# Cosmology from the kinetic polarized Sunyaev Zel’dovich effect

Selim C. Hotinli  William H. Miller III Department of Physics and Astronomy,
Johns Hopkins University, Baltimore, MD 21218, USA  Gilbert P. Holder
Astronomy Department, University of Illinois at Urbana-Champaign, 1002 W.
Green Street, Urbana, IL 61801, USA  Department of Physics, University of
Illinois Urbana-Champaign, 1110 W. Green Street, Urbana, IL 61801, USA
Matthew C. Johnson  Perimeter Institute for Theoretical Physics, 31 Caroline
St N, Waterloo, ON N2L 2Y5, Canada  Department of Physics and Astronomy, York
University, Toronto, ON M3J 1P3, Canada  Marc Kamionkowski  William H. Miller
III Department of Physics and Astronomy, Johns Hopkins University, Baltimore,
MD 21218, USA

###### Abstract

The cosmic microwave background (CMB) photons that scatter off free electrons
in the large-scale structure induce a linear polarization pattern proportional
to the remote CMB temperature quadrupole observed in the electrons’ rest
frame. The associated blackbody polarization anisotropies are known as the
polarized Sunyaev Zel’dovich (pSZ) effect. Relativistic corrections to the
remote quadrupole field give rise to a non-blackbody polarization anisotropy
proportional to the square of the transverse peculiar velocity field; this is
the kinetic polarized Sunyaev Zel’dovich (kpSZ) effect. In this paper, we
forecast the ability of future CMB and galaxy surveys to detect the kpSZ
effect, finding that a statistically significant detection is within the reach
of planned experiments. We further introduce a quadratic estimator for the
square of the peculiar velocity field based on a galaxy survey and CMB
polarization. Finally, we outline how the kpSZ effect is a probe of cosmic
birefringence and primordial non-Gaussianity, forecasting the reach of future
experiments.

##  I Introduction

The next generation of CMB surveys such as the ongoing Simons Observatory
(Aguirre  _et al._ , 2018; Lee  _et al._ , 2019), the upcoming CMB-S4
(Abazajian  _et al._ , 2016a, 2022), and the futuristic CMB-HD (Sehgal  _et
al._ , 2019; Aiola  _et al._ , 2022), and galaxy surveys such as DESI
(Aghamousa  _et al._ , 2016), the Vera Rubin Observatory (VRO) (LSST Science
Collaboration  _et al._ , 2009), and perhaps then MegaMapper (Schlegel  _et
al._ , 2019) promise to greatly expand our ability to infer the fundamental
characteristics of the Universe. One exciting new opportunity is to use the
CMB as a back-light, observing the effects on the CMB of scattering and
lensing by the intervening large-scale structure. These interactions include
the thermal, kinetic and polarized Sunyaev Zel’dovich effects (Zeldovich and
Sunyaev, 1969; Zel’Dovich, 1970; Sunyaev and Zeldovich, 1980, 1972; Sazonov
and Sunyaev, 1999; Itoh  _et al._ , 1998; Challinor  _et al._ , 2000;
Kamionkowski and Loeb, 1997; Chluba  _et al._ , 2013), and the integrated
Sachs-Wolfe effects (Sachs and Wolfe, 1967), which includes the moving-lens
effect (Birkinshaw and Gull, 1983; Cooray, 2002; Hotinli  _et al._ , 2019a,
2021a), and weak-gravitational lensing (for a review see e.g. (Lewis and
Challinor, 2006)). Each of these effects can provide new insights and
constraints on cosmological models.

When CMB photons scatter from electrons, they acquire a specific pattern of
linear polarization proportional to the remote CMB temperature quadrupole seen
by the electron in its rest frame. The dominant contributions to the remote
quadrupole field are primordial in nature, arising from the Sachs Wolfe and
Doppler effects at decoupling. In the late-Universe, the next most significant
contribution is from the integrated Sachs Wolfe effect. These contributions
all give rise to a blackbody polarization signal, and are described in detail
in e.g. (Deutsch  _et al._ , 2018). Primordial gravitational waves can also
contribute to the remote quadrupole field (Alizadeh and Hirata, 2012; Deutsch
_et al._ , 2017, 2019), giving rise to an additional blackbody polarization
signal.

Another contribution to the remote quadrupole field arises from the
relativistic Doppler effect. This ‘kinematic’ quadrupole (Sunyaev and
Zeldovich, 1980, 1972; Baumann  _et al._ , 2003; Cooray and Baumann, 2003; Hu,
2000; Shimon  _et al._ , 2006; Bunn, 2006; Yasini and Pierpaoli, 2016) gives
rise to a non-blackbody polarization anisotropy, which we refer to as the
kinematic polarized Sunyaev Zel’dovich (kpSZ) effect, whose study is the focus
of this work 111As described in detail in Ref. Yasini and Pierpaoli (2016),
there are additional frequency-dependent secondary polarization anisotropies
associated with the relativistic mixing of the primordial dipole and octupole
into the locally observed quadrupole of a moving scatterer. These effects are
generally smaller than the pSZ and kpSZ effects considered here, although they
can dominate in specific regions of the sky and at high frequencies Yasini and
Pierpaoli (2016).. While subdominant to the polarization anisotropies
associated with recombination, reionization, and the pSZ effect, the kpSZ
effect has a unique frequency dependence which allows it to be separated using
multi-frequency CMB experiments. The kpSZ effect has been studied previously
as a signal in e.g. Refs. (Roebber and Holder, 2014; Yasini and Pierpaoli,
2016; Fidler and Ringeval, 2017) and as a bias on the pSZ measurement in e.g.
Ref. (Hall and Challinor, 2014). Here, we provide a detailed calculation of
the $E$\- and $B$-mode signals induced by the kpSZ up to non-linear order and
assess the prospects to detect this signal through cross-correlating the CMB
with templates of the kinematic quadrupole that can be reconstructed from
measurements of galaxies. We highlight two distinct methods for detection:
first, through cross-correlating the CMB maps on degree scales with kpSZ
templates from galaxy surveys, and next, through reconstructing the remote
kinetic quadrupole with pSZ tomography (Alizadeh and Hirata, 2012; Deutsch
_et al._ , 2019, 2017, 2018; Lavaux  _et al._ , 2004; Amblard and White,
2005). The latter technique has been shown to be a promising direction for
multitudes of cosmological analyses, adding to the prospects to detect
gravitational tensor perturbations Deutsch  _et al._ (2019) and improving the
constraints on the mean optical depth to reionization Meyers  _et al._ (2018),
for example.

More generally, the reconstruction of large-scale cosmological fluctuations
via cross-correlating imprints of scattering and lensing effects on the CMB
with tracers of structure Dvorkin and Smith (2009); Dvorkin  _et al._ (2009);
Smith and Ferraro (2017); Kamionkowski (2009); Yadav  _et al._ (2009);
Gluscevic  _et al._ (2009); Alizadeh and Hirata (2012); Deutsch  _et al._
(2018, 2017); Meyers  _et al._ (2018); Hotinli  _et al._ (2019a, 2021a,
2021b); Hotinli and Johnson (2022); Cayuso  _et al._ has a wide range of
applications. These include the reconstruction of large-scale velocity fields
of matter via measurements of the kinetic Sunyaev-Zel’dovich (kSZ) effect
(Zeldovich and Sunyaev, 1969; Zel’Dovich, 1970; Sunyaev and Zeldovich, 1980,
1972; Sazonov and Sunyaev, 1999), which is anticipated to provide competitive
constraints on local-type primordial non-Gaussianity Deutsch  _et al._ (2017);
Smith  _et al._ (2018); Münchmeyer  _et al._ (2018), deviations from general
relativity Zhang and Johnson (2015), specific forms of primordial isocurvature
Hotinli  _et al._ (2019b), sources of CMB anomalies Cayuso and Johnson (2019),
and the reionization history Alvarez  _et al._ (2021); Hotinli and Johnson
(2022). Here, we focus on the kpSZ effect and describe possible ways to
utilize this signal for cosmological inference.

We begin in Sec. II by reproducing the calculation of the kpSZ effect,
presenting the $E$\- and $B$-mode polarization signal, including contributions
from the inhomogeneous optical depth at low redshift. In Sec. III we forecast
the ability of future CMB experiments to detect the kpSZ effect using two
techniques. First, we demonstrate that on large angular scales the kpSZ effect
can be detected by cross-correlating CMB polarization with a template
constructed using a Rubin LSST-like galaxy survey. Next, we introduce a
quadratic estimator for the square of the transverse velocity field based on
the statistically anisotropic cross-correlation of CMB polarization with a
galaxy survey. Either technique can be used to obtain a detection with signal-
to-noise greater than 2 using CMB-S4, with improvements expected for
experiments with greater resolution (e.g. CMB-HD) or additional frequency
channels (e.g. PIXIE). In Sec. IV, we demonstrate that the kpSZ effect can be
used to constrain cosmic birefringence and primordial non-Gaussianity, with
constraints that scale well with noise and resolution of future experiments.
We conclude in Sec.V.

##  II Kinetic Polarized Sunyaev Zel’dovich Signal

###  II.1 The kinetic quadrupole

Here, we calculate the kinetic quadrupole signal following similar
calculations for the pSZ signal as in e.g. Refs. (Meyers  _et al._ , 2018;
Hall and Challinor, 2014). The Stokes parameters $Q$ and $U$ that are measured
in CMB observations in a coordinate frame centred on the observer can be
projected onto a 2-sphere with unit vectors
$\\{\hat{\boldsymbol{\theta}},-\hat{\boldsymbol{\phi}}\\}$ in the plane
perpendicular to line of sight direction $\hat{\boldsymbol{n}}$, using
polarization vectors
$\boldsymbol{e}_{\pm}(\hat{\boldsymbol{n}})=(\hat{\boldsymbol{\theta}}\mp
i\hat{\boldsymbol{\phi}})/\sqrt{2}$. The resulting complex CMB polarization
along the line of sight takes the form of an integral in comoving distance
$\chi$,

$\displaystyle(Q\pm iU)(\hat{\boldsymbol{n}})=\int_{0}^{\chi_{\star}}{\rm
d}\chi\
\dot{\tau}(\chi\hat{\boldsymbol{n}})e^{-\tau(\chi)}\,_{\pm}p(\chi\hat{\boldsymbol{n}})\,,$
(1)

where $\chi$ is the comoving distance along our past light cone, $\chi=0$
today, $\chi_{*}$ is the comoving distance to the surface of last scattering,
and $\tau$ is the optical depth along the line of sight. We write the optical
depth at redshift $z$ along $\hat{\boldsymbol{n}}$ as
$\dot{\tau}(\chi\hat{\boldsymbol{n}})={\rm d}\tau/{\rm
d}\chi(\chi\hat{\boldsymbol{n}})$ where ${\rm d}\tau/{\rm
d}\chi(\chi\hat{\boldsymbol{n}})=\sigma_{T}an_{e}(\chi\hat{\boldsymbol{n}})$
and $\sigma_{T}$ is the Thomson cross section, $a$ is the scale factor,
$n_{e}$ is the free electron number density. The remote quadrupole field
projected along the line of sight $\,{}_{\pm}p(\chi\hat{\boldsymbol{n}})$ is
given by:

$\,{}_{\pm}p(\chi\hat{\boldsymbol{n}})\equiv\sum_{m=-2}^{m=2}\Theta_{2m}(\chi\hat{\boldsymbol{n}})\,_{\mp
2}Y_{\ell m}(\hat{\boldsymbol{n}})$ (2)

where $\Theta_{2m}(\chi\hat{\boldsymbol{n}})$ are the $\ell=2$ moments of the
CMB temperature anisotropies observed at the position
$\chi\hat{\boldsymbol{n}}$ along our past light cone. Although our perspective
will mainly be that the optical depth and remote quadrupole are continuous
fields, it is sometimes helpful to consider the contribution from a single
scatterer (e.g. a cluster) with optical depth $\tau$

$\displaystyle(Q\pm
iU)^{(e)}(\hat{\boldsymbol{n}})=\tau\,_{\pm}p(\hat{\boldsymbol{n}})\,,$ (3)

where we omit the $\chi$ dependence in what follows. Isolating the quadrupole
field

$\,{}_{\pm}p(\hat{\boldsymbol{n}})=\frac{1}{\tau}(Q\pm
iU)^{(e)}(\hat{\boldsymbol{n}})\ ,$ (4)

we can expand this into spin-2 spherical harmonics on a shell of constant
$\chi$ as

$\,{}_{\pm}p(\hat{\boldsymbol{n}})=\sum\limits_{\ell m}\,{}_{\mp}a^{p}_{\ell
m}\,{}_{\mp 2}Y_{\ell m}(\hat{\boldsymbol{n}})\ .$ (5)

Note that $(Q+iU)(\hat{\boldsymbol{n}})$ in our formalism is spin $-2$ due to
the choice of coordinate frame centred at the observer with scattered photon
travelling in the $-\hat{\boldsymbol{n}}$ direction.

