[7]G^ #1,#2_#3,#4(#5 #6— #7)

# Cosmology with the moving lens effect

Selim C. Hotinli  Department of Physics & Astronomy, Johns Hopkins University,
Baltimore, MD 21218, USA  Kendrick M. Smith  Perimeter Institute for
Theoretical Physics, 31 Caroline St N, Waterloo, ON N2L 2Y5, Canada  Mathew S.
Madhavacheril  Perimeter Institute for Theoretical Physics, 31 Caroline St N,
Waterloo, ON N2L 2Y5, Canada  Marc Kamionkowski  Department of Physics &
Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA

###### Abstract

Velocity fields can be reconstructed at cosmological scales from their
influence on the correlation between the cosmic microwave background and
large-scale structure. Effects that induce such correlations include the
kinetic Sunyaev Zel’dovich (kSZ) effect and the moving-lens effect, both of
which will be measured to high precision with upcoming cosmology experiments.
Galaxy measurements also provide a window into measuring velocities from the
effect of redshift-space distortions (RSDs). The information that can be
accessed from the kSZ or RSDs, however, is limited by astrophysical
uncertainties and systematic effects, which may significantly reduce our
ability to constrain cosmological parameters such as $f\sigma_{8}$. In this
paper, we show how the large-scale transverse-velocity field, which can be
reconstructed from measurements of the moving-lens effect, can be used to
measure $f\sigma_{8}$ to high precision.

##  I Introduction

Next generation cosmic microwave background (CMB) experiments such as the
Simons Observatory (SO) (Ade  _et al._ , 2019; Lee  _et al._ , 2019) and
CMB-S4 (Abazajian  _et al._ , 2016), and galaxy surveys such as DESI
(Aghamousa  _et al._ , 2016) and the Vera Rubin Observatory (VRO) (LSST
Science Collaboration  _et al._ , 2009) will generate a wealth of new data
with unprecedented precision on small scales. Correlations between CMB
anisotropies and the galaxy density carry valuable cosmological information
about the largest scales, creating novel opportunities for inference (e.g.
Seljak, 2009; Zhang and Johnson, 2015; Banerjee  _et al._ , 2018; Schmittfull
and Seljak, 2018; Modi  _et al._ , 2017; Deutsch  _et al._ , 2018a; Münchmeyer
_et al._ , 2019; Cayuso and Johnson, 2020; Pan and Johnson, 2019; Hotinli  _et
al._ , 2019). These correlations that are induced by interactions of CMB
photons with the intervening large-scale structure include the thermal and
kinetic Sunyaev Zel’dovich effects (Zeldovich and Sunyaev, 1969; Zel’Dovich,
1970; Sunyaev and Zeldovich, 1980, 1972; Sazonov and Sunyaev, 1999) and the
integrated Sachs-Wolfe effects (Sachs and Wolfe, 1967), which includes the
moving-lens effect (Birkinshaw and Gull, 1983). Among these, the kinetic
Sunyaev Zel’dovich (kSZ) and the moving-lens effects depend on the peculiar
velocities of cosmological structures.

A key product of velocity measurements is the combination of the linear-theory
growth rate $f$ and the amplitude $\sigma_{8}$ of matter fluctuations on the
scale of $8h^{-1}$Mpc. In principle, $f\sigma_{8}$ can be determined by
measurements of the kSZ effect (e.g. Deutsch  _et al._ , 2018b, a, 2019; Smith
_et al._ , 2018; Münchmeyer  _et al._ , 2019; Madhavacheril  _et al._ , 2019;
Contreras  _et al._ , 2019; Cayuso and Johnson, 2020; Hotinli and Johnson,
2020; Giri and Smith, 2020), or by measurements of redshift-space distortions
(RSDs) on galaxy surveys (Kaiser, 1987). The cosmological information
available from the kSZ effect, however, is limited by the degeneracy of this
signal with the optical depth of galaxies (except in the case of primordial
non-Gaussianity Münchmeyer  _et al._ (2019)). While fast radio burst (FRB)
sources may provide a way to break some of these biases by measuring the
galaxy-electron correlation (Madhavacheril  _et al._ , 2019), the prospect of
localising enough FRBs in the near future is challenging.

Furthermore, recent studies show evidence of anisotropic selection effects
(e.g. Obuljen  _et al._ , 2020) for galaxy RSD measurements. Such effects are
degenerate with redshift-space distortions, in the sense that the large-scale
bias of the galaxy field becomes $(b_{g}\\!+\\!b_{\rm rsd}~{}\\!f\mu^{2})$,
rather than simply $(b_{g}\\!+\\!f\mu^{2})$, where
$\mu\\!=\\!\hat{\bm{r}}\cdot\hat{\bm{k}}$. Here, $\hat{\bm{r}}$ is the unit
vector in the radial direction, $\hat{\bm{k}}$ is the three-dimensional
Fourier (unit) wavevector and $b_{\rm rsd}$ is a new bias parameter, defined
such that $b_{\rm rsd}\\!=\\!1$ in the absence of anisotropic selection
effects.111According to Ref. (Obuljen  _et al._ , 2020), $(b_{\rm rsd}-1)$ can
be of order $0.1-0.2$ in subcatalogs of SDSS. Anisotropic selection effects
could significantly degrade measurements of $f\,\sigma_{8}$ from future galaxy
surveys, due to degeneracy with $b_{\rm rsd}$. In this paper, we propose a
complementary method for measuring $f\,\sigma_{8}$ in which all degeneracies
with astrophysical bias parameters are broken.

We will show that the temperature fluctuations sourced by the transverse
velocities of large-scale structure (the moving-lens effect (Birkinshaw and
Gull, 1983)) can be used to measure $f\sigma_{8}$ and to break the kSZ
optical-depth degeneracy. These transverse velocities are inferred by cross-
correlation of the effect they induce in the CMB with galaxy surveys (e.g.
Hotinli  _et al._ , 2019; Yasini  _et al._ , 2019; Hotinli  _et al._ , 2021).
The moving lens effect sourced by individual objects is expected to be small
and can be easily confused with other effects. However, combining the signal
from many objects allows a tomographic reconstruction of the transverse-
velocity field on large scales, analogous and complementary to kSZ tomography,
which provides the radial-velocity fields (e.g. Deutsch  _et al._ , 2018b;
Smith  _et al._ , 2018). Note that the transverse-velocities measured from
moving-lens tomography will be biased similar to the case of kSZ tomography.
The moving-lens bias $b_{\perp}$ can be measured through galaxy-galaxy or
galaxy-CMB lensing, however, since $b_{\perp}$ is proportional to the galaxy-
matter cross-correlation, in contrast to the kSZ bias $b_{\parallel}$ which is
proportional to the galaxy-electron cross-correlation, which is difficult to
measure independently without many localised FRB sources (Madhavacheril  _et
al._ , 2019).

Small-scale CMB anisotropies are expected to be dominated by the kSZ effect
due to Compton scattering of the CMB photon off free electrons that have non-
zero peculiar velocities with respect to the CMB rest frame. This induces a
shift in the CMB temperature in the direction of the free-electron radial
velocity. The moving-lens effect is due to time-evolving gravitational
potentials on the line of sight. This induces additional temperature
anisotropies in the CMB, via the non-linear ISW (or Rees-Sciama) effect. This
signal, which is a dipole pattern oriented along the object’s transverse
velocity, is expected to be smaller than kSZ. Even so, it will be measured
with high signal-to-noise with upcoming CMB experiments (Hotinli  _et al._ ,
2019; Yasini  _et al._ , 2019; Hotinli  _et al._ , 2021).

In what follows, we demonstrate that the transverse velocities from the
moving-lens effect, when combined with galaxy measurements, can provide
competitive constraints on the $f\sigma_{8}$ parameter, comparable to the
scenario where the RSD bias is measured externally to high precision. This
parameter is useful for studying a large range of physics, including dark
energy (Linder, 2005), modified gravity (Linder and Cahn, 2007) and the
effects of neutrino mass (Bond and Szalay, 1983). Precision measurement of
$f\sigma_{8}$ also allows using the kSZ to learn about astrophysics, such as
the characteristics of the electron density profiles around halos, by breaking
the degeneracy between the electron-scattering optical depth and the growth
rate (Smith  _et al._ , 2018).

This paper is organised as follows: We begin with a description of kSZ and
moving-lens tomography in Section II. We show the dependence of the velocity
field on cosmology and discuss the anticipated RSD bias in Section III. We
introduce a formalism for the three-velocity in Section IV. We introduce a
Fisher analysis in Section III.2. We provide results from our forecasts using
experimental specifications anticipated for next generation CMB and galaxy
surveys in Section V. We conclude with a discussion in Section VI. Throughout,
we use the standard $\Lambda$CDM model with cosmological parameters
$\\{\Omega_{b}\,h^{2},\,\Omega_{\rm cdm}h^{2},\,A_{s},\,n_{s},\,\tau\\}$ set
equal to $\\{0.022,0.12,{2.2\\!\times\\!10^{-9}},3.04,0.965,0.06\\}$,
respectively.

##  II Velocity reconstruction

###  II.1 kSZ tomography

Figure 1: CMB signal and foregrounds at the frequency $\bm{\nu=150}$ GHz.
Total (lensed) CMB signal is shown with the black solid line. We show the
anticipated post-component separation noise (including foreground residuals)
for three experiments; CMB-S4 (S4), Simons Observatory (SO) and CMB-HD (HD);
along with the most significant foreground components (before foreground
cleaning) in temperature maps (shown at 150 GHz). These are the cosmic
infrared background (CIB) the thermal Sunyaev Zel’dovich effect (tSZ), late-
time kSZ effect (due to free energetic electrons in intergalactic mediums
where $z\in[0,6]$), the kSZ signal from patchy-reionization and the moving-
lens (ML) effect.

Kinetic Sunyaev Zel’dovich tomography aims to extract cosmological information
from the kSZ effect by using measurements of the CMB and a tracer of the
electron density, such as a galaxy survey, to reconstruct the radial-velocity
field. The temperature anisotropy induced by the kSZ effect from a large-scale
structure (LSS) shell at redshift $z=z_{*}$ is

$\displaystyle\Theta_{\rm kSZ}(\bm{\theta})=K(z_{*})\int_{0}^{L}{\rm
d}r\,q_{\parallel}(\bm{x}),$ (1)

where $\bm{x}\equiv\chi_{\star}\bm{\theta}+r\hat{\bm{r}}$, $\chi_{\star}$ is
the conformal distance to the LSS shell, $\hat{\bm{r}}$ is the unit vector in
the radial direction, $\bm{\theta}$ is the angular direction on the sky,
$\Theta(\bm{\theta})=\Delta T(\bm{\theta})/\bar{T}$ is the fractional CMB
temperature fluctuation, and

$\displaystyle K(z)=-\sigma_{T}n_{e,0}x_{e}(z)e^{-\tau(z)}(1+z)^{2}\,,$ (2)

is the radial weight function with units of $1/$Mpc. Here, $\tau(z)$ is the
optical depth at redshift $z$, $\sigma_{T}$ is the Thomson cross-section,
$n_{e}$ is the free electron number-density and
$q_{\parallel}(\bm{x})=\delta_{e}(\bm{x})v_{\parallel}(\bm{x})$ is the
electron-momentum field, projected onto the radial direction. Most of the
cosmological information is contained in the radial-velocity field
$v_{\parallel}$ while $n_{e}$ depends primarily on astrophysics and non-linear
large-scale structure; see Ref. Smith  _et al._ (2018) for a detailed
discussion of this point. We demonstrate the anticipated late-time kSZ signal
in Fig. 1, along with other foregrounds and the forecasted noise levels for
next-generation CMB experiments. The component separated CMB noise due to
foreground residuals, including the cosmic infrared background (CIB), radio
sources, the thermal Sunyaev Zel’dovich effect (tSZ), and the kSZ signal are
shown in Fig. 1 for the Simons Observatory, CMB-S4 and CMB-HD experimental
specifications (these were produced using the
orphics222github.com/msyriac/orphics code as described in Appendix A). We use
the electron model described in Ref. (Battaglia, 2016).

