# Uncorrelated Compensated Isocurvature Perturbations from kSZ Tomography

Neha Anil Kumar<EMAIL_ADDRESS>William H. Miller III Department of Physics
and Astronomy, Johns Hopkins University Baltimore, MD 21218, USA  Selim C.
Hotinli  William H. Miller III Department of Physics and Astronomy, Johns
Hopkins University Baltimore, MD 21218, USA  Marc Kamionkowski  William H.
Miller III Department of Physics and Astronomy, Johns Hopkins University
Baltimore, MD 21218, USA

###### Abstract

Compensated isocurvature perturbations (CIPs) are relative density
perturbations in which a baryon-density fluctuation is accompanied by a dark-
matter-density fluctuation such that the total-matter density is unperturbed.
These fluctuations can be produced primordially if multiple fields are present
during inflation, and therefore they can be used to differentiate between
different models for the early Universe. Kinetic Sunyaev-Zeldovich (kSZ)
tomography allows for the reconstruction of the radial-velocity field of
matter as a function of redshift. This technique can be used to reconstruct
the total-matter-overdensity field, independent of the galaxy-density field
obtained from large-scale galaxy surveys. We leverage the ability to measure
the galaxy- and matter-overdensity fields independently to construct a
minimum-variance estimator for the primordial CIP amplitude, based on a mode-
by-mode comparison of the two measurements. We forecast that a configuration
corresponding to CMB-S4 and VRO will be able to detect (at $2\sigma$) a CIP
amplitude $A$ (for a scale-invariant power spectrum) as small as $A\simeq
5\times 10^{-9}$. Similarly, a configuration corresponding to SO and DESI will
be sensitive to a CIP amplitude $A\simeq 1\times 10^{-7}$. These values are to
be compared to current constraints $A\leq{\cal O}(0.01)$.

##  I Introduction

Improving our understanding of the statistical characteristics of the
primordial density fluctuations of our Universe is one of the primary goals of
upcoming large-scale structure surveys and cosmic-microwave-background (CMB)
experiments. The current observations of the small-amplitude $[{\cal
O}(10^{-5})]$ temperature and polarization fluctuations in the CMB are
consistent with Gaussian adiabatic fluctuations, as predicted by single-field
models of inflation. Nevertheless, the search for small deviations from
adiabaticity or Gaussianity remains a promising direction of research that can
allow us to effectively distinguish between different models of inflation and
determine the number of degrees of freedom governing the dynamics of the early
Universe [e.g., 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12].

One such deviation that is particularly difficult to probe with CMB data alone
is the class of isocurvature perturbations that leave the total matter density
unchanged [13, 14, 15]. These compensated isocurvature perturbations (CIPs)
may arise in various models of inflation with multiple fields [16, 17, 14, 18,
13, 19, 20] and also during baryogenesis [21]. In the multi-field models, the
CIP fluctuations may be fully correlated with the adiabatic perturbation,
completely uncorrelated, or (most generally) somewhere in between.
Specifically, uncorrelated CIPs are a characteristic of the baryogenesis model
[21].

Because CIPs leave the total matter distribution unchanged, they give rise to
no CMB fluctuations at linear order. Instead, they induce higher-order effects
on the CMB power spectrum [22, 23, 24, 25, 26], and the CMB trispectrum [27,
28, 29]. On small distance scales, the effects of CIPs may be manifest in CMB
spectral distortions [30, 31] or the recombination history [32]. Because these
higher-order effects are harder to measure, they are rather poorly constrained
by the CMB data, with the recent constraints allowing for fairly large-
amplitude CIPs. However, there are various other prospects to probe different
models of CIPs. For example, the effects of CIPs on baryon acoustic
oscillations have been studied in Ref. [33, 34, 35]. The effects of CIPs on
21-cm fluctuations were considered in Ref. [14], and their implications for
the velocity acoustic oscillations [36, 37] in the 21-cm power spectrum are
discussed in Ref. [38]. Finally, Refs. [39, 40] assessed the sensitivity of
the galaxy clustering to the amplitude of CIPs through the measurement of the
scale-dependent galaxy bias induced due to CIPs.

Here, we study the prospects to use kinetic Sunyaev-Zeldovich (kSZ) tomography
to seek uncorrelated CIPs. kSZ tomography [41, 42, 43, 44, 45, 46, 47] allows
for the reconstruction of the line-of-sight component of the peculiar-velocity
field in a 3-dimensional volume. This is accomplished by cross-correlating the
peculiar-velocity-induced temperature fluctuation (the kSZ effect [48, 49, 50,
51, 52]), in a CMB map, with a large-scale galaxy survey, allowing for a
measurement of the kSZ contribution as a function of redshift. Given that the
total-matter field can be reconstructed from the velocity field, kSZ
tomography provides the ideal arena for testing models, like those with CIPs,
in which baryons and dark matter may be set apart from each other. In recent
work [53], the improvement coming from the inclusion of this independent
tracer was explored for models of correlated CIPs in which the CIP is fully
correlated with the adiabatic perturbation.

The above methodology allows us to compare the kSZ-tomography-based matter
reconstruction field to galaxy survey data to obtain excellent constrains on
the amplitude of the CIP power spectrum. In fact, because these two tracers of
the large scale matter distribution are obtained independently, we can
construct an estimator that compares the amplitude of the galaxy-density
fluctuation with that of the matter-density fluctuation for each Fourier
amplitude. This estimator is thus not cosmic-variance limited and can, in
principle (in the limit of perfect measurements), probe an arbitrarily small
CIP amplitude. Given that the estimator works on a mode-by-mode basis, it also
works for correlated CIPs, although it does not capitalize upon additional
effects induced by the correlation [53].

In this paper we explain the construction of this estimator and make forecasts
on the sensitivity of kSZ tomography to the CIP power-spectrum amplitude $A$.
We construct the estimator assuming that the CIP is a primordial perturbation
field. We note that the CIP amplitude is, strictly speaking, degenerate with a
CIP bias that relates the CIP perturbation to the galaxy-density perturbation
it induces. This CIP bias is, however, expected to be of order unity and can
be obtained from simulations [39, 40]. Furthermore, in the event that the
effects of CIPs are detected in the CMB, their effects in kSZ tomography can
then be used to establish the CIP bias.

In our forecasts, we consider two baseline experiment configurations:
‘baseline 1’ matching the expected specifications of the Vera Rubin
Observatory (VRO) [54] and CMB-S4 [55], and ‘baseline 2’ corresponding to the
Dark Energy Spectroscopic Instrument (DESI) [56] and Simons Observatory (SO)
[57, 58]. We find that baseline 1 results in a sensitivity of
$\sigma_{\hat{A}}\approx 2.3\times 10^{-9}$, where the errors represent the
root-variance with which the CIP power spectrum amplitude $A$ can be
determined. Similarly, we forecast that the expected sensitivity of baseline
2, based on our minimum variance estimator, is $\sigma_{\hat{A}}\approx
5.4\times 10^{-8}$. These results indicate that it may be possible to probe
CIP perturbations with an amplitude comparable to the amplitude of the
primordial power spectrum $A_{s}$. More specifically, we find a relative
uncertainty of $\sigma_{\hat{A}}/A_{s}\approx 1.0$ and
$\sigma_{\hat{A}}/A_{s}\approx 25$ for each of the baselines, respectively,
where we use the value of $A_{s}$ quoted by the Planck 2018 CMB analysis [26].

This paper is organized as follows: In Section II we introduce our
parameterization of the CIP model and in Section III we derive the minimum-
variance estimator for the CIP amplitude. We detail the relevant models for
the noise and power-spectra used in our analysis in Section IV. We then
present our results in Section V. For all our analysis we adopt the
$\Lambda$CDM Cosmology as the fiducial model with the following parameter
values, taken from Planck 2018 [26]: reduced Hubble constant $h=0.67$, baryon
density parameter $\Omega_{b}=0.049$, cold-dark-matter density parameter
$\Omega_{\rm{cdm}}=0.264$, spectral index $n_{s}=0.965$ and amplitude of the
primordial scalar power spectrum $A_{s}=2.2\times 10^{-9}$. These forecasts
represent a considerable improvement over current constraints $A\lesssim 0.01$
from the CMB [26, 29] and galaxy clusters [15, 29], although should be viewed
as complementary to the cluster constraint which probes wavenumbers primarily
around $k\sim 0.1$ Mpc-1 as opposed to Hubble scales $k\sim 10^{-4}$ Mpc-1.

##  II Compensated isocurvature perturbations

###  II.1 Definitions and conventions

We define the CIP field $\Delta(\vec{x})$ to be the primordial fractional
baryon overdensity through

$\rho_{b}(\vec{x},z)=\bar{\rho}_{b}(z)\left[1+\Delta(\vec{x})\right],$ (1)

which is then accompanied by a compensating dark-matter underdensity,

$\rho_{c}(\vec{x},z)=\bar{\rho}_{c}(z)\left[1-f_{b}\Delta(\vec{x})\right].$
(2)

Here $\bar{\rho}_{b}(z)$ and $\bar{\rho}_{c}(z)$ are respectively the mean
baryon and dark-matter densities at redshift $z$, and $f_{b}$ is the ratio
$\Omega_{b}/\Omega_{c}$ today. These defining relations are understood to be
valid at sufficiently early times such that the dark matter and baryons have
not moved significantly, either due to non-linear evolution at late times, or
before recombination due to tight-coupling of baryons to photons. Therefore,
this set-up leads to a modulation of the relative fraction of baryons and dark
matter on large scales, while keeping the total matter density fixed.

