# Cosmic microwave background spectral distortions
from Rayleigh scattering at second order

Atsuhisa Ota1<EMAIL_ADDRESS>1HKUST Jockey club Institute for Advanced Study,
The Hong Kong University of Science and Technology, Clearwater Bay, Hong Kong,
P.R.China

###### Abstract

Cosmic microwave background (CMB) spectral distortion from Rayleigh scattering
is calculated for the first time in rigorous second-order cosmological
perturbation theory. The new spectral distortion is sensitive to acoustic
dissipation at $10^{-2}<k{\rm Mpc}/h<1$, which slightly extends the scale
constrained by the CMB anisotropies. The spectral shape is different from
either temperature perturbations or any other traditional spectral distortions
from Compton scattering, such as $y$ and $\mu$. The new spectral distortion is
not formed in the late Universe, unlike the thermal Sunyaev-Zel’dovich effect
degenerated with the primordial $y$ distortions since photons must be hot for
Rayleigh scattering. Therefore, ideal measurements can distinguish the signal
from the other effects and extract new information during recombination.
Assuming cosmological parameters consistent with the recent CMB anisotropy
measurements, we find the new spectral distortion is $6.5\times
10^{-3}$Jy/str, which is one order of magnitude smaller than the currently
proposed target sensitivity range of voyage 2050.

Keywords

##  I Introduction

Spectral distortions of the cosmic microwave background (CMB) are potential
cosmological observables for the next decades Zeldovich:1969ff ;
Sunyaev:1970eu ; Hu:1992dc . Although cosmologists have prioritized the
anisotropy measurements to spectrum measurements since COBE FIRAS reported the
almost ideal black body spectrum in 1996 Fixsen:1996nj , several observational
projects are envisioned or ongoing targeting tiny spectral distortions
Kogut:2011xw ; Kogut:2019vqh ; PRISM:2013fvg , including extreme measurements
proposed for voyage 2050 Chluba:2019kpb ; Chluba:2019nxa . The spectral
distortions are formed due to various non-equilibrium phenomena in the
Universe. For example, acoustic dissipation of short-scale cosmological
perturbations Hu:1994bz ; Chluba:2012gq ; Chluba:2012we ; Dent:2012ne ;
Ota:2014hha ; Ota:2018zwm , atomic processes during recombination Wong:2005yr
; Sunyaev:2009qn ; Chluba:2015gta ; Chluba:2008aw , and energy injection from
unknown sectors Chluba:2009uv ; Chluba:2011hw ; Chluba:2013pya ;
Slatyer:2015kla ; Ali-Haimoud:2021lka ; Kumar:2018rlf ; Acharya:2019owx ;
Choi:2017kzp  can introduce the distortions. There is no universal
prescription for non-equilibrium physics, so considering all possible spectral
distortions and their cosmological implications in depth must be
indispensable.

Photons interact with free electrons via Thomson scattering during
recombination. While we normally ignore electrons bounded in atoms, those are
also scattered, which is known as Rayleigh scattering. Rayleigh scattering is
also elastic as Thomson scattering, but the cross-section is frequency-
dependent. This is because the bounded electrons are more likely to interact
with photons whose energy is close to the energy gaps $h\nu_{i}$ of the
electron states. The Rayleigh cross section is Taylor-expanded and behaves as
$(\nu/\nu_{i})^{4}$ during recombination.

Ref. Yu:2001gw  showed that Rayleigh scattering of neutral hydrogen induces
additional scattering, which changes the damping scale in the CMB
anisotropies. Ref. Lewis:2013yia  included neutral helium and polarization,
and they presented a precise numerical calculation of the CMB angular power
spectrum. Then, Ref. Alipour:2014dza  developed a moment expansion for the
frequency dependence and estimated the back reaction of Rayleigh scattering to
baryon acoustic oscillation. Recently, the Rayleigh anisotropies are also
considered for constraining primordial non-Gaussianity and neutrino mass in
Refs. Beringue:2020wxw ; Coulton:2020oxw . Thus, previous works on Rayleigh
scattering are about the linear anisotropies.

While the photon spectrum is considered to be a local Planck in the early
Universe, a spectral distortion is secondarily sourced by Silk damping of
small-scale anisotropies Pitrou:2009bc ; Chluba:2012gq ; Renaux-Petel:2013zwa
; Pitrou:2014ota ; Ota:2016esq ; Pitrou:2019hqg . Physically speaking,
acoustic oscillation of the viscous fluid leads to friction heat at second
order, and the energy release distorts the fiducial Planck spectrum when the
Universe is out of chemical equilibrium. Including Rayleigh scattering, the
acoustic source can also be frequency-dependent, and then Rayleigh scattering
at second order will introduce a different spectral distortion. This paper
computes the spectral distortion generated by second-order Rayleigh scattering
for the first time. Based on a rigorous framework of the general relativistic
Boltzmann equation, we compute the spectral distortion due to Rayleigh
scattering and discuss the size, detectability, and ability to constrain
cosmological models.

We organize the paper as follows. In section II, we provide the Rayleigh
scattering cross-section for neutral hydrogen, helium, and singly ionized
helium numerically. Then we review the first-order Rayleigh anisotropies in
section III. Section IV is the main part of the paper. We derive the spectral
distortion from Rayleigh scattering at second order. We discuss several
possible cosmological constraints on the new spectral distortions in Section
V. Finally, we conclude the paper in Section VI.

##  II Scattering cross section

Table 1: Expansion factors for Rayleigh scattering. The neutral hydrogen and the neutral helium parameters are taken from Ref. Alipour:2014dza , and singly ionized helium is obtained as $c^{(\rm HeII)}_{2i+4}=c^{(\rm H)}_{2i+4}/2^{8+4i}$ as discussed in Ref. Fi_k_2016 . $2i+4$ | $c^{(\rm HI)}_{2i+4}$ | $c^{(\rm HeI)}_{2i+4}$ | $c^{(\rm HeII)}_{2i+4}$
---|---|---|---
4 | 1.265625 | 0.120798 | 0.00494385
6 | 3.738281 | 0.067243 | 0.000912666
8 | 8.813931 | 0.031585 | 0.00013449
10 | 19.153795 | 0.014153 | 0.0000182665
12 | 39.923032 | 0.006226 | 2.3796$\times 10^{-6}$

During recombination, singly ionized helium (HeII) is first produced and then
neutral helium (HeI) is formed. In the last stage of recombination, the
remaining electrons are captured in neutral hydrogen (HI) atoms. Rayleigh
scattering starts after singly ionized helium appears at $z\sim 6000$. The
non-relativistic Rayleigh scattering cross section including those three
species is expanded into Alipour:2014dza

$\displaystyle\sigma_{\rm R}=\sigma_{\rm T}\sum_{j={\rm
HI,HeI,HeII}}\sum_{i=0}^{\infty}c_{2i+4}^{(j)}\Lambda^{2i+4}x^{2i+4},$ (1)

where $\sigma_{\rm T}$ is the Thomson scattering cross section, and we
introduced

$\displaystyle x$ $\displaystyle\equiv\frac{h\nu}{k_{\rm B}T_{\rm
CMB}}=0.0176119\left(\frac{\nu}{\rm GHz}\right),$ (2) $\displaystyle\Lambda$
$\displaystyle\equiv\frac{k_{\rm B}T_{\rm CMB}}{a{\rm
Ry}}=\frac{1.72663}{a}\times 10^{-5},$ (3)

with the Rydberg constant Ry, the Boltzmann constant $k_{\rm B}$, the present
CMB temperature $T_{\rm CMB}$, the Planck constant $h$, the comoving frequency
$\nu$, and the scale factor $a$. $c_{2i+4}^{(j)}$ are summarized in Tab. 1.
The expansion parameters for HI and HeI are taken from Tab.1 of Ref.
Alipour:2014dza . As discussed in Ref. Fi_k_2016 , the Rayleigh scattering
cross-section for hydrogen-like atoms of atomic number $Z$ is given as
$c^{(\rm HI)}_{2i+4}/Z^{8+4i}$, and we consider HeII as a hydrogen-like atom
of atomic number $Z=2$. The differential optical depth is expanded as

