# CMB-HD as a Probe of Dark Matter on Sub-Galactic Scales

Amanda MacInnis  Physics and Astronomy Department, Stony Brook University,
Stony Brook, NY 11794  Neelima Sehgal  Physics and Astronomy Department, Stony
Brook University, Stony Brook, NY 11794

###### Abstract

We show for the first time that high-resolution CMB lensing observations can
probe structure on sub-galactic scales. In particular, a CMB-HD experiment can
probe out to $k\sim 55\leavevmode\nobreak\ h$/Mpc, corresponding to halo
masses of about $10^{8}M_{\odot}$. Over the range $0.005\leavevmode\nobreak\
h$/Mpc$\leavevmode\nobreak\ <k<\leavevmode\nobreak\ 55\leavevmode\nobreak\
h$/Mpc, spanning four orders of magnitude, the total lensing signal-to-noise
ratio (SNR) from the temperature, polarization, and lensing power spectra is
greater than 1900. CMB-HD gains most of the lensing SNR at small scales from
the temperature power spectrum, as opposed to the lensing spectrum. These
lensing measurements allow CMB-HD to distinguish between cold dark matter
(CDM) and non-CDM models that change the matter power spectrum on sub-galactic
scales. We also find that CMB-HD can distinguish between baryonic feedback
effects and non-CDM models due to the different way each impacts the lensing
signal. The kinetic Sunyaev-Zel’dovich (kSZ) power spectrum further constrains
non-CDM models that deviate from CDM on the smallest scales CMB-HD measures.
For example, CMB-HD can detect 1 keV warm dark matter (WDM) at 30$\sigma$, or
rule out about 7 keV WDM at 95% CL, in a $\Lambda$WDM$+N_{\rm{eff}}+\sum
m_{\nu}+m_{\rm{WDM}}+{\rm{log_{10}}}T_{\rm{AGN}}+A_{\rm{kSZ}}+n_{\rm{kSZ}}$
model; here $T_{\rm{AGN}}$ characterizes the strength of the feedback, and
$A_{\rm{kSZ}}$ and $n_{\rm{kSZ}}$ allow freedom in the amplitude and slope of
the kinetic Sunyaev-Zel’dovich power spectrum. We make the CMB-HD Fisher code
used here publicly available, and note that it can be modified to use any non-
CDM model that changes the matter power spectrum.

##  I Introduction

Mapping the matter distribution in the Universe is a key goal of modern
cosmology. Mapping the matter distribution on large scales probes the
evolution of structure, which in turn yields knowledge of the Universe’s
components and history. Comparison of large-scale structure measurements with
measurements from expansion-rate probes also informs us about the nature of
gravity and dark energy. Mapping the matter distribution on small-scales, on
the other hand, has become a critical probe of dark matter properties. Such
measurements may be the only way to probe dark matter if dark matter does not
interact with standard model particles.

Measuring the gravitational lensing of the Cosmic Microwave Background (CMB)
is a powerful way to measure the matter distribution [1, 2, 3]. One advantage
of this method is that it probes dark matter directly via gravitational
lensing, instead of relying on baryonic tracers (such as stars, galaxies, or
gas). Using the CMB as the background light source that undergoes lensing also
has a number of unique advantages: 1) the source light is from a well-known
redshift, 2) the unlensed properties of the source light are well understood,
3) the source light is behind all structure in the Universe, and 4) the source
light on small scales is a smooth gradient field that retains the same
gradient when viewed with infinite resolution.

While the first direct detections of CMB lensing occurred relatively recently
[4, 5, 6], CMB lensing measurements have rapidly increased in signal-to-noise
ratio (SNR) on large scales [7, 8, 9, 10], as well as on the modest scales of
galaxy clusters [11], and galaxies [12]. The large-scale measurements in
particular are now routinely included when obtaining the tightest constraints
on cosmological parameters [e.g., 13, 14].

The kinetic Sunyaev-Zel’dovich (kSZ) power spectrum is another measurement CMB
observatories offer, which is sensitive to the distribution of dark matter on
small scales [15]. The kSZ effect arises when CMB photons scatter off
electrons in ionized gas in the Universe and obtain a Doppler shift due to the
bulk radial velocity of the gas [16, 17, 18]. Similar to CMB lensing, the kSZ
spectrum can be measured both from the CMB temperature power spectrum and from
the trispectrum [19]. Both the lensing and kSZ effects are frequency-
independent, and can be separated from frequency-dependent foregrounds by CMB
experiments that observe at many different frequencies. While the kSZ power
spectrum is routinely fit for in CMB power spectrum measurements [20, 21, 22],
it has yet to be directly measured via the trispectrum; recent efforts to
measure the kSZ trispectrum have yielded upper limits [23, 24]. Measurements
of the kSZ effect are expected to improve rapidly with future CMB experiments
[25].

In this work we quantify the ability of CMB-HD [26, 27] to measure the
gravitational lensing of the CMB and the kSZ power spectrum. We give special
attention to the ability of CMB-HD to probe sub-galactic scales, which is a
new regime for CMB observations. The matter distribution on sub-galactic
scales is of particular interest for understanding the nature of dark matter.
We also focus on the ability of CMB-HD measurements to distinguish between the
effect of baryonic feedback (which can move around the distribution of matter
on small-scales), and the effect of dark matter that differs from cold dark
matter (CDM). Distinguishing between these two scenarios is necessary to probe
the behavior of dark matter alone. Lastly, we quantify the extent to which
varying the standard $\Lambda$CDM parameters in addition to extensions (e.g.
$N_{\rm{eff}}$ and $\sum m_{\nu}$) impacts constraints on dark matter
properties. Given the enormous precision with which CMB-HD can measure the
primordial CMB, CMB lensing over four orders of magnitude in scale, and the
kSZ power spectrum, we examine whether CMB-HD could adopt a holistic approach
of varying all relevant parameters simultaneously.

In Section II, we briefly summarize the key results of this work. In Section
III we discuss how we calculate covariance matrices, CMB and lensing spectra,
lensing SNRs, the effect of non-CDM models, the kSZ spectrum, and the Fisher
matrix. In Section IV, we present our results, and in Section V, we discuss
and conclude.

##  II Summary of Key Results

Below we briefly summarize the key results.

  * •

We forecast the SNR for measuring lensing from the CMB-HD $TT$, $TE$, $EE$,
$BB$, and $\kappa\kappa$ power spectra and show the results in Table 3. We
find that the total lensing SNR for all the spectra combined is 1947, and that
more SNR comes from the temperature, as opposed to the lensing, power spectra.

  * •

The top panel of Figure 6 shows the breakdown of the lensing SNR by wavenumber
$k$ (in $h$/Mpc) and by redshift. It shows that smaller-scale signals
originate from lower redshifts. It also shows that CMB-HD lensing measurements
would span $0.005\leavevmode\nobreak\ h$/Mpc$<k<55\leavevmode\nobreak\ h$/Mpc,
over four orders of magnitude in scale, with a SNR in the several hundreds for
most of the $k$ bins shown. The bottom panel of Figure 6 further divides into
$\ell$ bins, and indicates that CMB-HD would probe lensing well into the non-
linear regime.

  * •

In Figure 2, we show forecasted CMB-HD constraints on the linear matter power
spectrum following [28] and [29]. We find that CMB-HD can measure the matter
power spectrum out to $k\sim\leavevmode\nobreak\ 55\leavevmode\nobreak\
h$/Mpc, corresponding to halos of $10^{8}\leavevmode\nobreak\ M_{\odot}$.

  * •

We show in Table 5 the forecasted constraints on the warm dark matter (WDM)
mass ($m_{\rm{WDM}}$) using CMB-HD power spectra plus DESI baryon acoustic
oscillation (BAO) data [30]. We find that $m_{\rm{WDM}}$ is most degenerate
with the slope and amplitude of the kSZ power spectrum, $n_{\rm{kSZ}}$ and
$A_{\rm{kSZ}}$, and that $m_{\rm{WDM}}$ constraints are significantly degraded
when these parameters are allowed to be free. This degeneracy is shown in
Figures 7 and 8.

  * •

We find that varying the $\Lambda$CDM parameters, the number of relativistic
species ($N_{\rm{eff}}$), and the sum of the neutrino masses ($\sum m_{\nu}$)
negligibly impacts $m_{\rm{WDM}}$ constraints, as shown in Table 5. From
Figures 7 and 8, we see that the most notable, but still mild, degeneracies
are with $n_{\mathrm{s}}$ and $N_{\rm{eff}}$.

  * •

As shown in Figures 7 and 8 and Table 5, marginalizing over baryonic feedback
effects (${\rm{log_{10}}}T_{\rm{AGN}}$) does not degrade constraints on
$m_{\rm{WDM}}$; this is due to the substantially different way baryonic
effects and warm dark matter alter the matter power spectrum, shown via the
difference in the CMB lensing signal in Figure 4.

Figure 1: We show the significance with which CMB-HD plus DESI BAO can measure
a given warm dark matter (WDM) mass, quantified by the signal-to-noise ratio
(SNR) $m_{\mathrm{WDM}}/\sigma(m_{\mathrm{WDM}})$. The blue curve shows the
SNR obtained from CMB-HD lensing data alone, while the orange curve uses both
CMB lensing and kSZ data. We assume a $\Lambda$WDM$+N_{\rm{eff}}+\sum
m_{\nu}+m_{\rm{WDM}}+{\rm{log_{10}}}T_{\rm{AGN}}$ model when only using
lensing data, and add kSZ parameters $A_{\rm{kSZ}}$ and $n_{\rm{kSZ}}$ to the
model when adding kSZ data; all parameters are varied simultaneously. We use
the Warm&Fuzzy code of [31] to calculate the non-linear WDM transfer function
(Eq. 11). Given the significant challenge of modelling this transfer function
even for WDM models, the plot above is illustrative of what CMB-HD can
measure; however, exact SNRs will depend on modelling details. We find that
CMB-HD can detect WDM with a mass of about 7 keV at the 95% C.L. when
including kSZ measurements, or 3.5 keV WDM from lensing alone.

  * •

We find that CMB-HD plus DESI BAO can detect 1 keV WDM at 30$\sigma$, or rule
out about 7 keV WDM at 95% C.L., in a $\Lambda$WDM$+N_{\rm{eff}}+\sum
m_{\nu}+{\rm{log_{10}}}T_{\rm{AGN}}+A_{\rm{kSZ}}+{n_{\rm{kSZ}}}+m_{\rm{WDM}}$
model, which we illustrate in Figure 1.

  * •

We make public the code to reproduce the results in this
work111https://github.com/CMB-HD/hdPk. We note that it can be modified to use
any non-CDM model that changes the non-linear matter power spectrum.

Figure 2: Top panel: We illustrate how CMB-HD lensing measurements can probe
the linear matter power spectrum. We show in red the forecasted errors; these
are obtained from the CMB-HD lensing SNR in each comoving wavenumber bin,
equating the inverse of this lensing SNR to the fractional error bar on the
linear matter power spectrum (see Section III.4.1 for SNR calculation and Eq.
27 in Section IV.1). We place the error bars on the theoretical prediction for
the linear matter power spectrum (black curve) at the center of each
wavenumber bin. We also indicate on the top $x$-axis the corresponding mass
scale today for the comoving wavenumbers on the bottom $x$-axis, using Eq. 28.
Of note is that the CMB-HD lensing constraints span four orders of magnitude
in wavenumber, $0.005\leavevmode\nobreak\ h$/Mpc$\leavevmode\nobreak\
<k<\leavevmode\nobreak\ 55\leavevmode\nobreak\ h$/Mpc, and 12 orders of
magnitude in mass scale, going down to $10^{8}M_{\odot}$. With the exception
of the last bin, the error bars are smaller than the band center markers. We
also include for reference constraints already measured and derived by [28]
based on the method of [29]: Lyman-$\alpha$ data [32] from the eBOSS DR14
release [33] (purple); cosmic shear data from DES Y1 [34] (brown); galaxy
clustering data for luminous red galaxies (LRG) from the SDSS DR7 release [35]
(pink); and Planck 2018 CMB temperature (blue), polarization (yellow), and
lensing (green) data [36, 7]. We also add constraints derived from high-
redshift measurements of the UV galaxy luminosity function (HST UV LF) from
[37] (teal). Note that this is not an exhaustive summary of current
constraints, but illustrative of what has already been achieved. We include
data that more directly measures the matter power spectrum, as opposed to data
that measures the halo mass function that is then used to infer the linear
matter power spectrum, which requires first marginalizing over cosmology,
baryonic physics, and sample variance, and modeling very nonlinear scales.
Bottom panel: We compare the error bars on the matter power spectrum from each
data set as a fraction of the total power. We see that CMB-HD achieves sub-
percent fractional errors in the range $0.03\leavevmode\nobreak\ h/$Mpc
$<\leavevmode\nobreak\ k\leavevmode\nobreak\ <\leavevmode\nobreak\ 10$
$h/$Mpc.

##  III Methods

We describe below the methods to calculate covariance matrices (Sections III.1
and III.2), signal spectra (Section III.3), lensing SNRs (Section III.4), the
effects of non-CDM models (Section III.5), the kSZ spectrum (Section III.6),
and Fisher matrices (Section III.7).

###  III.1 CMB-HD Covariance Matrix

To calculate the CMB-HD covariance matrix we follow the procedure in [38]. In
Table 1, we list the experimental configuration assumed. To summarize, we
include CMB and CMB lensing power spectra in the multipole range of
$\ell\in[30,20000]$ for $TT$, $TE$, $EE$, $BB$, and $\kappa\kappa$, and
calculate the $5n_{\mathrm{bin}}\times 5n_{\mathrm{bin}}$ covariance matrix,
$\mathbb{C}$, as in [38]. In addition, we extend the $\ell_{\mathrm{max}}$ for
$TT$ to 40,000, which differs from [38]. We calculate the covariance matrix,
SNR, and fisher matrix for $TT$ in the range of $\ell\in[20000,40000]$
separately, as described below.

####  III.1.1 Noise on CMB Power Spectra

We assume CMB-HD will observe 60% of the sky, and that the 90 and 150 GHz
channels will be used for the parameter analysis (while the other five
frequency channels will be used to reduce foreground contamination). We list
the instrument white noise levels and beam full width at half-maximum (FWHM)
at 90 and 150 GHz for CMB-HD in Table 1 [27]. We assume that the temperature
and polarization noise are uncorrelated.

We assume that CMB-HD will measure multipoles of $\ell\geq 1000$, and that CMB
data in the range $30\leq\ell<1000$ will come from measurements by precursor
CMB surveys. Since $TT$, $TE$, and $EE$ will already be sample-variance
limited over 60% of the sky for $30\leq\ell<1000$ from precursor surveys such
as the Simons Observatory (SO) [39], we extend the CMB-HD multipole range down
to $\ell_{\mathrm{min}}=30$ for these spectra since CMB-HD will have full
overlap with these surveys. For the $BB$ power spectrum, we use the expected
noise from the Advanced Simons Observatory (ASO) in the range
$\ell\in[30,1000)$. We provide the expected instrumental noise levels in
temperature and the beam FWHM at 90 and 150 GHz for ASO in the last row of
Table 1.

We also include the effect of residual extragalactic foregrounds in the
temperature data by adding the expected residual foreground power spectra for
CMB-HD to the temperature instrument noise spectra at 90 and 150 GHz. The
residual power spectra from the thermal Sunyaev-Zel’dovich (tSZ) effect, the
cosmic infrared background (CIB), and radio sources are obtained from [40],
and we include the full power spectrum of the kinetic Sunyaev-Zel’dovich (kSZ)
effect (with both reionization and late-time contributions) using a template
from [41] for the late-time part, and from [42, 43] for the reionization part,
normalized as described in Section III.6.222The model for the kSZ power
spectrum used in this work differs from that in [40] and [38], by including
both the late-time and reionization kSZ contributions, as opposed to just the
latter. The coadded noise spectra, $N_{\ell}^{XY}$, for
$XY\in\\{TT,TE,EE,BB\\}$ are obtained by calculating the beam-deconvolved
instrumental noise power spectrum for each frequency channel (90 and 150 GHz),
adding the residual foreground power to the beam-deconvolved instrumental
noise for $TT$ for each frequency, and coadding the spectra of the two
frequencies using inverse-noise weighting.

