# Small-scale CMB anisotropies induced by the primordial magnetic fields

Teppei Minoda,11footnotetext: Corresponding author. Kiyotomo Ichiki  and
Hiroyuki Tashiro

###### Abstract

The primordial magnetic fields (PMFs) produced in the early universe are
expected to be the origin of the large-scale cosmic magnetic fields. The PMFs
are considered to leave a footprint on the cosmic microwave background (CMB)
anisotropies due to both the electromagnetic force and gravitational
interaction. In this paper, we investigate how the PMFs affect the CMB
anisotropies on smaller scales than the mean-free-path of the CMB photons. We
solve the baryon Euler equation with Lorentz force due to the PMFs, and we
show that the vector-type perturbations from the PMFs induce the CMB
anisotropies below the Silk scale as $\ell>3000$. Based on our calculations,
we put a constraint on the PMFs from the combined CMB temperature anisotropies
obtained by Planck and South Pole Telescope (SPT). We have found that the
highly-resolved temperature anisotropies of the SPT 2017 bandpowers at
$\ell\lesssim 8000$ favor the PMF model with a small scale-dependence. As a
result, the Planck and SPT’s joint-analysis puts a constraint on the PMF
spectral index as $n_{B}<-1.14$ at 95$\%$ confidence level (C.L.), and this is
more stringent compared with the Planck-only constraint $n_{B}<-0.28$. We show
that the PMF strength normalized on the co-moving 1 Mpc scale is also tightly
constrained as $B_{1\mathrm{Mpc}}<1.5$ nG with Planck and SPT at 95% C.L.,
while $B_{1\mathrm{Mpc}}<3.2$ nG only with the Planck data at 95% C.L. We also
discuss the effects on the cosmological parameter estimate when including the
SPT data and CMB anisotropies induced by the PMFs.

##  1 Introduction

The origin and evolution of large-scale cosmic magnetic fields are still open
questions in modern cosmology. Magnetic fields in galaxies are measured to be
$B_{\mathrm{gal}}\sim 10^{-6}$–$10^{-5}~{}\mathrm{G}$ through many kinds of
observations, such as the synchrotron radiation, Faraday rotation, Zeeman
effect, and polarization from aligned dust grains (e.g., a detailed review for
magnetic fields in galaxies is Ref. [1]). Such $\mu$G magnetic fields in
galaxies are considered to be achieved by the galactic dynamo process related
to the galactic rotational motion [2]. However, the galactic dynamo process
requires a seed field, and the mechanisms to generate the seed magnetic field
remain unknown. On the other hand, low-density regions of the Universe, the
so-called cosmic voids, are also magnetized as $B\gtrsim 10^{-18}$–$10^{-15}$
G, as suggested by observations of $\gamma$-ray and ultra-high-energy cosmic-
rays [3, 4, 5]. Such magnetic fields in cosmic voids could be originated in a
cosmological phenomenon in the early universe.

A lot of physical processes in the early universe predict the generation of
magnetic fields, for example, inflation [6, 7, 8], phase transition [9, 10,
11], topological defects [12, 13], and the Harrison mechanism [14]. Such
magnetic fields generated in the early universe are called the “Primordial
Magnetic Fields” (PMFs), and the PMFs can solve both of the above problems;
the origin of the galactic and intergalactic magnetic fields. Therefore it is
important to observationally investigate the existence and the nature of the
PMFs. Since the PMFs leave different observational signatures, we can put
different constraints on the PMFs by comparing them with observational data.
We obtain many observational constraints on the PMFs from the abundance of
light elements produced by the Big Bang nucleosynthesis [15, 16], the
temperature and polarization anisotropies of the cosmic microwave background
(CMB) [17, 18, 19], magnetic reheating before the recombination epoch [20],
the galaxy population [21], Lyman alpha data [22, 23], the cluster abundance
[24], and the 21-cm global signal [25].

In this paper, we revisit the small-scale CMB anisotropy induced by the PMFs
and give updated constraints on the PMFs from CMB angular power spectra
obtained from the Planck and the latest South Pole Telescope (SPT) data. Many
authors have studied the small-scale CMB anisotropies induced by the PMFs well
[26, 27, 28]. CMB anisotropies induced by the PMFs do not suffer the
exponential Silk damping and can dominate over the inflationary anisotropies
below the Silk scale. Also, it is shown that the anisotropies decrease at
smaller scales [29]. Many authors have studied the CMB anisotropies induced by
the PMFs well [26, 27, 28]. One of their important features arises below the
photon diffusion scale, the so-called Silk scale. Although the photon
diffusion affects the CMB anisotropies by the PMF, their power spectrum does
not exponentially damp but decreases in the power-law shape [29]. However, in
this paper, we will show that their shape is totally different below the
photon mean-free-path scale. The PMFs can survive even much below the Silk
scale [30, 31]. Below the mean-free-path scale, the survived PMFs can
accelerate the baryon fluid motions decoupled from the photons. As a result,
on such small scales, although the photon diffusion decreases the CMB
anisotropies once, significant CMB anisotropies can be created again by the
Doppler effect of the baryon fluid velocities driven by the PMFs. We
demonstrate this enhancement on small scales by solving the baryon Euler
equation analytically with simple assumptions.

It is expected that this enhancement provides the impact on observational
constraints on the PMFs. Nowadays the SPT can reveal the CMB anisotropy
angular spectrum up to $\ell\sim 8000$ [32] in terms of the multipole $\ell$,
which corresponds to the scales below the mean-free-path scales. Therefore we
also investigate how much such small-scale CMB anisotropies affect the
constraints on the PMFs by performing a MCMC analysis. We provide the first
PMF constraint with both temperature and polarization anisotropies from Planck
and SPT including scales below the mean-free-path scale.