For a scatterer at rest, the remote quadrupole field receives contributions
from the Sachs Wolfe effect, primordial Doppler effect, integrated Sachs Wolfe
effect, and potentially primordial gravitational waves. When the scatterer has
a non-zero peculiar velocity, relativistic abberation in the rest frame of the
scatterer will lead to frequency-dependent Doppler leakage of adjacent
multipoles into the quadrupole. The dominant effect is the leakage of the
monopole into the quadrupole (see Ref. Yasini and Pierpaoli (2016) for
contributions from the dipole and octupole), which can be understood as
follows. In the electron rest frame, the CMB black-body temperature gets a
relativistic correction as
$T^{\prime}=Tf(x)(1-{\boldsymbol{v}}_{e}\cdot\hat{\boldsymbol{n}}_{e})/\sqrt{1-|\boldsymbol{v}_{e}|^{2}}$
where ${\boldsymbol{v}}_{e}$ is the electron peculiar velocity, $f(x)$
captures the frequency dependence of the effect, $T$ is the CMB black-body
temperature at the CMB rest frame, where the aberration of the CMB
fluctuations vanishes and we defined $\hat{\boldsymbol{n}}_{e}$ in observer’s
frame as the direction of the photon incoming on the electron.

The relativistic correction to the quadrupole can be calculated as equal to
$-(1/10)\,g(x)\beta_{\perp}^{2}$ where
$\beta_{\perp}^{2}=|\boldsymbol{v}_{e}|^{2}(1-\hat{\boldsymbol{n}}_{e}\cdot\hat{\boldsymbol{n}})$,
$\beta_{\perp}$ is the transverse velocity amplitude 222Note that the
longitudinal velocity contribution will get a relativistic correction $v/c$
when transformed back to CMB frame and is sub-dominant.,
$g(x)=(x/2)\coth(x/2)$ and $x=h\nu/k_{B}T$ is the dimensionless frequency of
the photon in CMB rest frame.

We can write the kinetic contribution to the remote quadrupole field as

$\displaystyle{\,{}_{\pm}}p(\chi\hat{\boldsymbol{n}})=-\frac{T_{\rm
CMB}}{10}\left(\frac{x}{2}\coth{\frac{x}{2}}\right)\left(V_{\theta}(\chi\hat{\boldsymbol{n}})\mp
iV_{\phi}(\chi\hat{\boldsymbol{n}})\right)^{2}\,,$ (6)

where we projected the velocity transverse to the line of sight
${\boldsymbol{v}}_{\perp}(\hat{\boldsymbol{n}})$ at each $\chi$, an ordinary
2-vector field in 2-sphere, onto the polarization basis we use
$\boldsymbol{e}_{\pm}(\hat{\boldsymbol{n}})$, with definitions
$V_{\theta}:={\boldsymbol{v}}_{\perp}\cdot\hat{\boldsymbol{\theta}}$ and
$V_{\phi}:={\boldsymbol{v}}_{\perp}\cdot\hat{\boldsymbol{\phi}}$. The
transverse velocity can be decomposed into a gradient and a curl part as

$\displaystyle{\boldsymbol{v}}_{\perp}(\hat{\boldsymbol{n}})=\boldsymbol{\nabla}\Upsilon(\hat{\boldsymbol{n}})+\boldsymbol{\nabla}\times\varpi(\hat{\boldsymbol{n}}).$
(7)

We define the commonly used spin raising and lowering operators as

$\displaystyle\partial\kern-3.00003pt\raise
1.80832pt\hbox{\tiny\textbackslash}\,_{s}\eta$
$\displaystyle=-\sin^{s}\\!\theta(\partial_{\theta}+i\mathrm{cosec}\theta\\!\
\partial_{\phi})(\sin^{-s}\\!\theta\,_{s}\eta)$
$\displaystyle\overline{\partial\kern-3.00003pt\raise
1.80832pt\hbox{\tiny\textbackslash}}\,_{s}\eta$
$\displaystyle=-\sin^{s}\\!\theta(\partial_{\theta}{-}i\mathrm{cosec}\theta\\!\
\partial_{\phi})(\sin^{-s}\\!\theta\,_{s}\eta)\,,$ (8)

which are useful to define higher spin harmonics in terms of standard spin-
weighted spherical harmonics, for example,

$\,{}_{1}Y_{\ell m}$ $\displaystyle=\ \ \ \partial\kern-3.00003pt\raise
1.80832pt\hbox{\tiny\textbackslash}\left(\sqrt{\frac{(\ell-1)!}{(\ell+1)!}}Y_{\ell
m}\right)\,,$ (9) $\,{}_{-1}Y_{\ell m}$
$\displaystyle=-\overline{\partial\kern-3.00003pt\raise
1.80832pt\hbox{\tiny\textbackslash}}\left(\sqrt{\frac{(\ell-1)!}{(\ell+1)!}}Y_{\ell
m}\right)\,.$ (10)

Moreover, using Eq. (7) and Eq. (II.1), we can show that

$\displaystyle{\boldsymbol{v}}_{\perp}\cdot\boldsymbol{e}_{-}$
$\displaystyle=-\partial\kern-3.00003pt\raise
1.80832pt\hbox{\tiny\textbackslash}(\Upsilon-i\varpi)=v_{\theta}+iv_{\phi}\,,$
(11) $\displaystyle{\boldsymbol{v}}_{\perp}\cdot\boldsymbol{e}_{+}$
$\displaystyle=-\overline{\partial\kern-3.00003pt\raise
1.80832pt\hbox{\tiny\textbackslash}}(\Upsilon+i\varpi)=v_{\theta}-iv_{\phi}\,.$
(12)

We can expand the fields $\Upsilon$ and $\varpi$ in spherical harmonics as

$\displaystyle\Upsilon(\hat{\boldsymbol{n}})$ $\displaystyle=\sum\limits_{\ell
m}\sqrt{\frac{(\ell-1)!}{(\ell+1)!}}V^{G}_{\ell m}Y_{\ell
m}(\hat{\boldsymbol{n}})\,,$ (13) $\displaystyle\varpi(\hat{\boldsymbol{n}})$
$\displaystyle=\sum\limits_{\ell
m}\sqrt{\frac{(\ell-1)!}{(\ell+1)!}}V^{C}_{\ell m}Y_{\ell
m}(\hat{\boldsymbol{n}})\,,$ (14)

which then lets us write

$\displaystyle(V_{\theta}\pm
iV_{\phi})(\hat{\boldsymbol{n}})=\sum\limits_{\ell m}({{\pm}}V^{G}_{\ell
m}+iV^{C}_{\ell m})(\hat{\boldsymbol{n}})\,_{\pm 1}Y_{\ell
m}(\hat{\boldsymbol{n}})\,.$ (15)

Under parity transformation, the transverse velocity transforms as
$\boldsymbol{v}(\hat{\boldsymbol{n}})\rightarrow-\boldsymbol{v}(-\hat{\boldsymbol{n}})$
while the transverse components transform as $(V_{\theta}\pm
iV_{\phi})(\hat{\boldsymbol{n}})\rightarrow-(V_{\theta}\mp
iV_{\phi})(-\hat{\boldsymbol{n}})$. Taking the parity transformation
properties of the spin-1 harmonics, one finds that in order to satisfy the
parity transformation properties of the velocity, the expansion coefficients
must transform as $V^{G}_{\ell m}\rightarrow(-1)^{\ell}V^{G}_{\ell m}$ and
$V^{C}_{\ell m}\rightarrow(-1)^{\ell+1}V^{C}_{\ell m}$, i.e. with electric and
magnetic parities respectively. Finally, note that in the presence of only
scalar perturbations, we expect only the gradient term to be present, i.e. we
will approximate $V^{C}_{\ell m}\approx 0$.

The transverse velocity potential $\Upsilon$ projected into the 2-sphere
perpendicular to line-of-sight $\hat{\boldsymbol{n}}$ can be calculated from
the velocity field ${\boldsymbol{v}}(\chi\hat{\boldsymbol{n}})$ by taking a
projected derivative $\nabla_{\hat{\boldsymbol{n}}}$ of the velocity
potential, and is related to the covariant derivative on the sphere as
$\nabla_{\hat{\boldsymbol{n}}}=(1/\chi)\nabla$. The Fourier transform of
$\Upsilon$ can then be written as

$\displaystyle\Upsilon(\chi\hat{\boldsymbol{n}})=-\int\frac{{\rm
d}^{3}\boldsymbol{k}}{(2\pi)^{3}}\frac{v(\boldsymbol{k},\chi)}{k\chi}e^{ik\chi\hat{\boldsymbol{k}}\cdot\hat{\boldsymbol{n}}}\,,$
(16)

where
$v({\boldsymbol{k},\chi})=\hat{\boldsymbol{k}}\cdot{\boldsymbol{v}}(\chi\hat{\boldsymbol{n}})$,
and we have reintroduced the conformal time dependence. The coefficient
$V^{G}_{\ell m}$ defined above can be found by expanding the exponential in
Eq. (16) as

$\displaystyle\exp(ik\chi\hat{\boldsymbol{k}}\cdot{\hat{\boldsymbol{n}}})=4\pi\sum\limits_{\ell
m}i^{\ell}j_{\ell}(k\chi)Y_{\ell m}(\hat{\boldsymbol{n}})Y_{\ell
m}^{*}(\hat{\boldsymbol{k}})\,,$ (17)

and isolating the spherical harmonic $Y_{\ell m}({\hat{\boldsymbol{n}}})$, we
get (Hall and Challinor, 2014; Bunn, 2006)

$\displaystyle V^{G}_{\ell m}=-4\pi i^{\ell}\sqrt{\ell(\ell+1)}\int\frac{{\rm
d}^{3}\boldsymbol{k}}{(2\pi)^{3}}{v(\boldsymbol{k},{\chi})}\frac{j_{\ell}(k\chi)}{k\chi}Y_{\ell
m}^{*}(\hat{\boldsymbol{k}})\ .$ (18)

We now use the expression for $V^{G}_{\ell m}$ to calculate the harmonic
coefficient of the polarization sourced by the kinematic quadrupole and its
spectrum $p_{\ell m}$. Using Eqs. (5), (6) and (15), we can write

$\displaystyle\sum_{\ell m}\,{}_{\pm}p_{\ell m}\,{}_{\pm 2}Y_{\ell
m}(\hat{\boldsymbol{n}})=f(x)\sum_{\ell
m}\sum_{\ell^{\prime}m^{\prime}}V^{G}_{\ell
m}V^{G}_{\ell^{\prime}m^{\prime}}\,{}_{\pm 1}Y_{\ell
m}(\hat{\boldsymbol{n}})\,_{\pm
1}Y_{\ell^{\prime}m^{\prime}}(\hat{\boldsymbol{n}})\,.$ (19)

where $f(x)=-(1/10)g(x)T_{\rm CMB}$ and we use notations as $\,{}_{\pm}p_{\ell
m}=\,_{\pm}a_{\ell m}^{p}$ for brevity. Some useful spherical harmonics
relations are

$\displaystyle\int{\rm d}^{2}\hat{\boldsymbol{n}}\,_{s}Y_{\ell
m}(\hat{\boldsymbol{n}})\,_{s}Y_{\ell^{\prime}m^{\prime}}^{\star}(\hat{\boldsymbol{n}})=\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}}\,,$
(20)

and

$\displaystyle\int\\!\\!{\rm
d}^{2}\hat{\boldsymbol{n}}\,_{s_{1}}\\!Y_{\ell_{1}m_{1}}(\hat{\boldsymbol{n}})\,_{s_{2}}\\!Y_{\ell_{2}m_{2}}$
$\displaystyle(\hat{\boldsymbol{n}})\,_{s_{3}}Y_{\ell_{3}m_{3}}(\hat{\boldsymbol{n}})$
(21) $\displaystyle=$
$\displaystyle\sqrt{\frac{(2\ell_{1}+1)(2\ell_{2}+1)(2\ell_{3}+1)}{4\pi}}\begin{pmatrix}\ell_{1}&\ell_{2}&\ell_{3}\\\
m_{1}&m_{2}&m_{3}\end{pmatrix}\begin{pmatrix}\ell_{1}&\ell_{2}&\ell_{3}\\\
\\!-s_{1}&\\!-s_{2}&\\!-s_{3}\end{pmatrix}\,.$

We multiply both sides of Eq. (19) by the spin-2 spherical harmonic $\,{}_{\mp
2}Y_{LM}^{*}(\hat{\boldsymbol{n}})$ and integrate over ${\rm
d}^{2}\hat{\boldsymbol{n}}$. Using the relations defined in Eqs. (20) and (21)
and $\,{}_{\mp 2}Y_{LM}^{*}(\hat{\boldsymbol{n}})=\,_{\mp
2}Y_{L-M}(\hat{\boldsymbol{n}})$, we get

$\,{}_{\pm}p_{\ell
m}\\!=\\!f(x)\sum_{\ell^{\prime}m^{\prime}}\sum_{\ell^{\prime\prime}m^{\prime\prime}}V^{G}_{\ell^{\prime}m^{\prime}}V^{G}_{\ell^{\prime\prime}m^{\prime\prime}}\,{}_{\pm}\mathcal{I}_{\ell^{\prime}\ell^{\prime\prime}\ell}^{m^{\prime}m^{\prime\prime}m}\,,$
(22)

where we defined

$\,{}_{\pm}\mathcal{I}_{\ell^{\prime}\ell^{\prime\prime}\ell}^{m^{\prime}m^{\prime\prime}m}=\sqrt{\frac{(2\ell^{\prime}+1)(2\ell^{\prime\prime}+1)(2\ell+1)}{4\pi}}\begin{pmatrix}\ell^{\prime}&\ell^{\prime\prime}&\ell\\\
m^{\prime}&m^{\prime\prime}&-m^{\prime}\end{pmatrix}\begin{pmatrix}\ell^{\prime}&\ell^{\prime\prime}&\ell\\\
\pm 1&\pm 1&\mp 2\end{pmatrix}\,.$ (23)

