#  Is cosmic birefringence due to dark energy or dark matter? A tomographic
approach

Hiromasa Nakatsuka  ICRR, University of Tokyo, Kashiwa, 277-8582, Japan
Toshiya Namikawa  Kavli IPMU (WPI), UTIAS, University of Tokyo, Kashiwa,
277-8583, Japan  Eiichiro Komatsu  Max Planck Institute for Astrophysics,
Karl-Schwarzschild-Str. 1, 85748 Garching, Germany  Kavli IPMU (WPI), UTIAS,
University of Tokyo, Kashiwa, 277-8583, Japan

###### Abstract

A pseudoscalar “axionlike” field, $\phi$, may explain the $3\sigma$ hint of
cosmic birefringence observed in the $EB$ power spectrum of the cosmic
microwave background (CMB) polarization data. Is $\phi$ dark energy or dark
matter? A tomographic approach can answer this question. The effective mass of
dark energy field responsible for the accelerated expansion of the Universe
today must be smaller than $m_{\phi}\simeq 10^{-33}$ eV. If $m_{\phi}\gtrsim
10^{-32}$ eV, $\phi$ starts evolving before the epoch of reionization and we
should observe different amounts of birefringence from the $EB$ power spectrum
at low ($l\lesssim 10$) and high multipoles. Such an observation, which
requires a full-sky satellite mission, would rule out $\phi$ being dark
energy. If $m_{\phi}\gtrsim 10^{-28}$ eV, $\phi$ starts oscillating during the
epoch of recombination, leaving a distinct signature in the $EB$ power
spectrum at high multipoles, which can be measured precisely by ground-based
CMB observations. Our tomographic approach relies on the shape of the $EB$
power spectrum and is less sensitive to miscalibration of polarization angles.

##  I Introduction

A pseudoscalar “axionlike” field is a candidate for dark matter and dark
energy in the Universe [1, 2]. Like pion in the standard model of elementary
particles and fields, a pseudoscalar can couple to the electromagnetic tensor
$F_{\mu\nu}$ and its Hodge dual $\tilde{F}^{\mu\nu}$ via a Chern-Simons term
in the Lagrangian density, ${\cal L}\supset-\frac{1}{4}g\phi
F_{\mu\nu}\tilde{F}^{\mu\nu}$ [3, 4], where $g$ is the axion-photon coupling
constant. This term rotates the plane of linear polarization of photons as
they travel through space filled with $\phi$ [5, 6, 7].

Such a rotation produces non-zero odd-parity $TB$ and $EB$ power spectra of
the cosmic microwave background (CMB) polarization fields, which vanish in the
standard cosmological model [8]. This effect is often referred to as “cosmic
birefringence,” as it resembles birefringence in a material (see Ref. [9] for
a review).

The plane of linear polarization of CMB photons rotates clockwise on the sky
by an angle $\beta=\frac{1}{2}g\int_{t_{\mathrm{LSS}}}^{t_{0}}dt~{}d\phi/dt$,
where $d\phi/dt$ is the total derivative of $\phi$ along the photon
trajectory, and the subscripts “0” and “LSS” denote the present day and the
last scattering surface of CMB photons, respectively. The CMB is an ideal
target for measuring $\beta$, as it is proportional to the path length of
photons when $\phi$ is evolving slowly.

Cosmic birefringence can be caused by $\phi$ of dark energy [10, 11] and dark
matter [12, 13], as well as by possible signatures of quantum gravity [14,
15]. How can we tell the origin? The effective mass of $\phi$,
$m_{\phi}^{2}\equiv d^{2}V/d\phi^{2}$, is the key parameter, where $V(\phi)$
is the potential. The field does not change very much when $m_{\phi}\lesssim
H(t)$, where $H(t)$ is the Hubble expansion rate at a time $t$; thus, $\phi$
would be dark energy today if $m_{\phi}\lesssim H_{0}\equiv H(t_{0})\simeq
10^{-33}$ eV. The $\phi$ field with mass greater than this value would
constitute a fraction of dark matter in the Universe today.

A tantalizing hint for $\beta$ has been found in the $EB$ power spectrum of
the Planck mission with the statistical significance exceeding $3\sigma$ [16,
17, 18]. If confirmed with higher statistical significance in future, it would
have profound implications for the fundamental physics behind dark energy and
dark matter, as well as for quantum gravity. Anticipating such a discovery, in
this paper we show how to determine $m_{\phi}$ using a tomographic approach to
cosmic birefringence.

There are two epochs in which linear polarization of the CMB was generated:
(1) the epoch of recombination of hydrogen atoms and the subsequent decoupling
of photons from plasma at a redshift of $z_{\mathrm{rec}}\simeq 1090$ [19];
and (2) the epoch of reionization of hydrogen atoms at $z_{\mathrm{rei}}\simeq
7$ [20]. The CMB photons that were last-scattered at these epochs would
experience different amounts of cosmic birefringence [21, 22], which changes
the relative amplitudes of the $EB$ power spectrum at low ($l\lesssim 10$) and
high multipoles. We can use this $l$ dependence to infer $\beta$ from $z\simeq
z_{\mathrm{rec}}$ to $z_{\mathrm{rei}}$ and that from $z\simeq
z_{\mathrm{rei}}$ to $0$, i.e., tomography. For example, if $m_{\phi}\gtrsim
10^{-31}$ eV, we should not detect the reionization bump in the $EB$ spectrum
at $l\lesssim 10$ [23]. Such an observation requires a full-sky satellite
mission like LiteBIRD [24] and would rule out $\phi$ being dark energy.

The formula $\beta=\frac{1}{2}g\int_{t_{\mathrm{LSS}}}^{t_{0}}dt~{}d\phi/dt$
assumes an instantaneous last scattering at $t_{\mathrm{LSS}}$, but a finite
duration of last scattering leaves unique signatures in the CMB power spectrum
[12]. If $m_{\phi}\gtrsim 10^{-28}$ eV, $\phi$ starts oscillating during or
earlier than the recombination epoch, modifying the $EB$ power spectrum at
high $l$. This effect can be measured precisely by ground-based CMB
observations such as Simons Observatory [25], South Pole Observatory [26], and
CMB-S4 [27], which opens up new scientific opportunities for CMB experiments.

In this paper, we solve the Boltzmann equation coupled with the equation of
motion (EoM) for $\phi$, assuming that $\phi$ is either a dark energy field or
a “spectator” field with negligible energy density. The energy density of
$\phi$ therefore does not enter the Friedmann equation explicitly. We show how
the shape of the $EB$ power spectrum depends on $m_{\phi}$, and provide a
forecast for future constraints on $m_{\phi}$ and $g\phi$.