This paper is organized as follows: In section 2, we introduce the statistical
property of the PMFs and discuss the CMB angular power spectra sourced by the
stress-energy tensor of the PMFs. Next, we focus on the CMB spectra induced by
the PMFs below the photon mean-free-path scale in section 3. Then we describe
the method of our MCMC analysis with the Planck and SPT data that we use to
constrain the PMF model parameters in section 4. We show our new constraint on
the PMF strength and give some uncertainties on the results in section 5.
Finally we summarize in section 6.

##  2 CMB anisotropies created by the PMFs

###  2.1 Statistical properties of the PMFs

Since the primordial plasma has high conductivity, the ideal
Magnetohydrodynamics (MHD) is valid to provide the evolution of the PMFs.
Besides, the back-reaction from the primordial plasma motion on the PMFs is
described as a higher-order effect in the linear cosmological perturbation
theory, and we can neglect them in this paper. Therefore, the PMF evolution
can be described adiabatically. In these assumptions, following the
cosmological expansion, the PMFs evolve as
$\bm{B}(\bm{x},t)=\bm{B}_{0}(\bm{x})/a^{2}(t)$, where $a(t)$ is the scale
factor at time $t$, which is normalized in $a(t_{0})=1$ at the present epoch,
$t_{0}$.

Next, we assume that the PMFs are statistically homogeneous and isotropic
Gaussian random fields. In this case, the two-point correlation function of
the PMFs in Fourier space can be written with the power spectrum $P_{B}(k)$ as

$\langle
B_{i}(\bm{k})B^{*}_{j}(\bm{k}^{\prime})\rangle=\delta_{\mathrm{D}}(\bm{k}-\bm{k}^{\prime})(\delta_{ij}-\hat{k}_{i}\hat{k}_{j})P_{B}(k)~{}.$
(2.1)

Here $\hat{k}_{i}$ is the $i$-th component of the unit vector of $\bm{k}$, and
the Fourier component of the PMFs is defined by $\bm{B}(\bm{k})=\int
d^{3}x~{}\mathrm{e}^{i\bm{k}\cdot\bm{x}}\bm{B}(\bm{x})$ . In Eq. (2.1), we
assume no helicity of the PMFs.

We are interested in the magnetic field strength in real space. Therefore, it
is convenient to introduce the magnetic field strength smoothed on
$\lambda=1~{}\mathrm{Mpc}$ scale, $B_{1\mathrm{Mpc}}$. This smoothed magnetic
field strength is related to the amplitude of the PMF power spectrum as

$B_{1\mathrm{Mpc}}^{2}=\int\cfrac{d^{3}k}{(2\pi)^{3}}~{}\mathrm{e}^{-k^{2}\lambda^{2}}P_{B}(k)~{},$
(2.2)

where we choose a Gaussian function for the smoothing window function.

For simplicity, we assume that the power spectrum of the PMFs has a power-law
shape as

$P_{B}(k)=S_{0}(k/k_{\mathrm{n}})^{n_{B}}~{},$ (2.3)

where $k_{\mathrm{n}}$ is the wave number for the normalization scale and we
take $k_{\mathrm{n}}\equiv 2\pi~{}\mathrm{Mpc}^{-1}$. By combining equations
(2.2) and (2.3), we can relate the amplitude of the PMF power spectrum $S_{0}$
to the smoothed PMF strength $B_{\mathrm{1Mpc}}$ as

$B_{1\mathrm{Mpc}}^{2}=\cfrac{S_{0}}{(2\pi)^{n_{B}+2}(1~{}\mathrm{Mpc})^{3}}~{}\Gamma\left(\cfrac{n_{B}+3}{2}\right)~{}.$
(2.4)

In this study, we do not focus on any specific PMF generation mechanism which
determines the spectrum of the PMFs. Instead, we evaluate the CMB anisotropies
by taking three free parameters, $B_{\mathrm{1Mpc}},n_{B},$ and $\eta_{B}$ for
characterizing our PMF model. Here, $\eta_{B}$ is the conformal time at which
the PMFs are generated. Although it does not appears in the power spectrum of
the PMFs, $\eta_{B}$ is an important parameter in the calculation of the PMFs.
We will discuss how $\eta_{B}$ affects the CMB anisotropies later.

The PMFs can create the CMB anisotropies through generating metric
perturbations by their own energy and stress gravitationally and altering the
primordial plasma motion through their electromagnetic interaction. These
effects can be described in the Einstein-Boltzmann equation system with their
energy-momentum tensors. We can write the energy-momentum tensor of the PMFs
in Fourier space, ${T_{B}}^{\mu}_{~{}\nu}$, as

$\displaystyle{T_{B}}^{0}_{~{}0}(\bm{k},t)$ $\displaystyle=-\cfrac{1}{8\pi
a^{4}(t)}\int\cfrac{d^{3}k^{\prime}}{(2\pi)^{3}}B^{a}(\bm{k}^{\prime})B_{a}(\bm{k}-\bm{k}^{\prime})\equiv-\rho_{\gamma}\Delta_{B}~{},$
(2.5) $\displaystyle{T_{B}}^{0}_{~{}i}(\bm{k},t)$ $\displaystyle=0~{},$ (2.6)
$\displaystyle{T_{B}}^{i}_{~{}j}(\bm{k},t)$ $\displaystyle=\cfrac{1}{4\pi
a^{4}(t)}\int\cfrac{d^{3}k^{\prime}}{(2\pi)^{3}}\left[\cfrac{1}{2}B^{a}(\bm{k}^{\prime})B_{a}(\bm{k}-\bm{k}^{\prime})\delta^{i}_{~{}j}-B^{i}(\bm{k}^{\prime})B_{j}(\bm{k}-\bm{k}^{\prime})\right]$
$\displaystyle\equiv
p_{\gamma}(\Delta_{B}\delta^{i}_{~{}j}-{\Pi_{B}}^{i}_{~{}j})~{}.$ (2.7)

Here we have defined the dimensionless parameters $\Delta_{B}$ and
${\Pi_{B}}^{i}_{~{}j}$ with the energy density $\rho_{\gamma}$ and pressure
$p_{\gamma}$ of CMB photons.

