Constraint on Primordial Magnetic Fields In the Light of ARCADE 2 and EDGES Observations
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Constraint on Primordial Magnetic Fields In the Light of ARCADE 2 and EDGES Observations
1, 2, ∗
Pravin Kumar Natwariya
1
Physical Research Laboratory, Theoretical Physics Division, Ahmedabad 380 009, India
2
Department of Physics, Indian Institute of Technology, Gandhinagar 382 424, India
(Dated: August 4, 2020)
Abstract
We study the constraints on primordial magnetic fields (PMFs) in the light of Experiment to Detect the Global
Epoch of Reionization Signature (EDGES) low-band observation and Absolute Radiometer for Cosmology,
Astrophysics and Diffuse Emission (ARCADE 2). In the presence of PMFs, 21 cm differential brightness tem-
perature can modify due to the heating of the gas by decaying magnetic fields. ARCADE 2 observation detected
excess radio radiation in the frequency range 3-90 GHz. Using the ARCADE 2 and EDGES observations, we
arXiv:2007.09938v2 [astro-ph.CO] 3 Aug 2020
find the upper constraint, at the length scale of 1 Mpc, on the primordial magnetic field B1 Mpc . 53.3 pG for the
nearly scale-invariant PMFs using 10% of observed excess radio radiation. However, taking into account the
heating effects due to x-ray and VDKZ18 (Venumadhav et al. 2018), the upper constraint on the strength of the
primordial magnetic fields can further be lowered to B1 Mpc . 37 pG.
Keywords: EDGES observation, ARCADE 2 observation, Magnetohydrodynamics, 21 cm signal, cosmic background radia-
tion, first stars
I. INTRODUCTION blackbody spectrum at large wavelength [6, 7]. This radio
radiation is larger than the observed radio count [8]. In the
The 21 cm signal, due to the hyperfine transition between Ref. [7], authors show an absorption signal having a large
1S singlet and triplet states of the neutral hydrogen atom, is amplitude of ∼ −1.1 K only using 10% of the observed excess
a treasure trove to provide an insight into the period when radiation by ARCADE 2. As dark-matter annihilation can in-
the galaxies and first stars formed. Recently, the EDGES col- crease the gas temperature, it can erase the 21 cm absorption
laboration observed an absorption signal in the redshift range signal. Still, dark-matter annihilation can be considered in the
15 . z . 20. It is nearly two times more than the theoretical presence of possible radio radiation excess [9]. To explain this
prediction based on the ΛCDM framework cosmological sce- observed excess radiation, several attempts have been made in
narios [1, 2]. During the cosmic dawn, in the standard cosmo- the literature for various cosmological scenarios. Authors of
logical scenario, the temperature of the gas (T gas ) and cosmic the Ref. [10], calculate stimulated emission of Bose (axion)
microwave background radiation (CMBR), T CMB , varies adi- stars and argue that it can give a large contribution to the radio
abatically. T gas and T CMB varies with the redshift as ∝ (1 + z)2 background, and it can also possibly explain EDGES and AR-
and ∝ (1 + z) respectively, and temperatures of both the gas CADE 2 observations. In the redshift range z ≈ 30 to 10, ac-
and CMBR found to be ∼ 6.7 K and ∼ 49.1 K at the redshift cretion onto the first intermediate-mass Black Holes can also
z = 17 respectively (for example see the Ref. [3–5]). EDGES produce a radio radiation [11]. Radio background around the
observation reported that the best fitting 21 cm model yields Cosmic down can be produced in other cosmological scenar-
an absorption profile centered at 78±1 MHz and in symmetric ios such as active galactic nuclei [12], by considering popu-
“U” shaped form having an amplitude of −0.5+0.2 −0.5 K with 99%
lation III stars [13], and dark matter annihilation [14–16] (for
confidence intervals [1]. the detailed review see the Refs. [8, 9, 17–25]).
Lack of ability of the standard theoretical scenarios to ex- Presence of decaying magnetohydrodynamics (MHD) can
plain the 21 cm signal reported by EDGES collaboration sug- heat the gas above the 6.7 K at z = 17 and even it can erase
gests a likelihood of new physics. To explain the EDGES ob- the absorption signal [5, 26, 27]. Still, the global 21 cm sig-
servation, for the best fitting amplitude at the centre of the “U” nal reported by EDGES collaboration can be explained by
profile, either the cosmic background radiation temperature considering the possible early excess of radio radiation [7].