The (inverse) noise on the velocity reconstruction is given by (Smith  _et
al._ , 2018)

$\displaystyle\frac{1}{N_{\parallel}(\bm{k}_{L})}=\frac{K_{*}^{2}}{\chi_{*}^{2}}\int\\!\\!\frac{k_{s}{\rm
d}k_{s}}{2\pi}\\!\left(\\!\frac{P_{ge}(k_{s})^{2}}{P^{\rm
obs}_{gg}(k_{s})C_{\ell}^{TT,\rm
obs}}\\!\right)_{\\!\\!\\!\ell=k\chi_{*}}\\!\\!\\!\\!\,,\\!\\!\,\,\,\,\,\,$
(3)

where $\bm{k}$ is the three-dimensional Fourier wavevector and the integral is
over small-scale Fourier modes $k_{S}$. We represent large-scale modes with an
‘$L$’ subscript. Here, $P_{ge}(k)$ is the power-spectrum of the galaxy-
electron correlation, $P^{\rm obs}_{gg}(k)$ is the observed galaxy power
spectrum and $C_{\ell}^{TT,\rm obs}$ is the observed CMB spectrum. We show the
reconstruction noise from kSZ tomography in Fig. 2.

###  II.2 Moving-lens tomography

Gravitational potentials that evolve in time induce a temperature modulation
on the CMB known as the integrated Sachs-Wolfe (ISW) effect, which has the
form

$\displaystyle\Theta_{\rm ML}(\bm{\theta})=-2\int_{0}^{L}{\rm
d}r\,\dot{\Phi}(\bm{x})\,,$ (4)

where $\Phi$ is the gravitational potential. The ISW effect can be sourced by
peculiar velocities of potentials, leading to the moving-lens effect, which
has the form

$\displaystyle\begin{split}\Theta_{\rm ML}(\bm{\theta})=-2\int_{0}^{L}{\rm
d}r\,\bm{\nabla}_{\\!\perp}\\!{\Phi}(\bm{x})\cdot\bm{v}_{\perp}(\bm{x})\,,\end{split}$
(5)

where $\bm{v}_{\perp}(\bm{x})$ is the peculiar (comoving) transverse velocity.
Our focus is on the large-scale velocity field, where we anticipate that the
true velocity is linear and curl-free, such that we can define a transverse
velocity potential $\Upsilon(\bm{x})$ as
$\bm{v}_{\perp}(\bm{x})=\bm{\nabla}_{\perp}{\Upsilon}(\bm{x})$. We demonstrate
the anticipated moving lens (ML) signal in Fig. 1. The ML signal is calculated
using the velocity-reconstruction pipeline described in Ref. (Cayuso  _et al._
, ) for the redshift range $z\in\\{0.1,5\\}$.

Moving-lens tomography can be used to measure the large-scale transverse-
velocity field. In the limit $\bm{k}\ll\bm{k}_{S},\bm{\ell}/\chi_{*}$, we find
the noise on the velocity reconstruction to be

$\displaystyle\frac{1}{N^{\hat{\Upsilon}\hat{\Upsilon}}(\bm{k}_{L})}\\!$
$\displaystyle=$
$\displaystyle\\!\frac{2\mu_{\perp}^{2}k_{L}^{2}}{\chi_{*}^{2}}\\!\int\\!\frac{k^{3}_{S}\,{\rm
d}k_{S}}{2\pi}\\!\left(\\!\frac{P_{\Phi\Phi}(k_{S})^{2}}{P^{\rm
obs}_{\Phi\Phi}(k_{S})C_{\ell}^{\Theta\Theta,\rm
obs}}\\!\right)_{\\!\\!\\!\ell=k_{S}\chi_{*}}\\!\\!\\!\\!\\!\\!,$

where $\mu_{\perp}=\sqrt{1-\mu^{2}}$, $\mu=\bm{\hat{k}}\cdot\hat{\bm{r}}$,
$P_{\Phi\Phi}^{\rm obs}(k,\mu)=P_{\Phi\Phi}(k)+N_{\Phi\Phi}(k,\mu)$, and

$\displaystyle
N_{\Phi\Phi}(k,\mu)={[(1+z)\rho_{m,0}/2]^{2}}/[{k^{4}\,b_{g}^{2}\,n_{g}\,W(k,\mu)^{2}}]\,,\,\,\,\,\,$
(7)

with the photo-$z$ term,

$\displaystyle W(k,\mu)=\exp\\{-\mu^{2}k^{2}\sigma_{z}^{2}/2H_{*}^{2}\\}\,,$
(8)

where $\sigma_{z}$ is the photo-$z$ error and $H_{*}$ is the Hubble parameter
evaluated at redshift $z_{*}$. Here, $\rho_{m,0}=3\Omega_{M}H_{0}^{2}$,
$\Omega_{M}$ is the matter fraction and $H_{0}$ is the Hubble constant. We
show the shot noise on the gravitational potential in Fig. 2 for the galaxy
surveys we consider, together with the reconstruction noise from moving-lens
tomography.

Figure 2: (Left) Comparison of the velocity signal to reconstruction noise at
redshift $z=1$. The power spectrum of the norm $|\bm{v}|$ of the 3-velocity is
shown with solid black curves. Coloured lines correspond to reconstruction-
noise spectra, calculated from the anticipated moving-lens (ML) and kSZ
signals. Two galaxy-survey configurations are shown; these are DESI and Rubin
Observatory Year 10 (VROY10). The CMB-S4 experimental specifications are used
for all noise spectra. The two subplots correspond to nearly-transverse
($\mu=0.1$) and nearly-radial ($\mu=0.9$) velocity reconstruction. (Right) The
gravitational potential signal (solid black line) is shown together with the
shot noise for the galaxy surveys we consider in this paper. In all plots, the
rise in the noise on small-scales for Rubin is due to scatter from photometric
redshifts.

##  III Cosmology from velocities

###  III.1 Cross-correlation signal

Where linear theory is valid, the reconstructed velocity fields are
proportional to the cosmic growth rate. They satisfy

$\displaystyle\hat{v}_{\parallel}$ $\displaystyle=$ $\displaystyle
b_{\parallel}\,\mu\frac{faH}{k}\delta_{m}(\bm{k})\,,$ (9)
$\displaystyle\hat{v}_{\perp}$ $\displaystyle=$ $\displaystyle
b_{\perp}\sqrt{1-\mu^{2}}\frac{faH}{k}\delta_{m}(\bm{k})\,,$ (10)

where $H$ is the Hubble parameter, $a$ is the scale factor,
$\delta_{m}(\bm{k})$ is the matter overdensity, $f={\rm d}\ln D(a)/{\rm d}\ln
a$ is the growth rate and the bias parameters $b_{\parallel}$ and $b_{\perp}$
are due to the mismatch between $P_{ge}^{\rm true}$ and $P_{ge}^{\rm fid}$,
and similarly for $P_{gm}$. Here, $D(a)$ is the linear-theory growth factor
for the matter spectrum that parametrises the time evolution of the matter
power-spectra, $P_{mm}(a)=D^{2}(a)P_{mm}(a=1)$. The combination of the galaxy
and the velocity satisfies

$\displaystyle P_{gg}(k,\mu)$ $\displaystyle=$ $\displaystyle(b_{g}+b_{\rm
rsd}f\mu^{2})^{2}P_{mm}(k)\,,$ (11) $\displaystyle P_{gv}(k,\mu)$
$\displaystyle=$ $\displaystyle b\left(\frac{faH}{k}\right)(b_{g}+b_{\rm
rsd}f\mu^{2})P_{mm}(k)\,,$ (12) $\displaystyle P_{vv}(k,\mu)$ $\displaystyle=$
$\displaystyle b^{2}\left(\frac{faH}{k}\right)^{2}P_{mm}(k)\,,$ (13)

where $b\in\\{b_{\parallel}\mu,b_{\perp}\\!\sqrt{1-\mu^{2}}\\}$ for the radial
and the transverse velocities, respectively. Here, $b_{g}$ is the linear
galaxy bias, $P_{gv}$ is the galaxy-velocity cross power spectrum and $P_{gg}$
is the galaxy auto power spectrum. Above, we included the RSD effect (Kaiser,
1987) $f\mu^{2}$ with a bias factor $b_{\rm rsd}$, which we discuss next.
These equations demonstrate the dependence of observables on cosmological
parameters.

###  III.2 Redshift space distortions

In the absence of measurement biases (and on large-scales) the effect of RSDs
is given by the Kaiser formula, which states that the galaxy profile $u_{g}$
is modified as

$\displaystyle u_{g}(\bm{k})\rightarrow\left(1+\beta\mu^{2}\right)u_{g}(k)\,,$
(14)

where $\beta=f/b_{g}$. Equivalently, the galaxy-density field in Fourier space
takes the form

$\displaystyle\delta_{g}(\bm{k})=(b_{g}+\mu^{2}f)\delta_{m}(\bm{k})+{\rm
noise}\,.$ (15)

Extracting cosmological information from $f\delta_{m}(\bm{k})$ assumes that
one can calibrate a Fourier mode measured from the galaxy density field to the
comoving $\bm{k}$, subject only to the knowledge of the distance scales,
$D_{a}(z)$ and $H(z)$, for $k_{\parallel}$ and $k_{\perp}$, respectively. This
can be done by using the BAO signature and the Alcock-Paczynski (Alcock and
Paczynski, 1979) method, internally for a galaxy survey, for example. In
reality, however, the galaxies are not randomly distributed and their
orientations show correlations with their local large-scale structure
environment (tidal fields, density, dust, etc.).

The appearance of the galaxy depends on the line-of-sight, expressed in the
galaxy’s frame of reference. This introduces an anisotropic selection function
when measuring galaxy clustering, which depends on the intrinsic alignment of
galaxies with its local environment (Hirata, 2009). In this work, we represent
such effects with an RSD bias as

$\displaystyle\delta_{g}(\bm{k})=(b_{g}+b_{\rm
rsd}\mu^{2}f)\delta_{m}(\bm{k})+{\rm noise}\,.$ (16)

Evidence for an RSD bias has recently been found at $5\sigma$ in Ref. (Obuljen
_et al._ , 2020) where authors find $b_{\rm rsd}$ variations on the order of
$10-20\%$ within the BOSS CMASS and LOWZ samples (Albareti  _et al._ , 2017).
Next, we demonstrate the improvement on cosmological constraints from the
moving-lens effect in the presence of an RSD bias.

##  IV Fisher analysis

Following Ref. (Smith  _et al._ , 2018), we use a simplified ‘snapshot’
geometry throughout this paper to represent the observed Universe, which we
take to be a 3D box with comoving side length $L$ and volume $V=L^{3}$, at a
constant redshift $z_{*}=1$ corresponding to a distance $\chi_{*}\simeq 3400$
Mpc.

The optical-depth degeneracy can be represented on large scales with a bias on
the radial-velocity measurement, in the form,

$\displaystyle{\hat{v}}_{\parallel}(\bm{k})=$
$\displaystyle\,b_{\parallel}v_{\parallel}^{\rm true}(\bm{k})$
$\displaystyle+\big{[}\mbox{noise with power spectrum
$N_{\parallel}(\bm{k})$}\,\big{]}\,,$

where ${v}^{\rm true}_{\parallel}$ is the true radial-velocity amplitude. The
radial velocity bias $b_{\parallel}$ depends only on the small-scale modelling
of the electron-galaxy correlation. It can be calculated as (Smith  _et al._ ,
2018):

$\displaystyle b_{\parallel}=\frac{\int{\rm d}k_{S}F(k_{S})P_{ge}^{\rm
true}(k_{S})}{\int{\rm d}k_{S}F(k_{S})P_{ge}^{\rm fid}(k_{S})}\,,$ (18)

where

$\displaystyle F(k_{S})=k_{S}\frac{P_{ge}^{\rm fid}(k_{S})}{P_{gg}^{\rm
obs}(k_{S})}\frac{1}{C_{\ell=k_{S}\chi}^{TT,\rm obs}}\,,$ (19)

which depends on the difference between the true galaxy-electron correlation,
$P_{ge}^{\rm true}$, and the fiducial model, $P_{ge}^{\rm fid}$.

A similar bias can also appear in the transverse-velocity reconstruction, as:

$\displaystyle{\hat{v}}_{\perp}(\bm{k})=\,$ $\displaystyle
b_{\perp}v_{\perp}^{\rm true}(\bm{k})$ $\displaystyle+\big{[}\mbox{noise, with
power spectrum $N_{\perp}(k,\mu)$}\big{]}\,,$

where ${v}^{\rm true}_{\perp}$ is the true transverse-velocity amplitude. For
the moving-lens reconstruction, this bias is due to the uncertain cross-
correlation between the gravitational potential and the density tracer. For a
galaxy field tracer, for example, this bias can be written as:

$\displaystyle b_{\perp}=\frac{\int{\rm d}k_{S}F(k_{S})P_{\Phi g}^{\rm
true}(k_{S})}{\int{\rm d}k_{S}F(k_{S})P_{\Phi g}^{\rm fid}(k_{S})}\,,$ (21)

where $F(k_{S})=k_{S}{P_{\Phi g}^{\rm fid}(k_{S})}/[{P_{gg}^{\rm
obs}(k_{S})}{C_{\ell=k_{S}\chi}^{TT,\rm obs}}]\,.$ Here, $P_{\Psi g}^{\rm
true}(k)$ is the true galaxy-potential correlation and $P_{\Psi g}^{\rm
fid}(k)$ is the fiducial model.

