# Geometrical constraints on curvature from galaxy-lensing cross-correlations

Yufei Zhang<EMAIL_ADDRESS>CAS Key Laboratory for Research in
Galaxies and Cosmology, Department of Astronomy, University of Science and
Technology of China, Hefei, Anhui, 230026, People’s Republic of China  School
of Astronomy and Space Science, University of Science and Technology of China,
Hefei, Anhui, 230026, People’s Republic of China  Wenjuan Fang
<EMAIL_ADDRESS>Corresponding author. CAS Key Laboratory for Research in
Galaxies and Cosmology, Department of Astronomy, University of Science and
Technology of China, Hefei, Anhui, 230026, People’s Republic of China  School
of Astronomy and Space Science, University of Science and Technology of China,
Hefei, Anhui, 230026, People’s Republic of China

###### Abstract

Accurate constraints on curvature provide a powerful probe of inflation.
However, curvature constraints based on specific assumptions of dark energy
may lead to unreliable conclusions when used to test inflation models. To
avoid this, it is important to obtain constraints that are independent on
assumptions for dark energy. In this paper, we investigate such constraints on
curvature from the geometrical probe constructed from galaxy-lensing cross-
correlations. We study comprehensively the cross-correlations of galaxy with
magnification, measured from type Ia supernovae’s brightnesses (“$g\kappa^{\rm
SN}$”), with shear (“$g\kappa^{\rm g}$”), and with CMB lensing (“$g\kappa^{\rm
CMB}$”). We find for the LSST and Stage IV CMB surveys, “$g\kappa^{\rm SN}$” ,
“$g\kappa^{\rm g}$” and “$g\kappa^{\rm CMB}$” can be detected with signal-to-
noise ratio $S/N=104,\ 2291,\ 1842$ respectively. When combined with
supernovae Hubble diagram (“SN”) to constrain curvature, we find galaxy-
lensing cross-correlation becomes increasingly important with more degrees of
freedom allowed in dark energy. Without any priors, we obtain error on
$\Omega_{K}$ of $0.723$ from “SN + $g\kappa^{\rm SN}$”, $0.0417$ from “SN +
$g\kappa^{\rm g}$”, and $0.04$ from “SN + $g\kappa^{\rm g}$ \+ $g\kappa^{\rm
CMB}$” for the LSST and Stage IV CMB surveys. The last one is more competitive
than a Stage IV BAO survey (“BAO”). When galaxy-lensing cross-correlations are
added to the combined probe of “SN + BAO + CMB”, where “CMB” stands for Planck
measurement for the CMB acoustic scale, we obtain constraint on $\Omega_{K}$
of $0.0013$, which is a factor of 7 improvement from “SN + BAO + CMB”. We
study improvements in these results from increasing the high redshift
extension of supernovae.

##  I Introduction

The Universe’s curvature is one of its fundamental properties, and deserves to
be accurately measured. More importantly, stringent constraints on curvature
also provide powerful probes of early Universe physics such as inflation Guth
(1981); Linde (1982); Albrecht and Steinhardt (1982). Though predictions of
the inflationary scenario have been shown to be consistent with accurate
measurements of the cosmic microwave background (CMB) anisotropies Planck
Collaboration  _et al._ (2020a), details of the scenario remain to be
uncovered by further measurement results including measurement of the
curvature.

In inflation models with a large number of e-foldings, curvature is predicted
to be undetectable, i.e., the magnitude of $\Omega_{K}$—the curvature density
parameter—is below $10^{-5}$, the measurement limit from local fluctuations in
the spatial curvature within our Hubble volume. While if ordinary slow-roll
inflation is preceded by false vacuum decay, potentially observable open
curvature can be produced Gott (1982); Bucher  _et al._ (1995); Yamamoto  _et
al._ (1995); Freivogel  _et al._ (2006); Bousso  _et al._ (2015).
Specifically, the analyses by Kleban and Schillo (2012) and Guth and Nomura
(2012) find that future detection of closed curvature at the level of
$|\Omega_{K}|\;\hbox to0.0pt{\lower 2.5pt\hbox{$\sim$}\hss}\raise
1.5pt\hbox{$>$}\;10^{-4}$ will exclude eternal inflation and pose challenges
to the inflationary scenario, while detection of open curvature at the same
level will suggest that false vacuum decay happened before the observable
inflation. Therefore, accurate measurement of curvature provides a powerful
tool to probe inflation.

Current constraints on curvature come mainly from measurements of the
Universe’s geometry. Due to severe degeneracy between curvature and dark
energy, most of these constraints adopt simple assumptions for dark energy.
For example, by assuming dark energy to be the cosmological constant, the
Planck collaboration obtains $\Omega_{K}\simeq 0.0007\pm 0.0019$ from their
measurements of the CMB temperature, polarization, reconstructed lensing, and
external measurements of the baryon acoustic oscillations (BAO) Planck
Collaboration  _et al._ (2020b), which is probably the best precision we can
achieve today under this assumption. However, since the nature of dark energy
remains a mystery Albrecht  _et al._ (2006); Frieman  _et al._ (2008);
Caldwell and Kamionkowski (2009); Weinberg  _et al._ (2013), one cannot reach
a decisive conclusion when using these constraints to test inflation models.
Actually, with the same assumption for dark energy, the Planck team found
their measurements for the CMB temperature and polarization alone prefers a
closed Universe otherwise Planck Collaboration  _et al._ (2020b), see also Di
Valentino  _et al._ (2020); Handley (2021). While the real reason for this
apparent discrepancy remains to be clarified, the situation also stresses the
importance of obtaining stringent constraints on curvature that are
independent on the uncertainties in our knowledge about dark energy.

In this paper, we investigate the constraints on curvature that are
independent on the “unknown” properties of dark energy by utilizing
geometrical cosmology probes. Specifically, besides the most extensively
explored probing techniques such as type Ia supernovae, BAO, and CMB, we study
the constraints from the cross-correlation between galaxy (tracers of large-
scale structure) distribution and signatures of weak-lensing, for which we
comprehensively consider the magnification of the brightnesses of background
standard candles such as type Ia supernovae, cosmic shear measured from the
distorted images of background galaxies, and the remapping of the CMB fields.

The galaxy-weak lensing cross-correlation itself certainly involves
information on structure growth. However, by comparing the signals for the
same lensing galaxies but sources at different redshifts, pure geometric
information can be extracted Jain and Taylor (2003); Hu and Jain (2004); Zhang
_et al._ (2005). With more and more ambitious cosmological surveys starting or
to start operation, especially the planned stage IV dark energy experiments
such as the LSST LSST Science Collaboration  _et al._ (2009), CSST Zhan
(2011); Gong  _et al._ (2019), the stage IV CMB experiment Abazajian  _et al._
(2016), this geometrical probe is drawing more attention these days. For
example, ratios of distances have recently been measured from real surveys
with high significance using the cross-correlation between galaxy and cosmic
shear (the so-called galaxy-galaxy lensing), and the cross-correlation between
galaxy and CMB lensing Miyatake  _et al._ (2017); Prat  _et al._ (2019). At
the same time, the cross-correlation between galaxy and supernovae
magnification itself has not been conclusively detected with current data yet.
For recent trials, see Smith  _et al._ (2014); Mitra (2018); Sakakibara  _et
al._ (2019). The opportunities with future surveys such as the LSST should be
wide-ranging, see Scovacricchi  _et al._ (2017) and our investigations in Sec.
III.1 below.