T R & 104 K for the standard T gas evolution or T gas . 3.2 K In the present work, we consider decaying magnetohydrody-
in the absence of any non-standard evolution of the T R , i.e. namics and constraint the present-day strength of primordial
T R = T CMB [1]. In the standard scenarios, background ra- magnetic fields (PMFs). Observations suggest that the mag-
diation is assumed to be the cosmic microwave background netic fields (MFs) are present on the length scale of galax-
(CMB). Recently, the Absolute Radiometer for Cosmology, ies to the clusters. Recently in the Ref. [28], authors show
Astrophysics and Diffuse Emission (ARCADE 2) collabora- that PMFs can be used as a remedy to resolve the tension be-
tion, a double-nulled balloon-borne instrument with seven ra- tween different observations. The present-day amplitude of
diometers, detected the excess radio radiation. It agrees with these MFs is constrained from the big bang nucleosynthesis,
CMBR at low wavelength but significantly deviates from a formation of structures and cosmic microwave background
anisotropies and polarization [27, 29, 30]. Strength of these
MFs can be a few µG in the intergalactic medium [31, 32].
Authors of the Ref. [26], put a upper constraint on PMFs
∗ pravin@prl.res.in strength B1Mpc . 10−10 G at the length-scale of 1 Mpc by2
considering T gas . T CMB so that, PMFs can not erase the ab- II. 21 CM DIFFERENTIAL BRIGHTNESS
sorption signal in the redshift range 15 . z . 20. Planck TEMPERATURE
2015 results put upper constraints on PMFs of the order of
the ∼ 10−9 G for different cosmological scenarios [33]. The After the recombination, the baryon number density mostly
authors of the Ref. [34], in the context of EDGES observa- dominated by the neutral hydrogen (NHI ) and some fraction
tion, put an upper and lower constraint on the PMFs to be of residual free electrons (Xe = Ne /NH ) and protons (X p =
6 × 10−3 nG and 5 × 10−4 nG respectively. Also, the lower N p /NH ). Here, Ne , N p and NH are number density of free
bound on the present-day strength of PMFs found in Refs. electrons, protons and hydrogen nuclei respectively. The hy-
[35–37]. Subsequently, in the Ref. [31], authors put a lower perfine interaction in neutral hydrogen atom splits it’s ground
bound on the strength of intergalactic magnetic fields of the state into 1S triplet (n1 ) and singlet (n0 ) hyperfine levels. The
order of 3 × 10−16 G using Fermi observations of TeV blazars. Relative number density of hydrogen atom in triplet (n1 ) and
Upper constraint on the PMFs at the end of big bang nucle- singlet (n0 ) state is characterized by spin temperature (T S ),
osynthesis found to be 2 × 109 G [38]. Presence of strong
PMFs can modify the present-day relic abundance of He4 and n1 g1
= × exp (−2πν10 /T S ) , (1)
other light elements. Therefore, Using the current observa- n0 g0
tion of light element abundances, present-day MFs can be here, g1 and g0 are statistical degeneracy of triplet and sin-
constrained [27, 39–41]. Generation of the magnetic fields in glet states respectively and ν10 = 1420 MHz = 1/(21 cm)
the early Universe for the various cosmological scenarios has is corresponding frequency for hyperfine transition. The spin
been studied in the earlier literature (for example see Refs. temperature depends on collisions between hydrogen atoms,
[35, 42–45]). It is to be noted that decaying MHD has been absorption/emission of background radiation and Ly-α radia-
studied in several literatures. In these works, the authors con- tion emitted from the first stars. Therefore, the spin tempera-
sider the decay of the PMFs by ambipolar diffusion and tur- ture can be defined by requiring equilibrium balance between
bulent decay [5, 26, 27, 46]. Ambipolar diffusion of mag- the populations of triplet and singlet state [2, 51–53],