$\displaystyle\dot{\tau}=\dot{\tau}^{(0)}+x^{4}\dot{\tau}^{(4)}+x^{6}\dot{\tau}^{(6)}+x^{8}\dot{\tau}^{(8)}+\cdots,$
(4)

where $\dot{\tau}^{(0)}=-n_{\rm e}\sigma_{\rm T}a$ with the free electron
number density $n_{\rm e}$ and

$\displaystyle\dot{\tau}^{(2i+4)}=\dot{\tau}^{(0)}\sum_{j={\rm
HI,HeI,HeII}}\frac{n_{j}}{n_{\rm e}}c_{2i+4}^{(j)}\Lambda^{2i+4}.$ (5)

$n_{j}$ is the number density of $j$. We find the number density of ions by
modifying a recombination code RECFAST Seager:1999bc . By default, RECFAST
provides hydrogen ionization fraction $x_{\rm H0}$, helium ionization fraction
$x_{\rm He0}$, electron ionization fraction $x_{\rm e}$, and helium mass
fraction $f_{\rm He}$. Using those parameters, we get

$\displaystyle\frac{n_{\rm HI}}{n_{\rm e}}$ $\displaystyle=x_{\rm
e}^{-1}(1-x_{\rm H0}),$ (6) $\displaystyle\frac{n_{\rm HeI}}{n_{\rm e}}$
$\displaystyle=x_{\rm e}^{-1}f_{\rm He}(1-x_{\rm He0}),$ (7)

where $n_{\rm H}$ and $n_{\rm He}$ are the total number densities of hydrogen
and helium. Moreover, eliminating the number density of doubly ionized helium
$n_{\rm HeIII}$ from

$\displaystyle x_{\rm He0}$ $\displaystyle=\frac{n_{\rm HeII}}{n_{\rm
He}}+\frac{n_{\rm HeIII}}{n_{\rm He}},$ (8) $\displaystyle x_{\rm e}$
$\displaystyle=1-\frac{n_{\rm HI}}{n_{\rm H}}+\frac{n_{\rm HeII}}{n_{\rm
H}}+2\frac{n_{\rm HeIII}}{n_{\rm H}},$ (9)

we find

$\displaystyle\frac{n_{\rm HeII}}{n_{\rm e}}$ $\displaystyle=x_{\rm
e}^{-1}x_{\rm H0}+2x_{\rm e}^{-1}f_{\rm He}x_{\rm He0}-1.$ (10)

With Eqs. (5), (6) and (7), we reproduced the Rayleigh scattering differential
optical depth in Refs. Lewis:2013yia  and Alipour:2014dza  for $\eta>100$Mpc,
and we can see a new feature in $\eta<100$Mpc in our Fig 1 with Eq. (10).
While the number fraction of HeII is smaller than HI, the differential optical
depth is comparable to HI because photons are hotter at higher redshift.
Integrating the differential optical depth at 875GHz, we find
$\tau^{(4)}x^{4}\sim 0.8$, i.e., $80\%$ of photons scatter on $\eta<50$Mpc.
This fact implies that Rayleigh scattering at $\eta<50$Mpc might be important
for small scale. While we mainly use RECFAST in our calculation, we also
reproduced Fig. 1 using another efficient modern numerical recombination code
HyRec Ali-Haimoud:2010hou . We crosschecked the new effect of singly ionized
helium Rayleigh scattering.

Figure 1: The differential optical depth multiplied by $x^{n}$ for various
frequencies. The solid and dashed curves imply 857GHz and 357GHz,
respectively. The peak at $\eta\lesssim 100{\rm Mpc}$ is Rayleigh scattering
of the singly ionized helium, which is bigger than the peak of neutral
hydrogen.

##  III Rayleigh scattering at first order

We have reviewed background thermodynamics in the previous section. Now we are
ready to compute the evolution of the linear Rayleigh anisotropies.

###  III.1 Boltzmann equation

We will solve the Boltzmann equation for the photon phase-space distribution
function (PDF) in the presence of Thomson and Rayleigh scattering with other
cosmological matter contents and the Einstein equation. It is quite difficult
to solve the nonlinear partial differential equations in a general setup, so
we perturbatively solve the equation order by order on top of a homogenous and
isotropic Friedman background.

The Universe is in local chemical equilibrium in the early epoch, but photon
production processes decouple as the Universe expands. The blackbody
photosphere is around the redshift $2\times 10^{6}$ when double Compton
scattering becomes inefficient. Thomson scattering and its relativistic
correction are dominant afterward until the end of recombination. The photon
fluid is defused due to the scattering, and deviation from the Planck
distribution is introduced. Therefore, the solution to the photon Boltzmann
equation should be written as a local Planck distribution with small spectral
distortions. Indeed, we immediately find the background solution is a
homogeneous and isotropic Planck distribution (See, e.g., Dodelson:2003ft )

$\displaystyle f_{\rm pl}(x)\equiv\frac{1}{e^{x}-1},$ (11)

with $x$ defined in Eq. (2). We perturb the Planck distribution as $f=f_{\rm
pl}(xe^{-\Theta})$ or equivalently $T_{\rm CMB}$ as $T_{\rm CMB}e^{\Theta}$,
with the temperature perturbation $\Theta$. Taylor expanding $f_{\rm
pl}(xe^{-\Theta})$ with respect to $\Theta$ up to first order, we find

$\displaystyle f_{\rm pl}(xe^{-\Theta})$ $\displaystyle=f_{\rm
pl}(x)+\Theta\mathcal{G}^{(0)}(x)+\mathcal{O}(\Theta^{2}),$ (12)

where we defined

$\displaystyle\mathcal{G}^{(0)}(x)\equiv-x\frac{\partial}{\partial x}f_{\rm
pl}(x).$ (13)

$\Theta$ is frequency independent and is a function of photon momentum
direction $\mathbf{n}$, position $\mathbf{x}$ and conformal time $\eta$, for
Thomson scattering. $\mathcal{G}^{(0)}$ is the frequency shape of the first
order temperature shift. Rayleigh scattering shifts the spectrum by
$x^{2n+4}$. Hence, the temperature perturbation for Rayleigh scattering
carries additional powers of $x$. The linearized spectrum including Rayleigh
scattering can be parameterized by Alipour:2014dza

$\displaystyle f=f_{\rm
pl}+\mathcal{G}^{(0)}\Theta^{(0)}+\mathcal{G}^{(4)}\Theta^{(4)}+\cdots,$ (14)

where we include the power of $x$ into

$\displaystyle\mathcal{G}^{(n)}\equiv x^{n}\mathcal{G}^{(0)}.$ (15)

The dots imply the Rayleigh scattering corrections of $x^{n>4}$. Rayleigh
scattering happens at most 1/10 times during cosmic history for the major
frequencies so that the frequency cascade typically happens just once per
photon, and the Rayleigh scattered photons are mostly Thomson scattered during
the rest of cosmic history. Therefore, hereafter we truncate series expansions
for $x$ at this order unless otherwise stated. Since $\mathcal{G}^{(4)}$
perfectly characterizes the frequency dependence at this order, we do not have
to consider many samples in the frequency space for calculations
Alipour:2014dza . The Liouville term up to first order in the cosmological
perturbation is written as

$\displaystyle\mathcal{L}[f]=\left(\Theta^{(0)^{\prime}}-(\ln
x)^{\prime}\right)\mathcal{G}^{(0)}+\mathcal{G}^{(4)}\Theta^{(4)^{\prime}}.$
(16)

A prime is an ordinary derivative with respect to conformal time. The time
derivatives of functions of $x$ are higher-order perturbations as $(\ln
x)^{\prime}$ is first order in metric perturbations Ma:1995ey . The linear
collision term for Thomson scattering is given as (See, e.g., Dodelson:2003ft
)