Exp. | $f_{\mathrm{sky}}$ | Freq., | Temp. Noise, | FWHM, | Multipole Range
---|---|---|---|---|---
GHz |  $\mu$K-arcmin | arcmin | $\ell,L$
HD | 0.6 | 90 | 0.7 | 0.42 | TT: [30, 40000]
150 | 0.8 | 0.25 | TE, EE, $\kappa\kappa$: [30, 20000]
|  |  | BB: [1000, 20000]
ASO | 0.6 | 90 | 3.5 | 2.2 | BB: [30,1000)
150 | 3.8 | 1.4
Table 1: We list white noise levels and beam full-width at half-maximum (FWHM)
at 90 and 150 GHz for CMB-HD (“HD”) and the Advanced Simons Observatory
(“ASO”), along with the assumed sky fraction ($f_{\mathrm{sky}}$) that each
experiment will observe, and the multipole ranges of the power spectra from
each experiment we use in our forecasts. The map noise levels given are for
temperature; the noise levels for polarization are higher by a factor of
$\sqrt{2}$. The CMB-HD experimental specifications are from [27].

####  III.1.2 Noise on CMB Lensing Spectra

The gravitational lensing of the primary CMB introduces coupling between modes
of the lensed CMB on different scales. This lensing-induced mode coupling can
be reconstructed from pairs of CMB maps that are filtered to isolate the non-
Gaussian effects of lensing, forming a quadratic estimator [1, 2, 44]. The
power spectrum of this estimator depends on the four-point function of the CMB
maps used to construct it, which has two components [45]. The non-Gaussian
part of the four-point function contains the CMB lensing power spectrum, but
the Gaussian component introduces a bias to the lensing power spectrum, which
arises from the primordial CMB signal and the instrumental noise. This
Gaussian bias, referred to as the $N^{(0)}$ bias, can be removed using a
realization-dependent (RDN0) subtraction technique [46]; however, it will
still act as a source of noise on the estimated CMB lensing power spectrum.

We use the lensing noise $N_{\ell}^{\kappa\kappa}$ from [38], which was
calculated using two kinds of quadratic estimators. We briefly discuss the
calculation here, and refer the reader to [38] for more details.

The first estimator is the “standard” quadratic estimator from [1, 2, 44],
which we refer to as the H&O estimator. It is designed to minimize the
variance of the lensing reconstruction. We use the CLASS delens package
[47]333https://github.com/selimhotinli/class_delens to calculate
$N_{\ell}^{\kappa\kappa}$ for the $TT$, $TE$, $EE$, $EB$, and $TB$ H&O
estimators, which uses an iterative delensing technique to reduce their noise
levels. This calculation includes the residual extragalactic foreground power
for temperature spectra mentioned above, and the ASO instrumental noise in
polarization on scales $\ell<1000$.

The second estimator is the quadratic estimator of [48], which we refer to as
the HDV estimator. It is designed to more optimally reconstruct small scales.
Here, as in [38] and [40], we also use it to reduce bias to the lensing
reconstruction on small-scales that arises due the presence of non-Gaussian
foregrounds. This is done by filtering one CMB map to isolate large scales
($\ell<2000$), and the other CMB map to isolate only small scales
($\ell\in[5000,20000]$) (see [40] for details). We replace the noise from the
H&O $TT$ estimator on scales $\ell>5000$ with the noise from the HDV $TT$
estimator, derived from the simulation-based covariance matrix of [40]. These
simulations include the anticipated residual extragalactic foregrounds for
CMB-HD, which are correlated with the non-Gaussian CMB lensing map [40].

The final lensing noise power spectrum $N_{\ell}^{\kappa\kappa}$ is a minimum-
variance combination of the lensing power spectrum noise curves from the $TT$,
$TE$, $EE$, $EB$, and $TB$ lensing estimators.

####  III.1.3 Construction of lensed and delensed covariance matrices

We follow a similar approach as [38] to calculate the covariance matrices for
the lensed and delensed CMB spectra and CMB lensing spectrum in the multipole
range $\ell\in[30,20000]$. Each of the lensed and delensed covariance matrices
is a $5n_{\mathrm{bin}}\times 5n_{\mathrm{bin}}$ matrix, since five spectra
are used in the forecasts. We denote each block by
$\mathbb{C}_{\ell_{1}\ell_{2}}^{XY,WZ}$, where $XY$ and $WZ$ can each be one
of $TT$, $TE$, $EE$, $BB$, and $\kappa\kappa$. We summarize the calculation
below, which is discussed in more detail in [38], [47], [49], and [40].

The blocks of the covariance matrices are calculated analytically using the
CLASS delens package [47] (unless otherwise specified). This calculation
accounts for Gaussian and non-Gaussian terms (i.e., diagonal and off-diagonal
elements, respectively), including:

  * •

The Gaussian variance between different CMB spectra, given by

$\mathbb{C}_{\ell_{1}\ell_{2},\mathrm{G}}^{XY,WZ}=\frac{\delta_{\ell_{1}\ell_{2}}f_{\mathrm{sky}}^{-1}}{2\ell_{1}+1}\left[C_{\ell_{1}}^{XW,\mathrm{tot}}C_{\ell_{1}}^{YZ,\mathrm{tot}}+C_{\ell_{1}}^{XZ,\mathrm{tot}}C_{\ell_{1}}^{YW,\mathrm{tot}}\right],$
(1)

where $C_{\ell}^{UV}$ refers to a lensed or delensed CMB power spectrum for
$UV\in\\{TT,TE,EE,BB\\}$, and
$C_{\ell}^{UV,\mathrm{tot}}=C_{\ell}^{UV}+N_{\ell}^{UV}$ is the sum of the
signal and noise spectra (including residual foregrounds for the latter, as
described above in Section III.1.1).

  * •

The Gaussian variance of the CMB lensing power spectrum, given by

$\mathbb{C}_{\ell_{1}\ell_{2},\mathrm{G}}^{\phi\phi,\phi\phi}=\delta_{\ell_{1}\ell_{2}}\frac{2f_{\mathrm{sky}}^{-1}}{2\ell_{1}+1}\left(C_{\ell_{1}}^{\phi\phi}+N_{\ell_{1}}^{\phi\phi}\right)^{2},$
(2)

where $C_{\ell}^{\kappa\kappa}=[\ell(\ell+1)]^{2}C_{\ell}^{\phi\phi}/4$, and
the noise $N_{\ell}^{\kappa\kappa}=[\ell(\ell+1)]^{2}N_{\ell}^{\phi\phi}/4$ on
the CMB lensing power spectrum is described in Section III.1.2.

  * •

The non-Gaussian lensing-induced covariances between different modes of the
CMB spectra (e.g., $TT\times TT$, $TT\times EE$, etc.) given by

$\mathbb{C}_{\ell_{1}\ell_{2},\mathrm{NG}}^{XY,WZ}=\sum_{\ell}\frac{\partial
C_{\ell_{1}}^{XY}}{\partial
C_{\ell}^{\phi\phi}}\frac{2f_{\mathrm{sky}}^{-1}}{2\ell+1}\left(C_{\ell}^{\phi\phi}\right)^{2}\frac{\partial
C_{\ell_{1}}^{WZ}}{\partial C_{\ell}^{\phi\phi}},$ (3)

where $XY$ and $WZ$ are each one of $TT$, $TE$, $EE$.

  * •

The non-Gaussian lensing-induced covariances between $BB$ and any of the CMB
spectra, given by

$\begin{split}\mathbb{C}_{\ell_{1}\ell_{2},\mathrm{NG}}^{XY,BB}&=(1-\delta_{\ell_{1}\ell_{2}})\sum_{\ell}\left[\frac{\partial
C_{\ell_{1}}^{XY}}{\partial
C_{\ell}^{XY,\mathrm{u}}}\mathbb{C}_{\ell\ell,\mathrm{G}}^{XY,EE;\mathrm{u}}\frac{\partial
C_{\ell_{2}}^{BB}}{\partial C_{\ell}^{EE,\mathrm{u}}}\right]\\\
&+\sum_{\ell}\frac{\partial C_{\ell_{1}}^{XY}}{\partial
C_{\ell}^{\phi\phi}}\frac{2f_{\mathrm{sky}}^{-1}}{2\ell+1}\left(C_{\ell}^{\phi\phi}\right)^{2}\frac{\partial
C_{\ell_{1}}^{BB}}{\partial C_{\ell}^{\phi\phi}},\end{split}$ (4)

where

$\mathbb{C}_{\ell\ell,\mathrm{G}}^{XY,WZ;\mathrm{u}}=\frac{f_{\mathrm{sky}}^{-1}}{2\ell+1}\left[C_{\ell}^{XW,\mathrm{u}}C_{\ell}^{YZ,\mathrm{u}}+C_{\ell}^{XZ,\mathrm{u}}C_{\ell}^{YW,\mathrm{u}}\right],$
(5)

for $XY\in\\{TT,TE,EE,BB\\}$, and $C_{\ell}^{UV,\mathrm{u}}$ is the unlensed
CMB spectrum (or, if $UV=BB$, we replace it with the unlensed $EE$ spectrum,
$C_{\ell}^{EE,\mathrm{u}}$).444In [38] and [47], Eq. 4 was used for all the
off-diagonal CMB $\times$ CMB covariances by setting $BB$ to any
$WZ\in\\{TT,TE,EE,BB\\}$. However, [49] claim that, when $XY$ and $WZ$ are
$\in\\{TT,TE,EE\\}$, then the first term in Eq. 4 is cancelled by other
higher-order corrections. This claim seems to be supported by unpublished
preliminary tests with simulations we have seen. Thus we limit $WZ$ to only
$BB$ in Eq. 4 in this work.

  * •

The non-Gaussian covariances between the CMB spectra and the CMB lensing
spectrum,

$\mathbb{C}_{\ell_{1}\ell_{2},\mathrm{NG}}^{\phi\phi,XY}=\frac{2f_{\mathrm{sky}}^{-1}}{2\ell_{1}+1}\left(C_{\ell_{1}}^{\phi\phi}\right)^{2}\frac{\partial
C_{\ell_{2}}^{XY}}{\partial C_{\ell_{1}}^{\phi\phi}}.$ (6)

  * •

The non-Gaussian covariances of the CMB lensing spectrum,
$\kappa\kappa\times\kappa\kappa$, for small scales ($\ell_{1},\ell_{2}>5000$)
given by the simulation-based covariance matrix from [40].

We additionally calculate the lensing covariance from modes that are larger
than the survey size (i.e. super-sample covariance (SSC)) [50, 51]. This SSC
term is given by [50, 51]

$\mathbb{C}_{\ell_{1}\ell_{2},\mathrm{SSC}}^{XY,WZ}=\frac{\partial\ell_{1}^{2}C_{\ell_{1}}^{XY}}{\partial\ln\ell_{1}}\frac{\sigma_{\kappa}^{2}}{\ell_{1}^{2}\ell_{2}^{2}}\frac{\partial\ell_{2}^{2}C_{\ell_{2}}^{WZ}}{\partial\ln\ell_{2}},$

where $XY$ and $WZ$ are each one of $\\{TT,TE,EE,BB\\}$. Here
$\sigma_{\kappa}^{2}$ is the variance of the lensing convergence field within
the observed patch of sky given by

$\sigma_{\kappa}^{2}=\frac{1}{A^{2}}\sum_{\ell m}\left|W_{\ell
m}\right|^{2}C_{\ell}^{\kappa\kappa},$

where $W_{\ell m}$ is the harmonic transform of the window function, and $A$
is the sky area observed in radians. We find that including this SSC term in
our covariance matrices does not change any parameter forecasts due to the
large sky area of CMB-HD ($f_{\mathrm{sky}}=60\%$). Thus, we do not include it
in our final covariance matrices, as also done in [38].

The total covariance for the CMB $\times$ CMB blocks is then the sum of Eq. 1
and either 3 or 4,
$\mathbb{C}_{\ell_{1}\ell_{2}}^{XY,WZ}=\mathbb{C}_{\ell_{1}\ell_{2},\mathrm{G}}^{XY,WZ}+\mathbb{C}_{\ell_{1}\ell_{2},\mathrm{NG}}^{XY,WZ}$.
We assemble the full five-by-five block covariance matrix by joining the
CMB$\times$CMB blocks with the CMB$\times$lensing (Eq. 6) and
lensing$\times$lensing blocks (Eq. 2 plus the simulation-based covariance
matrix from [40]).

We also compute a separate covariance matrix for the lensed and delensed $TT$
spectrum in the multipole range $\ell\in[20000,40000]$. We assume this
covariance matrix is diagonal, which is a good approximation given the large
noise levels on these small scales.555We confirm that this is true by
calculating the lensing SNR for $TT$ in the adjacent multipole range of
$\ell\in[15000,20000]$ using either the full covariance matrix (i.e. including
off-diagonal elements) or a diagonal-only covariance matrix, and find that the
SNRs are the same in both cases. We use Eq. 1 to calculate these diagonal
elements. For the SNRs presented in Section IV, we use this diagonal
covariance matrix to calculate the SNRs for $TT$ in the range
$20,000<\ell<40,000$, and add the resulting ratios in quadrature to the ratios
calculated from all spectra in the range $30\leavevmode\nobreak\
<\leavevmode\nobreak\ \ell\leavevmode\nobreak\ <\leavevmode\nobreak\ 20,000$.
We describe in Section III.7 how we include the covariance matrix for $TT$ on
scales $20,000<\ell<40,000$ in our parameter forecasts.

###  III.2 DESI Covariance Matrix

To forecast parameters from the combination of CMB-HD and DESI BAO, we also
estimate the DESI covariance matrix. We simulate a DESI dataset of BAO
measurements taken over 14,000 square degrees of the sky from the baseline
galaxy survey ($z\in[0.65,1.85]$) and bright galaxy survey ($z\in[0.05,0.45]$)
[30]. The BAO data consists of the distance ratio measurements
$r_{\mathrm{s}}/d_{V}(z)$ at redshift $z$, where $r_{\mathrm{s}}$ is the
comoving sound horizon at the end of the baryon drag epoch and the combined
distance measurement $d_{V}(z)$ is defined as [52]

$d_{V}(z)\equiv\left[D_{M}(z)^{2}\frac{cz}{H(z)}\right]^{1/3}.$ (7)

Here $D_{M}(z)=(1+z)d_{A}(z)$, $d_{A}(z)$ is the angular diameter distance to
redshift $z$, and $H(z)$ is the expansion rate at redshift $z$. We use CAMB to
calculate the theoretical $r_{\mathrm{s}}/d_{V}(z_{i})$ at each redshift
$z_{i}$. We follow the same approach as in [38] to form the covariance matrix
for the DESI BAO data, by assuming it is diagonal and propagating the
fractional errors on $r_{\mathrm{s}}/d_{A}(z_{i})$ and
$r_{\mathrm{s}}H(z_{i})$ forecasted by [30] into an uncertainty $\sigma_{i}$
on $r_{\mathrm{s}}/d_{V}(z_{i})$ at each redshift $z_{i}$. The diagonal
elements of the covariance matrix are then taken to be $\sigma_{i}^{2}$.

###  III.3 Calculating CMB and CMB Lensing Signal Spectra

While CAMB [53, 54] can calculate the lensed CMB and the CMB lensing power
spectra in a $\Lambda$CDM model (with the option to include the effects of
baryonic feedback using the HMcode2020 model [55]), it does not calculate
these spectra for an arbitrary non-CDM model that changes the non-linear
matter power spectrum. Thus, we calculate the CMB lensing power spectrum
outside of CAMB as described below, and then input this into CAMB to get the
lensed and delensed CMB power spectra. We show in Appendix A that for a
$\Lambda$CDM model, our calculation agrees with the CAMB-only calculation to
within 0.3% for $\ell\leq 20,000$, and to within 0.8% for $\ell\leq 40,000$.

To calculate the lensing power spectrum we follow [3] and use the Limber
approximation to obtain

$C_{\ell}^{\phi\phi}=4\int_{0}^{\chi_{s}}d\chi\left(\frac{\chi_{s}-\chi}{\chi^{2}\chi_{s}}\right)^{2}P_{\Psi}\left(k=\frac{\ell+1/2}{\chi},z(\chi)\right).$
(8)

Here $k\approx(\ell+1/2)/\chi(z)$, where $k$ is the comoving wavenumber in
Mpc-1,666Note that we plot the comoving wavenumber $k$ in units of $h$ Mpc-1
in the figures throughout this work as is standard. and $\chi(z)$ is the
comoving radial distance to redshift $z$ in Mpc [3, 56].
$\chi_{s}=\chi(z_{s})$ is the comoving distance to the last scattering surface
at redshift $z_{s}\approx 1100$. The CMB lensing convergence power spectrum
$C_{\ell}^{\kappa\kappa}$ is related to this by

$C_{\ell}^{\kappa\kappa}=\frac{[\ell(\ell+1)]^{2}}{4}C_{\ell}^{\phi\phi}.$ (9)

The power spectrum $P_{\Psi}(k,z)$ of the three-dimensional gravitational
potential $\Psi(\mathbf{k},z)$ is related to the non-linear matter power
spectrum $P_{\mathrm{m}}(k,z)$ by [56, 57, 3]

$P_{\Psi}(k,z)=\left(\frac{3\Omega_{\mathrm{m}}H_{0}^{2}}{2c^{2}}\right)^{2}\frac{(1+z)^{2}}{k^{4}}P_{\mathrm{m}}(k,z),$
(10)

where $\Omega_{\mathrm{m}}$ and $H_{0}$ are the matter density and Hubble rate
today. Thus, $C_{\ell}^{\phi\phi}$ can also be expressed as an integral over
$P_{\mathrm{m}}(k,z)$.