Gravitational lensing uniquely probes the angular diameter distance from the
lens (rather than from observers at $z=0$) to the source, in addition to the
distances to the lens and to the source. It is known that the three distances
altogether provide a pure metric probe for curvature Bernstein (2006); Rasanen
_et al._ (2014), see Fig. 1. In particular, Bernstein (2006) proposed to
obtain model-independent constraints on curvature by using the geometrical
probe constructed from galaxy-galaxy lensing. However, their analysis done in
real space adopts an oversimplified assumption that observables along
different line of sights are completely independent. In this paper, without
directly dealing with the correlations between observables along different
line of sights, we perform an analysis in Fourier space that automatically
takes into account the correlations. In addition, we extend the analysis to
include galaxy-supernovae magnification and galaxy-CMB lensing cross-
correlations. We notice that cosmological constraints from the cross-
correlation between galaxy and supernovae magnification have been barely
explored, if not completely none. We forecast the dark energy independent
constraints on curvature from galaxy-weak lensing cross-correlations for Stage
IV dark energy experiments, and see whether the desired accuracy level of
$10^{-4}$ can be reached when they are combined with other popular geometrical
probes.

Figure 1:  The comoving angular diameter distances to the lens $r_{L}$, to the
source $r_{S}$, and from the lens to the source $r_{LS}$ provide a probe of
curvature altogether. Shown in this plot is the dependence on $\Omega_{K}$ for
the ratio of $(r_{L}+r_{LS})$ to $r_{S}$, while $r_{L}$, $r_{S}$ fixed at
$1500$ and $3000h^{-1}$Mpc respectively. Here, we have used the approximation
for $r_{LS}$ when $|\Omega_{K}|\ll 1$ Peebles (1993).

The rest of this paper is organized as follows. In Sec. II, we present the
formulas we use to forecast the pure geometrical constraints from galaxy-
lensing cross-correlations. In Sec. III, we forecast the constraints on
curvature with different assumptions for dark energy from the geometrical
probe of galaxy-lensing cross-correlations and the ultimate combination with
supernovae, BAO and CMB. We discuss our results in Sec. IV and summarize in
Sec. V.

##  II Theoretical Calculations

In this section, we present the theoretical calculations to forecast the pure
geometrical constraints from galaxy-lensing cross-correlations. To break the
parameter degeneracies, we will combine the constraints with those from the
supernovae Hubble diagram, which is available from the same supernovae survey
used to measure the galaxy-supernovae cross-correlation, and the calculation
of which is given in Sec II.1. There are great similarity among the three
types of galaxy-lensing cross-correlations. The differences are mainly in
source redshifts and noises for measuring the lensing signals. Thus, we
elaborate the calculations for the galaxy-supernovae cross-correlation, which
are much less presented in the literature, while briefly mention those for the
other two. We note our calculations in Sec.II.2.1 applies to other types of
standard candles as well.

###  II.1 Type Ia Supernovae

The apparent magnitude of a supernova at redshift $z_{i}$ is given by

$m_{i}=5\log_{10}\left[H_{0}d_{L}(z_{i})\right]+\mathcal{M}+\epsilon_{i},$ (1)

We introduce $\mathcal{M}_{i}$ as a quantity involving the supernova’s
intrinsic luminosity and the Hubble constant $H_{0}$, and separate
$\mathcal{M}_{i}$ into its mean $\mathcal{M}$ and statistical variation
$(\mathcal{M}_{i}-\mathcal{M})$. The latter is included in $\epsilon_{i}$,
which we use to represent the total variation in $m_{i}$. The luminosity
distance $d_{L}$ is related to the comoving angular diameter distance $r$ by
$d_{L}=(1+z)r$, while $r$ is given as a function of the comoving radial
distance $\chi$ by

$r(\chi)=\frac{1}{\sqrt{-\Omega_{K}}H_{0}}\sin{\left[\sqrt{-\Omega_{K}}H_{0}\chi\right]},$
(2)

with $\chi$ calculated by $\chi=\int_{0}^{z}dz^{\prime}/H(z^{\prime})$.

We assume the total number of supernovae discovered from a supernovae survey
is $N_{\rm tot}$, and their redshift distribution is $dn/dz$, which we
normalize to be 1. To forecast parameter constraints from a supernovae-alone
probe — the supernovae Hubble diagram, following LSST Science Collaboration
_et al._ (2009), we neglect all possible correlations between the apparent
magnitudes of different supernovae, and for the variance in an individual
supernova’s apparent magnitude $\sigma_{m}^{2}$, we assume it is totally due
to statistical variation of the supernova’s intrinsic luminosity, and take it
to be a constant. We note that correlations between different supernovae
induced by various systematic effects and other types of statistical
variations should be taken into account for a more accurate forecast. We then
use the Fisher matrix technique to forecast the anticipated constraints, which
is constructed as

$F_{\alpha\beta}=N_{\rm tot}\int
dz\frac{dn}{dz}\frac{1}{\sigma_{m}^{2}}\frac{\partial\bar{m}(z)}{\partial
p_{\alpha}}\frac{\partial\bar{m}(z)}{\partial p_{\beta}}.$ (3)

Here, besides the cosmological parameters, our parameter set also includes
$\mathcal{M}$ which involves both the mean of the supernovae’s intrinsic
brightness and $H_{0}$. In obtaining constraints on the cosmological
parameters, we marginalize over $\mathcal{M}$. It can be seen that the
parameter constraints will be inversely proportional to $N_{\rm tot}^{1/2}$.

###  II.2 Galaxy-lensing cross-correlation

####  II.2.1 Galaxy-supernovae cross-correlation

In the previous section, we have considered the total variation in an
individual supernova’s brightness at a given redshift comes only from its
intrinsic luminosity. In reality, an additional type of variation will be
introduced by the magnification effect of gravitational lensing by matter
distribution in the foreground Frieman (1996); Wambsganss  _et al._ (1997);
Holz (1998); Metcalf (1999). Some supernovae are magnified, and some
demagnified. In the limit of weak lensing, $\epsilon_{i}$ in Eq. (1) will have
an additional term of $-5/(\ln{10})\kappa_{i}$, besides
$(\mathcal{M}_{i}-\mathcal{M})$. Here, $\kappa_{i}$ is the lensing convergence
for a source at the supernova’s location. This magnification effect can be
statistically measured by cross-correlating the supernovae’s brightnesses with
the distribution of large-scale structure tracers in their foreground such as
galaxies (see e.g., Williams and Song (2004); Ménard and Dalal (2005); Cooray
_et al._ (2006)), which we study in this section.