The scalar-vector-tensor decomposition is powerful to analyze the PMF effect
on the cosmological perturbation quantities including the CMB anisotropies.
Although $\Delta_{B}$ is a scalar quantity, the PMF anisotropic stress tensor
${\Pi_{B}}^{i}_{~{}j}$ can be decomposed into the scalar $\Pi_{B}^{(0)}$,
vector $\Pi_{B}^{(\pm 1)}$, and tensor part $\Pi_{B}^{(\pm 2)}$ (for detailed
discussion, see, e.g., Refs. [33, 34]). Therefore, through the Einstein
equation, $\Delta_{B}$ and $\Pi_{B}^{(0)}$ can generate the scalar type of the
metric perturbations and $\Pi_{B}^{(\pm 1)}$ and $\Pi_{B}^{(\pm 2)}$ can
create the vector and tensor types of those, respectively. The PMFs can source
all types of metric perturbations.

Besides the metric perturbation, the PMFs can alter the motion of charged
particles because of the electromagnetic interaction, and generate
perturbations on the baryon plasma through the Lorentz force. The Lorentz
force term, $L_{i}$, which appears in the Boltzmann equation of the baryon
plasma (see e.g., Ref. [35]), can be related to the magnetic energy density
and anisotropic stress. Thus the Lorentz force term is also decomposed into
the scalar-type $L^{(\mathrm{S})}=2\Pi_{B}^{(0)}/9-\Delta_{B}/3$ with the
scalar-type values, $\Delta_{B}$ and $\Pi_{B}^{(0)}$, and the vector-type
$L_{i}^{(\mathrm{V})}=ik(\Pi_{B}^{(+1)}\hat{k}_{(i}\mathrm{e}_{j)}^{+}+\Pi_{B}^{(-1)}\hat{k}_{(i}\mathrm{e}_{j)}^{-})\hat{k}_{j}$
with the vector-type value $\Pi_{B}^{(\pm 1)}$. Here,
$\mathbf{e}^{\pm}\equiv-i/\sqrt{2}(\mathbf{e}^{1}\pm i\mathbf{e}^{2})$ are the
helicity bases with complex orthonormal bases, $\mathbf{e}^{1}$ and
$\mathbf{e}^{2}$, which are perpendicular to $\bm{k}$ [33, 34].

For the complete set of the Einstein-Boltzmann equations with the PMFs in the
cosmological linear perturbation theory, we refer the readers to Ref. [36].

###  2.2 Impact of the PMFs on the CMB spectra

In order to study the evolution of the cosmological perturbations, we solve
the Boltzmann and Einstein equations with some initial conditions including
the PMF parameters. In practice, we set the initial condition on the super-
horizon scale and well after the neutrino decoupling. Here, due to the impact
of neutrino fluid on the metric perturbations, there are two initial
conditions for perturbations created by the PMFs. After the neutrino
decoupling, neutrinos freely stream, and then they can create non-zero
anisotropic stress to compensate that of the PMFs. On the other hand, before
the neutrino decoupling, neutrinos tightly couple with the photon-baryon
fluid. Therefore, neutrinos have negligible anisotropic stress and cannot
compensate that of PMFs. This difference between before and after the neutrino
decoupling brings two modes of the perturbations; passive and compensated
modes.

The passive mode arises due to the anisotropic stress of the PMFs before the
neutrino decoupling. For the scalar and tensor types, the non-zero anisotropic
stress of the PMF gives rise to the passive mode of the metric perturbations
on both super- and sub-horizon scales soon after the generation of the PMFs.
When neutrinos decouple from the photon-baryon fluid, the anisotropic stress
of the PMFs is compensated by free-streaming neutrinos, as mentioned before.
The source of the metric perturbations vanishes and the perturbation growth
halts after that. As a result, the induced perturbation can exist as a
constant perturbation on the super-horizon scales, similarly to an
inflationary adiabatic perturbation. Therefore, the evolution after the
horizon entry is the same as in case of the inflationary adiabatic
perturbation. The amplitude of these types depends on
$\ln(\eta_{\nu}/\eta_{B})$, with the conformal time of the neutrino decoupling
$\eta_{\nu}$ and that of the PMF generation $\eta_{B}$ [34]. On the other
hand, although the vector-type perturbations are also generated as the passive
mode, they decay quickly after the neutrino decoupling as same as the
inflationary adiabatic vector-type perturbations do. Therefore the CMB
anisotropies arise from the scalar and tensor-type perturbation for the
passive mode. The angular power spectrum of the CMB anisotropies due to the
passive mode are proportional to $\langle\Pi_{B}^{(0)*}\Pi_{B}^{(0)}\rangle$
for the scalar-type contribution and $\langle\Pi_{B}^{(\pm 2)*}\Pi_{B}^{(\pm
2)}\rangle$ for the tensor-type contribution.