This equation suggests that the quadrupole coefficient $\,{}_{\pm}p_{LM}$
satisfies the reality condition, i.e. $X^{*}_{\ell m}=X_{\ell-m}$, if
$V^{G\,*}_{\ell m}=V^{G}_{\ell-m}$ does as well.333Seeing this also requires
the symmetry relation
$\displaystyle\begin{pmatrix}\ell_{1}&\ell_{2}&\ell_{3}\\\
m_{1}&m_{2}&m_{3}\end{pmatrix}=(-1)^{\ell_{1}+\ell_{2}+\ell_{3}}\begin{pmatrix}\ell_{1}&\ell_{2}&\ell_{3}\\\
-m_{1}&-m_{2}&-m_{3}\end{pmatrix}$ (24) which returns a factor
$(-1)^{2(\ell_{1}+\ell_{2}+\ell_{3})}$ for $p^{*}_{\ell m}$. The latter
equality can be seen to be true by observing Eq. (13) and noting that we take
$V^{C}_{\ell m}=0$. We define the variance of the coefficients in the usual
way,

$\displaystyle\langle X_{\ell
m}Y^{\star}_{\ell^{\prime}m^{\prime}}\rangle=C^{XY}_{\ell}\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}}\,.$
(25)

Next, we calculate the kinematic quadrupole auto-spectra $\langle p_{\ell
m}p^{*}_{\ell^{\prime}m^{\prime}}\rangle$ which can be found by using the
reality of $p_{LM}^{*}$ in order to satisfy Eq. (25) as

$\displaystyle\begin{split}\langle\,_{\pm}p_{\ell
m}\,{}_{\pm}p^{*}_{\ell^{\prime}m^{\prime}}\rangle&=f^{2}(x)\sum\limits_{\ell_{1}m_{1}}\sum\limits_{\ell_{1}^{\prime}m_{1}^{\prime}}\sum\limits_{\ell_{2}m_{2}}\sum\limits_{\ell_{2}^{\prime}m_{2}^{\prime}}\langle
V^{G}_{\ell_{1}m_{1}}V^{G}_{\ell_{1}^{\prime}m_{1}^{\prime}}V^{G\,*}_{\ell_{2}m_{2}}V^{G\,*}_{\ell_{2}^{\prime}m_{2}^{\prime}}\rangle\,\,_{\pm}\mathcal{I}_{\ell_{1}\ell_{1}^{\prime}\ell}^{m_{1}m_{1}^{\prime}m}\,{}_{\pm}\mathcal{I}_{\ell_{2}\ell_{2}^{\prime}\ell^{\prime}}^{m_{2}m_{2}^{\prime}m^{\prime}}\end{split}$
(26)

where we collected the frequency dependent terms assuming equal frequencies.
We write the four-point function defined in the second line in Eq. (26) as

$\displaystyle\langle V^{G}_{\ell_{1}m_{1}}$ $\displaystyle
V^{G}_{\ell_{1}^{\prime}m_{1}^{\prime}}V^{G\,*}_{\ell_{2}m_{2}}V^{G\,*}_{\ell_{2}^{\prime}m_{2}^{\prime}}\rangle$
$\displaystyle=\langle
V^{G}_{\ell_{1}m_{1}}V^{G}_{\ell_{1}^{\prime}m_{1}^{\prime}}\rangle\langle
V^{G\,*}_{\ell_{2}m_{2}}V^{G\,*}_{\ell_{2}^{\prime}m_{2}^{\prime}}\rangle+\langle
V^{G}_{\ell_{1}m_{1}}V^{G\,*}_{\ell_{2}^{\prime}m_{2}^{\prime}}\rangle\langle
V^{G\,*}_{\ell_{2}m_{2}}V^{G}_{\ell_{1}^{\prime}m_{1}^{\prime}}\rangle+\langle
V^{G}_{\ell_{1}m_{1}}V^{G\,*}_{\ell_{2}m_{2}}\rangle\langle
V^{G\,*}_{\ell_{2}^{\prime}m_{2}^{\prime}}V^{G}_{\ell_{1}^{\prime}m_{1}^{\prime}}\rangle\,.$

Note that the first term in the second line of Eq. (II.1) is proportional to
$\langle p_{LM}^{*}\rangle\langle p_{L^{\prime}M^{\prime}}^{*}\rangle$ and
only contributes to the monopole444This is due to the relation
$\displaystyle\sum\limits_{m}(-1)^{-m}\begin{pmatrix}\ell&\ell&L\\\
m&-m&0\end{pmatrix}=(-1)^{\ell}\sqrt{2\ell-1}\delta_{L0}$ (28) which we assume
to be correct for the reality condition we have chosen. Traditionally there is
a $(-1)^{m}$ term that is involved that one must apply to both coefficients
and the spherical harmonics functions. In cosmology literature the definition
we used is more common., i.e. vanishes for modes $\\{L,L^{\prime}\\}\geq 1$.
For second and third terms of Eq. (II.1), which are identical, the identities

$\displaystyle\sum\limits_{mm^{\prime}}\begin{pmatrix}\ell&\ell^{\prime}&L\\\
m&m^{\prime}&M\end{pmatrix}\begin{pmatrix}\ell&\ell^{\prime}&L^{\prime}\\\
m&m^{\prime}&M^{\prime}\end{pmatrix}=(2L+1)^{-1}\delta_{LL^{\prime}}\delta_{MM^{\prime}}\,,$
$\displaystyle\begin{split}\begin{pmatrix}\ell_{1}&\ell_{2}&\ell_{3}\\\
s_{1}&s_{2}&s_{3}\end{pmatrix}&\begin{pmatrix}\ell_{1}&\ell_{2}&\ell_{3}\\\
s_{1}^{\prime}&s_{2}^{\prime}&s_{3}^{\prime}\end{pmatrix}=\\!\frac{1}{2}\\!\int_{-1}^{1}\\!{\rm
d}(\cos{\theta})d^{\ell_{1}}_{s_{1}s_{1}^{\prime}}(\theta)d^{\ell_{2}}_{s_{2}s_{2}^{\prime}}(\theta)d^{\ell_{3}}_{s_{3}s_{3}^{\prime}}(\theta)\,,\end{split}$

where $d$ is the Wigner-$d$ matrix (Wigner, 1931), and
$d^{\ell}_{s_{1}s_{1}^{\prime}}=d^{\ell}_{-s_{1}-s_{1}^{\prime}}$ are useful.
Using these, we find the correlation function $\langle
p_{LM}p_{L^{\prime}M^{\prime}}^{*}\rangle=\delta_{LL^{\prime}}\delta_{MM^{\prime}}C^{pp}_{L}$
as

$\displaystyle C_{\ell}^{pp}=4\pi f^{2}(x)\int_{-1}^{1}{\rm
d}(\cos\theta)\xi^{VV}(\theta)\xi^{VV}(\theta)d^{\ell}_{22}(\theta)\,,$

where we defined

$\displaystyle\xi^{VV}(\theta,\chi)\equiv\sum\limits_{\ell}\frac{(2\ell+1)}{4\pi}C_{\ell}^{VV}d_{11}^{\ell}(\theta)\,,$
(32)

and

$\displaystyle C_{\ell}^{VV}=4\pi\ell(\ell+1)\int\frac{k^{2}{\rm
d}k}{2\pi^{2}}P_{v}(k,\chi)\left[\frac{j_{\ell}(k\chi)}{k\chi}\right]^{2}\,,$
(33)

where $P_{v}(k)$ is the dimensionless power spectrum of the three-dimensional
velocity. Having calculated the kinematic quadrupole signal, we turn to
estimating its variance in the presence of primary CMB polarization and noise.

###  II.2 Contribution to the CMB

The $E$\- and $B$\- mode CMB polarization satisfy

$\displaystyle E_{\ell m}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\,_{+2}a_{\ell m}+\,_{-2}a_{\ell m})$ (34)
$\displaystyle B_{\ell m}$ $\displaystyle=$
$\displaystyle\frac{1}{2i}(\,_{+2}a_{\ell m}-\,_{-2}a_{\ell m})$ (35)

where

$\,{}_{\pm 2}a_{\ell m}=\int{\rm d}^{2}\hat{\boldsymbol{n}}\,(Q\pm
iU)(\hat{\boldsymbol{n}})\,_{\pm 2}Y^{*}_{\ell m}(\hat{\boldsymbol{n}})\,.$
(36)

Next, we expand the optical depth with an homogeneous and anisotropic part
$\dot{\tau}(\chi\hat{\boldsymbol{n}})=\sigma_{T}a(\chi)\bar{n}_{e}(\chi)[1+\delta_{e}(\chi\hat{\boldsymbol{n}})]$
where $\bar{n}_{e}(\chi)$ is the mean electron density at comoving distance
$\chi$. The homogeneous part gives

$\,{}_{\pm 2}a_{\ell m}$ $\displaystyle=$ $\displaystyle f(x)\int{\rm
d}\chi\dot{\bar{\tau}}(\chi)\sum_{\ell^{\prime}m^{\prime}}\sum_{\ell^{\prime\prime}m^{\prime\prime}}V^{G}_{\ell^{\prime}m^{\prime}}V^{G}_{\ell^{\prime\prime}m^{\prime\prime}}\,{}_{\pm
1}Y_{\ell^{\prime}m^{\prime}}(\hat{\boldsymbol{n}})\,_{\pm
1}Y_{\ell^{\prime\prime}m^{\prime\prime}}(\hat{\boldsymbol{n}})\,_{\pm
2}Y_{\ell m}^{*}(\hat{\boldsymbol{n}})$ (37) $\displaystyle=$ $\displaystyle
f(x)\int{\rm
d}\chi\dot{\bar{\tau}}(\chi)\sum_{\ell^{\prime}m^{\prime}}\sum_{\ell^{\prime\prime}m^{\prime\prime}}V^{G}_{\ell^{\prime}m^{\prime}}V^{G}_{\ell^{\prime\prime}m^{\prime\prime}}\,{}_{\pm}\mathcal{I}_{\ell^{\prime}\ell^{\prime\prime}\ell}^{m^{\prime}m^{\prime}m}\,.$
(38)

Using Eqs. (34) and (35), the kpSZ induced CMB polarization satisfies

$\displaystyle E_{\ell m}^{(0)}$ $\displaystyle=$ $\displaystyle f(x)\int{\rm
d}\chi\dot{\bar{\tau}}(\chi)\sum_{\ell^{\prime}m^{\prime}}\sum_{\ell^{\prime\prime}m^{\prime\prime}}\frac{1}{2}[1+(-1)^{\ell^{\prime}+\ell^{\prime\prime}+\ell}]V^{G}_{\ell^{\prime}m^{\prime}}V^{G}_{\ell^{\prime\prime}m^{\prime\prime}}\,{}_{\pm}\mathcal{I}_{\ell^{\prime}\ell^{\prime\prime}\ell}^{m^{\prime}m^{\prime\prime}m}\,,$
(39) $\displaystyle B_{\ell m}^{(0)}$ $\displaystyle=$ $\displaystyle
f(x)\int{\rm
d}\chi\dot{\bar{\tau}}(\chi)\sum_{\ell^{\prime}m^{\prime}}\sum_{\ell^{\prime\prime}m^{\prime\prime}}\frac{1}{2}[1-(-1)^{\ell^{\prime}+\ell^{\prime\prime}+\ell}]V^{G}_{\ell^{\prime}m^{\prime}}V^{G}_{\ell^{\prime\prime}m^{\prime\prime}}\,{}_{\pm}\mathcal{I}_{\ell^{\prime}\ell^{\prime\prime}\ell}^{m^{\prime}m^{\prime\prime}m}\,.$
(40)

where $(0)$ superscript denotes the polarization induced by the homogeneous
optical depth. The CMB spectra can be calculated as

$\displaystyle C_{\ell}^{EE,\,(0)}$ $\displaystyle=$ $\displaystyle
f(x)f(x^{\prime})\int_{-1}^{1}{\rm
d}\cos(\theta)\sum_{\ell^{\prime}\ell^{\prime\prime}}[1+(-1)^{\ell^{\prime}+\ell^{\prime\prime}+\ell}]f^{VV}_{\ell^{\prime}\ell^{\prime\prime}}d_{11}^{\ell^{\prime}}(\theta)d_{11}^{\ell^{\prime\prime}}(\theta)d_{22}^{\ell}(\theta)$
(41) $\displaystyle C_{\ell}^{BB,\,(0)}$ $\displaystyle=$ $\displaystyle
f(x)f(x^{\prime})\int_{-1}^{1}{\rm
d}\cos(\theta)\sum_{\ell^{\prime}\ell^{\prime\prime}}[1-(-1)^{\ell^{\prime}+\ell^{\prime\prime}+\ell}]f^{VV}_{\ell^{\prime}\ell^{\prime\prime}}d_{11}^{\ell^{\prime}}(\theta)d_{11}^{\ell^{\prime\prime}}(\theta)d_{22}^{\ell}(\theta)$
(42)

where

$\displaystyle
f^{VV}_{\ell\ell^{\prime}}=\frac{(2\ell+1)(2\ell^{\prime}+1)}{4\pi}\int{\rm
d}\chi\dot{\bar{\tau}}(\chi)\int{\rm
d}\chi^{\prime}\dot{\bar{\tau}}(\chi^{\prime})C_{\ell}^{VV}(\chi,\chi^{\prime})C_{\ell^{\prime}}^{VV}(\chi,\chi^{\prime})$
(43)

Figure 1: The power spectra of the CMB linear-polarization signals. The blue
(red) curve is the lensed primary CMB $E$-mode ($B$-mode) polarization. The
green and olive-color lines correspond to the polarized Sunyaev Zel’dovich
(pSZ) effect due to the primary quadrupole on the $E$\- and $B$-mode
fluctuations, respectively. Note that the E-mode pSZ at low-$\ell$ is simply
the ‘reionization bump’. The purple solid and pink dotted lines correspond to
the kinetically-induced Sunyaev Zel’dovich polarization (kpSZ) signal on the
$E$\- and $B$-mode fluctuations due to peculiar transverse velocities of the
electrons respectively. Figure 2: Contributions to the $B$-mode polarization
power spectra of the CMB. The red curve is the lensed primary CMB $B$-mode
polarization. The olive-color dashed line correspond to the $B$-mode
contribution from the polarized Sunyaev Zel’dovich (pSZ) effect due to the
primary quadrupole. The purple dotted and dashed-dotted lines tensor
contributions to the $B$-mode spectra for the tensor-to-scalar ratio
satisfying $r=\\{10^{-2},10^{-4}$}, respectively. The teal solid line is the
contribution due to patchy reionization. The salmon solid line correspond to
the $B$-mode contribution from the kpSZ signal due to peculiar transverse
velocities of electrons.

