宇宙初期磁場と 21cm シグナル
12

3Kiyotomo Ichiki , Maresuke Shiraishi , Hiroyuki Tashiro
12KMI, Nagoya University, JAPANINFN, Sezione di Padova, Italy

3Arizona State University, USAABSTRACT
Recent discoveries of magnetic fields in intergalactic void regions and in high redshift galaxies may indicate that large scale magnetic fields
have a primordial origin. If primordial magnetic fields were present soon after the recombination epoch, they would induce density fluctuations
on the one hand and dissipate their energy into the primordial gas on the other, and thereby significantly alter the thermal history of theuniverse. Here we consider both the effects and calculate the brightness temperature fluctuations of the 21cm line using simple Monte-Carlosimulations because analytic manipulation is difficult due to complicated non-linearities. We find that fluctuations of 21cm line from theenergy dissipation appear only on very small scales and those from the density fluctuations always dominate on the observationally relevant
angular scales probed by SKA.

Tγ + yk Tgas,Ts =1 + ykThe power spectra are directly estimated from the simulations.

(3)

10000

(4)

|(∇ × B) × B|2,Γ(x) =2216π χρb xi(1)where ρb is baryon energy density and χ = 3.5 ×1013cm3g−1s−1 denotes the drag coefficient. The evolutions of the hydrogen gas temperature Tgas and ionization fraction xi is given by [4]dTgas
xi 8ργ σTΓ, (2)= −2HTgas +(Tγ − Tgas) +dt
1 + xi 3mec
1.5kB nH

where kB , ργ , H, xi, ne me and σT denote the Boltzmann constant, photon energy density, the Hubble parameter, the ionization fraction, electron number density and mass, and cross section of Thomson scattering,respectively.

where z is the redshift corresponding to the frequencyof observation, 1 + z = ν21/ν, Ts(z) is the spin temperature and τ (z) is the optical depth of the IGM atz.

Tgas

100

101
10100

1000

10000

100000z

Figure 1: Typical time evolutions of IGM. Different colors
correspond to different pixels in the simulation box. Primordial magnetic field parameters are taken as Bλ = 3.0nG and nB = 0.0.

0.00013231e-06

PBm
0.01

0.0001

1e-061101000.1110100-1k [Mpc ]Figure 4: power spectra of 21cm signal at z1,2 = 20 for
magnetic field strengths ranging from Bλ = 0.001 nG to 1nG, with spectral index nB = −2.9 and 0.0. See text below
for details.

10000

1nG hist11nG hist21nG hist30.1nG hist10.1nG hist20.1nG hist30.01nG hist10.01nG hist20.01nG hist30.001nG hist10.001nG hist20.001nG hist3100010010101001000

10000100000zFigure 2: Same as Fig. 1, but for spin temperature.

We generate seed magnetic fields in k-space with 5123
grid whose power spectrum is parameterized by apower law with strength smoothed on λ = 1 Mpc (Bλ)
and power-law spectral index (nB ) asDE
P(k)∗ (p) = (2π)3 B
B0i(k)B0j
δij − k̂ik̂j δ (3)(k − p) ,2(5)(2π)nB +5 Bλ2 nB
PB (k) = n +3 n +3 k ,Γ( B2 )kλ B

The power spectrum of 21 cm brightness temperature
mainly has three parts. First is the temperature fluctu
ation term that is proportional to δ (T̄s − T̄γ )/T̄s n̄HI(and dented by Ps in Fig. 4), second is the density fluctuation term proportional to (T̄s − T̄γ )/T̄sδnHI where
δnHI is the density fluctuations that originate fromthe standard adiabatic mode (denoted by PmCDM), andthird is the density fluctuations from the primordial
magnetic fields (dented by PmB). In Fig. 4, we separately plot these three contributions. We find thatmagnetic fields with nano Gauss levels significantly enhance the power over the wide range of scales throughthe density fluctuation term, because they realize that(T̄s − T̄γ )/T̄s|PMF ∼ 1 ≫ (T̄s − T̄γ )/T̄s|no PMF andgive larger density fluctuations especially on smallscales. If we consider the case with nB = 0.0, even
weaker magnetic fields with strength as small as Bλ =10−3 nG can amplify the standard signal by three orders of magnitude (see the right panel in the figure).
We also find that the temperature fluctuation term cangive comparable contributions only at smallest scales
for nearly scale invariant magnetic fields (nB = −2.9;
the left panel in the figure). The contributions, however, are always subdominant for magnetic fields withbluer spectrum, as shown in the right panel in Fig 4.