$\displaystyle\mathcal{C}[f]$
$\displaystyle=\dot{\tau}^{(0)}\left(f-f_{0}+x\frac{\partial f}{\partial
x}V+\frac{1}{2}f_{2}P_{2}\right),$ (17)

where $V\equiv\mathbf{v}\cdot\mathbf{n}$ with the baryon bulk velocity
$\mathbf{v}$, the Legendre polynomial $P_{\ell}$ of the cosine between
$\mathbf{n}$ and $\mathbf{v}$, and the multiple coefficient $f_{\ell}$.
Following Refs. Yu:2001gw ; Lewis:2013yia  we obtain the linear Rayleigh
collision term by replacing $\dot{\tau}^{(0)}$ in Eq. (17) with Eq. (4).
Substituting Eqs. (14) and (4) into (17), we find

$\displaystyle\mathcal{C}[f]=$
$\displaystyle\dot{\tau}^{(0)}\mathcal{A}^{(0)}\mathcal{G}^{(0)}+\dot{\tau}^{(0)}\mathcal{A}^{(4)}\mathcal{G}^{(4)}+\dot{\tau}^{(4)}\mathcal{A}^{(0)}\mathcal{G}^{(4)},$
(18)

where we introduced

$\displaystyle\mathcal{A}^{(n)}$
$\displaystyle\equiv\Theta^{(n)}-\Theta_{0}^{(n)}-\delta_{n0}V+\frac{1}{2}\Pi^{(n)}P_{2},$
(19) $\displaystyle\mathcal{A}_{\rm P}^{(n)}$
$\displaystyle\equiv\Theta^{(n)}_{\rm P}-\frac{1}{2}\Pi^{(n)}(1-P_{2}).$ (20)

We included polarization by replacing $\Theta_{2}$ with
$\Pi\equiv\Theta_{2}+\Theta_{\rm P0}+\Theta_{\rm P2}$. $\Theta_{\rm P}$ is
polarization temperature perturbation, which corresponds to
$G_{\gamma}=4\Theta_{\rm P}$ in Ref. Ma:1995ey .

From Eqs. (18) and (16), the linear order part of the Boltzmann equation
$\mathcal{L}[f]=\mathcal{C}[f]$ can be obtained as

$\displaystyle\Theta^{(0)^{\prime}}-(\ln x)^{\prime}$
$\displaystyle=\dot{\tau}^{(0)}\mathcal{A}^{(0)},$ (21)
$\displaystyle\Theta^{(4)^{\prime}}$
$\displaystyle=\dot{\tau}^{(0)}\mathcal{A}^{(4)}+\dot{\tau}^{(4)}\mathcal{A}^{(0)}.$
(22)

The first equation is the standard Thomson scattering Boltzmann equation. The
second equation is the first moment of Rayleigh scattering proportional to
$\mathcal{G}^{(4)}$. The gravitational redshift $(\ln x)^{\prime}$ does not
introduce the Rayleigh anisotropies. The above derivation straightforwardly
applies to the linear Rayleigh polarization.

###  III.2 Boltzmann Hierarchy equations

Linear scalar perturbations associated with the photon PDF are decomposed into

$\displaystyle\Theta^{(n)}$ $\displaystyle=(4\pi)\sum_{\ell
m}i^{\ell}\int_{\mathbf{k}}e^{i\mathbf{k}\cdot\mathbf{x}}Y^{\star}_{\ell
m}(\hat{k})Y_{\ell
m}(\mathbf{n})\tilde{\Theta}_{\ell}^{(n)}\zeta_{\mathbf{k}},$ (23)
$\displaystyle V$
$\displaystyle=(4\pi)\sum_{m}i\int_{\mathbf{k}}e^{i\mathbf{k}\cdot\mathbf{x}}Y^{\star}_{1m}(\hat{k})Y_{1m}(\mathbf{n})\tilde{V}_{1}\zeta_{\mathbf{k}},$
(24)

where a tilde implies the transfer function in Fourier space,
$\int_{\mathbf{k}}\equiv\int(2\pi)^{-3}d^{3}k$ and $\zeta_{\mathbf{k}}$ is the
initial Gaussian stochastic variable that satisfies

$\displaystyle\langle\zeta_{\mathbf{k}}\zeta_{\mathbf{k}^{\prime}}\rangle=(2\pi)^{3}\delta_{\rm
D}^{(3)}(\mathbf{k}+\mathbf{k}^{\prime})\frac{2\pi^{2}}{k^{3}}\mathcal{P}_{\zeta}(k).$
(25)

$\mathcal{P}_{\zeta}$ contains the details of inflationary cosmology.
Expanding the above equations into the Legendre coefficients, we get the
Boltzmann hierarchy equations for Rayleigh scattering Lewis:2013yia ;
Alipour:2014dza

$\displaystyle\dot{\tilde{\Theta}}^{(4)}_{0}$
$\displaystyle=-k\tilde{\Theta}^{(4)}_{1},$ (26)
$\displaystyle\dot{\tilde{\Theta}}^{(4)}_{1}$
$\displaystyle=\dot{\tau}^{(0)}\tilde{\Theta}^{(4)}_{1}+\frac{k}{3}\left(\tilde{\Theta}^{(4)}_{0}-2\tilde{\Theta}^{(4)}_{2}\right)+\dot{\tau}^{(4)}\tilde{\Theta}^{(0)}_{1g},$
(27) $\displaystyle\dot{\tilde{\Theta}}^{(4)}_{2}$
$\displaystyle=\frac{k}{5}\left(2\tilde{\Theta}^{(4)}_{1}-3\tilde{\Theta}^{(4)}_{3}\right)+\dot{\tau}^{(4)}\left(\tilde{\Theta}^{(0)}_{2}-\frac{\tilde{\Pi}^{(0)}}{10}\right)+\dot{\tau}^{(0)}\left(\tilde{\Theta}^{(4)}_{2}-\frac{\tilde{\Pi}^{(4)}}{10}\right),$
(28) $\displaystyle\dot{\tilde{\Theta}}^{(4)}_{\ell\geq 3}$
$\displaystyle=\frac{k}{2\ell+1}\left(\ell\tilde{\Theta}^{(4)}_{\ell-1}-(\ell+1)\tilde{\Theta}^{(4)}_{\ell+1}\right)+\dot{\tau}^{(4)}\tilde{\Theta}^{(0)}_{\ell}+\dot{\tau}^{(0)}\tilde{\Theta}^{(4)}_{\ell},$
(29)

where we introduce the photon-baryon velocity difference
$\tilde{\Theta}^{(0)}_{1g}\equiv\tilde{\Theta}^{(0)}_{1}-\tilde{V}_{1}$. The
polarization hierarchy equations are Lewis:2013yia ; Alipour:2014dza

$\displaystyle\dot{\tilde{\Theta}}^{(4)}_{\rm P,0}$
$\displaystyle=-k\tilde{\Theta}^{(4)}_{{\rm
P,}1}+\dot{\tau}^{(0)}\left(\tilde{\Theta}^{(4)}_{{\rm
P}0}-\frac{\tilde{\Pi}^{(4)}}{2}\right)+\dot{\tau}^{(4)}\left(\tilde{\Theta}^{(0)}_{{\rm
P}0}-\frac{\tilde{\Pi}^{(0)}}{2}\right),$ (30)
$\displaystyle\dot{\tilde{\Theta}}^{(4)}_{\rm P,1}$
$\displaystyle=\dot{\tau}^{(0)}\tilde{\Theta}^{(4)}_{{\rm
P,}1}+\frac{k}{3}\left(\tilde{\Theta}^{(4)}_{{\rm
P,}0}-2\tilde{\Theta}^{(4)}_{{\rm
P,}2}\right)+\dot{\tau}^{(4)}\tilde{\Theta}^{(0)}_{{\rm P,}1},$ (31)
$\displaystyle\dot{\tilde{\Theta}}^{(4)}_{\rm P,2}$
$\displaystyle=\frac{k}{5}\left(2\tilde{\Theta}^{(4)}_{{\rm
P,}1}-3\tilde{\Theta}^{(4)}_{{\rm
P,}3}\right)+\dot{\tau}^{(4)}\left(\tilde{\Theta}^{(0)}_{{\rm
P,}2}-\frac{\tilde{\Pi}^{(0)}}{10}\right)+\dot{\tau}^{(0)}\left(\tilde{\Theta}^{(4)}_{{\rm
P,}2}-\frac{\tilde{\Pi}^{(4)}}{10}\right),$ (32)
$\displaystyle\dot{\tilde{\Theta}}^{(4)}_{{\rm P},\ell\geq 3}$
$\displaystyle=\frac{k}{2\ell+1}\left(\ell\tilde{\Theta}^{(4)}_{{\rm
P,}\ell-1}-(\ell+1)\tilde{\Theta}^{(4)}_{{\rm
P,}\ell+1}\right)+\dot{\tau}^{(4)}\tilde{\Theta}^{(0)}_{{\rm
P,}\ell}+\dot{\tau}^{(0)}\tilde{\Theta}^{(4)}_{{\rm P,}\ell}.$ (33)