We take the true velocity field to be curl-free, i.e.

$v_{j}^{\rm true}(\bm{k})=i{\hat{k}}_{j}s(\bm{k})\,,$ (22)

where $s(\bm{k})$ is a scalar field with power spectrum
$P_{vv}(k)=(faH/k)^{2}P_{mm}(k)$. To avoid covariance matrices which are not
full rank, we combine $\hat{v}_{\parallel}$ and $\hat{v}_{\perp}$ into a
“unified” reconstruction $\hat{v}_{i}$ (a three-vector). Then, the two-point
function of $\hat{v}_{i}$ can be written as the sum of signal and noise
contributions from the velocity, as:

$\big{\langle}\hat{v}_{i}(\bm{k})\,\hat{v}_{j}(\bm{k}^{\prime})^{*}\big{\rangle}=\Big{(}S_{ij}(\bm{k})+N_{ij}(\bm{k})\Big{)}\,(2\pi)^{3}\delta^{3}(\bm{k}-\bm{k}^{\prime})\,.$
(23)

The average (shown with brackets) is over realisations of the three-velocity.

Given a wavenumber $\bm{k}_{i}$ and line-of-sight direction ${\hat{r}}_{i}$,
one natural choice of basis for the 3-by-3 matrices $S_{ij}$ and $N_{ij}$ is
the “radial/tangential” basis, where ${\hat{r}}_{i}$ is a basis vector:

${\hat{r}}_{i}=\left(\begin{array}[]{c}1\\\ 0\\\ 0\end{array}\right)\hskip
28.45274pt\bm{k}=\left(\begin{array}[]{c}k\mu\\\ k\sqrt{1-\mu^{2}}\\\
0\end{array}\right)\,.\hskip 28.45274pt$ (24)

In what follows, we add the galaxy field to Eq. (23), promoting
$S_{ij}(\bm{k})$ and $N_{ij}(\bm{k})$ to 4-by-4 matrices.

In the radial/tangential basis, the noise covariance [including the galaxy
field, $g(\bm{k})$] can be read off from Eqs. (IV) and (IV):

$\displaystyle N_{ij}(\bm{k})$ (25) $\displaystyle={\rm
diag}\\{N_{\parallel}(k,\mu),N_{\perp}(k,\mu),N_{\perp}(k,\mu),1/[W^{2}(k,\mu)n_{g}]\\}\,,$

where $n_{g}$ is the galaxy number-count density in a given box. To compute
the signal covariance, we combine Eqs. (IV) and (IV) to get

${\hat{v}}_{i}(\bm{k})=B_{ij}(\bm{k})v_{j}^{\rm
true}(\bm{k})+(\mbox{noise})\,,$ (26)

where the 4-by-4 bias matrix $B_{ij}(\bm{k})$ is given by

$\displaystyle B_{ij}(\bm{k})={\rm
diag}\\{b_{\parallel},b_{\perp},b_{\perp},b_{g}\\}\,.$ (27)

By Eq. (22), the signal covariance of $v_{i}^{\rm true}$ is

$\big{\langle}v_{i}^{\rm true}(\bm{k})v_{j}^{\rm
true}(\bm{k}^{\prime})^{*}\big{\rangle}={\hat{k}}_{i}{\hat{k}}_{j}P_{v}(k)\,(2\pi)^{3}\delta^{3}(\bm{k}-\bm{k}^{\prime})\,,$
(28)

and the signal covariance of ${\hat{v}}_{i}$ is

$\displaystyle S_{ij}=c_{i}c_{j}P_{vv}(k)\,,$ (29)

where

$\displaystyle c_{i}$ $\displaystyle=$
$\displaystyle(c_{\parallel},c_{\perp},0,c_{g})\,,$ (30) $\displaystyle
c_{\parallel}$ $\displaystyle\equiv$ $\displaystyle b_{\parallel}\mu\,,$ (31)
$\displaystyle c_{\perp}$ $\displaystyle\equiv$ $\displaystyle
b_{\perp}\sqrt{1-\mu^{2}}\,,$ (32) $\displaystyle c_{g}$ $\displaystyle\equiv$
$\displaystyle(b_{g}+b_{\rm rsd}f\mu^{2})\,{k}/({faH})\,.$ (33)

Given parameters $\pi_{1},\cdots,\pi_{N}$, the $N$-by-$N$ Fisher matrix is

$F_{ab}=\frac{V}{2}\int\frac{d^{3}\bm{k}}{(2\pi)^{3}}\,\mbox{Tr}\left[C^{-1}(\bm{k})\frac{\partial
S(\bm{k})}{\partial\pi_{a}}C^{-1}(\bm{k})\frac{\partial
S(\bm{k})}{\partial\pi_{b}}\right]\,,$ (34)

where $C=S+N$ is the total covariance. We can simplify a little by noting that
the signal and noise covariance matrices in Eqs. (IV) and (29) only depend on
$\bm{k}$ through $(k,\mu)$. This lets us reduce the 3D integral to a 2D
integral,

$\displaystyle F_{ab}=\frac{V}{2}$
$\displaystyle\int_{0}^{\infty}\frac{k^{2}\,dk}{4\pi^{2}}\int_{-1}^{1}d\mu$
(35) $\displaystyle\times$
$\displaystyle\mbox{Tr}\left[C^{-1}(k,\mu)\frac{\partial
S(k,\mu)}{\partial\pi_{a}}C^{-1}(k,\mu)\frac{\partial
S(k,\mu)}{\partial\pi_{b}}\right]\,.$

In our forecasts we consider the following 6 parameters:
$\\{f\sigma_{8},\,b_{g}\sigma_{8},\,b_{\parallel},\,b_{\perp},\,b_{\rm
rsd}\\}$.

##  V Results

Figure 3: Constraints on $\bm{f\sigma_{8}}$ from moving-lens tomography and
galaxy clustering. Coloured lines correspond to constraints from the
reconstructed transverse-velocity fields, and their cross-correlation with the
galaxy field on large scales (with no RSD bias prior). A range of CMB
experiment noise levels ($\Delta_{T}$) are shown and two beam sizes,
appropriate for CMB-S4 and CMB-HD. We scale the noise levels for all of the
frequency bands of these experiments. The ILC calculation is discussed in
Appendix A. Blue (orange) markers show anticipated noise levels of CMB-S4
(CMB-HD). We use Rubin Observatory Year 10 (VROY10) for moving-lens
tomography. Note that here the radial-velocity reconstruction only marginally
improves the constraints on $f\sigma_{8}$: the constraints are dominated by
the correlation of the transverse-velocity field and galaxy clustering. We
consider a $1\%$ prior on the transverse-velocity bias [defined in Eq. (21)],
from an external measurement of the matter-galaxy correlation on small scales,
such as galaxy and lensing correlations, for example. The $1\%$ constraints on
the transverse-velocity bias falls near the scenario where $b_{\perp}$ is
fixed and not marginalised. Finally, we show constraints on the $f\sigma_{8}$
parameter from measurements of the RSD effect, using the spectroscopic DESI
galaxy survey. The dotted gray horizontal lines corresponds to $5\%-1\%$
priors on the RSD bias from top to bottom. The solid gray horizontal line
correspond to fixing $b_{\rm rsd}$ to 1. Recent studies show evidence for the
case $b_{\rm rsd}\neq 1$ (Obuljen  _et al._ , 2020).  Figure 4: Constraints on
$\bm{b_{\parallel}}$ from combination of moving-lens tomography, kSZ
tomography and galaxy clustering. Coloured lines correspond to constraints
from the reconstructed transverse- and radial-velocity fields, and their
cross-correlation with the galaxy field on large scales (with no RSD bias
prior); similar to Figure 3, with $1\%$ prior on the transverse-velocity bias.
We use Rubin Observatory Year 10 (VROY10) survey for kSZ and moving-lens
tomography. Gray lines correspond to constraints on $b_{\parallel}$ from
measurements of the RSD effect and kSZ tomography, using the spectroscopic
DESI galaxy survey and the CMB-S4 experimental specifications. The dotted gray
lines corresponds to $5\%-1\%$ priors on the RSD bias from top to bottom. The
solid grey line correspond to fixing $b_{\rm rsd}$ to 1.

We consider a range of CMB experiments with varying noise, including SO and
CMB-S4, as well as the futuristic CMB-HD survey. The galaxy-survey (shot)
noise depends on the number-density function, $n_{g}(z)$. We consider three
LSS experiments: DESI, Vera Rubin Observatory Year 1 (VROY1) and Year 10
(VROY10). The parameters for VRO are based on (Hearin  _et al._ , 2012; The
LSST Dark Energy Science Collaboration  _et al._ , 2018) and those for DESI
are based on (DESI Collaboration  _et al._ , 2016). For DESI, we take a 116
Gpc3 box with galaxy survey density $n_{g}=1.4\times 10^{-4}/$Mpc3 and
$\sigma_{z}=0$. For VROY1, we take a 113.4 Gpc3 box, with galaxy survey
density $n_{g}=6.9\times 10^{-3}/$Mpc3. For VROY10, we take a 180Gpc3 box,
with galaxy survey density $n_{g}=1.2\times 10^{-2}/$Mpc3. For both VROY1 and
VROY10, we consider photo-$z$ errors satisfying $\sigma_{z}=0.06$ at $z=1$.
For $\\{$DESI, VROY1, VROY10$\\}$, we consider a galaxy bias of
$\\{1.5,1.7,1.6\\}$, respectively. We take all the boxes to be centered at
$z=1$. For all forecasts, we set the maximum wavenumber we include in our
analysis to $k_{\rm max}=0.1/$Mpc.

The total CMB power spectrum gets contributions from weak gravitational
lensing, the kSZ effect (both from reionization and late times) and the
experimental noise, which we take to satisfy

$\displaystyle
N_{\ell}=\Delta_{T}^{2}\exp\left[\frac{\ell(\ell+1)\theta^{2}_{\rm FWHM}}{8\ln
2}\right]\,.$ (36)

Note that when forecasting the CMB noise for these experiments, we also
include the contribution to the power spectra from residual foregrounds after
foreground cleaning we discuss in Appendix A. Fig. 1 demonstrates how the CMB
noise and different residual foregrounds compare with the CMB signal.

Table 1: The detection signal-to-noise (SNR) of the (reconstructed) velocity and galaxy-density cross-correlation: $P_{\hat{v}g}(k)$. Velocities are reconstructed from the kSZ and moving-lens (ML) tomography for various CMB and LSS experiments. VROY1 and VROY10 refer to Vera Rubin Observatory Year 1 and 10 respectively. kSZ SNR |  | CMB |  | ML SNR |  | CMB |
---|---|---|---|---|---|---|---
LSS |  SO | S4 | HD | LSS |  SO | S4 | HD
DESI | 231 | 414 | 1170 | DESI | 7 | 8 | 62
VROY1 | 116 | 210 | 669 | VROY1 | 16 | 28 | 85
VROY10 | 123 | 228 | 775 | VROY10 | 20 | 37 | 112

In Table 1, we show the detection signal-to-noise ratio (SNR) of the
reconstructed velocity and galaxy density cross-correlation $P_{\hat{v}g}(k)$
from our experimental configurations. The velocities are reconstructed either
form the moving-lens (ML) effect or the kSZ tomography. We find similar SNR
forecasts compared to previous results (Hotinli  _et al._ , 2019, 2021).
Nevertheless, note that our calculation of the transverse velocity noise
differs from earlier work by the use of the simplified box formalism
here.333Our forecasts for the moving-lens SNR differs from Refs. (Hotinli  _et
al._ , 2019, 2021), in various ways: First, Refs (Hotinli  _et al._ , 2019,
2021) included a map-based reconstruction on the two-sphere and with 11
redshift bins in the range $z\in[0,3]$, unlike our single-box formalism which
is simpler but potentially more optimistic than the former. Furthermore,
forecasts in Refs (Hotinli  _et al._ , 2019, 2021) did not include the adverse
effects of residual foregrounds after foreground cleaning (except the effects
of kSZ and weak lensing), while including delensed CMB spectra with ideal
noise. Delensing noticeably improves the detection prospects of reconstruction
at the low-noise limit, hence important for experiments like CMB-S4 and CMB-
HD, for example. Despite these differences, we find matching SNR forecasts for
CMB-S4 and SO.

