_{d}\delta_{d}+\rho_{b}\delta_{b})=0,$ (22)

where, $\rho_{d}$ is the cold dark matter density. $\delta_{\rm b}$ and
$\delta_{\rm d}$ are the density contrast of the baryon and cold dark matter,
respectively. In equation (21), the Lorentz force averaged over the coherence
length scale of the magnetic fields are represented by the quantity $S(t)$ and
it is defined as Minoda et al. (2017)

$\displaystyle S(t,{\bf
x})=\frac{\nabla\cdot\left[(\nabla\times\mathbf{B}(t,\mathbf{x}))\times\mathbf{B}(t,\mathbf{x})\right]}{4\pi\rho_{\mathrm{b}}(t)a^{2}(t)},$
(23)

The absence of the source term in the DM equations from the magnetic fields
signifies the absence of direct coupling of DM with the magnetic fields. In
the range of interest of redshift $10<z<1100$, density perturbation
$\delta_{b}\leq 10^{-6}-10^{-5}$ is very small. In this scenario, analytic
solution of above equations for $\delta_{b}$ gives (Minoda et al. (2017):

$\displaystyle\delta_{b}=\frac{2S(t)}{15H^{2}(t)}\left[\left\\{3A\right.\right.+2A^{-\frac{3}{2}}-\left.15\ln
A\right\\}\frac{\Omega_{b}}{\Omega_{m}}+15\ln A$
$\displaystyle+30\left(1-\frac{\Omega_{b}}{\Omega_{m}}\right)A^{-\frac{1}{2}}-\left.\left(30-\frac{25\Omega_{b}}{\Omega_{m}}\right)\right].$
(24)

The scale factor at the time of recombination is given by $a_{\rm
rec}\propto(1+z_{\rm rec})^{-1}$ and quantity $A=a/a_{\rm rec}$. The
additional contributes from the density fluctuation generated by the magnetic
fields to the baryon temperature is given by
$dT_{b}^{\delta}/dt=-T_{b}d(ln(1+\delta_{b}))/dt$ reduced to following
equation

$\displaystyle\frac{dT_{b}^{\delta}}{dz}=\frac{T_{b}}{1+\delta_{b}}\frac{d\delta_{b}}{dz}.$
(25)

A positive sign in front of the equation indicates that gas cools due to the
growing density fluctuations induced by the magnetic fields with increase of
the redshift.

(a)

(b)

Figure 3: DM annhilation and BDM+Annihilation: Evolution of $T_{b}$ and
$x_{e}$ for the various combination of the parameters for the case of DM
annhilation only and case when BDM+DM annihilation is considered. In the
current figure, we set $\sigma_{0}=3.0\times 10^{-45}$ m2, $n=-4.0$ for the
scenario where we consider BDM interaction.

(a)

(b)

Figure 4: DM annhilation and PMF: Evolution of $T_{b}$ at fixed values of
$B_{0}=1\rm{nG}$,$n_{B}=-1.0$ and $\lambda_{B}=1.0$ Mpc and in presence of DM
annhilation with many possible combinations of the channels and annhilation
cross-sections is shown in figures (4a) and (4b), respectively.

(a)

(b)

Figure 5: BDM+PMF+Annihilation: Baryon and DM temperature for two values of
$f_{\chi,{\rm int}}$ at fix values of $f_{\chi,{\rm ann}}$ and $f_{\rm eff}$.
The light green color between the two Tb curves and the hatched area between
the two Td curves shows the area for for $f_{\chi,{\rm int}}=0.001$ and $1$.
The results are consistent to the result obtained in (Liu & Slatyer, 2018).

(a)

(b)

Figure 6: BDM+PMF+DM annihilation: Here we have shown dependence of baryon
temperature on the $f_{\chi,{\rm int}}$ and $\sigma_{0}$. In left figure,
countour plot of $T_{b}$ in $\sigma_{0}-f_{\chi,{\rm int}}$ plain at redshift
$z=20$ is given. In the right panel, we show the contour plot of $T_{b}$ in
$z-f_{\chi,{\rm int}}$ plain for $\sigma_{0}=3.0\times 10^{-45}$ m2. In both
cases, we set $B_{0}=0.003$ nG, $n_{B}=-2.9$, $\lambda_{B}=0.25$ Mpc,
$m_{\chi}=0.1$ GeV, $p_{\rm ann}=4.4\times 10^{-7}$ m3 sec-1 Kg-1,
$f_{\chi,{\rm ann}}=0.6$ and $f_{\rm eff}=0.4$.

(a)

(b)

Figure 7: DM decay only Evolution of $T_{b}$ and $x_{e}$ for the various
combination of the parameters. Here we compare baryon temperature for two DM
decay channels with the standard baryon temperature evolution.

(a)

(b)

Figure 8: DM decay+PMF: Evolution of $T_{b}$ at fixed values of
$B_{0}=1\rm{nG}$, $n_{B}=-1.0$ and $\lambda_{B}=1.0$ Mpc and in presence of DM
decay with many possible combinations of the channels and decay time is shown
in figures (8a) and (8b) respectively.

(a)

(b)

Figure 9: DM decay+BDM: Evolution of $T_{b}$ at fixed values of
$B_{0}=0.003\rm{nG}$, $n_{B}=-1.0$ and $\lambda_{B}=1.0$ Mpc and in presence
of DM decat with many possible combinations of the channels and decay time is
shown in figures (8a) and (8b) respectively.

(a)

(b)

Figure 10: DM decay+PMF+BDM: Evolution of $T_{b}$ at fixed values of
$B_{0}=0.003\rm{nG}$, $n_{B}=-1.0$ and $\lambda_{B}=1.0$ Mpc and in presence
of DM decat with many possible combinations of the channels and decay time is
shown in figures (8a) and (8b) respectively.

###  2.5 Effect of the Baryon-DM interaction

EDGES collaboration detected a global 21 cm absorption signal centered at 78
MHz at a redshift of $z\approx 17$, which is 2.5 times larger than the
predicted from the standard $\Lambda$CDM model Bowman et al. (2018). After
detecting this signal, various attempts have been made to explain the detected
EDGES anomaly Moroi et al. (2018); Fraser et al. (2018). Since DM is colder
than the gas in the dark ages, gas particles can be cooled by elastic
scattering of the baryons by the DM particles Fialkov et al. (2014); Tashiro
et al. (2014); Barkana et al. (2018). One of the crucial aspects of this
consideration is to understand the behavior and structure of the DM particles,
whose nature is, however, still not known. The assumption that baryon and DM
interact opens up a new area of research, where it is one of our best chances
to understand these particles from a different angle.