where kλ = 2π/λ. We then calculate the thermal
history at every pixel for a realization of Γ(x), andestimate the power spectrum of 21cm signal.

y [Mpc]1nG hist11nG hist21nG hist30.1nG hist10.1nG hist20.1nG hist30.01nG hist10.01nG hist20.01nG hist30.001nG hist10.001nG hist20.001nG hist30.01

PCDMm1-1

10000

1000PBm
CDM
Pm

CDMPm
PBm
Bλ = 0.001nG: Ps

100k [Mpc ]y [Mpc]The energy dissipation rate due to ambipolar diffusion
is given byPCDM
m

10.1

Monte Carlo Simulation
Ambipolar Diffusion

100

P21(k, z1 = z2 = 20) [mK Mpc ]Ts(z) − Tγ (z)(1 − e−τ (z)),δT21 =1+zBλ = 0.01nG: Ps

CDM

PmPBm
Bλ = 0.1nG: Ps

2

where yk is the kinetic coupling strength. For a given
frequency, the differential brightness temperature isgiven by

10000
Bλ = 1nG: Ps

P21(k, z1 = z2 = 20) [mK Mpc ]Recent discoveries of magnetic fields in galaxies at highredshift [1], and in void regions [2], may support thehypothesis that the seed fields are primordial origin.
If this is the case, the primordial magnetic fields haveinfluenced many kinds of cosmological processes, suchas the Big Bang Nucleosynthesis, Cosmic Microwave
Background Anisotropies, and the formation of large
scale structure of the universe (see [3] and references
therein). Recently, Planck collaboration placed limits
on primordial magnetic fields as Bλ < 3.4 nG andnB < 0 from the temperature anisotropies at large
and small angular scales.
In this paper, we consider the effect of primordial magnetic fields on the evolution of cosmological perturbations after the cosmological recombination. In particular, we investigate the thermal history of the primordial hydrogen gas in the universe by taking into
account the heat injection due to the ambipolar diffusion of the magnetic fields [4]. We improve the analysis
in [4], taking the fluctuations of the heat injection into
account using Monte Carlo simulations.
The heat injection from the magnetic fields into theweakly ionized primordial gas will leave unique signature in the future 21cm observations. If the heating
rises the gas temperature and hence the spin temperature high above the background CMB temperature,the 21cm signal comes as emission even at redshift
z & 20, while the signal is expected to be absorption
in the standard thermal history of the universe.

Result (Power Spectrum)

TsIntroduction

Since the spin temperature of the IGM is determinedby the balance among absorption of CMB photons,
thermal collisional excitation, we can obtain the spintemperature from1098765432
101098765432
10xe

T210.03

14
121086
4
20-2

0.025
0.02
0.015
0.01
0.005
0Bx

Ts
2.5
2
1.5
1
0.5
0-0.5
-1
-1.5
-2
-2.5
-3

90
85
80
75
70
65
6055

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10x [Mpc]x [Mpc]Figure 3: Realization of (the x-component of) primordial
magnetic fields [nG] (bottom left) and corresponding ionization fraction (top left), spin temperature [K] (bottom center) and 21cm brightness temperature [mK] (top center),with parameters Bλ = 1 [nG] and nB = −2.9. The heating is dominated by small scale structure. It can be seenthat spin temperature and ionization fraction fluctuations
are positively correlated.

Summary
We consider the heat injection into IGM in highredshift due to the ambipolar diffusion of primordial
magnetic fields. Using simple Monte Carlo simulations we successfully estimate the 21cm power spectrum, and find that nG magnetic fields can significantly amplify the cosmological signal by up to threeorders of magnitude at z = 20. The dominant contribution is coming from the homogeneous heating on topof density fluctuations of the standard adiabatic modeon large scales, and that generated from primordial
magnetic fields on small scales.

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arXiv:0807.3347.
[2] K. Takahashi, M. Mori, K. Ichiki, S. Inoue, and H. Takami,
ApJ 771, L42, L42 (Jul. 2013), arXiv:1303.3069.
[3] D. G. Yamazaki, T. Kajino, G. J. Mathews, and K. Ichiki,Phys. Rep. 517, 141 (Aug. 2012), arXiv:1204.3669.
[4] S. K. Sethi and K. Subramanian, MNRAS 356, 778 (Jan.
2005), arXiv:astro-ph/0405413.