We show forecasts on $f\sigma_{8}$ in Fig. 3. We find that moving-lens
tomography, together with the galaxy density, can constrain $f\sigma_{8}$ to
high precision, comparable to the scenario where the RSD bias is known up to
few percent accuracy. For SO and VROY10, improvements on $f\sigma_{8}$
relative to RSD can be achieved if the RSD bias is uncertain at the level of
$\sigma(b_{\rm rsd})>5\%$. Using S4 and VROY10 further improves the benefit
from using the transverse velocities, while constraints from CMB-HD and VROY10
improves upon the scenario where the RSD bias is known up to around $1-2\%$
accuracy.

Note that in our forecasts we considered a $1\%$ prior on the transverse
velocity field. This can potentially be provided from external measurements of
the galaxy-matter cross-correlation on small scales (from cross-correlation of
galaxy-lensing measurements, for example) as defined in Eq. (21). For upcoming
experiments, we find $\lesssim 1\%$ prior on the transverse-velocity bias
recovers the cosmic-variance limit with the transverse-velocity bias fixed as
$b_{\perp}=1$.

Fig. 4 shows the constraints on $b_{\parallel}$ from a combination of moving-
lens tomography, kSZ tomography, and galaxy clustering. Moving-lens
measurement of $f\sigma_{8}$ removes the degeneracy between the radial-
velocity bias $b_{\parallel}$ and $f\sigma_{8}$, suffered by the radial
velocities reconstructed with kSZ tomography. This allows kSZ to measure
directly $b_{\parallel}$, allowing the possibility of astrophysical inference.

In Fig. 5, we show the ratio of the errors on $f\sigma_{8}$ obtained from the
DESI RSDs and from forecasts combining VRO galaxies and the moving-lens
tomography. The RSDs are taken to be biased for both calculations with a
varying prior assumption on $\sigma(b_{\rm rsd})$ (shown on the $x$ axis). Our
results suggest that moving-lens tomography improves the $f\sigma_{8}$
constraints for $\sigma(b_{\rm rsd})/b_{\rm rsd}\gtrsim{\rm few}\%$. Note that
in the absence of prior knowledge on the RSD bias [or for $\sigma(b_{\rm
rsd})\sim 5\%$ or worse], moving-lens tomography may play an essential role in
our ability to constrain cosmology from velocity reconstruction. So far, the
only other observable that can constrain $f\sigma_{8}$ (together with the kSZ
tomography) is the FRB dispersion measurements, which allow the breaking of
the optical-depth degeneracy of the kSZ signal (Madhavacheril  _et al._ ,
2019). Our forecasts suggest the moving-lens effect may provide a constraining
power comparable to measurement of $\sim\\!10^{5}$ FRBs, depending on the
uncertainty on the FRB dispersion measurement (Madhavacheril  _et al._ ,
2019).

Figure 5: The fractional improvement in $\bm{\sigma(f\sigma_{8})}$ from the
moving-lens effect, relative to the constraint from biased galaxy RSDs. The
ratio of constraints from DESI RSDs (with varying RSD bias prior) and the
combined measurement of biased galaxy RSDs and transverse velocities is shown
for various experimental configurations.

##  VI Discussion

In this work, we have explored one of the cosmological applications of
reconstructing the transverse-velocity field from the moving-lens effect.
Velocity fields are direct tracers of the combination of cosmic growth rate
and the amplitude of matter fluctuations and contain valuable cosmological
information. While velocity measurements from the kSZ and the moving-lens
effects are biased due to uncertain electron-galaxy and matter-galaxy cross-
correlation on small-scales, respectively, the latter degeneracy can be broken
by relatively robust measurements such as correlations of the galaxy density
and (galaxy or CMB) weak lensing. Overcoming the former “optical depth”
degeneracy, however, is likely more difficult. Galaxy clustering is also
sensitive to cosmic growth due to the RSD effect (Kaiser, 1987), but future
measurements may be biased by selection effects. Using moving-lens
reconstructions that combine the Vera Rubin Observatory and CMB surveys on the
other hand provides strong constraints on cosmological parameters such as
$f\sigma_{8}$ at the level of $\lesssim 5\%$, $3\%$ and $2\%$ for the Simons
Observatory, CMB-S4 and CMB-HD experiments respectively.

We focused on systematics in the RSD effect, but moving forward, a more
detailed analysis involving the modelling of all relevant effects will give
further insight into the benefit of moving-lens tomography. These effects can
be separated into two categories. First, the theory predictions from our
models are often gauge dependent and defined in comoving coordinates, rather
than in redshift space. Mapping from the theory to gauge-independent
observables (such as the galaxy power spectrum or CMB temperature) requires a
multitude of correction terms on top of the density fluctuations, which take
into account the photon geodesics in an inhomogeneous Universe. These include
general relativistic effects such as Doppler (magnification) terms, potential
(Sachs-Wolfe, integrated Sachs-Wolfe, time delay) terms and weak lensing, as
well as anisotropies in the mapping from real space to redshift space induced
by peculiar velocities (RSDs). Second, galaxy clustering measurements are
subject to selection effects, where detecting particular galaxies with various
properties may be more or less likely than on average. These effects include
tidal alignments (Hirata, 2009), lensing magnification (Hui  _et al._ , 2007),
and redshift-evolution biases (e.g. Alonso  _et al._ , 2015). These biases can
complicate the mapping between theory and observed quantities and should be
explored in future work.

The upcoming decade will host a wealth of new and high-quality data. These
developments will open many novel opportunities for cosmological inference. In
particular, using the CMB as a cosmological backlight, secondary fluctuations
induced by the interaction between CMB photons and large-scale structure allow
new methods like transverse-velocity reconstruction from the moving-lens
effect, as described in this paper. These new observables provide the
different tracers that can potentially improve the constraints on cosmological
parameters, beyond the limits imposed by selection effects, astrophysical
uncertainties, and cosmic variance.

##  VII Acknowledgements

We thank J. Colin Hill for comments and suggestions on the ILC noise
calculation. SCH is supported by the Horizon Fellowship from Johns Hopkins
University. SCH was also supported by a postdoctoral grant from Imperial
College London and a Perimeter Visiting Fellowship. SCH thanks Matthew Johnson
and Juan Cayuso for useful conversations. MK was supported by NSF Grant No.
1818899 and the Simons Foundation. Research at Perimeter Institute is
supported in part by the Government of Canada through the Department of
Innovation, Science and Industry Canada and by the Province of Ontario through
the Ministry of Colleges and Universities.