To explain the 21-cm absorption signal by EDGES collaboration at redshift
$z\approx 17.2$, a non-standard Coulomb interaction cross-section $\sigma({\rm
v})=\sigma_{0}\,({\rm v}/c)^{n}$ (here ${\rm v}$ is the relative velocity
between the DM and baryons) between the DM and baryons is considered Bowman et
al. (2018); Barkana et al. (2018). The exponent $n$ in the above relation is a
parameter and $n=-1$ correspondence to Yukawa potential, $n=-2$ is given for
the case when the DM particles have an electric dipole moment and for a
millicharge DM particles $n=-4$. Heat transfer between the DM and baryons
occurs by two processes: i). one by the temperature difference between the
baryons and the DM particles, ii). second one due to the drag force acting
between the two particles. The second force act as resistive force and tend to
become zero with time. The drag force per unit mass between the DM and baryons
at the kinetic decoupling is quantified by the rate of change of the bulk
velocity of the DM fluid with respect to the baryons. The net energy transfer
due to the interaction from DM to baryon is therefore written as Dvorkin et
al. (2014); Tashiro et al. (2014); Muñoz et al. (2015); Muñoz & Loeb (2017);
Barkana et al. (2018)

$\displaystyle\dot{Q}_{b}=\frac{\rho_{d}\mu_{b}\,k_{B}\,c^{-n}\,\sigma_{0}e^{-r^{2}/2}}{(m_{d}+m_{b})^{2}}\left(\frac{k_{B}T_{b}}{m_{b}}+\frac{k_{B}T_{d}}{m_{d}}\right)^{\frac{n+1}{2}}\,$
$\displaystyle\times\frac{2^{\frac{5+n}{2}}\Gamma\left(3+\frac{n}{2}\right)}{\sqrt{\pi}}(T_{d}-T_{b})+\frac{\rho_{d}}{\rho_{d}+\rho_{b}}\frac{m_{d}\,m_{b}}{m_{d}+m_{b}}\,|{\rm
v}\,\frac{d\textbf{v}}{dt}|,$

Similarly, we can write the expression for the energy exchange from baryons to
DM is given by just exchanging $b\leftrightarrow d$. First-term in the above
equation on the right-hand side represents the heat transfer due to the finite
temperature difference between the baryons and the DM particles. However, the
second term is given for the effects of drag force. In above equation (2.5)
$\rho_{m}$ is the total matter mass density, and $m_{b}$, $m_{d}$ are the mass
of the baryon and dark matter particles, respectively. The variables $r$ is
defined as $r\equiv\text{v}/u_{\rm th}$ and $u_{\rm th}$ is given by: $u_{\rm
th}^{2}=\left(\frac{k_{b}T_{b}}{m_{b}}+\frac{k_{b}T_{d}}{m_{d}}\right)$. Also
$\mu_{b}\simeq m_{H}(n_{H}+4n_{He})/(n_{H}+n_{He}+n_{e})$. For large values of
$n$, DM and baryons are coupled until the scattering rate is lower than the
Hubble expansion rate of the universe. For example, when $n=0$, the decoupling
time scale of the baryon-DM is given by $\tau_{\rm
BDM}\equiv(a/R_{\chi})(m_{d}+m_{H})/m_{d}$. $R_{\chi}$ is the baryon-DM
momentum exchange rate, and it is given in reference Xu et al. (2018). In this
case, the temperature of the baryons and the DM particles remain the same when
$H\tau_{\rm BDM}>1$. We have also assumed that the DM is non-relativistic and
its velocity decreases only adiabatically with $v\propto(1+z)$ after the
decoupling epoch. Therefore, the evolution equation of the baryons and the DM
are following

$\displaystyle\frac{dT_{b_{\rm BDM}}}{dz}$ $\displaystyle=$
$\displaystyle-\frac{2\,\dot{Q}_{b}}{3k_{B}(1+z)H}$ (27)
$\displaystyle\frac{dT_{d_{\rm BDM}}}{dz}$ $\displaystyle=$
$\displaystyle-\frac{2\,\dot{Q}_{d}}{3k_{B}(1+z)H}$ (28)

These two equations represent the change in temperature of the baryons and DM
due to baryon-DM interactions. For this case, $f_{\chi,{\rm int}}$ dependence
of these equations are shown in reference (Venumadhav et al., 2018; Muñoz et
al., 2015).

###  2.6 Decay/annihilation of DM particles

Figure 11: BDM+$B_{0}$+Decay: In this case, we have considered following
parameters: $B_{0}=0.003$ nG, $n_{B}=-2.9$, $\lambda_{B}=0.25$ Mpc,
$m_{\chi}=0.1$ GeV, $\sigma_{0}=3\times 10^{-45}~{}\text{m}^{2}$,
$\tau=2\times 10^{25}$ sec, $f_{\chi,{\rm eff}}=0.6$,
$f_{\chi,\text{dec}}=0.6$. In this plot, we have only considered redshift
$z=10-200$. At higher redshift, we have found that there are no visible effect
on the baryon temperature.

It is known that both stable and unstable standard model particles and their
antiparticles are found in the exotic decay of particles. The unstable
products eventually decay into stable particles, such as electrons, photons,
protons, and neutrinos. These particles and their antiparticles lose energy
either adiabatically or interact with the particles before or after the
recombination epoch and transfer energy to the photons and light element atoms
(for example, electrons and nuclei). In the present work, we will limit
ourselves to the injection of electron-positron or photons from the decay of
DM ($\chi\rightarrow e^{+}e^{-}$ or $\chi\rightarrow\gamma\gamma$) and from
the annihilation of the DM ($\chi\chi\rightarrow e^{+}e^{-}$ or
$\chi\chi\rightarrow\gamma\gamma$) with a monochromatic energy spectrum (for
the other possible decay channels, see reference Slatyer (2016b)). We have
ignored the energy deposition from the processes where protons/antiprotons and
neutrinos are decay products. Since neutrinos are invisible to the medium
because of their tiny mass and interaction cross-section, they carry a small
fraction of the total output energy. Similarly, protons and antiprotons inject
very small energy into the medium. Hence we have ignored the decay channels,
where protons/antiprotons are main decay products Weniger et al. (2013); Liu
et al. (2016, 2020); Cang et al. (2020). The $\gamma\gamma$ final state of the
decay of sterile neutrinos from the decay mode $\gamma\nu$ is relevant to the
unstable gravitinos with mass below the $W$ gauge bosons Liu & Slatyer (2018).
It is thus clear that the chosen channels are mostly dictated by simplicity
since, in most decay models, a continuous spectrum of standard model particles
is emitted. Our formalism in the present work applies to any final state with
a generic $e^{\pm}$ and photons $\gamma$. Hence the results obtained are
mostly indicative and do not represent the actual, more complicated energy
injected process from the decay/annihilation of the DM particles.