The compensated modes are generated by the PMFs after the neutrino decoupling.
Since the PMF anisotropic stress is canceled by the neutrino free streaming
motion after the neutrino decoupling, there is no contribution of the PMF
anisotropic stress to the metric perturbation. As a result, the metric
perturbations are not generated at the leading order on super-horizon scales
in the compensated modes. In this respect, the compensated mode is similar to
the isocurvature perturbation. With no initial perturbation on super-horizon
scales, the PMFs can induce the perturbation sourced by the energy-stress
tensors including Lorentz force acting on the baryon fluid on sub-horizon
scales. Therefore, the compensated mode is important on the small-scale CMB
anisotropies. Although some fraction of the CMB anisotropies due to the
compensated modes is erased by the Silk damping effect, the CMB anisotropies
on smaller scales than the Silk scale can be created by the compensated mode.
This is because the PMFs can survive and continue to source the baryon
velocity perturbations to create CMB anisotropies even below the Silk damping
scale. We discuss the behavior of such small-scale CMB anisotropies in section
3. The compensated mode includes the scalar-, vector-, and tensor-type
perturbations. All of them can generate the CMB anisotropies. In the scalar
type, the amplitude of the CMB angular power spectrum is proportional to
$\langle\Delta_{B}^{*}\Delta_{B}\rangle$,
$\langle\Pi_{B}^{(0)*}\Pi_{B}^{(0)}\rangle$, and
$\langle\Delta_{B}^{*}\Pi_{B}^{(0)}\rangle$. In the vector and tensor types,
$\langle\Pi_{B}^{(\pm 1)*}\Pi_{B}^{(\pm 1)}\rangle$ and $\langle\Pi_{B}^{(\pm
2)*}\Pi_{B}^{(\pm 2)}\rangle$ appear on the CMB angular power spectrum,
respectively.

In figure 1, we summarize the CMB temperature-temperature (TT) auto-power
spectra induced by the PMFs for three dominant contributions, namely, the
passive and compensated scalar modes, and the compensated vector mode. We
neglect the tensor-type contribution because the created CMB anisotropies are
much smaller than the other types contributions on small scales. (e.g., see
Fig. 2 in Ref. [34]). It is clearly seen that the compensated vector mode is
dominant for $\ell\gtrsim 4000$. When plotting figure 1, we fix the PMF
generation epoch at the grand unified theory (GUT) phase transition, as
$\eta_{\nu}/\eta_{B}=10^{17}$. We also plot the measurements of the CMB
temperature anisotropies obtained by Planck and SPT in figure 1. The
comparison with the current observation status tells us that small-scale CMB
anisotropies due to the compensated vector perturbations are crucial to obtain
the constraint on the PMFs because the compensated vector perturbations have a
significant blue-tilted angular power spectrum on small scales. In the next
section, we represent how such a blue-tilted spectrum is created by the
compensated vector perturbations.

Figure 1:  The top thin solid line is the primary CMB temperature angular
power spectrum, which is due to the adiabatic perturbation created by the
inflation mechanism. We show the magnetically induced power spectra for three
magnetic modes, the passive scalar mode (sca P) with the thick dashed line,
the compensated scalar mode (sca C) with the thin dashed line, and the
compensated vector mode (vec C) with the dash-dotted line. In the plots, we
take the PMF parameters as $B_{\mathrm{1Mpc}}=4.0$ nG and $n_{B}=-2.5$ for the
left panel, and $B_{\mathrm{1Mpc}}=3.0$ nG and $n_{B}=-1.0$ for the right
panel. The compensated scalar and tensor modes are not shown because they have
relatively smaller amplitudes on large multipoles. We also show the total
magnetic contribution as the thick solid line.

##  3 The compensated vector mode on small scales

In the compensated vector mode, the PMF can create the perturbations of the
divergence-free baryon velocity and metric. Therefore, the CMB temperature
anisotropies are created from the Doppler and integrated Sachs-Wolfe (ISW)
effect with these perturbations [37]. The observable temperature anisotropies
can be written as

$\Theta(\eta_{0},k,\hat{\bm{n}})=\Bigl{[}\bm{v}_{\mathrm{b}}(\eta,k)\cdot\hat{\bm{n}}\Bigr{]}^{\eta_{\mathrm{rec}}}_{\eta_{0}}-\int^{\eta_{\mathrm{rec}}}_{\eta_{0}}d\eta~{}\bm{\dot{V}}(\eta,k)\cdot\hat{\bm{n}}~{},$
(3.1)

where $\hat{\bm{n}}$ is a line-of-sight unit vector, and $\eta_{\mathrm{rec}}$
denotes the recombination time. Also, $\bm{v}_{\mathrm{b}}(\eta)$ is
divergence-free baryon velocity perturbation, and $\bm{V}(\eta)$ is the gauge-
invariant metric perturbation in vector mode as defined in Ref. [38].
Therefore, in order to calculate the CMB anisotropies from the compensated
vector mode, we need to solve the Euler equation of baryons for
$\bm{v}_{\mathrm{b}}(\eta)$ and Einstein equation for $\bm{V}(\eta)$222In
general, besides them, we also need to take into account the Boltzmann
equations of photons, dark matter, and neutrinos. However, the impact of these
components is negligible in our interested case. Therefore we do not consider
their evolutions.. However, we do not discuss the evolution of the vector
potential $\bm{V}(\eta)$ because its contribution to the CMB anisotropies via
the ISW effect is negligible on small scales. In other words, the enhancement
of the CMB anisotropies on small scales comes from the scale-dependence of the
baryon velocity at the recombination,
$\bm{v}_{\mathrm{b}}(\eta_{\mathrm{rec}},k)$. In order to confirm this
statement, we plot the fully numerical solution for
$\bm{v}_{\mathrm{b}}(\eta_{\mathrm{rec}},k)$ calculated by the Boltzmann code
[18] in figure 2. This shows that the scale dependence of the baryon velocity
can be divided into three different scales: the large scale $k\lesssim
k_{S}\approx 0.14$ Mpc-1, the intermediate scale $k_{S}\lesssim k\lesssim
k_{\mathrm{mfp}}\approx 0.38~{}$Mpc-1, and the small scale $k\gtrsim
k_{\mathrm{mfp}}$. Here $k_{S}$ and $k_{\mathrm{mfp}}$ denote the Silk damping
scale and the mean-free-path scale of the CMB at the recombination epoch,
respectively.