Specifically, we consider the cross-correlation between the following two
observables,

$\displaystyle\delta_{n}^{2D}(\vec{\theta})$ $\displaystyle\equiv$
$\displaystyle\int dzW_{L}(z)\delta_{n}(z,\vec{\theta}),$ (4) $\displaystyle
M(\vec{\theta})$ $\displaystyle\equiv$ $\displaystyle\int
dzW_{S}(z)m(z,\vec{\theta}),$ (5)

where $\delta_{n}^{2D}$ and $\delta_{n}$ are the two and three-dimensional
galaxy overdensities; $W_{L}$ and $W_{S}$ are the redshift selection functions
for the galaxies (“Lens”) and supernovae (“Source”) respectively, both of
which have been normalized to be 1, i.e., $\int W(z)dz=1$; $M(\vec{\theta})$
represents the average apparent magnitude for the selected supernovae whose
angular positions are within a solid angle $d^{2}\theta$
($d^{2}\theta\rightarrow 0$) around $\vec{\theta}$. Intrinsic fluctuations in
supernovae’s brightnesses drop out in the cross-correlation, and by a Fourier
transform, we get the following cross-correlation power spectrum,

$C_{\ell}^{LS}=-\frac{15}{2\ln(10)}\Omega_{m}H_{0}^{2}\int
dzW_{L}(z)\frac{g_{S}(z)}{ar^{2}}P_{gm}(k=\frac{\ell}{r},z),$ (6)

where $\Omega_{m}$ is the density parameter for matter, $a$ is the scale
factor, $P_{gm}$ is the galaxy-matter power spectrum, and $g_{S}(z)$ is given
by,

$g_{S}(z)=\int
dz^{\prime}W_{S}(z^{\prime})\frac{r(\chi)r(\chi^{\prime}-\chi)}{r(\chi^{\prime})}\Theta(\chi^{\prime}-\chi),$
(7)

where $\Theta$ is the Heaviside step function. In our derivation for Eq (6),
we have used the Limber approximation Limber (1954); Kaiser (1992).

Galaxy-supernovae cross-correlation potentially provides a pure geometrical
probe of the Universe, as is the case for galaxy-galaxy lensing Jain and
Taylor (2003); Zhang  _et al._ (2005). A direct way to see this is by taking
the limit that the foreground galaxies are all selected to be at a single
redshift, i.e., $W_{L}(z)\rightarrow\delta(z-z_{L})$, then we have
$C_{\ell}^{LS}/C_{\ell}^{LS^{\prime}}\rightarrow
g_{S}(z_{L})/g_{S^{\prime}}(z_{L})$, i.e., the ratio of the cross-correlations
between these galaxies and supernovae selected with different selection
functions, $W_{S}(z)$ and $W_{S^{\prime}}(z)$, probes a pure geometrical
quantity. Furthermore, in the limit of $W_{S}(z)\rightarrow\delta(z-z_{S})$
and $W_{S^{\prime}}(z)\rightarrow\delta(z-z_{S^{\prime}})$,
$C_{\ell}^{LS}/C_{\ell}^{LS^{\prime}}\rightarrow
r(\chi_{S}-\chi_{L})r(\chi_{S^{\prime}})/r(\chi_{S})r(\chi_{S^{\prime}}-\chi_{L})$,
which directly probes the ratio of the angular diameter distances.

In this section, we investigate the pure geometrical probe constructed from
the galaxy-supernovae cross-correlation. We consider a survey of both
supernovae and galaxies, and assume the former to be observed to a maximum
redshift $z_{\rm max}$, and the latter to a redshift beyond that. For
measurements of the cross-correlation, galaxies at $z>z_{\rm max}$ would be of
no use, since their cross-correlation with the supernovae would vanish. We
divide both the supernovae and galaxies from $z=0$ to $z=z_{\rm max}$ into
redshift bins of equal size $\Delta z$. Hereafter, we use “S” as the index for
the supernova (source) redshift bins, and “L” for the galaxy (lens) redshift
bins. The measured cross power spectra $C_{\ell}^{LS}$ and
$C_{\ell^{\prime}}^{L^{\prime}S^{\prime}}$ have the following covariance,

${\rm{Cov}}(C_{\ell}^{LS},C_{\ell^{\prime}}^{L^{\prime}S^{\prime}})=\frac{\delta_{\ell\ell^{\prime}}^{K}}{2\ell\Delta\ell
f_{\rm
sky}}\left(C_{\ell}^{SS^{\prime}}C_{\ell}^{LL^{\prime}}+C_{\ell}^{LS^{\prime}}C_{\ell}^{L^{\prime}S}\right),$
(8)

where $\delta^{K}$ is the Kronecker delta, $\Delta\ell$ is the size of the
multipole bin used to measure $C_{\ell}^{LS}$, and $f_{\rm sky}$ is the
fraction of sky covered by the survey. Both the auto power spectra for the
supernovae brightnesses $C_{\ell}^{SS^{\prime}}$ and for the galaxy
distribution $C_{\ell}^{LL^{\prime}}$ have two contributions: one from large-
scale structure (“LSS”), and the other from shot noise (“shot”), given by

$\displaystyle C_{\ell,\rm{shot}}^{SS^{\prime}}$ $\displaystyle=$
$\displaystyle\delta_{SS^{\prime}}^{K}\sigma_{m}^{2}/\bar{n}_{S}^{2D},$ (9)
$\displaystyle C_{\ell,\rm{LSS}}^{SS^{\prime}}$ $\displaystyle=$
$\displaystyle(\frac{15}{2\ln(10)}\Omega_{m}H_{0}^{2})^{2}\int
dz\frac{d\chi}{dz}$ (10)
$\displaystyle\times\frac{g_{S}(z)g_{S^{\prime}}(z)}{a^{2}r^{2}}P_{mm}(k=\frac{\ell}{r},z),$
$\displaystyle C_{\ell,\rm{shot}}^{LL^{\prime}}$ $\displaystyle=$
$\displaystyle\delta_{LL^{\prime}}^{K}/\bar{n}_{L}^{2D},$ (11) $\displaystyle
C_{\ell,\rm{LSS}}^{LL^{\prime}}$ $\displaystyle=$ $\displaystyle\int
dzW_{L}(z)W_{L^{\prime}}(z)(\frac{d\chi}{dz})^{-1}r^{-2}$ (12)
$\displaystyle\times P_{gg}(k=\frac{\ell}{r},z),$

where $\sigma_{m}^{2}$, as in Sec. II.1, is the variance of an individual
supernova’s brightness due to its intrinsic luminosity,
$\bar{n}^{2D}_{S},\bar{n}^{2D}_{L}$ are the mean angular number densities for
supernovae in the “S”th redshift bin and galaxies in the “L”th redshift bin
respectively, while $P_{mm}$ and $P_{gg}$ are the matter and galaxy power
spectra in turn. Note $C_{\ell,\rm{LSS}}^{SS^{\prime}}$ is the power spectrum
of the E-mode shear except for a constant factor of $(5/\ln(10))^{2}$, see
e.g., Zhang  _et al._ (2005); Fang and Haiman (2007).