The additional particles and photons injection have three main contributions:
i). the number of non-thermal particles grows; however, their average energy
density decreases due to the interaction with the relic photons, ii). when the
energy density of the non-thermal particles reaches keV scale, they start
interacting with the atoms (mostly hydrogen atoms) and induce Lyman-$\alpha$
excitation of the hydrogen, which in results a higher number of electrons in
$n=2$ states than the ground state of hydrogen and hence increased ionization
fraction Galli et al. (2013), and iii). can directly heat the gas. The first
two effects change the ionization fraction, and the last terms lead to a
change in the baryon temperature. It has been shown that the injected energy
is often not absorbed on-the-spot; instead, it is redshift away before being
deposited Galli et al. (2009); Liu et al. (2016). Therefore, the annihilation
or decay of the DM into the standard model (SM) particles inject energy
(denoted by, $\Gamma_{\rm dep}=\frac{dE}{dV\,dt}|_{\rm dep}$) into the
universe and leads to an additional ionization (‘$i$’), excitation
(‘$\alpha$’) and heating (‘$h$’) of the medium. The most critical parameters
that affect the energy injection from the decay/annihilation are, a).
decay/annihilation channels of the DM particles, b). mass of the DM particles,
and c). the redshift at which the energy injections occur. The injected energy
is parameterized in terms of the redshift as

$\Gamma^{\rm dep}_{c}=f_{c}(z)\,\frac{dE}{dV\,dt}\bigg{|}_{\rm inj}.$ (29)

Here the dimensionless function, $f_{c}(z)$ represents the energy deposition
efficiency in the medium from a relevant channels Shull & van Steenberg
(1985); Chen & Kamionkowski (2004); Galli et al. (2009); Finkbeiner et al.
(2012); Slatyer (2013); Poulin et al. (2015); Slatyer (2016a, b); Liu et al.
(2016); Slatyer & Wu (2017). The subscript ‘$c$’ represents three different
channels and is given by ‘$c=\\{i,\alpha,h\\}$’. This function encodes all the
physics of the energy deposition accounted for all additional standard model
electromagnetic processes due to the annihilation/decay in IGM 111Usually this
function $f_{c}$ depends on not only $z$ but also on ionization fraction
$x_{e}$, i.e, $f_{c}(z,x_{e})$. However, for energy injection $\geq 10$ MeV,
$f_{c}(z,x_{e})\simeq g(z)\chi_{c}(x_{e})$ is a good approximation. This
correspondence to a factorization between the high energy process (which
determine the function $g(z)$) and low energy processes, responsible for the
absorption (determined by function $\chi_{c}(x_{e})$) Galli et al. (2013);
Slatyer (2016a).. In references Finkbeiner et al. (2012); Giesen et al.
(2012), it has been shown that, up to first-order approximation, redshift
dependence of the function $f_{c}(z)$ can be ignored, as current CMB data are
sensitive to energy injection over a relatively narrow range of redshift
($z\sim 1000-600$) Lambiase et al. (2021). Therefore, $f_{c}(z)$ can be
replaced by a constant number $f_{\rm eff}$ and for sake of simplicity, we
have approximated $f_{c}(z)=f_{\rm eff}$ a constant efficiency factor over the
entire redshift range. This assumption means that the injected energy is
instantly deposited in the intergalactic medium (IGM). Normally the $f_{\rm
eff}$ depends on the mass of DM and the EM branching ratio, i.e., decay
channels. As the DM annihilation or decay into SM particles injects energy
into the medium, an additional ionization, excitation, and heating occur. The
rate of energy injection from these processes is given by

$\frac{dE}{dVdt}\bigg{|}_{\rm inj}=\begin{cases}(1+z)^{3}\frac{f_{\chi,{\rm
dec}}}{f_{\chi_{,\rm in}}^{-1}}\Omega_{\rm
DM}c^{2}\rho_{c}\frac{Exp(-\frac{t}{\tau})}{\tau},&\text{for decay}\\\
(1+z)^{6}\left(\frac{f_{\chi,{\rm ann}}^{2}}{f_{\chi,{\rm
in}}^{-1}}\right)\Omega_{\rm DM}^{2}c^{2}\rho_{c}^{2}\frac{\langle\sigma
v\rangle}{m_{\chi}},&\text{for annihilation}\end{cases}$ (30)

where $f_{\chi,{\rm dec}}$ and $f_{\chi,{\rm ann}}$ are the decay/annihilation
fractions by mass of the DM particles, $\Omega_{\rm DM}$ is the present-day
total abundance of cold DM, $\rho_{c}$ is the critical density of the
universe, $\tau$ is the lifetime of the decaying particle (related to the
decay rate $\Gamma_{d}$ by $\tau=\Gamma_{d}^{-1}$). $m_{\chi}$ is the DM
particle mass, and $\langle\sigma v\rangle$ is the thermally averaged self-
annihilation cross-section. If annihilation occurs between two indistinct
particles, an additional factor of $1/2$ for Majorana fermions and $1/4$ for
the Dirac fermions should be multiplied. In the present work, we only consider
that annihilating particle is indistinguishable. $f_{\chi}=1$ represents the
complete decay/annihilation of the DM particles. For the DM annihilation, we
have defined a new more condense efficiency parameter $p_{\rm ann}=f_{\rm
eff}\langle\sigma v\rangle/m_{\chi}$. It is important to note that we have not
considered the structure formation. In reference Slatyer (2013) authors have
shown that at redshift $z\leq 30$ there might be measurable effects on the
21-cm line of hydrogen from the structure formation Furlanetto et al. (2006);
Valdés et al. (2007). In this case, halo contributes to additional factors in
equations (30) as erfc$\left(\frac{1+z}{1+z_{h}}\right)\bar{f}_{h}$, where
$z_{h}$ is the redshift at which halo is formed and
$\bar{f}_{h}=\frac{200}{3}(1+z_{h})^{3}f_{\rm NFW}$ is the normalization
factor obtained from N-body structure formation simulations Giesen et al.
(2012). Change in the temperature of the universe due to the energy deposition
from the DM decay/annihilation is given by

$\displaystyle\frac{dT_{b_{\rm
dec/ann}}}{dz}=-\frac{1}{H(z)(1+z)}\frac{2f_{h}(z)}{3k_{B}n_{H}(1+f_{He}+x_{e})}\frac{dE}{dVdt}\bigg{|}_{\rm
inj}.$ (31)

The heating term on the right-hand side of the equation (31) comes when a non-
zero DM decay/annihilation and function $f_{h}(z)$ is the dimensionless
efficiency function which parameterizes the extra heat injected to increase
the gas temperature. The contribution to the ionization fraction from the DM
decay/annihilation is given by

$\displaystyle\frac{dx_{e}^{\rm dec/ann}}{dz}=-\frac{1}{(1+z)H(z)}I_{X}(z).$
(32)

The term on the right hand side of equation (32) represents the ionization
rate due to extra injection of energetic particles and it can be split into
two terms as: $I_{X}=I_{X_{i}}+I_{X_{\alpha}}$ Chen & Kamionkowski (2004);
Poulin et al. (2017); Slatyer & Wu (2017); Chluba (2010). Here $I_{X_{i}}$ and
$I_{X_{\alpha}}$ represent the hydrogen ionization from the ground state
($n=0$) and by additional Ly-$\alpha$ photons, which boosts the population at
the $n=2$ state and hence increases the photoionization by CMB photons. These
two terms $I_{X_{i}}$ and $I_{X_{\alpha}}$ are defined as

$\displaystyle I_{X_{i}}$ $\displaystyle=$
$\displaystyle-\frac{1}{n_{H}(z)E_{i}}\frac{dE}{dV\,dt}\bigg{|}_{{\rm dep},i}$
(33) $\displaystyle I_{X_{\alpha}}$ $\displaystyle=$
$\displaystyle-\frac{1-C}{n_{H}(z)E_{\alpha}}\frac{dE}{dV\,dt}\bigg{|}_{{\rm
dep},\alpha}$ (34)