The tomographic approach can also mitigate partially the artificial rotation
angle, $\alpha$, by miscalibration of polarization angles of detectors and
other instrumental effects [28, 29, 30, 31, 32]. The artificial rotation
affects the $EB$ power spectrum at all multipoles equally, whereas the
tomographic approach relies on the $l$-dependent effect. For example, the
Galactic foreground emission experiences only a negligible amount of
birefringence, and we can use the different $l$ dependence of the foreground
and CMB power spectra to determine $\alpha$ and $\beta$ simultaneously [33].
As we show in this paper, the difference between recombination and
reionization signals can probe $10^{-32}~{}\mathrm{eV}\lesssim
m_{\phi}\lesssim 10^{-31}$ eV (see Ref. [23] for an earlier, more qualitative
study), and details of the shape of the high-$l$ $EB$ power spectrum can probe
$m_{\phi}\gtrsim 10^{-28}$ eV.

The rest of the paper is organized as follows. In Sec. II, we present the
Boltzmann equation and the EoM for $\phi$ [21, 12, 34, 35]. Our approach is
different from Ref. [36], which did not solve the EoM. In Sec. III, we solve
these equations to calculate the $EB$ power spectrum, and show new features
that are important for cosmic birefringence tomography. In Sec. IV, we
forecast expected constraints on the axion parameters for experiments similar
to LiteBIRD [24], Simons Observatory [25], and CMB-S4 [27]. We discuss
possible improvements for our calculation in Sec. V and conclude in Sec. VI.

We use the Friedmann-Lemaître-Robertson-Walker spacetime with a metric tensor
given by $a^{2}(\eta)\mathrm{diag}(-1,\mathbf{1})$, where $a(\eta)$ is the
scale factor of the expansion of the Universe. We use the conformal time,
$\eta$, as time coordinates unless noted otherwise. We focus on the
homogeneous axion background, $\phi(\eta)$, and ignore inhomogeneity in
$\phi$.

##  II Boltzmann equation for isotropic cosmic birefringence

###  II.1 Setup

We work with the Lagrangian density of axion electrodynamics given by [3, 4]

$\displaystyle\mathcal{L}=-\frac{1}{2}(\partial_{\mu}\phi)^{2}-V(\phi)-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{4}g\phi
F_{\mu\nu}\tilde{F}^{\mu\nu}.$ (1)

The dispersion relation of photons is given by
$\omega_{\pm}^{2}=k^{2}\left(1\mp g\phi^{\prime}/k\right)$, where
$\omega_{\pm}$ is the angular frequency of $\pm$ helicity states [5, 6, 7].
The $+$ and $-$ states correspond to the right and left circular-polarization
modes, respectively, in right-handed coordinates with the $z$ axis taken in
the direction of propagation of photons. The prime denotes the derivative with
respect to $\eta$.

In the WKB limit where $\phi$ varies slowly so that $\omega_{\pm}$ is much
larger than the time evolution of $\phi$, i.e.,
$|\omega_{\pm}^{\prime}|/\omega_{\pm}^{2}\ll 1$, the rotation of the plane of
linear polarization from $\eta$ to the present time is written as [5, 6, 7]

$\displaystyle\beta(\eta)\equiv-\int^{\eta_{0}}_{\eta}\mathrm{d}\eta_{1}~{}\frac{\omega_{+}-\omega_{-}}{2}=\frac{g}{2}\left[\phi(\eta_{0})-\phi(\eta)\right],$
(2)

where $\eta_{0}$ is the conformal time today. Here, we use the CMB convention
for the position angle of linear polarization, i.e., $\beta>0$ is a clockwise
rotation in the sky in right-handed coordinates with the $z$ axis taken in the
direction of observer’s lines of sight. The EoM for $\phi$ is

$\displaystyle\phi^{\prime\prime}+2\frac{a^{\prime}}{a}\phi^{\prime}+a^{2}m_{\phi}^{2}\phi=0,$
(3)

for $V(\phi)=m_{\phi}^{2}\phi^{2}/2$.

The field does not evolve very much when $H(\eta)=a^{\prime}/a^{2}\gg
m_{\phi}$. We choose the initial conditions at $\eta_{\mathrm{in}}$ such that
$H(\eta_{\mathrm{in}})\gg m_{\phi}$, $\phi^{\prime}(\eta_{\mathrm{in}})=0$,
and $\phi(\eta_{\mathrm{in}})=\phi_{\mathrm{in}}$. We do not include the
energy density of $\phi$ in the Friedmann equation explicitly, but use
$H(\eta)$ derived from a flat $\Lambda$ cold dark matter ($\Lambda$CDM) model.
This approximation is valid when the energy density of $\phi$ is negligible
(e.g., a tiny fraction of dark matter) or the axion mass is so small
($m_{\phi}\lesssim 10^{-33}$ eV) that it behaves as dark energy. The axion
field with a tiny energy fraction can still induce a sizable amount of
birefringence [37].

The EoM is a linear equation for $\phi$. We therefore introduce a function,
$f(\eta)\equiv\phi(\eta)/\phi_{\mathrm{in}}$, which satisfies the same EoM as
in Eq. 3 with the initial condition $f(\eta_{\mathrm{in}})=1$. The
birefringence angle is given by

$\displaystyle\beta(\eta)=\frac{g\phi_{\mathrm{in}}}{2}\left[f(\eta_{0})-f(\eta)\right].$
(4)

###  II.2 Boltzmann equation

We work with parity eigenstates of CMB polarization, $E$ and $B$ modes, which
have even and odd parity, respectively [38, 39]. In the standard cosmological
model, the $E$ and $B$ modes are uncorrelated due to parity symmetry. Cosmic
birefringence violates parity symmetry and leads to a correlation between $E$
and $B$ modes [8].