netic fields is important in neutral medium as it is inversely
proportional to free-electron fraction (Xe ) and Xe ∼ 10−4 af- T R−1 + xα T α−1 + xc T gas
−1
ter redshift z . 100 [5, 27, 47]. Magnetic energy dissipation T S−1 = . (2)
1 + xα + xc
into gas, due to ambipolar diffusion, happens because of rela-
tive velocity between ion and neutral components of gas [48]. Here, T gas is the kinetic temperature of the gas and T R =
After the recombination (z ∼ 1100), the radiative viscosity T r (ν) (1 + z) is the background radiation temperature reported
of fluid dramatically decreases, and velocity perturbations are by ARCADE 2 observation [6, 7, 9, 22],
no longer damped, and tangled magnetic fields having length !β
scale smaller than the magnetic Jeans length can dissipate via ν
T r (ν) = T 0 + ξ T c , (3)
another mode–turbulent decay [5, 27, 49]. Magnetic heating ν0
of the gas due to the turbulent decay decreases with redshift
but later when ionization fraction decreases, heating increases where, the T 0 = 2.729 ± 0.004 K, T c = 1.19 ± 0.14 K
due to ambipolar diffusion [5, 27]. In the present work, we with a reference frequency ν0 = 1 GHz, spectral index
argue that the presence of early excess of radio radiation can β = −2.62 ± 0.04 and ξ represents the excess fraction of
be used as a probe for present-day PMFs strength in the Uni- radiation. Authors of the Ref. [6], sets ξ to be unity. For
verse, in the light of the absorption signal reported by EDGES the 21-cm differential brightness temperature ν is taken to be
collaboration. 1420/(1 + z) MHz. T α ≈ T gas is the colour temperature due
to Lyα radiation from the first star [51, 54]. xc and xα are
collisional and Wouthuysen-Field (WF) coupling coefficients
respectively [51, 54–56],
T 10 C10 T 10 P01
xc = , xα = , (4)
T R A10 T R A10
The present work is divided into the following sections: In here, T 10 = 2 π ν10 = 5.9 × 10−6 eV and C10 = Ni k10 iH
is col-
section (II), we discuss the 21 cm signal due to the hyperfine lision deexcitation rate. i stands for hydrogen atom, electron
iH
transition between triplet and singlet state of the neutral hy- and proton. k10 is the spin deexcitation specific rate coeffi-
drogen atom. We also discuss 21 cm differential brightness cient, due to collisions of species i with hydrogen atom [2].
temperature due to the deviation of spin temperature from the P01 = 4Pα /27 and Pα is scattering rate of Lyα radiation [2].
background radiation temperature. In section (III), the evolu- A10 = 2.86 × 10−15 sec−1 is the Einstein coefficient for spon-
tion of the gas temperature and ionization fraction in the pres- taneous emission from triplet to singlet state.
ence of decaying PMFs is discussed. Next, in section (IV), The 21 cm differential brightness temperature is given by
we consider the effects on the IGM temperature due to first [1, 52, 57],
stars. In section (V), we discuss our results and obtain upper #1/2
0.15 1 + z Ωb h2
" ! !
TR
constraint on the present day strength of PMFs in the absence T 21 ≈ 23xHI 1− mK , (5)
and presence of x-ray and VDKZ18 heating [50]. Ωm h2 10 0.02 TS3
here, xHI = NHI /NH is the neutral hydrogen fraction. For this here, ργ = ar T CMB
4
, ar = 7.57×10−16 J m−3 K−4 is the radiation
work, we consider the following values for the cosmological density constant, σT is the Thomson scattering cross-section
parameters: Ωm = 0.31, Ωb = 0.048, h = 0.68, σ8 = 0.82 and me is the mass of electron. Change in the electron fraction
and n s = 0.97 [58]. As T 21 ∝ (T S − T R ), there can be three with redshift [3, 4, 46, 62],
scenarios. If T S = T R then T 21 = 0 and there will not be
4 RLyα + 4 Λ2s,1s
3 1
any signal. For the case when T S > T R , emission spectra can dXe 1
=
be observed, and when T S < T R , it leaves an imprint of ab- dz H(1 + z) βB + 34 RLyα + 41 Λ2s,1s