To calculate the CMB lensing power spectrum for a non-CDM model, we apply a
transfer function to the non-linear matter power spectrum, defined as

$T^{2}_{\mathrm{non-lin}}(k,z)=\frac{P_{\mathrm{m}}^{\mathrm{Non-
CDM}}(k,z)}{P_{\mathrm{m}}^{\mathrm{CDM}}(k,z)}.$ (11)

Given the relationship between $P_{\Psi}$ and $P_{\mathrm{m}}$ in Eq. 10, this
is equivalent to changing $P_{\Psi}(k,z)$ to
$T^{2}(k,z)P_{\Psi}^{\mathrm{CDM}}(k,z)$ in the integral of Eq. 8. We use CAMB
to calculate $P_{\Psi}^{\mathrm{CDM}}(k,z)$ with the accuracy settings
described in [38].777We use the 2016 version of HMCode [58] to calculate the
non-linear $P_{\Psi}^{\mathrm{CDM}}(k,z)$, and use the single-parameter
baryonic feedback model of HMCode2020 [55] to calculate
$P_{\Psi}^{\mathrm{CDM+baryons}}(k,z)$ when including baryonic feedback. For
consistency we calculate $C_{\ell}^{\phi\phi}$ from Eq. 8 for all models,
including $\Lambda$CDM and $\Lambda$CDM plus baryonic feedback.

After we obtain $C_{\ell}^{\kappa\kappa}$, we provide it to CAMB so that CAMB
can calculate the lensed $TT$, $TE$, $EE$, and $BB$ CMB power spectra from the
corresponding unlensed spectra. The delensed power spectra are calculated in
the same way as the lensed power spectra, except we replace
$C_{\ell}^{\kappa\kappa}$ with the residual lensing power spectrum
$C_{\ell}^{\kappa\kappa,\mathrm{res}}$, given by [59, 38]

$C_{\ell}^{\kappa\kappa,\mathrm{res}}=C_{\ell}^{\kappa\kappa}\left(1-\frac{C_{\ell}^{\kappa\kappa}}{C_{\ell}^{\kappa\kappa}+N_{\ell}^{\kappa\kappa}}\right).$
(12)

Here, $N_{\ell}^{\kappa\kappa}$ is the expected noise on the lensing power
spectrum discussed above.

###  III.4 Calculating Lensing Signal-to-Noise Ratios

In order to quantify how well CMB-HD will be able to measure the gravitational
lensing of the CMB, we use the SNR statistic defined as

$\frac{S}{N}=\sqrt{\sum_{ij}\left(\Delta
C_{\ell}\right)_{i}\mathbb{C}^{-1}_{ij}\left(\Delta C_{\ell}\right)_{j}}.$
(13)

Here $\Delta C_{\ell}$ is a one-dimensional vector holding the difference
between the two sets of model power spectra being considered, in this case
lensed and unlensed spectra (with the unlensed $BB$ and $\kappa\kappa$ spectra
set to zero). Each spectrum is binned into $n_{\mathrm{bin}}$ bins, and the
elements of $\Delta C_{\ell}$ are $\Delta C_{\ell}=(\Delta
C_{\ell_{1}}^{TT},\dots,\Delta C_{\ell_{n_{\mathrm{bin}}}}^{TT},\Delta
C_{\ell_{1}}^{TE},\dots,\Delta C_{\ell_{n_{\mathrm{bin}}}}^{\kappa\kappa})$,
in the order $TT$, $TE$, $EE$, $BB$, $\kappa\kappa$, where $\ell_{b}$ is the
center of the $b^{\mathrm{th}}$ bin. The sum in Eq. 13 is taken over bin
indices $i$ and $j$ from $1$ to $5n_{\mathrm{bin}}$. $\mathbb{C}$ is the
$5n_{\mathrm{bin}}\times 5n_{\mathrm{bin}}$ covariance matrix for the lensed
spectra, with $n_{\mathrm{bin}}\times n_{\mathrm{bin}}$ blocks of
$\mathbb{C}_{\ell_{b}\ell_{b^{\prime}}}^{XY,WZ}$, where $XY$ and $WZ$ can each
be one of $TT$, $TE$, $EE$, $BB$, or $\kappa\kappa$. The blocks are arranged
in the same order as the spectra. This SNR statistic can also be applied to an
individual spectrum $XY\in\\{TT,TE,EE,BB,\kappa\kappa\\}$ using the difference
vector $\Delta C_{\ell}^{XY}$ of length $n_{\mathrm{bin}}$ and the
$n_{\mathrm{bin}}\times n_{\mathrm{bin}}$ covariance matrix block
$\mathbb{C}^{XY,XY}$.

####  III.4.1 Lensing Signal-to-Noise Ratio per $k$ Mode

We follow the procedure discussed above, using Eq. 13 and the covariance
matrix described in Section III.1, to calculate the SNRs given in Table 3 in
Section IV.

However, for visualization purposes, it is useful to see what comoving
wavenumber $k$ in units of $h$ Mpc-1 the CMB-HD lensing signal is sensitive to
(as we show in Figures 2 and 6). Below, we summarize how we approximate this,
giving full details in Appendix B.

The lensing signal in the CMB maps provides a measurement of the matter
distribution, integrated along the line of sight over all redshifts from
recombination until today. As discussed in Section III.3, we use the
relationship $k\approx(\ell+1/2)/\chi(z)$ to translate the lensing measured
from the CMB power spectra on a given angular scale (multipole $\ell$) to a
measurement of the matter distribution on a given physical scale (wavenumber
$k$) at a given redshift $z$. Therefore, to approximate the lensing SNR from a
given wavenumber bin, we first need to calculate the SNR per multipole in each
redshift bin.

We calculate the contribution to the lensing signal from a given redshift bin
of width $\Delta z=0.5$, for redshifts $z\in[0,6]$.888While the lensing signal
is from all redshifts $z\in$ [0, 1100], we only go to $z=6$ in Figures 2 and 6
to keep our redshift bins small and ease the conversion from $\ell$ to $k$ at
a given redshift. We note that the range $z\in[6,1100]$ contributes only 15%
extra lensing signal. The $i^{\mathrm{th}}$ redshift bin ranges from
$z_{i,\mathrm{min}}$ to $z_{i,\mathrm{max}}$, centered at $z_{i}$. We
integrate Eq. 8 from $\chi(z_{i,\mathrm{min}})$ to $\chi(z_{i,\mathrm{max}})$
and then use Eq. 9 to calculate $C_{\ell}^{\kappa\kappa,z_{i}}$, which
contains the lensing signal from only the $i^{\mathrm{th}}$ redshift bin. We
use CAMB to obtain the lensed CMB spectra $C_{\ell}^{UV,z_{i}}$, for
$UV\in\\{TT,TE,EE,BB\\}$, that were lensed by this
$C_{\ell}^{\kappa\kappa,z_{i}}$. We then use the set of $C_{\ell}^{XY,z_{i}}$,
for $XY\in\\{TT,TE,EE,BB,\kappa\kappa\\}$, in Eqs. 1 and 2 to calculate the
diagonal elements of the covariance matrices for the five spectra,
$\left(\sigma_{\ell}^{XY,z_{i}}\right)^{2}$. In Eqs. 1 and 2, $N_{\ell}^{XY}$
is the noise on the power spectrum, which we assume is the same for each
redshift bin. The lensing SNR per multipole $\ell$ from the $i^{\mathrm{th}}$
redshift bin is then defined as

$\left(\frac{S}{N}\right)^{z_{i}}_{\ell}=\sqrt{\sum_{XY}\left[\frac{\Delta
C_{\ell}^{XY,z_{i}}}{\sigma_{\ell}^{XY,z_{i}}}\right]^{2}},$ (14)

where $\Delta C_{\ell}^{XY,z_{i}}=C_{\ell}^{XY,z_{i}}-\tilde{C}_{\ell}^{XY}$
is the difference between the lensed and unlensed spectra, and the sum is
taken over $XY\in\\{TT,TE,EE,BB,\kappa\kappa\\}$.

We then consider wavenumber $k$ bins, uniformly spaced in $\log_{10}(k)$. For
a given wavenumber bin $j$ ranging from $k_{j,\mathrm{min}}$ to
$k_{j,\mathrm{max}}$ and centered at $k_{j}$, we use the approximation
$k\approx(\ell+1/2)/\chi(z)$ to find the multipole range that corresponds to
that $j$ bin for each redshift bin $i$. Thus we calculate
$\ell_{\mathrm{min}}(z_{i},k_{j})=\chi(z_{i})k_{j,\mathrm{min}}-1/2$ and
$\ell_{\mathrm{max}}(z_{i},k_{j})=\chi(z_{i})k_{j,\mathrm{max}}-1/2$. We then
calculate the SNR within the $j^{\mathrm{th}}$ wavenumber bin and
$i^{\mathrm{th}}$ redshift bin as

$\left(\frac{S}{N}\right)^{z_{i}}_{k_{j}}=\sqrt{\sum_{\ell=\ell_{\mathrm{min}}(z_{i},k_{j})}^{\ell_{\mathrm{max}}(z_{i},k_{j})}\left[\left(\frac{S}{N}\right)^{z_{i}}_{\ell}\right]^{2}},$
(15)

adding the SNRs in quadrature. We sum over redshift bins to estimate the total
SNR within the $j^{\mathrm{th}}$ wavenumber bin, by

$\left(\frac{S}{N}\right)_{k_{j}}=\sqrt{\sum_{i}\left[\left(\frac{S}{N}\right)^{z_{i}}_{k_{j}}\right]^{2}}.$
(16)

While Eq. 16 underestimates the noise by neglecting the off-diagonal elements
of the covariance matrix, it also underestimates the signal; when summing over
redshift bins by adding in quadrature, the lensing cross-terms between
redshift bins are neglected in the squared signal (see Appendix B). These two
effects coincidentally nearly cancel, yielding a reasonable approximation to
the SNR. For example, using Eq. 13 to calculate the lensing SNR from all the
spectra yields 1947 (see Table 3 in Section IV), while using Eq. 14 and
summing over all redshifts (to $z=1100$) and multipoles gives 1798, which is
only slightly lower.999When using only $z\in[0,6]$ as shown in Figure 6, Eq.
13 yields 1632, while Eq. 14 yields 1585 after summing over redshifts and
multipoles, as shown in Table 6 in Appendix B.

###  III.5 Including non-CDM Models

Models that are not purely CDM, but that have varying properties of the dark
matter particle or include baryonic effects, yield non-linear matter power
spectra that differ from that of CDM on small scales. We include this effect
by calculating the non-linear transfer function given by Eq. 11. While our
formalism (and the public code we provide) allow for any non-linear transfer
function, there are relatively few provided in the literature for alternate
dark matter models.

A non-linear transfer function for warm dark matter (WDM) is provided by [60]
as

$T^{2}_{\mathrm{non-lin}}(k,z)=\left[1+(\alpha k)^{\nu l}\right]^{-s/\nu},$
(17)

where $\nu=3$, $l=0.6$, $s=0.4$, and $\alpha$ is given by

$\alpha(m_{\mathrm{WDM}},z)=0.0476\left(\frac{1\leavevmode\nobreak\
\mathrm{keV}}{m_{\mathrm{WDM}}}\right)^{1.85}\left(\frac{1+z}{2}\right)^{1.3}.$
(18)

However, [60] find this fitting function to be accurate only for redshifts
$z<3$ from comparison with simulations.

Since we need to integrate over the matter power spectrum all the way back to
recombination in Eq. 8, we need a transfer function that extends to higher
redshifts. Thus we calculate a non-linear transfer function from the
Warm&Fuzzy code [31], which extends to all redshifts. We obtain
$P_{\mathrm{m}}^{\mathrm{CDM}}(k,z)$ and $P_{\mathrm{m}}^{\mathrm{WDM}}(k,z)$
for various WDM masses ranging from 1 to 10 keV keeping the other cosmological
parameters fixed.101010When calculating the WDM transfer function with the
Warm&Fuzzy code, we fix the cosmological parameters to
$\Omega_{\mathrm{m}}=0.31377$, $\Omega_{\mathrm{b}}=0.04930$, $h=0.6736$,
$n_{\mathrm{s}}=0.9649$ and $\sigma_{8}=0.811$, from [13] and also given in
Table 2. We then apply the computed transfer function to the power spectrum
$P_{\Psi}(k,z)$ from CAMB to obtain $C_{\ell}^{\kappa\kappa}$ from Eq. 8, and
to the kSZ power spectrum, described below in Section III.6.

Figure 3: We show transfer functions, defined by Eq. 11, for a 1 keV warm dark
matter (WDM) model. The solid curves show the transfer function for the non-
linear matter power spectrum, calculated from the Warm&Fuzzy code [31] at 15
evenly-spaced redshifts in the range $z\in[0,\leavevmode\nobreak\ 14]$. At
redshifts $z\gtrsim 14$, the non-linear transfer function converges to the
transfer function for the linear matter power spectrum, given by Eq. 19 [61]
and shown as the magenta dotted curve. In this work, we use the WDM transfer
function from Warm&Fuzzy for redshifts $z\in[0,\leavevmode\nobreak\ 14]$, and
the linear transfer function at higher redshifts. For comparison, we also show
the non-linear transfer function from [60], given by Eq. 17, for redshifts in
the range $z\in[0,3]$ (dashed-dotted curves), which is the regime found to
have a good match to simulations [60]. We also show as small-dotted curves an
extension of this model to $z=14$, and note that it does not converge to the
linear transfer function at high $z$ (nor was it expected to be accurate in
this regime). We emphasize that we do not advocate for any particular model in
this work, and acknowledge the complexity of modelling non-linear dynamics and
matching to simulations.

We show the transfer function from [31] for a WDM mass of 1 keV as the solid
curves in Figure 3 for different redshifts. As noted by [31], at high
redshifts (e.g., $z>14$) the non-linear transfer function approaches the
transfer function for the WDM linear matter power spectrum from [61, 62], for
the range of $k$-modes considered in this work. We show this linear transfer
function as the magenta dotted curve in Figure 3, which is given by the
fitting function

$T^{2}_{\mathrm{lin}}(k,z)=\left[1+(\alpha_{\mathrm{lin}}k)^{2\nu_{\mathrm{lin}}}\right]^{-10/\nu_{\mathrm{lin}}},$
(19)

from [61, 62], where $\nu_{\mathrm{lin}}=1.12$ and $\alpha_{\mathrm{lin}}$ is
given by

$\alpha_{\mathrm{lin}}(m_{\mathrm{WDM}})=0.049\left(\frac{1\leavevmode\nobreak\
\mathrm{keV}}{m_{\mathrm{WDM}}}\right)^{1.11}\left(\frac{\Omega_{\mathrm{WDM}}}{0.25}\right)^{0.11}\left(\frac{0.7}{h}\right)^{1.22}.$
(20)

Here $\alpha_{\mathrm{lin}}$ is in units of $h^{-1}$ Mpc, $k$ is in units of
$h$ Mpc-1, $m_{\mathrm{WDM}}$ is the WDM mass in units of keV,
$\Omega_{\mathrm{WDM}}$ is the WDM density parameter today, and $h$ is the
reduced Hubble constant. Thus, we calculate the non-linear transfer function
at 15 evenly-spaced redshifts in the range $z\in[0,\leavevmode\nobreak\ 14]$
using the Warm&Fuzzy code, and use the linear fit from Eq. 19 at higher
redshifts. We show this in Figure 3 for a 1 keV WDM mass, and also compare to
the non-linear transfer function of [60] given by Eq. 17. We show as dashed-
dotted curves the non-linear transfer function of Eq. 17 for $z\leq 3$, where
[60] claims agreement with simulations at the 2% level. We also show as faint
dotted curves this transfer function for $3<z\leq 14$, which does not converge
to the linear transfer function of Eq. 19, but is also not expected to be
accurate in this regime.