The hierarchy equations for Thomson scattering are unchanged as Eq. (21) does
not contain Rayleigh scattering. The Rayleigh anisotropies and polarization
are sourced by $\tilde{\Theta}_{1g}^{(0)}$, $\tilde{\Theta}_{\ell>2}^{(0)}$
and $\tilde{\Theta}_{{\rm P},\ell}^{(0)}$, which are non-vanishing only on
sub-horizion scale and are linear gauge invariants. The Boltzmann hierarchy
equations for Rayleigh scattering are independent of the metric perturbations.
From the above two facts, the Rayleigh anisotropies are gauge independent, and
we can use the above hierarchy equations in any gauge. We truncate the
Boltzmann hierarchy with the traditional method in Ref. Ma:1995ey . In this
paper, we numerically evaluate the Boltzmann hierarchy by modifying a publicly
available linear Boltzmann solver CLASS Blas:2011rf  with cosmological
parameters in Ref. Aghanim:2018eyx .

###  III.3 Tight-coupling approximation

The Thomson Boltzmann hierarchy is a stiff system as it contains
$\dot{\tau}^{(0)}\tilde{\Theta}^{(0)}_{1g}$. Before last scattering,
$\dot{\tau}^{(0)}$ is large, $\tilde{\Theta}_{1}$ and $\tilde{V}_{1}$ are the
almost same size; therefore, we need extra precision for both
$\tilde{\Theta}_{1}$ and $\tilde{V}_{1}$ to get sufficiently accurate
$\dot{\tau}^{(0)}\tilde{\Theta}^{(0)}_{1g}$. For efficient calculations, we
directly find the evolution of $\tilde{\Theta}_{1g}$ by subtracting the Euler
equations for photons and baryons. The equation is expanded into a series of
$1/\dot{\tau}^{(0)}$ and CLASS truncates the series expansion at second order
in $1/\dot{\tau}^{(0)}$ Blas:2011rf . The multipole of $\ell>3$ is
$\mathcal{O}(1/\dot{\tau}^{(0)3})$ and we drop them during the tight coupling
approximation. This simplification significantly reduces the number of
equations to solve.

The Rayleigh anisotropies are produced by $\tilde{\Theta}^{(0)}_{1g}$ and
higher multipoles $\tilde{\Theta}^{(0)}_{\ell\geq 2}$. In contrast to the
Thomson hierarchy, the Rayleigh hierarchy is at most
$\mathcal{O}(1/\dot{\tau}^{(0)2})$. Up to second order in tight-coupling
approximation, we find

$\displaystyle\tilde{\Theta}_{1}^{(4)}\approx-\frac{\dot{\tau}^{(4)}}{\dot{\tau}^{(0)}}\tilde{\Theta}^{(0)}_{1g},$
(34)

and for $\ell>1$,

$\displaystyle\tilde{\Theta}_{\ell}^{(4)}\approx-\frac{\dot{\tau}^{(4)}}{\dot{\tau}^{(0)}}\tilde{\Theta}^{(0)}_{\ell}.$
(35)

Similarly, polarization satisfies

$\displaystyle\tilde{\Theta}_{{\rm
P},\ell}^{(4)}\approx-\frac{\dot{\tau}^{(4)}}{\dot{\tau}^{(0)}}\tilde{\Theta}^{(0)}_{{\rm
P},\ell}.$ (36)

For simplicity, we also consider

$\displaystyle\tilde{\Theta}_{0}^{(4)}=-k\eta\tilde{\Theta}_{1}^{(4)}.$ (37)

The expansion scheme for the tight-coupling approximation is nicely reviewed
in Ref. Blas:2011rf , and we use the same method. After the tight coupling
approximation, we fully evolve the Rayleigh Boltzmann hierarchy. Adding the
new hierarchy equations in the Boltzmann code is straightforward.

###  III.4 Back reaction

Back reaction of Rayleigh scattering appears in the Einstein equation and
Euler equation for baryons because those equations contain $x$ integrals of
the photon distribution function Alipour:2014dza . The photon density
perturbation is given by

$\displaystyle\delta_{\gamma}$ $\displaystyle=\frac{\int
x^{3}dx\mathcal{G}^{(0)}(\tilde{\Theta}^{(0)}_{0}+\tilde{\Theta}^{(4)}_{0}x^{4}+\cdots)}{\int
x^{3}dxf_{\rm pl}}$
$\displaystyle=4\left(\tilde{\Theta}^{(0)}_{0}+\frac{I_{4}}{I_{0}}\tilde{\Theta}^{(4)}_{0}+\cdots\right),$
(38)

where we have defined

$\displaystyle I_{n}\equiv\int
dxx^{n+3}\mathcal{G}^{(0)}=\zeta(n+4)\Gamma(n+5).$ (39)

Similarly, we get photon fluid velocity divergence $\theta_{\gamma}$, photon
shear $\sigma_{\gamma}$, the higher order multipoles $F_{\gamma\ell}$ as

$\displaystyle\theta_{\gamma}$
$\displaystyle=3k\left(\tilde{\Theta}^{(0)}_{1}+\frac{I_{4}}{I_{0}}\tilde{\Theta}^{(4)}_{1}+\cdots\right),$
(40) $\displaystyle\sigma_{\gamma}$
$\displaystyle=2\left(\tilde{\Theta}^{(0)}_{2}+\frac{I_{4}}{I_{0}}\tilde{\Theta}^{(4)}_{2}+\cdots\right),$
(41) $\displaystyle F_{\gamma,\ell}$
$\displaystyle=4\left(\tilde{\Theta}^{(0)}_{\ell}+\frac{I_{4}}{I_{0}}\tilde{\Theta}^{(4)}_{\ell}+\cdots\right).$
(42)

We use the above quantities for the right hand side of the Einstein equation.
The Euler equation for baryons contains the integrated collision term

$\displaystyle\dot{\tau}^{(0)}(\tilde{\Theta}_{1}^{(0)}-\tilde{V}_{1})=\frac{\int
x^{3}\left[\dot{\tau}^{(0)}(\tilde{\Theta}_{1}^{(0)}-\tilde{V}_{1})\right]\mathcal{G}^{(0)}dx}{4\int
x^{3}f_{\rm pl}dx}.$ (43)

Including Rayleigh scattering, we modify the Thomson optical depth and
$\tilde{\Theta}_{1}^{(0)}$ and we get

$\displaystyle\dot{\tau}^{(0)}\left(\tilde{\Theta}^{(0)}_{1}+\frac{I_{4}}{I_{0}}\tilde{\Theta}^{(4)}_{1}-\tilde{V}_{1}\right)+\frac{I_{4}}{I_{0}}\dot{\tau}^{(4)}(\tilde{\Theta}^{(0)}_{1}-\tilde{V}_{1}).$
(44)

The back reaction is typically the order of
$\mathcal{O}(I_{4}/I_{0}\cdot\dot{\tau}^{(4)}/\dot{\tau}^{(0)})$.