Same as before, we use the Fisher matrix technique to forecast parameter
constraints from galaxy-supernovae cross-correlation. Since statistical
isotropy implies that different multipoles are uncorrelated, the Fisher matrix
can be written as a sum of contributions from different multipoles,

$F_{\alpha\beta}=\sum_{\ell}\sum_{\begin{subarray}{c}(LS),\\\
(L^{\prime}S^{\prime})\end{subarray}}\frac{\partial C_{\ell}^{LS}}{\partial
p_{\alpha}}({\rm Cov}_{\ell})^{-1}\frac{\partial
C_{\ell}^{L^{\prime}S^{\prime}}}{\partial p_{\beta}},$ (13)

where $(LS)$ or $(L^{\prime}S^{\prime})$ labels distinct cross power spectra.
${\rm Cov}_{\ell}$ is the subblock of the full covariance matrix (Eq. 8) for
all the cross power spectra with multipole $\ell$. We note its inverse is
proportional to $2\ell\Delta\ell f_{\rm sky}$, hence $F_{\alpha\beta}\propto
f_{\rm sky}$, and the parameter constraints will be proportional to $f_{\rm
sky}^{-1/2}$.

To extract the pure geometrical constraints, we choose our redshift bins to be
narrow enough such that the following approximation (under the limit of
$\Delta z\rightarrow 0$) to the cross power spectrum holds to a good accuracy,

$\displaystyle C_{\ell}^{LS}$ $\displaystyle\approx$
$\displaystyle-\frac{15}{2\ln(10)}\Omega_{m}H_{0}^{2}\frac{r(\chi_{S}-\chi_{L})}{r(\chi_{S})r(\chi_{L})}\frac{1}{a(z_{L})}$
(14) $\displaystyle\times
P_{gm}(k=\frac{\ell}{r(\chi_{L})},z_{L})\Theta(\chi_{S}-\chi_{L}),$

where $\chi_{S}=\chi(z_{S})$, $\chi_{L}=\chi(z_{L})$, with $z_{S}$, $z_{L}$
representing the mean redshifts of the narrow supernova and galaxy bins
respectively. With this approximation, we can easily separate geometrical
information ($r(\chi_{S}-\chi_{L})/r(\chi_{S})$) from what remains whose
prediction typically involves uncertainties in galaxy bias and matter power
spectrum in the nonlinear regime, which we hereafter denote as
$C_{\ell}^{gm}(z_{L})$. Next, we take the $C^{gm}$s at different multipoles
and redshifts also as parameter entries for the Fisher matrix, and marginalize
over them for the final geometrical constraints on the cosmological
parameters. Including these extra parameters significantly increases the
dimension of the Fisher matrix, hence increases the difficulty for its
inversion. However, from Eq. (13), we find that $F_{\alpha\beta}=0$, when
$p_{\alpha}$ and $p_{\beta}$ correspond to $C^{gm}$ at different multipoles.
This feature greatly simplifies the inversion of the Fisher matrix with the
method of “inversion by partitioning” Press  _et al._ (1992).

####  II.2.2 Galaxy-galaxy lensing

Compared to galaxy-supernovae cross-correlation, galaxy-galaxy lensing Tyson
_et al._ (1984) can be detected with a stronger significance and to a higher
redshift, for it is much easier to observe a large number of galaxies to a
high redshift than to observe supernovae. In this section, we consider the
pure geometrical probe from galaxy-galaxy lensing.

While galaxy-supernovae cross-correlation is the correlation between the
distribution of a foreground galaxy population and magnifications in the
background supernovae’s brightnesses, galaxy-galaxy lensing is the correlation
between the former and distortions in the background galaxies’ images caused
by weak lensing - the cosmic shear field. To be explicit, galaxy-galaxy
lensing is the cross-correlation between $\delta_{n}^{2D}$, given by Eq. (4),
and $\Gamma^{i}$, given by the following,

$\Gamma^{i}(\vec{\theta})=\int dzW_{S}(z)\gamma^{i}(z,\vec{\theta}),$ (15)

where $\gamma^{i}$ is the shear field estimated through measurements of
background galaxies’ ellipticities, and “$i$” labels the two shear components.
Note, $W_{S}(z)$ here is the selection function for the “sources”, i.e.,
background galaxies. Similar to $M(\vec{\theta})$, $\Gamma^{i}(\vec{\theta})$
represents the average of the shear estimated from background galaxies whose
angular positions are within a solid angle $d^{2}\theta$
($d^{2}\theta\rightarrow 0$) around $\vec{\theta}$.

The shear field can be decomposed into a curl-free E-mode component and a
divergence-free B-mode component. With scalar perturbations alone, only the
E-mode shear exists, which is equivalent to the lensing convergence.
Therefore, we only need to study the cross-correlation between galaxy and the
E-mode shear, which we recognize to be exactly the same as the galaxy-
supernovae cross-correlation, except for a factor of $5/\ln(10)$.

In this section, we consider a weak lensing survey. As before, we divide the
galaxies into redshift bins of equal size $\Delta z$, and denote the galaxy-
galaxy lensing power spectrum as $C_{\ell}^{LS}$. Different from before, “S”
here labels the redshift bin for the “source” galaxies. The expression for the
covariance of $C_{\ell}^{LS}$ and $C_{\ell^{\prime}}^{L^{\prime}S^{\prime}}$
remains the same as before, except the following differences: (1) $C_{\ell,\rm
LSS}^{SS^{\prime}}$ has not a factor of $(5/\ln(10))^{2}$; (2)
$C_{\ell,\rm{shot}}^{SS^{\prime}}$ is now given by

$C_{\ell,\rm{shot}}^{SS^{\prime}}=\frac{\delta_{SS^{\prime}}^{K}}{\bar{n}_{S}^{2D}}\int
dzW_{S}(z)\gamma_{\rm rms}^{2}(z),,\\\ $ (16)

where $\bar{n}_{S}^{2D}$ is the mean angular number density for source
galaxies in the “S”th redshift bin, and $\gamma_{\rm rms}$ is the rms of shear
in each component from galaxies’ intrinsic ellipticities. In the end, we use
the same method as in Sec II.2.1 to forecast the pure geometrical constraints
on cosmological parameters from galaxy-galaxy lensing.