In general, the two transfer functions are in rough agreement for $z<3$.
Figure 13 in [63], shows the non-linear transfer function from simulations at
$z=0$ for a 1 keV WDM model, and compares that to the transfer function of
[60]. From simulations, [63] find a 2% suppression in power at $k=10h$ Mpc-1,
whereas the transfer function from [60] predicts 0.7% and the one from [31]
predicts 3.7%. We therefore emphasize how sensitive these transfer functions
are to details of modelling. For example, for 5 keV WDM, at $z=0.5$ and
$k=10h$ Mpc-1, the suppression predicted by [60] is 0.01% and by [31] is 0.1%.
However, [60] only claims a 2% agreement with simulations.

Figure 4: We show the fractional change in the power spectra, $\Delta
C_{\ell}/C_{\ell}$, for a 1 keV WDM model versus a CDM-only model (red and
orange) and for a CDM+baryons model versus a CDM-only model (blue and green).
The spectra shown are the CMB temperature spectrum ($C_{\ell}^{TT}$, shown as
solid curves) and the CMB lensing spectrum ($C_{\ell}^{\kappa\kappa}$, shown
as dashed curves). The $C_{\ell}^{TT}$ spectra only include the effect of CMB
lensing, not the kSZ effect; however, the matter power spectrum would show a
similar behavior via the kSZ spectrum. We also plot the CMB-HD error bars on
$\Delta C_{\ell}^{TT}/C_{\ell}^{TT}$, which we place on the CDM-only model.
While both WDM and CDM+baryons models result in a suppression of power
relative to a CDM-only model, the shape of the suppression differs for the two
models, allowing CMB-HD to distinguish between them.

We view the transfer functions discussed above as illustrative of the effects
of alternate dark matter models, as opposed to definitive, and all WDM
constraints presented below should be viewed with the understanding that they
can change with updated transfer functions. We also make public the code used
for the calculations throughout this work111111https://github.com/CMB-HD/hdPk,
which can be easily modified to use another non-linear transfer function in
place of the one described above.

To include the effects of baryonic feedback on the non-linear matter
distribution, we use the single-parameter baryonic feedback model of
HMCode2020 [55] within CAMB to calculate the non-linear power spectrum,
$P_{\Psi}^{\mathrm{CDM+baryons}}(k,z)$, directly (instead of applying a
transfer function). This single-parameter model connects the various feedback
parameters of hydrodynamic simulations to a single “AGN temperature”
parameter, ${\rm{log_{10}}}(T_{\rm{AGN}}/{\rm{K}})$, which characterizes the
overall strength of the feedback in the simulations. We then follow the
procedure described in Section III.3 to calculate the power spectra with this
model.

We show in Figure 4 the change in the $TT$ or lensing $\kappa\kappa$ power
spectrum from a 1 keV WDM or a CDM+baryonic feedback model, relative to a CDM-
only model. For the $TT$ spectrum, we only show the change due to the
difference in the lensing effect; the change in the kSZ effect would show a
similar behavior. We see both WDM and CDM+baryonic feedback models result in a
suppression of power relative to a CDM-only model, however, the shape of the
suppression differs considerably between the two models.

###  III.6 Modelling the kinetic Sunyaev-Zel’dovich Effect

Since the kSZ effect is frequency-independent, it is harder to separate from
the lensed CMB. Thus we include the kSZ power spectrum in our analysis, and
assume that it will need to be fit for along with the other cosmological
parameters.121212Parameters for frequency-dependent residual foreground levels
will also need to be fit for in a full power spectrum analysis, however, we
assume with the seven frequency channels of CMB-HD, those other parameters
will be well constrained. We do not include measurements of the kSZ
trispectrum in this work, but note that such measurements are an additional
way to separate the kSZ signal from the lensed CMB, and would yield additional
constraining power [19].

To model the kSZ power spectrum, we use a template for the late-time kSZ power
spectrum from [41]131313The results are similar if we instead use the late-
time kSZ template from [64]. and for the reionization contribution from [42,
43]. We extend these templates to $\ell_{\mathrm{max}}=40,000$ using a linear
fit to each template in the tail end of their existing range. We follow the
approach of [65] by normalizing the full kSZ spectrum
$D_{\ell}^{\mathrm{kSZ}}\equiv\ell(\ell+1)C_{\ell}^{\mathrm{kSZ}}/2\pi$ to
equal 1 $\mu$K2 at $\ell=3000$. Denoting the normalized template by
$C_{\ell}^{\mathrm{kSZ},0}$, we model the kSZ power spectrum in a CDM-only
model by,

$C_{\ell}^{\mathrm{kSZ;CDM}}=\left(\frac{\ell}{\ell_{0}}\right)^{n_{\mathrm{kSZ}}}A_{\mathrm{kSZ}}C_{\ell}^{\mathrm{kSZ},0},$
(21)

where $\ell_{0}=3000$. This model has two free parameters: $A_{\mathrm{kSZ}}$,
which is the amplitude of the kSZ power spectrum at $\ell_{0}=3000$, and
$n_{\mathrm{kSZ}}$, which is its slope. We use fiducial values of
$A_{\mathrm{kSZ}}$ = 1 and $n_{\mathrm{kSZ}}$ = 0 in this work, and allow
these parameters to be free in the parameter analysis described below. Fitting
with a similar kSZ template in the ACT DR4 power spectrum analysis, [21] found
$A_{\mathrm{kSZ}}<1.8$ at 95% CL. We assume that measurements of the thermal
Sunyaev-Zel’dovich (tSZ) power spectrum will constrain the large-scale shape
of the kSZ power spectrum and $\ell_{0}$ pivot, since tSZ and kSZ trace the
same large-scale structure; thus we do not free $\ell_{0}$.

Figure 5: Top panel: We show the CMB temperature power spectrum,
$C_{\ell}^{TT}$, including both the effects of lensing and the kSZ power
spectra, calculated as described in Sections III.3, III.5, and III.6. We show
$C_{\ell}^{TT}$ with our fiducial kSZ parameters ($A_{\mathrm{kSZ}}=1$ and
$n_{\mathrm{kSZ}}=0$) in a fiducial CDM-only model (blue solid curve) and in a
5 keV WDM model (pink solid curve). We also show $C_{\ell}^{TT}$ for a CDM-
only model when we vary the kSZ slope (i.e. $n_{\mathrm{kSZ}}=\pm 0.001$;
green and purple dashed curves). The different $C_{\ell}^{TT}$ are
indistinguishable by eye. Bottom panel: We show the fractional change in the
$C_{\ell}^{TT}$ spectra described above, relative to the fiducial model. We
see that changing the slope of the kSZ power spectrum in a CDM-only model does
not reproduce the suppression due to WDM. We also show the forecasted CMB-HD
error bars on $\Delta C_{\ell}^{TT}/C_{\ell}^{TT}$ (in red).

To model the kSZ power spectrum for WDM, we approximate the suppression of the
kSZ power spectrum by applying the WDM transfer function defined in Eq. 11 and
discussed in Section III.5 to the kSZ power. We evaluate this transfer
function at a fixed redshift $z_{0}=0.5$ to simplify the analysis, assuming
the late-time kSZ dominates the kSZ spectrum. We relate the comoving
wavenumber $k$ to a multipole $\ell$ with the approximation
$k\approx(\ell+1/2)/\chi(z_{0})$. For each WDM mass we define
$T^{2}_{\ell}=T^{2}_{\mathrm{non-
lin}}\bigl{(}k\approx(\ell+1/2)/\chi(z_{0})\bigr{)}$ and obtain the kSZ power
spectrum in a WDM-only model by,

$C_{\ell}^{\mathrm{kSZ;WDM}}=T^{2}_{\ell}C_{\ell}^{\mathrm{kSZ;CDM}}.$ (22)

We note that assuming all the kSZ signal is from the relatively low-redshift
of $z_{0}=0.5$ is conservative. The kSZ signal from higher redshifts, such as
the contribution from reionization, will experience a larger suppression in
power due to WDM, as can be seen from Figure 3.

We do not model the impact of baryonic feedback for the kSZ power spectrum,
even though the change in the matter power spectrum due to baryonic effects
will impact it. This again is a choice to simplify the analysis, and is made
because baryonic feedback already has such a large impact on the lensing
signal in the modest multipole range of $\ell\sim 10,000$ where CMB-HD
measurements are very constraining, as shown in Figure 4. Thus, baryonic
feedback parameters are already constrained well enough to not be degenerate
with other parameters (see Section IV.2 below). In addition, tSZ power
spectrum measurements will further inform the level of baryonic feedback.
However, fully modelling the kSZ effect, incorporating both a redshift-
dependent non-CDM transfer function and baryonic effects, is one area where
the modelling presented here could be extended.

We show in the top panel of Figure 5 the CMB temperature power spectrum,
$C_{\ell}^{TT}$, including both the effect of lensing and the kSZ power
spectrum. We show this with the fiducial kSZ spectrum given by Eq. 21 for a
CDM-only model with the kSZ parameters $n_{\mathrm{kSZ}}=0$ and
$A_{\mathrm{kSZ}}=1$ (blue solid curve). We also show this for kSZ spectra
with $n_{\mathrm{kSZ}}=\pm 0.001$ (purple and green dashed curves). We compare
this to $C_{\ell}^{TT}$ for a 5 keV WDM-only model with the kSZ spectrum given
by Eq. 22 and with fiducial kSZ parameters ($n_{\mathrm{kSZ}}=0$ and
$A_{\mathrm{kSZ}}=1$); the WDM model modifies both the lensing and kSZ in this
$C_{\ell}^{TT}$ (pink solid curve). Since these spectra are indistinguishable
by eye, in the bottom panel we show the fractional difference between these
different $C_{\ell}^{TT}$ and our fiducial $C_{\ell}^{TT}$, along with the
forecasted CMB-HD error bars on $\Delta C_{\ell}^{TT}/C_{\ell}^{TT}$ (in red).
We see that the suppression due to WDM cannot be easily mimicked by varying
the slope of the kSZ power spectrum.

###  III.7 Calculating the Fisher Matrix

We construct a Fisher matrix to forecast parameter constraints for CMB-HD.
While we do not use a Markov chain Monte Carlo method (MCMC) in this work,
which could be more accurate but also more time consuming, in [38] we verified
that our Fisher forecasts were consistent with the results obtained from an
MCMC method. To calculate the Fisher matrix, we vary the parameters discussed
below.

For the CDM-only model, we vary the six $\Lambda$CDM parameters: the physical
density parameters for baryons, $\Omega_{\mathrm{b}}h^{2}$, and for cold dark
matter (CDM), $\Omega_{\mathrm{c}}h^{2}$; the amplitude of the comoving
primordial curvature power spectrum via the parameter
$\ln\left(10^{10}A_{\mathrm{s}}\right)$, and the scalar spectral index
$n_{\mathrm{s}}$, defined at the pivot scale $k_{0}=0.05$ Mpc-1; the optical
depth to reionization, $\tau$; and the
cosmoMC141414https://cosmologist.info/cosmomc/readme.html approximation to the
angular sound horizon at recombination, $100\theta_{\mathrm{MC}}$. We also
vary the effective number of relativistic species $N_{\mathrm{eff}}$, the sum
of the neutrino masses $\sum m_{\nu}$, and the amplitude and slope of the kSZ
power spectrum, $A_{\mathrm{kSZ}}$ and $n_{\mathrm{kSZ}}$ (see Section III.6).
We refer to this as a $\Lambda$CDM + $N_{\mathrm{eff}}$ \+ $\sum m_{\nu}$ \+
$A_{\mathrm{kSZ}}$ \+ $n_{\mathrm{kSZ}}$ model. We adopt the Planck baseline
cosmological parameter constraints as our fiducial model [13].

When we consider the case where all of the dark matter is warm, rather than
cold, (the WDM-only model), we vary the ten parameters listed above, now with
the CDM density parameter $\Omega_{\mathrm{c}}h^{2}$ replaced by the WDM
density parameter $\Omega_{\mathrm{w}}h^{2}$, and additionally vary the mass
of the WDM particle, $m_{\mathrm{WDM}}$. We refer to this eleven-parameter
model as a $\Lambda$WDM + $N_{\mathrm{eff}}$ \+ $\sum m_{\nu}$ \+
$m_{\mathrm{WDM}}$ \+ $A_{\mathrm{kSZ}}$ \+ $n_{\mathrm{kSZ}}$ model. In the
figures and tables in Section IV, we use $\Omega_{\mathrm{d}}h^{2}$ as a
general dark matter density parameter, to indicate either
$\Omega_{\mathrm{c}}h^{2}$ or $\Omega_{\mathrm{w}}h^{2}$.

In both of the cases above, we also consider the effects of baryonic feedback,
and vary the feedback parameter $\log_{10}(T_{\mathrm{AGN}}/\mathrm{K})$ of
[55]. For a model with baryonic feedback, we also include a 0.06% prior on
$\log_{10}(T_{\mathrm{AGN}}/\mathrm{K})$, which is expected from tSZ
measurements and was also included in [38]. In Table 2, we list the set of
varied parameters, their fiducial values, and the step sizes used in the
Fisher matrix calculation described below. All forecasts include a prior on
the optical depth of $\sigma(\tau)=0.007$ from Planck [13].

Parameter | Fiducial value | Step size
---|---|---
$\Omega_{\mathrm{b}}h^{2}$ |  $0.02237$ | 1%
$\Omega_{\mathrm{d}}h^{2}$ |  $0.1200$ | 1%
$\ln(10^{10}A_{\mathrm{s}})$ |  $3.044$ | 0.3%151515The step size of 0.3% for $\ln(10^{10}A_{\mathrm{s}})$ corresponds to an approximate step size of 1% on $A_{\mathrm{s}}$.
$n_{\mathrm{s}}$ |  $0.9649$ | 1%
$\tau$ |  $0.0544$ | 5%
$100\theta_{\mathrm{MC}}$ |  $1.04092$ | 1%
$N_{\mathrm{eff}}$ |  $3.046$ | 5%
$\sum m_{\nu}$ [eV] |  $0.06$ | 10%
$\log_{10}\left(T_{\mathrm{AGN}}/\mathrm{K}\right)$ | 7.8 | 0.05
$m_{\mathrm{WDM}}$ [keV] | [1, 10] | 10%
$A_{\mathrm{kSZ}}$ | 1 | 0.1
$n_{\mathrm{kSZ}}$ | 0 | 0.01
Table 2: We list the cosmological parameters used in the Fisher forecasts,
their fiducial values, and the step sizes used when varying the parameter
values to compute the Fisher matrix. The fiducial values for the first six
rows are taken from the Planck baseline cosmological parameters [13], and
their step sizes are taken as a fraction of their fiducial values. The
fiducial value of the baryonic feedback parameter
$\log_{10}\left(T_{\mathrm{AGN}}/\mathrm{K}\right)$ is taken from [55]. We
also list the range of values of the warm dark matter mass,
$m_{\mathrm{WDM}}$, that we consider, and two parameters giving flexibility to
the kSZ power spectrum ($A_{\mathrm{kSZ}}$ and $n_{\mathrm{kSZ}}$). The step
sizes for the baryonic feedback and kSZ parameters are absolute (i.e., not a
relative fraction of their fiducial value). Note that we use
$\Omega_{\mathrm{d}}h^{2}$ to refer to the physical dark matter density
parameter, for either cold or warm dark matter.

To form a Fisher matrix for the CMB or BAO data, we assume a Gaussian
likelihood function $\mathcal{L}(\hat{\vec{d}}|\vec{\theta})$ for the data
vector $\hat{\vec{d}}$ given a set of theoretical model parameters
$\vec{\theta}$,

$-2\ln\mathcal{L}\left(\hat{\vec{d}}|\vec{\theta}\right)=\sum_{ij}\Delta
d_{i}(\vec{\theta})\mathbb{C}^{-1}_{ij}\Delta d_{j}(\vec{\theta}),$ (23)

where $\Delta d_{i}(\vec{\theta})$ is the $i^{\mathrm{th}}$ element of
$\Delta\vec{d}(\vec{\theta})=\hat{\vec{d}}-\vec{d}(\vec{\theta})$,
$\vec{d}(\vec{\theta})$ is the theoretical model of the data as a function of
the parameters $\vec{\theta}$, and $\mathbb{C}^{-1}_{ij}$ is the $(i,j)$
element of the inverse covariance matrix for the data.