In this paper, we consider only scalar-mode perturbations and ignore tensor
modes. The evolution of linear polarization of CMB photons follows the
Boltzmann equation [19]. We expand Stokes parameters of linear polarization,
$Q$ and $U$, in Fourier space with the wave vector $\mathbf{q}$. We define the
cosine between $\mathbf{q}$ and the photon propagation direction as
$\mu\equiv\mathbf{q}\cdot\mathbf{k}/(qk)$. We then write the Boltzmann
equation for the Fourier coefficients of $Q\pm iU$, ${}_{\pm
2}\Delta_{P}(\eta,q,\mu)$, as [9]

$\displaystyle{}_{\pm 2}\Delta_{P}^{\prime}+iq\mu~{}{}_{\pm 2}\Delta_{P}=$
$\displaystyle\tau^{\prime}\left[-{}_{\pm
2}\Delta_{P}+\sqrt{\frac{6\pi}{5}}~{}{}_{\pm 2}Y_{l}^{0}(\mu)\Pi\right]$
$\displaystyle\pm 2i\beta^{\prime}~{}{}_{\pm 2}\Delta_{P},$ (5)

where ${}_{\pm 2}Y^{m}_{l}$ is the spin-2 spherical harmonics, $\Pi(\eta,q)$
is the polarization source term [38], $\beta^{\prime}\equiv g\phi^{\prime}/2$
gives $\beta(\eta)=\int_{\eta}^{\eta_{0}}d\eta_{1}\beta^{\prime}(\eta_{1})$ in
Eq. 2, and $\tau^{\prime}\equiv a(\eta)n_{e}(\eta)\sigma_{T}$ is the
differential optical depth with the Thomson scattering cross section
$\sigma_{T}$ and the number density of electrons $n_{e}$.

With the $\mu$ dependence of ${}_{\pm 2}\Delta_{P}$ expanded in spin-2
spherical harmonics,

$\displaystyle{}_{\pm 2}\Delta_{P}(\eta,q,\mu)$
$\displaystyle=\sum_{l}i^{-l}\sqrt{4\pi(2l+1)}~{}{}_{\pm
2}\Delta_{P,l}(\eta,q)~{}{}_{\pm 2}Y^{0}_{l}(\mu),$ (6)

the formal solution for the Boltzmann equation is [21]

$\displaystyle{}_{\pm
2}\Delta_{P,l}(\eta_{0},q)=-\frac{3}{4}\sqrt{\frac{(l+2)!}{(l-2)!}}\int^{\eta_{0}}_{0}$
$\displaystyle\mathrm{d}\eta~{}\tau^{\prime}e^{-\tau(\eta)}\Pi\frac{j_{l}(x)}{x^{2}}$
$\displaystyle\times\exp\left[\pm 2i\beta(\eta)\right],$ (7)

where $j_{l}(x)$ is the spherical Bessel function with $x=q(\eta_{0}-\eta)$
and
$\tau(\eta)=\int_{\eta}^{\eta_{0}}\mathrm{d}\eta_{1}\tau^{\prime}(\eta_{1})$.

Cosmic birefringence induces an imaginary part of ${}_{\pm
2}\Delta_{P,l}(\eta_{0},q)$, which leads to $B$ modes. We write the
coefficients of $E$ and $B$ modes as [38]

$\displaystyle\Delta_{E,l}(q)\pm i\Delta_{B,l}(q)\equiv-{}_{\pm
2}\Delta_{P,l}(\eta_{0},q).$ (8)

Using Eqs. (7) and (8), the CMB polarization power spectrum is given by

$\displaystyle C_{l}^{XY}=4\pi\int\mathrm{d}(\ln
q)~{}\mathcal{P}_{s}(q)\Delta_{X,l}(q)\Delta_{Y,l}(q),$ (9)

where $\mathcal{P}_{s}(q)$ is the primordial scalar curvature power spectrum
and $X,~{}Y=E$ or $B$. While cosmic birefringence modifies all polarization
modes, we focus on the $EB$ power spectrum since it is more sensitive than
$TB$ for the current and future generations of CMB experiments with low
polarization noise.

We implement Eq. 7 in the CLASS code [40, 41] and calculate $C_{l}^{EB}$ with
the best-fitting Planck 2018 cosmological parameters for a flat $\Lambda$CDM
model [42]. One significant change made to the code is the treatment of $B$
modes induced by scalar perturbations. Cosmic birefringence transfers a part
of scalar $E$ modes into $B$ modes, and we need to compute $\Delta_{B,l}$
which vanishes otherwise.

###  II.3 Axion mass and the visibility function

Cosmic birefringence tomography relies on two epochs in which CMB polarization
was generated. In Fig. 1, we show the visibility function, the probability
density of photons being last scattered, defined by
$g_{\mathrm{vis}}(\eta)\equiv\sigma_{T}a(\eta)n_{e}(\eta)e^{-\tau(\eta)}$, as
a function of redshift with the thermal history obtained from the RECFAST code
[43, 44, 45]. As expected, the visibility function has the largest value at
the recombination and photon decoupling epoch, $z_{\mathrm{rec}}\simeq 1090$.
The second peak appears at $z_{\mathrm{rei}}\simeq 7$.

Figure 1:  The visibility function as a function of redshift $z$. The orange
and blue regions show the recombination and reionization epochs, respectively.
The reionization is included using the tanh model with $z_{\mathrm{rei}}=7.82$
[46, 42].

In Fig. 2, we compare the evolution of $\phi$ with the epochs of recombination
and reionization. The axion with $m_{\phi}=10^{-28.0}$ eV (green line) starts
evolving significantly before recombination, which experiences a reduction in
the amount of birefringence [12, 13].

The axion with $m_{\phi}=10^{-30.3}$ eV (blue line) starts evolving after
recombination but has decayed before reionization; thus, very little cosmic
birefringence would occur after reionization. The axion with
$m_{\phi}=10^{-31.2}$ eV (red line) starts oscillating during reionization,
and that with $m_{\phi}=10^{-32.3}$ eV (black line) evolves only after
reionization. The amplitude of the reionization bump in $C_{l}^{EB}$ is
therefore sensitive to $10^{-32}~{}\mathrm{eV}\lesssim m_{\phi}\lesssim
10^{-31}$ eV.

Figure 2:  Evolution of $\phi$ for $m_{\phi}=10^{-28.0}$ (green), $10^{-30.3}$
(blue), $10^{-31.2}$ (red), and $10^{-32.3}$ eV (black). The shaded regions
show recombination and reionization epochs as in Fig. 1.

For $m_{\phi}\lesssim 10^{-32}$ eV we expect $C_{l}^{EB}$ to be scaled by a
single $\beta$ at all $l$. The polarization modes [Eq. 8] are simply given by
$\Delta_{E,l}\pm i\Delta_{B,l}=e^{\pm 2i\beta}(\tilde{\Delta}_{E,l}\pm
i\tilde{\Delta}_{B,l})$, where the tildes denote the values before cosmic
birefringence. Then, the polarization power spectra after birefringence are
given by [47, 21]

$\displaystyle C_{l}^{EE}$
$\displaystyle=\cos^{2}(2\beta)\tilde{C}_{l}^{EE}+\sin^{2}(2\beta)\tilde{C}_{l}^{BB},$
(10) $\displaystyle C_{l}^{BB}$
$\displaystyle=\cos^{2}(2\beta)\tilde{C}_{l}^{BB}+\sin^{2}(2\beta)\tilde{C}_{l}^{EE},$
(11) $\displaystyle C_{l}^{EB}$
$\displaystyle=\frac{1}{2}\sin(4\beta)(\tilde{C}_{l}^{EE}-\tilde{C}_{l}^{BB}).$
(12)

In this case, $C_{l}^{EB}\propto C_{l}^{EE}$ when we ignore the primordial $B$
modes, and $\beta$ is degenerate with the instrumental miscalibration angle
$\alpha$ [29, 30]. The tomography approach breaks this degeneracy by using the
change in shape of $C_{l}^{EB}$ induced by either $m_{\phi}\gtrsim 10^{-32}$
eV [23] or the Galactic foreground [33].