sorption spectra. 21 cm signal evolution can be described as:
× NH Xe2 αB − 4(1 − Xe ) βB e−E21 /TCMB ,
after recombination (z ∼ 1100) to z ∼ 200, gas and cosmic
background radiation shares same temperature and maintain (8)
thermal equilibrium due to the Compton scattering. There- here, αB is the case-B recombination coefficient and βB is the
fore, T 21 = 0 and signal is not observed. After z ∼ 200 until photo-ionization rate. E21 = 2π/λLyα , λLyα = 121.5682 ×
z ∼ 40, gas decouples from background radiation and temper- 10−9 meter is the hydrogen Lyα rest wavelength [3]. Λ2s,1s =
ature falls as T gas ∝ (1+z)2 . It implies early absorption spectra 8.22 sec−1 is the two photon decay rate of hydrogen and
of 21 cm signal. Nevertheless, this signal is not observed due RLyα = 3N (1−X
8πH
)λ3
is the Lyα photon escape rate [62]. Heat-
H e
to the poor sensitivity of radio antennas. The sensitivity falls Lyα
ing rate per unit volume due to the ambipolar diffusion (Γambi )
dramatically below 50 MHz. After z ∼ 40 to the formation of
and turbulence decay (Γturb ) is given by [5, 27],
the first star, number density and temperature of gas are very
small, hence, xc → 0. Therefore, there is no signal [52, 59]. (1 − Xe ) E 2B fL (nB + 3)
After the first star formation, gas couples to the Lyα radiation Γambi ≈ , (9)
γ Xe (MH Nb )2 Ld2
emitted from the first star by Wouthuysen-Field (WF) effect
[54, 60]. Therefore, xα 1, xc and absorption spectra can 1.5 m [ln(1 + ti /td )]m
Γturb = H EB ,
be seen. After z ∼ 15, x-ray emitted from active galactic nu- [ln(1 + ti /td ) + 1.5 ln{(1 + zi )/(1 + z)}]m+1
clei (AGN) starts to heat the gas and emission spectra can be (10)
seen [52]. EDGES collaboration observed an absorption sig-
nal centered at 78 ± 1 MHz corresponds to redshift z = 17.2 here, E B = B2 /(8π), B = B0 (1+z)2 , B0 is the present day mag-
and reported T 21 = −0.5+0.2 netic field strength, m = 2(nB +3)/(nB +5), fL (x) = 0.8313 (1−
−0.5 K [1].
1.020 × 10−2 x) x1.105 , γ = 1.9 × 1014 (T gas /K)0.375 cm3 /g/s
the coupling coefficient, zi = 1088 is the redshift when heat-
ing starts due the magnetic fields, MH is the mass of Hydro-
III. EVOLUTION OF THE GAS TEMPERATURE IN THE gen atom, Nb is the number density of baryons and ti /td ≈
PRESENCE OF PMFS
14.8 (nG/B0 ) (Mpc−1 /kd ). Coherence length scale of the mag-
netic field, Ld = 1/[kd (1 + z)], is constrained by damping
In the presence of decaying magnetohydrodynamics effects, length scale of Alfvén wave. MFs having length scale smaller
the gas temperature can increase. It can even increase above than Ld , are strongly damped by the radiative-viscosity and it
the background radiation and can erase the 21 cm absorption is given by [5, 27, 49, 63, 64],
signal [5, 26, 27, 49]. Therefore, present-day PMFs strength !
nG
can be constrained by the EDGES observed 21 cm signal in kd ' 286.91 Mpc−1 . (11)
the presence of excess radiation reported by ARCADE 2 [1, B0
6, 7, 22]. In the presence of turbulent decay and ambipolar
diffusion, thermal evolution of the gas with redshift can be
IV. HEATING OF THE IGM DUE TO BACKGROUND
written as [5, 27, 48, 49, 61],
RADIATION
dT gas T gas Γc
=2 + (T gas − T CMB ) After the first star formation (z ∼ 30), their background ra-
dz 1 + z (1 + z) H
2 diation starts to heat the intergalactic medium (IGM) [53, 56,
− (Γturb + Γambi ) , (6) 65–69]. Authors of the Ref. [50], suggests that the kinetic
3 Ntot (1 + z) H temperature of the gas can increase due the background radi-
ation even in the absence of x-ray heating. The Lyα photons
Here, Ntot = NH (1+ fHe +Xe ), fHe = 0.079 and T CMB = T 0 (1+
due to first stars intermediate the energy transfer between the
z) is the cosmic microwave background (CMB) temperature.
thermal motions of the hydrogen and background radiation.