On small scales the fluctuations of the electron density become large.
Assuming cross correlations between $\tau(\chi\hat{\boldsymbol{n}})$ and the
velocity are subdominant compared to auto-correlations of
$\tau(\chi\hat{\boldsymbol{n}})$ and $V^{G}(\chi\hat{\boldsymbol{n}})$, we can
calculate the polarization signals as

$\displaystyle E_{\ell m}^{(1)}\simeq-{\sigma_{T}}\int{\rm
d}\chi_{e}a_{e}\bar{n}_{e}(\chi_{e})\sum\limits_{\ell^{\prime}m^{\prime}}\sum\limits_{\ell^{\prime\prime}m^{\prime\prime}}$
$\displaystyle\frac{1}{2}\left(1+(-1)^{\ell^{\prime}+\ell^{\prime\prime}+\ell}\right)\,_{\pm}\mathcal{J}_{\ell^{\prime}\ell^{\prime\prime}\ell}^{m^{\prime}m^{\prime\prime}m}\delta_{\ell^{\prime\prime}m^{\prime\prime}}(\chi_{e})p_{\ell^{\prime}m^{\prime}}(\chi_{e})\,,\,\,$
(44)

and

$\displaystyle B_{\ell m}^{(1)}\simeq-{\sigma_{T}}\int{\rm
d}\chi_{e}a_{e}\bar{n}_{e}(\chi_{e})\sum\limits_{\ell^{\prime}m^{\prime}}\sum\limits_{\ell^{\prime\prime}m^{\prime\prime}}$
$\displaystyle\frac{1}{2}\left(1-(-1)^{\ell^{\prime}+\ell^{\prime\prime}+\ell}\right)\,_{\pm}\mathcal{J}_{\ell^{\prime}\ell^{\prime\prime}\ell}^{m^{\prime}m^{\prime\prime}m}\delta_{\ell^{\prime\prime}m^{\prime\prime}}(\chi_{e})p_{\ell^{\prime}m^{\prime}}(\chi_{e})\,,\,\,$
(45)

where

$\,{}_{\pm}\mathcal{J}_{\ell^{\prime}\ell^{\prime\prime}\ell}^{m^{\prime}m^{\prime\prime}m}=\sqrt{\frac{(2\ell^{\prime}+1)(2\ell^{\prime\prime}+1)(2\ell+1)}{4\pi}}\begin{pmatrix}\ell&\ell^{\prime}&\ell^{\prime\prime}\\\
m^{\prime}&m^{\prime\prime}&-m\end{pmatrix}\begin{pmatrix}\ell&\ell^{\prime}&\ell^{\prime\prime}\\\
-2&2&0\end{pmatrix}\,.$ (46)

The total $E$\- and $B$-mode power-spectra for the kpSZ signal

$\displaystyle C_{\ell}^{EE,\,\rm
kpSZ}=C_{\ell}^{EE\,(0)}+C_{\ell}^{EE\,(1)}\,,$ (47) $\displaystyle
C_{\ell}^{BB,\,\rm kpSZ}=C_{\ell}^{BB\,(0)}+C_{\ell}^{BB\,(1)}\,,$ (48)

are shown in Fig. 1 together with the $E$\- and $B$-mode pSZ signals from the
primordial quadrupole and the lensed primary spectra.555Note that unlike pSZ,
the kpSZ $B$-modes are sourced both at linear and non-linear orders, and are
dominated by the linear contributions. The kpSZ effect remains sub-dominant
compared to the $E$-mode spectrum induced by the pSZ effect on most scales,
and is lower than the contribution due to patchy reionization by over an order
of magnitude on all scales. These effects can be a limiting factor in $B$-mode
searches for small $r$ as shown in Fig. 2, depending on the effectiveness of
CMB delensing, see e.g. Refs. (Knox and Song, 2002; Kesden  _et al._ , 2002;
Seljak and Hirata, 2004; Smith  _et al._ , 2012; Abazajian  _et al._ , 2016a,
2019, 2020; Carron  _et al._ , 2017; Aghanim  _et al._ , 2020; Green  _et al._
, 2017; Hotinli  _et al._ , 2021c). Throughout this work, we set the
cosmological parameters
$\\{\Omega_{c}h^{2},\Omega_{b}h^{2},\theta_{s},\tau,A_{s},n_{s}\\}$ to
$\\{0.1197,0.022,0.0104,0.06,2.196\\!\times\\!10^{-9},0.9655\\}$,
respectively, where $\Omega_{c}h^{2}$ ($\Omega_{b}h^{2}$) is the physical cold
dark matter (baryon) density, $\theta_{s}$ is the angle subtended by acoustic
scale, $\tau$ is the Thomson optical depth to recombination, $A_{s}$ is the
primordial scalar fluctuation amplitude and $n_{s}$ is the primordial scalar
fluctuation slope.

##  III Detecting the signal

| Beam FWHM | Noise RMS
---|---|---
|  | ($\mu$K-arcmin)
| S4 | HD | S4 | HD
39 GHz | $5.1^{\prime}$ | $36.3^{\prime\prime}$ | 12.4 | 3.4
93 GHz | $2.2^{\prime}$ | $15.3^{\prime\prime}$ | 2.0 | 0.6
145 GHz | $1.4^{\prime}$ | $10.0^{\prime\prime}$ | 2.0 | 0.6
225 GHz | $1.0^{\prime}$ | $6.6^{\prime\prime}$ | 6.9 | 1.9
280 GHz | $0.9^{\prime}$ | $5.4^{\prime\prime}$ | 16.7 | 4.6
Table 1: Inputs to ILC noise for the baseline CMB configurations: The beam and temperature noise RMS parameters are chosen to roughly match CMB-S4 and CMB-HD. We model the CMB noise as shown in Eq. (59). In both cases, we account for the degradation due to Earth’s atmosphere by defining the CMB noise choose $\ell_{\rm knee}=100$ and $\alpha_{\rm knee}=-3$. The polarization noise satisfy $\Delta_{E}=\Delta_{B}=\sqrt{2}\Delta_{T}$. Frequency (GHz) | 20 | 27 | 39 | 93 | 145 | 225 | 278
---|---|---|---|---|---|---|---
Beam FWHM (arcmin) | 11.0 | 8.4 | 5.8 | 2.5 | 1.6 | 1.1 | 1.0
noise $T$($\mu K$-arcmin) | 9.31 | 4.60 | 2.94 | 0.45 | 0.41 | 1.29 | 3.07
noise E ($\mu K$-arcmin) | 13.16 | 6.50 | 4.15 | 0.63 | 0.59 | 1.83 | 4.34
noise $B$ ($\mu K$-arcmin) | 13.16 | 6.50 | 4.15 | 0.63 | 0.59 | 1.83 | 4.34
Table 2: Inputs to ILC noise for the CMB-S4 V3R0-25 configuration. For CMB-HD,
we take the same configuration with the improved CMB-HD beam given on Table I.
We model the CMB noise as shown in Eq. (59) and take $\ell_{\rm knee}=100$ and
$\alpha_{\rm knee}=-3$ for all frequency bins and observables.

Figure 3: The kpSZ signal and the background. The CMB $E$\- and $B$-mode
spectra are shown with blue and red solid lines respectively. The former
contains the primary pSZ contribution the $E$-mode spectrum, which is shown
with the green solid line. The $B$-mode contribution form the pSZ effect is
shown with the olive-color line on the right plot. The ILC-cleaned foregrounds
for the kpSZ measurement (the primary CMB and the pSZ signals) are shown with
the dot-dashed brown and dotted gray lines for CMB-S4 and CMB-HD like surveys,
respectively, using the V3R0-25 configuration. The teal-colored dot-dashed
lines correspond to the ILC-cleaned noise curves anticipated from the PIXIE
experiment. The kpSZ signal on the $E$\- and $B$-mode spectra are shown with
the purple and pink solid lines, respectively. The ILC-cleaning is performed
by appropriating the frequency dependence of the kpSZ signal to the CMB
primary, the pSZ signal, and the noise (both white and pink) as described in
the text.

###  III.1 ILC-cleaned kpSZ maps

The frequency dependence of the kpSZ signal provides a window into removing
some of the lensed primary CMB and the pSZ signals – both of which are black-
body – that are dominant in comparison to the kpSZ. The multi-frequency
information can be used to clean contributions to the temperature maps other
than the kpSZ signal using a standard harmonic-space internal linear
combination (ILC) procedure. We write the covariance between the de-beamed CMB
at different frequencies as a matrix satisfying

$\displaystyle\boldsymbol{C}_{\ell}=\tilde{C}_{\ell}^{EE,\,\rm
kpSZ}\boldsymbol{e}\boldsymbol{e}^{\dagger}+\boldsymbol{g}^{-1}_{\rm
kpSZ}{C}_{\ell}^{FG}+(\boldsymbol{B}^{-1}\boldsymbol{g}^{-1}_{\rm
kpSZ}\boldsymbol{N})_{\ell}$ (49)

where $(\boldsymbol{g}^{-1}_{\rm kpSZ})_{ij}=1/[g(x_{i})g(x_{j})]$ where
$g(x_{i})$ is the frequency-dependent coefficient of the kpSZ signal and
$\tilde{C}^{EE,\,\rm kpSZ}_{\ell}$ is the frequency-independent component of
the kpSZ signal introduced in Eq. (6), i.e. $\tilde{C}^{EE,\,\rm
kpSZ}_{\ell}={C}^{EE,\,\rm kpSZ}_{\ell}/[g(x)]^{2}$. Here, ${C}_{\ell}^{FG}$
consists of the non-kpSZ contribution (foregrounds for kpSZ) to the CMB (the
primary lensed blackbody CMB and pSZ signals) and
$(\boldsymbol{B}^{-1}\boldsymbol{N})_{\ell}$ is the de-beamed instrumental
noise covariance (assumed diagonal).

For the contributions to the signal in the millimeter wavelength, we consider
the black-body late-time kSZ, clustered CIB, Poisson CIB and tSZ foregrounds
following Ref. (Madhavacheril  _et al._ , 2017). We omit including the
correlation between CIB and tSZ. We include the radio sources using the flux-
limit-dependent radio-source power model from Ref. (Lagache  _et al._ , 2020)
in the 39, 93 and 145 GHz channels. We assume flux limits of 10, 7 and 10 mJy
(2, 1 and 1 mJy) for CMB-S4 (CMB-HD), respectively in those channels. Finally,
we include the kSZ signal from reionization following Ref. (Park  _et al._ ,
2013) and the lensed CMB black-body contribution using CAMB (Lewis  _et al._ ,
2000).