## References

  * Ade  _et al._ (2019) Peter Ade, James Aguirre, Zeeshan Ahmed, Simone Aiola, Aamir Ali, David Alonso, Marcelo A. Alvarez, Kam Arnold, Peter Ashton, Jason Austermann, and et al., “The Simons Observatory: science goals and forecasts,” J. Cosmology Astropart. Phys 2019, 056 (2019), arXiv:1808.07445 [astro-ph.CO] .
  * Lee  _et al._ (2019) Adrian Lee, Maximilian H. Abitbol, Shunsuke Adachi, Peter Ade, James Aguirre, Zeeshan Ahmed, Simone Aiola, Aamir Ali, David Alonso, Marcelo A. Alvarez, and et al., “The Simons Observatory,” in  _Bulletin of the American Astronomical Society_ , Vol. 51 (2019) p. 147, arXiv:1907.08284 [astro-ph.IM] .
  * Abazajian  _et al._ (2016) Kevork N. Abazajian _et al._ , “CMB-S4 Science Book, First Edition,” arXiv e-prints , arXiv:1610.02743 (2016), arXiv:1610.02743 [astro-ph.CO] .
  * Aghamousa  _et al._ (2016) Amir Aghamousa _et al._ (DESI), “The DESI Experiment Part I: Science,Targeting, and Survey Design,” (2016), arXiv:1611.00036 [astro-ph.IM] .
  * LSST Science Collaboration  _et al._ (2009) LSST Science Collaboration _et al._ , “LSST Science Book, Version 2.0,” ArXiv e-prints (2009), arXiv:0912.0201 [astro-ph.IM] .
  * Seljak (2009) Uroš Seljak, “Extracting Primordial Non-Gaussianity without Cosmic Variance,” Phys. Rev. Lett. 102, 021302 (2009), arXiv:0807.1770 [astro-ph] .
  * Zhang and Johnson (2015) Pengjie Zhang and Matthew C. Johnson, “Testing eternal inflation with the kinetic Sunyaev Zel’dovich effect,” J. Cosmology Astropart. Phys 2015, 046 (2015), arXiv:1501.00511 [astro-ph.CO] .
  * Banerjee  _et al._ (2018) Arka Banerjee, Bhuvnesh Jain, Neal Dalal, and Jessie Shelton, “Tests of neutrino and dark radiation models from galaxy and CMB surveys,” J. Cosmology Astropart. Phys 2018, 022 (2018), arXiv:1612.07126 [astro-ph.CO] .
  * Schmittfull and Seljak (2018) Marcel Schmittfull and Uroš Seljak, “Parameter constraints from cross-correlation of CMB lensing with galaxy clustering,” Phys. Rev. D 97, 123540 (2018), arXiv:1710.09465 [astro-ph.CO] .
  * Modi  _et al._ (2017) Chirag Modi, Martin White, and Zvonimir Vlah, “Modeling CMB lensing cross correlations with CLEFT,” J. Cosmology Astropart. Phys 2017, 009 (2017), arXiv:1706.03173 [astro-ph.CO] .
  * Deutsch  _et al._ (2018a) Anne-Sylvie Deutsch, Emanuela Dimastrogiovanni, Matthew C. Johnson, Moritz Münchmeyer, and Alexandra Terrana, “Reconstruction of the remote dipole and quadrupole fields from the kinetic Sunyaev Zel’dovich and polarized Sunyaev Zel’dovich effects,” Phys. Rev. D 98, 123501 (2018a), arXiv:1707.08129 [astro-ph.CO] .
  * Münchmeyer  _et al._ (2019) Moritz Münchmeyer, Mathew S. Madhavacheril, Simone Ferraro, Matthew C. Johnson, and Kendrick M. Smith, “Constraining local non-Gaussianities with kinetic Sunyaev-Zel’dovich tomography,” Phys. Rev. D 100, 083508 (2019), arXiv:1810.13424 [astro-ph.CO] .
  * Cayuso and Johnson (2020) Juan I. Cayuso and Matthew C. Johnson, “Towards testing CMB anomalies using the kinetic and polarized Sunyaev-Zel’dovich effects,” Phys. Rev. D 101, 123508 (2020), arXiv:1904.10981 [astro-ph.CO] .
  * Pan and Johnson (2019) Zhen Pan and Matthew C. Johnson, “Forecasted constraints on modified gravity from Sunyaev-Zel’dovich tomography,” Phys. Rev. D 100, 083522 (2019), arXiv:1906.04208 [astro-ph.CO] .
  * Hotinli  _et al._ (2019) Selim C. Hotinli, James B. Mertens, Matthew C. Johnson, and Marc Kamionkowski, “Probing correlated compensated isocurvature perturbations using scale-dependent galaxy bias,” Phys. Rev. D100, 103528 (2019), arXiv:1908.08953 [astro-ph.CO] .
  * Zeldovich and Sunyaev (1969) Ya. B. Zeldovich and R. A. Sunyaev, “The Interaction of Matter and Radiation in a Hot-Model Universe,” Ap&SS 4, 301–316 (1969).
  * Zel’Dovich (1970) Y. B. Zel’Dovich, “Gravitational instability: an approximate theory for large density perturbations.” A&A 500, 13–18 (1970).
  * Sunyaev and Zeldovich (1980) R. A. Sunyaev and I. B. Zeldovich, “Microwave background radiation as a probe of the contemporary structure and history of the universe,” ARA&A 18, 537–560 (1980).
  * Sunyaev and Zeldovich (1972) R. A. Sunyaev and Ya. B. Zeldovich, “The Observations of Relic Radiation as a Test of the Nature of X-Ray Radiation from the Clusters of Galaxies,” Comments on Astrophysics and Space Physics 4, 173 (1972).
  * Sazonov and Sunyaev (1999) S. Y. Sazonov and R. A. Sunyaev, “Microwave polarization in the direction of galaxy clusters induced by the CMB quadrupole anisotropy,” MNRAS 310, 765–772 (1999), arXiv:astro-ph/9903287 [astro-ph] .
  * Sachs and Wolfe (1967) R. K. Sachs and A. M. Wolfe, “Perturbations of a Cosmological Model and Angular Variations of the Microwave Background,” ApJ 147, 73 (1967).
  * Birkinshaw and Gull (1983) M. Birkinshaw and S. F. Gull, “A test for transverse motions of clusters of galaxies,” Nature 302, 315–317 (1983).
  * Deutsch  _et al._ (2018b) Anne-Sylvie Deutsch, Matthew C. Johnson, Moritz Münchmeyer, and Alexandra Terrana, “Polarized Sunyaev Zel’dovich tomography,” J. Cosmology Astropart. Phys 2018, 034 (2018b), arXiv:1705.08907 [astro-ph.CO] .
  * Deutsch  _et al._ (2019) Anne-Sylvie Deutsch, Emanuela Dimastrogiovanni, Matteo Fasiello, Matthew C. Johnson, and Moritz Münchmeyer, “Primordial gravitational wave phenomenology with polarized Sunyaev Zel’dovich tomography,” Phys. Rev. D 100, 083538 (2019), arXiv:1810.09463 [astro-ph.CO] .
  * Smith  _et al._ (2018) Kendrick M. Smith, Mathew S. Madhavacheril, Moritz Münchmeyer, Simone Ferraro, Utkarsh Giri, and Matthew C. Johnson, “KSZ tomography and the bispectrum,” arXiv e-prints , arXiv:1810.13423 (2018), arXiv:1810.13423 [astro-ph.CO] .
  * Madhavacheril  _et al._ (2019) Mathew S. Madhavacheril, Nicholas Battaglia, Kendrick M. Smith, and Jonathan L. Sievers, “Cosmology with kSZ: breaking the optical depth degeneracy with Fast Radio Bursts,” arXiv e-prints , arXiv:1901.02418 (2019), arXiv:1901.02418 [astro-ph.CO] .
  * Contreras  _et al._ (2019) Dagoberto Contreras, Matthew C. Johnson, and James B. Mertens, “Towards detection of relativistic effects in galaxy number counts using kSZ tomography,” J. Cosmology Astropart. Phys 2019, 024 (2019), arXiv:1904.10033 [astro-ph.CO] .
  * Hotinli and Johnson (2020) Selim C. Hotinli and Matthew C. Johnson, “Reconstructing large scales at cosmic dawn,” (2020), arXiv:2012.09851 [astro-ph.CO] .
  * Giri and Smith (2020) Utkarsh Giri and Kendrick M. Smith, “Exploring KSZ velocity reconstruction with $N$-body simulations and the halo model,” arXiv e-prints , arXiv:2010.07193 (2020), arXiv:2010.07193 [astro-ph.CO] .
  * Kaiser (1987) Nick Kaiser, “Clustering in real space and in redshift space,” MNRAS 227, 1–21 (1987).
  * Obuljen  _et al._ (2020) Andrej Obuljen, Will J. Percival, and Neal Dalal, “Detection of anisotropic galaxy assembly bias in BOSS DR12,” J. Cosmology Astropart. Phys 2020, 058 (2020), arXiv:2004.07240 [astro-ph.CO] .
  * Hotinli  _et al._ (2019) Selim C. Hotinli, Joel Meyers, Neal Dalal, Andrew H. Jaffe, Matthew C. Johnson, James B. Mertens, Moritz Münchmeyer, Kendrick M. Smith, and Alexander van Engelen, “Transverse Velocities with the Moving Lens Effect,” Phys. Rev. Lett. 123, 061301 (2019), arXiv:1812.03167 [astro-ph.CO] .
  * Yasini  _et al._ (2019) Siavash Yasini, Nareg Mirzatuny, and Elena Pierpaoli, “Pairwise Transverse Velocity Measurement with the Rees-Sciama Effect,” ApJ 873, L23 (2019), arXiv:1812.04241 [astro-ph.CO] .
  * Hotinli  _et al._ (2021) Selim C. Hotinli, Matthew C. Johnson, and Joel Meyers, “Optimal filters for the moving lens effect,” Phys. Rev. D 103, 043536 (2021), arXiv:2006.03060 [astro-ph.CO] .
  * Linder (2005) Eric V. Linder, “Cosmic growth history and expansion history,” Phys. Rev. D 72, 043529 (2005), arXiv:astro-ph/0507263 [astro-ph] .
  * Linder and Cahn (2007) Eric V. Linder and Robert N. Cahn, “Parameterized beyond-Einstein growth,” Astroparticle Physics 28, 481–488 (2007), arXiv:astro-ph/0701317 [astro-ph] .
  * Bond and Szalay (1983) J. R. Bond and A. S. Szalay, “The collisionless damping of density fluctuations in an expanding universe,” ApJ 274, 443–468 (1983).
  * Battaglia (2016) N. Battaglia, “The tau of galaxy clusters,” J. Cosmology Astropart. Phys 2016, 058 (2016), arXiv:1607.02442 [astro-ph.CO] .
  * (39) Juan Cayuso, Richard Bloch, Selim C. Hotinli, and Matthew Johnson, “Upcoming,” .
  * Alcock and Paczynski (1979) C. Alcock and B. Paczynski, “An evolution free test for non-zero cosmological constant,” Nature 281, 358 (1979).
  * Hirata (2009) Christopher M. Hirata, “Tidal alignments as a contaminant of redshift space distortions,” MNRAS 399, 1074–1087 (2009), arXiv:0903.4929 [astro-ph.CO] .
  * Albareti  _et al._ (2017) Franco D. Albareti _et al._ , “The 13th Data Release of the Sloan Digital Sky Survey: First Spectroscopic Data from the SDSS-IV Survey Mapping Nearby Galaxies at Apache Point Observatory,” ApJS 233, 25 (2017), arXiv:1608.02013 [astro-ph.GA] .
  * Hearin  _et al._ (2012) A. P. Hearin, A. R. Zentner, and Z. Ma, “General requirements on matter power spectrum predictions for cosmology with weak lensing tomography,” J. Cosmology Astropart. Phys 4, 034 (2012), arXiv:1111.0052 .
  * The LSST Dark Energy Science Collaboration  _et al._ (2018) The LSST Dark Energy Science Collaboration, Rachel Mandelbaum, Tim Eifler, Renée Hložek, Thomas Collett, Eric Gawiser, and others Scolnic, “The LSST Dark Energy Science Collaboration (DESC) Science Requirements Document,” arXiv e-prints , arXiv:1809.01669 (2018), arXiv:1809.01669 [astro-ph.CO] .
  * DESI Collaboration  _et al._ (2016) DESI Collaboration, Amir Aghamousa, Jessica Aguilar, Steve Ahlen, Shadab Alam, Lori E. Allen, Carlos Allende Prieto, James Annis, Stephen Bailey, Christophe Balland, and et al., “The DESI Experiment Part I: Science,Targeting, and Survey Design,” arXiv e-prints , arXiv:1611.00036 (2016), arXiv:1611.00036 [astro-ph.IM] .
  * Hui  _et al._ (2007) Lam Hui, Enrique Gaztañaga, and Marilena Loverde, “Anisotropic magnification distortion of the 3D galaxy correlation. I. Real space,” Phys. Rev. D 76, 103502 (2007), arXiv:0706.1071 [astro-ph] .
  * Alonso  _et al._ (2015) David Alonso, Philip Bull, Pedro G. Ferreira, Roy Maartens, and Mário G. Santos, “Ultra-large-scale Cosmology in Next-generation Experiments with Single Tracers,” ApJ 814, 145 (2015), arXiv:1505.07596 [astro-ph.CO] .
  * Abazajian  _et al._ (2019) Kevork Abazajian _et al._ , “CMB-S4 Science Case, Reference Design, and Project Plan,” arXiv e-prints , arXiv:1907.04473 (2019), arXiv:1907.04473 [astro-ph.IM] .
  * Madhavacheril  _et al._ (2017) Mathew S. Madhavacheril, Nicholas Battaglia, and Hironao Miyatake, “Fundamental physics from future weak-lensing calibrated Sunyaev-Zel’dovich galaxy cluster counts,” Phys. Rev. D 96, 103525 (2017), arXiv:1708.07502 [astro-ph.CO] .
  * Dunkley  _et al._ (2013) J. Dunkley, E. Calabrese, J. Sievers, _et al._ , “The Atacama Cosmology Telescope: likelihood for small-scale CMB data,” J. Cosmology Astropart. Phys 2013, 025 (2013), arXiv:1301.0776 [astro-ph.CO] .
  * Lagache  _et al._ (2020) G. Lagache, M. Béthermin, L. Montier, P. Serra, and M. Tucci, “Impact of polarised extragalactic sources on the measurement of CMB B-mode anisotropies,” A&A 642, A232 (2020), arXiv:1911.09466 [astro-ph.CO] .
  * Lewis  _et al._ (2000) Antony Lewis, Anthony Challinor, and Anthony Lasenby, “Efficient Computation of Cosmic Microwave Background Anisotropies in Closed Friedmann-Robertson-Walker Models,” ApJ 538, 473–476 (2000), arXiv:astro-ph/9911177 [astro-ph] .
  * Park  _et al._ (2013) Hyunbae Park, Paul R. Shapiro, Eiichiro Komatsu, Ilian T. Iliev, Kyungjin Ahn, and Garrelt Mellema, “The Kinetic Sunyaev-Zel’dovich Effect as a Probe of the Physics of Cosmic Reionization: The Effect of Self-regulated Reionization,” ApJ 769, 93 (2013), arXiv:1301.3607 [astro-ph.CO] .

##  Appendix A Internal Linear Combination noise curves

| Beam FWHM | Noise RMS
---|---|---
|  | ($\mu$K-arcmin)
| SO | S4 | HD | SO | S4 | HD
39 GHz | $5.1^{\prime}$ | $5.1^{\prime}$ | $36.3^{\prime\prime}$ | 36 | 12.4 | 3.4
93 GHz | $2.2^{\prime}$ | $2.2^{\prime}$ | $15.3^{\prime\prime}$ | 8 | 2.0 | 0.6
145 GHz | $1.4^{\prime}$ | $1.4^{\prime}$ | $10.0^{\prime\prime}$ | 10 | 2.0 | 0.6
225 GHz | $1.0^{\prime}$ | $1.0^{\prime}$ | $6.6^{\prime\prime}$ | 22 | 6.9 | 1.9
280 GHz | $0.9^{\prime}$ | $0.9^{\prime}$ | $5.4^{\prime\prime}$ | 54 | 16.7 | 4.6
Table 2: Inputs to ILC noise:  The beam and noise RMS parameters we assume for
survey configurations roughly corresponding to Simons Observatory (baseline),
CMB-S4 and CMB-HD. We do not include the 30 GHz channel and do not include
atmospheric noise since the kSZ and moving lens information is primarily in
scales $\ell>2000$.

We describe here the procedure used to analytically calculate the effective
total noise in CMB maps for the Simons Observatory (SO), CMB-S4 and CMB-HD
surveys, guided by the Internal Linear Combination (ILC) algorithm that would
be used in a realistic analysis. Our approach here is approximate and the
resulting noise curves are only meant to be roughly representative of the
forecast performance of these surveys; we do however find 10-20% agreement of
the SO and CMB-S4 results with the official noise curves in Ade  _et al._
(2019) and Abazajian  _et al._ (2019), respectively. In particular, for SO, we
find better than 10% agreement with the official forecasts for almost all
relevant multipoles. We employ this analytic approach (as opposed to the
simulation-based work in Ade  _et al._ (2019) and Abazajian  _et al._ (2019))
so as to be able to easily propagate the results of varying the white noise
RMS in the CMB experiments.

We assume that each of these surveys measures the millimeter sky at 39, 93,
145, 225 and 280 GHz with beam FWHM and white noise RMS as shown in Table 2.
We do not include large-scale atmospheric noise but do not expect this to make
a difference for our forecasts given the availability of Planck data at 44,
100, 143 and 217 GHz as well as the fact that most of the kSZ and moving lens
information is derived from small scales with $\ell>2000$.