The CIP perturbation $\Delta(\vec{x})$ is a realization of a random field with
power spectrum $P_{\Delta\Delta}(k)=AF(k)$, which we have written in terms of
an amplitude $A$ and fiducial $k$ dependence $F(k)$. A canonical choice for
the $k$ dependence is the scale-invariant power spectrum $F(k)=1/k^{3}$. In
this case the CIP variance, smoothed in spheres of radius $R$, is [24]:

$\Delta_{\rm
rms}^{2}(R)=\frac{1}{2\pi^{2}}\int\,k^{2}\,dk\,\left[3j_{1}(kR)/(kR)\right]^{2}P_{\Delta\Delta}(k),$
(3)

where $j_{1}(x)$ is the spherical Bessel function. If we take $R$ to be the
CMB scale considered in Ref. [24], then $\Delta_{\rm rms}^{2}\simeq A/4$. The
current constraints to this amplitude (for uncorrelated CIPs) are $\Delta_{\rm
rms}^{2}=0.0037^{+0.0016}_{-0.0021}$ from Planck [26], $\Delta_{\rm
rms}^{2}\lesssim 0.012$ (95% CL) from the WMAP trispectrum [29], and
$\Delta_{\rm rms}^{2}\lesssim 0.006$ from baryon fractions in galaxy clusters
[15, 29].

###  II.2 CIPs and the galaxy perturbation

Following Ref. [40], the linear-order expression for the fractional galaxy-
density perturbation at comoving position $\bm{x}$ and redshift $z$ can be
written

$\delta_{g}(\bm{x},z)=b_{g}(z)\delta_{m}(\bm{x},z)+b_{\rm
CIP}(z)\Delta(\bm{x}),$ (4)

where $b_{g}(z)$ is the usual linear galaxy bias, and $b_{\rm CIP}(z)$ is a
CIP bias that parametrizes the contribution of the CIP to the galaxy-density
perturbation. The fractional matter-density perturbation
$\delta_{m}(\bm{x},z)$ is taken to be the large-scale matter perturbation
which grows proportional to the linear-theory growth factor. Given that the
CIP generates no gravitational-potential perturbation, $\Delta(\bm{x})$ will
remain approximately constant on large distance scales and so has no redshift
dependence.

The relation between $\Delta(\bm{x})$ and $\delta_{g}(\bm{x},z)$,
parameterized by the CIP bias $b_{\rm CIP}(z)$, can be obtained through
simulations. This bias is determined by two competing effects: (1) the effect
on the halo mass function, which decreases with increasing $\Delta$; and (2)
the ratio of the stellar mass to the halo mass, which increases with $\Delta$.
Simulation results for $b_{\rm CIP}$ depend on whether the galaxies are
selected by halo mass or stellar mass. Further details can be found in Refs.
[39, 40].

Given Eq. (4), the galaxy power spectrum for uncorrelated CIPs will be

$P_{gg}(k,z)=b_{g}^{2}(z)P_{mm}(k,z)+b_{\rm CIP}^{2}(z)P_{\Delta\Delta}(k).$
(5)

Thus, the CIPs show up as an additional contribution to the galaxy power
spectrum. In principle (and in practice), the CIP contribution $b_{\rm
CIP}^{2}(z)P_{\Delta\Delta}(k)$ to the galaxy power spectrum can be inferred
by comparing the observed galaxy power spectrum to the matter power spectrum
obtained from the peculiar-velocity field determined from kSZ tomography.
However, the measurements of both of the power spectra, $P_{gg}(k)$ and
$P_{mm}(k)$, are cosmic-variance limited i.e.; they are both independently
limited by the number of Fourier modes of the galaxy and velocity fields that
can be obtained with high signal to noise. Therefore, using the above model to
constrain the CIP amplitude will be limited by the effects of cosmic variance
on each of the measured power spectra.

##  III Minimum-variance estimator

With kSZ tomography, the CIP perturbation amplitude can be obtained on a mode-
by-mode basis, under (relative) cosmic-variance cancellation. In Fourier
space, the estimator for the amplitude $\Delta_{\bm{k}}$ is then

$\widehat{\Delta_{\bm{k}}}=\left(\widehat{\delta_{g,\bm{k}}}-b_{g}\widehat{\delta_{m,\bm{k}}}\right)/b_{\rm
CIP},$ (6)

where the overhat denotes an estimator, and we have dropped any redshift
dependence for ease of notation. This estimator has a variance (under the null
hypothesis $\Delta=0$),

$P_{\Delta\Delta}^{N}(\bm{k})=b_{\rm
CIP}^{-2}\left[{\left\langle\left|\Delta_{\bm{k}}\right|^{2}\right\rangle}\right]=b_{\rm
CIP}^{-2}\left[N_{gg}(k)+b_{h}^{2}N_{mm}(k)\right],$ (7)

where $N_{gg}(k)$ and $N_{mm}(k)$ are the noise contributions to the galaxy
and matter power spectra, respectively.

The detectability of CIPs can be assessed by determining the error
$\sigma_{\widehat{A}}$ with which the amplitude $A$ for the CIP power spectrum
can be measured. The minimum-variance estimator $\widehat{A}$ for the
amplitude is then obtained by adding the estimators from each Fourier mode
with inverse-variance weighting:

$\widehat{A}=b_{\rm
CIP}^{2}\sigma_{\widehat{A}}^{2}\sum_{\bm{k}}\frac{\left|\widehat{\delta_{g,\bm{k}}}-b_{g}\widehat{\delta_{m,\bm{k}}}\right|^{2}/F(k)}{2\left[P^{N}_{\Delta\Delta}(\bm{k})/F(k)\right]^{2}}.$
(8)

Here,

$\sigma_{\widehat{A}}^{2}=b_{\rm
CIP}^{-4}\left[\frac{1}{2}\sum_{\bm{k}}\left[F(k)/P_{\Delta\Delta}^{N}(\vec{k})\right]^{2}\right]^{-1},$
(9)

is the variance with which the CIP amplitude $A$ can be determined. Since this
method relies on measurements of $\widehat{\delta_{g,\bm{k}}}$ and
$\widehat{\delta_{m,\bm{k}}}$, we no longer have two independent terms
carrying the cosmic-variance limitations. Therefore, using this estimator
method, we can decrease the effects of sample variance and increase
sensitivity to the CIP amplitude, in comparison to the methodology presented
below Eq. (5).

##  IV Noise Models

We model the noise in the galaxy auto-power spectrum assuming that the primary
contribution comes from galaxy shot noise along with photo-$z$ errors.
Photo-$z$ errors can be implemented by a convolution of the galaxy density
field with a Gaussian kernel in the radial direction. The galaxy noise power
spectrum is then given by:

$N_{gg}(k,\mu)=\frac{1}{W^{2}(k,\mu)n_{\rm gal}},$ (10)

where $n_{\rm gal}$ is the average galaxy number density of the specific
survey, and Gaussian kernel $W(k,\mu)$ is defined as

$W^{2}(k,\mu)=e^{-k^{2}\mu^{2}\sigma^{2}(z)/H^{2}(z)},$ (11)

with redshift scattering $\sigma(z)$.

The noise in the independently-calculated matter-overdensity field is derived
from the kSZ velocity reconstruction noise. As shown in Ref. [46], the noise
in the kSZ-tomography-based reconstruction of the velocity field is given by

$N_{vv}(k_{L},\mu_{L})=\mu_{L}^{-2}\frac{2\pi\chi_{*}^{2}}{K_{*}^{2}}\Bigg{[}\int
dk_{S}\frac{k_{S}P_{ge}^{\rm NL}(k_{S})^{2}}{P_{gg}^{\rm NL}(k_{S})\
C_{\ell=k_{S}\chi_{*}}^{\text{tot}}}\Bigg{]}^{-1},$ (12)

where $\chi_{*}$ refers to the comoving distance to the redshift of
consideration $z_{*}$, $k_{L}$ refers to the long-wavelength mode, $k_{S}$
refers to the short-wavelength mode, and $\mu_{L}$ refers to the angle of the
large-scale mode with respect to the line of sight, i.e.,
$\mu_{L}=\hat{\bm{k}}_{L}\cdot\hat{\bm{n}}$. Furthermore, $P_{gg}^{\rm
NL}(k_{S},\mu_{S})$ refers to the small-scale galaxy-galaxy auto-power
spectrum and $P_{ge}^{\rm NL}(k_{S},\mu_{S})$ is the small-scale galaxy-
electron power spectrum. Finally, in the above equation we use the radial
weight function $K_{*}$ given by

$K_{*}\equiv-
T_{\text{CMB}}\sigma_{T}\bar{n}_{e,0}e^{-\tau(\chi_{*})}(1+z_{*})^{2},$ (13)

where $\bar{n}_{e,0}$ is the mean electron density today, and $\tau$ is the
optical depth. It is important to note that the velocity reconstruction noise
is independent of of the magnitude of $k_{L}$.