##  IV Rayleigh scattering at second order

Next, we consider Rayleigh scattering at second order. To be more precise, we
are interested in the spectral distortion of the average CMB spectrum, so we
only consider the angular averaged part at second order.

###  IV.1 Collision integral

The monopole component of the Thomson scattering collision term up to second
order is given as (see e.g., Ota:2016esq  for a derivation)

$\displaystyle\mathcal{C}_{0}[f]=$
$\displaystyle-\dot{\tau}^{(0)}\left[V\left(3+x\frac{\partial}{\partial
x}\right)f\right]_{0}$
$\displaystyle-\dot{\tau}^{(0)}\left[V^{2}\left(x^{2}\frac{\partial^{2}}{\partial
x^{2}}+4x\frac{\partial}{\partial x}\right)f\right]_{0},$ (45)

where the subscript 0 implies the average with respect to the photon direction
$\mathbf{n}$. The rest of the second-order collision terms are dropped after
the angular average. The second-order collision term includes the relativistic
correction of the bulk motion, which introduces the isotropic spectral
distortions at second order (the Doppler shift).

Since Thomson scattering is an elastic scattering of a photon and an electron,
it must rigorously preserve the number of photons for any $f$. In general, the
covariant derivative of the photon number flux $N^{\mu}$ is related to the
collision term as (see, e.g., Ota:2017jte )

$\displaystyle\nabla_{\mu}N^{\mu}=c_{1}\int x^{2}dx\mathcal{C}_{0}[f],$ (46)

where $c_{1}$ is a numerical factor of the Jacobian for the perturbed
spacetime. Eq. (45) implies $x^{2}\mathcal{C}_{0}[f]$ reduces to a total
derivative for any $f$ and thus the number flux conservation is always
satisfied.

Rayleigh scattering is also elastic and must conserve the number of photons.
However, the straightforward replacement of the differential optical depth in
Eq. (45) does not guarantee the number conservation. The prescription does not
work at second order simply because we ignored the incoming photon frequency
dependence of Rayleigh scattering 2018MNRAS.473..457R . The non-relativistic
limit Rayleigh scattering does not exchange photon energy so that the energies
of incoming and outgoing photons are always the same, but this is not the case
when including relativistic corrections.

Derivative operators with respect to $x$ in the collision integral appear when
integrating by parts, the Taylor expanded the delta function responsible for
the energy conservation of the scattering process. To be more specific, Eq,
(45) is obtained after integrating $\bar{x}$ out from the following type of
integrals Ota:2016esq

$\displaystyle\int
d\bar{x}\dot{\tau}^{(0)}\left[f(\bar{x},\bar{\mathbf{n}})-f(x,\mathbf{n})\right]x^{n}\frac{\partial^{n}}{\partial\bar{x}^{n}}\delta_{\rm
D}(\bar{x}-x).$ (47)

To properly account for $\bar{x}$ dependence of the Rayleigh scattering, we
include the Rayleigh cross section by replacing the differential optical depth
in Eq. (47) as

$\displaystyle\dot{\tau}^{(0)}\to\dot{\tau}^{(0)}+\bar{x}^{4}\dot{\tau}^{(4)}+\cdots.$
(48)

With this prescription, substituting the first order solution (14) and Eq
.(48) into the second order Thomson collision integral, Eq. (45) yields

$\displaystyle C_{0}[f]=$
$\displaystyle-\dot{\tau}^{(0)}\left[V^{2}-V\Theta^{(0)}\right]_{0}\mathcal{Y}^{(0)}$
$\displaystyle+\dot{\tau}^{(0)}\left[V\Theta^{(4)}\right]_{0}\mathcal{Y}^{(4)}$
$\displaystyle-\dot{\tau}^{(4)}\left[V^{2}-V\Theta^{(0)}\right]_{0}\mathcal{Y}^{(4)},$
(49)

where we introduced

$\displaystyle\mathcal{Y}^{(0)}$
$\displaystyle\equiv\left(-x\frac{\partial}{\partial
x}\right)\mathcal{G}^{(0)}-3\mathcal{G}^{(0)},$ (50)
$\displaystyle\mathcal{Y}^{(n)}$ $\displaystyle\equiv
x^{n}\mathcal{Y}^{(0)}-n\mathcal{G}^{(n)}.$ (51)

Note that the second-order correction to $f$ turns into third-order correction
in Eq. (49). The first term proportional to $\mathcal{Y}^{(0)}$ is the source
term for the standard $y$ spectral distortion from Thomson scattering
Chluba:2012gq . $\mathcal{Y}^{(4)}$ is the new spectral distortion. Both
spectral distortions preserve the number of photons as we have

$\displaystyle\int x^{2}dx\mathcal{Y}^{(n)}=0.$ (52)

###  IV.2 Liouville term

The first-order solution fixes the second-order collision term Eq. (49). Our
next step is to find second-order Liouville equation consistent with Eq. (49).
For Thomson scattering, second-order ansatz can be written as Pitrou:2009bc ;
Chluba:2012gq ; Renaux-Petel:2013zwa ; Pitrou:2014ota ; Ota:2016esq ;
Pitrou:2019hqg

$\displaystyle f_{\rm
pl}\left(xe^{-\Theta^{(0)}}\right)+y^{(0)}\mathcal{Y}^{(0)}.$ (53)

where we ignored $\mu$ spectral distortion since we only consider
recombination epoch when $\mu$ is not produced. The first term is nonlinearly
perturbed Planck distribution, and the second term is the $y$ distortion that
we cannot include in the temperature. Eq. (53) will be generalized to

$\displaystyle f_{\rm
pl}\left(xe^{-\Theta^{(0)}-x^{4}\Theta^{(4)}}\right)+y^{(0)}\mathcal{Y}^{(0)}+y^{(4)}\mathcal{Y}^{(4)}.$
(54)

Similar to Thomson scattering, $y^{(0)}$ and $y^{(4)}$ represent the spectral
distortion that cannot be included in the frequency dependent temperature
perturbation $\Theta^{(0)}+x^{4}\Theta^{(4)}+\cdots$. Expanding this equation
up to second order, we find

$\displaystyle f=$ $\displaystyle f_{\rm
pl}+\Theta^{(0)}\mathcal{G}+\Theta^{(4)}\mathcal{G}^{(4)}$
$\displaystyle+\frac{3}{2}\Theta^{(0)2}\mathcal{G}^{(0)}+3\Theta^{(0)}\Theta^{(4)}\mathcal{G}^{(4)}$
$\displaystyle+\frac{1}{2}\Theta^{(0)2}\mathcal{Y}^{(0)}+\Theta^{(0)}\Theta^{(4)}\mathcal{Y}^{(4)}$
$\displaystyle+y^{(0)}\mathcal{Y}^{(0)}+y^{(4)}\mathcal{Y}^{(4)},$ (55)

where we redefine second-order temperature perturbations as
$\Theta^{(4)}\to\Theta^{(4)}-4y^{(4)}$ for notational simplicity. Taking the
time derivative of Eq. (55), we find

$\displaystyle
f^{\prime}=\left(1+3\Theta^{(0)}\right)\left(\Theta^{(0)^{\prime}}-(\ln
x)^{\prime}\right)\mathcal{G}^{(0)}$
$\displaystyle+\left[\left(1+3\Theta^{(0)}\right)\Theta^{(4)^{\prime}}+3\Theta^{(4)}\left(\Theta^{(0)^{\prime}}-(\ln
x)^{\prime}\right)\right]\mathcal{G}^{(4)}$
$\displaystyle+[y^{(0)^{\prime}}+\Theta^{(0)}\left(\Theta^{(0)^{\prime}}-(\ln
x)^{\prime}\right)]\mathcal{Y}^{(0)}$
$\displaystyle+[y^{(4)^{\prime}}+\Theta^{(0)}\Theta^{(4)^{\prime}}+\Theta^{(4)}\left(\Theta^{(0)^{\prime}}-(\ln
x)^{\prime}\right)]\mathcal{Y}^{(4)},$ (56)

where we used the chain rule:

$\displaystyle\mathcal{G}^{(n)^{\prime}}$ $\displaystyle=-(\ln
x)^{\prime}\left(\mathcal{Y}^{(n)}+3\mathcal{G}^{(n)}\right).$ (57)

We do not have to consider $\mathcal{Y}^{(n)^{\prime}}$ since this term turns
into a third-order correction that we ignore.