For CMB-HD, the data vector $\hat{\vec{d}}$ is the set of binned CMB spectra
(which could be lensed or delensed and includes the kSZ spectra) and the CMB
lensing power spectrum. The covariance matrix is the binned covariance given
by Eqs. 1 to 6, and the sum is taken over the bin centers. Each binned power
spectrum is given by

$C_{\ell_{b}}^{XY}=\sum_{\ell}M_{b\ell}C_{\ell}^{XY},$ (24)

and the binned blocks of the covariance matrix are given by

$\mathbb{C}_{\ell_{b}\ell_{b}^{\prime}}^{XY,WZ}=\sum_{\ell\ell^{\prime}}M_{b\ell}\mathbb{C}_{\ell\ell^{\prime}}^{XY,WZ}M_{b^{\prime}\ell^{\prime}}^{T},$
(25)

where $M_{b\ell}$ is the binning matrix, $\ell_{b}$ represents the bin center
of the $b^{\mathrm{th}}$ bin, and $XY$ and $WZ$ can each be one of
$\\{TT,TE,EE,BB,\kappa\kappa\\}$.

For DESI, the data vector is the set of distance ratio measurements
$r_{s}/d_{V}(z_{j})$, with a covariance matrix given by
$\mathbb{C}_{ij}=\delta_{ij}\sigma_{i}^{2}$ as discussed in Section III.2, and
the sum is taken over the redshifts $z_{i}$.

The Fisher matrix $F$ describes how the likelihood changes when the parameters
$\vec{\theta}$ are varied. Assuming that the likelihood is maximized for some
fiducial set of parameters $\vec{\theta}_{0}$ and Taylor expanding about this
point, the elements of the Fisher matrix are given by

$\begin{split}F_{\alpha\beta}&\equiv\left.-\left\langle\frac{\partial^{2}\ln\mathcal{L}}{\partial\theta_{\alpha}\partial\theta_{\beta}}\right\rangle\right|_{\vec{\theta}_{0}}\\\
&=\sum_{ij}\left.\left(\frac{\partial
d_{i}}{\partial\theta_{\alpha}}\mathbb{C}_{ij}^{-1}\frac{\partial
d_{j}}{\partial\theta_{\beta}}\right)\right|_{\vec{\theta}_{0}}.\end{split}$
(26)

Here the indices $\alpha$ and $\beta$ correspond to two parameters in
$\vec{\theta}$. We use CAMB to evaluate the derivatives numerically for the
cosmological and baryonic feedback parameters (first nine rows of Table 2),
varying each parameter up or down by its step size while holding the other
parameters fixed. To calculate the derivatives for $A_{\mathrm{kSZ}}$ and
$n_{\mathrm{kSZ}}$, we similarly vary these parameters by the step sizes given
in Table 2. For the WDM models, we vary the WDM mass in the transfer function
of Eq. 11; this transfer function adjusts both the lensing effect and the kSZ
spectrum. We change our fiducial model for each WDM mass considered (in the
range 1 to 10 keV), keeping all other parameters fixed to their fiducial
values. We then calculate derivatives for $m_{\mathrm{WDM}}$ by varying
$m_{\mathrm{WDM}}$ around that fiducial mass by 10% of its value.161616We have
confirmed that the parameter constraints are stable when the step sizes are
varied slightly. We find a $1\sigma$ uncertainty that is smaller than the step
size for all parameters except $\tau$ and $\sum m_{\nu}$, and have verified
the CDM + baryons Fisher errors are consistent with those from an MCMC in
[38].

The inverse of the Fisher matrix gives the covariance matrix for the
parameters, such that
$\left(F^{-1}\right)_{\alpha\beta}=\mathrm{cov}\left(\theta_{\alpha},\theta_{\beta}\right)$;
the uncertainty $\sigma_{\alpha}$ on the parameter $\theta_{\alpha}$ is found
from its variance, $\sigma_{\alpha}^{2}=\left(F^{-1}\right)_{\alpha\alpha}$.
We apply a prior from Planck [13] on the optical depth $\tau$ of
$\sigma(\tau)=0.007$ by adding its inverse variance $1/\sigma^{2}(\tau)$ to
the element $F_{\tau\tau}^{\mathrm{CMB}}$ of the CMB Fisher matrix.

The Fisher matrix for the combination of CMB-HD and DESI is formed by taking
the sum of their Fisher matrices, $F=F^{\mathrm{CMB}}+F^{\mathrm{BAO}}$. To
calculate $F^{\mathrm{CMB}}$ for CMB-HD, we use Eq. 26 to calculate a Fisher
matrix from the $TT,TE,EE,BB,\kappa\kappa$ power spectra in the range
$\ell\in(30,20000)$ with the (binned) covariance matrix given by Eqs. 1 to 6.
We then calculate a second Fisher matrix from the $TT$ power spectrum in the
range $\ell\in(20000,40000)$, with a diagonal covariance matrix given by Eq.
1. We take the sum of these two Fisher matrices to obtain $F^{\mathrm{CMB}}$.

##  IV Results

###  IV.1 CMB-HD Lensing from $TT$, $TE$, $EE$, $BB$, and $\kappa\kappa$

We show in Table 3 the SNR with which CMB-HD can measure the CMB lensing
signal. As described in Section III.4, we use Eq. 13 where $\Delta C_{\ell}$
is the difference between lensed and unlensed spectra. In the first five rows,
we give the SNR for each of the lensed CMB spectra $XY\in\\{TT,TE,EE,BB\\}$
and the CMB lensing $\kappa\kappa$ spectrum, over the multipole range
$\ell\in[30,20000]$. We find roughly equal contributions to the lensing SNR
from the lensed $TT$ power spectrum and the lensing $\kappa\kappa$ power
spectrum, with the SNR from $TT$ (SNR=1571) being slightly higher than from
$\kappa\kappa$ (SNR=1287). We also find considerable SNR from the lensed $BB$
power spectrum alone (SNR=493). In the sixth row, we show the SNR for $TT$ in
the range $\ell\in(20000,40000]$ (SNR=79); while this is modest compared to
the SNR from lower multipoles, it is still almost double the final Planck
lensing SNR [7]. In the last row of Table 3, we give the total SNR from all
the spectra. We account for correlations between the spectra by using the full
$5n_{\mathrm{bin}}\times 5n_{\mathrm{bin}}$ covariance matrix, described in
Section III.1, to calculate the SNR from all spectra in the range
$\ell\in[30,20000]$. We obtain our final lensing SNR by adding in quadrature
this value to the SNR from $TT$ in the range $\ell\in(20000,40000]$. We find a
total lensing SNR of 1947.

Spectra | Lensing SNR
---|---
$TT$-only $\in(30,20000)$ | 1571
$TE$-only $\in(30,20000)$ | 161
$EE$-only $\in(30,20000)$ | 276
$BB$-only $\in(30,20000)$ | 493
$\kappa\kappa$-only $\in(30,20000)$ | 1287
$TT$-only $\in(20000,40000)$ | 79
All $TT+TE+EE+BB+\kappa\kappa$ | 1947
Table 3: We list the total signal-to-noise ratio (SNR) with which CMB-HD can
measure the CMB lensing signal in a CDM-only model. The first six rows list
the SNR values obtained only from the given spectrum
$XY\in\\{TT,TE,EE,BB,\kappa\kappa\\}$ in the given multipole range. We use Eq.
13 to calculate the SNR, with $\Delta C_{\ell}$ being the difference between
the lensed and unlensed power spectrum. For the first five rows,
$\mathbb{C}^{XY,XY}$ is given by the covariance matrix including non-Gaussian
terms described in Section III. For the sixth row, we approximate the small-
scale TT covariance matrix as diagonal, given by Eq. 1, due to the high noise
level on these scales. The last row gives the total SNR from the combined
spectra listed in the first six rows. We calculate the total SNR from the
first five spectra in the range $\ell\in(30,20000)$ from Eq. 13 using the full
$5\times 5$ block covariance matrix, and then combine this value with the $TT$
SNR from $\ell\in(20000,40000)$ by summing the two values in quadrature.

In Figure 6, we illustrate which comoving scales and redshifts contribute to
the CMB-HD lensing SNR given in the last row of Table 3. From Eq. 8, we see
that lensing from a range of comoving wavenumbers, $k$, integrated over all
redshifts, contribute to lensing at a given multipole, $\ell$. Thus we can
only approximate the contribution to the lensing SNR for each redshift and
wavenumber bin, as described in Section III.4.1 and Appendix B.

In the top panel of Figure 6, we plot a histogram of the lensing SNR per
comoving wavenumber bin, $k$, and further break this down into the
contribution from each redshift bin for $z\in[0,6]$. We see on large scales
that the lensing SNR is derived from relatively high redshifts ($z>2$), while
on smaller scales of $k<1$ $h$ Mpc-1 most of the SNR is from $z<2$. We find
approximate lensing SNRs of SNR $\approx 330$ for $k\gtrsim 5$ $h$ Mpc-1 and
SNR $\approx 85$ for $k\gtrsim 10$ $h$ Mpc-1.

In the bottom panel of Figure 6, we calculate the lensing SNR per multipole
bin of width $\Delta\ell=30$ and redshift bin, using Eq. 14. We plot it at the
wavenumber corresponding to the multipole bin center at that redshift. We
indicate the SNR value by both the height of the bin and by its color, and
also indicate a few multipoles as black dotted lines. We show as the purple
dashed line the transition between linear and non-linear scales at each
redshift, which we derive by finding the wavenumber at which the linear matter
power spectrum starts to be suppressed by more than 1% relative to the non-
linear matter power spectrum in a CDM-only model.

From this figure, we see three peaks in the SNR at different scales. This is
from the contribution to the SNR from different spectra: $\kappa\kappa$ on the
large scales (peak below $\ell=1000$), $BB$ on intermediate scales (peak just
above $\ell=1000$, which is the minimum multipole where we assume CMB-HD will
measure $BB$), and $TT$ on small scales (peak near $\ell=5000$). While the
lensing measured from the $TT$ power spectrum on angular scales $\ell>20,000$
provides a relatively small contribution to the total lensing SNR, as shown in
Table 3, it allows CMB-HD to probe the matter distribution out to scales
corresponding to $k\approx 55$ $h$ Mpc-1. We also note that above $\ell>5000$
there are many multipole bins, so while each may not have high SNR, added
together they are substantial, as indicated in the top panel of Figure 6.

Figure 6: Top panel: We illustrate the approximate breakdown of the CMB-HD
lensing SNR by comoving wavenumber and redshift for $z<6$. (Note that the
total SNR over all redshifts is 1947; for $z\leq 6$, the SNR is 1632.) Bottom
panel: We further breakdown the lensing SNR per multipole bin. The SNR
increases at $\ell=1000$ due to the inclusion of the CMB-HD $BB$ spectrum
which we assume has a minimum multipole of $\ell=1000$. We also indicate a few
multipoles (black dotted curves), and the linear and non-linear regimes
(purple dashed curve). This figure was inspired by Figure 2 in [66].

In Figure 2, we illustrate how well CMB-HD lensing measurements can be used to
measure the matter power spectrum. We place the CMB-HD data points in Figure 2
(shown in red) at the theoretical prediction (from CAMB) for the linear matter
power spectrum $P_{\mathrm{m}}^{\mathrm{lin}}(k)$. We calculate lensing SNRs
per $k$ bin the same way as in the top panel of Figure 6, this time doubling
the number of $k$ bins. Since this SNR arises from the non-linear matter
distribution on a given scale, we equate
$(S/N)_{k_{j}}=P_{\mathrm{m}}^{\mathrm{non-
lin}}(k_{j})/\sigma(P_{\mathrm{m}}^{\mathrm{non-lin}}(k_{j}))$. Following the
approach of [28], we use this to obtain the error bar on the linear matter
power spectrum,

$\begin{split}\sigma(P_{\mathrm{m}}^{\mathrm{lin}}(k_{j}))&=P_{\mathrm{m}}^{\mathrm{lin}}(k_{j})\left[\frac{\sigma(P_{\mathrm{m}}^{\mathrm{non-
lin}}(k_{j}))}{P_{\mathrm{m}}^{\mathrm{non-lin}}(k_{j})}\right]\\\
&=\frac{P_{\mathrm{m}}^{\mathrm{lin}}(k_{j})}{(S/N)_{k_{j}}}.\end{split}$ (27)

We note that in the non-linear HMCode [58] used in this work to obtain
$P_{\mathrm{m}}^{\mathrm{non-lin}}(k)$, there is no mixing of wavenumbers
between the linear and non-linear matter power spectra (the model assumes a
one-to-one correspondence); thus Eq. 27 is reasonable for illustrative
purposes.

We follow the method of [67] to place the mass scale on the upper $x$-axis of
Figure 2. We assume $k=2\pi/\lambda$, where $\lambda$ is the diameter of a
halo on scales corresponding to the comoving wavenumber $k$, and calculate the
mass by

$M=\frac{4\pi}{3}\rho_{\mathrm{m}}\left(\frac{\pi}{k}\right)^{3},$ (28)

where $\rho_{\mathrm{m}}$ is the total density in matter today.

The data points for the other experiments in Figure 2 are taken from
[28]171717https://github.com/marius311/mpk_compilation, which derives them
following the method of [29]: the Lyman-$\alpha$ data (purple points) are
derived from the 1D transmitted flux power spectrum [32] from the eBOSS DR14
release [33]; the cosmic shear data (brown points) are derived from the DES Y1
measurements of the cosmic shear two-point correlation function [34]; the
galaxy clustering data (pink points) are derived from measurements of the halo
power spectrum using luminous red galaxies (LRG) from the SDSS DR7 release
[35]; the CMB data are derived from the Planck 2018 temperature (blue points),
polarization (yellow points), and lensing (green points) power spectra [36,
7]. We also include constraints derived from high-redshift measurements of the
UV galaxy luminosity function from [37]. The external data included in Figure
2 does not represent the most recent measurements available (e.g. [68, 21, 8,
10, 69, 22, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81]) since we mainly
use the pre-computed data points derived by [28]. However, we make public the
code used to generate Figure 2, and any other data sets can be readily added.

###  IV.2 Parameter Forecasts

We use the Fisher matrix method described in Section III.7, to forecast
parameter constraints from CMB-HD delensed $TT$, $TE$, $EE$, $BB$ and lensing
$\kappa\kappa$ power spectra combinaed with DESI BAO data. In all cases we
apply a prior of $\sigma(\tau)=0.007$ from Planck [13].

In Figures 7 and 8 we show forecasts for a 1 keV WDM model, varying the full
set of parameters shown in Table 2. Figure 7 shows a subset of these
parameters, highlighting negligible degeneracy between $m_{\mathrm{WDM}}$ and
$\log_{10}\left(T_{\mathrm{AGN}}/\mathrm{K}\right)$, and a slight degeneracy
between $m_{\mathrm{WDM}}$, $n_{\mathrm{s}}$, and $N_{\mathrm{eff}}$. We see
$m_{\mathrm{WDM}}$ is most degenerate with $A_{\mathrm{kSZ}}$,
$n_{\mathrm{kSZ}}$.

In Table 4, we compare parameter forecasts for a CDM model, with and without
baryonic feedback, and a 1 keV WDM model with baryonic feedback. We find that
including baryonic feedback plus an SZ prior has negligible impact on
parameter constraints. For the WDM model, we find a small increase in
$\sigma(n_{s})$, and significant increases in $\sigma(A_{\mathrm{kSZ}})$ and
$\sigma(n_{\mathrm{kSZ}})$ compared to the CDM case, since these parameters
are most degenerate with $m_{\mathrm{WDM}}$.