Using the late-time, linearized, continuity-equation-based relation between
the peculiar-velocity field and matter-overdensity field, we can write the
noise in the matter reconstruction as

$N_{mm}(k_{L},\mu)=\frac{k_{L}^{2}}{(faH)_{*}^{2}}N_{vv}(k_{L},\mu),$ (14)

where $f$ refers to the linear growth rate $d\ln{G}/d\ln{a}$, $H$ is the
Hubble parameter, and $a$ is the scale factor at the redshift of interest.
Since $N_{vv}$ is independent of the magnitude of $k_{L}$, the above relation
implies that the noise in the reconstructed matter power spectrum is
proportional to $k_{L}$; i.e., the noise is lowest on the largest scales.

The small-scale galaxy-galaxy and galaxy-electron power spectra appearing in
Eq. (12) are calculated within the halo model including the halo occupation
distribution (HOD) [59, 60]. The specific modelling assumptions and parameter
values used to construct the small-scale spectra can be found in Appendix A of
Ref. [61]. To ensure that the computed small-scale spectra under the HOD model
are consistent with the assumed experiment specifications, we use the
following prescription. In the HOD model, the galaxy sample is specified by
imposing a particular threshold stellar mass $m_{\star}^{\rm thresh}$ of
observable galaxies. For each configuration, we choose an $m_{\star}^{\rm
thresh}$ such that the total predicted number-density of observed galaxies
matches the number density for the given experiment.

Finally, in order to complete the model of the velocity-reconstruction noise,
we define the CMB contribution as follows. The total CMB contribution
$C_{\ell}^{\rm tot}$, appearing in Eq. (12), is assumed to be

$C_{\ell}^{\text{tot}}=C_{\ell}^{TT}+C_{\ell}^{\text{kSZ-late-
time}}+N_{\ell},$ (15)

where $C_{\ell}^{TT}$ is the lensed CMB temperature power spectrum,
$C_{\ell}^{\text{kSZ-late-time}}$ is the low-redshift contribution to kSZ, and
finally $N_{\ell}$ is the instrumental-noise power spectrum of the CMB map,
which is modelled as

$N(\ell)=s^{2}\text{exp}\Bigg{[}\frac{\ell(\ell+1)\theta_{\rm FWHM}^{2}}{8\
\text{ln}2}\Bigg{]}.$ (16)

Here, $s$ labels the sensitivity of the instrument and $\theta_{\rm FWHM}$ is
the resolution. We do not include a contribution from atmospheric noise since
it is expected to be subdominant to the instrument and kSZ contributions at
the relevant high multipoles of $\ell>3000$.

##  V Results

In this section, we provide forecasts for two different experimental
configurations, choosing a fixed, fiducial set of values for the survey
parameters to model the noise expected in each case. We then present the
dependence of $\sigma_{\hat{A}}$ on the survey parameters by varying each
independently, to better establish direction for improvements to future
surveys.

###  V.1 Baseline Forecasts

Figure 1: Left: $\sigma_{\hat{A}}$ as a function of $k_{\text{min}}$ for each
of the baselines. The lower cut-off for $k_{\rm min}$, on the left, is defined
by the survey volume $V$. For lower $k_{\rm min}$ baseline 1 performs better
due to VRO’s higher $n_{\rm gal}$. For larger values of $k_{\rm min}$,
photo-$z$ errors dominate the noise in baseline 1 and baseline 2 performs
better, in comparison. Right: $\sigma_{\hat{A}}$ as a function of $n_{\rm
gal}$ for each of the baselines. The significant increase in sensitivity to
the CIP power-spectrum amplitude $\hat{A}$ with increasing galaxy number
density occurs to the lowered shot-noise which allows for a better
reconstruction of the large scale galaxy and matter overdensity modes.
Baseline 2 performs better at lower $n_{\rm gal}$ due to spectroscopic
redshift measurements, however the increased CMB noise for this configuration
causes results to plateau at larger $n_{\rm gal}$.

We forecast future sensitivity to the amplitude of the CIP by evaluating Eq
(9) for two experimental configurations: (1) a high galaxy number density,
photometric survey similar to VRO [54] along with a CMB experiment with
specifications that match CMB-S4 [55], and (2) a low galaxy number density,
spectroscopic survey like DESI [56] with a CMB experiment like SO [57]. The
set of experimental survey parameters used in our calculation have been taken
from Ref. [61], and are summarized in Table 1.

Table 1: Baseline configurations for the cross-correlated CMB and LSS experiments. Values for baseline 1 match the specifications of the VRO survey and CMB-S4. The values for baseline 2 are similar to those expected for DESI and SO. The chosen values for the CIP bias are taken from Table 1 of Ref. [40]. The survey volumes are the same across the two configurations to emphasize the dependence of the results on galaxy number density and photo-$z$ errors.  |  | baseline 1 | baseline 2
---|---|---|---
redshift | $z$ | 1.0 | 1.0
survey volume | $V$ | 100 $\ \text{Gpc}^{3}$ | 100$\ \text{Gpc}^{3}$
halo bias | $b_{h}$ | 1.6 | 1.6
galaxy density | $n_{\rm{gal}}$ | $10^{-2}\ \ \text{Mpc}^{-3}$ | $2\times 10^{-4}\ \ \text{Mpc}^{-3}$
photo-$z$ error | $\sigma_{z}$ | 0.06 | -
threshold mass | $m_{\star}^{\rm thresh}$ | $10^{9.5}\ M_{\odot}$ | $10^{11}\ M_{\odot}$
CIP bias | $b_{\rm CIP}$ | 0.32 | 0.40
CMB resolution | $\theta_{\rm FWHM}$ | 1.5 arcmin | 1.5 arcmin
CMB sensitivity | $s$ | 1 $\ \mu\rm{K}-$arcmin | 5 $\ \mu\rm{K}-$arcmin

It is important to note that the CIP bias $b_{\rm CIP}$ is degenerate with the
CIP amplitude. Despite this degeneracy, in our constructed estimator $\hat{A}$
and the associated variance $\sigma_{\hat{A}}$ we continue to treat $A$ and
$b_{\rm CIP}$ as separate parameters to clearly establish the dependence of
$\sigma_{\hat{A}}$ on the chosen value of the bias. The exact value $b_{\rm
CIP}$ can be computed using simulations, and is expected to be of order unity,
as presented in Refs. [39, 40]. To remain consistent with our previous
definitions, for these forecasts, we fix the value of $b_{\rm CIP}$, assuming
that the galaxy samples are selected by a threshold stellar mass $m^{\rm
thresh}_{\star}$. The $m^{\rm thresh}_{\star}$ values are chosen to match the
predicted galaxy number density of each survey and are consistent with the
small-scale galaxy power spectra used to compute the velocity reconstruction
noise for each experimental configuration. The assumed value of
$m_{\star}^{\rm thresh}$ for each survey along with the corresponding value of
$b_{\rm CIP}$, estimated from the results in [39, 40], have also been included
in Table 1.

Instead of discretely summing over the Fourier modes, to compute
$\sigma_{\widehat{A}}$, we evaluate Eq. (9) in the continuous limit as
follows:

$\begin{split}\sigma_{\widehat{A}}^{2}&=b_{\rm
CIP}^{-4}\left[\frac{V}{2}\int\frac{dk^{3}}{(2\pi)^{3}}\left(\frac{F(k)}{P_{\Delta\Delta}^{N}(\bm{k})}\right)^{2}\right]^{-1},\\\
&=b_{\rm CIP}^{-4}\left[\frac{V}{2}\int_{k_{\rm min}}^{k_{\rm
max}}\int_{-1}^{1}\frac{k^{2}dk\
d\mu}{(2\pi)^{2}}\left(\frac{F(k)}{P_{\Delta\Delta}^{N}(\bm{k})}\right)^{2}\right]^{-1},\end{split}$
(17)

where we have accounted for the fact that the variance $P_{\Delta\Delta}^{N}$
is only dependent on $k$ and $\mu$, with the latter being induced by the kSZ
based velocity reconstruction and the inclusion of photo-$z$ errors. For our
forecasts, we adopt the canonical choice $F(k)=1/k^{3}$. The integral over
Fourier modes is performed from a lower limit
$k_{\text{min}}\equiv\pi/V^{1/3}$, restricted by the survey volume $V$, to an
upper limit $k_{\text{max}}\approx 10^{-1}\ \text{Mpc}^{-1}$.