##  III Cosmic birefringence from recombination and reionization

###  III.1 Toy example

To build an intuitive understanding of the full numerical result for
$C_{l}^{EB}$ due to evolving $\phi$, we first study a toy example in which
$\beta(z)$ integrated from $z$ to the present time [Eq. 2] changes abruptly:

$\displaystyle\beta(z)=\begin{cases}0&\quad\mathrm{for}\quad z=0\\\
\beta_{\mathrm{rei}}&\quad\mathrm{for}\quad 0<z\leq 10\\\
\beta_{\mathrm{rec}}&\quad\mathrm{for}\quad 10<z\end{cases},$ (13)

where $\beta_{\mathrm{rei}}$ and $\beta_{\mathrm{rec}}$ are piecewise constant
angles integrated out to $z=10$ and recombination, respectively.

In the top panel of Fig. 3 we show $C_{l}^{EB}$ for
$\\{\beta_{\mathrm{rei}}[\mathrm{deg}],~{}\beta_{\mathrm{rec}}[\mathrm{deg}]\\}=\\{1,~{}1\\}$
(red), $\\{0.5,~{}1\\}$ (green), and $\\{0,~{}1\\}$ (blue). All lines coincide
at $l\gtrsim 20$ because $\beta_{\mathrm{rec}}$ is the same. As
$\beta_{\mathrm{rei}}$ decreases, the reionization bump of $C_{l}^{EB}$ also
decreases. However, the reionization bump does not disappear even for
$\beta_{\mathrm{rei}}=0$.

Figure 3:  The $EB$ power spectrum from piecewise constant angles given in Eq.
13: $\beta=\beta_{\mathrm{rec}}$ for $z>10$ and $\beta=\beta_{\mathrm{rei}}$
for $z\leq 10$. The top and bottom panels show $l(l+1)C_{l}^{EB}/(2\pi)$ and
the effective angles given in Eq. 16, respectively.

The shape of $C_{l}^{EB}$ can be understood as follows. The $E$ and $B$ modes
are written as

$\displaystyle\Delta_{E,l}\pm i\Delta_{B,l}$
$\displaystyle=\sum_{x=\mathrm{rei,~{}rec}}e^{\pm
2i\beta_{x}}(\tilde{\Delta}_{E,l}^{(x)}\pm i\tilde{\Delta}_{B,l}^{(x)}).$ (14)

Ignoring the primordial $B$ modes, $C_{l}^{EB}$ is given by

$\displaystyle C_{l}^{EB}=$
$\displaystyle\frac{1}{2}\sin(4\beta_{\mathrm{rec}})\tilde{C}_{l}^{E\mathrm{rec},E\mathrm{rec}}+\frac{1}{2}\sin(4\beta_{\mathrm{rei}})\tilde{C}_{l}^{E\mathrm{rei},E\mathrm{rei}}$
$\displaystyle+\sin\left[2(\beta_{\mathrm{rei}}+\beta_{\mathrm{rec}})\right]\tilde{C}_{l}^{E\mathrm{rei},E\mathrm{rec}},$
(15)

where $\tilde{C}_{l}^{Ex,Ey}$ is the cross power spectrum of
$\tilde{\Delta}_{E,l}^{(x)}$ and $\tilde{\Delta}_{E,l}^{(y)}$ with $x,~{}y=$
rei, rec. The first term dominates at $l\gtrsim 20$. The second term produces
the reionization bump at $l\lesssim 10$. The third term was overlooked in Ref.
[23]. The cross correlation of reionization and recombination $E$ modes
induces a small reionization bump even when the rotation angle is zero at
reionization. This effect appears in the blue line of Fig. 3 where a small
bump is seen at $l\lesssim 10$. Therefore, there is always some $C_{l}^{EB}$
at low $l$.

When the rotation angle depends on time, $C_{l}^{EB}$ is no longer given by
Eq. 12. We thus define an effective angle at each $l$ as

$\displaystyle\beta_{l}^{\mathrm{(eff)}}\equiv\frac{1}{4}\arcsin\left(\frac{2x_{l}}{1+x_{l}^{2}}\right)\quad\mathrm{with}\quad{x_{l}=\frac{C_{l}^{EB}}{C_{l}^{EE}}},$
(16)

which is defined so as to reproduce Eq. 12 for the simplest case. Note that
$\beta_{l}^{\mathrm{(eff)}}\simeq C_{l}^{EB}/(2C_{l}^{EE})$ for
$C_{l}^{EB}/C_{l}^{EE}\ll 1$. In the bottom panel of Fig. 3, we show effective
angles for the piecewise constant angles given in Eq. 13. The effective angle
reproduces $\beta_{l}^{\mathrm{(eff)}}=\beta_{\mathrm{rec}}$ for $l\gg 20$,
where the recombination contribution dominates. It converges to
$\beta_{\mathrm{rei}}$ for $l\ll 20$, where the reionization contribution
dominates.

###  III.2 $EB$ power spectrum from axion dynamics

Figure 4:  Same as Fig. 3, but for axion dynamics shown in Fig. 2.

In the left panel of Fig. 4 we present the full Boltzmann solution for
$C_{l}^{EB}$ with axion dynamics shown in Fig. 2. In the right panel we show
the corresponding $\beta^{\mathrm{(eff)}}_{l}$. We use
$g\phi_{\mathrm{in}}/2=-1~{}\mathrm{deg}\simeq-0.01745$, which determines the
overall amplitude of $\beta$ via Eq. 4.

####  III.2.1  Reionization bump as a probe of $m_{\phi}\lesssim 10^{-31}$ eV

We first study $m_{\phi}\lesssim 10^{-30}$ eV. As $\phi$ starts evolving well
after the recombination epoch, the only difference appears in the reionization
bump. We find the largest amplitude for $m_{\phi}\simeq 10^{-32}$ eV, for
which $\phi$ starts evolving only after reionization. The amplitude decreases
as $m_{\phi}$ increases; however, the reionization bump does not disappear
even for $m_{\phi}=10^{-30.3}$ eV, as explained in Sec. III.1. We can
therefore probe $m_{\phi}$ that falls between the black and blue lines in the
left panel of Fig. 4.