Following the ILC method in harmonic space, we estimate the ILC-cleaned signal
as

$\displaystyle\hat{E}_{\ell m}=\boldsymbol{w}^{\dagger}_{\ell
m}\boldsymbol{E}_{\ell m}$ (50)

where $\boldsymbol{w}_{\ell m}$ are weights that minimize the variance of the
resulting signal satisfying

$\displaystyle\boldsymbol{w}_{\ell
m}=\frac{(\boldsymbol{C}_{\ell})^{-1}\boldsymbol{e}}{\boldsymbol{e}^{\dagger}(\boldsymbol{C})^{-1}\boldsymbol{e}}\,,$
(51)

and the ensemble averaged $E$-mode power spectrum (same for the $B$-mode) of
the cleaned map is

$\displaystyle C_{\ell}^{EE,\,\rm cleaned}=\tilde{C}_{\ell}^{EE,\,\rm
kpSZ}+\boldsymbol{w}_{\ell}^{\dagger}[\boldsymbol{g}^{-1}_{\rm
kpSZ}{C}_{\ell}^{FG}+(\boldsymbol{B}^{-1}\boldsymbol{g}^{-1}_{\rm
kpSZ}\boldsymbol{N})_{\ell}]\boldsymbol{w}_{\ell}\,.$ (52)

We demonstrate the cleaned $E$-mode polarization spectra in Fig. 3 for CMB-S4
and CMB-HD like survey specifications (dot-dashed lines). We find that
although the primary CMB and pSZ signals are removed significantly from the
cleaned $E$\- and $B$-maps, some residual from these dominant foregrounds
still remain and the CMB background on the kpSZ measurement is still
$\sim\mathcal{O}(10^{2})$ larger than the experimental white-noise. As the
number of frequency bins plays an important role in reducing the dominant
blackbody contribution, the proposed PIXIE experiment (Kogut  _et al._ , 2016,
2014) with 15GHz wide frequency bins within the range 30GHz-3THz can
potentially provide much more significant reduction. Using the experimental
specifications of PIXIE, we find the blackbody signal can be removed almost
perfectly from the CMB allowing PIXIE to provide detector-noise limited
measurement of the kpSZ signal up to scales $\ell\sim 100$ allowed by its 2.6
degree beam. In what follows we provide forecasts for the ‘direct-detection’
prospects for the kpSZ effect using CMB-S4, CMB-HD and PIXIE experiments. In
addition to these upcoming LiteBIRD (Matsumura  _et al._ , 2014; Hazumi  _et
al._ , 2019) and proposed ESA Voyage 2050 (Basu  _et al._ , 2021; Delabrouille
_et al._ , 2021) surveys can also potentially improve the prospects of
detecting this effect directly on large scales by cross-correlating with
velocity templates.

###  III.2 Direct detection

Figure 4: The kpSZ signal and noise. The redshift-binned polarization signals
reconstructed from a galaxy survey shown together with the cross-correlation
signal between the CMB. The two lines show the anticipated CMB variance after
ILC-cleaning as discussed in the text. The third and fourth (identical) lines
from the top are the total kpSZ signal in the CMB maps. We find the cross-
correlation signal between the polarization maps reconstructed from a galaxy
survey at a given redshift bin and the CMB is comparable to the polarization
signal autocorrelation reconstructed from a galaxy survey on scales of
interest for detection. Here, the galaxy survey is taken to match the
experimental specifications of VRO. The ILC-cleaned CMB lines correspond to
the noise estimates from the ultra-deep V3R0-25 CMB-S4 configuration. We show
the signal from the fourth galaxy redshift bin and we separate the redshift
range $z\in[0.2,6]$ into 32 redshift bins.

Figure 5: Detection prospects of the kpSZ signal from cross-correlating ILC-
cleaned kpSZ maps with velocity templates. We use beams matching the
experimental specifications of the CMB-S4 and the CMB-HD surveys. The kinetic
quadrupole is reconstructed using Vera Rubin Observatory. We show the SNR as a
function of the maximum reconstructed velocity multipole $L_{\rm max}$ from
the galaxy survey.

Figure 6: Detection prospects of the kpSZ signal from cross-correlating ILC-
cleaned kpSZ maps with velocity templates. We use CMB experimental
specifications matching the proposed PIXIE survey. Figure otherwise similar to
Fig. 5

One of the most promising directions for a first detection of the kpSZ effect
is via cross-correlating the degree-scale kinetic quadrupole reconstructed
from a galaxy survey with the $E$\- and $B$-mode polarizations measured from
the CMB. Although the lensed primary CMB is orders of magnitude larger than
the kpSZ signal, the frequency dependence of the latter can be used to
increase the prospects to detect this signal as we discussed in the previous
section.

The cross-correlation of the kpSZ $E$\- and $B$-mode fluctuations in the CMB
and the reconstructed field satisfies the same equalities in Eqs. (41) and
(42) with

$\displaystyle f^{VV}_{\ell\ell^{\prime}}\rightarrow f^{V\hat{V}_{\rm
gal}^{(z_{i})}}_{\ell\ell^{\prime}}=\frac{(2\ell+1)(2\ell^{\prime}+1)}{4\pi}\int{\rm
d}\chi\dot{\bar{\tau}}(\chi)\int{\rm
d}\chi^{\prime}{W}_{i}(\chi^{\prime})\dot{\bar{\tau}}(\chi^{\prime})C_{\ell}^{V\hat{V}_{\rm
gal}}(\chi,\chi^{\prime})C_{\ell^{\prime}}^{V\hat{V}_{\rm
gal}}(\chi,\chi^{\prime})\,,$

and

$\displaystyle C_{\ell}^{V\hat{V}_{\rm
gal}}=\frac{[C_{\ell}^{Vg}(\chi,\chi^{\prime})]^{2}}{C_{\ell}^{gg}(\chi^{\prime})+N_{\ell}^{gg}(\chi^{\prime})}\,,$
(54)

where $C_{\ell}^{Vg}(\chi,\chi^{\prime})$ is the velocity-galaxy cross-
correlation, $W_{i}(\chi)$ is the window function set equal to zero outside
the redshift range of the $i$th bin, and
$C_{\ell}^{gg}(\chi^{\prime})+N_{\ell}^{gg}(\chi^{\prime})$ is the observed
galaxy auto-spectrum at comoving distance $\chi^{\prime}$. We define the the
galaxy-galaxy power spectrum computed using the Halo Model following Refs.
(Smith  _et al._ , 2018; Münchmeyer  _et al._ , 2018) and use the ReCCO
simulation software (Cayuso  _et al._ , ) for calculating the galaxy-galaxy,
galaxy-velocity and velocity-velocity spectra. We model the photometric
redshift errors $\sigma_{z}=\sigma_{0}(1+z)$ with $\sigma_{0}=0.3$. We assume
shot noise is uncorrelated between redshift bins, and compute the number
density per bin assuming the galaxy number density per square arcmin satisfies

$\displaystyle
n(z)=\frac{n_{g}}{2z_{0}}\left(\frac{z}{z_{0}}\right)^{2}\exp\left({-\frac{z}{z_{0}}}\right)$
(55)

with $n_{g}=40/({\rm arcmin})^{2}$ and $z_{0}=0.3$.

We calculate the total signal-to-noise ratio (SNR) from measurements of $E$\-
and $B$-mode CMB polatization by approximating the likelihood as Gaussian

$\displaystyle{\rm SNR}^{2}=$
$\displaystyle\sum_{\ell\ell^{\prime};PXY}C_{\ell}^{\hat{P}^{\rm
gal}_{X}P}\textbf{cov}^{-1}\left(\tilde{C}^{\hat{P}^{\rm
gal}_{X}P}_{\ell},\tilde{C}^{\hat{P}^{\rm
gal}_{Y}P}_{\ell^{\prime}}\right)C_{\ell^{\prime}}^{\hat{P}^{\rm
gal}_{Y}P}\,,$ (56)

where $P\in\\{E,B\\}$ indicates the CMB polarization, the $\\{X,Y\\}$ indices
run over redshift bins. Here, the fields with hats refer to polarization maps
reconstructed from the galaxy survey in a given redshift bin, those without a
hat refer to the polarization maps measured from the CMB, the tilde refers to
spectra including noise, and the covariance is given by

$\displaystyle\textbf{cov}\left(\tilde{C}^{\hat{P}^{\rm
gal}_{X}P}_{\ell},\tilde{C}^{\hat{P}^{\rm
gal}_{Y}P}_{\ell^{\prime}}\right)=\frac{\delta_{\ell\ell^{\prime}}}{2\ell+1}f_{\mathrm{sky}}^{-1}\left(\tilde{C}_{\ell}^{\hat{P}^{\rm
gal}_{X}\hat{P}^{\rm
gal}_{Y}}\tilde{C}_{\ell}^{PP}+\tilde{C}_{\ell}^{\hat{P}^{\rm
gal}_{X}P}\tilde{C}_{\ell}^{\hat{P}^{\rm gal}_{Y}P}\right)\,.$ (57)

To assess the detectability of the kpSZ effect, we take a null hypothesis a
scenario in which there is no signal or noise in the CMB cross with the
galaxy-derived polarization ($\tilde{C}_{\ell}^{\hat{P}^{\rm gal}_{X}P}=0$)
when calculating the covariance matrix. We also take to have noise diagonal in
the redshift bins ($N_{\ell}^{\hat{P}^{\rm gal}_{X}\hat{P}^{\rm
gal}_{Y}}=\delta_{\hat{X}\hat{Y}}N_{\ell}^{\hat{P}^{\rm gal}_{X}}$). Our
covariance matrix has the form

$\displaystyle\textbf{cov}\left(\tilde{C}^{\hat{P}^{\rm
gal}_{X}P}_{\ell},\tilde{C}^{\hat{P}^{\rm
gal}_{Y}P}_{\ell^{\prime}}\right)=\begin{pmatrix}\tilde{C}_{\ell}^{\hat{P}^{\rm
gal}_{1}\hat{P}^{\rm gal}_{1}}\tilde{C}_{\ell}^{PP}&{C}_{\ell}^{\hat{P}^{\rm
gal}_{1}\hat{P}^{\rm
gal}_{2}}\tilde{C}_{\ell}^{PP}&\ldots&{C}_{\ell}^{\hat{P}^{\rm
gal}_{1}\hat{P}^{\rm gal}_{N_{\rm bins}}}\tilde{C}_{\ell}^{PP}\\\
{C}_{\ell}^{\hat{P}^{\rm gal}_{2}\hat{P}^{\rm
gal}_{1}}\tilde{C}_{\ell}^{PP}&\tilde{C}_{\ell}^{\hat{P}^{\rm
gal}_{2}\hat{P}^{\rm
gal}_{2}}\tilde{C}_{\ell}^{PP}&\ldots&{C}_{\ell}^{\hat{P}^{\rm
gal}_{2}\hat{P}^{\rm gal}_{N_{\rm bins}}}\tilde{C}_{\ell}^{PP}\\\
\vdots&\vdots&\ddots&\vdots\\\ {C}_{\ell}^{\hat{P}^{\rm gal}_{N_{\rm
bins}}\\!\\!\\!\hat{P}^{\rm
gal}_{1}}\tilde{C}_{\ell}^{PP}&{C}_{\ell}^{\hat{P}^{\rm gal}_{N_{\rm
bins}}\\!\\!\\!\hat{P}^{\rm
gal}_{2}}\tilde{C}_{\ell}^{PP}&\ldots&\tilde{C}_{\ell}^{\hat{P}^{\rm
gal}_{N_{\rm bins}}\\!\\!\\!\hat{P}^{\rm gal}_{N_{\rm
bins}}}\tilde{C}_{\ell}^{PP}\end{pmatrix}\,.$ (58)

We model the CMB noise with the white and pink noise components as

$\displaystyle
N_{\ell}^{PP}=\Delta_{P}^{2}\exp\left[\ell(\ell+1)\frac{\theta_{\rm
FWHM}^{2}}{8\log(2)}\right][1+(\ell_{\rm knee}/\ell)^{\alpha_{\rm knee}}]$
(59)

where we set the polarization noise RMS $\Delta_{P}$, the beam full-width
half-maximum (FWHM) resolution, and the $\\{\alpha_{\rm knee},\ell_{\rm
knee}\\}$ parameters which capture the effect of the degradation due to
Earth’s atmosphere to match upcoming experiments as defined in Tables I and
II.

We display the kpSZ signal and noise for direct detection in Fig. 4, where the
redshift-binned polarization signals reconstructed from a galaxy survey shown
together with the cross-correlation signal between the CMB. We demonstrate the
contributions to the total detection SNR from different maximum multipoles
included in the analysis on Fig. 5 for CMB-S4 and CMB-HD surveys. We find most
of the SNR is contributed from scales satisfying $\ell\in[50,200]$ and that
the contribution to the SNR is dominated by the $B$-mode polarization maps.
Our forecast suggest that both CMB-S4 and CMB-HD may reach SNR values of $\sim
2$, and possibly greater for large sky fraction. In Fig. 6 we show results
from using the PIXIE survey specifications. The large number of frequency bins
and the absence of atmospheric foregrounds allow accessing the largest scale
polarization modes $\ell\lesssim 100$ almost up to the detector noise
precision, suggesting a ‘direct’ direction with ${\rm SNR}\gtrsim 3$ may be
possible.

###  III.3 The kpSZ quadratic-estimator variance

Detecting the kpSZ from the polarization maps directly on large scales may be
difficult due to atmospheric noise, unaccounted foregrounds and effects of
masking, for example. An alternative method for measuring large-scale
cosmological fluctuations is through using the non-Gaussian information on the
small-scale CMB maps, together with maps of the galaxy distribution in
tomographic redshift bins. This technique has been shown to be a very
promising avenue to cosmological inference for the upcoming cosmology
experiments. Here we go over the pSZ quadratic estimator formalism and
evaluate the prospects to detect the large-scale kinematic quadrupole through
pSZ tomography.

The minimum variance estimator for the remote quadrupole field can be
reconstructed using observed maps of the CMB polarization and the fluctuations
of the optical depth
$\delta\tau(\hat{\boldsymbol{n}})=\sigma_{T}n_{e}a\delta_{e}(\hat{\boldsymbol{n}})$.
The estimator is already calculated in Refs. (Alizadeh and Hirata, 2012;
Meyers  _et al._ , 2018; Deutsch  _et al._ , 2017, 2018, 2019). Here, we
shortly review the method presented in Ref. (Meyers  _et al._ , 2018). Note we
are using the flat-sky approximation, which we anticipate to agree well with a
full-sky estimator that is calculated in Ref. (Alizadeh and Hirata, 2012), as
this is also the case for lensing estimators (Okamoto and Hu, 2003).
Parameters $\boldsymbol{\ell}$ and $\boldsymbol{L}$ introduced in this section
are two-dimensional Fourier wavevectors.