Through our analysis we find that for the configuration of VRO and CMB-S4,
$\sigma_{\hat{A}}\approx 2.3\times 10^{-9}$ which corresponds to a relative
sensitivity of $\sigma_{\hat{A}}/A_{s}\approx 1.0$, where $A_{s}$ is the
amplitude of the primordial power spectrum. Similarly, for the configuration
of DESI and an SO-like CMB experiment, we find that $\sigma_{\hat{A}}\approx
5.4\times 10^{-8}$ with a relative uncertainty of
$\sigma_{\hat{A}}/A_{s}\approx 25$. For these relative uncertainty estimates,
we use the value $A_{s}=2.2\times 10^{-9}$ determined by the most recent
Planck 2018 CMB analysis [26].

###  V.2 Experiment Parameter Variations

In order to assess which experimental limitations have the most significant
impact on our ability to measure the CIP power spectrum amplitude, we isolate
the effects of certain experimental parameters from Table 1 by varying each
individually and holding all other elements of the configuration constant. The
results of these variations are discussed below.

First, to highlight the scales that most prominently contribute to the signal,
we plot the value of $\sigma_{\hat{A}}$ as a function of the smallest
measurable Fourier mode $k_{\rm min}$. This variation corresponds to changing
the largest recoverable wave number from survey volume $V$, and directly
impacts the lower limit of ‘summation’ evaluated via Eq. (17). The results for
both baselines have been presented in Fig. 1 (left). The displayed results
indicate that the inclusion of larger scales increases survey sensitivity to
the CIP power spectrum amplitude. This is an expected result, not only because
the CIP signal is largest at small $k$ [since we have chosen $F(k)\sim
1/k^{3}$ for this analysis], but also because the noise in the reconstructed
matter overdensity field is smallest on largest scales [see Eq. (14)]. At
lower values of $k_{\rm min}$, baseline 1 performs better than baseline 2,
likely due to the lowered shot noise (higher $n_{\rm gal}$). However, baseline
2 performs better at higher $k_{\rm min}$, where the effects of shot noise are
minimised and the photo-$z$ errors become dominant in the baseline 1
estimates.

Next, we focus on highlighting the effects of increasing galaxy number density
$n_{\rm gal}$ on the value of $\sigma_{\hat{A}}$. The results for this
variation for each of the baselines (holding all other experimental parameters
constant, for each individual baseline) can be seen in Figure 1 (right). The
results displayed show that an increasing galaxy number density allows for
higher survey sensitivity to the CIP power spectrum amplitude $A$. This
behaviour is a direct result of the fact that a higher galaxy number density
equates to a lower shot noise, which not only allows for the measurement of
the larger scale galaxy modes but also decreases the matter reconstruction
noise (through a lower overall $N_{vv}$). The two curves are relatively
parallel for $10^{-4}\ {\rm Mpc}^{-3}<n_{\rm gal}<10^{-2}\ {\rm Mpc}^{-3}$,
with baseline 2 performing better in this region due to spectroscopic redshift
measurements. However the baseline 1 (VRO+CMB-S4) performs better at a higher
galaxy number density, while the results from baseline 2 (DESI+CMB-SO)
plateau, likely due to the difference in CMB-resolutions.

What is more interesting to analyze is the effect of galaxy number density on
the relation between $\sigma_{\hat{A}}$ and $k_{\rm min}$. Figure 2 displays
$\sigma_{\hat{A}}$ as a function of $k_{\rm min}$ for different values of
$n_{\rm gal}$. For these curves we assume the baseline 1 configuration for all
other survey parameters and keep the value of $b_{\rm CIP}$ fixed. The
displayed results indicate that a higher galaxy number density results in a
steeper decrease of $\sigma_{\hat{A}}$ with decreasing $k_{\rm min}$ i.e., a
higher $n_{\rm gal}$ allows for a greater order-of-magnitude improvement in
$\sigma_{\hat{A}}$ with a fixed increase in survey volume. This effect is
particularly evident for $10^{-3}\ {\rm Mpc}^{-1}<k_{\rm min}<10^{-2}\ {\rm
Mpc}^{-1}$. The black dashed line, labelled ‘No Noise’, portrays the
dependence of $\sigma_{\hat{A}}$ on $k_{\rm min}$ in the absence of shot noise
($n_{\rm gal}\rightarrow\infty$) and photo-$z$ errors. In this ideal case, we
see that $\sigma_{\hat{A}}$ approximately scales as $k_{\rm min}^{3.5}$. This
behaviour is explained by the chosen model for $P_{\Delta\Delta}(k)$ [with
$F(k)=1/k^{3}$] along with the $k^{2}$ scale dependence of the matter
reconstruction noise [Eq. (14)].

Figure 2: $\sigma_{\hat{A}}$ as a function of $k_{\text{min}}$ for various
values of $n_{\rm gal}$ (in ${\rm Mpc}^{-3}$). The results indicate that an
increasing galaxy number density improves the steepness of decrease in
$\sigma_{\hat{A}}$ with decreasing $k_{\rm min}$, most evidently for $10^{-3}\
{\rm Mpc}^{-1}<k_{\rm min}<10^{-2}\ {\rm Mpc}^{-1}$. The dashed line displays
the ‘ideal’ case in which $n_{\rm gal}$ is infinitely large i.e., the shot
noise is zero.

On the contrary, assuming the same baseline configuration as above, we found
that varying $k_{\rm min}$ between $10^{-3}\ {\rm Mpc}^{-1}$ and $10^{-2}\
{\rm Mpc}^{-1}$ has a minimal impact on the steepness of the dependence of
$\sigma_{\hat{A}}$ on the galaxy number density. That is, even though a
decreased $k_{\rm min}$ improves sensitivity to the CIP power-spectrum
amplitude, a fixed increase in the galaxy number density consistently leads to
a fixed order-of-magnitude improvement in $\sigma_{\hat{A}}$ for $10^{-3}\
{\rm Mpc}^{-1}<k_{\rm min}<10^{-2}\ {\rm Mpc}^{-1}$. The results only
significantly diverge for $n_{\rm gal}>10^{-1.75}\ {\rm Mpc}^{-3}$, which is
likely due to the shot noise becoming sub-dominant at these higher values of
galaxy number density.

Finally, to highlight the effects of CMB noise on survey sensitivity to $A$,
for each of the discussed baseline configurations, we varied the CMB telescope
sensitivity $s$ and resolution $\theta_{\rm FWHM}$ individually, holding all
other experimental parameters constant. We found that, once again, the
difference in galaxy number density across the two baselines severely impacts
the order of magnitude improvement in $\sigma_{\hat{A}}$, given a fixed
improvement in CMB noise parameters. We vary the CMB sensitivity from $0.25\
\mu$K-arcmin to $10\ \mu$K-arcmin and find a steady increase in
$\sigma_{\hat{A}}$ by a factor of 3 for the baseline 1 configuration and a
factor of 1.1 for the baseline 2 configuration. Similarly, we vary CMB
telescope resolution from 0.1 arcmin to 10 arcmin and find a relatively steady
increase in both cases, by a factor of 250 for the baseline 1 configuration
and a factor of 40 for baseline 2. This indicates that, at a higher value of
$n_{\rm gal}$, surveys are more sensitive to increases in CMB instrument
noise.

For completeness, we also varied the photo-$z$ error assumed for baseline 1,
holding all other experiment parameters fixed. Varying the value of
$\sigma_{z}$ from 0.0 to 2.0 resulted in an increase in $\sigma_{\hat{A}}$ by
a factor of 3.5. This minimal effect from increasing photo-$z$ is expected,
given that we are primarily reliant on signal from the largest scales for the
measurement. We also varied the assumed value of the Gaussian halo bias
$b_{h}$ for both the baseline configurations to conclude that its effect on
survey sensitivity to $\sigma_{\hat{A}}$ is negligible.

##  VI Discussion

In this paper, we forecast future survey sensitivity to the amplitude of the
CIP power spectrum $A$, assuming that the compensated perturbations are
sourced primordially. The compensated nature of these isocurvature
perturbations causes CIPs to contribute only at second order to the CMB,
leading to poor constraints that allow for the CIP amplitude to be over 5
orders of magnitude larger than that of the primordial adiabatic perturbation.
In contrast, the CIP amplitude is expected to contribute at leading order to
the galaxy overdensity field [see Eq. (4)], making it a valuable statistic to
investigate the CIP amplitude. Therefore, in our work, we construct a minimum
variance estimator that compares the amplitude of the galaxy density
fluctuation to the independently obtained matter overdensity amplitude, on a
mode-by-mode basis. We show that leveraging the ability to measure the matter
over-density field using kSZ tomography, independently of the galaxy over-
density field, allows one to probe CIP amplitudes as small as that of the
primordial adiabatic perturbation, under sample variance cancellation.