####  II.2.3 Galaxy-CMB lensing cross-correlation

The Universe’s large-scale structure gravitationally deflects the photons of
CMB as well, and thus perturbs the CMB power spectra. The weak lensing
convergence for the CMB can be reconstructed from the various lensed CMB power
spectra using the minimum variance estimator, which minimizes the
reconstruction noise Okamoto and Hu (2003). For cosmic shear, the source
galaxies are typically distributed across a relative broad range of redshift,
and the signals are then averaged over this distribution. However, the CMB
photons originate from a very narrow range of comoving distance, thus the
source redshift distribution can be approximated as a Dirac $\delta$-function
with the value of redshift known to a very precise level. For the cross-
correlation between galaxy and CMB lensing, the expressions for the cross
power $C_{\ell}^{LS}$ and the covariance between $C_{\ell}^{LS}$ and
$C_{\ell^{\prime}}^{L^{\prime}S}$ (“S” here labels the redshift bin for the
“source” of CMB) remain the same as for the galaxy-galaxy lensing, except
$C_{\ell,\rm{shot}}^{SS}$ is now replaced by the CMB lensing reconstruction
noise, whose expression is given by, e.g. Eq.(42) in Okamoto and Hu (2003).

##  III curvature constraints

In this section, we present the constraints on curvature from the geometrical
probe constructed from galaxy-lensing cross-correlations. In Sec III.1, we
forecast the constraints from the galaxy-lensing cross-correlations in
combination with the supernovae Hubble diagram. We make our forecast for
fiducial surveys mimicking the LSST and Stage IV CMB experiment. In addition
to detecting a large number of type Ia supernovae, the LSST will measure both
the galaxy distribution and cosmic shear field at the same time. Hence, three
of the probes discussed in the last section, i.e., the supernovae Hubble
diagram, galaxy-supernovae cross-correlation, and galaxy-galaxy lensing, will
be available from the LSST, while the galaxy-CMB lensing cross-correlation can
be measured from the overlapping area between the LSST and Stage IV CMB
experiment. (We assume the Stage IV CMB survey overlaps completely with the
LSST.) In Sec III.2, we add in BAO and CMB to further tighten the constraints.
In Sec III.3, we study improvements in the constraints from increasing the
high redshift extension of supernovae while keeping their total number fixed.

###  III.1 Combination of galaxy-lensing cross-correlation with supernovae
Hubble diagram

The supernovae Hubble diagram probes the angular diameter distance from a
given redshift to an observer on the earth, i.e., at $z=0$, while the galaxy-
lensing cross-correlation can additionally probe the angular diameter
distances from a given redshift (the sources’ redshift) to observers at all
intermediate redshifts with $z\neq 0$ (the lenses’ redshifts). Hence, the
latter provides important complementary information.

The LSST is about to survey a sky area of approximately 20,000 deg2 for a
duration of 10 years starting by 2022 Ivezić  _et al._ (2019). It will detect
about half a million type Ia supernovae to a redshift slightly beyond $z=1$,
see Fig. 2 for the supernovae’s redshift distribution LSST Science
Collaboration  _et al._ (2009). In the following, we assume the total number
of supernovae $N_{\rm tot}$ to be $4\times 10^{5}$ Ivezić  _et al._ (2019),
and the rms of their intrinsic brightnesses $\sigma_{m}$ to be $0.1$ LSST
Science Collaboration  _et al._ (2009). At the same time, the LSST will
observe galaxies at an average angular number density of $50$ arcmin-2, which
is the so-called “gold” sample of the LSST galaxies, the redshift distribution
of which is shown as the solid line in Fig. 2 LSST Science Collaboration  _et
al._ (2009). Among these galaxies, we assume about $60\%$ of them can be used
for shear measurement Ivezić  _et al._ (2019), hence, for galaxy-galaxy
lensing, the number density of source galaxies is 30 arcmin-2, and we assume
$\gamma_{\rm rms}$ has a redshift-independent value of $0.28$ LSST Science
Collaboration  _et al._ (2009). For the Stage IV CMB experiment, we assume a 1
arcmin beam and $1\mu\rm K$ arcmin noise Abazajian  _et al._ (2016), and
assume it covers the LSST survey area.

Figure 2:  Normalized redshift distributions of type Ia supernovae (histogram)
and galaxies (solid line) for the LSST.

For the galaxy-lensing cross-correlations, we choose a redshift bin size of
$\Delta z=0.1$ for the type Ia supernovae, lens and source galaxies. We note
that the recent analysis by Prat  _et al._ (2019) chooses a bin size of
$\Delta z=0.15$ for the lens galaxies, and finds the correction from the
narrow tracer bin approximation to be negligible compared to their measurement
errors. For galaxy-supernovae cross-correlation, we have 11 supernovae bins
from $z=0$ to $z_{\rm max}=1.1$ – the highest supernova redshift, and this
gives us 55 ($10\times 11/2$) distinct cross power spectra. For galaxy-galaxy
lensing, we cut off the galaxy distribution at $z_{\rm max}=4$ in our
numerical calculation, which includes $\sim 99\%$ of all the galaxies, and we
have $780$ distinct cross power spectra. This also gives us 40 galaxy-CMB
lensing cross power spectra. In Fig. 3, we explicitly show the power spectra
for the galaxy-supernovae cross-correlation (“$g\kappa^{\rm SN}$”, black) and
galaxy-galaxy lensing (“$g\kappa^{\rm g}$”, red) for the same redshift bins of
lenses and sources. Note, the former has been divided by a factor of
$5/\ln(10)$, hence the two curves agree. The cross power spectrum for the
galaxy and CMB lensing for the same lens galaxies is also shown as the green
curve. The error bands including both sample variance and shot noise are
forecasted according to Eq. (8), with appropriate adaptions for $g\kappa^{\rm
g}$ and $g\kappa^{\rm CMB}$ as discussed in Sec. II. For the auto power
spectrum of CMB lensing and the reconstructed noises for the Stage IV CMB
experiment as considered here, we refer the readers to Schmittfull and Seljak
(2018).

For CMB lensing, we use the public available package $quicklens$ qui  to do
lensing reconstruction from the T, E and B modes of CMB, and we cut off the
multipoles at $\ell=3000$, due to the difficulty of cleaning temperature
foregrounds at $\ell>3000$ for ground-based CMB experiments. For $g\kappa^{\rm
SN}$ and $g\kappa^{\rm g}$, since we marginalize over parameters directly
describing the galaxy-matter power spectrum, there is no need to worry about
nonlinear effects and baryonic effects on small scales, we use information on
angular scales up to $\ell\sim 10^{5}$ Zhang  _et al._ (2005). Of course, not
much information can be obtained from modes with high enough multipoles due to
significant noise.