Figure 2:  The scale-dependent baryon velocity perturbation and the photon
dipole moment at the recombination are shown by the solid and dashed lines,
respectively. The model parameters of the PMFs are fixed with
$B_{\mathrm{1Mpc}}=3.0$ nG and $n_{B}=-1.0$.

For large and intermediate scales as $k\lesssim k_{\mathrm{mfp}}$, the
evolution of the vector perturbation has been analytically studied with the
tight-coupling approximation, $\bm{v}_{\mathrm{b}}\simeq\bm{v}_{\gamma}$, in
Refs. [39, 33]. According to their results, for the larger scales than the
Silk scale (or on the earlier phase of the evolution), although the Lorentz
force acts on the baryon fluid and increases the baryon velocity, the photon
velocity (i.e., dipole moment, $\bm{v}_{\gamma}$) catches up baryons quickly
because of the tight coupling due to the Compton scattering. As a result, the
baryon velocity and the photon dipole moment are growing with wavenumber (or
time) together for $k\lesssim k_{S}\approx 0.1$ Mpc-1. However, for the
intermediate scales, the effect of non-zero mean-free-path for photons is not
negligible. The non-zero mean-free-path loses the tight coupling and affects
the perturbation evolution as the photon viscosity in the photon-baryon
plasma. As a result, the baryon and photon velocity perturbations,
$\bm{v}_{\mathrm{b}}$ and $\bm{v}_{\gamma}$, starts to decay.

As shown in figure 2, there exists a weak coupling between baryons and photons
below the Silk scale and they continue to decrease for a while. However, when
the photon mean-free-path scale is larger than the perturbation scale, the
baryon-photon coupling no longer holds. After the decoupling, the baryon
velocity starts to grow due to the PMFs again. Now let us obtain the
analytical expression of the baryon velocity evolution on such small scales.
Below the Silk scale, the photon dipole moment becomes small more quickly than
the baryon velocity. Neglecting the photon dipole moment, we can write the
Euler equation of the baryon fluid in the presence of the PMFs in

$\left(\frac{\partial}{\partial
t}+H+\cfrac{\tau^{\prime}}{R}\right)\bm{v}_{\mathrm{b}}=-\cfrac{\rho_{\gamma}\bm{L}^{(V)}}{2a\rho_{\mathrm{b}}}~{},$
(3.2)

in the non-relativistic limit [30, 40]. Here $\rho_{\mathrm{b}}$ and
$\rho_{\gamma}$ are the energy densities of baryons and photons respectively,
$R\equiv 3\rho_{\mathrm{b}}/4\rho_{\gamma}$ is the total energy and pressure
ratio of baryons and photons, and $\tau^{\prime}\equiv
an_{e}\sigma_{\mathrm{T}}$ is the photon opacity. The left-hand side of
equation (3.2) includes the Hubble expansion, Compton scattering, and the
right-hand side represents the Lorentz force due to the PMFs. Before the
recombination epoch, we can neglect the Hubble expansion term, in comparison
with the Compston scattering term. Solving equation (3.2), we obtain

$\displaystyle\bm{v}_{\mathrm{b}}(k,a)=$
$\displaystyle\bm{v}_{\mathrm{b,i}}\exp\left\\{\cfrac{\tau^{\prime}}{2aHR}\left[1-y_{\mathrm{i}}^{2}(a)\right]\right\\}$
$\displaystyle-\cfrac{\rho_{\gamma}\bm{L}^{(V)}}{4aH\rho_{\mathrm{b}}}\left[\operatorname{Ei}\left(-\left.\cfrac{\tau^{\prime}}{2aHR}\right|_{\mathrm{i}}\right)-\operatorname{Ei}\left(-\cfrac{\tau^{\prime}}{2aHR}\right)\right]\exp\left(\cfrac{\tau^{\prime}}{2aHR}\right),$
$\displaystyle\hskip 244.6937pt(a\lesssim a_{\mathrm{rec}}),$ (3.3)

where $\operatorname{Ei}(x)\equiv\int^{x}_{-\infty}e^{t}/t~{}dt$ is the
exponential integral, $y_{\mathrm{i}}(a)\equiv a/a_{\mathrm{i}}$ and
$y_{\mathrm{rec}}(a)\equiv a/a_{\mathrm{rec}}$ are the scale factors
normalized at the initial time and at the recombination time, and the values
with the subscript “i” and “rec” represent the ones at the initial time and at
the recombination time, respectively.

Around the recombination epoch, the Hubble expansion term starts to dominate
the Compton scattering term in equation (3.2). Therefore, neglecting the
Compton scattering term, we get

$\displaystyle\bm{v}_{\mathrm{b}}(k,a)=$
$\displaystyle\bm{v}_{\mathrm{b,rec}}~{}y_{\mathrm{rec}}^{-1}(a)+\left(\left.\cfrac{\rho_{\gamma}\bm{L}^{(V)}}{H\rho_{\mathrm{b}}}\right|_{\mathrm{rec}}-\cfrac{\rho_{\gamma}\bm{L}^{(V)}}{H\rho_{\mathrm{b}}}\right)a^{-1},\hskip
36.98857pt(a\gtrsim a_{\mathrm{rec}})~{}.$ (3.4)

In equations (3.3) and (3.4), the first terms on the right hand side
represents the decaying terms due to the Compoton scattering and the Hubble
expansion, respectively. On the other hand, the second terms in both equations
are the inhomogeneous solution which comes from the PMF source term, i.e.,
from the right hand side in equation (3.2). In figure 3, we plot the
evolutions of the baryon velocity fluctuation from equations (3.3) and (3.4)
in the dashed line and one from the full numerical calculation in the solid
line. Here we set the initial time at $a_{\mathrm{i}}=10^{-5}$. The figure
tells us that our solutions agree with the full numerical calculation in
particular before the recombination epoch.