Figure 2: Left: the Fourier space window function
$I_{n}/I_{0}\cdot\mathcal{W}^{(X)}$ for the spectral distortions at $z=100$.
The dashed curves imply the negative sign.
$\mathcal{W}^{(0,4)}\approx-2\mathcal{W}^{(4,0)}$ so that the window function
is negative during the tight coupling regime, but $\mathcal{W}^{(4,0)}>0$
dominates after hydrogen recombination. Right: the spectral distortion source
$I_{n}/I_{0}\cdot\mathcal{S}^{(X)}$ for the primordial power spectrum of
spectral index $n_{s}=0.96$. Similarly,
$\mathcal{S}^{(0,4)}\approx-2\mathcal{S}^{(4,0)}$ is established for high
redshifts, but $\mathcal{S}^{(4,0)}>0$ dominates after hydrogen recombination.

###  IV.3 Boltzmann equation

Equating Eqs. (49) and (56), with the linear collision terms (21) and (22), we
find

$\displaystyle\Theta^{(0)^{\prime}}_{0}=$
$\displaystyle-3\dot{\tau}^{(0)}[\Theta^{(0)}\mathcal{A}^{(0)}]_{0},$ (58)
$\displaystyle\Theta^{(4)^{\prime}}_{0}=$
$\displaystyle-3\dot{\tau}^{(0)}[\Theta^{(0)}\mathcal{A}^{(4)}]_{0}-3\dot{\tau}^{(4)}[\Theta^{(0)}\mathcal{A}^{(0)}]_{0}$
$\displaystyle-3\dot{\tau}^{(0)}[\Theta^{(4)}\mathcal{A}^{(0)}]_{0},$ (59)
$\displaystyle y^{(0)^{\prime}}_{0}=$
$\displaystyle-\dot{\tau}^{(0)}[\Theta^{(0)}\mathcal{A}^{(0)}]_{0}-\dot{\tau}^{(0)}\left[V^{2}-V\Theta^{(0)}\right]_{0},$
(60) $\displaystyle y^{(4)^{\prime}}_{0}=$
$\displaystyle-\dot{\tau}^{(0)}[\Theta^{(4)}\mathcal{A}^{(0)}]_{0}-\dot{\tau}^{(0)}[\Theta^{(0)}\mathcal{A}^{(4)}]_{0}$
$\displaystyle+\dot{\tau}^{(0)}\left[V\Theta^{(4)}\right]_{0}-\dot{\tau}^{(4)}[\Theta^{(0)}\mathcal{A}^{(0)}]_{0}$
$\displaystyle-\dot{\tau}^{(4)}\left[V^{2}-V\Theta^{(0)}\right]_{0}.$ (61)

The source terms for $y^{(0)}_{0}$ and $y^{(4)}_{0}$ are given as products of
the linear scalar perturbations. The ensemble and angular average of a product
of linear perturbations is evaluated as

$\displaystyle\langle[AB]_{0}\rangle=$ $\displaystyle\int d\ln
k\mathcal{P}_{\zeta}\sum_{\ell}(2\ell+1)\tilde{A}_{\ell}\tilde{B}_{\ell}.$
(62)

Using Eqs. (23), (24), (60), (61), and (62), we find

$\displaystyle\langle y^{(0)}_{0}\rangle^{\prime}$ $\displaystyle=\int d\ln
k\mathcal{P}_{\zeta}\mathcal{K}^{(0)},$ (63) $\displaystyle\langle
y^{(4)}_{0}\rangle^{\prime}$ $\displaystyle=\int d\ln
k\mathcal{P}_{\zeta}\left[\mathcal{K}^{(0,4)}+\mathcal{K}^{(4,0)}\right],$
(64)

where we defined

$\displaystyle-\frac{\mathcal{K}^{(0)}}{\dot{\tau}^{(0)}}=$ $\displaystyle
3\tilde{\Theta}^{(0)2}_{1g}-\frac{\tilde{\Theta}^{(0)}_{2}\tilde{\Pi}^{(0)}}{2}+\sum_{\ell=2}^{\infty}(2\ell+1)\tilde{\Theta}^{(0)2}_{\ell},$
(65) $\displaystyle-\frac{\mathcal{K}^{(0,4)}}{\dot{\tau}^{(0)}}=$
$\displaystyle
6\tilde{\Theta}^{(4)}_{1}\tilde{\Theta}^{(0)}_{1g}-\frac{\tilde{\Theta}^{(4)}_{2}\tilde{\Pi}^{(0)}}{2}-\frac{\tilde{\Theta}^{(0)}_{2}\tilde{\Pi}^{(4)}}{2}$
$\displaystyle+2\sum_{\ell=2}^{\infty}(2\ell+1)\tilde{\Theta}^{(4)}_{\ell}\tilde{\Theta}^{(0)}_{\ell}$
(66) $\displaystyle\mathcal{K}^{(4,0)}=$
$\displaystyle\frac{\dot{\tau}^{(4)}}{\dot{\tau}^{(0)}}\mathcal{K}^{(0)}.$
(67)

The gauge invariance of spectral distortions is critical as the spectral
distortions are physical observables. The kernel function for the traditional
$y$ spectral distortion is composed of $\tilde{\Theta}_{1g}^{(0)}$ and
$\tilde{\Theta}^{(0)}_{\ell\geq 2}$. Those variables are gauge invariant at
linear order, so $y$ distortion is also gauge invariant Chluba:2012gq . The
new spectral distortion is sourced by $\tilde{\Theta}_{1g}^{(0)}$,
$\tilde{\Theta}^{(0)}_{\ell\geq 2}$, and $\tilde{\Theta}^{(4)}_{\ell\geq 1}$.
While $\tilde{\Theta}^{(0)}_{1}$ is gauge dependent,
$\tilde{\Theta}^{(4)}_{1}$ is not since it is sourced by
$\tilde{\Theta}_{1g}^{(0)}$. Therefore, the new spectral distortion is also
gauge invariant.

Energy release at high redshift is thermalized due to the energy transfer by
high energy electrons. $y$ and our new spectral distortions are realized when
the thermalization effect is weak. Roughly speaking, the spectral dependence
of the energy release after $z=6\times 10^{4}$ is not changed anymore, and
Ref. Chluba:2013vsa  found a precise energy branching ratio

$\displaystyle\mathcal{J}_{y}(z)=\frac{1}{1+\left(\frac{1+z}{6.0\times
10^{4}}\right)^{2.58}},$ (68)

and, $\mathcal{J}_{y}\mathcal{K}^{(X)}$ contributes to the new spectral
distortion. Integrating Eq. (64) with respect to time, we finally get

$\displaystyle\langle y^{(4)}_{0}\rangle=\int_{k_{\rm min}}^{k_{\rm max}}d\ln
k\mathcal{P}_{\zeta}\left(\mathcal{W}^{(0,4)}+\mathcal{W}^{(4,0)}\right),$
(69)

where we introduced the window function

$\displaystyle\mathcal{W}^{(X)}(k,\eta)$
$\displaystyle\equiv\int^{\eta}_{0}d\bar{\eta}\mathcal{J}_{y}\mathcal{K}^{(X)}.$
(70)

The Fourier space window functions $\mathcal{W}^{(X)}$ are presented in the
left panel of Fig. 2. In the figure, we normalized the window function by the
energy fraction of the new spectral distortion