The two constants $E_{i}=13.6$ eV and $E_{\alpha}=E_{21}=10.2$ eV are the
average ionization per hydrogen atom and the Ly-$\alpha$ energy respectively.
Parameter $\mathcal{C}$ is known as the inhibition factor, and it represents
the number of excitation events that contributes to the ionization (each event
deposits energy equals 10.2 eV). For a direct ionization $\mathcal{C}=0$. In
the present work, we neglect the extra heat injected from the Helium
ionization as the Helium ionization is sub-dominant Galli et al. (2013);
Slatyer (2016a) and hence it will not affect the results significantly.
Therefore, the contribution of the ionization and excitation is given by

$\frac{dx_{e}^{\rm
dec/ann}}{dz}=\frac{1}{H(z)(1+z)}\left[\frac{f_{i}(z)}{E_{i}n_{H}}+\frac{(1-\mathcal{C})f_{\alpha}(z)}{E_{\alpha}n_{H}}\right]\frac{dE}{dVdt}\bigg{|}_{\rm
inj},$ (35)

here $f_{i}(z)$ and $f_{\alpha}(z)$ are related to the ionization and
excitation efficiency factors respectively. Thus, energy deposition from the
DM annihilation/decay will increase the ionization fraction and IGM
temperature after the recombination epoch. Another important effect that can
also affect the ionization and thermal history is DM clustering after the
formation of halos. In the cold DM scenario, DM clusters into halos due to
gravity at various ranges of masses; for example: of size $10^{-11}-10^{-3}$
M⊙ for WIMPs to DM clusters halos of size $10^{15}$ M⊙ Bringmann (2009). It
has been shown that the annihilation rate is boosted late due to DM halos and
the DM decay maintains a steady energy injection rate Cang et al. (2020). In
spite of this we have confined our study only to redshift $z\simeq 10$ and
avoided the effects of the DM halo formation.

###  2.7 Complete set of equations

In the previous subsections (2.1-2.6), we have first discussed standard
thermal and ionization history of the universe. Later, we presented a brief
discussion on the heating of the gas temperature from the magnetic heating and
DM annihilation/decay along with the energy exchange between the baryons and
the DM particles due to a non-standard interaction between baryons the DM
particles. After considering all these effects, the baryon temperature is
$T_{b}=T_{b}^{\rm sta}+T_{b}^{\rm cool}+T_{b}^{\delta}+T_{b}^{\rm
mag}+T_{b}^{\rm BDM}+T_{b}^{\rm dec/ann}$, and $x_{e}=x_{e}^{\rm
sta}+x_{e}^{\rm dec/ann}$. Therefore, the final equations, we have used
following equations to study the thermal and ionization history of the
universe

$\displaystyle\frac{dT_{b}}{dz}$ $\displaystyle=$
$\displaystyle\frac{dT_{b}^{\rm sta}}{dz}+\frac{dT_{b}^{\rm
cool}}{dz}+\frac{dT_{b}^{\delta}}{dz}+\frac{dT_{b}^{\rm mag}}{dz}$ (36)
$\displaystyle+$ $\displaystyle\frac{dT_{b}^{\rm BDM}}{dz}+\frac{dT_{b}^{\rm
dec/ann}}{dz},$ $\displaystyle\frac{dT_{d}}{dz}$ $\displaystyle=$
$\displaystyle\frac{2T_{d}}{(1+z)}-\frac{1}{1.5k_{B}n_{H}\,(1+z)H}\frac{d{Q_{d}}}{dt}\,,$
(37) $\displaystyle\frac{d\text{v}}{dz}$ $\displaystyle=$
$\displaystyle\frac{\text{v}}{(1+z)}+\frac{D(\text{v})}{(1+z)H}\,,$ (38)

In equation (36), first and second term on the right hand side represents the
effects of standard Compton scattering and cooling from the bremsstrahlung,
collisional excitation, recombination and collisional ionization cooling. The
third term is solely coming from the density perturbation created by the
magnetic fields. The fourth, fifth and sixth terms in equation (36),
represents the terms related to the magnetic heating, BDM interaction and the
decay/annihilation respectively. The ionization equation under the present
study is given by

$\displaystyle\frac{dx_{e}}{dz}=\frac{dx_{e}^{\rm sta}}{dz}+\frac{dx_{e}^{\rm
dec/ann}}{dz}$ (39)

Here the two terms on the right hand side are given in equations (3) and (35).
In our numerical simulation, we have used equations (36)-(38), along with the
magnetic energy density evolution equation (15) and the ionization fraction
evolution equation (39) to understand the thermal and ionization history of
our universe in IGM. In our work, $B_{0}$, $n_{B}$, $\lambda_{B}$,
$\sigma_{0}$, $m_{\chi}$, $\langle\sigma v\rangle/m_{\chi}$ and $\tau$ are
used as a parameter. In different cases, different parameter combinations have
been used to solve these equations.

##  3 Result and Discussion

In the present work, our primary motivation is to understand the thermal and
ionization history of the IGM between redshift $1100-10$. To avoid the effect
of structure formation, we limit our study upto redshift $z\sim 10$. We have
computed the evolution of temperature, ionization fraction, and $y-$parameter
of the tSZ effect to understand the processes involved. We have included the
following effects in addition to the standard terms related to Thomson
Scattering and the adiabatic cooling:

  * •

cooling effects like bremsstrahlung, collisional excitation, recombination and
collisional ionization cooling,

  * •

magnetic decay from the ambipolar diffusion and turbulent process,

  * •

effects from the interaction of baryons to the DM particles (BDM processes),

  * •

processes involving the decay or annihilation of the DM particles.

In particular, the BDM interaction with interaction cross-section of the form
$\sigma=\sigma_{0}({\rm v}/c)^{n}$ and decay/annihilation channels
$\chi\rightarrow(e^{+}e^{-},~{}\text{or}~{}\gamma\gamma)$,
$\chi\chi\rightarrow(~{}e^{+}e^{-},~{}\text{or}~{}\gamma\gamma)$ are
considered in the present work. Important thing to be noted here is that the
energy deposition efficiency function $f_{c}(z)$ is constant over the redshift
of interest. The assumption is made to simplify our numerical simulations.
Here $B_{0}$, $n_{B}$, $\sigma_{0}$, $n$, $f_{\chi,{\rm int}}$, $f_{\chi,{\rm
ann}}$, $f_{\chi,{\rm dec}}$, $p_{\rm ann}=f_{\rm eff}\langle\sigma
v\rangle/m_{d}$ and $\tau$ are considered as a parameter. For our convenience,
following values are used $B_{0}=0.003$, $1.0$ nG, $n_{B}=-2.9$ and $-1.0$,
$n=-4$ and $\lambda_{B}=0.25,1.0$ Mpc. In the upcoming subsections, we briefly
explain the role of various effects on the temperature of baryons and DM and
on the ionization fraction. In subsection (3.1), the temperature and
ionization fraction of the baryons along with the DM temperature evolution is
presented. In this subsection, we also discuss the role of partial
decay/annihilation on the temperature of the baryons. Subsection (3.2)
contains the impact of various effects on the baryon temperature and hence the
CMB anisotropy during redshift 1100-10. Finally, subsection (3.3) holds a
discussion on the bound on the magnetic fields from the decay/annihilation of
the DM particles in our present context.