CMB temperature anisotropies on small scales are created by the Doppler effect
in the recombination epoch as discussed earlier. Since the second term in
equation (3.3) is dominant around the recombination epoch, the $k$-dependence
of the CMB anisotropies comes from $\bm{L}^{(V)}$,

$\mathcal{D}_{l}\propto|{\bm{v}}_{b}|^{2}_{\rm
rec}\propto|\bm{L}^{(V)}|^{2}\propto k^{2}\Pi^{2}(k)\propto k^{2n+8}.$ (3.5)

Therefore, the temperature anisotropies are strongly enhanced on smaller
scales than the photon mean-free-path scales. This k-dependence also appears
in the temperature anisotropies on large scales as shown in Ref. [29]. Because
of this fact, the angular power spectrum also have the same $\ell$-dependence
on both large scales, $\ell\lesssim 1000$, and small scales, $\ell\gtrsim
6000$.

Figure 3:  Numerical solution for the baryon velocity and the analytical
solution that we have obtained. After the baryon velocity damps due to the
photon viscous effect, our analytical solution is in good agreement with the
numerical solution for $10^{-5}\lesssim a\lesssim 10^{-3}$. Here we fix
$k=1000$ Mpc-1 and the PMF parameters are the same as in figure 2.

##  4 Data Analysis with Planck and SPT

The aim of this paper is to obtain the constraint on the PMFs from the new
small-scale CMB anisotropies obtained by SPT. We use the Markov Chain Monte
Carlo (MCMC) method in order to provide the constraint on the PMF model
parameters, $(B_{\rm 1Mpc},n_{B},\tau_{B})$. For the MCMC analysis, we use
publicly available numerical codes, MagCAMB and MagCosmoMC, developed by Zucca
et al. [18]. MagCAMB is the code to solve the linearized Boltzmann and
Einstein equations with the PMFs. Based on CAMB [41] and its modified version
[34], MagCAMB can calculate the CMB angular power spectra for all
contributions sourced by the PMFs mentioned in the previous section.
MagCosmoMC is developed, based on CosmoMC [42]. It enables us to explore
cosmological parameters and the PMF model parameters with the foreground and
data calibration parameters.

We investigate how the small-scale measurement of the CMB anisotropy by SPT
improves the constraint on the PMF parameters, compared to the previous
constraints by Planck data [17, 18]. However, it is difficult to constraint
the standard cosmological parameters because they are well constrained by the
CMB data on larger scales than the SPT scales. Therefore, we use the combined
data for Planck 2015 and SPTpol 2015/2017. Planck 2015 data includes the
low-$\ell$ ($2\leq\ell\leq 29$) TT, EE, TE, and BB power spectra data, and
high-$\ell$ ($30\leq\ell\leq 2508$) TT, TE, and EE power spectra [43]. SPTpol
2015 data includes the BB power spectra in $300\leq\ell\leq 2300$ for three
spectral combinations, 95 GHz $\times$ 95 GHz, 95 GHz $\times$ 150 GHz, and
150 GHz $\times$ 150 GHz [44], and SPTpol 2017 data contains the TT, TE, and
EE power spectra in $50\leq\ell\leq 8000$ for the frequency band, 150 GHz
$\times$ 150 GHz [32]. We note that Zucca et al. [18] has already studied the
constraint on the PMFs with the Planck and SPTpol 2015 BB bandpowers. In this
work, we update the constraint on the PMFs by adding small-scale TT, TE, and
EE data from SPTpol 2017.

One problem is that the SPT 2017 bandpowers are partially overlapped with the
Planck power spectra for $50\leq\ell\leq 2508$, which corresponds to acoustic
oscillation scales. We used both Planck and SPT for these multipole scales,
and this possibly leads to erroneously tighter constraints on the PMF
parameters. However, this acoustic oscillation region is mainly determined by
the standard cosmological parameters, and we can expect that the constraint on
the PMF parameters is not affected much by using the duplicated CMB data on
these scales.

We show the CMB temperature anisotropies measured by Planck and SPT in Figure
1. The PMF contributions to the CMB angular power spectra with
$B_{\mathrm{1Mpc}}=4.0~{}\text{nG},n_{B}=-2.5$ are plotted in the left panel,
and those with $B_{\mathrm{1Mpc}}=3.0~{}\text{nG},n_{B}=-1.5$ are shown in the
right panel. Both of these parameter combinations are allowed by the previous
constraint from Planck collaboration [17] with a 95% confidence level (C.L.).
However, we expect that the high-$\ell$ CMB spectrum measured by SPT can
constrain large $n_{B}$ as shown in Figure 1. Clearly, the CMB spectra
observed by SPT have tiny errors on small-scales as $\ell\gtrsim 2000$, and
the effect on such small-scale CMB anisotropy is dominated by the compensated
vector mode, which is relevant for the parameter estimation using the SPT
data. Finally, the blue-tilted PMF spectra (this corresponds to a large
spectral index $n_{B}$) that enhance the small-scale CMB anisotropies are
tightly constrained by the high-$\ell$ CMB measurement of SPT.