To make this statement more quantitative, we define the ratio of the effective
angles for low and high $l$ as

$\displaystyle\frac{\sum_{l=2}^{10}\beta^{\mathrm{(eff)}}_{l}/9}{\sum_{l=11}^{500}\beta^{\mathrm{(eff)}}_{l}/490}.$
(17)

In Fig. 5 we find that the ratio is sensitive to the change in mass over
$10^{-32}~{}\mathrm{eV}\lesssim m_{\phi}\lesssim 10^{-31}$ eV. We thus
conclude that this is the range of $m_{\phi}$ we can probe using the relative
amplitudes of the reionization bump and the high-$l$ $EB$ power spectrum.

Figure 5:  Ratio of the effective cosmic birefringence angles from
reionization and recombination defined in Eq. 17. The ratio is sensitive to
the change in the axion mass over $10^{-32}~{}\mathrm{eV}\lesssim
m_{\phi}\lesssim 10^{-31}$ eV.

####  III.2.2  High-$l$ features as a probe of $m_{\phi}\gtrsim 10^{-28}$ eV

Figure 6:  The $EB$ power spectrum for large $m_{\phi}$. The left panel shows
$l(l+1)C_{l}^{EB}/(2\pi)$ for $m_{\phi}=10^{-28.8}$ (black), $10^{-27.9}$
(red), $10^{-27.8}$ (blue), and $10^{-27.7}$ eV (green). The dotted vertical
lines show the positions of the third peak. The right panel shows axion
dynamics for each $m_{\phi}$. The shaded region shows the recombination epoch
as in Fig. 1.

The axion starts oscillating during recombination for $m_{\phi}\gtrsim
10^{-28}$ eV. Photons last-scattered at different times experience different
amounts of rotation of the plane of linear polarization, which can lead to
partial cancellation of cosmic birefringence [12, 48, 13] as well as to
complex features in $C_{l}^{EB}$ at high $l$ that can be used to probe
$m_{\phi}\gtrsim 10^{-28}$ eV in a completely new way.

In the left panel of Fig. 6 we show $C_{l}^{EB}$ for axion dynamics shown in
the right panel. There are two effects on $C_{l}^{EB}$: the overall amplitude
and shape.

We first discuss the amplitude, which changes dramatically depending on the
value of $\phi$ during recombination. For $m_{\phi}=10^{-28.8}$ eV, $\phi$ is
nearly constant, which results in $C_{l}^{EB}\propto C_{l}^{EE}$ except for
the reionization bump. For $m_{\phi}=10^{-27.9}$ eV, $\phi$ starts evolving
during recombination, resulting in a smaller $C_{l}^{EB}$. For
$m_{\phi}=10^{-27.8}$ eV, $\phi$ averaged over recombination is tiny,
resulting in a highly suppressed $C_{l}^{EB}$. For $m_{\phi}=10^{-27.7}$ eV,
$\phi$ averaged over recombination is negative, hence $C_{l}^{EB}<0$.

In the previous work that did not solve the Boltzmann equation, the amplitude
of $C_{l}^{EB}$ has been calculated by averaging $\phi$ over the visibility
function [13, 48, 37]:

$\displaystyle\braket{\beta}$
$\displaystyle\equiv\frac{g}{2}\left[\phi(\eta_{0})-\braket{\phi}_{\mathrm{LSS}}\right],$
(18)

where

$\displaystyle\braket{\phi}_{\mathrm{LSS}}$
$\displaystyle\equiv\frac{\int^{\eta_{z=10}}_{0}\mathrm{d}\eta~{}g_{\mathrm{vis}}(\eta)\phi(\eta)}{\int^{\eta_{z=10}}_{0}\mathrm{d}\eta~{}g_{\mathrm{vis}}(\eta)},$
(19)

with $\eta_{z=10}$ being a conformal time at $z=10$. Since we focus on the
rotation angle from recombination, the average is limited to $z>10$ with
$\int^{\eta_{z=10}}_{0}\mathrm{d}\eta~{}g_{\mathrm{vis}}(\eta)\simeq 0.95$. We
use $g_{\mathrm{vis}}(\eta)$ computed with CLASS as shown in Fig. 1.

In Fig. 7 we compare $\beta^{\mathrm{(eff)}}_{l}$ computed from the Boltzmann
equation and Eq. 18 for $m_{\phi}=10^{-27.9}$ eV (top panel) and
$m_{\phi}=10^{-27.8}$ eV (bottom). It is clear that
$\beta^{\mathrm{(eff)}}_{l}$ shows much more complex features than just the
average value $\braket{\beta}$ shown by the horizontal lines.

Figure 7:  The effective rotation angles from the Boltzmann equation (wiggly
lines) and $\braket{\beta}$ given in Eq. 18 (horizontal lines) for
$m_{\phi}=10^{-27.9}$ eV (top) and $10^{-27.8}$ eV (bottom).

How can we understand such complex dependence of $\beta^{\mathrm{(eff)}}_{l}$
(hence $C_{l}^{EB}$) on $m_{\phi}$? We find that the location of the acoustic
peaks in $C_{l}^{EB}$ for $m_{\phi}=10^{-27.9}$ eV shifts to higher $l$
compared to that for $m_{\phi}=10^{-28.8}$ eV (see the vertical dotted lines
in Fig. 6). This peak shift is the origin of the oscillating behavior of
$\beta^{\mathrm{(eff)}}_{l}$ at $l>100$.

The peak location is determined by $D_{A}/r_{s}$, where $r_{s}$ and $D_{A}$
are the sound horizon and angular diameter distance at last scattering,
respectively. In our fiducial cosmological model, $D_{A}/r_{s}\simeq(1+z_{\rm
LSS})^{1/2}$ where $z_{\rm LSS}$ is the redshift of last scattering. For
$m_{\phi}=10^{-27.9}$ eV, $\phi$ starts evolving before recombination, and the
$B$ modes are mainly generated in the early stage of recombination. In such
case, $D_{A}/r_{s}$ for the induced $B$ modes becomes effectively large and
the peaks shift to higher $l$. If $m_{\phi}=10^{-27.7}$ eV, the time when
polarization is mostly produced is close to that for $m_{\phi}=10^{-28.8}$ eV
and the peak locations are almost identical to those of the black line while
the amplitude is negative.