For the signal contributions in the millimeter sky, we include contributions
from tSZ, clustered CIB and conservative (large) levels of Poisson CIB
foregrounds at these frequencies as well as the black-body late-time kSZ
following the approach in Madhavacheril  _et al._ (2017), which is based on
fits to ACT data from Dunkley  _et al._ (2013), but we do not include the tSZ-
CIB correlation. We include radio sources in the 39, 93 and 145 GHz channels
using the flux-limit-dependent radio source power model from Lagache  _et al._
(2020), where for both SO and S4 we assume flux limits in those channels of
10, 7 and 10 mJy respectively. For HD, we assume lower flux limits of 2, 1 and
1 mJy respectively. In addition, we include the black-body lensed CMB
contribution calculated using CAMB Lewis  _et al._ (2000) as well as the
reionization kSZ signal from Park  _et al._ (2013). We do not include the
contribution from the moving lens signal itself since it is sub-dominant to
the sum of the above.

Given the above sky model involving signal components correlated across
frequencies indexed by $i$ (and uncorrelated beam-deconvolved noise), the
total covariance between two frequency channels at each multipole $\ell$ is
${\bm{C}}^{ij}_{\ell}$. The final minimum-variance (standard ILC) noise for
the black-body CMB+kSZ signal that includes all contributions (including the
sample variance in the CMB and kSZ) is then given by
$N_{\ell}=\left[\sum_{ij}\left({\bm{C}}^{-1}\right)^{ij}_{\ell}\right]^{-1}$.
These curves are shown in Figure 1.


# Generalization of interlayer tunneling models to cuprate superconductors
with charge density waves

Hércules Santana and E. V. L. de Mello  Instituto de Física, Universidade
Federal Fluminense, 24210-346 Niterói, RJ, Brazil<EMAIL_ADDRESS>
###### Abstract

At the beginning of cuprate superconductors, the interlayer tunneling (ILT)
and Lawrence-Doniach (L-D) models, which connect the CuO planes by Josephson
coupling, were considered the leading theoretical proposals for these
materials. However, measurements of the interlayer magnetic penetration depth
$\lambda_{c}$ yielded larger values than required by the ILT model. After the
discovery of planar stripes and incommensurate charge ordering, it was also
possible to consider Josephson coupling between these mesoscopic charge
domains or blocks. We show that the average intralayer is larger than the
interlayer coupling and comparable with the condensation energy, leading to a
superconducting transition by long-range phase order. Another consequence is
that the ratio $[\lambda_{c}/\lambda_{ab}]^{2}$ is related to the resistivity
ratio $\rho_{c}/\rho_{ab}$ near the superconducting transition temperature in
agreement with several measurements.

##  I Introduction

Several experiments on high-temperature cuprate superconductors (HTS) verified
their large anisotropic properties that arise mainly because of the much
smaller resistivity ($\rho_{\rm ab}$) along with the CuO layers. These facts
suggested that the ILTClem (1989, 1991); Chakravarty  _et al._ (1993);
Anderson (1995); Leggett (1996) or L-D modelsLawrence and Doniach (1971),
which describes a layered superconductor as a stack of Josephson-coupled
adjacent blocks or layers were ideal candidates to describe the HTS. The ILT
model of Anderson and collaboratorsChakravarty  _et al._ (1993); Anderson
(1995) consisted of non-Fermi liquid planar electrons and, in the
superconducting (SC) phase, interlayer tunneling of Cooper pairs. This
approach results in a strong decrease of the $c$-axis kinetic energy with a
concomitant increase of the condensation energy (the gain of free energy in
the SC state compared with the normal state).

On the experimental side, Shibauchi et alShibauchi  _et al._ (1994) measured
the magnetic penetration depth $\lambda_{\rm c}$ and the planar $\lambda_{\rm
ab}$ in single crystals of La2-xSrxCuO4 (LSCO) and found that $\lambda_{\rm
c}(T)$ was in good agreement with the L-D model. However, images of interlayer
Josephson vorticesMoler  _et al._ (1998) in single-layer compounds Tl2Ba2CuO6
yielded about 20 $\mu$m which is about 20 times the penetration depth
determined by the ILT modelLeggett (1996). This result was considered a strong
evidence against the ILT and L-D models to cupratesMoler  _et al._ (1998);
Anderson (1998) and they were abandoned as the leading general HTS theories.

On the other hand, over the years, a significant number of new experiments
with novel techniques and methods revealed properties not known when the ILT
was originally proposed, which opened new possibilities: In particular, charge
inhomogeneities in the form of stripes were discovered in underdoped Nd
substituted in LSCO by neutron scatteringTranquada  _et al._ (1995), which was
a key experimet to the detection of charge-ordering (CO) or charge density
waves (CDW) phenomena in HTS. Along this line, scanning tunneling microscopy
(STM) experiments made it possible to obtain atomically resolved mapsLang  _et
al._ (2002); Fischer  _et al._ (2007) on the energy-dependent local density of
states (LDOS). More recently, resonant x-ray diffraction (REX) revealed the
subtle variations of the CO wave length $\lambda_{CO}$ with the doping level
of several families of cupratesComin and Damascelli (2016).

To interpret their inhomogeneous STM data on underdoped Bi2Sr2CaCu2O8+δ
(Bi2212), Lang et alLang  _et al._ (2002) proposed a structure of mesoscopic
superconductors grains connected by Josephson coupling. This original proposal
was not generally accepted, mainly because there was no evidence of CO
instability near the optimal and in the overdoped region. However a few years
later, similar CO granular patterns were observed by STM near the optimal
valueWise  _et al._ (2008) and even in the overdoped regions Gomes  _et al._
(2007); Parker  _et al._ (2010); He  _et al._ (2014). Furthermore, a variety
of complementary experimental probes detected charge instability in all hole-
doped HTS familiesComin and Damascelli (2016) as well as in Nd-based electron-
dopedda Silva Neto  _et al._ . Recently, charge inhomogeneities have been
detected in overdoped LSCO up to at least $x=p=0.21$Wu  _et al._ (2017); Chen
_et al._ (2019); Fei  _et al._ (2019) and possibly up to $p=0.25$Miao  _et
al._ (2021). Therefore, the ubiquitous presence of CDW in all HTS compounds
suggested that they are intertwined with the SC phase and somehow related to
the SC interactionde Mello (2012); de Mello and Sonier (2014, 2017); de Mello
(2020).

To understand the way they intertwine we use the Cahn-Hilliard (CH) equation
that can simulate the observed CO wavelength $\lambda_{\rm CO}$ of different
materials employing a phase separation Ginzburg-Landau (GL) free energy. The
$V_{\rm GL}$ free energy can be tuned in different forms or shapes and acts as
a template for the CO or CDW while confining the charges in alternating hole-
rich and hole-poor domains. $V_{\rm GL}$ works like a surface potential that
binds the electrons in physical grains of a granular superconductor with the
difference that the CDW domains are of nanoscopic dimensions. But the Cooper
pairs coherence lengths in HTS are also of nanoscopic sizes, and they may be
formed by local hole pairs interaction mediated by $V_{\rm GL}$ modulations.

In this scenario, we calculate local SC amplitudes by a self-consistent
Bogoliubov-deGennes (BdG) approach. Akin to granular superconductors, there
are Josephson coupling between the nanoscopic charge domains that compete with
thermal disorder to promote long-range phase order at the SC critical
temperature $T_{\rm c}$de Mello (2020); Santana and de Mello (2022). We also
consider the planar Josephson coupling between the CO domains together with
interlayer coupling to formulate a generalization of the ILT and L-D models.

We mentioned above that the measurements and calculations of the penetration
depth $\lambda_{c}$ were important tests to the ILT theories. On the other
hand, the Josephson couplings are proportional to the local superfluid
densitiesSpivak and Kivelson (1991) that, in turn are proportional to the
inverse of the magnetic penetration depthBožović  _et al._ (2016), which is
our route to estimate $\lambda_{c}$ and $\lambda_{ab}$. Using the LSCO
calculations from Ref. de Mello, 2020 we reproduce several low temperatures
$(\lambda_{c}(p)/\lambda_{ab}(p))^{2}$ measurementsShibauchi  _et al._ (1994);
Panagopoulos  _et al._ (2000). We also demonstrate a new equation relating
this ratio to the resistivities $R^{c}_{\rm n}/R^{ab}_{\rm n}$ just above the
SC transition, which is easy to test experimentally and is in agreement with
several old measurementsKimura  _et al._ (1992); Nakamura and Uchida (1993).

##  II CDW Calculations

We mentioned that the CH phase separation method reproduces the observed
planar CDW, but its great advantage is the GL free energy map that provides a
scale to the pairing attraction. The starting point is the time-dependent
phase separation order parameter associated with the relative local electronic
density, $u({\bf r},t)=(p({\bf r},t)-p)/p$, where $p({\bf r},t)$ is the local
charge or hole density at a position ${\bf r}$ in the CuO plane. The CH
equation is based on the GL free energy expansion in terms of this (conserved)
order parameter $u$de Mello and da Silveira Filho (2005); de Mello  _et al._
(2009); de Mello (2012):

$f(u)={{\frac{1}{2}\varepsilon|\nabla u|^{2}+V_{\rm GL}(u,T)}},$ (1)

where $\varepsilon$ is the parameter that controls the charge modulations and
${V_{\rm GL}}(u,T)=-\alpha[T_{\rm PS}-T]u^{2}/2+B^{2}u^{4}/4+...$ is a double-
well potential that characterizes the two (hole-rich and hole-poor) local
charge densities of the CDW structure. The phase separation transition
temperature $T_{\rm PS}$ is assumed to be near the pseudogap instability at
$T^{*}(p)$.

An elegant way to derive the CH equation is through the continuity equation
for the local free energy current density ${\bf J}=M{\bf\nabla}(\delta
f/\delta u)$,Bray (1994)

$\displaystyle\frac{\partial u}{\partial t}$ $\displaystyle=$
$\displaystyle-{\bf\nabla.J}$ (2) $\displaystyle=$
$\displaystyle-M\nabla^{2}[\varepsilon^{2}\nabla^{2}u-\alpha^{2}(T)u+B^{2}u^{3}].$

The equation is non-linear and solved by a stable and fast finite difference
scheme with free boundary conditions, and we stop the simulation time $t$ when
a given CDW structure is reproduced and the solution $u({\bf r})$ or $p({\bf
r})$ are used in the SC calculations.

We have provided a detailed description of the CH simulations in several
previous worksde Mello (2020, 2021); Santana and de Mello (2022). Figure 1(a)
illustrates a typical  ${V_{\rm GL}}({\bf r},T)\equiv{V_{\rm GL}}(u({\bf
r}),T)$ low temperature solution for a $p=0.19$ LSCO compound. We can see that
${V_{\rm GL}}({\bf r},T)$ form an array of side-by-side potential minima that
hosts the alternating hole-rich and hole-poor charge densities domains (not
shown here, see many simulations in the supplemental material of Ref. Santana
and de Mello, 2022).

Figure 1:  a) A three-dimensional plot of the two-dimensional phase separation
potential ${V_{\rm GL}}({\bf r},T)$ viewed from just above the CuO plane.
Notice the array of similar potential wells that host rich and poor
alternating charge density domains $p({\bf r})$. b) Similar view of the SC
amplitudes $\Delta_{d}({\bf r})$ that also follows the same modulation
pattern, what is known as pair density waves.

The derived CDW density map that reproduces the measurements of a given
compound and the respective ${V_{\rm GL}}$ will be used in the BdG
calculations to obtain the SC properties in the next section.

##  III The BdG superconducting Calculations

To perform the BdG SC approach, we use two results from the CH calculations:

1- The CDW density map $p({\bf r})$.

2- The functional $V_{\rm GL}({\bf r})$ shown in Fig. 1(a)

As mentioned in the introduction, at low temperatures, $V_{\rm GL}({\bf r},T)$
constrains the planar charges in alternating hole-rich and hole-poor domains
forming the CDW structure. These alternating densities force the ions to
oscillate around new displaced positions as observed by x-ray diffractionChang
_et al._ (2012). They interact back with the holes, leading to a local
lattice-mediated hole-hole attraction  and Cooper pairs inside the CDW
domains, recalling that this possible because the SC coherence length $\xi$ is
in general shorter than $\lambda_{CO}$. This pairing interaction is dependent
on the CDW hole-rich and hole-poor local concentrations, and it is reasonable
to assume that it scales with the localization potential $V_{\rm GL}({\bf
r},T)$de Mello (2020).