###  3.1 Thermal and ionization history

In this subsection, we briefly discuss the evolution of baryon, DM
temperature, and the ionization fraction from redshift $z=1100-10$ in various
scenarios. Below in all figures, black dashed and solid black lines represent
the CMB temperature and baryon temperature in the standard scenario
respectively. Based on the effects considered, this subsection can be
separated into the following cases:

  1. 1.

BDM, $B_{0}$: In this scenario, the effects of the magnetic field and the BDM
interaction are only considered (see figures (1) and (2)). Solid and dashed
(except the black dashed line) lines in this plot represent the evolution of
baryon and dark matter temperature, respectively. In reference Pandey et al.
(2020), authors studied this scenario at a great length. Some of the most
relevant scenarios for the current project have been discussed here. In figure
(1), we fix $f_{\chi,{\rm int}}=1.0$, $\sigma_{0}=3.0\times 10^{-45}$ m2 and
$m_{\chi}=1.0$ GeV and compare the results for the BDM only case to the
results of the case when BDM+$B_{0}$ are present. We now show the role of
$f_{\chi,{\rm int}}$ in determining the baryon and DM temperatures in figure
(2). In figure (1a), the solid red line corresponding to $B_{0}=0.003$ nG and
$n_{B}=-2.9$ (nearly scale-invariant spectrum) and the solid green line (only
BDM) show almost identical behavior except for lower redshift where minor
effects of magnetic fields are seen. However, for $B_{0}=0.003$ nG and
$n_{B}=-1.0$, baryon temperature follows the standard pattern of the baryons
until redshift $z\approx 30$, at which point it turns and later crosses the
CMB temperature. We have observed that the baryon temperature for $B_{0}\geq
1$ nG and $n_{B}=-2.9$ becomes more prominent at a very early stage, i.e., the
redshift at $z\simeq 800$. Even starting with the temperature $T_{d}\approx 0$
K, we can see that the temperature of the DM starts to increase as we move to
a smaller redshift. The BDM interaction caused the energy transfer from the
baryon to the DM. It is important to note that magnetic fields do not directly
affect DM temperature but rather magnetic fields through BDM interactions.
Figure (1b) contains the evolution of the ionization fraction of the neutral
hydrogen for various combinations of parameters. In this case, we have
compared the ionization fraction in the presence of magnetic fields and BDM
interaction to the standard scenario. It is visible from the plot that the
larger is the value of $B_{0}$, the larger is the ionization fraction.
Starting from fully ionized state at $z\approx 1100$, $x_{e}$ decreases
rapidly and attain $x_{e}\approx 10^{-4}$. However, extra energy injected from
the decaying magnetic field leads to a larger value of $x_{e}$ at a lower
redshift. In these plots, we have considered $f_{\chi,{\rm int}}=1$, which
means all DM particles are interacting with the baryons. This nature changes
when we assume a fraction of the total DM particles are interacting with the
baryons. This can be seen in figure (2). In the left panel of this figure,
(i.e. figure (2a)), we set $B_{0}=0.003$ nG, $\lambda_{B}=0.25$ Mpc,
$\sigma_{0}=3\times 10^{-45}$ m2, and DM mass $m_{\chi}=0.1$ GeV. It is
visible that the larger the value of $f_{\chi,{\rm int}}$, the more
significant will be the impact on the baryon temperature. The explanation for
this trend is that many DM particle interactions lead to more energy
extraction from baryons. In the case of BDM+$B_{0}$, we can observe that the
influence is stronger for bigger values of the $f_{\chi,{\rm int}}$ at lower
redshift (2b). At redshift $z=100(\text{and}~{}50)$, baryon temperature drops
from $T_{b}\approx 100(\text{and}~{}40)$ K to $10$ K and $2$ K respectively.
However, at high redshift, no significant effect of $f_{\chi,{\rm int}}$ is
found. A similar trend is noted in the DM temperature.

  2. 2.

BDM, $B_{0}$, Annihilation: In this case, we have chosen three cases: one when
only DM particles are annihilating (no BDM, no $B_{0}$), the second when we
include magnetic field effects in addition to DM annihilation, and finally
when we use BDM interaction+effects of magnetic fields+DM decay/annihilation.
Following annihilation channels are of our interest:
$\chi\chi\rightarrow\gamma\gamma~{}\text{or}~{}e^{+}e^{-}$. The baryon and DM
temperature evolution is explored in two scenarios in figure (3): when just DM
annihilation is considered and when DM annihilation+BDM interaction is
present. In figure (4), we add the effects of the magnetic fields and the DM
annihilation. Figure (5) shows the effects of BDM interaction+decaying
magnetic fields+DM annihilation. In this figure, we show $f_{\chi,{\rm int}}$
dependence of the baryon-DM temperature. In figure (3a), we plot the baryon
and DM temperature for various combinations of the parameter and the CMB
temperature together. In this case, we have considered $f_{\chi,{\rm
int}}=1.0$. Clearly, with the introduction of DM annihilation, additional
energy injection leads to a larger baryon temperature than the standard
scenario. Baryon temperate follows each other for the most part for a
particular annihilation channel. This behavior is also reflected in the plots
of the DM temperature (in figure (3a)) and the ionization fraction (in figure
(3b)). When the effect of the BDM interaction is incorporated, things start to
change. All other lines show a small effect from DM annihilation and BDM+DM
annihilation, with the exception of the annihilation channel
$\chi\chi\rightarrow\gamma\gamma$ with DM mass $m_{\chi}=0.1$ GeV (darker
green solid line). The darker green lines corresponding to BDM+annihilation
channel $\chi\chi\rightarrow\gamma\gamma$ with DM mass $m_{\chi}=0.1$ GeV show
a drastic change in the behavior at low redshift due to the fact that small
mass DM has smaller annihilation parameter $\langle\sigma{\rm
v}\rangle/m_{\chi}$, and hence lower energy injection from the decay of DM
particles (check the figure-5 of reference Liu & Slatyer (2018). However, BDM
interaction, in this case, will extract energy and lower the baryon
temperature further. Things get changed when we include the effect of magnetic
fields in addition to the DM annihilation (see figures (4) and (5)). In the
presence of DM annihilation, the the heating from the magnetic fields leads to
higher baryon temperature and ionization fraction at a given redshift than
when solely DM annihilates. It’s also worth noting that the ionization
fraction increases with the mass of the annihilating DM particles. Figure (5)
shows the dependence of the baryon and DM temperature on $f_{\chi,{\rm int}}$.
In figure (5a), there are two regions, one light green and another blue-shaded
one. The area between solid red ($f_{\chi,{\rm int}}=0.001$) and solid blue
($f_{\chi,{\rm int}}=1.0$) represents the region over which baryon temperature
varies with respect to $f_{\chi,{\rm int}}$ at a given redshift. Similar is
the behavior of the DM temperature, shown in the red-dashed line (for which
$f_{\chi,{\rm int}}=0.001$) and blue-dashed line (for which $f_{\chi,{\rm
int}}=1.0$) (see figure (5a)). In figure (5b), $f_{\chi,{\rm int}}$ dependence
of the baryon and DM temperate are shown. Clearly, there is no visible role of
$f_{\chi,{\rm int}}$ on the baryon or DM temperature at redshifts of $z=1000$
or even for $z=500$. At lower redshifts, however, the parameter $f_{\chi,{\rm
int}}$ has a stronger influence on both baryon and DM temperature. Baryon
temperature $T_{b}(z=100)\approx 150$ K and $T_{b}(z=50)=45$ K at
$f_{\chi,{\rm int}}=0.001$, drops to $T_{b}(z=100)=50$ K and $T_{b}(z=50)=3.9$
K for $f_{\chi,{\rm int}}=1.0$. Contour plots for the baryon temperature in
$\sigma_{0}-f_{\chi,{\rm int}}$ at redshift $z=20$ plane and $z-f_{\chi,{\rm
int}}$ plane in figures (6a) and (6b) respectively. Even at smaller values of
$f_{\chi,{\rm int}}$ but for a large values of $\sigma_{0}$, baryons
temperature at redshift $z\approx 20$, have larger values compared to the case
when $\sigma_{0}$ is small but large values of $f_{\chi,{\rm int}}$ (see the
figure (6a)). The in-set figure, in figure (6a) represent the contour plot of
$T_{b}$ for a smaller range of $\sigma_{0}$ and $f_{\chi,{\rm int}}$. Clearly
when BDM interaction and magnetic fields are present, larger effects happens
on baryon temperature in presence DM annihilation at small values of
$f_{\chi,{\rm int}}$ and larger values of $\sigma_{0}$. For a DM of mass
$m_{\chi}=0.1$ GeV, CMB bound on the BDM interaction cross-section
$\sigma_{0}\approx 3.0\times 10^{-45}~{}\text{m}^{2}$ gives a baryon
temperature $T_{b}\approx 3.0$ K at $f_{\chi,{\rm int}}\approx 0.1$ and
$T_{b}\approx 0.5$ K at $f_{\chi,{\rm int}}\approx 0.5$. It is also evident
that the at lower redshift, evolution of $T_{b}$ is highly dependent on the
values of $f_{\chi,{\rm int}}$ (see figure (6b)).