H ≡ H(z) is the Hubble parameter. At early times, T gas can
Authors claim that this correction to the kinetic temperature
depend on CMB due to Compton scattering. However, it will
of the gas is the order of (∼ 10%) at z = 17, in the absence
not be strongly affected by the comparatively small amount of
of x-ray heating. Following the above reference, the equation
energy in the non-thermal radio radiation. Therefore, T gas and
(6) will modify,
T α can be assumed independent of the excess radiation [7]. ΓC
is the Compton scattering rate, defined as, dT gas dT gas dT gas ΓR
= + − ,
dz dz dz (1 + z) (1 + fHe + Xe )
8σT ργ Ne [eq.(6)] x−ray
ΓC = , (7) (12)
3 me Ntot4
B0=7.9×10-2 nG
104 B0=5.6×10-2 nG 104
B0=3.5×10-2 nG
103 103
Tgas (K)
Tgas (K)
102 102
B0=4.3×10-1 nG
B0=3.0×10-1 nG
B0=1.9×10-1 nG
101 101
100 101 102 103 100 101 102 103
z z
(a) (b)
FIG. 1: The gas temperature evolution with redshift for different magnetic field strengths – solid lines. The blue dot-dashed line
indicates the T gas evolution in the absence of PMFs. The shaded region is corresponds to 21-cm absorption signal, 15 ≤ z ≤ 20,
reported by EDGES observation. Plot (1a) and (1b) represent the cases when excess radiations are 10% (ξ = 0.1) and 100%
(ξ = 1) respectively. The black and red dashed horizontal lines are corresponds to the T 21 ' −300 and −1000 mK respectively,
at redshift z = 17.2 considering T S ' T gas .
where, dT gas /dz [eq.(6)] stands for the gas temperature evolu- 100
tion represented in equation (6), and
" #
A10 TR
ΓR = xHI xR − 1 T 10 , (13) 10-1
2H TS
B0 (nG)
here, xR = 1/τ21 × [1 − exp(−τ21 )], and the 21 cm optical
depth τ21 = 8.1 × 10−2 xHI [(1 + z)/20]1.5 (10 K/T S ). And, 10-2
T 10 = 2πν10 = 0.0682 K. To include the x-ray heating of
T21≃-300 mK
the IGM, we consider the tanh parameterization [70–72]. In T21≃-1000 mK
the presence of x-ray radiation, the ionization fraction evolu- 10-3
tion with redshift will also change. For the present case, we 10 20 30 40 50 60 70 80 90 100
ξ—Radiation Excess (%)
consider the fiducial model, for x-ray heating and ionization
fraction evolution, motivated by Ref. [70]. FIG. 2: Black and red solid lines are corresponds to
T 21 ' −300 and −1000 mK at z = 17.2 , respectively. Black
solid line represents the upper constraints on present day
V. RESULT AND DISCUSSION
PMFs strength (B0 ) for different fraction of observed
radiation excess (ξ). Here, we consider T S ' T gas at redshift
To study the gas temperature evolution with redshift in the z = 17.2 .