Figure 7: We show forecasts for a subset of the parameters varied in the Fisher analysis. The blue dotted contours and curves show the parameter constraints in a $\Lambda$CDM + $N_{\mathrm{eff}}$ \+ $\sum m_{\nu}$ \+ $A_{\mathrm{kSZ}}$ \+ $n_{\mathrm{kSZ}}$ model assuming CDM. The red dotted contours and curves show the forecasts for a 1 keV $\Lambda$WDM + $N_{\mathrm{eff}}$ \+ $\sum m_{\nu}$ \+ $m_{\mathrm{WDM}}$ \+ $A_{\mathrm{kSZ}}$ \+ $n_{\mathrm{kSZ}}$ model. We also show forecasts for these two models when including the effects of baryonic feedback (cyan contours and curves for CDM, pink contours and curves for WDM), with the strength of the feedback characterized by $\log_{10}(T_{\mathrm{AGN}}/\mathrm{K})$; we place a prior on this parameter from the anticipated cross-correlation of CMB-HD SZ and CMB lensing measurements as done in [38]. For the WDM + baryons model, we find no degeneracy between $\log_{10}(T_{\mathrm{AGN}}/\mathrm{K})$ and the WDM mass, $m_{\mathrm{WDM}}$, indicating that they can be separately distinguished. Figure 8: We show the same forecasts as in Figure 7, now plotting the full set of varied parameters. For both CDM and WDM models, we marginalize over the baryonic feedback parameter, $\log_{10}(T_{\mathrm{AGN}}/\mathrm{K})$, with almost no increase in uncertainty on the other cosmological parameters (shown by the difference between dotted and solid contours of the same color). Table 4 lists the $1\sigma$ parameter uncertainties corresponding to this figure. Parameter |  $1\sigma$ Error from CMB-HD Delensed $TT$, $TE$, $EE$, $BB$, and $\kappa\kappa$ \+ DESI BAO
---|---
CDM | CDM + baryons + SZ prior | 1 keV WDM + baryons + SZ prior
$\Omega_{\mathrm{b}}h^{2}$ | 0.000025 |  0.000025 |  0.000025
$\Omega_{\mathrm{d}}h^{2}$ | 0.00041 |  0.00041 |  0.00041
$100\theta_{\mathrm{MC}}$ | 0.000060 |  0.000060 |  0.000060
$\ln\left(10^{10}A_{\mathrm{s}}\right)$ | 0.010 |  0.011 |  0.011
$n_{\mathrm{s}}$ | 0.0016 |  0.0016 |  0.0017
$\tau$ | 0.0055 |  0.0058 |  0.0058
$N_{\mathrm{eff}}$ | 0.015 |  0.016 |  0.016
$\sum m_{\nu}\leavevmode\nobreak\ $[eV] | 0.026 |  0.028 |  0.028
$A_{\mathrm{kSZ}}$ | 0.00076 |  0.00075 |  0.0012
$n_{\mathrm{kSZ}}$ | 0.00049 |  0.00049 |  0.0013
$\log_{10}(T_{\mathrm{AGN}}/\mathrm{K})$ | — |  0.0046 |  0.0046
$m_{\mathrm{WDM}}\leavevmode\nobreak\ $ [keV] | — |  — |  0.034
Table 4: Parameter forecasts from the combination of CMB-HD delensed $TT$, $TE$, $EE$, $BB$ and lensing $\kappa\kappa$ power spectra with DESI BAO data. We consider two dark matter models: one in which all the dark matter is cold (columns 2 and 3), and one in which all the dark matter is warm with a mass of 1 keV (column 4). We list the parameters in the first column, and the 1$\sigma$ uncertainty on each parameter in the following columns; a blank value indicates the parameter is not included in the model. For CDM, we consider a model without baryonic feedback (column 2) and including baryonic feedback, applying an SZ prior on the baryonic feedback parameter, $\log_{10}(T_{\mathrm{AGN}}/\mathrm{K})$ (column 3). We use the same baryonic feedback model and prior for the 1 keV WDM model. In all cases, we apply a prior on the optical depth, $\tau$, of $\sigma(\tau)=0.007$ from Planck [13]. We find that including baryonic feedback plus an SZ prior has negligible impact on parameter constraints. For the WDM model, we find a small increase in $\sigma(n_{s})$, and significant increases in $\sigma(A_{\mathrm{kSZ}})$ and $\sigma(n_{\mathrm{kSZ}})$ compared to the CDM case, since these parameters are most degenerate with $m_{\mathrm{WDM}}$. We see that $\sigma(n_{s})$ and $\sigma({N_{\rm{eff}}})$ are slightly larger than found in [38] due to the addition of the free kSZ parameters. |  $\sigma(m_{\mathrm{WDM}})$ from CMB-HD + DESI BAO
---|---
$m_{\mathrm{WDM}}$ |  $m_{\mathrm{WDM}}$ only free |  $m_{\mathrm{WDM}}$ \+ $A_{\mathrm{kSZ}}$ \+ $n_{\mathrm{kSZ}}$ | \+ $\Lambda$WDM | \+ $N_{\mathrm{eff}}$ \+ $\sum m_{\nu}$ | \+ baryons + SZ prior
1 keV | 0.0023 ($434.8\sigma$) | 0.026 ($38.5\sigma$) | 0.032 ($31.2\sigma$) | 0.034 ($29.4\sigma$) |  0.034 (29.4$\sigma$)
2 keV | 0.013 ($153.8\sigma$) | 0.20 ($10.0\sigma$) | 0.22 ($9.1\sigma$) | 0.22 ($9.1\sigma$) |  0.22 (9.1$\sigma$)
3 keV | 0.044 ($68.2\sigma$) | 0.30 ($10.0\sigma$) | 0.31 ($9.7\sigma$) | 0.31 ($9.7\sigma$) |  0.31 (9.7$\sigma$)
4 keV | 0.11 ($36.4\sigma$) | 0.56 ($7.1\sigma$) | 0.57 ($7.0\sigma$) | 0.58 ($6.9\sigma$) |  0.57 (7.0$\sigma$)
5 keV | 0.24 ($20.8\sigma$) | 1.1 ($4.5\sigma$) | 1.1 ($4.5\sigma$) | 1.1 ($4.5\sigma$) |  1.1 (4.5$\sigma$)
6 keV | 0.45 ($13.3\sigma$) | 1.9 ($3.2\sigma$) | 1.9 ($3.2\sigma$) | 1.9 ($3.2\sigma$) |  1.9 (3.2$\sigma$)
7 keV | 0.73 ($9.6\sigma$) | 3.2 ($2.2\sigma$) | 3.2 ($2.2\sigma$) | 3.3 ($2.1\sigma$) |  3.2 (2.2$\sigma$)
8 keV | 1.2 ($6.7\sigma$) | 4.8 ($1.7\sigma$) | 4.9 ($1.6\sigma$) | 5.0 ($1.6\sigma$) |  4.9 (1.6$\sigma$)
9 keV | 1.8 ($5.0\sigma$) | 7.3 ($1.2\sigma$) | 7.5 ($1.2\sigma$) | 7.5 ($1.2\sigma$) |  7.5 (1.2$\sigma$)
10 keV | 2.3 ($4.3\sigma$) | 11.0 ($0.9\sigma$) | 11.0 ($0.9\sigma$) | 11.0 ($0.9\sigma$) |  11.0 (0.9$\sigma$)
Table 5: We show the forecasted constraints on the WDM mass,
$m_{\mathrm{WDM}}$, from CMB-HD delensed $TT,TE,EE,BB$ and $\kappa\kappa$
power spectra plus DESI BAO. We list both the forecasted $1\sigma$ uncertainty
on $m_{\mathrm{WDM}}$ in units of keV and the significance with which CMB-HD
can measure this $m_{\mathrm{WDM}}$ (number in parentheses). We show these
constraints when only $m_{\mathrm{WDM}}$ is varied, when the kSZ parameters
are also varied, when the six $\Lambda$WDM parameters are additionally varied,
when $N_{\mathrm{eff}}$ and $\sum m_{\nu}$ is varied as well, and when the
baryonic feedback parameter, $\log_{10}(T_{\mathrm{AGN}}/\mathrm{K})$, is also
allowed to be free. We see that only varying $m_{\mathrm{WDM}}$ results in
much tighter constraints than when also varying the kSZ parameters. Varying
all the other parameters has minimal impact. CMB-HD can measure 1 keV WDM at
about $30\sigma$ significance and rule out about 7 keV WDM at $2\sigma$.

In Table 5, we list the forecasted $1\sigma$ uncertainty on the WDM mass,
$\sigma(m_{\mathrm{WDM}})$, for fiducial WDM models with masses ranging from 1
to 10 keV. We also quantify the significance with which we can measure a non-
zero WDM mass by taking the ratio $m_{\mathrm{WDM}}/\sigma(m_{\mathrm{WDM}})$.
For each WDM mass considered, we calculate this uncertainty for five different
cases: 1) freeing only $m_{\mathrm{WDM}}$ with all other parameters fixed to
their fiducial values; 2) additionally freeing the kSZ parameters,
$A_{\mathrm{kSZ}}$ and $n_{\mathrm{kSZ}}$; 3) additionally freeing the six
$\Lambda$WDM parameters (but fixing $N_{\mathrm{eff}}$ and $\sum m_{\nu}$); 4)
also freeing $N_{\mathrm{eff}}$ and $\sum m_{\nu}$; and 5) also freeing the
baryonic feedback parameter
$\log_{10}\left(T_{\mathrm{AGN}}/\mathrm{K}\right)$. As expected, our
constraints are strongest when we only vary the WDM mass. Varying the kSZ
parameters $A_{\mathrm{kSZ}}$ and $n_{\mathrm{kSZ}}$ significantly degrades
these constraints. We find that varying the cosmological parameters and
baryonic feedback minimally impacts constraints on the WDM mass once we allow
flexibility in the shape of the kSZ spectrum. In the scenario where all
parameters are freed, we find CDM-HD can measure 1 keV WDM mass with
$30\sigma$ significance, and can rule out about 7 keV WDM at the 95%
confidence level.

##  V Discussion

In this work, we find that CMB-HD can measure gravitational lensing with a
total significance of $1947\sigma$. To put this in context, current best
constraints on CMB lensing are $40\sigma$ from Planck [7] and $43\sigma$ from
ACT [8]. This lensing measurement would span $0.005\leavevmode\nobreak\
h$/Mpc$\leavevmode\nobreak\ <k<\leavevmode\nobreak\ 55\leavevmode\nobreak\
h$/Mpc, over four orders of magnitude in scale. The smallest scales probed by
CMB-HD would correspond to halo masses of about $10^{8}M_{\odot}$ today.

We find that most of the lensing SNR comes from the $TT$ spectrum, although
the $\kappa\kappa$ spectrum has a roughly equal contribution. We find that the
$BB$ power spectrum alone can measure CMB lensing with a significance of
almost $500\sigma$. On the smallest scales, most of the lensing SNR is from
the $TT$ spectrum, which suggests a different focus for efforts to remove
foreground contamination from lensing on these scales; it may be more
important to lower the foreground contribution to the $TT$ spectrum or
constrain this contribution well with multi-frequency observations, than to
remove non-Gaussian foreground contributions from the lensing trispectrum
($\kappa\kappa$).

Since the kSZ signal is frequency-independent, it is harder to separate this
foreground from the CMB than frequency-dependent foregrounds. Thus, we include
it in this work as part of the $TT$ spectrum. We find that including the kSZ
spectrum improves constraints on non-CDM models since the kSZ effect is a
dominant signal at large multipoles ($\ell>5000$), and non-CDM models impact
the matter power spectrum it traces. Figure 3 of [82] shows how the kSZ
spectrum from CMB-HD can also constrain fractional amounts of ultralight dark
matter, extending the work of [15].

We also find that baryonic feedback effects change the shape of the matter
power spectrum in ways that differ significantly from the change induced by
non-CDM models. From the difference in the CMB-HD lensing signal alone, we can
readily distinguish between them. This overcomes a significant challenge in
efforts to measure non-CDM models; one often cannot tell if a given small-
scale deviation from CDM is due to a non-CDM model or baryonic feedback
effects.

Current efforts to constrain non-CDM models have focused in large part on
ruling out WDM. These efforts use measurements of the Lyman-$\alpha$ forest,
Milky Way satellite galaxies, strong lensing of quasars, and stellar streams
[83, 84, 85, 86, 87, 88, 89, 90, 91, 92]. These constraints suggest roughly 5
keV WDM is ruled out at the 95% confidence level, with some spread due to
differing assumptions or techniques. Here we present an independent and
complimentary method with different systematics that can inform the particle
properties of dark matter. An advantage of this technique is that the CMB
lensing signal can be calculated theoretically from first principles, given
the non-linear matter power spectrum. In addition, the CMB lensing signal can
be measured in multiple different ways, which themselves have different
systematics, allowing for important cross checks. While the kSZ power spectrum
involves some astrophysics modelling, being sensitive to the product of the
free electron density and velocity, there are a few different external handles
on the kSZ effect from measurements of the tSZ power spectrum, the kSZ
trispectrum, and cross-correlation with spectroscopic galaxy surveys. We can
also see that limiting the flexibility in the kSZ power spectrum shape, with,
for example, priors from external measurements, would improve WDM constraints
considerably.

Future CMB-HD measurements will not only be able to probe the early Universe
via measurements of inflation and light relics, they will be a powerful and
complementary probe of structure in the Universe down to sub-galactic scales
and of dark matter particle properties.

###### Acknowledgements.

The authors thank Boris Bolliet, Anthony Challinor, Francis-Yan Cyr-Racine,
Rouven Essig, Gil Holder, Mathew Madhavacheril, David Marsh, Joel Meyers,
Julian Munoz, Blake Sherwin, and Anže Slosar for useful discussions. The
authors thank Miriam Rothermel for testing the code and Jupyter notebooks made
public with this paper, and Keyi Chen for an early version of the bottom panel
of Figure 6. AM and NS acknowledge support from DOE award number DE-SC0020441
and the Stony Brook OVPR Seed Grant Program. This research used resources of
the National Energy Research Scientific Computing Center (NERSC), a U.S.
Department of Energy Office of Science User Facility located at Lawrence
Berkeley National Laboratory, operated under Contract No. DE-AC02-05CH11231
using NERSC award HEP-ERCAPmp107.

##  Appendix A Comparison of Analytic Spectra Calculations to CAMB

While CAMB can calculate the CMB lensing power spectrum and the lensed and
delensed CMB $TT$, $TE$, $EE$, and $BB$ power spectra in a CDM-only or CDM +
baryons model, it cannot calculate this for an arbitrary non-CDM model. Thus,
to model the impact of non-CDM models that change the matter power spectrum,
we need to calculate the modified $C_{\ell}^{\kappa\kappa}$ outside of CAMB,
and then feed this $C_{\ell}^{\kappa\kappa}$ into CAMB to generate lensed and
delensed CMB $TT,TE,EE$, and $BB$ spectra. Since we compare results from non-
CDM models to those from CDM models, we calculate all of the CMB-HD power
spectra in this way for consistency. This ensures that any difference between
sets of power spectra is due to a difference in the model, rather than a
difference in the method used to calculate the spectra.

Figure 9: We show the ratio of delensed CMB $TT$, $EE$, $BB$ and CMB lensing
$\kappa\kappa$ power spectra, for a CDM model with $C_{\ell}^{\kappa\kappa}$
computed using Eq. 8 versus obtained directly from CAMB. In both cases, the
delensed CMB power spectra are calculated as described in Section III.3, using
the corresponding $C_{\ell}^{\kappa\kappa}$. Our calculations are consistent
with CAMB within 0.3% up to $\ell=20,000$ for all spectra, and within 0.8% for
$TT$ out to $\ell=40,000$ (latter not shown).

In Figure 9, we compare the CMB lensing spectrum, $C_{\ell}^{\kappa\kappa}$,
from Eq. 8 with the CAMB-only result for a CDM-only model to test the accuracy
our calculation. In our calculation of Eq. 8, we obtain $P_{\Psi}(k,z)$ from
CAMB initially. We use identical cosmological and accuracy parameters in both
cases (Eq. 8 versus CAMB only) and then take the ratio of their spectra (shown
in red). We also compare the delensed $TT$, $EE$, and $BB$ spectra obtained by
passing the residual lensing power given by Eq. 12 to CAMB, where the residual
lensing is calculated using either our calculation of
$C_{\ell}^{\kappa\kappa}$, or CAMB’s internal calculation. We find that our
calculation of the CMB and lensing power spectra agree with CAMB to within
0.3% out to $\ell=20,000$, which is the region that accounts for nearly all of
the CMB-HD lensing SNR, as shown in Table 3. Our calculation of
$C_{\ell}^{TT}$ differs from that of CAMB by at most 0.8% out to
$\ell=40,000$.