These features are important: we can use this complex dependence of
$C_{l}^{EB}$ to determine $m_{\phi}$ in a completely new manner. While the
overall amplitude $\braket{\beta}$ is degenerate with the miscalibration angle
$\alpha$ (unless we have access to $l\lesssim 10$), the $l$ dependence is not.
This is relevant for ground-based CMB experiments (Sec. IV.2).

##  IV Forecast

###  IV.1 Simultaneous determination of $\alpha$ and cosmic birefringence with
the reionization signal

We first consider the case in which we simultaneously constrain cosmic
birefringence and miscalibration angles, $\alpha$, using the reionization bump
in $C_{l}^{EB}$ [23]. As explained in Sec. III.2.1, the axion with
$m_{\phi}\gtrsim 10^{-32}$ eV changes relative amplitudes of $C_{l}^{EB}$ at
low and high $l$, which cannot be mimicked fully by $\alpha$ because it
affects all multipoles equally via Eq. 12 with $\beta\to\alpha$.

In Ref. [23], $C_{l}^{EB}$ was modeled as the sum of the reionization and
recombination contributions:

$\displaystyle C^{EB}_{l}=\sum_{x={\rm rei,rec}}2\beta_{x}(C^{EE,x,{\rm
lens}}_{l}-C^{BB,x,{\rm lens}}_{l})\,,$ (20)

where $C^{EE,x,{\rm lens}}_{l}$ and $C^{BB,x,{\rm lens}}_{l}$ are the lensed
$E$\- and $B$-mode power spectra, respectively. As shown in Eq. 15, however,
we cannot decompose $C_{l}^{EB}$ in this way. Therefore, we constrain the
axion parameters instead of $\beta_{x}$.

Assuming that the observed $E$\- and $B$ modes obey a multivariate Gaussian
distribution with zero mean, the Fisher information matrix is given by [49]

$\displaystyle{\mathbf{F}}_{ij}=f_{\rm
sky}\sum_{l=2}^{l_{\mathrm{max}}}\frac{2l+1}{2}\mathrm{Tr}\left({\mathbf{C}}^{-1}_{l}\dfrac{\partial{\mathbf{C}}_{l}}{\partial
p_{i}}{\mathbf{C}}^{-1}_{l}\dfrac{\partial{\mathbf{C}}_{l}}{\partial
p_{j}}\right)\bigg{|}_{p_{i}=p_{i,\rm fid}}\,,$ (21)

where $l_{\mathrm{max}}$ is the maximum multipole included in the analysis,
$f_{\rm sky}$ is a sky fraction used for the analysis, $p_{i}$ are the
parameters to be constrained, $p_{i,\rm fid}$ are the fiducial parameter
values, and the covariance matrix of the observed $E$\- and $B$ modes is given
by

$\displaystyle{\mathbf{C}}_{l}\equiv\begin{pmatrix}\widehat{C}_{l}^{EE}&\widehat{C}_{l}^{EB}\\\
\widehat{C}_{l}^{EB}&\widehat{C}_{l}^{BB}\end{pmatrix}\,.$ (22)

The covariance matrix contains the total power spectra, $\widehat{C}^{EE}$,
$\widehat{C}^{EB}$ and $\widehat{C}^{BB}$, including the lensed CMB, noise,
and Galactic foregrounds after component separation. The $1\,\sigma$
constraint on $p_{i}$ is given by
$\sigma(p_{i})=(\\{{\mathbf{F}}^{-1}\\}_{ii})^{1/2}$.

We consider two parameters, $p_{1}\equiv g\phi_{\rm in}/2$ and
$p_{2}\equiv\alpha$, and set their fiducial values to be $p_{i,\rm fid}\equiv
0$. As the $E$\- and $B$-mode spectra do not have a linear term of $p_{i}$ and
their derivatives with respect to $p_{i}$ at $p_{i,\rm fid}=0$ vanish,
$\partial\mathbf{C}_{l}/\partial p_{i}|_{p_{i}=0}$ contains only off-diagonal
elements, $\partial C^{EB}_{l}/\partial p_{i}|_{p_{i}=0}$. The Fisher matrix
simplifies to [23]

$\displaystyle{\mathbf{F}}_{ij}$ $\displaystyle=f_{\rm
sky}\sum_{l=2}^{l_{\mathrm{max}}}\frac{2l+1}{\widehat{C}_{l}^{EE}\widehat{C}_{l}^{BB}}\dfrac{\partial\widehat{C}^{EB}_{l}}{\partial
p_{i}}\dfrac{\partial\widehat{C}^{EB}_{l}}{\partial
p_{j}}\bigg{|}_{p_{i}=0}\,.$ (23)

We assume a LiteBIRD-like white noise ($2\mu$K-arcmin), angular resolution
($30$ arcmin), $f_{\rm sky}=0.7$, and the residual Galactic foregrounds
obtained by Ref. [50]. We set $l_{\mathrm{max}}=500$ because of the angular
resolution.

The impact of gravitational lensing on $C_{l}^{EB}$ would be negligible in the
Fisher matrix. Lensing does not create $C_{l}^{EB}$ but only distorts small-
scale polarization anisotropies. To see this, we first note that $C_{l}^{EB}$
is approximately given by Eq. 15. As discussed in Ref. [51], the two
operators, lensing and birefringence, commute, and lensing replaces
$C_{l}^{EE}$ in Eq. 15 with
$C_{l}^{EE,\mathrm{lens}}-C_{l}^{BB,\mathrm{lens}}$. As
$C_{l}^{BB,\mathrm{lens}}\ll C_{l}^{EE,\mathrm{lens}}$ at $l\ll 5000$ and
$C_{l}^{EE}$ is modified by lensing only at high $l$, we ignore the
gravitational lensing effect on $C_{l}^{EB}$.

Figure 8:  The expected $1\,\sigma$ error contours in the two-dimensional
parameter space ($\alpha$ and $g\phi_{\mathrm{in}}/2$ in units of degrees).
The axion masses are $m_{\phi}=10^{-32}$ eV (top) and $10^{-30}$ eV (bottom).
We show the results from the full $l$ range (black), reionization bump ($2\leq
l\leq 20$; orange dashed), and high multipoles ($21\leq l\leq 500$; green dot-
dashed). We assume a LiteBIRD-like experiment without delensing.

Fig. 8 shows the expected error contours in the two-dimensional parameter
space, $g\phi_{\rm in}/2$ and $\alpha$, for a given $m_{\phi}$. For
$m_{\phi}=10^{-32}$ eV the reionization bump in $C_{l}^{EB}$ is close to its
maximum amplitude as explained in Sec. III.2.1. In this case $C_{l}^{EB}$
becomes close to that of the miscalibration angle, and $g\phi_{\rm in}/2$ and
$\alpha$ are strongly degenerate. For $m_{\phi}=10^{-30}$ eV the reionization
bump is suppressed and the degeneracy is reduced. We thus confirm and make
more precise the result of Ref. [23].