  3. 3.

BDM, $B_{0}$, Decay: In the case of DM decay, we have considered following
decay channels: $\chi\rightarrow e^{-}e^{+}~{}\text{or}~{}\gamma\gamma$. Again
we have subdivided this scenario: (i). Only DM decay, (ii). Decay+$B_{0}$,
(iii). Decay+BDM and (iv). Decay+$B_{0}$+BDM. In figures (7)-(10), baryon and
DM temperature along with the ionization fractions are shown in the above
mentioned scenarios. Figure (11) shows the contour plot of the baryon
temperature in $z-f_{\chi,{\rm int}}$ plain. The effects of DM decay on baryon
temperature (and ionization fraction) are compared in figures (7a) (and (7b))
respectively to scenarios where only BDM interaction is present and to
standard baryon temperature (and ionization fraction) evolution. Figure (8)
shows the evolution of baryon temperature and the ionization fraction when DM
decay and the magnetic fields are present. In figure (7a), it is clear that
the DM decay injects additional energy, which in turn enhances the baryon
temperature. Up to $z\approx 800$, the temperature of baryons follows that of
the CMB but subsequently exceeds it at lower redshifts. It is also worth
noting that, regardless of the decay channels, the temperature of baryons
remains higher than that of the CMB for redshift $z<800$ and even can reach
$4000$ K at redshift 10. Figure (7b) shows that the ionization fraction
declines at first, then rises to a nearly fully ionized state, starting from a
fully ionized state at $z\approx 1100$. When the effects of magnetic fields
are turned on, again, the baryon temperature remains higher than the CMB
temperature for the majority of the time for selected magnetic field settings
(see figure (8a)). The red dashed lines show the baryon temperature evolution
when only magnetic fields are considered (see figure (8a)). Except for $z<17$,
when we turn on the DM decay together with $B0$, the influence of magnetic
fields in the absence of DM decay (red dashed line) dominates most of the
time.

Similarly, in figure (8b), the ionization fraction is always higher in case of
decay of the DM particles. It is to be noted here that the universe started
with a fully ionized state at redshift $z=1100$, and during the dark ages, it
degrees drastically. However, due to the additional energy injection from the
decay of the DM particles, it begins to reionize at lower redshift $z$ and
becomes fully ionized at redshift $z\leq 10$. Because the effect of structure
formation has not yet been factored into the present work, it’s interesting to
examine the behavior of baryon temperature and ionization fraction in the IGM
when DM particles are decaying or annihilating. This has been left for our
future work. We have compared the baryon temperature and ionization fraction
in the case of DM decay with the case of BDM+DM decay in figure (9). In figure
(9a), the solid and dashed lines indicate the evolution of the baryon
temperature and DM temperature with respect to redshift. As discussed above,
the BDM interaction extracts heat, and hence in the absence of DM decay,
baryon temperature remains lower than $T_{b,{\rm std}}$ (solid blue line).
Except in the redshift region $20<z<800$, baryon temperature remains the same
in the presence of BDM interaction (solid sky-blue line) and the absence of
BDM interaction (solid red line) for decay channel $\chi\rightarrow
e^{-}e^{+}$. In this case, BDM interaction extracts the heat and lowers the
baryon temperature. Because both the solid red and solid sky-blue lines merge
at redshift $z<20$, it suggests that BDM interaction has little effect at
lower redshifts for the chosen set of parameters. As can be observed in plot
(9a), these characteristics are the same for all other channels. The evolution
of the ionization fraction for the given set of parameters is depicted in
figure (9b) in two cases: when the sole influence of DM decay is evaluated and
when both DM decay and BDM interaction are considered. The discussion from
plot (8b) applies to this plot as well. In figure (10), dependence of baryon
temperature on $f_{\chi,{\rm int}}$ is compared from BDM$+B_{0}+$Decay to the
standard scenario. Baryon temperature and CMB temperature exhibit a crossover
for two channels $\chi\rightarrow e^{-}e^{+}$ and $\gamma\gamma$ at redshifts
$z\approx 80$ and $250$, respectively, in the left panel of this figure (see
figure (10a). Here we have fixed the value of $f_{\chi,{\rm int}}$. The
dependence of baryon temperature on $f_{\chi,{\rm int}}$ at a given redshift
is shown in figure (10b). Clearly baryon and DM temperature have similar
dependence on $f_{\chi,{\rm int}}$. At a large redshift, irrespective of decay
channels, there is no visible dependence of the baryon and DM temperature on
the $f_{\chi,{\rm int}}$ similar to the annihilation case. As we go from large
values of $f_{\chi,{\rm int}}$ to smaller values, baryon and DM temperature
decreases. However, in the case of annihilation, we have seen that the
behavior is the opposite. The temperature of both baryons and DM rises at
first, then settles at a particular value at lower values of $f_{\chi,{\rm
int}}$ (see figure (5a)). A contour plot of baryon temperature in a redshift
$z-f_{\chi,{\rm int}}$ plane is depicted in figure (11) for a particular set
of parameters (the values of parameters are given in caption). Several color
zones represent the temperature of baryons. The temperature of baryons is more
dependent at lower redshifts and larger values of $f_{\chi,{\rm int}}$. The
baryon temperature has the least influence on $f_{\chi,{\rm int}}$ in the
redshift range $\sim 10-100$. This range is exciting when considering the
effects of structure formation or DM haloes.