If we define the total $S/N$ square for the galaxy-lensing cross-correlation
as

$\left(\frac{S}{N}\right)^{2}\equiv\sum_{\ell}\sum_{\begin{subarray}{c}(LS),\\\
(L^{\prime}S^{\prime})\end{subarray}}C_{\ell}^{LS}({\rm
Cov}_{\ell})^{-1}C_{\ell}^{L^{\prime}S^{\prime}},$ (17)

we find for the LSST, the galaxy-supernovae cross-correlation has $S/N=103.8$,
while the galaxy-galaxy lensing has $S/N=2291.2$, about $22$ times larger due
to its wider redshift coverage and smaller shot noise (see Table 1). While for
the LSST and Stage IV CMB experiment, the galaxy-CMB lensing cross-correlation
has $S/N=1842.3$.

Figure 3:  The power spectra for the galaxy-supernovae cross-correlation (“$g\kappa^{\rm SN}$”, black), galaxy-galaxy lensing (“$g\kappa^{\rm g}$”, red), and galaxy-CMB lensing cross-correlation (“$g\kappa^{\rm CMB}$”, green). Error bands are forecasted for the LSST and Stage IV CMB experiment. We have divided the galaxy-supernovae cross power and its errors by a factor of $5/\ln(10)$. The lens galaxies are in the redshift bin of z:[0.4, 0.5], while the supernovae and source galaxies are in z:[0.5, 0.6]. The error bands for the galaxy-CMB lensing cross power are plotted up to $\ell=3000$. Table 1:  Total $S/N$ for detecting the galaxy-lensing cross-correlation from the LSST and Stage IV CMB experiments. | S/N
---|---
$g\kappa^{\rm SN}$ | 103.8
$g\kappa^{\rm g}$ | 2291.2
$g\kappa^{\rm CMB}$ | 1842.3

Throughout this paper, we choose our fiducial cosmological model to be the
flat $\Lambda$CDM model with $\Omega_{\Lambda}=0.689$ Planck Collaboration
_et al._ (2020b); when calculating $P_{\rm mm}$, we further choose
$\\{\Omega_{b}h^{2},\Omega_{m}h^{2},\sigma_{8},n_{s}\\}$=$\\{0.022,0.142,0.81,0.967\\}$
Planck Collaboration  _et al._ (2020b), and adopt the Smith et al prescription
Smith  _et al._ (2003) to account for the effects of non-linear evolution;
when calculating $P_{gg}$ and $P_{gm}$, we assume a simple linear bias of
$b=1$.

Degeneracy between dark energy parameters and curvature is expected, since
they both affect the Universe’s expansion rate, hence the comoving radial
distance, though curvature has an additional effect on the comoving angular
diameter distance. Therefore, the constraints on curvature will depend on
specific assumptions about dark energy, characterized by assumptions for its
equation of state $w$ which determines the evolution of its energy density.

For comparison, we forecast the constraints on curvature for a wide range of
choices of $w$. We first consider the following common choices: (1) $w$ is
fixed to be (-1), i.e., dark energy is the cosmological constant; (2) $w$ is a
constant whose value needs to be determined; (3) $w$ varies with time as
$w=w_{0}+w_{a}(1-a)$, with the values of $w_{0}$, $w_{a}$ to be determined.
From the first to the third choice, the degrees of freedom in $w$ is
increasing. Next, we consider a fourth more general parametrization where $w$
is a binned function parametrized by its values in $N$ equal-sized $a$ bins
from $a=0$ to $a=1$, which we denote as $w_{i}$, with $i=1,...,N$. Each
$w_{i}$ is allowed to vary independently. This parametrization potentially can
enclose all possible models for $w$, hence the constraints on curvature
obtained with this choice after marginalizing over all dark energy parameters
can be considered as independent of models of dark energy. In the following,
we will study the constraints on curvature with $N=$10, 20, and 50, and the
results when $N=50$ will be our most dark energy independent constraints on
curvature.

The constraints on curvature with the above assumptions for dark energy
forecasted for the LSST and Stage IV CMB experiments are shown in Table 2. In
different columns, we show the constraints from different probes, with the
second column showing the constraints from the supernovae Hubble diagram alone
(“SN”), and the third and fourth columns giving the constraints from the
combinations of “SN” with “$g\kappa^{\rm SN}$” and “$g\kappa^{\rm g}$”
respectively. The last column gives the constraints from the combinations of
“SN” with “$g\kappa^{\rm g}$” and “$g\kappa^{\rm CMB}$”. We do not show the
constraints from either “$g\kappa^{\rm SN}$”, “$g\kappa^{\rm g}$” or
“$g\kappa^{\rm CMB}$” alone, because they are less interesting due to severe
parameter degeneracy Bernstein (2006). In different sections, we show the
constraints with different assumptions for dark energy. From the first to the
third sections, we show in turn the constraints with our first three choices
of $w$, while from the fourth to the last, we give the constraints for the
fourth choice with $N=10,\ 20,\ 50$ respectively. Note, from the top to the
bottom sections, the uncertainty in our knowledge about $w$ is increasing.

Table 2: Parameter constraints forecasted for the LSST and Stage IV CMB experiments. From left to right, the columns are constraints from the supernovae Hubble diagram (“SN”), its combination with galaxy-supernovae cross-correlation (“SN + $g\kappa^{\rm SN}$”), and combination with galaxy-galaxy lensing (“SN + $g\kappa^{\rm g}$”), and combination with galaxy-galaxy lensing and galaxy-CMB lensing(“SN + $g\kappa^{\rm g}$ \+ $g\kappa^{\rm CMB}$”). Note, our constraints from “$g\kappa^{\rm SN}$” , “$g\kappa^{\rm g}$” and ”$g\kappa^{\rm CMB}$”are pure geometrical. From top to bottom, the constraints in different sections are based on dark energy parametrizations with more degrees of freedom in its equation of state $w$: the first three sections are for (1) $w=-1$, (2) $w=$const, (3) $w=w_{0}+w_{a}(1-a)$; while the last three sections assume $w$ is a binned function parametrized by its values in $N$ bins from $a=0$ to $a=1$, with $N=10,\ 20,\ 50$ respectively. parameters | SN | SN+$g\kappa^{\rm SN}$ | SN+$g\kappa^{\rm g}$ | SN+$g\kappa^{\rm g}$+$g\kappa^{\rm CMB}$
---|---|---|---|---
$\Omega_{K}$ | 0.00815 | 0.00815 | 0.00659 | 0.00166
$\Omega_{\Lambda}$ | 0.00568 | 0.00568 | 0.00463 | 0.00142
$\Omega_{K}$ | 0.0701 | 0.0698 | 0.00874 | 0.00200
$\Omega_{\Lambda}$ | 0.0720 | 0.0717 | 0.00882 | 0.00396
$w$ | 0.0825 | 0.0822 | 0.0123 | 0.0101
$\Omega_{K}$ | 0.408 | 0.356 | 0.0151 | 0.00214
$\Omega_{\Lambda}$ | 0.517 | 0.451 | 0.00886 | 0.00792
$w_{0}$ | 0.515 | 0.449 | 0.0156 | 0.0101
$w_{a}$ | 1.28 | 1.12 | 0.173 | 0.107
$\Omega_{K}$ | 1.16 | 0.614 | 0.0359 | 0.0343
$\Omega_{\Lambda}$ | 1.30 | 0.702 | 0.0735 | 0.0709
$\Omega_{K}$ | 8.22 | 0.722 | 0.0407 | 0.0389
$\Omega_{\Lambda}$ | 9.19 | 1.32 | 0.175 | 0.170
$\Omega_{K}$ | 9.72 | 0.723 | 0.0417 | 0.0400
$\Omega_{\Lambda}$ | 8.68 | 2.35 | 0.391 | 0.384