##  Appendix B Approximating the Lensing Signal-to-Noise Ratio per k Mode

Figure 6 illustrates how a given comoving physical scale at a given redshift
$z$ contributes to the total lensing SNR listed in Table 3. This information
is also used to visualize the CMB-HD constraints on the matter power spectrum
shown in Fig. 2. As discussed in Section III.4.1, we can only approximate how
much lensing comes from a specific comoving wavenumber ($k$) and redshift
($z$) since measurable angular multipoles ($\ell$) of lensing are drawn from
an integral over $k$ and $z$. The two approximations we make are 1) assuming
the covariance matrix is diagonal to obtain a lensing SNR value for each
redshift and wavenumber bin, and 2) summing the SNRs in quadrature over
redshift bins to obtain the total lensing SNR per wavenumber bin. The first
approximation underestimates the noise by neglecting the off-diagonal elements
of the covariance matrix. The second approximation underestimates the signal
by ignoring the lensing cross-terms between the redshift bins when adding SNRs
from redshift bins in quadrature. These two effects nearly cancel, yielding a
reasonable approximation, as will discuss show below. We note that this
approximation procedure is only used to create Figures 2 and 6; we use Eq. 13
with the full covariance matrix described in Section III.1 to calculate the
total lensing SNR values presented in Table 3.

We begin with a set of theoretical lensed and unlensed CMB power spectra,
$C_{\ell}^{XY}$ and $\tilde{C}_{\ell}^{XY}$, respectively, and with a set of
CMB-HD noise curves $N_{\ell}^{XY}$, for
$XY\in\\{TT,TE,EE,BB,\kappa\kappa\\}$. We first calculate a SNR per multipole,
$(S/N)_{\ell}$. We do this by considering the unbinned version of Eq. 13 for a
single spectrum, $XY\in\\{TT,TE,EE,BB,\kappa\kappa\\}$:

$\left(\frac{S}{N}\right)^{XY}=\sqrt{\sum_{\ell\ell^{\prime}}(\Delta
C_{\ell}^{XY})(\mathbb{C}^{XY,XY})^{-1}_{\ell\ell^{\prime}}(\Delta
C_{\ell^{\prime}}^{XY})},$ (29)

where the sum is taken from $\ell_{\mathrm{min}}$ to $\ell_{\mathrm{max}}$.
Assuming the covariance matrix is diagonal,
$\mathbb{C}^{XY,XY}_{\ell\ell^{\prime}}=\delta_{\ell\ell^{\prime}}\left(\sigma_{\ell}^{XY}\right)^{2}$,
this reduces to

$\left(\frac{S}{N}\right)^{XY}=\sqrt{\sum_{\ell}\left(\frac{\Delta
C_{\ell}^{XY}}{\sigma_{\ell}^{XY}}\right)^{2}},$ (30)

which allows us to define a SNR per multipole $\ell$,

$\left(\frac{S}{N}\right)_{\ell}^{XY}=\frac{\left|\Delta
C_{\ell}^{XY}\right|}{\sigma_{\ell}^{XY}}.$ (31)

Approximating the full $5\times 5$ block covariance matrix as diagonal, with
blocks
$\mathbb{C}_{\ell\ell^{\prime}}^{XY,WZ}=\delta_{XY,WZ}\delta_{\ell\ell^{\prime}}\left(\sigma_{\ell}^{XY}\right)^{2}$,
yields a total SNR given by

$\frac{S}{N}=\sqrt{\sum_{XY}\sum_{\ell}\left(\frac{\Delta
C_{\ell}^{XY}}{\sigma_{\ell}^{XY}}\right)^{2}}=\sqrt{\sum_{\ell}\sum_{XY}\left[\left(\frac{S}{N}\right)_{\ell}^{XY}\right]^{2}}.$
(32)

The total lensing SNR per multipole $\ell$ is then

$\left(\frac{S}{N}\right)_{\ell}=\sqrt{\sum_{XY}\left[\left(\frac{S}{N}\right)_{\ell}^{XY}\right]^{2}}.$
(33)

Next we calculate an approximate lensing SNR per multipole and per redshift.
We consider redshifts in the range $0<z<6$, using 12 redshift bins of width
$\Delta z=0.5$. The contribution $C_{\ell}^{\phi\phi,z_{i}}$ to the CMB
lensing power spectrum from the $i^{\mathrm{th}}$ redshift bin, ranging from
$z_{i,\mathrm{min}}$ to $z_{i,\mathrm{max}}$ and centered at $z_{i}$, is
calculated by integrating Eq. 8 from $\chi(z_{i,\mathrm{min}})$ to
$\chi(z_{i,\mathrm{max}})$. This lensing power spectrum is then used to lens
the unlensed CMB $TT$, $TE$, $EE$, and $BB$ power spectra as described in
Section III.3, resulting in a set of five spectra $C_{\ell}^{XY,z_{i}}$ with a
lensing signal arising only from $z_{i,\mathrm{min}}<z<z_{i,\mathrm{max}}$. We
calculate an approximate covariance matrix for each redshift bin $i$, again
only including elements along the diagonal. The diagonals of each covariance
matrix are calculated for each multipole $\ell$, so for the $i^{\mathrm{th}}$
redshift bin we define
$\mathbb{C}^{XY,WZ;z_{i}}_{\ell\ell^{\prime}}=\delta_{XY,WZ}\delta_{\ell\ell^{\prime}}\left(\sigma_{\ell}^{XY,z_{i}}\right)^{2}$,
where

$\begin{split}\left(\sigma_{\ell}^{XY,z_{i}}\right)^{2}=\frac{f_{\mathrm{sky}}^{-1}}{(2\ell+1)}\Bigl{[}&\left(C_{\ell}^{XX,z_{i}}+N_{\ell}^{XX}\right)\left(C_{\ell}^{YY,z_{i}}+N_{\ell}^{YY}\right)\\\
&+\left(C_{\ell}^{XY,z_{i}}+N_{\ell}^{XY}\right)^{2}\Bigr{]},\end{split}$ (34)

$C_{\ell}^{UV,z_{i}}$ is the power spectrum including lensing only from the
$i^{\mathrm{th}}$ redshift bin, and $N_{\ell}^{UV}$ is the noise on the power
spectrum, which we assume is the same for each redshift bin. (This is the same
as Eq. 1 for $XY=WZ$, which results in Eq. 2 when $XY=\phi\phi$, but replacing
each $C_{\ell}^{\mathrm{tot}}=C_{\ell}+N_{\ell}$ with
$C_{\ell}^{z_{i}}+N_{\ell}$). We then define the lensing SNR per multipole
$\ell$ in the $i^{\mathrm{th}}$ redshift bin as

$\left(\frac{S}{N}\right)^{z_{i}}_{\ell}=\sqrt{\sum_{XY}\left[\left(\frac{S}{N}\right)_{\ell}^{XY,z_{i}}\right]^{2}},$
(35)

where

$\left(\frac{S}{N}\right)_{\ell}^{XY,z_{i}}=\frac{\left|C_{\ell}^{XY,z_{i}}-\tilde{C}_{\ell}^{XY}\right|}{\sigma_{\ell}^{XY,z_{i}}},$
(36)

and $\tilde{C}_{\ell}^{XY}$ is the unlensed power spectrum.

We use the above to calculate an approximate lensing SNR per redshift and
comoving wavenumber $k$. The wavenumber $k$ corresponding to the multipole
$\ell$ at the redshift $z_{i}$ is given by $k\approx(\ell+1/2)/\chi(z_{i})$.
To make the bottom panel of Fig. 6, we first bin the lensed spectra
$C_{\ell}^{XY,z_{i}}$, unlensed spectra $\tilde{C}_{\ell}^{XY}$, and diagonal
covariance matrices $\mathbb{C}^{XY,XY;z_{i}}_{\ell\ell^{\prime}}$ using
uniform $\Delta\ell=30$ binning. We denote the upper and lower multipoles of
the $b^{\mathrm{th}}$ multipole bin by $\ell_{b,\mathrm{min}}$ and
$\ell_{b,\mathrm{max}}$, respectively, and the bin center by $\ell_{b}$. This
results in a SNR per redshift bin $i$ and per multipole bin $b$,
$(S/N)^{z_{i}}_{\ell_{b}}$. We then assign this lensing SNR value to the
corresponding wavenumber range, from
$k_{\mathrm{min}}(\ell_{b},z_{i})=(\ell_{b,\mathrm{min}}+1/2)/\chi(z_{i})$ to
$k_{\mathrm{max}}(\ell_{b},z_{i})=(\ell_{b,\mathrm{max}}+1/2)/\chi(z_{i})$.

In the top panel of Fig. 6, we plot the lensing SNR per wavenumber bin, summed
over the redshift bins in the range $0\leq z\leq 6$. We consider 11 bins in
wavenumber $k$, evenly spaced in $\log_{10}(k)$, denoting the upper and lower
edges of the $j^{\mathrm{th}}$ bin by $k_{j,\mathrm{min}}$ and
$k_{j,\mathrm{max}}$, respectively, and the bin center by $k_{j}$. For the
$i^{\mathrm{th}}$ redshift bin and $j^{\mathrm{th}}$ wavenumber bin, we apply
the approximation $k\approx(\ell+1/2)/\chi(z)$ to calculate the corresponding
multipole range,
$\ell_{\mathrm{min}}(z_{i},k_{j})=\chi(z_{i})k_{j,\mathrm{min}}-1/2$ to
$\ell_{\mathrm{max}}(z_{i},k_{j})=\chi(z_{i})k_{j,\mathrm{max}}-1/2$. The
lensing SNR in the $i^{\mathrm{th}}$ redshift bin and $j^{\mathrm{th}}$
wavenumber bin is given by summing $(S/N)_{\ell}^{z_{i}}$ in quadrature over
this multipole range

$\left(\frac{S}{N}\right)^{z_{i}}_{k_{j}}=\sqrt{\sum_{\ell=\ell_{\mathrm{min}}(z_{i},k_{j})}^{\ell_{\mathrm{max}}(z_{i},k_{j})}\left[\left(\frac{S}{N}\right)^{z_{i}}_{\ell}\right]^{2}}.$
(37)

We then sum over the redshift bins

$\left(\frac{S}{N}\right)_{k_{j}}=\sqrt{\sum_{i}\left[\left(\frac{S}{N}\right)^{z_{i}}_{k_{j}}\right]^{2}}$
(38)

to obtain a lensing SNR for each wavenumber bin $j$.

For each wavenumber bin shown in the top panel of Fig. 6, we show the
contribution from each redshift bin to the total lensing SNR in that
wavenumber bin. We quantify this by taking the ratio
$[(S/N)^{z_{i}}_{k_{j}}]^{2}/[(S/N)_{k_{j}}]^{2}$, which gives the fractional
contribution of the $i^{\mathrm{th}}$ redshift bin to the total lensing SNR in
the $j^{\mathrm{th}}$ wavenumber bin, and then multiplying this by the total
lensing SNR $(S/N)_{k_{j}}$ in that wavenumber bin.

To obtain the error bars in Fig. 2, we follow the same procedure to obtain a
SNR value in each wavenumber bin $j$, $(S/N)_{k_{j}}$, but instead use 22
wavenumber bins. We place the data points on the theoretical linear matter
spectrum at the $k$-bin centers, and use

$\begin{split}\sigma(P_{\mathrm{m}}^{\mathrm{lin}}(k_{j}))&=\frac{P_{\mathrm{m}}^{\mathrm{lin}}(k_{j})}{(S/N)_{k_{j}}}\end{split}$
(39)

to calculate the CMB-HD error bars.

| Lensing SNR
---|---
|  $z\in[0,1100]$ | $z\in[0,6]$
Spectra | correct | approx. | correct | approx.
$TT$-only | 1573 | 850 | 1353 | 770
$TE$-only | 161 | 103 | 133 | 90
$EE$-only | 276 | 149 | 221 | 126
$BB$-only | 493 | 1044 | 386 | 906
$\kappa\kappa$-only | 1287 | 1178 | 1029 | 1037
All | 1947 | 1798 | 1632 | 1585
Table 6: We compare the CMB-HD lensing SNR obtained from Eq. 13 (columns
labelled ”correct”) and from Eq. 42 (equivalent to Eq. 14 after summing over
the redshifts and multipoles), where we apply the approximations described in
Appendix B (columns labelled ”approx.”). We use either the full redshift range
$z\in[0,1100]$ back to recombination, or the range $z\in[0,6]$ (used in
Figures 2 and 6). We list the lensing SNRs for each individual spectrum
$XY\in\\{TT,TE,EE,BB,\kappa\kappa\\}$ in the first five rows, and for the
combination of all the spectra in the last row, calculated over the multipole
ranges listed in Table 1. The SNRs obtained from Eq. 13 use the full
covariance matrix (or the $XY\times XY$ block for individual spectra),
including off-diagonal elements, with spectra obtained by integrating Eq. 8
over the given redshift range, as described in Section III.3. When using Eq.
42, we calculate the spectra and a corresponding diagonal covariance matrix
within each redshift bin, and then take a sum over redshifts. Note that the
first column contains the same information as Table 3. We emphasize that the
approximate lensing SNR calculations are only used in Figures 2 and 6 for
illustrative purposes.

Approximations approximately cancelling: In general, neglecting the off-
diagonal components of a covariance matrix will result in overestimating the
total SNR. However, summing the SNR from redshift bins in quadrature also
underestimates the SNR. We see the latter from

$\begin{split}\frac{S}{N}&=\sqrt{\sum_{\ell}\left[\left(\frac{S}{N}\right)_{\ell}\right]^{2}}\\\
&=\sqrt{\sum_{\ell}\left(\frac{C_{\ell}-\tilde{C}_{\ell}}{\sigma_{\ell}}\right)^{2}}\\\
&=\sqrt{\sum_{\ell}\left(\frac{\sum_{i}\Delta
C_{\ell}^{z_{i}}}{\tilde{\sigma}_{\ell}+\sum_{i}\Delta\sigma_{\ell}^{z_{i}}}\right)^{2}}.\end{split}$
(40)

where we have assumed a diagonal covariance matrix with

$\begin{split}\tilde{\sigma}_{\ell}+\Delta\sigma_{\ell}^{z_{i}}&=\sqrt{\frac{2}{f_{\mathrm{sky}}(2\ell+1)}}\left(\tilde{C}_{\ell}+\Delta
C_{\ell}^{z_{i}}+N_{\ell}\right).\end{split}$ (41)

Here $i$ sums over redshift, and $\tilde{C}_{\ell}$ is the unlensed CMB. If
instead we add in quadrature the SNR from each redshift bin we have

$\begin{split}\sqrt{\sum_{i}\sum_{\ell}\left[\left(\frac{S}{N}\right)^{z_{i}}_{\ell}\right]^{2}}&=\sqrt{\sum_{i}\sum_{\ell}\left(\frac{C_{\ell}^{z_{i}}-\tilde{C}_{\ell}}{\tilde{\sigma}_{\ell}+\Delta\sigma_{\ell}^{z_{i}}}\right)^{2}}\\\
&=\sqrt{\sum_{\ell}\sum_{i}\left(\frac{\Delta
C_{\ell}^{z_{i}}}{\tilde{\sigma}_{\ell}}\right)^{2}}\end{split}$ (42)

in the limit where
$\tilde{\sigma}_{\ell}\gg\sum_{i}\Delta\sigma_{\ell}^{z_{i}}$, or equivalently
$\tilde{C}_{\ell}\leavevmode\nobreak\ +\leavevmode\nobreak\
N_{\ell}\gg\sum_{i}\Delta C_{\ell}^{z_{i}}$. In this limit, Eq. 42 is less
than Eq. 40. However, we find that the two approximations mentioned above
roughly cancel, as we show in Table 6, and that the agreement between
calculations is close. We show this in Table 6 for both the case where we sum
over all redshifts $z\in[0,1100]$ and for $z\in[0,6]$ (used in Figures 2 and
6).