###  3.2 Thermal SZ effect induced by the decay/annihilation of DM and the
y-parameter

In this section, we discuss the tSZ effect for various possible scenarios. The
$y-$ parameter is a quantity that gives information on the distortion in the
CMB spectrum. We compare the $y-$ parameter for the standard scenario with the
case when we include the effects of magnetic fields, BDM interactions, and
decaying magnetic fields via ambipolar diffusion and turbulent decay. We also
include the effect of background density perturbations from the magnetic
fields and the cooling via Bremsstrahlung, collisional excitation,
recombination, and collisional ionization cooling. The $y-$parameter on a sky
plane in direction $\hat{n}$ is defined as

$y(\hat{n})=\int
dz\frac{k_{B}\sigma_{T}}{m_{e}c}\frac{\left[x_{e}n_{H}(T_{b}-T_{\rm
CMB})\right]|_{\hat{n},z}}{(1+z)H(z)}.$ (40)

Since we are mainly interested in studying the background evolution, we have
only the average quantity in the sky plane. From the equation (40), we can see
that the $y-$parameter not only depends on ionization fraction and background
baryon number density, but it depends on the difference of baryon temperature
with the CMB temperature at a given redshift. In the previous subsection of
the result and discussion section, we have seen that the decay/annihilation
could have significantly affected at low redshift. What is important here is
that the behavior of the value of $y-$ parameter depends on the difference
between baryon temperature and the CMB temperature at a given redshift in a
particular direction of the sky plane. Therefore, given the sign of this
difference, the $y-$parameter changes its sign from positive to negative and
vice versa. In figure (12), we have shown the evolution of absolute values of
$y-$parameter with respect to $z$. In this plot, we have given a color code
for each case.

Figure 12: Baryon and Dark matter temperature in presence of baryon dark
matter interaction and magnetic fields but no decay/annihilation processes.
Here we assume one to one interaction is between the baryons and dark matter
particles (i.e. $f_{\chi_{,\rm in}}=1$.). Also coherence scale of the magnetic
fields are $\lambda_{B}=0.25$ Mpc. Color of the lines are explained in the
legends. This plot is similar to the result obtained in reference.

In this plot, red-colored lines denote the case when we have only considered
annihilation. The blue color lines represent the annihilation of the DM
particles in the presence of the background magnetic field. The decay case is
shown in light blue (only decay) and green colors (with decay and background
magnetic fields). When background magnetic fields are present, we have chosen
$B_{0}=1.0$ nG and $n_{B}=-2.9$ at a magnetic coherence length scale
$\lambda_{B}=1.0$ Mpc. In the plots shown in figure (12), on y-axis, we have
considered absolute value of the $y-$parameter and on x-axis it is redshift
$z$. We considered the absolute value of $y-$parameter since we chose the log-
log scale, and any negative parameter value can not give a real value. So
absolute value of the $y-$parameter is the best choice here. In the above
plot, any kink in the plots represents the redshift at which the sign of the
difference $T_{b}-T_{\rm CMB}$ changes the sign. It is clear from the plot
that decay of the dark matter can contribute to a large distortion of the CMB
compared to the annihilation at low redshift. Inverse Compton scattering by
high-energy electrons in galaxy clusters causes CMB-spectrum distortion, in
which low-energy CMB photons acquire an average energy increase during the
collision with high-energy cluster electrons. The redshift of the kink in case
of decay depends on the chosen parameters. In references (Minoda et al., 2017;
Chluba et al., 2015; Tashiro & Sugiyama, 2011), authors have shown the impact
of the magnetic fields on the absolute values of mean $y-$parameter. In this
case the value of the parameter at $z\approx 10$ redshift for a quasi-scale-
invariant PMF heating, leads to a maximum value of $y-$parameter: $|y|\sim
1.1\times 10^{-7}$ ($95\%$ confidence level). However in the case of BDM
interaction$+B_{0}$ can contribute no more than $|y|\leq 10^{-9}$ (Pandey et
al., 2020). Although measurements using a PIXIE-like experiment (Kogut et al.,
2011) may achieve this sensitivity, the reionization and structure building
processes alone cause a far higher distortion ($|y|\sim 10^{-7}-10^{-6}$) (Hu
et al., 1994; Refregier et al., 2000). In a magneticum Pathfinder simulations
(Komatsu et al., 2011), state of the art cosmological hydrodynamic simulations
of large cosmological volume of $(896~{}\text{Mpc}~{}h^{-1})^{3}$ predicts the
mean fluctuating Compton $y-$parameter of $1.18\times 10^{-6}$ which is well
within the values obtained from the PLANCK data, i.e., $|y|\leq 2.2\times
10^{-6}$ (Khatri & Sunyaev, 2015; Dolag et al., 2016). Other best fitted value
from the COBE/FIRAS measurement is $|y|<5.5\times 10^{-5}$ Fixsen et al.
(1996). Thus, regardless of the decay/annihilation channels of the DM
particles, the maximum value of the mean $y-$parameter in the current scenario
is well within the observational limits set by experimental data such as
PLANCK, FIRAS, PIXIE, and others.

###  3.3 Dark photons and the magnetic fields

In this subsection, we determine the strength of magnetic field strength if we
assume that these fields are originated from the decay/annihilation of the DM
particles. In equation (36), in the absence of any background magnetic field,
the second and third terms will not contribute to the thermal and ionization
history of IGM. However, there may be a possible scenario when the
decay/annihilation of the dark matter particles may contribute to the ordinary
photons. In this case, if we compare the third and last term in equation (36),
for the decay of DM particles,

$B_{0}(z)^{2}=\frac{f_{\rm eff}}{1+f_{\rm
He}+x_{e}(z)}(1+z)^{3}\frac{f_{\chi,{\rm dec}}}{f_{\chi,{\rm
in}}}\left(\frac{8\pi\Omega_{\rm DM}c^{2}\rho_{c}{\rm
ln}(2)}{3mH_{0}(z)}\right)\frac{1}{\tau}$ (41)

Similarly in the case of annihilation, we can obtain the formula

(42)

Therefore, we can see that the value of $B_{0}$ has a redshift depending on
$z$, and it can be easily evaluated if we know the ionization fraction of the
hydrogen atoms and the decay or annihilation channels.