###  IV.2 Constraining the axion mass

Next, we consider joint constraints on $m_{\phi}$, $g\phi_{\rm in}/2$, and
$\alpha$. Since $C_{l}^{EB}$ depends on $m_{\phi}$ nonlinearly, the Fisher
matrix formalism, in which the errors are estimated from curvature of the
posterior distribution around the fiducial value, does not provide accurate
results. We thus use the likelihood analysis. We define $\chi^{2}$ as

$\displaystyle\chi^{2}(\mathbf{p})=f_{\rm sky}\sum_{l=l_{\rm min}}^{l_{\rm
max}}(2l+1)\frac{[\widehat{C}^{EB}_{l}-C^{EB,\rm
th}_{l}(\mathbf{p})]^{2}}{\widehat{C}^{EE}_{l}\widehat{C}^{BB}_{l}}\,,$ (24)

where $C^{EB,\rm th}_{l}$ is a theoretical model for the $EB$ power spectrum
and is given as the sum of the contributions from cosmic birefringence and
$\alpha$. For a given observed $\widehat{C}^{EB}_{l}$ we compute $\chi^{2}$
for each parameter set, $\mathbf{p}=(m_{\phi},g\phi_{\rm in}/2,\alpha)$, and
obtain the posterior distribution,
$P(\mathbf{p}|\widehat{C}^{EB})\propto\exp\left[-\chi^{2}(\mathbf{p})/2\right]$.

We consider specifications similar to LiteBIRD [24], Simons Observatory (SO;
[25]), and CMB-S4 [27]. For SO we assume $l_{\rm min}=100$, $l_{\rm
max}=2500$, $6\mu$K-arcmin white noise, $1$ arcmin Gaussian beam, $f_{\rm
sky}=0.4$, and no foregrounds. We also assume $30\%$ residual lensing-induced
$B$ modes after delensing [52]. For CMB-S4 we assume the same beam,
$l_{\mathrm{min}}$, $l_{\mathrm{max}}$, and $f_{\rm sky}$, but with
$1\mu$K-arcmin noise and $10\%$ residual lensed $B$ modes [53].

Fig. 9 shows the expected $1$ and $2$ $\sigma$ contours on $\log_{10}m_{\phi}$
and $g\phi_{\rm in}/2$. We consider the LiteBIRD-like experiment with
specifications given in Sec. IV.1. The fiducial axion mass is
$m_{\phi,\mathrm{fid}}=10^{-30}\,$eV and $g\phi_{\rm in}/2$ is obtained from
$\beta_{\rm rec}=0.35\,$deg [16, 17, 18] via Eq. 18. The fiducial value of
$\alpha$ is set to zero. We marginalize the posterior distribution over
$\alpha$ using a prior distribution obtained from calibration of instruments.
Specifically, we use a Gaussian prior,
$\exp[-\alpha^{2}/(2\sigma_{\alpha}^{2})]$, with $\sigma_{\alpha}=0.5$ deg
(top) and 0.1 deg (bottom). The former precision is achieved already for
calibration of the current generation of CMB experiments [54, 55, 56], whereas
the latter can be achieved by employing new calibration strategy [57, 58, 59,
60].

Figure 9:  The expected 1 and 2 $\sigma$ error contours on $\log_{10}m_{\phi}$
and $|g\phi_{\rm in}/2|$ (in units of degrees) for
$m_{\phi,\mathrm{fid}}=10^{-30}$ eV. The miscalibration angle is marginalized
over using a Gaussian prior with $\sigma_{\alpha}=0.5$ deg (top) and 0.1 deg
(bottom). The fiducial value of $g\phi_{\rm in}/2$ is chosen to give
$\beta_{\rm rec}=0.35\,$deg (red dots). The black dashed lines show
$|g\phi_{\rm in}/2|$ for $\beta_{\rm rec}=0.35\,$deg at each $m_{\phi}$. We
assume a LiteBIRD-like experiment.

The black dashed lines show the values of $|g\phi_{\rm in}/2|$ giving
$\beta_{\rm rec}=0.35\,$deg for each $m_{\phi}$. The spikes in
$m_{\phi}>10^{-28}$ eV occur when $C_{l}^{EB}$ at high $l$ becomes highly
suppressed (see the blue line in Fig. 6). That is to say, we need a larger
value of $|g\phi_{\mathrm{in}}/2|$ to compensate for suppression of
$C_{l}^{EB}$ by a small value of $\braket{\phi}_{\mathrm{LSS}}$ [Eq. 19].

We find that the prior on $\alpha$ tightens the constraint on the overall
amplitude parameter ($g\phi_{\mathrm{in}}$) significantly, but $m_{\phi}$ can
be constrained almost independently of the prior. We also find the same trend
when removing the prior on $\alpha$ entirely. This is because the information
on $m_{\phi}$ comes from the shape of $C_{l}^{EB}$. For
$m_{\phi,\mathrm{fid}}=10^{-30}$ eV the reionization bump is already at its
minimum (see the blue line in Fig. 4). Therefore, the shape of $C_{l}^{EB}$
can tell us that $m_{\phi}$ is greater than $10^{-32}$ eV, but cannot tell how
large it is until $m_{\phi}$ is so large that it affects the shape at high
$l$. This explains an upper bound, $m_{\phi}\lesssim 10^{-28}$ eV.

The “islands” of parameter space seen in $m_{\phi}>10^{-28}$ eV are allowed
because the peak locations of $C_{l}^{EB}$ for the respective $m_{\phi}$ in
the islands happen to coincide with those for the fiducial mass of $10^{-30}$
eV, while the amplitudes are adjusted by varying $g\phi_{\mathrm{in}}$. The
islands shrink when $g\phi_{\mathrm{in}}$ is constrained by a tighter prior on
$\alpha$.

For $m_{\phi,\mathrm{fid}}=10^{-28}$ eV the shape of $C_{l}^{EB}$ at high $l$
becomes quite different from that of $\alpha$, which enables us to determine
$m_{\phi}$. However, the LiteBIRD-like experiment cannot determine such a
large $m_{\phi}$ accurately because of the limited angular resolution, giving
only discrete islands of constraints (Fig. 10). Therefore, it gives
effectively a lower bound for $m_{\phi}$ almost independently of
$\sigma_{\alpha}$.

Figure 10:  Same as Fig. 9 but for $m_{\phi,\mathrm{fid}}=10^{-28}$ eV.