By comparing the constraints in different sections for each probe, we find
that when there are more degrees of freedom in dark energy’s equation of
state, the constraints on curvature get weaker, consistent with one’s
expectation. Our strongest constraints are hence obtained when $w=-1$, and the
weakest ones obtained when $w$ is a binned function with 50 bins in $a$. It
can also be easily seen that the constraints from the combined probes of
either “SN + $g\kappa^{\rm SN}$” or “SN + $g\kappa^{\rm g}$” are better than
“SN” alone, due to the important complimentary information provided by the
galaxy-lensing cross-correlation which breaks the degeneracy between curvature
and dark energy parameters.

It is interesting to find that the improvements in the curvature constraints
by adding in “$g\kappa^{\rm SN}$” or “$g\kappa^{\rm g}$” get more significant
when dark energy has more degrees of freedom, with the improvements being
minimal when dark energy is known to be the cosmological constant and maximal
when $w$ is parametrized by 50 $w_{i}$s. This highlights the importance of
galaxy-lensing cross-correlation in obtaining dark energy independent
constraints on curvature. Moreover, the constraints from the combination of
“SN + $g\kappa^{\rm g}$” are better than from “SN + $g\kappa^{\rm SN}$”. For
$N=50$, the improvement factor is 233 by adding in “$g\kappa^{\rm g}$” to
“SN”, while 13 when adding in “$g\kappa^{\rm SN}$”. This is expected as
compared to “$g\kappa^{\rm SN}$”, “$g\kappa^{\rm g}$” has a wider redshift
coverage and lower shot noise. To tighten the constraints more, we add in
“$g\kappa^{\rm CMB}$” to “SN + $g\kappa^{\rm g}$” in the last column, we do
not add “$g\kappa^{\rm SN}$” here to avoid the strong correlations between
“$g\kappa^{\rm SN}$” and “$g\kappa^{\rm g}$” . We notice, the improvements by
adding in “$g\kappa^{\rm CMB}$” to “SN + $g\kappa^{\rm g}$” get milder when
dark energy has more degrees of freedom, ranging from 4 when dark energy is
known to be the cosmological constant to 1.04 when $N=50$.

Finally, we find that though the curvature constraints keep getting weaker
when $w$ is allowed to have more degrees of freedom, the constraints from the
combined probes of “SN + $g\kappa^{\rm g}$ \+ $g\kappa^{\rm CMB}$” do not
degrade much when we take $w$ to be a binned function and increase the number
of bins from 10 to 20 to 50. As discussed in Witzemann  _et al._ (2018), a
limited number of bins that is equally-spaced in scale factor is enough for
parameter constraints to converge. Therefore, we hereafter quote the
constraints with $w$ a binned function with 50 bins in $a$ as our final “dark
energy independent” constraints.

To summarize, we obtain dark energy independent constraints on $\Omega_{K}$ of
$0.723$ from the combination of “SN + $g\kappa^{\rm SN}$” , $0.0417$ from “SN
+ $g\kappa^{\rm g}$” for the LSST, and $0.04$ from “SN + $g\kappa^{\rm g}$ \+
$g\kappa^{\rm CMB}$” for the LSST and Stage IV CMB experiments. We find the
galaxy-lensing cross-correlation plays a significant role in obtaining these
results. It improves the curvature constraints by breaking the degeneracy
between curvature and the dark energy parameters from “SN” alone. If we know
dark energy to be the cosmological constant, we are able to get much tighter
constraints of $0.00815$ for “SN + $g\kappa^{\rm SN}$”, $0.00659$ for “SN +
$g\kappa^{\rm g}$”, and $0.00166$ for “SN + $g\kappa^{\rm g}$ \+ $g\kappa^{\rm
CMB}$”. We note that in obtaining these results, we do not apply any priors.
These results do not reach the desired accuracy level of $10^{-4}$, so below
we add other geometrical probes to improve the constraints on curvature
further.

###  III.2 Adding in BAO and CMB

In this section, we are interested in tightening the constraints on curvature
further by combining with other geometrical probes. Specifically, we include
the probes utilizing the standard ruler of sound horizon at recombination $s$.
Measurements of the CMB anisotropies can probe the angular size extended by
the sound horizon at recombination $\theta_{*}$, hence the angular diameter
distance to recombination $r_{*}(=s/\theta_{*})$. Late-time BAO measurements
can also probe the sound horizon111Strictly speaking, late-time BAO
measurements probe the sound horizon at the end of the baryon drag epoch. We
here neglect the small difference following Weinberg  _et al._ (2013). through
its imprints on matter distribution. Its extensions in the transverse
direction ($\delta\theta=s/r$) and the line-of-sight direction ($\delta z=sH$)
probe the late-time angular diameter distance $r$ and Hubble expansion rate
$H$ respectively. In this section, we include these two probes in our
forecast.

For the CMB constraints, we simply incorporate the Planck measurement for
$\theta_{*}$, which is at an accuracy level of $\sim 0.03\%$, and is very
stable to changes in the assumed cosmology Planck Collaboration  _et al._
(2020b). We calculate the sound horizon and redshift of recombination
according to the fitting formulas given in Hu (2005). These two quantities are
determined by the baryon and matter densities in the Universe, which
themselves are well measured from morphology of the CMB anisotropy power
spectrum. Therefore, we adopt the Planck constraints on $\Omega_{m}h^{2}$ and
$\Omega_{b}h^{2}$ as priors for our calculation.

For the BAO constraints, we follow Weinberg  _et al._ (2013) and consider a
Stage IV BAO experiment that maps $25\%$ of the full sky from $z=0$ to $z=3$
with errors $\sim 80\%$ larger than the linear theory sample variance errors
(to account for non-negligible shot noise and non-linear degradation of the
BAO signal). Such an experiments can be collectively achieved by the BAO
programs that are currently ongoing or under design, such as the programs from
Euclid euc , WFIRST wfi , and DESI des . We adopt the forecasted covariance
matrix for the measured quantities of $r/s$ and $sH$ from Weinberg  _et al._
(2013), and then use the Fisher matrix to derive constraints on the
cosmological parameters.