presence of primordial magnetic field dissipation, we use the
code recfast++1 [3–5]. In this code we include the correction
function for recombination physics suggested by the authors
[73]. In FIG. (1), we plot the evolution of the gas temperature and 6.05 × 101 K. These black and red dashed lines are ob-
with the redshift. In these plots, we do not consider x-ray and tained for T 21 ' −300 and −1000 mK respectively at z = 17.2
VDKZ18 heating, due to the first stars, of the gas. We take in- for ξ = 0.1 . The redshift z = 17.2 is corresponds to the fre-
finite Lyα coupling, i.e. xα → ∞, it implies T S ' T gas . Now, quency ν ' 78 MHz, i.e. the best fit reported by EDGES col-
using equation (5), we can get 21 cm differential brightness laboration. As we increase the magnetic field strength from
temperature for different fraction of radiation excess. In plot B0 = 3.5 × 10−2 nG, the gas temperature rises because the
(1a), we take ξ = 0.1 i.e. 10% radiation excess. The black Γambi ∝ B40 and Γturb ∝ B20 . Hence, increasing the MFs strength
and red dashed horizontal lines are corresponds to 1.87 × 102 the gas temperature rises. The green solid line corresponds to
EDGES best fit for 21 cm signal, i.e. T 21 = −500 mK at
78 MHz. Therefore, using the upper bound on T 21 by EDGES
observation, we get the upper limit on B0 to be 7.9 × 10−2 nG
1 http://www.jb.man.ac.uk/˜jchluba/Science/CosmoRec/ for ξ = 0.1 . In FIG. (1b), the black and red dashed lines are
Recfast++.html corresponds to 1.82 × 103 and 5.90 × 102 K respectively. For5
this case, we consider excess radiation 100%, i.e. ξ = 1 . By 0
increasing ξ we get more window to increase T gas because T R
rises as we grow radiation excess observed by ARCADE 2. -200
Therefore, using the ARCADE 2 observed radiation excess,
and upper constraint on T 21 by EDGES observation, we get -400
T21 (mK)
the constraint on B0 to be: B0 . 4.3 × 10−1 nG by considering B0 = 11 pG
ξ to be unity [6]. We consider the primordial magnetic field -600
B0
B0
B0
= 20.8 pG
= 22.8 pG
= 55.7 pG
power spectrum, PB (k) = AknB for L ≥ Ld and PB (k) = 0 B0
B0
= 78 pG
= 84 pG
B0 = 14 pG
for L < Ld , as a power law in k space. Therefore, the PMFs -800
amplitude smoothed over the length scale λ is [5, 64, 74],
-1000
Z 3
√ nB +3
d k 2 10 15 20 25 30
B2λ = P B,0 (k) exp(−k λ
2 2
) = B20 , (14) z
kd λ
(2π)3
FIG. 3: 21 cm differential brightness temperature with
here, we take A0 = 2π2 2(nB +3)/2 /Γ (nB + 3)/2 (1/kd )nB +3 B20 .
h i
redshift for different strength of primordial magnetic fields.
We consider nearly scale invariant spectral index, nB = −2.9. Black, blue and red solid (dot-dashed) lines represent the
Using equation (14), we can get the amplitude of PMFs case when radio radiation excess is only 10%, 50% and
(B1 Mpc ) at the scale of λ = 1 Mpc. Therefore, for ξ = 1 and 100% respectively. The red doted line corresponds to ξ = 1
0.1 we get the upper bound on B1 Mpc to be 31.6 × 10−2 and and 20% fiducial fraction of Lyα coupling. The Magenta and
53.3 × 10−3 nG, respectively. Using the Planck+WP+highL green dashed horizontal lines represent the upper (-300 mK)
likelihood, Planck collaboration reported the upper bound and lower (-1000 mK) constraints on T 21 by EDGES
on the magnetic field amplitude smoothed over the scale of observation, respectively. Here, we consider the finite Lyα
1 Mpc: B1 Mpc < 3.4 nG with the 95% confidence level for the coupling (xα ).
spectral index, nB < 0 [74]. In FIG. (2), we plot present day
upper bound on the magnetic field strength B0 for the different
excess radiation fraction. As we increase the ξ, T R increases constraints on 21 cm differential brightness temperature, T 21 ,
and it provides a window to increase the allowed B0 . It is be- by EDGES observation, respectively. The minimum of solid
cause T 21 ∝ (1 − T R /T S ). Therefore, we can increase the up- and dot-dashed (doted) lines correspond to T 21 = −1000 and
per bound for the temperature. Here, we take the infinite Lyα -300 mK, respectively. As T 21 ∝ (1 − T R /T S ), therefore, when
coupling, i.e. xα → ∞, and it implies T S ' T gas . The black we increase the value of ξ the T R rises, and we get more win-
solid line represents the upper constraint on B0 . We get upper dow to increase the magnetic field strength. For 10% and 50%