Ground-based experiments such as SO- and S4-like experiments have better
sensitivity to large $m_{\phi}$. In Fig. 11 we show the expected constraints
for SO ($m_{\phi,\mathrm{fid}}=10^{-28}$ eV). The constraints tighten
significantly compared to the LiteBIRD-like case. Some degeneracy between
$g\phi_{\rm in}$ and $m_{\phi}$ still exist for $\sigma_{\alpha}=0.5$ deg (top
panel): the shift of peak locations can be absorbed partially by a combination
of the rescaled $C_{l}^{EB}$ and $\alpha$. The degeneracy is eliminated when a
tighter prior is used (bottom panel). With CMB-S4 we can determine $m_{\phi}$
precisely, independent of $\alpha$ (see Fig. 12).

Figure 11:  Same as Fig. 10 but for an SO-like experiment.  Figure 12:  Same
as the top panel of Fig. 11 but for a CMB-S4-like experiment. The result for
$\sigma_{\alpha}=0.1$ deg is similar.

As SO and CMB-S4 cannot measure the reionization bump, they can only place an
upper bound on $m_{\phi}$ if the fiducial mass is $10^{-30}\,$eV. Thus,
LiteBIRD and ground-based experiments are highly complementary.

##  V Discussion

We have made some simplifying assumptions in our calculation of $C_{l}^{EB}$.
First, we did not include the energy density of $\phi$ in the Friedmann
equation explicitly. This assumption can be justified to some extent. For
$m_{\phi}\lesssim 10^{-33}$ eV, $\phi$ acts as dark energy and its energy
density is included approximately as a cosmological constant in the Friedmann
equation for $\Lambda$CDM cosmology. For $10^{-32}~{}\mathrm{eV}\lesssim
m_{\phi}\lesssim 10^{-25.5}~{}\mathrm{eV}$, $\phi$ acts as a small fraction of
dark matter today with the density parameter $\Omega_{\phi}h^{2}\lesssim
0.006$ [61], which may be ignored for the current study. However, the change
in shape of $C_{l}^{EB}$ at high $l$ is a subtle effect, which can be
influenced quantitatively when $\Omega_{\phi}$ is included in the Friedmann
equation. We leave the full treatment for future work.

Second, we did not vary cosmological parameters when calculating the expected
future constraints on the axion parameters. This assumption can also be
justified, as the effect of $\phi$ on $C_{l}^{EB}$ is distinct from the
cosmological parameter dependence of the parity-even temperature and
polarization power spectra. Nevertheless, there may still be some subtle
correlation between the cosmological parameters and the axion parameters,
which should be accounted for when the axion energy density is included in the
Friedmann equation.

We now discuss possible future extensions of the calculation. We did not
include inhomogeneity in $\phi$, which causes anisotropic polarization
rotation [62, 63, 64]. While there is no evidence for anisotropic
birefringence [65, 66, 67, 68], it seems natural to expect discovery of
anisotropy if the $3\sigma$ hint of isotropic birefringence [16, 17, 18] is
confirmed with higher statistical significance in future. Thus, incorporating
anisotropic birefringence in the Boltzmann equation [35, 48] would be a
natural next step.

We ignored the gravitational lensing effect on $C_{l}^{EB}$. The lensing would
smear the acoustic peaks of $C_{l}^{EB}$ and enhance the small-scale power,
but these effects would not be degenerate with cosmic birefringence.
Nevertheless, for completeness, the impact of the lensing effect will be
included in future work.

We have so far focused on cosmic birefringence by a single axion field, but it
is entirely possible that cosmic birefringence is induced by multiple fields
[15, 69]. In this case, the time evolution of $\beta(\eta)$ becomes more
interesting, which can be constrained by the tomographic approach. In this
paper we considered two epochs, reionization and recombination, during which
the polarization is efficiently generated. Other sources of polarization
include the polarized Sunyaev-Zeldovich effect in clusters of galaxies, the
so-called remote quadrupole [70, 71, 72, 73]. The polarization is generated
after the epoch of reionization, and we can in principle use such large-scale
polarization signals to probe $\beta(\eta)$ in a late-time Universe.

##  VI Conclusion

In this paper, we solved the Boltzmann equation coupled with the EoM for an
axionlike field $\phi$ to calculate the detailed shape of the $EB$ power
spectrum of the CMB due to cosmic birefringence. There are two critical axion
masses: (1) $m_{\phi}\gtrsim 10^{-32}$ eV, for which relative amplitudes of
the reionization bump ($l\lesssim 10$) and the high-$l$ power spectrum are
modified; and (2) $m_{\phi}\gtrsim 10^{-28}$ eV, for which the evolution of
$\phi$ during recombination yields complex features (such as a shift in the
locations of acoustic peaks) at high $l$. Such a change in shape cannot be
mimicked fully by the miscalibration angle $\alpha$, offering a powerful probe
of $m_{\phi}$. In Ref. [23], this phenomenon was called “cosmic birefringence
tomography,” as it allows us to measure the time evolution of $\phi$.

Probing the first critical mass requires a full-sky coverage by a satellite
mission such as LiteBIRD [24], whereas the second one can be probed by ground-
based experiments [25, 26, 27]. The important application of tomography is to
distinguish whether $\phi$ is dark energy or (a fraction of) dark matter
today. A convincing detection of the relative amplitude change of the low- and
high-$l$ power spectrum by LiteBIRD would rule out $\phi$ being dark energy.
Ground-based experiments can constrain the value of $m_{\phi}$, especially at
$\gtrsim 10^{-28}$ eV, almost independently of $\alpha$. Together they can
discover new physics and provide new scientific opportunities for CMB
experiments [9].

Finally, the $EB$ data at high $l$ from on-going ground-based CMB experiments
such as Polarbear [74], Atacama Cosmology Telescope [75], and South Pole
Telescope [76] may already set interesting constraints on $m_{\phi}$.

###### Acknowledgements.

We thank K. Murai, I. Obata, and M. Shiraishi for discussion and comments.
This work was supported in part by JSPS KAKENHI Grant No. JP19J21974 (H.N.),
No. JP20H05850 (E.K.) and No. JP20H05859 (T.N. and E.K.), Advanced Leading
Graduate Course for Photon Science (H.N.), the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) under Germany’s Excellence Strategy -
EXC-2094 - 390783311 (E.K.), and the European Union’s Horizon 2020 research
and innovation programme under the Marie Skłodowska-Curie grant agreement No.
101007633 (E.K.). The Kavli IPMU is supported by World Premier International
Research Center Initiative (WPI), MEXT, Japan.

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