We use the minimum-variance estimator to forecast that a survey configuration
corresponding to CMB-S4 and VRO results in a sensitivity of
$\sigma_{\hat{A}}\approx 2.3\times 10^{-9}$. Similarly, a configuration
corresponding to SO and DESI results in a sensitivity of
$\sigma_{\hat{A}}\approx 5.4\times 10^{-8}$. These sensitivities correspond to
relative uncertainties of $\sigma_{\hat{A}}/A_{s}\approx 1.0$ and
$\sigma_{\hat{A}}/A_{s}\approx 25$ for each of the combinations, respectively,
where $A_{s}$ represents the amplitude of the primordial power spectrum. For
these forecasts, we assume a fixed value for the CIP bias $b_{\rm CIP}$ for
each configuration, drawing from the simulation-based results presented in
Refs. [39, 40]. Although the CIP bias is, strictly speaking, perfectly
degenerate with the CIP perturbation amplitude, we choose not to consolidate
these two parameters into a single amplitude term to make explicit the
dependence of $\sigma_{\hat{A}}$ on the value of $b_{\rm CIP}$. Furthermore,
since this dependence is just a factor of scale, it is straightforward to map
the sensitivities quoted in this paper to a different value of $b_{\rm CIP}$
or to a constraint on a consolidated amplitude parameter $b_{\rm
CIP}^{2}\times A$.

The dramatic improvement in sensitivity to CIPs derives from the possibility,
enabled by kSZ tomography, to measure the galaxy and total-matter fields
independently and thereby circumvent the cosmic-variance limit in many other
probes. Thus, even one very well measured Fourier mode allows the CIP to be
probed. Our results indicate, moreover, that the sensitivity comes primarily
from measurements at the largest scales, a consequence largely of the $k$
dependence of the relation between the total-matter perturbation and the
peculiar velocity probed by the kSZ effect. We thus conclude that in order for
the promising statistical errors forecast here to be achieved, systematic
effects that might affect measurement of galaxy-density and CMB-temperature
perturbations on the largest distance scales must be well under control. We
also surmise that relativistic effects will need to be included in the
analysis.

The sensitivity to the CIP amplitude we forecast here compare well (within a
factor of $\sim 4$) with Ref. [53], where authors evaluated the prospects to
probe correlated CIP fluctuations similarly with kSZ tomography. Most recent
upper limits on CIPs amplitude are provided by the scale-dependent mass-to-
light ratio from measurements of BAOs [62, 34], which are comparable to the
constraints from the CMB [24], of the order
$\sigma_{A}\sim\mathcal{O}(10^{-4})$. These constraints compare also with
forecast sensitivities on the BAO phase shift induced by spatially varying
correlated CIP fluctuations explored in Ref. [35]. More recently, Ref. [38]
proposed using measurements of the velocity acoustic oscillations (VAOs)
during cosmic dawn [36, 37] to probe both correlated and uncorrelated CIPs
fluctuations at a sensitivity reaching $\sigma_{A}\sim\mathcal{O}(10^{-5})$ in
the foreseeable future. These studies find that the sensitivity of the kSZ
tomography studied here and in Ref. [53] will likely remain orders of
magnitude better compared to that of CMB, BAO and the VAO signals.

Constraining the CIP amplitude at higher order will not only allow for a
better understanding of whether baryon and CDM fluctuations trace the matter
density but also will help rule out different, non-trivial models of many-
field inflation. In fact, to accurately probe signatures of deviations from
adiabaticity and Guassianity of the early Universe, accounting for CIPs may be
essential. For example, Ref. [40] shows that, depending on the degree of
correlation of the CIP with the primordial adiabatic perturbation, the CIP
signal may exactly match the scale dependent signal from the $f_{\rm NL}$ term
when probing scale dependent bias for signatures of primordial non-Gaussianity
in the single field scenario. In the curvaton scenario, depending on the
correlation coefficient assumed between the CIP and the inflaton or the
curvaton, we would expect similar degeneracies to arise when using the galaxy
bias to simultaneously probe $f_{\rm NL}$ and $\tau_{\rm NL}$. Similar
degeneracies are may also affect the fidelity of lensing data extracted from
the CMB, due to similarities between the effects of lensing and CIPs on the
CMB two-point statistics [23].

This emphasizes the importance of considering CIPs to make unbiased
measurements of early Universe characteristics. Although we do not consider
the effects of non-Gaussianities in our current estimator construction and
make a simple set of forecasts under the null hypothesis, we highlight the
effectiveness of kSZ tomography as a probe for early universe cosmology. When
considering more complicated models including the CIP, we expect cross-
correlation tools such as the kSZ tomography, multi-tracer analysis with
different populations of galaxies and haloes, CMB lensing and many others to
be essential in obtaining tighter constraints under sample variance
cancellation and breaking degeneracies across the varying signatures of the
inflationary Universe.

###### Acknowledgements.

We would like to thank Gabriela Sato-Polito for helpful discussions. This work
was supported by NSF Grant No. 2112699 and the Simons Foundation. SCH is
supported by the Horizon Fellowship at Johns Hopkins University.