We wish to construct an estimator for the quadrupole field $\hat{p}$ from
observed maps of the CMB polarization $(Q(\hat{\boldsymbol{n}})\pm
iU(\hat{\boldsymbol{n}}))^{\mathrm{obs}}$, and maps of the local optical depth
$\delta\tau^{\mathrm{ext}}(\hat{\boldsymbol{n}},\eta)$ observed by some
external survey, at small scales,

$\displaystyle{\,{}_{\pm}}\hat{p}(\hat{\boldsymbol{n}},\eta)\sim\left[(Q(\hat{\boldsymbol{n}})\pm
iU(\hat{\boldsymbol{n}}))^{\mathrm{obs}}\delta\tau(\hat{\boldsymbol{n}},\eta)^{\mathrm{ext}}\right]_{f}\,,$
(60)

filtered in order to minimize the variance of the estimator, similar to CMB
lensing quadratic estimator (Hu and Okamoto, 2002). The observed CMB
polarization will contain contribution that is primordial in its origin
$(Q(\hat{\boldsymbol{n}})\pm iU(\hat{\boldsymbol{n}}))^{\mathrm{prim}}$,
present in the absence of free electrons at late times; as well as
contributions from Thomson scattering,
${\,{}_{\pm}}p^{\mathrm{pSZ}}(\hat{\boldsymbol{n}},\eta)\delta\tau^{\mathrm{ext}}(\hat{\boldsymbol{n}},\eta)$
which include polarization sourced by both primordial and kinematic
quadrupoles; as well as instrumental noise.

The remote quadrupole estimators can be found as

$\displaystyle\hat{p}^{E}(\hat{\boldsymbol{n}},\eta)$ (61)
$\displaystyle=N^{EE}(\boldsymbol{L})\\!\\!\int\frac{{\rm
d}^{2}\boldsymbol{\ell}}{(2\pi)^{2}}\delta\tau^{\rm
ext}(\boldsymbol{L}-\boldsymbol{\ell})f^{E}(\boldsymbol{\ell},\boldsymbol{L})[E^{\rm
obs}(\boldsymbol{\ell})\cos^{2}(2(\varphi_{\boldsymbol{L}}\\!-\\!\varphi_{\boldsymbol{\ell}}))+B^{\rm
obs}(\boldsymbol{\ell})\sin^{2}(2(\varphi_{\boldsymbol{L}}\\!-\\!\varphi_{\boldsymbol{\ell}}))]\,,$

and

$\displaystyle\hat{p}^{B}(\hat{\boldsymbol{n}},\eta)$ (62)
$\displaystyle=N^{BB}(\boldsymbol{L})\\!\\!\int\frac{{\rm
d}^{2}\boldsymbol{\ell}}{(2\pi)^{2}}\delta\tau^{\rm
ext}(\boldsymbol{L}-\boldsymbol{\ell})f^{B}(\boldsymbol{\ell},\boldsymbol{L})[E^{\rm
obs}(\boldsymbol{\ell})\sin^{2}(2(\varphi_{\boldsymbol{L}}\\!-\\!\varphi_{\boldsymbol{\ell}}))+B^{\rm
obs}(\boldsymbol{\ell})\cos^{2}(2(\varphi_{\boldsymbol{L}}\\!-\\!\varphi_{\boldsymbol{\ell}}))]\,,$

where we defined

$\displaystyle{p}_{\pm}(\boldsymbol{\ell},\eta)=(p^{E}\mp
ip^{B})(\boldsymbol{\ell},\eta)e^{\mp 2i\phi_{\boldsymbol{\ell}}}\,.$ (63)

The filters that minimize the variance can be found as

$\displaystyle f^{E}(\boldsymbol{\ell},\boldsymbol{L})=[C_{\ell}^{EE,\rm
obs}\cos^{2}(2(\varphi_{\boldsymbol{L}}\\!-\\!\varphi_{\boldsymbol{\ell}}))+C_{\ell}^{BB,\rm
obs}\sin^{2}(2(\varphi_{\boldsymbol{L}}\\!-\\!\varphi_{\boldsymbol{\ell}}))]^{-1}$
(64)

and

$\displaystyle f^{B}(\boldsymbol{\ell},\boldsymbol{L})=[C_{\ell}^{EE,\rm
obs}\sin^{2}(2(\varphi_{\boldsymbol{L}}\\!-\\!\varphi_{\boldsymbol{\ell}}))+C_{\ell}^{BB,\rm
obs}\cos^{2}(2(\varphi_{\boldsymbol{L}}\\!-\\!\varphi_{\boldsymbol{\ell}}))]^{-1}\,.$
(65)

The estimated variances of the reconstructed field of remote quadrupoles on
large scales are then (Meyers  _et al._ , 2018),

$N^{EE}(\boldsymbol{L})\\!=\\!\\!\Big{[}\\!\int\\!\\!\frac{{\rm
d}^{2}{\boldsymbol{\ell}}}{(2\pi)^{2}}\frac{(C_{|{\boldsymbol{\ell}}-{\boldsymbol{L}}|}^{\delta\tau\,g})^{2}}{C_{|{\boldsymbol{\ell}}-{\boldsymbol{L}}|}^{gg\,\rm
obs}}\left(C^{EE,\mathrm{obs}}_{\ell}\cos^{2}\left[2\left(\varphi_{\boldsymbol{L}}\\!-\\!\varphi_{\boldsymbol{\ell}}\right)\right]+\\!C^{BB,\mathrm{obs}}_{\ell}\sin^{2}\left[2\left(\varphi_{\boldsymbol{L}}\\!-\\!\varphi_{\boldsymbol{\ell}}\right)\right]\right)^{-1}\Big{]}^{-1},\\\
$ $N^{BB}(\boldsymbol{L})\\!=\\!\\!\Big{[}\\!\int\\!\\!\frac{{\rm
d}^{2}{\boldsymbol{\ell}}}{(2\pi)^{2}}\frac{(C_{|{\boldsymbol{\ell}}-{\boldsymbol{L}}|}^{\delta\tau\,g})^{2}}{C_{|{\boldsymbol{\ell}}-{\boldsymbol{L}}|}^{gg\,\rm
obs}}\left(C^{EE,\mathrm{obs}}_{\ell}\sin^{2}\left[2\left(\varphi_{\boldsymbol{L}}\\!-\\!\varphi_{\boldsymbol{\ell}}\right)\right]+\\!C^{BB,\mathrm{obs}}_{\ell}\cos^{2}\left[2\left(\varphi_{\boldsymbol{L}}\\!-\\!\varphi_{\boldsymbol{\ell}}\right)\right]\right)^{-1}\Big{]}^{-1},\\\
$

and $N^{EB}(\boldsymbol{L})=0$ due to symmetry. The $C^{\mathrm{obs}}$
indicates the ILC-cleaned CMB spectrum including the instrumental noise, and
$C_{|{\boldsymbol{\ell}}-{\boldsymbol{L}}|}^{\delta\tau g}$ is cross-
correlation between the optical depth fluctuations and and externally inferred
map of galaxies. More details on derivation can be found in (Meyers  _et al._
, 2018).

In order to evaluate the detection prospects of the kinetic pSZ effect from
tomography, we cross-correlate the large-scale kinetic quadrupole
reconstructed from maps of the CMB (and a density tracer) on small scales,
with that inferred directly from a galaxy survey. We assume that the latter
method provides a precise enough measurement of the large-scale density that
we can infer the large-scale transverse velocity up to galaxy shot noise,
which should be a reasonable approximation for the high number densities of
galaxies expected in the surveys we are considering. We calculate the total
signal-to-noise ratio similar to before as

$\displaystyle{\rm SNR}^{2}=$
$\displaystyle\sum_{\ell\ell^{\prime};p;XYWZ}C_{\ell}^{\hat{p}^{\rm
gal}_{X}\hat{p}_{Y}}\textbf{cov}^{-1}\left(\tilde{C}^{\hat{p}^{\rm
gal}_{X}\hat{p}_{Y}}_{\ell},\tilde{C}^{\hat{p}^{\rm
gal}_{W}\hat{p}_{Z}}_{\ell^{\prime}}\right)C_{\ell^{\prime}}^{\hat{p}^{\rm
gal}_{W}\hat{p}_{Z}}\,,$ (66)

where $p\in\\{p^{E},p^{B}\\}$ are the reconstructed $E$\- and $B$-mode
quadrupoles, the indices run over redshift bins, the fields with hats refer to
the remote-quadrupole field reconstructed from the CMB, those without a hat
refer to the quadrupole reconstructed from the galaxy distribution, the tilde
refers to spectra including noise, and the covariance is given by

$\displaystyle\textbf{cov}\left(\tilde{C}^{\hat{p}^{\rm
gal}_{X}\hat{p}_{Y}}_{\ell},\tilde{C}^{\hat{p}^{\rm
gal}_{W}\hat{p}_{Z}}_{\ell^{\prime}}\right)=\frac{\delta_{\ell\ell^{\prime}}}{2\ell+1}f_{\mathrm{sky}}^{-1}\left(\tilde{C}_{\ell}^{\hat{p}^{\rm
gal}_{X}\hat{p}^{\rm
gal}_{W}}\tilde{C}_{\ell}^{\hat{p}_{Y}\hat{p}_{Z}}+\tilde{C}_{\ell}^{\hat{p}^{\rm
gal}_{X}\hat{p}_{Z}}\tilde{C}_{\ell}^{\hat{p}^{\rm
gal}_{Y}\hat{p}_{W}}\right)\,.$ (67)

To assess the detectability of the kpSZ effect, we take – similar to before –
a null hypothesis scenario in which there is no signal in the CMB-
reconstructed quadrupole, which we also take to have noise diagonal in the
redshift bins
($\tilde{C}_{\ell}^{\hat{X}\hat{Y}}=\delta_{\hat{X}\hat{Y}}N_{\ell}^{\hat{X}}$),
and no signal or noise in the cross with the galaxy-derived quadrupole
($\tilde{C}_{\ell}^{\hat{X}^{\rm gal}\hat{Y}}=0$) when calculating the
covariance matrix.

Figure 7: (left) The remote kinematic quadrupole power-spectra (solid blue
line) shown together with the estimated reconstruction noise for combination
of the VRO survey and two CMB surveys [CMB-S4 (dotted green) and CMB-HD
(dashed red)] on small scales. We show the results for the fourth tomographic
redshift bin where we take 32 bins spanning the range $z\in[0.2,6.0]$ with the
same radial size in comoving distance. (Top right) The redshift dependence of
reconstruction noise (normalised to its minimum value) for the same redshift
bin. (Bottom right) The cumulative SNR (normalised to its maximum value) for
the same redshift bin as a function of the maximum multipole of the quadrupole
reconstructed from the galaxy survey. Figure 8: The total detection SNR from
the kpSZ tomography. The dash-dotted green lines assume a CMB experiment whose
specifications match the futuristic CMB-HD survey. The dash-dotted blue lines
corresponds to the CMB-S4 survey. For both experiments, the upper darker-
coloured dash-dotted lines correspond to assuming electron fluctuations trace
dark matter on all scales, while lower lighter-coloured dash-dotted lines
correspond to the standard ‘AGN’ electron gas profile model. We take the
baseline experimental configurations shown in Table 1. In both cases we take
$f_{\rm sky}=0.5$ and show the SNR as a function of the maximum multipole
included in the quadropole reconstruction. The top (bottom) red dotted line
correspond to SNR value of 3 (1). Improving the beam of the CMB experiment
allows better access to the non-Gaussian information on small-scales.

We estimate the signal-to-noise ratio (SNR) of the reconstructed quadrupole
assuming a cosmology consistent with the latest results from Planck (Aghanim
_et al._ , 2018). We perform the ILC cleaning taking into account the
frequency-dependence of the kpSZ signal as described in Sec. III.1. We take
two sets of baseline noise values for the CMB experiment which we describe in
Table 1 matching specifications anticipated from the CMB-S4 and the CMB-HD
experiments. Note that the RMS noise values shown in Table 1 are for
temperature. We calculate the noise levels for polarization as
$\Delta_{E}=\Delta_{B}=\sqrt{2}\Delta_{T}$ and model the CMB noise as in Eq.
(59). We described our modelling of the galaxy field in Sec. III.2. For the
large-scale structure experiment we take the specifications of VRO (Abell  _et
al._ , 2009) and take 32 redshift bins in the range $z\in[0.2,6.0]$ with
constant comoving bin width. Throughout, we model the electron distribution
either by assuming the ‘AGN’ gas profile from Ref. (Battaglia, 2016) using the
ReCCO software (Cayuso  _et al._ , ), or by assuming electrons trace dark
matter on all scales. In the former case, physical processes such as AGN
feedback, result in baryonic matter not tracing dark matter inside of halos,
reducing the statistical power of pSZ tomography using galaxy tracers.