$\displaystyle\frac{\int x^{3}dx\mathcal{Y}^{(n)}}{\int
x^{3}dx\mathcal{Y}^{(0)}}=\frac{\int x^{3}dx\mathcal{G}^{(n)}}{\int
x^{3}dx\mathcal{G}^{(0)}}=\frac{I_{n}}{I_{0}}.$ (71)

Assuming standard power law initial power spectrum

$\displaystyle\mathcal{P}_{\zeta}=A_{\zeta}\left(\frac{k}{k_{0}}\right)^{n_{s}-1},$
(72)

with $k_{0}{\rm Mpc}/h=0.05$ and $n_{s}=0.96$, we integrate the Fourier
momentum in Eq. (64) and find

$\displaystyle\langle y^{(4)}_{0}\rangle$
$\displaystyle=\int_{z}^{\infty}d\ln(1+z)\left(\mathcal{S}^{(0,4)}+\mathcal{S}^{(4,0)}\right),$
(73)

where

$\displaystyle\mathcal{S}^{(X)}(z)$ $\displaystyle\equiv\int_{k_{\rm
min}}^{k_{\rm max}}d\ln
k\mathcal{P}_{\zeta}\mathcal{J}_{y}\frac{\mathcal{K}^{(X)}}{H}.$ (74)

The source functions $\mathcal{S}^{(X)}$ in units of $A_{\zeta}$ as a function
of the redshift are shown in the right panel of Fig. 2.

###  IV.4 Acoustic dissipation

Rayleigh scattering is an additional photon scattering that makes photon-
baryon coupling stronger. Hence, Rayleigh scattering decreases the multipole
anisotropy during the tight coupling regime. As a result, the energy released
from acoustic oscillation is reduced. Thus, the new spectral distortion is a
negative contribution to the $y$ distortion during the tight coupling regime.
Combining Eqs. (34), (35), (36) and (66), we find

$\displaystyle\langle
y^{(4)}_{0}\rangle^{\prime}\sim-\frac{\dot{\tau}^{(4)}}{\dot{\tau}^{(0)}}\langle
y^{(0)}_{0}\rangle^{\prime},$ (75)

which is consistent with both panels in Fig. 2. Including Rayleigh scattering
effectively implies increasing baryons for Thomson scattering, and we also
confirmed $\partial y^{(0)}/\partial\Omega_{b}<0$ during recombination.

After hydrogen recombination, Thomson scattering is less efficient. Then,
Rayleigh scattering adds extra diffusion, which causes a positive spectral
distortion. Energy contribution from the final recombination stage is dominant
because hydrogen Rayleigh scattering is most significant. For $n_{s}=0.96$ and
$A_{\zeta}=2.196\times 10^{-9}$, the main contribution comes from $k{\rm
Mpc}/h\sim 0.1$, and the total new spectral distortion has the same sign as
$y$. We found a new spectral distortion from acoustic dissipation during
recombination is $y^{(4)}_{\rm ac.}\mathcal{Y}^{(4)}=4\times 10^{-3}$Jy/str
around $\nu=300$GHz and 600GHz in Fig. 4. We also confirmed that the $y$
distortion decreases by $0.1\%$, including the backreaction effect. Those
corrections are one order of magnitude smaller than the sensitivity range of
ESA voyage 2050 Chluba:2019kpb ; Chluba:2019nxa .

###  IV.5 Reionization

Figure 3: The $y$ distortion window functions for $z\leq 100$ and $z>100$. The
window function for $z\leq 100$ is amplified because the baryon velocity is
growing with the dark matter sector. Such a reionization enhancement does not
happen for Rayleigh scattering.

After hydrogen recombination, both Thomson scattering and Rayleigh scattering
are inefficient. Photons free-stream while baryons evolve like dark matter
after recombination. Reionization introduces additional $y$ spectral
distortion since the source term is enhanced drastically as
$V_{1}\gg\Theta_{1}^{(0)}$ at a late time. $y$ distortion from the
reionization is five times bigger than recombination. The Fourier space window
functions for $z\leq 100$ and $z>100$ are shown in Fig. 3. Including
reionization contribution $y^{(0)}_{\rm reio}=2.4\times 10^{-8}$, we find the
total spectral distortion $y^{(0)}=2.9\times 10^{-8}$ at present. Rayleigh
scattering will not happen at a late time since the photon energy is too
small; therefore, $\mathcal{S}^{(4,0)}$ is negligible for reionization. The
Rayleigh anisotropies produced during recombination can also be Thomson
scattered after reionization. However, the Rayleigh dipole $\Theta_{1}^{(4)}$
is not enhanced, unlike $\Theta_{1g}^{(0)}$ since it does not have the baryon
velocity. Hence, $\mathcal{S}^{(0,4)}$ is also negligible. We evaluated the
spectral distortion from Rayleigh scattering after reionization, and we found
that it is 0.1% of the recombination one. Thus, Rayleigh scattering will not
generate secondary spectral distortions during reionization.

###  IV.6 Compton cooling

So far, we have ignored the relativistic correction $T_{\rm e}/m_{\rm e}$ to
the Boltzmann equation. This term leads to the relativistic correction to the
isotropic part of the photon distribution function independently from second-
order perturbations. The relativistic correction to the Thomson scattering
collision integral is (see e.g., Ota:2016esq  for a derivation)

$\displaystyle-\dot{\tau}^{(0)}\frac{T_{\rm e}}{m_{\rm
e}}\left(x^{2}\frac{\partial^{2}}{\partial x^{2}}+4x\frac{\partial}{\partial
x}\right)f$ $\displaystyle-\dot{\tau}^{(0)}\frac{T_{\rm CMB}(1+z)}{m_{\rm
e}}\left[x^{2}\frac{\partial}{\partial x}+4x\right]f(1+f),$ (76)

and for $f=f_{\rm pl}$, Eq. (76) turns into

$\displaystyle-\dot{\tau}^{(0)}\left[\frac{T_{\rm e}}{m_{\rm e}}-\frac{T_{\rm
CMB}(1+z)}{m_{\rm e}}\right]\mathcal{Y}^{(0)}.$ (77)

The $y$ distortion from this term is called Compton cooling as the electron
temperature is always slightly lower than the photon’s, and photons are cooled
via Compton scattering Chluba:2011hw . Including reionization, we found
$y_{\rm e}^{(0)}=2.74\times 10^{-10}$, which is subdominant to acoustic
dissipation. With the same prescription as Sec. IV.1, we found
$\mathcal{Y}^{(0)}$ is replaced with $\mathcal{Y}^{(4)}$ for Rayleigh
scattering and find the following additional contribution

$\displaystyle-\dot{\tau}^{(4)}\left[\frac{T_{\rm e}}{m_{\rm e}}-\frac{T_{\rm
CMB}(1+z)}{m_{\rm e}}\right]\mathcal{Y}^{(4)}.$ (78)

Since $\tau^{(4)}$ is suppressed during reionization, the Compton cooling via
Rayleigh scattering only happens during recombination. The cooling effect for
Rayleigh scattering is estimated as $y^{(4)}_{\rm e}\mathcal{Y}^{(4)}\sim
2.5\times 10^{-3}$Jy/str. As Rayleigh scattering introduces an additional
cooling effect, the spectral distortion has the same sign as the standard
cooling effect. $y^{(0)}_{\rm ac}$ is sourced during recombination, but
$y^{(0)}_{\rm e}$ is produced only at the end of recombination. As a result,
we find $y^{(0)}_{\rm e}\ll y^{(0)}_{\rm ac}$. On the other hand, both
$y^{(4)}_{\rm ac}$ and $y^{(4)}_{\rm e}$ are sourced at the end of
recombination; those signals are comparable. The total $y^{(4)}$ increases by
about 60% after including the cooling effect.