##  5 Results and Discussion

First, we show the constraint on the PMF parameters, ($B_{1\mathrm{Mpc}}$,
$n_{B}$), derived from MagCosmoMC with the Planck and SPT data in figure 4.
For comparison, we plot the constraint with only the Planck data, which is the
same analysis as done in the previous works [17, 18]. We have found the
normalized PMF strength and the PMF spectral index can be constrained more
tightly with the SPT data. For the constraint with Planck and SPT, the MCMC
analysis consumes computing time dozens of times longer than for the Planck
only. This is because the calculation includes the CMB anisotropies up to
$\ell\sim 8000$ and, on small scales, there arises a strong degeneracy between
the PMF and foreground parameters. In figure 4, our analysis with the Planck
and SPT does not reach the convergence of the MCMC analysis. As one of the
most popular methods to evaluate the convergence, the MCMC analyses usually
impose the so-called Gelman-Rubin diagnostic, $R-1<0.01$, with $R$ being the
square root of the ratio of the marginalized variance of all chains and that
for each chain. Our analysis with the Planck and SPT in figure 4 finds $R>10$.
Therefore the 2D color contour in Figure 4 is not a smooth shape. However, the
marginalized one-dimensional constraints on $B_{1\mathrm{Mpc}}$ and $n_{B}$
seem to have smooth probability distribution functions in Figure 4. Besides,
we found that, after reaching the current level of the constraint in Figure 4,
the upper limit on the PMF parameters does not change for a long time during
the calculation. Therefore we conclude that the constraint on the PMF
parameters in Figure 4 is not far from the result that would be obtained from
the converged MCMC analysis.

Figure 4:  The constraint on the magnetic field strength $B_{1\mathrm{Mpc}}$
and the spectral index of the PMFs $n_{B}$. The thick and thin color region
stands for 68% and 95% confidence level (C.L.), respectively.

The upper limits for $B_{\mathrm{1Mpc}}$ and $n_{B}$ and the best-fitted
values for $\tau_{B}$ from Planck and SPT data are presented in table 1. The
PMF amplitude smoothed on 1 Mpc is constrained as $B_{\mathrm{1Mpc}}<1.52$ nG
for a 95 % confidence level (C.L.) when we include the SPT bandpowers for MCMC
analysis, while the Planck 2015 data only put the upper limit as
$B_{\mathrm{1Mpc}}<3.18$ nG. The constraint on the magnetic spectral index is
also improved, as $n_{B}<-1.14$ for Planck and SPT data and $n_{B}<-0.28$ for
only Planck data. We find the high-$\ell$ SPT temperature anisotropy data
favor the nearly scale-invariant PMF spectrum, as discussed in section 4. On
the other hand, the change on $\eta_{\nu}/\eta_{B}$ mainly affects the
amplitudes of the passive mode perturbation, and consequently, it influences
the large-scale CMB anisotropies. Therefore we have an almost unchanged
constraint on the $\eta_{\nu}/\eta_{B}$ when adding the small-scale SPT data
for the parameter estimation. In table 1, although we put the best-fitted
value for $\eta_{\nu}/\eta_{B}$, we do not show the error-bars for the Planck
and SPT case. This is because we have used duplicated information of CMB
angular power spectra for $50\leq\ell\leq 2500$, and may we underestimate the
error, as explained in section 4.

Table 1:  The best-fitted values for the epoch of the PMF generation, $\eta_{B}$, and the upper limits for $B_{\mathrm{1Mpc}}$ and $n_{B}$ with 95% C.L. only from the Planck data, and those from the Planck and SPT data are shown.  Parameters | PMF
---|---
Planck | Planck + SPT
$B_{\mathrm{1Mpc}}$ |  $<$ 3.182 |  $<$ 1.515
$n_{B}$ |  $<$ -0.28 |  $<$ -1.14
$\log\left(\eta_{\nu}/\eta_{B}\right)$ | $9.92^{+2.09}_{-5.92}$ | 9.41

Next, we discuss the impact of the PMF parameter on the determination of the
$\Lambda$CDM cosmological parameter. In figure 5, we compare the $\Lambda$CDM
cosmological parameter constraint with only Planck data and that with Planck
and SPT TT/TE/EE, and BB data, including the PMF parameters. Our results
indicate that the cosmological parameter estimation can be biased when
considering the PMF effects and the SPT high-$\ell$ data. In particular,
Thomson optical depth $\tau$ and the amplitude of the primordial power
spectrum for the scalar sector $A_{s}$ decrease by 1-2 $\sigma$ compared with
the Planck only analysis. The SPT TT data favor a nearly scale-invariant
spectrum of the PMFs as shown in figure 4. In this case, the passive tensor
mode significantly enhances the low-$\ell$ EE power spectrum. To compensate
for the enhancement on large scales by the PMFs, the MCMC chooses the small
optical depth $\tau$, which also contributes to the signals on large scales of
the CMB EE power spectrum.

As dedicated in section 4, the measured ranges of the TT/TE/EE spectra which
we have used are overlapped for Planck and SPT for $50\lesssim\ell\lesssim
2500$. However, this is not harmful to our results because the CMB spectra for
these angular scales are almost determined only with the cosmological
parameters, and are not affected by the PMF contribution very much. Actually,
we have confirmed that the best-fitted values of the cosmological parameters
are not changed for Planck and Planck+SPT, when excluding the PMFs for our
analyses ($\Lambda$CDM case). For similar reasons, we do not perform
BICEP2/Keck-Planck joint analysis [17, 18].

Figure 5:  The constraint on the magnetic field strength $B_{1\mathrm{Mpc}}$
and the spectral index of the PMFs $n_{B}$. The thick and thin color region
stands for 68% and 95% confidence level, respectively. The red (RUN-
PlanckOnly) and blue (RUN-SPT-TEB-MH) contours indicate the constraints from
Planck 2015 data and those from both of Planck and SPT data, respectively.