## References

  * Baumann and Green [2012] D. Baumann and D. Green, Signatures of Supersymmetry from the Early Universe, Phys. Rev. D 85, 103520 (2012), arXiv:1109.0292 [hep-th] .
  * Assassi  _et al._ [2012] V. Assassi, D. Baumann, and D. Green, On Soft Limits of Inflationary Correlation Functions, JCAP 11, 047, arXiv:1204.4207 [hep-th] .
  * Chen and Wang [2012] X. Chen and Y. Wang, Quasi-Single Field Inflation with Large Mass, JCAP 09, 021, arXiv:1205.0160 [hep-th] .
  * Noumi  _et al._ [2013] T. Noumi, M. Yamaguchi, and D. Yokoyama, Effective field theory approach to quasi-single field inflation and effects of heavy fields, JHEP 06, 051, arXiv:1211.1624 [hep-th] .
  * Arkani-Hamed and Maldacena [2015] N. Arkani-Hamed and J. Maldacena, Cosmological Collider Physics (2015), arXiv:1503.08043 [hep-th] .
  * Lee  _et al._ [2016] H. Lee, D. Baumann, and G. L. Pimentel, Non-Gaussianity as a Particle Detector, JHEP 12, 040, arXiv:1607.03735 [hep-th] .
  * Kumar and Sundrum [2018] S. Kumar and R. Sundrum, Heavy-Lifting of Gauge Theories By Cosmic Inflation, JHEP 05, 011, arXiv:1711.03988 [hep-ph] .
  * An  _et al._ [2018a] H. An, M. McAneny, A. K. Ridgway, and M. B. Wise, Quasi Single Field Inflation in the non-perturbative regime, JHEP 06, 105, arXiv:1706.09971 [hep-ph] .
  * An  _et al._ [2018b] H. An, M. McAneny, A. K. Ridgway, and M. B. Wise, Non-Gaussian Enhancements of Galactic Halo Correlations in Quasi-Single Field Inflation, Phys. Rev. D 97, 123528 (2018b), arXiv:1711.02667 [hep-ph] .
  * Baumann  _et al._ [2018] D. Baumann, G. Goon, H. Lee, and G. L. Pimentel, Partially Massless Fields During Inflation, JHEP 04, 140, arXiv:1712.06624 [hep-th] .
  * Kumar and Sundrum [2019] S. Kumar and R. Sundrum, Seeing Higher-Dimensional Grand Unification In Primordial Non-Gaussianities, JHEP 04, 120, arXiv:1811.11200 [hep-ph] .
  * Anninos  _et al._ [2019] D. Anninos, V. De Luca, G. Franciolini, A. Kehagias, and A. Riotto, Cosmological Shapes of Higher-Spin Gravity, JCAP 04, 045, arXiv:1902.01251 [hep-th] .
  * Gordon and Lewis [2003] C. Gordon and A. Lewis, Observational constraints on the curvaton model of inflation, Phys. Rev. D 67, 123513 (2003), arXiv:astro-ph/0212248 .
  * Gordon and Pritchard [2009] C. Gordon and J. R. Pritchard, Forecasted 21 cm constraints on compensated isocurvature perturbations, Phys. Rev. D 80, 063535 (2009), arXiv:0907.5400 [astro-ph.CO] .
  * Holder  _et al._ [2010] G. P. Holder, K. M. Nollett, and A. van Engelen, On Possible Variation in the Cosmological Baryon Fraction, Astrophys. J. 716, 907 (2010), arXiv:0907.3919 [astro-ph.CO] .
  * Linde and Mukhanov [1997] A. D. Linde and V. F. Mukhanov, Nongaussian isocurvature perturbations from inflation, Phys. Rev. D 56, R535 (1997), arXiv:astro-ph/9610219 .
  * Sasaki  _et al._ [2006] M. Sasaki, J. Valiviita, and D. Wands, Non-Gaussianity of the primordial perturbation in the curvaton model, Phys. Rev. D 74, 103003 (2006), arXiv:astro-ph/0607627 .
  * Lyth  _et al._ [2003] D. H. Lyth, C. Ungarelli, and D. Wands, The Primordial density perturbation in the curvaton scenario, Phys. Rev. D 67, 023503 (2003), arXiv:astro-ph/0208055 .
  * Langlois and Riazuelo [2000] D. Langlois and A. Riazuelo, Correlated mixtures of adiabatic and isocurvature cosmological perturbations, Phys. Rev. D 62, 043504 (2000), arXiv:astro-ph/9912497 .
  * He  _et al._ [2015] C. He, D. Grin, and W. Hu, Compensated isocurvature perturbations in the curvaton model, Phys. Rev. D 92, 063018 (2015), arXiv:1505.00639 [astro-ph.CO] .
  * De Simone and Kobayashi [2016] A. De Simone and T. Kobayashi, Cosmological Aspects of Spontaneous Baryogenesis, JCAP 08, 052, arXiv:1605.00670 [hep-ph] .
  * Muñoz  _et al._ [2016] J. B. Muñoz, D. Grin, L. Dai, M. Kamionkowski, and E. D. Kovetz, Search for Compensated Isocurvature Perturbations with Planck Power Spectra, Phys. Rev. D 93, 043008 (2016), arXiv:1511.04441 [astro-ph.CO] .
  * Heinrich  _et al._ [2016] C. H. Heinrich, D. Grin, and W. Hu, Lensing Bias to CMB Measurements of Compensated Isocurvature Perturbations, Phys. Rev. D 94, 043534 (2016), arXiv:1605.08439 [astro-ph.CO] .
  * Smith  _et al._ [2017] T. L. Smith, J. B. Muñoz, R. Smith, K. Yee, and D. Grin, Baryons still trace dark matter: probing CMB lensing maps for hidden isocurvature, Phys. Rev. D 96, 083508 (2017), arXiv:1704.03461 [astro-ph.CO] .
  * Valiviita [2017] J. Valiviita, Power Spectra Based Planck Constraints on Compensated Isocurvature, and Forecasts for LiteBIRD and CORE Space Missions, JCAP 04, 014, arXiv:1701.07039 [astro-ph.CO] .
  * Akrami  _et al._ [2020] Y. Akrami _et al._ (Planck), Planck 2018 results. X. Constraints on inflation, Astron. Astrophys. 641, A10 (2020), arXiv:1807.06211 [astro-ph.CO] .
  * Grin  _et al._ [2011a] D. Grin, O. Dore, and M. Kamionkowski, Compensated Isocurvature Perturbations and the Cosmic Microwave Background, Phys. Rev. D 84, 123003 (2011a), arXiv:1107.5047 [astro-ph.CO] .
  * Grin  _et al._ [2011b] D. Grin, O. Dore, and M. Kamionkowski, Do baryons trace dark matter in the early universe?, Phys. Rev. Lett. 107, 261301 (2011b), arXiv:1107.1716 [astro-ph.CO] .
  * Grin  _et al._ [2014] D. Grin, D. Hanson, G. P. Holder, O. Doré, and M. Kamionkowski, Baryons do trace dark matter 380,000 years after the big bang: Search for compensated isocurvature perturbations with WMAP 9-year data, Phys. Rev. D 89, 023006 (2014), arXiv:1306.4319 [astro-ph.CO] .
  * Haga  _et al._ [2018] T. Haga, K. Inomata, A. Ota, and A. Ravenni, Exploring compensated isocurvature perturbations with CMB spectral distortion anisotropies, JCAP 08, 036, arXiv:1805.08773 [astro-ph.CO] .
  * Chluba and Grin [2013] J. Chluba and D. Grin, CMB spectral distortions from small-scale isocurvature fluctuations, Mon. Not. Roy. Astron. Soc. 434, 1619 (2013), arXiv:1304.4596 [astro-ph.CO] .
  * Lee and Ali-Haïmoud [2021] N. Lee and Y. Ali-Haïmoud, Probing small-scale baryon and dark matter isocurvature perturbations with cosmic microwave background anisotropies, Phys. Rev. D 104, 103509 (2021), arXiv:2108.07798 [astro-ph.CO] .
  * Soumagnac  _et al._ [2016] M. T. Soumagnac, R. Barkana, C. G. Sabiu, A. Loeb, A. J. Ross, F. B. Abdalla, S. T. Balan, and O. Lahav, Large-Scale Distribution of Total Mass versus Luminous Matter from Baryon Acoustic Oscillations: First Search in the Sloan Digital Sky Survey III Baryon Oscillation Spectroscopic Survey Data Release 10 (2016), arXiv:1602.01839 [astro-ph.CO] .
  * Soumagnac  _et al._ [2018] M. T. Soumagnac, C. G. Sabiu, R. Barkana, and J. Yoo, Large scale distribution of mass versus light from Baryon Acoustic Oscillations: Measurement in the final SDSS-III BOSS Data Release 12 (2018), arXiv:1802.10368 [astro-ph.CO] .
  * Heinrich and Schmittfull [2019] C. Heinrich and M. Schmittfull, BAO Modulation as a Probe of Compensated Isocurvature Perturbations, Phys. Rev. D 100, 063503 (2019), arXiv:1904.00024 [astro-ph.CO] .
  * Muñoz [2019a] J. B. Muñoz, Robust Velocity-induced Acoustic Oscillations at Cosmic Dawn, Phys. Rev. D 100, 063538 (2019a), arXiv:1904.07881 [astro-ph.CO] .
  * Muñoz [2019b] J. B. Muñoz, Standard Ruler at Cosmic Dawn, Phys. Rev. Lett. 123, 131301 (2019b), arXiv:1904.07868 [astro-ph.CO] .
  * Hotinli  _et al._ [2021] S. C. Hotinli, T. Binnie, J. B. Muñoz, B. R. Dinda, and M. Kamionkowski, Probing compensated isocurvature with the 21-cm signal during cosmic dawn, Phys. Rev. D 104, 063536 (2021), arXiv:2106.11979 [astro-ph.CO] .
  * Barreira  _et al._ [2020a] A. Barreira, G. Cabass, D. Nelson, and F. Schmidt, Baryon-CDM isocurvature galaxy bias with IllustrisTNG, JCAP 02, 005, arXiv:1907.04317 [astro-ph.CO] .
  * Barreira  _et al._ [2020b] A. Barreira, G. Cabass, K. D. Lozanov, and F. Schmidt, Compensated Isocurvature Perturbations in the Galaxy Power Spectrum, JCAP 07, 049, arXiv:2002.12931 [astro-ph.CO] .
  * Zhang and Pen [2001] P.-J. Zhang and U.-L. Pen, Deprojecting Sunyaev-Zeldovich statistics, Astrophys. J. 549, 18 (2001), arXiv:astro-ph/0007462 .
  * Ho  _et al._ [2009] S. Ho, S. Dedeo, and D. Spergel, Finding the Missing Baryons Using CMB as a Backlight (2009), arXiv:0903.2845 [astro-ph.CO] .
  * Shao and Fang [2016] J. Shao and T. Fang, The kinetic Sunyaev-Zel’dovich tomography II: probing the circumgalactic medium, Mon. Not. Roy. Astron. Soc. 458, 3773 (2016), arXiv:1602.08932 [astro-ph.CO] .
  * Zhang and Stebbins [2011] P. Zhang and A. Stebbins, Confirmation of the Copernican Principle at Gpc Radial Scale and above from the Kinetic Sunyaev Zel’dovich Effect Power Spectrum, Phys. Rev. Lett. 107, 041301 (2011), arXiv:1009.3967 [astro-ph.CO] .
  * Munshi  _et al._ [2016] D. Munshi, I. T. Iliev, K. L. Dixon, and P. Coles, Extracting the late-time kinetic Sunyaev–Zel’dovich effect, Mon. Not. Roy. Astron. Soc. 463, 2425 (2016), arXiv:1511.03449 [astro-ph.CO] .
  * Smith  _et al._ [2018] K. M. Smith, M. S. Madhavacheril, M. Münchmeyer, S. Ferraro, U. Giri, and M. C. Johnson, KSZ tomography and the bispectrum (2018), arXiv:1810.13423 [astro-ph.CO] .
  * Cayuso  _et al._ [2021] J. Cayuso, R. Bloch, S. C. Hotinli, M. C. Johnson, and F. McCarthy, Velocity reconstruction with the cosmic microwave background and galaxy surveys (2021), arXiv:2111.11526 [astro-ph.CO] .
  * Sunyaev and Zeldovich [1980a] R. A. Sunyaev and Y. B. Zeldovich, The Velocity of clusters of galaxies relative to the microwave background. The Possibility of its measurement, Mon. Not. Roy. Astron. Soc. 190, 413 (1980a).
  * Zeldovich and Sunyaev [1969] Y. B. Zeldovich and R. A. Sunyaev, The Interaction of Matter and Radiation in a Hot-Model Universe, Astrophys. Space Sci. 4, 301 (1969).
  * Zeldovich [1970] Y. B. Zeldovich, Gravitational instability: An Approximate theory for large density perturbations, Astron. Astrophys. 5, 84 (1970).
  * Sunyaev and Zeldovich [1980b] R. A. Sunyaev and Y. B. Zeldovich, Microwave background radiation as a probe of the contemporary structure and history of the universe, Ann. Rev. Astron. Astrophys. 18, 537 (1980b).
  * Sazonov and Sunyaev [1999] S. Y. Sazonov and R. A. Sunyaev, Microwave polarization in the direction of galaxy clusters induced by the CMB quadrupole anisotropy, Mon. Not. Roy. Astron. Soc. 310, 765 (1999), arXiv:astro-ph/9903287 .
  * Hotinli  _et al._ [2019] S. C. Hotinli, J. B. Mertens, M. C. Johnson, and M. Kamionkowski, Probing correlated compensated isocurvature perturbations using scale-dependent galaxy bias, Phys. Rev. D 100, 103528 (2019), arXiv:1908.08953 [astro-ph.CO] .
  * Abell  _et al._ [2009] P. A. Abell _et al._ (LSST Science, LSST Project), LSST Science Book, Version 2.0 (2009), arXiv:0912.0201 [astro-ph.IM] .
  * Abazajian  _et al._ [2016] K. N. Abazajian _et al._ (CMB-S4), CMB-S4 Science Book, First Edition (2016), arXiv:1610.02743 [astro-ph.CO] .
  * Aghamousa  _et al._ [2016] A. Aghamousa _et al._ (DESI), The DESI Experiment Part I: Science,Targeting, and Survey Design (2016), arXiv:1611.00036 [astro-ph.IM] .
  * Ade  _et al._ [2019] P. Ade _et al._ (Simons Observatory), The Simons Observatory: Science goals and forecasts, JCAP 02, 056, arXiv:1808.07445 [astro-ph.CO] .
  * Abitbol  _et al._ [2019] M. H. Abitbol _et al._ (Simons Observatory), The Simons Observatory: Astro2020 Decadal Project Whitepaper, Bull. Am. Astron. Soc. 51, 147 (2019), arXiv:1907.08284 [astro-ph.IM] .
  * Leauthaud  _et al._ [2012] A. Leauthaud _et al._ , New constraints on the evolution of the stellar-to-dark matter connection: a combined analysis of galaxy-galaxy lensing, clustering, and stellar mass functions from z=0.2 to z=1, Astrophys. J. 744, 159 (2012), arXiv:1104.0928 [astro-ph.CO] .
  * Leauthaud  _et al._ [2011] A. Leauthaud, J. Tinker, P. S. Behroozi, M. T. Busha, and R. Wechsler, A theoretical framework for combining techniques that probe the link between galaxies and dark matter, Astrophys. J. 738, 45 (2011), arXiv:1103.2077 [astro-ph.CO] .
  * Anil Kumar  _et al._ [2022] N. Anil Kumar, G. Sato-Polito, M. Kamionkowski, and S. C. Hotinli, Primordial trispectrum from kSZ tomography (2022), arXiv:2205.03423 [astro-ph.CO] .
  * Soumagnac  _et al._ [2016] M. T. Soumagnac, R. Barkana, C. G. Sabiu, A. Loeb, A. J. Ross, F. B. Abdalla, S. T. Balan, and O. Lahav, Large-Scale Distribution of Total Mass versus Luminous Matter from Baryon Acoustic Oscillations: First Search in the Sloan Digital Sky Survey III Baryon Oscillation Spectroscopic Survey Data Release 10, Phys. Rev. Lett. 116, 201302 (2016), arXiv:1602.01839 [astro-ph.CO] .