We use this interaction as a nearest neighbor potential attraction in an
extended Hubbard model to calculate the local SC amplitudes $\Delta_{d}({\bf
r}_{i})$. This is done by a self-consistent approach that keeps the $p({\bf
r}_{i})$ CDW structure constant from the beginning to the end of the
calculations following several different experimentsWise  _et al._ (2008);
Gomes  _et al._ (2007); Parker  _et al._ (2010); Chang  _et al._ (2012); Miao
_et al._ (2021). This is achieved by changing the local chemical potential
$\mu({\bf r}_{i})$ at each iteration until the $\Delta_{d}({\bf r}_{i})$
amplitudes converge and the density map $p({\bf r}_{i})$ is preserved. At the
end of the calculations we obtain the original CDW map and the local $d$-wave
amplitudes $\Delta_{d}({\bf r}_{i})$ with the same charge modulations
($\lambda_{\rm CO}(p)$) what is known as pair density waves (PDW)de Mello and
Sonier (2017). This is shown in Fig. 1(b) for the same compound of Fig. 1(a).

The $\Delta_{d}({\bf r}_{i},p,T)$ local spatial variations imply that global
properties like the condensation energies, critical temperatures, and inter-
intralayer Josephson coupling, are a function of the average SC amplitudesde
Mello (2020, 2021); Santana and de Mello (2022) given by:

${\left<\Delta_{d}(p,T)\right>}=\sum_{i}^{N}\Delta_{d}({\bf r}_{i},p,T)/N,$
(3)

where the sum, like in the case of ${\left<V_{\rm GL}(p)\right>}$, is over the
N unit cells of a single CuO plane.

##  IV Josephson Coupling Calculations

We mentioned in the introduction that the CDW structure shown in Fig. 1(a)
with its charge domains bounded by the $V_{\rm GL}({\bf r}_{i},p,T)$ potential
has some similarities with granular materials. In this case, the charges are
bounded to the physical grains by the surface potential and local
superconductivity may arise in the interior. Long-range order or supercurrents
are a consequence of Josephson tunneling between the grainsKetterson and Song
(1999). Although the HTS crystals are not granular in a structural sense, the
ubiquitous CDW in these materials led us to suggestde Mello (2012) that they
may form an array of mesoscopic Josephson junctions.

Under this assumption, the SC transition develops in two steps when the
temperature decreasesKetterson and Song (1999): Firstly, the order parameters
with local amplitudes $\Delta_{d}({\bf r}_{i},p,T)$ and phases $\theta_{i}$
arise in each charge CDW domain $``i$”.  These localized amplitudes give rise
to local Josephson coupling $E_{\rm J}(r_{ij})$ that is proportional to th
local supercurrent or th lattice version of the local superfluid densitySpivak
and Kivelson (1991) $n_{\rm sc}$, and proportional to the local phase
stiffness.

Secondly, upon cooling more, the local phase stiffness increases and
eventually overcomes thermal disorder, which leads to a SC transition by long-
range phase order. Therefore, we emphasize that the SC critical temperature
$T_{\rm c}$ is determined by the competition between thermal disorder and the
average planar Josephson energy ${\left<E_{\rm J}(p,T)\right>}$.

These in-plane calculations are the fundamental pillars of the three-
dimensional LRO in the whole system, which we infer from transport
measurements. For low doping $p$, the $c$-direction resistivity $\rho_{c}$ is
$\approx 10^{3}-10^{6}$ larger than the $a$ or $b$-axis resistivity
$\rho_{ab}$, a behavior shared also by
${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2-x}{\mathrm{La}}_{x}{\mathrm{CuO}}_{6+\delta}$Ono
and Ando (2003); Komiya  _et al._ (2002). Despite this huge difference, it is
surprising that both $\rho_{c}(T)$, and $\rho_{ab}(T)$ fall to zero at the
same temperature ($T_{\rm c}$). We have recently argued that the mechanism
behind this puzzling behavior may be understood in terms of the planar and
weaker out-of-plane average Josephson couplingde Mello (2012), exactly like
the weakly coupled XY models.

As explained previouslyde Mello and Sonier (2014), even for $d$-wave
amplitudes, it is sufficient to use the Ambegaokar-Baratoff analytical
$s$-wave expressionAmbegaokar and Baratoff (1963) averaged over the plane:

${\left<E_{\rm
J}(p,T)\right>}^{X}=\frac{\pi\hbar{\left<\Delta_{d}(p,T)\right>}}{4e^{2}R^{X}_{\rm
n}(p)}{\rm tanh}\bigl{[}\frac{\left<\Delta_{d}(p,T)\right>}{2k_{\rm
B}T}\bigr{]}.$ (4)

Where $X=ab$ for planar and $X=c$ for interlayer coupling and $R^{X}_{\rm
n}(p,\sim T_{\rm c})$ is the corresponding normal state directional resistance
just above $T_{\rm c}$.  In our model of an array of Josephson junctions, the
current is composed of Cooper pairs tunneling between the CDW domains and by
normal carriers or quasiparticle planar currentBruder  _et al._ (1995). For a
d-wave HTS near Tc the supercurrent is dominantBruder  _et al._ (1995), which
justifies the use of the experimental $R_{\rm n}(T_{\rm c})$ between the
charge domains in Eq. 6.

As mentioned, thermal energy causes phase disorder and coherence is achievedde
Mello (2012, 2021) at ${\left<E_{\rm J}(p,T_{\rm c})\right>}^{ab}=k_{\rm
B}T_{\rm c}$. The smaller planar resistances yield larger $E_{\rm J}^{ab}$
that promote first LRO in the planes, but each plane “$i$” would have its own
SC phase $\theta_{i}$ if it was not for the weaker inter-plane $E_{\rm J}^{c}$
coupling. It is similar to a ferromagnet cooled down in the presence of a tiny
magnetic field causing all the moments to become aligned.

Thus, the weaker interlayer coupling ${\left<E_{\rm J}(p,T)\right>}^{c}$
connects the planes but leads to only a single-phase $\theta$ at $T\leq T_{\rm
c}$ in the whole system, and both $c$ and $ab$ resistivity drop off together
despite their orders of magnitude difference.

##  V Magnetic Penetration depth and resistivity

According to Eq. 4, due to the large difference in the directional
resistivities, we expect smaller superfluid densities $n_{\rm sc}$ along the
$c$-direction than along the plane, which is confirmed by the $ab$ and
$c$-axis penetration depth anisotropyShibauchi  _et al._ (1994); Panagopoulos
_et al._ (2000, 1999). We recall also that the square of the magnetic
penetration depth $\lambda$ is inversely proportional to the phase stiffness
$\rho_{\rm sc}$Božović  _et al._ (2016) that is proportional to the average
Josephson currentSpivak and Kivelson (1991).

Along these lines and in the frame of the L-D modelLawrence and Doniach
(1971), Shibauchi et alShibauchi  _et al._ (1994) successfully reproduced
their $c$-axis $\lambda_{c}(p)$ measurements. We extend here their approach to
account for Josephson current between the CDW charge domains in the CuO planes
and use thatde Mello (2020, 2021); Božović  _et al._ (2016)
$\lambda^{2}_{X}(p)\propto 1/{\left<E_{\rm J}(p,T)\right>}^{X}$. Therefore, we
may write:

$[\frac{\lambda_{c}(0)}{\lambda_{ab}(0)}]^{2}\propto[\frac{{\left<E_{\rm
J}(p,T)\right>}^{ab}}{{\left<E_{\rm
J}(p,T)\right>}^{c}}\propto\frac{R^{c}_{\rm n}}{R^{ab}_{\rm n}},$ (5)

and $R^{c}_{\rm n}=\rho_{c}(T_{\rm c})s$ where $s$ is the distance between the
CuO layers in double-plane LSCO crystals is approximately 6.6 $\AA$. This
means that $\lambda_{c}(p)$ is dominated by the interplane Josephson current
and, extending this idea, $\lambda_{ab}(p)$ is dominated by the planar average
coupling ${\left<E_{\rm J}(p,T_{\rm c})\right>}^{ab}$. Therefore, we may write
the planar resistivity $R^{ab}_{\rm n}=\rho_{ab}(T_{\rm c})\lambda_{\rm CO}$,
since $\lambda_{\rm CO}(p)$ is the distance between the planar CDW domains.
Therefore, in a general way,

$[\frac{\lambda_{c}(0)}{\lambda_{ab}(0)}]^{2}\propto\frac{R^{c}_{\rm
n}}{R^{ab}_{\rm n}}\propto\frac{\rho_{c}(T_{\rm c})s}{\rho_{ab}(T_{\rm
c})\lambda_{\rm CO}}.$ (6)

This expression gives the magnetic penetration depths out of the plane and
planar ratio in terms of similar resistivities ratio, the planar distance $s$,
and the CDW wavelength $\lambda_{\rm CO}$. Notice that there is not any
adjustable parameter in Eq. 6 and all quantities have previously been
measured.

Some HTS samples with similar doping have comparable resistivities like the La
and Y-based compounds studied by Ando et alAndo  _et al._ (2004).  When this
is the case the above equation shows why the low temperature ratios
$\lambda_{c}(p,0)/\lambda_{ab}(p,0)$ for different families of compounds have
similar valuesSchneider and Keller (2004). This is the case for a large number
of LSCO and HgBa2CuO4+x Panagopoulos  _et al._ (2000) samples and they are
also comparable with the measurements of $c$-axis grain-aligned orthorhombic
YBa2Cu3O7-δ (YBCO) with $\delta=0.0,0.3,$ and $0.43$Panagopoulos  _et al._
(1998). The quantitative explanation of these data and their connection with
the $\rho_{c}(T_{\rm c})/\rho_{ab}(T_{\rm c})$ is one of the main motivations
of our present calculations.

Figure 2:  Six LSCO $[\lambda_{c}(0)/\lambda_{ab}(0)]^{2}$ experimental points
Shibauchi  _et al._ (1994); Panagopoulos  _et al._ (2000) and one optimal
YBCO-aligned powder in magnetic field with 30% uncertaintyPanagopoulos  _et
al._ (1998) represented by the error bar. They are together with the
calculations from Eq. 6 which use the CDW wavelengthsComin and Damascelli
(2016) $\lambda_{CO}(p)$ and the respective resistivity ratiosKimura  _et al._
(1992); Nakamura and Uchida (1993) listed in Tables 1 and 2. For $p=0.09$ we
use the resistivities of $p=0.10\pm 0.01$ from Ref. Nakamura and Uchida, 1993
as explained in the text.

On the other hand, most of the data on directional $\lambda_{X}(0)$ and
$\rho_{X}(T_{\rm c})$ with the same doping $p$ were performed a long time ago
with LSCO crystals in order to understand the anisotropies in HTS. Nowadays
there are single crystals of many other materials but since the anisotropies
are already established these measurements are not remade. The difficulty to
fabricate single crystals in the earlier days of HTS is the reason why data on
other materials are practically nonexistent. In most cases, whenever there are
data on $\lambda_{c}(0)/\lambda_{ab}(0)$, it is not accompanied by
$\rho_{c}(T_{\rm c})/\rho_{ab}(T_{\rm c})$ that is needed by our Eq. 6.
Nevertheless, we list in Table 1 the available data on LSCO for
$(\lambda_{c}(0)/\lambda_{ab}(0))^{2}$Shibauchi  _et al._ (1994); Panagopoulos
_et al._ (2000) ratios and $\rho_{c}(T_{\rm c})/\rho_{ab}(T_{\rm c})$Kimura
_et al._ (1992); Nakamura and Uchida (1993). The case of
$(\lambda_{c}(0.09)/\lambda_{ab}(0.09))^{2}$, the resistivity data of
different groups have discrepant results and we used
$\rho_{c}/\rho_{ab}(0.10\pm 0.01)$ from Ref. Nakamura and Uchida, 1993 (marked
with a star). With these data and the respective CDW wavelengths
$\lambda_{CO}(p)$ that enters in Eq. 6 for the planar $R^{ab}_{\rm n}$, we
calculated the magnetic penetration depth ratio. The experimental results and
our estimates are plotted together for comparison in Fig. 2, and listed in
columns two and five of Table 1.