###  IV.7 Sunyaev-Zel’dovich effect

$y$ spectral distortion from recombination is useful observable for tests of
primordial perturbations at $0.1\leq k{\rm Mpc}/h\leq 100$ Chluba:2012gq ;
Chluba:2012we . However, $\mathcal{Y}^{(0)}$-type spectral dependence is
dominated by the thermal Sunyaev-Zel’dovich effect, i.e., the Compton
scattering between the CMB photons and hot electron gas in the late Universe.
The typical size of $y^{(0)}_{\rm tSZ}$ is $10^{-6}$ Refregier:2000xz , which
is 100 times bigger than the primordial signals even after including
reionization enhancement. The tSZ effect is indistinguishable from
reionization or recombination signals. The degeneration is a crucial
disadvantage of $y$ spectral distortion for primordial perturbation search.
While the new spectral distortion from Rayleigh scattering is weak, Rayleigh
scattering must not secondarily produce the signal after recombination because
the photons must be sufficiently hot. For clusters at $z<10$, the suppression
factor is below $10^{-8}$. Therefore, the new spectral distortions are
straightforwardly related to cosmology during recombination.

##  V Constraints on cosmology

Figure 4:  The $y$ spectral distortion and the new spectral distortion from
Rayleigh scattering with future experimental target sensitivities. The
spectral distortions are evaluated for cosmological parameters consistent with
the recent CMB anisotropy measurements.

We calculated the new spectral distortions for the standard power-law initial
spectrum and found the total spectral distortion, including the cooling
effect, is $6.5\times 10^{-3}$Jy/str. The size is about 0.7% of the $y$
spectral distortion from recombination and one order of magnitude smaller than
the target sensitivity range of voyage 2050 Chluba:2019kpb ; Chluba:2019nxa .
Observing the new spectral distortions for the standard cosmological scenario
with the almost scale invariant curvature perturbatins is quite difficult.
However, there are a few advantages of the new spectral distortions. The $y$
spectral distortion degenerates with the tSZ effect, which is $10^{3}$ times
bigger than the $y$ distortion from recombination. Therefore, we cannot use
$y^{(0)}$ for cosmological constraints about recombination even if the
sensitivity goes beyond $10^{-3}$Jy/str. The new spectral distortion is free
from the issue. We can extract the cosmological information from ideal
measurements of $y^{(4)}$ and crosscheck cosmology determined by other surveys
such as Planck Aghanim:2018eyx .

There are two peaks in the right panel of Fig. 2. The first negative peak
around $z\sim 6000$ is due to Rayleigh scattering of singly ionized helium
during the tight-coupling regime. The second positive peak is from neutral
hydrogen Rayleigh scattering. The late time peak dominates the total
distortion for the almost scale-invariant power spectrum because hydrogen
Rayleigh scattering is more efficient than singly ionized helium.

The scale dependence of the sign could be potentially useful to distinguish
several nonstandard scenarios. For example, let us consider a blue tilted
initial spectrum, i.e., $n_{s}>1$. As we have $\log_{10}[W^{(0,4)}(k{\rm
Mpc/h}=1.)/\mathcal{W}^{(4,0)}(k{\rm Mpc/h}=0.1)]\approx 3.4$, the negative
part dominates for the blue spectrum with $n_{s}\gtrsim 4.4$. However, such a
highly blue spectrum also introduces detectable spectral distortions like
$\mu$ spectral distortions Cabass:2016giw . Several inflationary models
predict peaks in the curvature power spectrum that can be approximated by the
lognormal spectrum

$\displaystyle\mathcal{P}_{\zeta}=\frac{A_{\zeta}}{\sqrt{2\pi}\Delta}e^{-\frac{\left(\ln
k-\ln k_{0}\right)^{2}}{2\Delta^{2}}},$ (79)

which is often considered for primordial black hole formation and associated
induced gravitational waves Pi:2020otn . Consider the peak located at
$k_{0}{\rm Mpc}/h=2.$, with $A_{\zeta}=10^{-4}$ and $\Delta=10^{-3}$. We
cannot exclude such a lognormal spectrum from the current temperature
anisotropy measurements since the CMB anisotropies at $k{\rm Mpc}/h>0.1$ is
exponentially suppressed due to Silk damping. In this setup, we find
$y^{(0)}_{\rm ac}=1.\times 10^{-6}$, which is comparable to the tSZ effect, so
we cannot distinguish the tSZ effect from $y_{\rm ac.}^{(0)}$. For the same
parameters, we find the new spectral distortion is $3.\times 10^{-2}$Jy/str
with the opposite sign, which is in the target range of voyage 2050.

##  VI Conclusions

Towards future precise CMB intensity spectral measurements Kogut:2011xw ;
Kogut:2019vqh ; PRISM:2013fvg ; Chluba:2019kpb ; Chluba:2019nxa , we
considered a new spectral distortion from Rayleigh scattering during
recombination. There are a few works on the Rayleigh anisotropies Yu:2001gw ;
Lewis:2013yia ; Alipour:2014dza ; Beringue:2020wxw ; Coulton:2020oxw , but all
of them are about the linear anisotropies, and thus distortions to the angular
averaged spectrum have not been considered. In this work, we expanded the
photon Boltzmann equation to second-order cosmological perturbations in the
presence of Thomson scattering and Rayleigh scattering, and we show that the
second-order effect introduces a new monopole spectral distortion. The
generation mechanism is similar to the traditional $y$ spectral distortion
from the relativistic corrections to the Thomson scattering. We show that
Rayleigh scattering also introduces the frequency-dependent acoustic
dissipation and Compton cooling effect. The spectral shape is different from
the other traditional spectral distortions. Spectral distortions from atomic
processes during recombination were studied at the background level in the
literature, but those works do not include Rayleigh scattering since diffusion
introduced by second order perturbations is mandatory for the present
consideration Wong:2005yr ; Sunyaev:2009qn ; Chluba:2015gta ; Chluba:2008aw .

Fig. 4 shows several $y$ and new spectral distortions obtained in this work.
We found the new spectral distortion, including both acoustic dissipation and
the cooling effect, is $6.5\times 10^{-3}$Jy/str for the cosmological
parameters determined by Ref. Aghanim:2018eyx , which is one order of
magnitude smaller than the envisioned target sensitivity range of voyage 2050.
Thus, the spectral distortion from the second-order Rayleigh scattering is
tiny, and it will not be easy to detect the signal soon.

Even if the new spectral distortion is tiny, it can be useful. The scale
invariant primordial density perturbations at $0.01\leq k{\rm Mpc}\leq 100$
yields $y\sim 10^{-9}$, which is the detectable size in the next generation of
spectral measuremets. However, the Sunyaev-Zel’dovich effect is known as the
same spectral dependence as the $y$ distortion; Compton scattering of the CMB
photons by high energy electron gas in the late time clusters is three orders
of magnitude bigger than the primordial $y$ distortion. In principle, the sky
averages degenerate. The new spectral distortion is not produced in the late
Universe because CMB photons are too cold for Rayleigh scattering after
recombination. In the longer term, the new spectral distortion can be
distinguishable from other signals and can be more useful than the traditional
$y$ distortion for recombination studies. We also considered several scale-
dependent primordial power spectra for the new spectral distortion, and we
showed that there is a unique window for constraining the large initial
perturbations at $k{\rm Mpc}/h\sim 1$.

Rayleigh scattering adds new electrons interacting with photons; therefore,
the photon-electron coupling becomes stronger. As a result, the multipole
anisotropies are suppressed, and then the acoustic dissipation is reduced
during the tight coupling regime. Thus, additional interacting species
suppress the acoustic source in the early Universe. Such a response will be
universal; therefore, spectral distortions will be useful for constraining
unknown interacting dark sectors.

We considered a rigorous framework of second-order Boltzmann equations for
isotropic spectral distortions. Full second-order CMB, including Rayleigh
scattering, will be more complicated but important to subtract the intrinsic
nonlinearity for constraining the primordial non-Gaussianity in the Rayleigh
anisotropies. The extension to the full second-order CMB will be discussed
elsewhere.

###### Acknowledgements.

The author would like to thank William R. Coulton for his useful discussions.

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