## References

  * Hu [2001a] W. Hu, Astrophysical Journal Letters 557, L79 (2001a), arXiv:astro-ph/0105424 .
  * Hu and Okamoto [2002] W. Hu and T. Okamoto, Astrophys. J. 574, 566 (2002), arXiv:astro-ph/0111606 .
  * Lewis and Challinor [2006] A. Lewis and A. Challinor, Phys. Rept. 429, 1 (2006), arXiv:astro-ph/0601594 .
  * Das  _et al._ [2011] S. Das _et al._ , Phys. Rev. Lett. 107, 021301 (2011), arXiv:1103.2124 [astro-ph.CO] .
  * van Engelen  _et al._ [2012] A. van Engelen _et al._ , Astrophys. J. 756, 142 (2012), arXiv:1202.0546 [astro-ph.CO] .
  * Ade  _et al._ [2014] P. A. R. Ade _et al._ (Planck), Astron. Astrophys. 571, A17 (2014), arXiv:1303.5077 [astro-ph.CO] .
  * Aghanim  _et al._ [2020a] N. Aghanim _et al._ (Planck), Astron. Astrophys. 641, A8 (2020a), arXiv:1807.06210 [astro-ph.CO] .
  * Qu  _et al._ [2023] F. J. Qu _et al._ (ACT), (2023), arXiv:2304.05202 [astro-ph.CO] .
  * Madhavacheril  _et al._ [2024] M. S. Madhavacheril _et al._ (ACT), Astrophys. J. 962, 113 (2024), arXiv:2304.05203 [astro-ph.CO] .
  * Pan  _et al._ [2023] Z. Pan _et al._ (SPT), Phys. Rev. D 108, 122005 (2023), arXiv:2308.11608 [astro-ph.CO] .
  * Baxter  _et al._ [2015] E. J. Baxter _et al._ , Astrophys. J. 806, 247 (2015), arXiv:1412.7521 [astro-ph.CO] .
  * Madhavacheril  _et al._ [2015] M. Madhavacheril _et al._ (ACT), Phys. Rev. Lett. 114, 151302 (2015), [Addendum: Phys.Rev.Lett. 114, 189901 (2015)], arXiv:1411.7999 [astro-ph.CO] .
  * Aghanim  _et al._ [2020b] N. Aghanim _et al._ (Planck), Astronomy & Astrophysics 641, A6 (2020b), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO] .
  * Adame  _et al._ [2024] A. G. Adame _et al._ (DESI), (2024), arXiv:2404.03002 [astro-ph.CO] .
  * Farren  _et al._ [2022] G. S. Farren, D. Grin, A. H. Jaffe, R. Hložek, and D. J. E. Marsh, Phys. Rev. D 105, 063513 (2022), arXiv:2109.13268 [astro-ph.CO] .
  * Zeldovich and Sunyaev [1969] Y. B. Zeldovich and R. A. Sunyaev, Astrophysics and Space Science 4, 301 (1969).
  * Sunyaev and Zeldovich [1970] R. A. Sunyaev and Y. B. Zeldovich, Astrophysics and Space Science 7, 3 (1970).
  * Sunyaev and Zeldovich [1972] R. A. Sunyaev and Y. B. Zeldovich, Comments on Astrophysics and Space Physics 4, 173 (1972).
  * Smith and Ferraro [2017] K. M. Smith and S. Ferraro, Phys. Rev. Lett. 119, 021301 (2017), arXiv:1607.01769 [astro-ph.CO] .
  * Aghanim  _et al._ [2020c] N. Aghanim _et al._ (Planck), Astron. Astrophys. 641, A5 (2020c), arXiv:1907.12875 [astro-ph.CO] .
  * Choi  _et al._ [2020] S. K. Choi _et al._ (ACT), JCAP 12, 045 (2020), arXiv:2007.07289 [astro-ph.CO] .
  * Balkenhol  _et al._ [2023] L. Balkenhol _et al._ (SPT-3G), Phys. Rev. D 108, 023510 (2023), arXiv:2212.05642 [astro-ph.CO] .
  * Raghunathan  _et al._ [2024] S. Raghunathan _et al._ (SPT-3G, SPTpol), (2024), arXiv:2403.02337 [astro-ph.CO] .
  * MacCrann  _et al._ [2024] N. MacCrann _et al._ (ACT), (2024), arXiv:2405.01188 [astro-ph.CO] .
  * Jain  _et al._ [2023] D. Jain, T. R. Choudhury, S. Raghunathan, and S. Mukherjee, (2023), arXiv:2311.00315 [astro-ph.CO] .
  * Sehgal  _et al._ [2019] N. Sehgal _et al._ , (2019), arXiv:1906.10134 [astro-ph.CO] .
  * Aiola  _et al._ [2022] S. Aiola _et al._ (CMB-HD), (2022), arXiv:2203.05728 [astro-ph.CO] .
  * Chabanier  _et al._ [2019a] S. Chabanier, M. Millea, and N. Palanque-Delabrouille, Mon. Not. Roy. Astron. Soc. 489, 2247 (2019a), arXiv:1905.08103 [astro-ph.CO] .
  * Tegmark and Zaldarriaga [2002] M. Tegmark and M. Zaldarriaga, Phys. Rev. D 66, 103508 (2002), arXiv:astro-ph/0207047 .
  * Aghamousa  _et al._ [2016] A. Aghamousa _et al._ (DESI), (2016), arXiv:1611.00036 [astro-ph.IM] .
  * Marsh [2016] D. J. E. Marsh, (2016), arXiv:1605.05973 [astro-ph.CO] .
  * Chabanier  _et al._ [2019b] S. Chabanier _et al._ , JCAP 07, 017 (2019b), arXiv:1812.03554 [astro-ph.CO] .
  * Abolfathi  _et al._ [2018] B. Abolfathi _et al._ , Astrophysical Journal Supplement 235, 42 (2018), arXiv:1707.09322 [astro-ph.GA] .
  * Troxel  _et al._ [2018] M. A. Troxel _et al._ (DES), Phys. Rev. D 98, 043528 (2018), arXiv:1708.01538 [astro-ph.CO] .
  * Reid  _et al._ [2010] B. A. Reid _et al._ , Monthly Notices of the Royal Astronomical Society 404, 60 (2010), arXiv:0907.1659 [astro-ph.CO] .
  * Aghanim  _et al._ [2020d] N. Aghanim _et al._ (Planck), Astron. Astrophys. 641, A1 (2020d), arXiv:1807.06205 [astro-ph.CO] .
  * Sabti  _et al._ [2022] N. Sabti, J. B. Muñoz, and D. Blas, Astrophys. J. Lett. 928, L20 (2022), arXiv:2110.13161 [astro-ph.CO] .
  * MacInnis  _et al._ [2023] A. MacInnis, N. Sehgal, and M. Rothermel, (2023), arXiv:2309.03021 [astro-ph.CO] .
  * Ade  _et al._ [2019] P. Ade _et al._ (Simons Observatory), JCAP 02, 056 (2019), arXiv:1808.07445 [astro-ph.CO] .
  * Han and Sehgal [2022] D. Han and N. Sehgal, Phys. Rev. D 105, 083516 (2022), arXiv:2112.02109 [astro-ph.CO] .
  * Battaglia  _et al._ [2010] N. Battaglia, J. R. Bond, C. Pfrommer, J. L. Sievers, and D. Sijacki, Astrophys. J. 725, 91 (2010), arXiv:1003.4256 [astro-ph.CO] .
  * Smith  _et al._ [2018] K. M. Smith, M. S. Madhavacheril, M. Münchmeyer, S. Ferraro, U. Giri, and M. C. Johnson, (2018), arXiv:1810.13423 [astro-ph.CO] .
  * Park  _et al._ [2013] H. Park, P. R. Shapiro, E. Komatsu, I. T. Iliev, K. Ahn, and G. Mellema, Astrophys. J. 769, 93 (2013), arXiv:1301.3607 [astro-ph.CO] .
  * Okamoto and Hu [2003] T. Okamoto and W. Hu, Phys. Rev. D 67, 083002 (2003), arXiv:astro-ph/0301031 .
  * Hu [2001b] W. Hu, Phys. Rev. D 64, 083005 (2001b), arXiv:astro-ph/0105117 .
  * Namikawa  _et al._ [2013] T. Namikawa, D. Hanson, and R. Takahashi, Monthly Notices of the Royal Astronomical Society 431, 609 (2013), arXiv:1209.0091 [astro-ph.CO] .
  * Hotinli  _et al._ [2022] S. C. Hotinli, J. Meyers, C. Trendafilova, D. Green, and A. van Engelen, J. Cosmol. Astropart. Phys. 04, 020 (2022), arXiv:2111.15036 [astro-ph.CO] .
  * Hu  _et al._ [2007] W. Hu, S. DeDeo, and C. Vale, New J. Phys. 9, 441 (2007), arXiv:astro-ph/0701276 .
  * Benoit-Lévy  _et al._ [2012] A. Benoit-Lévy, K. M. Smith, and W. Hu, Phys. Rev. D 86, 123008 (2012), arXiv:1205.0474 [astro-ph.CO] .
  * Manzotti  _et al._ [2014] A. Manzotti, W. Hu, and A. Benoit-Lévy, Phys. Rev. D 90, 023003 (2014), arXiv:1401.7992 [astro-ph.CO] .
  * Motloch and Hu [2019] P. Motloch and W. Hu, Phys. Rev. D 99, 023506 (2019), arXiv:1810.09347 [astro-ph.CO] .
  * Eisenstein  _et al._ [2005] D. J. Eisenstein _et al._ (SDSS), Astrophys. J. 633, 560 (2005), arXiv:astro-ph/0501171 .
  * Howlett  _et al._ [2012] C. Howlett, A. Lewis, A. Hall, and A. Challinor, J. Cosmol. Astropart. Phys. 1204, 027 (2012), arXiv:1201.3654 [astro-ph.CO] .
  * Lewis  _et al._ [2000] A. Lewis, A. Challinor, and A. Lasenby, Astrophys. J. 538, 473 (2000), arXiv:astro-ph/9911177 .
  * Mead  _et al._ [2021] A. Mead, S. Brieden, T. Tröster, and C. Heymans, Monthly Notices of the Royal Astronomical Society 502, 1401 (2021), arXiv:2009.01858 [astro-ph.CO] .
  * Dodelson and Schmidt [2020] S. Dodelson and F. Schmidt,  _Modern Cosmology_ (2020).
  * Nguyen  _et al._ [2019] H. N. Nguyen, N. Sehgal, and M. Madhavacheril, Phys. Rev. D 99, 023502 (2019), arXiv:1710.03747 [astro-ph.CO] .
  * Mead  _et al._ [2016] A. Mead, C. Heymans, L. Lombriser, J. Peacock, O. Steele, and H. Winther, Monthly Notices of the Royal Astronomical Society 459, 1468 (2016), arXiv:1602.02154 [astro-ph.CO] .
  * Han  _et al._ [2021] D. Han _et al._ (ACT), J. Cosmol. Astropart. Phys. 01, 031 (2021), arXiv:2007.14405 [astro-ph.CO] .
  * Viel  _et al._ [2012] M. Viel, K. Markovič, M. Baldi, and J. Weller, Monthly Notices of the Royal Astronomical Society 421, 50 (2012), arXiv:1107.4094 [astro-ph.CO] .
  * Viel  _et al._ [2005] M. Viel, J. Lesgourgues, M. G. Haehnelt, S. Matarrese, and A. Riotto, Phys. Rev. D 71, 063534 (2005), arXiv:astro-ph/0501562 .
  * Bode  _et al._ [2001] P. Bode, J. P. Ostriker, and N. Turok, Astrophys. J. 556, 93 (2001), arXiv:astro-ph/0010389 .
  * Schneider  _et al._ [2012] A. Schneider, R. E. Smith, A. V. Macciò, and B. Moore, Monthly Notices of the Royal Astronomical Society 424, 684 (2012), arXiv:1112.0330 [astro-ph.CO] .
  * Omori [2022] Y. Omori, (2022), arXiv:2212.07420 [astro-ph.CO] .
  * Dunkley  _et al._ [2013] J. Dunkley _et al._ , JCAP 07, 025 (2013), arXiv:1301.0776 [astro-ph.CO] .
  * Green and Meyers [2021] D. Green and J. Meyers, (2021), arXiv:2111.01096 [astro-ph.CO] .
  * Hlozek  _et al._ [2012] R. Hlozek _et al._ , Astrophys. J. 749, 90 (2012), arXiv:1105.4887 [astro-ph.CO] .
  * Aiola  _et al._ [2020] S. Aiola _et al._ (ACT), JCAP 12, 047 (2020), arXiv:2007.07288 [astro-ph.CO] .
  * Dutcher  _et al._ [2021] D. Dutcher _et al._ (SPT-3G), Phys. Rev. D 104, 022003 (2021), arXiv:2101.01684 [astro-ph.CO] .
  * Amon  _et al._ [2022] A. Amon _et al._ (DES), Phys. Rev. D 105, 023514 (2022), arXiv:2105.13543 [astro-ph.CO] .
  * Secco  _et al._ [2022] L. F. Secco _et al._ (DES), Phys. Rev. D 105, 023515 (2022), arXiv:2105.13544 [astro-ph.CO] .
  * Asgari  _et al._ [2021] M. Asgari _et al._ (KiDS), Astron. Astrophys. 645, A104 (2021), arXiv:2007.15633 [astro-ph.CO] .
  * Li  _et al._ [2023a] S.-S. Li _et al._ , Astron. Astrophys. 679, A133 (2023a), arXiv:2306.11124 [astro-ph.CO] .
  * Li  _et al._ [2023b] X. Li _et al._ , Phys. Rev. D 108, 123518 (2023b), arXiv:2304.00702 [astro-ph.CO] .
  * Dalal  _et al._ [2023] R. Dalal _et al._ , Phys. Rev. D 108, 123519 (2023), arXiv:2304.00701 [astro-ph.CO] .
  * Rodríguez-Monroy  _et al._ [2022] M. Rodríguez-Monroy _et al._ (DES), Mon. Not. Roy. Astron. Soc. 511, 2665 (2022), arXiv:2105.13540 [astro-ph.CO] .
  * Semenaite  _et al._ [2022] A. Semenaite _et al._ (eBOSS), Mon. Not. Roy. Astron. Soc. 512, 5657 (2022), arXiv:2111.03156 [astro-ph.CO] .
  * Alam  _et al._ [2017] S. Alam _et al._ (BOSS), Mon. Not. Roy. Astron. Soc. 470, 2617 (2017), arXiv:1607.03155 [astro-ph.CO] .
  * de Belsunce  _et al._ [2024] R. de Belsunce, O. H. E. Philcox, V. Irsic, P. McDonald, J. Guy, and N. Palanque-Delabrouille, (2024), arXiv:2403.08241 [astro-ph.CO] .
  * Gilman  _et al._ [2022] D. Gilman, A. Benson, J. Bovy, S. Birrer, T. Treu, and A. Nierenberg, Mon. Not. Roy. Astron. Soc. 512, 3163 (2022), arXiv:2112.03293 [astro-ph.CO] .
  * Esteban  _et al._ [2023] I. Esteban, A. H. G. Peter, and S. Y. Kim, (2023), arXiv:2306.04674 [astro-ph.CO] .
  * Antypas  _et al._ [2022] D. Antypas _et al._ , (2022), arXiv:2203.14915 [hep-ex] .
  * Baur  _et al._ [2016] J. Baur, N. Palanque-Delabrouille, C. Yèche, C. Magneville, and M. Viel, JCAP 08, 012 (2016), arXiv:1512.01981 [astro-ph.CO] .
  * Garzilli  _et al._ [2021] A. Garzilli, A. Magalich, O. Ruchayskiy, and A. Boyarsky, Mon. Not. Roy. Astron. Soc. 502, 2356 (2021), arXiv:1912.09397 [astro-ph.CO] .
  * Gilman  _et al._ [2020] D. Gilman, S. Birrer, A. Nierenberg, T. Treu, X. Du, and A. Benson, Mon. Not. Roy. Astron. Soc. 491, 6077 (2020), arXiv:1908.06983 [astro-ph.CO] .
  * Hsueh  _et al._ [2020] J.-W. Hsueh, W. Enzi, S. Vegetti, M. Auger, C. D. Fassnacht, G. Despali, L. V. E. Koopmans, and J. P. McKean, Mon. Not. Roy. Astron. Soc. 492, 3047 (2020), arXiv:1905.04182 [astro-ph.CO] .
  * Banik  _et al._ [2021] N. Banik, J. Bovy, G. Bertone, D. Erkal, and T. J. L. de Boer, JCAP 10, 043 (2021), arXiv:1911.02663 [astro-ph.GA] .
  * Nadler  _et al._ [2021] E. O. Nadler _et al._ (DES), Phys. Rev. Lett. 126, 091101 (2021), arXiv:2008.00022 [astro-ph.CO] .
  * Villasenor  _et al._ [2023] B. Villasenor, B. Robertson, P. Madau, and E. Schneider, Phys. Rev. D 108, 023502 (2023), arXiv:2209.14220 [astro-ph.CO] .
  * Iršič  _et al._ [2024] V. Iršič _et al._ , Phys. Rev. D 109, 043511 (2024), arXiv:2309.04533 [astro-ph.CO] .
  * Hooper  _et al._ [2022] D. C. Hooper, N. Schöneberg, R. Murgia, M. Archidiacono, J. Lesgourgues, and M. Viel, JCAP 10, 032 (2022), arXiv:2206.08188 [astro-ph.CO] .
  * Keeley  _et al._ [2024] R. E. Keeley _et al._ , (2024), arXiv:2405.01620 [astro-ph.CO] .