Table 1:  Data and calculations for LSCO. The first column is hole density per unit cell. Second is the $(\lambda_{c}(0)/\lambda_{ab}(0))^{2}$ measurements of Shibauchi et alShibauchi  _et al._ (1994) and Panagopoulos et alPanagopoulos  _et al._ (2000). Third is the $\rho_{c}(T_{\rm c})/\rho_{ab}(T_{\rm c})$Kimura  _et al._ (1992); Nakamura and Uchida (1993) resistivity ratios. The fourth is $\lambda_{CO}$ in units of the lattice parameter $a_{0}\approx 3.78\AA$ measured by REXSComin and Damascelli (2016). The last column is the calculations from Eq. 6 with $s=6.6\AA$ that should match the magnetic penetration depth ratio of the second column. The case of $(\lambda_{c}(0.09)/\lambda_{ab}(0.09))^{2}$, the resistivity data of different groups conflict and we used the $p=0.10\pm 0.01$ resistivity ratio $\rho_{c}/\rho_{ab}$ from Ref. Nakamura and Uchida, 1993 (marked with a star).  Sample | $(\lambda_{c}(0)/\lambda_{ab}(0))^{2}$ | $\rho_{c}/\rho_{ab}(T_{\rm c})$ | $\lambda_{CO}(\AA)$ | $\rho_{c}s/\rho_{ab}\lambda_{CO}$
---|---|---|---|---
p = 0.09 | 625 | 2070* | 5.6 $a_{0}$ |  645*
p = 0.10 |  $\propto$ 564 | 1714 | 5.0 $a_{0}$ | 595
p = 0.12 | 441 | 1000 | 4.25 $a_{0}$ | 408
p = 0.15 | 196 | 433 | 3.9 $a_{0}$ | 193
p = 0.18 | 132 | 300 | 3.7 $a_{0}$ | 164
p = 0.20 | 100 | 200 | 3.6 $a_{0}$ | 97

There are also some data on $\lambda_{c}(0)/\lambda_{ab}(0)$ in magnetic
aligned powder of YBa2Cu3O7-δPanagopoulos  _et al._ (1998) (YBCO) that
contains some uncertainty up to 30% in the $c$-axis but provides good
estimates of this ratio. Similarly, we use new anisotropic $\rho_{c}(T_{\rm
c})/\rho_{ab}(T_{\rm c})$ optimal data on thin films of YBCO grown on an off-
axis cut SrTiO3 substrateHeine  _et al._ (2021). Combining these data, we can
apply Eq. 6 to this near optimal compound and the calculation is very close to
the YBCO experimental result as shown in Fig. 2 and in Table 2.

Table 2:  Data and calculations for optimal YBCO similar to Table 1. The uncertainty of 30% in $(\lambda_{c}(0)/\lambda_{ab}(0))^{2}$ is due to the $c$-axis alignment uncertainty of the powders. For YBCO, the distance between two planes is $s=5.84\AA$ and for optimal dopingComin and Damascelli (2016) $\lambda_{CO}=3.12a_{0}$. Sample | $(\lambda_{c}(0)/\lambda_{ab}(0))^{2}$ | $\rho_{c}(T_{\rm c})/\rho_{ab}(T_{\rm c})$ | $\lambda_{CO}(\AA)$ | $\rho_{c}s/\rho_{ab}\lambda_{CO}$
---|---|---|---|---
p = 0.15 | 81 $\pm$ 25 | 167 | 3.9 $a_{0}$ | 81.5

##  VI Conclusion

In this paper, we generalize the ideas of ILT and L-D models to account also
for Josephson coupling between the mesoscopic CDW domains or blocks. The in-
plane average Josephson energy is much larger than the interlayer coupling,
proportional to $1/\lambda_{ab}^{2}$ and to the thermal energy at the critical
temperature $T_{\rm c}$. Therefore the SC properties are almost entirely
dependent on the planar Josephson coupling in opposition to the old ILT model,
which depended on the interlayer Josephson tunneling. Our approach yields
values of $\lambda_{ab}$ at least one order of magnitude larger than
$\lambda_{c}$, which gives some insights why measurementsMoler  _et al._
(1998) of $\lambda_{c}$ gave much smaller condensation energy than the old ILT
predictionsLeggett (1996).

Furthermore, we derive a new equation relating the magnetic penetration depths
$(\lambda_{c}(0)/\lambda_{ab}(0))^{2}$ with the resistivities
$[\rho_{c}(T_{\rm c})s]/[\rho_{ab}(T_{\rm c})\lambda_{\rm CO}]$ (Eq. 6), which
is in agreement with many measurements without any adjustable parameter.
Nowadays there are better single crystals of many materials but since the
anisotropies are already established these combined
measurements($\rho_{X}(T_{\rm c})$ and $(\lambda_{X}(0)$) on a single sample
are not anymore explored. However, Eq. 6 provides new motivation for more
precise tests in future experiments with modern pristine HTS crystals.

##  VII acknowledgements

We acknowledge partial support from the Brazilian agencies CNPq and FAPERJ.

## References

  * Clem (1989) J. R. Clem, Physica C 162-164, 1137 (1989).
  * Clem (1991) J. R. Clem, Phys. Rev. B 43, 7837 (1991).
  * Chakravarty  _et al._ (1993) S. Chakravarty, A. Sudbo, P. W. Anderson, and S. Strong, Science , 337+ (1993), 5119.
  * Anderson (1995) P. Anderson, Science 268, 1154 (1995).
  * Leggett (1996) A. J. Leggett, Science 274, 587 (1996).
  * Lawrence and Doniach (1971) W. Lawrence and S. Doniach, in  _Proceedings of the Twelfth International Conference on Low Temperature Physics_ (Ed. by E. Kanda, Academic Press of Japan, Kyoto, Japan, 1971) p. 361.
  * Shibauchi  _et al._ (1994) T. Shibauchi, H. Kitano, K. Uchinokura, A. Maeda, T. Kimura, and K. Kishio, Phys. Rev. Lett. 72, 2263 (1994).
  * Moler  _et al._ (1998) K. A. Moler, J. R. Kirtley, D. Hinks, T. Li, and M. Xu, Science 279, 1193 (1998).
  * Anderson (1998) P. W. Anderson, Science 279, 1196+ (1998), 5354.
  * Tranquada  _et al._ (1995) J. M. Tranquada, B. J. Sternlieb, J. D. Axe, Y. Nakamura, and S. Uchida, Nature 375, 561 (1995).
  * Lang  _et al._ (2002) K. M. Lang, V. Madhavan, J. E. Hoffman, E. W. Hudson, H. Eisaki, S. Uchida, and J. C. Davis, Nature 415, 412 (2002).
  * Fischer  _et al._ (2007) O. Fischer, M. Kugler, I. Maggio-Aprile, C. Berthod, and C. Renner, Rev. Mod. Phys. 79, 353 (2007).
  * Comin and Damascelli (2016) R. Comin and A. Damascelli, Ann. Rev. of Cond. Mat. Phys. 7, 369 (2016).
  * Wise  _et al._ (2008) W. D. Wise _et al._ , Nature Physics 4, 696 (2008).
  * Gomes  _et al._ (2007) K. K. Gomes _et al._ , Nature 447, 569 (2007).
  * Parker  _et al._ (2010) C. V. Parker, P. Aynajian, E. H. da Silva Neto, A. Pushp, S. Ono, J. Wen, Z. Xu, G. Gu, and A. Yazdani, Nature 468, 677 (2010).
  * He  _et al._ (2014) Y. He, Y. Yin, M. Zech, A. Soumyanarayanan, M. M. Yee, T. Williams, M. C. Boyer, K. Chatterjee, W. D. Wise, I. Zeljkovic, T. Kondo, T. Takeuchi, H. Ikuta, P. Mistark, R. S. Markiewicz, A. Bansil, S. Sachdev, E. W. Hudson, and J. E. Hoffman, Science 344, 608 (2014).
  * (18) E. H. da Silva Neto, M. Minola, B. Yu, W. Tabis, M. Bluschke, D. Unruh, H. Suzuki, Y. Li, G. Yu, D. Betto, K. Kummer, F. Yakhou, N. B. Brookes, M. Le Tacon, M. Greven, B. Keimer, and A. Damascelli, .
  * Wu  _et al._ (2017) J. Wu, A. T. Bollinger, X. He, and I. Božović, Nature 547, 432 (2017).
  * Chen  _et al._ (2019) S.-D. Chen, M. Hashimoto, Y. He, D. Song, K.-J. Xu, J.-F. He, T. P. Devereaux, H. Eisaki, D.-H. Lu, J. Zaanen, and Z.-X. Shen, Science 366, 1099 (2019).
  * Fei  _et al._ (2019) Y. Fei, Y. Zheng, K. Bu, W. Zhang, Y. Ding, X. Zhou, and Y. Yin, Science China Physics, Mechanics & Astronomy 63, 227411 (2019).
  * Miao  _et al._ (2021) H. Miao, G. Fabbris, R. J. Koch, D. G. Mazzone, C. S. Nelson, R. Acevedo-Esteves, G. D. Gu, Y. Li, T. Yilimaz, K. Kaznatcheev, E. Vescovo, M. Oda, T. Kurosawa, N. Momono, T. Assefa, I. K. Robinson, E. S. Bozin, J. M. Tranquada, P. D. Johnson, and M. P. M. Dean, npj Quantum Materials 6, 31 (2021).
  * de Mello (2012) E. V. L. de Mello, Europhys. Lett. 99, 37003 (2012).
  * de Mello and Sonier (2014) E. V. L. de Mello and J. E. Sonier, J. Phys.: Condens. Matter 26, 492201 (2014).
  * de Mello and Sonier (2017) E. V. L. de Mello and J. E. Sonier, Phys. Rev. B 95, 184520 (2017).
  * de Mello (2020) E. V. de Mello, J. Phys.: Condens. Matter 32, 40LT02 (2020).
  * Santana and de Mello (2022) H. S. Santana and E. de Mello, Phys. Rev. B 105, 134513 (2022).
  * Spivak and Kivelson (1991) B. I. Spivak and S. A. Kivelson, Phys. Rev. B 43, 3740 (1991).
  * Božović  _et al._ (2016) I. Božović, X. He, J. Wu, and A. T. Bollinger, Nature 536, 309 (2016).
  * Panagopoulos  _et al._ (2000) C. Panagopoulos, J. R. Cooper, T. Xiang, Y. S. Wang, and C. W. Chu, Phys. Rev. B 61, R3808 (2000).
  * Kimura  _et al._ (1992) T. Kimura, K. Kishio, T. Kobayashi, Y. Nakayama, N. Motohira, K. Kitazawa, and K. Yamafuji, Physica C: Superconductivity 192, 247 (1992).
  * Nakamura and Uchida (1993) Y. Nakamura and S. Uchida, Phys. Rev. B 47, 8369 (1993).
  * de Mello and da Silveira Filho (2005) E. de Mello and O. T. da Silveira Filho, Physica A 347, 429 (2005).
  * de Mello  _et al._ (2009) E. V. L. de Mello, R. B. Kasal, and C. A. C. Passos, J. Phys.: Condens. Matter 21, 235701 (2009).
  * Bray (1994) A. Bray, Adv. Phys. 43, 357 (1994).
  * de Mello (2021) E. V. L. de Mello, J. of Phys.: Cond. Matter 33, 145503 (2021).
  * Chang  _et al._ (2012) J. Chang, E. Blackburn, T. Holmes, N. B. Christensen, J. Larsen, J. Mesot, R. Liang, D. A. Bonn, W. N. Hardy, A. Watenphul, M. V. Zimmermann, E. M. Forgan, and S. M. Hayden, Nature Physics 8, 871 (2012).
  * Ketterson and Song (1999) J. B. Ketterson and S. Song,  _Superconductivity_ (Cambridge University Press, London, 1999).
  * Ono and Ando (2003) S. Ono and Y. Ando, Phys. Rev. B 67, 104512 (2003).
  * Komiya  _et al._ (2002) S. Komiya, Y. Ando, X. F. Sun, and A. N. Lavrov, Phys. Rev. B 65, 214535 (2002).
  * Ambegaokar and Baratoff (1963) V. Ambegaokar and A. Baratoff, Phys. Rev. Lett. 10, 486 (1963).
  * Bruder  _et al._ (1995) C. Bruder, A. van Otterlo, and G. T. Zimanyi, Phys. Rev. B 51, 12904 (1995).
  * Panagopoulos  _et al._ (1999) C. Panagopoulos, J. L. Tallon, and T. Xiang, Phys. Rev. B 59, R6635 (1999).
  * Ando  _et al._ (2004) Y. Ando, S. Komiya, K. Segawa, S. Ono, and Y. Kurita, Phys. Rev. Lett. 93, 267001 (2004).
  * Schneider and Keller (2004) T. Schneider and H. Keller, New Journal of Physics 6, 144 (2004).
  * Panagopoulos  _et al._ (1998) C. Panagopoulos, J. R. Cooper, and T. Xiang, Phys. Rev. B 57, 13422 (1998).
  * Heine  _et al._ (2021) G. Heine, W. Lang, R. Rössler, and J. D. Pedarnig, Nanomaterials 11 (2021), 10.3390/nano11030675.