We compare the anticipated remote quadrupole signal from our fourth redshift
bin to the reconstruction noise on the left plot in Fig. 7. The upper right
plot in the same figure demonstrates the non-trivial dependence of the
reconstruction noise on the decreasing density fluctuations and increasing
visibility function $g(\eta)$ with increasing redshifts. The bottom right plot
of this figure shows that the dominant contribution to the SNR comes from
scales satisfying $\ell\sim 100$. In Fig. 8, we show the detection SNR from
kpSZ tomography by combining the information from our 32 redshift bins as
described above. We find a CMB-S4- (CMB-HD) like survey can reach the SNR
value of $\sim 2~{}(\sim 9)$ when combined with the VRO galaxy survey. Our
analysis suggests that detection with CMB-S4 will likely be difficult while
CMB-HD can potentially detect this signal to high SNR. Here, we omit
calculating the bias on the velocity reconstruction from the primordial
quadrupole assuming the ILC-cleaning will remove this significantly. Note also
that the reconstructed kinetic quadrupole templates discussed throughout this
work can potentially be used to reduce the contamination from kpSZ for the
primordial gravitational wave searches by means of subtracting the kpSZ effect
from CMB maps.

##  IV Cosmology from the kpSZ effect

Measurement of the kpSZ effect may open new avenues for cosmological
inference. Here, we give two possible applications and demonstrate the role of
future kpSZ measurements in improving our understanding of the fundamental
properties of the Universe.

###  IV.1 Detecting cosmic birefringence

If the parity symmetry predicted by the standard LCDM is broken due to new
physics, the plane of linear polarizations sourced due to pSZ effect can
rotate due to the photon’s coupling to the dark sector (Lue  _et al._ , 1999;
Carroll and Field, 1991; Harari and Sikivie, 1992). We model this cosmic-
birefringence effect on the $E$\- and $B$\- mode polarization due to kpSZ
effect as

$\displaystyle(Q^{\prime}\pm iU^{\prime})(\hat{\boldsymbol{n}})=(Q\pm iU)^{\rm
pkSZ}(\hat{\boldsymbol{n}})e^{\pm 2i\alpha}$ (68)

where the kpSZ superscript denotes the kinetic polarization signal sourced at
the electron locations. We omit showing this superscript in what follows. From
the above equation, the modulation of the polarization modes due to
birefringence sourced between their emission and their arrival to our
detectors is

$\displaystyle E_{\ell m}^{\prime}$ $\displaystyle=E_{\ell m}\cos 2\alpha-
B_{\ell m}\sin 2\alpha$ (69) $\displaystyle B_{\ell m}^{\prime}$
$\displaystyle=B_{\ell m}\cos 2\alpha+E_{\ell m}\sin 2\alpha\,.$ (70)

The remote kinetic quadrupole fields measured from the cross of the CMB
polarization and galaxies on small scales transforms similarly

$\displaystyle{p^{E}_{\ell m}}^{\prime}$ $\displaystyle=p^{E}_{\ell m}\cos
2\alpha-p^{B}_{\ell m}\sin 2\alpha$ (71) $\displaystyle{p^{B}_{\ell
m}}^{\prime}$ $\displaystyle=p^{B}_{\ell m}\cos 2\alpha+p^{E}_{\ell m}\sin
2\alpha\,.$ (72)

Like before, we assume templates of $E$\- and $B$-mode fluctuations induced by
the kpSZ effect can be reconstructed from a large-scale structure survey. The
set of observables from cross-correlations of the templates and the
reconstructed remote quadrupole satisfy

$\displaystyle C_{\ell}^{{p}^{E}_{i}{p}^{E,\rm gal}_{j}}$
$\displaystyle\simeq$ $\displaystyle C_{\ell}^{{p}^{E}_{i}{p}_{j}^{E,\rm
gal}}\cos(2(\alpha_{i}-\alpha_{\rm cal}))\,,$ (73) $\displaystyle
C_{\ell}^{{p}^{B}_{i}{p}^{B,\rm gal}_{j}}$ $\displaystyle\simeq$
$\displaystyle C_{\ell}^{{p}^{B}_{i}{p}^{B,\rm
gal}_{j}}\cos(2(\alpha_{i}-\alpha_{\rm cal}))\,,$ (74) $\displaystyle
C_{\ell}^{{p}^{B}_{i}{p}^{E,\rm gal}_{j}}$ $\displaystyle\simeq$
$\displaystyle C_{\ell}^{{p}^{E}_{i}{p}_{j}^{E,\rm
gal}}\sin(2(\alpha_{i}-\alpha_{\rm cal}))\,,$ (75) $\displaystyle
C_{\ell}^{{p}^{E}_{i}{p}^{B,\rm gal}_{j}}$ $\displaystyle\simeq$
$\displaystyle-C_{\ell}^{{p}^{B}_{i}{p}_{j}^{B,\rm
gal}}\sin(2(\alpha_{i}-\alpha_{\rm cal}))\,,$ (76)

where we parametrised the unknown CMB detector calibration with the parameter
$\alpha_{\rm cal}$. Here, we only use the cross-correlation signal and define
an ensemble information matrix as

$\boldsymbol{\mathcal{F}}_{ab}=\sum_{\ell\ell^{\prime};ijkl}({\partial
C_{\ell}^{{p}^{B}_{i}\hat{p}^{E,\rm
gal}_{j}}}/{\partial\boldsymbol{p}_{a}})\textbf{cov}^{-1}\left(\tilde{C}^{{p}^{B}_{i}\hat{p}^{E,\rm
gal}_{j}}_{\ell},\tilde{C}^{{p}^{B}_{k}\hat{p}^{E,\rm
gal}_{l}}_{\ell^{\prime}}\right)({\partial
C_{\ell^{\prime}}^{{p}_{k}\hat{p}^{E,\rm
gal}_{l}}}/{\partial\boldsymbol{p}_{b}})\,,$ (77)

where $\boldsymbol{p}=\\{\alpha_{\rm cal},\alpha\\}$ are the set of parameters
we consider, where we define $\alpha$ as the net rotation from a comoving
range of $\alpha\simeq 220$Mpc, assuming constant differential birefringence
at all redshifts, i.e. assuming constant ${\rm d}\alpha/{\rm d}\chi$.

We demonstrate the prospect of measuring the birefringence angle for two CMB
experiments in Fig. 9, assuming VRO experimental specifications for the galaxy
survey. We find that the kpSZ tomography can potentially detect rotation angle
per redshift bin of width $\Delta\chi\sim 220$Mpc to be $\sim 1$ degree with
future experiments. While these constraints may prove to be a marginal
improvement compared to current and upcoming analyses e.g. Refs. (Gruppuso
_et al._ , 2012; Minami  _et al._ , 2019; Minami and Komatsu, 2020a; Diego-
Palazuelos  _et al._ , 2022; Minami and Komatsu, 2020b; Abazajian  _et al._ ,
2016b), they may provide a complementary probe of birefringence by breaking
the calibration angle degeneracy through tomography, and by being subject to
different systematics than constraints based on the recombination and
reionization contributions to the CMB. Cross-correlation with the reionization
$E$-modes, as well as the pSZ effect from the primordial quadrupole may
potentially improve these constraints. Finally, note that the pSZ tomography
from the primordial quadrupole signal can also probe cosmic-birefringence due
to a similar degeneracy breaking. We expand on the observations made here in
Ref. (Lee  _et al._ , 2022). Note that we have not considered the so-called
‘optical-depth degeneracy’ due to mis-modelling of the electron profiles in
the absence of perfect knowledge of the electron distribution (see e.g. Ref.
(Smith  _et al._ , 2018) for an in-depth discussion). This leads to a
multiplicative bias on the reconstructed velocities. Here, we conjecture that
a combination of the reconstructed remote quadruple auto-correlation along
with cross-correlations with templates as discussed above, as well as the
potential measurements of electron profiles (e.g. from fast-radio burst
measurements (Madhavacheril  _et al._ , 2019)), can mitigate the optical depth
degeneracy.

Figure 9: Constraints on the birefringence angle per redshift bin of width
$\Delta\chi~{}\sim 220{\rm Mpc}$. We assume constant birefringence within each
redshift bin of equal width $\Delta\chi$ in comoving distance, i.e.
$\dot{\alpha}={\rm constant}$ where $\dot{\alpha}$ is the induced differential
birefringence at comoving distance $\chi$, and we define $\alpha\equiv\int{\rm
d}\chi\dot{\alpha}$ with the integral in comoving distance spanning the range
of each redshift bin. The blue-shaded region corresponds to constraints from
the baseline CMB-S4 experiment we decribe in Table I, but with varying RMS
noise value $\Delta_{T}$. The upper dark blue line corresponds to modelling
electrons with the ‘AGN’ model, while the lower light blue line assumes
electrons follow dark matter on all scales. The green-shaded region
corresponds to the constraints from a CMB-HD like experiment with upper and
lower lines defined similarly. We find that a CMB-S4 like experiment can
potentially constrain birefringence angles as a function of redshift at $\sim
10$ degree precision, while a CMB-HD like experiment can potentially reach
degree-level constraints. The green marker corresponds to constraints using
the current proposed experimental specifications of the CMB-HD experiment.

###  IV.2 Primordial non-Gaussianity

We now turn our attention to calculating the contribution to the remote
quadrupole power-spectrum in the presence of a significant trispectrum,
parameterized by an amplitude $\tau_{\rm NL}$, that can be sourced by models
that deviate from the standard single-field slow roll inflation. Since the
single-field slow roll inflation is just the simplest model, it is natural to
wonder whether there may be other fields coupled to the inflaton.
“Cosmological collider physics” (e.g. Baumann and Green, 2012; Assassi  _et
al._ , 2012; Chen and Wang, 2012; Pi and Sasaki, 2012; Noumi  _et al._ , 2013;
Arkani-Hamed and Maldacena, 2015; Gong  _et al._ , 2013; Lee  _et al._ , 2016;
Kehagias and Riotto, 2017; Kumar and Sundrum, 2018; An  _et al._ , 2018a, b;
Baumann  _et al._ , 2018; Kumar and Sundrum, 2019; Goon  _et al._ , 2019;
Anninos  _et al._ , 2019; Kumar and Sundrum, 2020; Hook  _et al._ , 2020)
refers to the effort to seek the signatures of these new fields in the
distribution of primordial density perturbations. One simple starting point in
this inquiry is the coupling of the inflaton $\phi(x)$ to some other scalar
$\psi(x)$ through an interaction $\propto\psi\phi^{2}$ (e.g., Ref. (Jeong and
Kamionkowski, 2012)). Such a coupling would enhance the inflaton (and thus
density, or curvature) four-point function (or trispectrum) without any
enhancement in the three-point function (or bispectrum) Tseliakhovich  _et
al._ (2010); Chen and Wang (2010); Suyama and Yokoyama (2011). A curvaton
model could also enhance the trispectrum without much contribution to the
bispectrum (Tseliakhovich  _et al._ , 2010; Smith and LoVerde, 2011). In many
models, the trispectrum takes a “local-model” form, which amounts
heuristically to a spatial modulation of the power-spectrum amplitude
(equivalent to the scalar fossil field discussed in Ref. Jeong and
Kamionkowski (2012)). The primordial trispectrum amplitude $\tau_{\rm NL}$ is
currently constrained to be $\tau_{\rm NL}\lesssim 2\\!\times\\!10^{3}$ at the
68% confidence level by Planck (Ade  _et al._ , 2014; Akrami  _et al._ ,
2019). There is not much room to improve upon this significantly with CMB
measurements due to Silk damping of the temperature fluctuations. There are
possible ways to seek $\tau_{\rm NL}$ in the trispectrum of the 21-cm
brightness temperature Cooray  _et al._ (2008), however, as well as the halo
bias Ferraro and Smith (2015); Yamauchi and Takahashi (2016); Sekiguchi  _et
al._ (2019) and the 3-point correlations between two CMB temperature and one
$\mu$ spectral distortion fluctuations Bartolo  _et al._ (2016); where the
expected sensitivities are $\tau_{\rm NL}\sim 50-100$.

Locally sourced density perturbations produce fluctuations that scale as
dictated by the Poisson equation,
$\delta(\boldsymbol{k},z)\\!\propto\\!k^{2}\Phi(\boldsymbol{k})$, or with
higher factors of $k$, hence are much less significant at large scales
compared to the primordial curvature fluctuations. Scale-dependence of
cosmological fluctuations observed on the largest scales hence provide an
opportunity to infer the initial conditions relatively unpolluted by the
complicated local dynamics. Here, we are interested in a specific signal, the
$\tau_{\rm NL}$-type non-Gaussianity, which boosts the signal from cross-
correlations of fluctuations that are quadratic in density, such as the remote
kinematic quadrupole field.