ideas follows the exposition of [5] closely, but we recount in detail for
clarity.

###### 4.7.3. PROPOSITION.

Let $(\mathbf{T}_{1},\mathbf{T}_{2})\in\mathcal{T_{D}}(M_{1},M_{2})$ where
$M_{1}=\\{1,\ldots,n\\}$. Let $U$ be a set of extended cycles in
$\mathbf{T}_{1}$ relative to $\mathbf{T}_{2}$ such that for every cycle $c$ in
$\mathbf{T}_{1}$ such that $c\subseteq c^{\prime}$ for some $c^{\prime}\in U$
we have $c\in OC(\mathbf{T}_{1})$. Let
$(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})=\mathbf{E}((\mathbf{T}_{1},\mathbf{T}_{2});U,L)$
Then
$(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})\sim_{GTD}(\mathbf{T}_{1},\mathbf{T}_{2})$.

###### Proof.

We show by induction on $m$ that
$(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})\sim_{m}(\mathbf{T}_{1},\mathbf{T}_{2})$
for all nonnegative integers $m$. When $m=0$, this follows from our extension
of Proposition 3.1.4. So assume $m>1$ and that
$(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})\sim_{m-1}(\mathbf{T}_{1},\mathbf{T}_{2})$.
Let $\Sigma$ be a sequence for $\Pi^{*}$ of length one. Arguing as in the
proof of 3.4.3, but using the results of Section 4.5, we get the desired
result. ∎

The above has two immediate consequences:

###### 4.7.4. COROLLARY.

Let $(\mathbf{T}_{1},\mathbf{T}_{2})\in\mathcal{T}_{D}(M_{1},M_{2})$ with
$M_{1}=\\{1,\ldots,n\\}$ and suppose that for some sequence $\Sigma$ for
$\Pi^{*}$ of length one we have
$\mathbf{T}_{\Sigma}^{L}((\mathbf{T}_{1},\mathbf{T}_{2}))=\\{Z_{1},Z_{2}\\}$
with $Z_{1}\neq Z_{2}$. If $c(3,\mathbf{T}_{1})$ and
$c(4,\mathbf{T}_{1})\in\textup{OC}(\mathbf{T}_{1})$, then
$Z_{1}\sim_{GTD}Z_{2}$.

###### 4.7.5. PROPOSITION.

Let $(\mathbf{T}_{1},\mathbf{T}_{2})\in\mathcal{T}_{D}(M_{1},M_{2})$ with
$M_{1}=\\{1,\ldots,n\\}$. Then
$\mathbf{S}(\mathbf{T}_{1},\mathbf{T}_{2})\sim_{GTD}(\mathbf{T}_{1},\mathbf{T}_{2})$.

###### 4.7.6. PROPOSITION.

Let $(\mathbf{T}_{1},\mathbf{T}_{2})$ and
$(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})$
$\in\mathcal{T}_{D}(M_{1},M_{2})$ with $M_{1}=\\{1,\ldots,n\\}$. If
$\mathbf{S}(\mathbf{T}_{1})=\mathbf{S}(\mathbf{T}_{1}^{\prime})$, then
$(\mathbf{T}_{1},\mathbf{T}_{2})\sim_{GTD}(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})$.

###### Proof.

We adapt the argument of Proposition 3.4.7. By Proposition 4.7.5, we may
assume that $\mathbf{T}_{1}$ and $\mathbf{T}_{1}^{\prime}$ are both special
and in fact equal to each other. We will show that
$(\mathbf{T}_{1},\mathbf{T}_{2})\sim_{m}(\mathbf{T}_{1},\mathbf{T}_{2}^{\prime})$
for all values of $m$ whenever $\mathbf{T}_{1}$ is special. This is clear when
$m=0$, so we assume that it holds for $m-1$ and argue by induction. Let
$\Sigma$ be a sequence for $\Pi^{*}$ of length one and suppose that

$T_{\Sigma}^{L}(\mathbf{T}_{1},\mathbf{T}_{2})=\\{Z_{11},Z_{12}\\}\text{ and
}T_{\Sigma}^{L}(\mathbf{T}_{1},\mathbf{T}_{2}^{\prime})=\\{Z_{21},Z_{22}\\}$

where $Z_{i1}$ may equal $Z_{i2}$ for either or both $i\in\\{1,2\\}$. Write
$Z_{ij}=(\mathbf{T}_{1}^{ij},\mathbf{T}_{2}^{ij})$. Directly from the
definitions, we find that, perhaps after reordering,
$\mathbf{S}(\mathbf{T}_{1}^{1j})=\mathbf{S}(\mathbf{T}_{1}^{2j})$ for $j=1$
and $2$. By Proposition 4.7.5, we have that
$\mathbf{S}(Z_{ij})\sim_{GTD}Z_{ij}$, and by induction,
$\mathbf{S}(Z_{1j})\sim_{m-1}\mathbf{S}(Z_{2j})$, implying that
$Z_{1j}\sim_{m-1}Z_{2j}$, and consequently, that
$(\mathbf{T}_{1},\mathbf{T}_{2})\sim_{m}(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})$.
∎

Then Remark 3.4.2 carries over to type $D$ with the same proof, as do
3.4.3–3.4.7, replacing the pair $\\{\alpha,\beta\\}$ wherever it occurs by
$\\{\alpha,X\\}$, where $\alpha$ is one of the simple roots
$\alpha_{1}^{\prime},\alpha_{2}$, or $\alpha_{4}$ and $X=C$ or $D$. One also
replaces the conditions $1\in
ec(2,\mathbf{T}_{1};\mathbf{T}_{2}),c(2,\mathbf{T}_{1})\in\textup{OC}^{*}(\mathbf{T}_{1})$
by $3\in ec((4,\mathbf{T}_{1};\mathbf{T}_{2}),c(4,\mathbf{T}_{1})\in
OC^{*}(\mathbf{T}_{1})$, respectively. Thus for any tableau pairs
$(\mathbf{T}_{1},\mathbf{T}_{2}),(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})$
in $\mathcal{T}_{D}(n,n)$ we have
$(\mathbf{T}_{1},\mathbf{T}_{2})\sim_{GTD}(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})$
whenever $\mathbf{S}(\mathbf{T}_{1})=\mathbf{S}(\mathbf{T}_{1}^{\prime})$.
Corresponding to Lemma 3.4.8 we have