bound by taking T 21 = −300 mK for the redshift z = 17.2. radio radiation excess we get the upper bound on the mag-
The region below the black solid line represents the allowed netic field strength to be 55.7 pG (B1 Mpc ≈ 37 pG) and 78 pG
region for the present day magnetic field strength for different (B1 Mpc ≈ 53 pG) respectively. For the case when ξ = 1, this
ξ. Here, we do not include the effects of first stars formation bound changes to 84 pG (B1 Mpc ≈ 57 pG). Considering Lyα
on bounds. coupling to its fiducial fraction, i.e. Aα to 20%, we get the
Next, we consider the effects of first stars, discussed in the upper constraint on B1 Mpc . 84 × 10−1 pG for ξ = 0.5 and
section (IV), on IGM gas temperature. Inclusion of IGM heat- 87 × 10−1 pG for ξ = 1; while for 10% fraction of observed
ing, due to x-ray and VDKZ18 [50], increases the gas temper- radio radiation excess we get B1 Mpc . 67 × 10−1 pG. The au-
ature quickly above the previous scenario, discussed in sec- thors of the Ref. [75], put a constraint on the upper bound of
tion (III), when only magnetic heating is included. There- PMFs strength to . 47 pG for scale-invariant PMFs by com-
fore, in the presence of finite Lyα coupling, the upper bound paring CMB anisotropies, reported by the WMAP and Planck,
on present-day strength of PMFs modifies. Following the with calculated CMB anisotropies. We consider nearly scale-
Refs. [70–72], we consider WF coupling coefficient, xα = invariant PMFs, i.e. nB = −2.9. For the 10% radiation excess,
2Aα (z) × (T 0 /T R ). Here, Aα (z) = Aα (1 + tanh[(zα0 − z)/∆zα ]), on the scale of 1 Mpc, the present-day amplitude of primor-
the step height Aα = 100, pivot redshift zα0 = 17 and duration dial magnetic fields can be constrained to, B1 Mpc . 37 pG for
∆zα = 2. The collisional coupling coefficient, xc = T 10 /T R × the nearly scale-invariant PMFs.
HH
(NH k10 )/A10 . After the inclusion of x-ray and VDKZ18 heat-
ing, the gas temperature remains > 10 K. Therefore, we can
HH
take k10 ≈ 3.1×10−11 (T gas /K)0.357 exp(−32 K/T gas ) cm3 /sec VI. CONCLUSIONS
for 10 K < T gas < 103 K. Further, to obtain figure (3), we
have included the VDKZ18 heating together with the x-ray In the present work, we put an independent upper con-
heating due to first stars after z . 30 and consider finite Lyα straint on the strength of the primordial magnetic fields us-
coupling. In FIG. (3), the black, blue and red lines are corre- ing the bound of EDGES observation on 21 cm differential
spond to radiation excess ξ = 0.1, 0.5 and 1 respectively. The brightness temperature in the presence of observed excess ra-
red doted line corresponds to ξ = 1 and 20% fiducial frac- dio radiation by ARCADE 2. We consider the nearly scale-
tion of Lyα coupling. Magenta and green dashed horizontal invariant PMFs, i.e. nB = −2.9. We find the upper constraint
lines represent the upper (-300 mK) and lower (-1000 mK) on B1 Mpc . 31.6 × 10−2 nG for ξ = 1; while considering 10%6
radiation excess, we get B1 Mpc . 53.3 pG. Here, it is to be for ξ = 0.5; while for 10% fraction of observed radio radiation
noted that, we do not consider x-ray and VDKZ18 heating ef- excess we get B1 Mpc . 67 × 10−1 pG. These upper limits on
fects on IGM [50, 70]; and we take infinite Lyα coupling, i.e. the strength of the primordial magnetic fields are also consis-
xα → ∞, therefore, T S ' T gas . After the inclusion of heating tent with the Planck observations [74, 75].
effects of the first stars on IGM temperature and considering
finite Lyα coupling [70], after z ∼ 30, these upper bounds on
the present-day strength of primordial magnetic fields mod- VII. ACKNOWLEDGMENTS
ify. For 10% and 50% radio radiation excess we get the upper
bound on the magnetic field strength to be B1 Mpc . 37 pG and The author would like to thank Prof. Jitesh R. Bhatt
. 53 pG, respectively. For the case when ξ = 1, this bound and Alekha C. Nayak for improving the presentation of the
changes to B1 Mpc . 57 pG. Considering Lyα coupling to its manuscript. The author would also like to thank Prof. Karsten
fiducial fraction, i.e. Aα to 20%, we get the upper constraint Jedamzik for his comments. All the computations are accom-
on B1 Mpc . 87×10−1 pG for ξ = 1 and B1 Mpc . 84×10−1 pG plished on the Vikram-100 HPC cluster at Physical Research
Laboratory, Ahmedabad.
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