Figure 13: Strength of the magnetic fields: Here we have assumed that there
are no background magnetic fields and that $f_{\rm eff}=1.0$, $f_{\chi,{\rm
ann}}=1.0$, $f_{\chi,{\rm dec}}=1.0$, $f_{\chi,{\rm in}}=1.0$, $n_{B}=-2.9$.

In the figure (13), we have shown the variation of $B_{0}$ as a function of
$z$ using equations (41) and (42). For simplicity, we choose $f_{\rm
eff}=1.0$, $f_{\chi,{\rm ann}}=1.0$, $f_{\chi,{\rm dec}}=1.0$, $f_{\chi,{\rm
in}}=1.0$, $n_{B}=-2.9$. At redshift $z\approx 1000$, in case of annihilation,
magnetic fields of strength is $B_{0}\approx 10^{-10}\mu G$ at a length scale
$\lambda_{B}\sim 10$ Kpc, which is equivalent to a present day comoving length
scale of $\sim$ Mpc. These magnetic fields at redshift of 10, are equivalent
to the field of the order of $\approx 10^{-15}$ G. Similarly for the decay
channels, any $B_{0}\approx 10^{-11}$ G at recombination (equivalent to
$\approx 10^{-16}$ G) at $z\approx 10$ at a length scale of $\sim 10$ Kpc
(comoving length $\sim$ Mpc) can be obtained. Therefore, above calculated
bound on the strength of the magnetic fields in the context of the DM
decay/annihilation through various possible channels, could also affect the
thermal and ionization history of IGM even in absence of any background
magnetic fields. To understand the decay/annihilation channels and hence
origin of the magnetic fields, we have briefly summarizes processes involves
below.

As we have discussed above that the dark photons and the ordinary photons mix
kinetically, the direct product of the groups corresponding to the two
photons, i.e., $U(1)^{\prime}\times U(1)$, eventually broken down to
$U(1)_{\rm em}$ Berezhiani et al. (1996); Berezhiani & Mohapatra (1995). In
this scenario, conversion of the dark photons to ordinary photons could
explain the observed magnetic fields at a large scale Berezhiani et al.
(2013). Before the recombination DM particles are strongly coupled to
electrons and through them to photons. Therefore DM particles cannot
participate in the structure formation. After the recombination epoch, when
the ionization fraction drops below $10^{-4}$, the collision time becomes
larger than the cosmological time. At this epoch, DM particles decouple from
the baryon-photon fluid and do not follow the baryon bulk flow. At this epoch,
the velocity of baryons and the DM particles are somewhat different. The drag
force exerted by the DM particles on the electrons is given by
$\vec{F}=e\Delta\vec{v}B_{F}$, where $\Delta\vec{v}$ is the relative velocity
and $B_{F}=\sigma_{e\chi}n_{\chi}\omega_{\chi}/e$. In the expression of the
force $\vec{F}$, $n_{\chi}$ and $\omega_{\chi}$ are the number density and the
mean energy density of the DM particles. The DM interaction with the protons
is damped by the factor $(m_{p}/m_{e})^{2}$. Therefore for a larger pressure
on the electrons from the DM than CMB photons, cross-section
$\sigma_{e\gamma}$ must be larger than $\sigma_{e\chi}$ at low momentum
transfer. For the case when $m_{\chi}\approx 10~{}\text{keV}<m_{e}$, it has
been shown that maximum values of $B_{0}$ could be of the order of $\sim
10^{-13}$ G in a galactic halos of size $10$ kpc Berezhiani et al. (2013).
However, for the case when $m_{\chi}>m_{e}$, the channel ($\chi\chi\rightarrow
e^{+}e^{-}$) could give a magnetic field of strength $\sim 10^{-15}$ G for
$m_{\chi}\simeq 1$ MeV and $\sim 10^{-12}$ G for $m_{\chi}\simeq 1$ GeV at
$\lambda_{B}\sim$ kpc scale. These channels at $\sim 1$ Mpc scale could also
generate a magnetic field of the order of $\sim 10^{-14}$ G.

##  4 Conclusion

The IGM’s thermal and ionization evolution is determined by the processes
involved in influencing the dynamics of the baryons and the DM particles in
IGM. Here we have included the following processes: heating from the magnetic
fields, energy injection from DM decay/annihilation in the presence of non-
standard interaction of baryon and the DM particles. To study the thermal and
ionization history of the IGM, we have solved coupled equations of the baryon
temperature $T_{b}$, DM temperature $T_{d}$, ionization fraction $x_{e}$ along
with the relative velocity ${\rm v}$ and the magnetic energy evolution
equations. We have also evolved the absolute value of the mean $y-$parameter
of the thermal SZ effect for redshift $z$ to study the CMB anisotropy. The
relative strength and nature of the magnetic field, on the one hand, and the
BDM interaction cross-section, on the other, determine the nature of the IGM’s
thermal and ionization history when BDM interaction and magnetic fields are
present. However, in the case when only DM decay or annihilation processes are
considered, thermal and ionization history is solely determined by the
lifetime of the DM particle or annihilation parameter $f_{\rm
eff}\langle\sigma v\rangle/m_{\chi}$. It’s worth noting that the energy
deposited by DM decay or annihilation is considered to be instantaneous.
However, this may not be the case, and the energy deposition’s dependency on
$z$ must be carefully studied. We also assume that only a fraction of the
total DM particles interact with the baryons. The interaction is so that DM
particles interact only with the charged baryons and have a non-zero small
electric charge. We show that these processes significantly impact the baryon
and DM temperature and the ionization fraction. This, in turn, leaves imprints
through the tSZ effect on the CMB distortion at small scales. We show that the
maximum value obtained for the absolute value of the mean $y-$parameter is
well within the values obtained from the observational data of PLANCK, FIRAS,
and PIXIE. We also calculate the magnetic field’s strength, which is generated
by the dark photon, a dark sector gauge field that is identical to the
ordinary photon but is associated with the dark sector. We show that at a
comoving scale of $\sim$ Mpc, the strength of the magnetic fields is of the
order of $10^{-15}~{}\mu$G for annihilation channels and $10^{-16}~{}\mu$G for
decay channels. In the current context, it is worthwhile to investigate the
role of several other probable channels of DM particle decay/annihilation on
the temperature and ionization history of the IGM. Especially at low redshift
during the structure formation, the decay/annihilation processes could impact
the gravitational collapse.

## Acknowledgments

A.K.P. acknowledges the facilities at I.C.A.R.D., Department of Physics and
Astrophysics, University of Delhi, Delhi, India. A.K.P. is financially
supported by Dr. D.S. Kothari Post-Doctoral Fellowship provided by U.G.C.,
Govt. of India, under the Grant No. D.S.K.P.D.F. Ref. No. F.$4-2/2006$
(BSR)/PH/$18-19/0070$. Prof. T. R. Seshadri was also instrumental in
motivating the author and providing valuable feedback and suggestions during
the early stages of the project.

## Data Availability Statement

This manuscript has no associated external data [Authors’ comment: This is a
theoretical study. We have not used any experimental data]. The datasets
generated during and/or analysed during the current study are available from
the corresponding author on reasonable request.

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