The (connected) $N$-point correlation function of the gravitational potential
is given by

$\displaystyle\langle\Phi(\boldsymbol{k}_{1})\\!\ldots\\!\Phi(\boldsymbol{k}_{N}\\!)\rangle_{c}\\!=\\!(2\pi)^{3}\delta_{D}(\boldsymbol{k}_{1\dots
N\\!})F_{\Phi}(\boldsymbol{k}_{1},\ldots,\boldsymbol{k}_{N}\\!),\ \ \ \ $ (78)

where $\boldsymbol{k}_{1\ldots N}=\sum_{1}^{N}\boldsymbol{k}_{i}$ and
$F_{\Phi}$ is a function that depends on the specific model of inflation. The
collapsed 4-point function is equivalent to the large-scale correlation of
small scale power, and therefore can be seen as directly related to the
2-point function of a quadratic perturbation, as the one in defined in Eq.
(22), which can be generally expressed as

$\displaystyle{\delta_{2}(\boldsymbol{x},z)=B(z)[\delta(\boldsymbol{x},z)]^{2}\,,}$
(79)

with 2-point correlations given by

$\displaystyle{\langle\delta_{2}(\boldsymbol{x},z)\delta_{2}(\boldsymbol{x}^{\prime},z^{\prime})\rangle\\!=\\!B(z)B^{\prime}(z^{\prime})\\!\langle[\delta(\boldsymbol{x},z)]^{2}[\delta(\boldsymbol{x}^{\prime},z)]^{2}\rangle\,,}$
(80)

and

$\displaystyle\begin{split}&\langle\delta_{2}(\boldsymbol{k}_{L},z)\delta_{2}^{\prime}(\boldsymbol{k}_{L}^{\prime},z^{\prime})\rangle=\\!B(z)B^{\prime}(z^{\prime})\\!\\!\\!\int_{\boldsymbol{k}_{1}}\\!\\!\int_{\boldsymbol{k}_{3}}\\!\\!\left\langle\delta(\boldsymbol{k}_{1},z)\delta(\boldsymbol{k}_{L}\\!-\\!\boldsymbol{k}_{1},z)\delta^{\prime}(\boldsymbol{k}_{3},z^{\prime})\delta^{\prime}(\boldsymbol{k}_{L}^{\prime}\\!-\\!\boldsymbol{k}_{3},z^{\prime})\right\rangle\,,\end{split}$
(81)

which can be seen as equal to the collapsed limit of the 4-point when
$\boldsymbol{k}_{1}\gg\boldsymbol{k}_{L}$ and
$\boldsymbol{k}_{3}\gg\boldsymbol{k}_{L}^{\prime}$. Note also we defined
$\int_{\boldsymbol{k}}\equiv\int[{\rm d}^{3}k/(2\pi)^{3}]$ and $B(z)$ as a
redshift dependent coefficient. Eqs. (80) and (81) show that two-point
function of the remote quadrupole field $C_{\ell}^{pp}$ is a collapsed 4-point
function.

In various multi-field inflation models Tseliakhovich  _et al._ (2010); Chen
and Wang (2010); Suyama and Yokoyama (2011), as well as some curvaton models
(Tseliakhovich  _et al._ , 2010; Smith and LoVerde, 2011), the observation
signature of additional fields are well approximated by a spatial modulation
of the power. In this case, the four-point function for the gravitational
potential becomes

$\displaystyle\lim\limits_{|\boldsymbol{k}_{12}|\rightarrow
0}F_{\Phi}(\boldsymbol{k}_{1},\ldots,\boldsymbol{k}_{4})=16\left({5}/{6}\right)^{2}\tau_{\rm
NL}P_{12}P_{1}P_{3},$ (82)

where $P_{i}=P_{\Phi}(\boldsymbol{k}_{i})$ and
$P_{ij}=P_{\Phi}(|\boldsymbol{k}_{i}+\boldsymbol{k}_{j}|)$. This defines the
non-Gaussian amplitude $\tau_{\rm NL}$ as commonly done in literature, and can
be directly related to the amplitude of the local bispectrum, $f_{\rm NL}$, in
the absence of additional degrees of freedom (see e.g., (Suyama and Yamaguchi,
2008; Enqvist and Nurmi, 2005; Byrnes  _et al._ , 2006; Battefeld and Easther,
2007; Vernizzi and Wands, 2006; Seery and Lidsey, 2007; Kogo and Komatsu,
2006; Choi  _et al._ , 2007; Enqvist and Takahashi, 2008; Huang and Wang,
2008; Chen  _et al._ , 2009; Byrnes and Choi, 2010; Enqvist  _et al._ , 2010;
Smidt  _et al._ , 2010)).

The contribution from primordial non-Gaussianity to the power spectrum of
$\delta_{2}(\boldsymbol{k})$ field takes the form

$\displaystyle\left\langle\delta_{2}(\boldsymbol{k}_{L},z)\delta_{2}(\boldsymbol{k}^{\prime}_{L},z)\right\rangle_{\rm
NG}\\!\\!=16$ $\displaystyle\,\left({5}/{6}\right)^{2}\\!\tau_{\rm
NL}\sigma^{4}P_{\Phi}(k_{L})\times(2\pi)^{3}\delta_{D}(\boldsymbol{k}_{L}+\boldsymbol{k}^{\prime}_{L})\,,\
\ $ (83)

where

$\displaystyle\sigma^{2}=B(z)\int{\rm d}[\ln
k/(4\pi)]\Delta^{2}_{\delta}({k},z)$ (84)

is the variance of the density fluctuation.

For the collapsed limit and the inflationary models described above,

$\displaystyle\begin{split}\langle{\,{}_{\pm}}p(\boldsymbol{k}_{L}){\,{}_{\pm}}p(\boldsymbol{k}_{L}^{\prime})\rangle_{\rm
NG}=f^{2}(x)(100/9)&\tau_{\rm
NL}\sigma_{v_{\perp}}^{4}P_{\Phi}(\boldsymbol{k}_{L})\times(2\pi)^{3}\delta_{D}(\boldsymbol{k}_{L}+\boldsymbol{k}_{L}^{\prime})\,.\
\ \end{split}$ (85)

In order to calculate the non-Gaussian contribution to $C_{\ell}^{pp}$, we use
the relation

$\displaystyle(\delta_{2})_{\ell m}=\int{\rm
d}^{2}\hat{\boldsymbol{n}}\,\delta_{2}(\chi\hat{\boldsymbol{n}})Y_{\ell
m}^{*}(\hat{\boldsymbol{n}})\,,$ (86)

to set

$\displaystyle(\delta_{2})_{\ell m}=$ $\displaystyle\int{\rm
d}^{2}\hat{\boldsymbol{n}}\int_{\boldsymbol{k}}\delta_{2}(\boldsymbol{k},z)\Big{[}4\pi\sum\limits_{\ell^{\prime}m^{\prime}}i^{\ell^{\prime}}j_{\ell^{\prime}}(k\chi)Y_{\ell^{\prime}m^{\prime}}(\hat{\boldsymbol{n}})Y_{\ell^{\prime}m^{\prime}}^{*}(\hat{\boldsymbol{k}})\Big{]}Y_{\ell
m}^{*}(\hat{\boldsymbol{n}})$ (87) $\displaystyle=$
$\displaystyle\int_{\boldsymbol{k}}\delta_{2}(\boldsymbol{k},z)\Big{[}\frac{16\pi^{2}}{2\ell+1}i^{\ell}j_{\ell}(k\chi)Y_{\ell
m}^{*}(\hat{\boldsymbol{n}})\Big{]}\,.$ (88)

Using Eq. (87), we write the non-Gaussian contribution to the collapsed
4-point function as

$\displaystyle\langle{\,{}_{\pm}}p_{\ell m}$
$\displaystyle{\,{}_{\pm}}p^{*}_{\ell^{\prime}m^{\prime}}\rangle_{\rm NG}$
(90)
$\displaystyle=\int_{\boldsymbol{k}}\int_{\boldsymbol{k}^{\prime}}\langle\delta_{p}(\boldsymbol{k},z)\delta_{p}(\boldsymbol{k}^{\prime},z)\rangle_{\rm
NG}\Big{[}\frac{16\pi^{2}}{2L+1}i^{L}j_{L}(k\chi)Y_{LM}^{*}(\hat{\boldsymbol{k}})\Big{]}\Big{[}\frac{16\pi^{2}}{2L^{\prime}+1}i^{L^{\prime}}j_{L^{\prime}}(k^{\prime}\chi)Y_{L^{\prime}M^{\prime}}^{*}(\hat{\boldsymbol{k}}^{\prime})\Big{]}$
$\displaystyle=f^{2}(x)(100/9)\tau_{\rm
NL}\,\sigma_{v_{\perp}}^{4}\\!\\!\int\frac{k^{2}{\rm
d}k}{(2\pi)^{3}}[16\pi^{2}j_{L}(k\chi)]^{2}\frac{4\pi}{(2L+1)^{3}}P_{\Phi}(k)\,,$

where

$\displaystyle
P_{\Phi}(k)=\frac{2\pi^{2}}{k^{3}}A_{s}\Big{(}\frac{k}{k_{\star}}\Big{)}^{1-n_{s}}\,.$
(91)

Here, $A_{s}=2.2\times 10^{-9}$ and $n_{s}=0.965$ for $k_{\star}=0.5/{\rm
Mpc}$. We define the transverse velocity variance as

$\displaystyle\sigma^{2}_{v_{\perp}}=\int{\rm d}[\ln
k/(4\pi)]W(k\chi)^{2}\Delta^{2}_{v_{\perp}}({k},z)$ (92)

which is equivalent to the RMS value of the bulk transverse velocity, defined
with a window function $W(k\chi)$ which sets the maximum Fourier wavenumber
$k$ of the integral. In what follows we will take the RMS velocity to satisfy
$\sigma_{\rm v_{\perp}}=2\\!\times\\!10^{-3}c\approx 600$km$/$s.

We show the non-Gaussian contribution to the kinetic remote quadrupole power-
spectra in Fig. 10. We find a CMB-S4 (CMB-HD) like experiment can detect
$\tau_{\rm NL}$ value of 400 (100) at $68\%$ confidence if electrons trace
matter fluctuations at all scales. Assuming an electron gas model, we find the
lowest value of $\tau_{\rm NL}$ that can be detected at $68\%$ confidence is
$\sim 1000$ ($\sim 300$) with CMB-S4 (CMB-HD). While these values are less
impressive compared to the statistical power of galaxy surveys (Ferraro and
Smith, 2015) or kSZ tomography (Anil Kumar  _et al._ , 2022), for example,
they are nevertheless an order of magnitude smaller than the current
constraints from the CMB, and can potentially play a role in validating
findings using different methods.

Figure 10: The enhancement of the kinematic remote quadrupole signal due to
$\tau_{\rm NL}$ non-Gaussianity. The dotted lines show the non-Gaussianity
signal for $\tau_{\rm NL}$ values of $\\{40,200,1000\\}$ from bottom to top.
We see on large-scales the non-Gaussian contribution to the kinematic remote
quadrupole power-spectra indeed dominates over the Gaussian contribution.
Figure 11: Constraints on the amplitude of the collapsed 4-point non-
Gaussianity, $\tau_{\rm NL}$ for a range of CMB experiment RMS values with
other experimental specifications matching the baseline configuration of Table
I. The upper dark blue line corresponds to modelling electron fluctuation
using the AGN model, while the lower light blue line assumes electrons follow
matter fluctuations on all scales. The green-shaded region corresponds to the
constraints from a CMB-HD like experiment with upper and lower lines defined
similarly. We find that a CMB-S4 like experiment can potentially constrain
primordial non-Gaussianity of the $\tau_{\rm NL}$ type down to $\tau_{\rm
NL}\sim 400$ precision, while a CMB-HD like experiment can potentially improve
these constraints by a factor of $\sim 4$. The green marker corresponds to
constraints using the current proposed experimental specifications of the CMB-
HD experiment.

##  V Discussion

The next generation of CMB experiments and galaxy surveys are poised to drive
a paradigm shift, in which a wide variety of secondary effects in the CMB are
detected at high enough significance to become powerful cosmological probes in
their own right. In this paper, we discussed the prospects for these
experiments to detect the kinetic polarized Sunyaev Zel’dovich (kpSZ) effect
using two different techniques. First, we demonstrated that on large (roughly
degree) angular scales it is possible to detect the kpSZ effect by using
multifrequency measurements of the CMB from CMB-S4, CMB-HD, or PIXIE in cross-
correlation with a transverse velocity template constructed using a Rubin
LSST-like galaxy survey. Next, we constructed a quadratic estimator for the
kinetic component of the quadrupole field based on the anisotropic small-
angular scale cross-correlation between CMB and a galaxy survey. We
demonstrated how the reconstructed remote kinetic quadrupole field can be used
to constrain cosmic birefringence and primordial non-Gaussianity.

In future work, it will be important to compare the forecasts presented here
to the analysis of simulations, where the full non-linear effects of gravity
are accounted for. The potential biasing of the reconstruction due to mis-
modelling of the electron density field, as well as the likely enhancement of
noise due to this foreground when using different methods such as real-space
matched filtering, also warrant further analysis. Despite the potential
hurdles, the outlook for detecting kpSZ with near-term experiments is
promising, and it may provide novel constraints on various cosmological
scenarios. The kpSZ effect is just one of a number of CMB secondaries, and the
detection of this signal will undoubtedly be an important milestone towards
the development of new cosmological observables and new methods to constrain
fundamental physics with future experiments.

##  VI Acknowledgements

We thank Jose Bernal Luis, Jens Chuluba, Simone Ferraro, Andrew H. Jaffe,
Nanoom Lee and Joel Meyers for useful conversations. SCH thanks Jose Bernal
Luis for useful discussions on the conceptualization of the non-Gaussian
$\tau_{\rm NL}$ signal and Andrew H. Jaffe for their contributions at the
early stages of this project. This work was started in part at the Aspen
Center for Physics, which is supported by National Science Foundation grant
PHY-1607611. SCH is supported by the Horizon Fellowship from Johns Hopkins
University. SCH also acknowledges the support of a grant from the Simons
Foundation at the Aspen Center for Physics. MCJ is supported by the National
Science and Engineering Research Council through a Discovery grant. This
research was supported in part by Perimeter Institute for Theoretical Physics.
Research at Perimeter Institute is supported by the Government of Canada
through the Department of Innovation, Science and Economic Development Canada
and by the Province of Ontario through the Ministry of Research, Innovation
and Science.

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