Before closing this section, we make some comments on other PMF effects on the
CMB anisotropies, which are not included in our analysis. Previous studies
predict the generation of the helical PMFs during inflation [45, 46, 47, 48].
While such helical PMFs can induce the parity-odd spectrum, TB and EB, the
contributions of the helical part of the PMFs to the parity-even TT, TE, EE,
and BB power spectra are always subdominant in comparison with the
contribution of the non-helical PMFs [17]. Therefore, adding the helical
component of the PMFs to parameter estimation would not significantly change
our constraint. The dissipation of magnetic fields before the recombination
epoch affects the thermal history and recombination history of the baryon gas
[30, 31, 49, 50]. Including this effect can improve the constraint on the PMFs
via increasing the energy density of CMB photons [20], creating $y$-type
distortion [51, 52], and changing the evolution of Thomson optical depth [53].
Including these effects can improve our constraint on the PMFs, in particular,
with a large spectral index. However, the dissipation of the PMFs is a strong
nonlinear process, and there is still a large uncertainty in the calculation
of these effects. Therefore we do not include the dissipation effect in our
analysis. The PMFs also can induce the non-Gaussian CMB anisotropies [54, 55].
The current constraint on the non-Gaussianity in the CMB anisotropies by
Planck observation provides the same order of our PMF constraint [17].
Moreover, the CMB polarization map should be altered by Faraday rotation, if
the PMFs exist. The upper limit on the PMF amplitude via Faraday rotation from
the Planck observation is much weaker than via the other effects described
here. The PMFs can also create a secondary contribution to the CMB temperature
anisotropy, which is the so-called thermal Sunyaev-Zel’dovich (tSZ) effect
[56]. This effect enhances the small-scale CMB anisotropy of
$\ell^{2}C_{\ell}\sim 10-100$ $\mu$K2 at $\ell\sim 10^{6}-10^{7}$ for sub-nano
Gauss PMFs. Of course, the CMB anisotropy on such scales cannot be measured
even with the present and on-going radio telescope including CMB-S4 [57, 58].
Therefore, we have not included the tSZ power spectrum from the PMFs into our
analysis.

##  6 Conclusion

In this paper, we have investigated the impact of the PMFs on small-scale CMB
anisotropies. Then we have provided the constraint on the PMFs from the CMB
temperature and polarization anisotropies observed by the latest Planck and
SPT data. If the PMFs exist, the stress-energy tensor of the PMFs induces the
additional metric perturbation besides the primordial curvature perturbation,
and the Lorentz force acts on the motion of primordial plasma. In the
analysis, we have assumed that the PMFs have a simple power-law spectrum, as
$P_{B}(k)\propto B_{1\mathrm{Mpc}}^{2}k^{n_{B}}$. We have also taken the
assumption of the ideal MHD approximation for the PMF evolution and neglected
the helicity of the PMFs. Under these assumptions, we have performed a
numerical calculation to solve the Einstein-Boltzmann equation system. To
obtain the CMB anisotropy angular power spectra with the PMFs, we consider the
time evolution of the scalar and tensor perturbations for the passive mode,
and the scalar, vector, and tensor perturbations for the compensated mode.
While the primary CMB anisotropies are rapidly suppressed for $\ell\gtrsim
1000$ due to the Silk damping effect, the compensated vector perturbation from
a blue-tilted PMFs significantly contributes to the small-scale CMB
anisotropies as we have shown in Figure 1. In this paper, we have given an
analytical solution for the Euler equation in the small-scale limit for the
first time. We also have shown that the derived baryon velocity perturbation
induces the dominant CMB anisotropies on $\ell\gtrsim 4000$, depending on the
spectral indices of the PMFs. Therefore, we have found that the nearly scale-
invariant PMFs are favored by the high-resolution measurement of the CMB
anisotropies with SPT.

Finally, we have obtained the constraint on the PMFs as
$B_{\mathrm{1Mpc}}<1.52$ nG and $n_{B}<-1.14$ for 95 % C.L. with the Planck
2015 and SPT 2015/2017 bandpowers. Our analysis improves the PMF constraint
compared with the Planck 2015 constraint. We have found that the cosmological
parameter estimation is biased when including the PMF parameters and SPT data.
The PMF induced perturbation with the small spectral index $n_{B}$ enhances
the low-$\ell$ polarization anisotropies, and this results in the smaller
Thomson optical depth $\tau$ and the primordial scalar power spectrum
amplitude $A_{s}$.

Finally, we mention the rigidity and future outlook about our constraint.
First, throughout this paper, we assume that the PMFs are frozen in, and they
evolve adiabatically. However, small scale velocity perturbations are reduced
by the Compton scattering as discussed in section 3. Therefore the PMFs also
might be suppressed to some degree by the back-reaction from such radiative
diffusion of baryon perturbations. For instance, a full MHD simulation
probably give a more accurate speculation into the cosmological perturbations
and CMB anisotropies including such non-linear effects. Moreover, on small
scales, the degeneracy between the PMF and foreground parameters arises and
brings an impact on the constraint on the PMFs. Subtracting the foreground is
important not only in the CMB constraint on the PMFs but also in understanding
galaxy cluster physics through the SZ effect [59], the CMB gravitational
lensing [60], and the reionization process with the CMB Doppler effect in the
patchy reionization [61]. In principle, multi-frequency observation might help
to remove the CMB foreground and improve our constraint. However, this is
beyond the scope of this paper, and we put these issues on the future works.

## Acknowledgments

This work is based on observations obtained with Planck
(http://www.esa.int/Planck), an ESA science mission with instruments and
contributions directly funded by ESA Member States, NASA, and Canada.
Numerical computations were in part carried out on Cray XC50 at Center for
Computational Astrophysics, National Astronomical Observatory of Japan. This
work was supported by JSPS (Japan Society for the Promotion of Science)
KAKENHI Grant Number 19J13898 (TM), 18K03616 (KI), and 17H01110 (HT).

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