###### 4.7.7. LEMMA.

Let $(\mathbf{T}_{1},\mathbf{T}_{2})\in\mathcal{T}_{D}(M_{1},M_{2})$ with
$M_{1}=\\{1,\ldots,n\\}$. Let $1<l<n$ and let $\bar{\mathbf{T}}_{1}$ be the
tableau obtained from $\mathbf{T}_{1}$ by removing the dominos labelled
$l+1,\ldots,n$. Let $\bar{P}$ be an extremal position in
$\bar{\mathbf{T}}_{1}$. Assume that there exists a tableau $\mathbf{T}$ having
the same shape as $\bar{\mathbf{T}}_{1}$, the $l$-domino in position
$\bar{P}$, and the same $n_{v}$ value as $n_{v}(\bar{\mathbf{T}}_{1})$ modulo
4. Then there is a sequence $\Sigma$ for
$\\{\alpha_{1}^{\prime},\alpha_{2},\ldots,\alpha_{l}\\}$ and a
$(\mathbf{T}_{3},\mathbf{T}_{2})\in
T_{\Sigma}^{L}((\mathbf{T}_{1},\mathbf{T}_{2}))$ such that
$P(l,\mathbf{T}_{3})=\bar{P}$ and, for $l+1\leq r\leq
n,P(r,\mathbf{T}_{3})=P(r,\mathbf{T}_{1})$.

###### Proof.

This is proved as in 3.4.8, using Lemma 4.6.8 instead of Theorem 3.2.2. ∎

Then 3.4.9–3.4.14 carry over to type $D$ with the same proofs. Our main result
extends Theorem 3.4.17 to type $D$.

###### 4.7.8. THEOREM.

Let
$(\mathbf{T}_{1},\mathbf{T}_{2}),(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})\in\mathcal{T}_{D}(M_{1},M_{2})$,where
$M_{1}=M_{2}=\\{1,\ldots,n\\}$. Then
$(\mathbf{T}_{1},\mathbf{T}_{2})\sim_{GTD}(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})$
if and only if
$\mathbf{S}(\mathbf{T}_{1})=\mathbf{S}(\mathbf{T}_{1}^{\prime})$.

###### Proof.

This is proved in the same way as Theorem 3.4.17, using the analogues of
Lemmas 3.4.15 and 3.4.16 for type $D$, except that (as noted above) Lemma
3.4.8 no longer holds for all tableau shapes. Note that this lemma is not
needed for the proof of Lemma 3.4.15 or 3.4.16, so we are reduced to one of
the cases in the proof of Theorem 3.4.17 not involving a closed cycle, so that
we have two tableaux of distinct special shape and need to show that they have
different generalized $\tau$-invariants. If Lemma 3.4.8 fails for one of the
tableau shapes, then (bearing in mind the remark after 4.6.5) we can argue as
in Case F of the proof of Theorem 3.4.17 to find a sequence of operators such
that when we apply it to both tableaux we get another pair of tableaux such
that when the largest domino is removed from each, the resulting pair of
tableaux have different generalized $\tau$-invariants. But then Lemma 3.4.9
shows that the same holds for the original pair of tableaux, as desired. ∎

##

We define the relations $\sim_{JLD}$ and $\sim_{JRD}$ on
$W^{\prime},\mathcal{S}^{\prime}(n,n),\mathcal{S}^{\prime\prime}(n,n)$ and
$\mathcal{T}_{D}(n,n)$ in exactly the same way for type $D$ as
$\sim_{JL},\sim_{JR}$ were defined for types $B$ and $C$ in Definition 3.5.1,
but using the notion of sequence in Definition 4.6.1. It is these relations,
when applied to $W^{\prime}$, that turn out to coincide with
$\sim_{L},\sim_{R}$, respectively, in the sense of Kazhdan and Lusztig [8].
When applied to $\mathcal{S}^{\prime}(n,n)$ or $\mathcal{S}"(n,n)$, on the
other hand, note that
$(\mathbf{L}(v),\mathbf{R}(v))\sim_{JL}(\mathbf{L}(w),\mathbf{R}(w))$ if and
only if $v\sim_{JR}w$, by the remark in Definition 4.3.1, and similarly with
$L$ and $R$ reversed.

###### 4.8.1. THEOREM.

Let
$(\mathbf{T}_{1},\mathbf{T}_{2}),(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})$
correspond to $w,w^{\prime}$ in the same coset of $W^{\prime}$ in $W$. Then
$(\mathbf{T}_{1},\mathbf{T}_{2})\sim_{JRD}(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})$
if and only if
$\mathbf{S}(\mathbf{T}_{1})=\mathbf{S}(\mathbf{T}_{1}^{\prime})$.

###### Proof.

This follows at once from Theorem 4.6.2 (and its easy converse). ∎

Now 3.5.4–3.5.6 carry over immediately to type $D$ with the obvious changes in
notation. In the statements of these results for type $D$, we assume
throughout that either $w_{1},w_{2}\in W^{\prime}$ or $w_{1},w_{2}\in W"$.
Then Definition 3.5.7 carries over word for word to type $D$ and we have
$I_{\lambda}(w_{1})\sim_{GTD}I_{\lambda}(w_{2})$ if and only if
$w_{1}\sim_{GTDR}w_{2}$. We have

###### 4.8.2. THEOREM.

Suppose all the simple factors of $\Delta_{\lambda}$ are of type $D$. Let
$I_{1},I_{2}\in$Prim${}_{\lambda}(U(\mathfrak{g}))$. Then
$I_{1}\sim_{GTD}I_{2}$ if and only if $I_{1}=I_{2}$.

We conclude with the definition of the operator $T_{\alpha_{1}^{\prime}C}^{L}$
of Definition 4.4.10 on the set $\mathcal{T}_{D}^{S}(n)$ of tableaux of
special shape relative to type $D$; we make the analogous definition for the
other operators defined there. If $C^{3}$ or $C^{4}$ is a subset of a tableau
$\mathbf{T}$ of special shape, then the cycle $c(4,\mathbf{T}_{1})$ is
necessarily closed; move through this cycle and replace the subset $C^{1}$
(resp. $C_{2}$) in the resulting tableau by $B_{1}^{1}$ (resp. $B_{1}^{2}$) to
obtain the single tableau $T_{\alpha_{1}^{\prime}C}(\mathbf{T})$. Otherwise
$C^{1}$ or $C^{2}$ is a subset of $\mathbf{T}$; replace it by $B^{1}$ or
$B^{2}$ to obtain a tableau $\mathbf{T}^{\prime}$.
if$c(4,\mathbf{T}^{\prime})$ is open, take
$T_{\alpha_{1}^{\prime}C}(\mathbf{T})$ to be the single tableau
$\mathbf{T}^{\prime}$. Finally, if $c(4,\mathbf{T})$ is closed,
$T_{\alpha_{1}^{\prime}C}(\mathbf{T})$ has two values, the first one being
$\mathbf{T}^{\prime}$ and the second one obtained from it by moving through
$c(4,\mathbf{T}^{\prime})$.

Then the classification theorem reads

###### 4.8.3. THEOREM.

Suppose that $\mathfrak{g}$ is of type $D_{n}$ and $\lambda\in\mathfrak{h}$ is
integral, regular, and anti-dominant. Then the map $cl$ from
Prim${}_{\lambda}(U(\mathfrak{g}))$ to $\mathcal{T}_{D}^{S}(n)$ sending
$I_{\lambda}(w)$ to $\mathbf{S}(\mathbf{L}(\delta(w)))$ is a bijection; it is
also the unique map from its domain to its range preserving $\tau$-invariants
and commuting with the operators $T_{\alpha\beta}$ and the operators of
Definition 4.4.3.

## References

  * [1] D. Barbasch and D. Vogan.  Primitive ideals and orbital integrals in complex classical groups,  Math. Ann. 259(2):153–199, 1982.
  * [2] N. Bourbaki.  Groupes et Algèbres de Lie, Chapitre 4,  Hermann, Paris, 1968.
  * [3] D. Garfinkle.  On the classification of primitive ideals for complex classical Lie algebras (I),  Compositio Math., 75(2):135–169, 1990.
  * [4] D. Garfinkle.  On the classification of primitive ideals for complex classical Lie algebras (II),  Compositio Math., 81(3):307–336, 1992.
  * [5] D. Garfinkle.  On the classification of primitive ideals for complex classical Lie algebras (III),  Compositio Math., 88:187–234, 1993.
  * [6] D. Garfinkle and D. A. Vogan, Jr.  On the structure of Kazhdan-Lusztig cells for branched Dynkin diagrams,  J. Algebra., 153(1):99–120, 1992.
  * [7] J. C. Jantzen.  Einhüllende Algebren halbeinfacher Lie-Algebren,  Ergebnisse der Mathematik and ihrer Grenzgebiete, Band 3,  Springer, Berlin, 1983.
  * [8] D. Kazhdan and G. Lusztig.  Representations of Coxeter groups and Hecke algebras,  Inv. Math., 53:165–184, 1979.