The constraints we obtained from BAO and CMB are shown in Table 3. By adding
BAO and CMB to our previous calculations we get stronger constraints, also
shown in Table 3. Here we only show the constraints obtained with the
assumption that dark energy is the cosmological constant (top section), and
with our most uncertain assumption for dark energy (i.e. $w$ is a binned
function with 50 equal-sized bins from $a=0$ to $a=1$, bottom section). Note,
whenever CMB is included, we apply a weak Gaussian prior with width $\Delta
w_{i}=10\sqrt{N}$ on all the $w_{i}$s Weinberg  _et al._ (2013).

Table 3:  Constraints from “BAO + CMB”, “SN + BAO + CMB” and “SN + $g\kappa^{\rm g}$ \+ $g\kappa^{\rm CMB}$ \+ BAO + CMB”(denoted as “All”). The upper section assumes dark energy is the cosmological constant, while the lower section assumes $w$ is a binned function parametrized by 50 $w_{i}$s - its values in $50$ bins from $a=0$ to $a=1$. For each $w_{i}$, a prior of $\Delta w_{i}=10\sqrt{N}(N=50)$ is applied. parameters | BAO+CMB | SN+BAO+CMB | All
---|---|---|---
$\Omega_{K}$ | 0.000644 | 0.000514 | 0.000496
$\Omega_{\Lambda}$ | 0.00288 | 0.000783 | 0.000782
$\Omega_{K}$ | 0.0227 | 0.00938 | 0.00130
$\Omega_{\Lambda}$ | 0.264 | 0.00939 | 0.00315

We focus on discussions about the “dark energy-independent constraints on
curvature”, which we simply refer to as “constraints on curvature” unless
otherwise explicitly stated in the following. From Table 3, it can be seen
that our constraint on curvature from “BAO + CMB” is better than that from “SN
+ $g\kappa^{\rm g}$ \+ $g\kappa^{\rm CMB}$”. Therefore, “BAO + CMB” is more
promising in constraining curvature in a dark-energy independent way than the
combination of “SN” and galaxy-lensing cross correlations.

At the same time, we also look at the constraint on curvature from BAO
alone222Planck priors on $\Omega_{m}h^{2}$ and $\Omega_{b}h^{2}$ are still
included here., which we find to be $0.0522$. This is slightly worse than the
constraint from “SN + $g\kappa^{\rm g}$” and “SN + $g\kappa^{\rm g}$ \+
$g\kappa^{\rm CMB}$”. Therefore, for Stage IV dark energy surveys, the
combination of “SN” and galaxy-lensing cross correlations provides a slightly
better probe of curvature than BAO.

To see the importance of galaxy-lensing cross correlations when they are added
to the combined probe of “SN + BAO + CMB”, we compare the 3rd and 4th columns
of Table 3, and find they are only mildly helpful when dark energy is the
cosmological constant, but can tighten the constraint on curvature by
approximately a factor of 7 when dark energy has a general parametrization.
Therefore galaxy-lensing cross-correlations play an important role in
extracting information on the Universe’s curvature in a dark energy
independent way.

Our ultimate constraint on curvature from the combination of the five
geometrical probes “SN + $g\kappa^{\rm g}$ \+ $g\kappa^{\rm CMB}$ \+ BAO +
CMB” now reaches $0.0013$. This is much better than either “SN + $g\kappa^{\rm
g}$ \+ $g\kappa^{\rm CMB}$” or “BAO + CMB”, which give curvature constraints
only at the level of $10^{-2}$, reflecting the strong complementarity of the
two combined probes.

To conclude, when BAO and CMB are included, the improvements on the curvature
constraints can be one order of magnitude. We find the dark energy independent
constraints on curvature now can be as tight as $\sim 1\times 10^{-3}$, but
still one order of magnitude away from the desired level of $10^{-4}$.
Moreover, even if dark energy is the cosmological constant, the constraint on
curvature from the combination of the five geometrical probes of “SN +
$g\kappa^{\rm g}$ \+ $g\kappa^{\rm CMB}$ \+ BAO + CMB” is only $\sim 0.0005$,
still larger than the target precision of $1\times 10^{-4}$. In the following
we explore how much we can gain on the curvature constraint by broadening the
redshift coverage of supernovae.

###  III.3 Increasing $z_{\rm max}$ of Supernovae

It is known that for a supernovae alone probe, a broader redshift coverage can
give better parameter constraints provided the total number of supernovae is
kept fixed Huterer and Turner (2001). This is because supernovae at different
redshifts usually lead to different degeneracy directions among the parameter
space, and a wider redshift coverage results in better complementarity. In
this section we investigate the possibility of tightening the curvature
constraints more with a wider redshift coverage of supernovae or standard
candles in general.

Specifically, we keep the total number of supernovae to be fixed at $4\times
10^{5}$, and assume their redshift distribution follows that of the LSST
galaxies but cuts off at $z_{\rm max}$. In the following, we will forecast the
constraints on curvature for two choices of $z_{\rm max}$: $z_{\rm max}=2$ and
$z_{\rm max}=3$. Supernovae at such high redshifts may be challenging to be
surveyed with ground-based telescopes like the LSST, but may be easier to
observe with future space-based ones. For example, the WFIRST mission is about
to find supernovae to $z_{\rm max}=1.7$, but with $N_{\rm tot}$ only $\sim
3000$ Spergel  _et al._ (2013), far less than $4\times 10^{5}$. Thus, we note
the forecasts we make in this section may be too optimistic for type Ia
supernovae surveys currently in plan. However, they may be more realistic for
other types of standard candles such as quasars and gamma-ray bursts, which
can be observed to much higher redshifts King  _et al._ (2014); Risaliti and
Lusso (2019).

The dark energy independent constraints on curvature obtained by extending
$z_{\rm max}$ from $z_{\rm max}=1.1$ to $z_{\rm max}=2$ and $z_{\rm max}=3$
are plotted in Fig. 4. It can be easily seen that the curvature constraints
from either “SN”, “SN+$g\kappa^{\rm SN}$”, “SN + $g\kappa^{\rm g}$ \+
$g\kappa^{\rm CMB}$ ”, or “SN + $g\kappa^{\rm g}$ \+ $g\kappa^{\rm CMB}$ \+
BAO + CMB” all get better when the supernovae’s redshift distribution can
reach a higher $z_{\rm max}$. We also find that the improvements in the
constraints by increasing $z_{\rm max}$ from $z_{\rm max}=1.1$ (Table 2) to
$z_{\rm max}=2$ are more significant than increasing it from $z_{\rm max}=2$
to $z_{\rm max}=3$. For the dark energy independent constraints on curvature
from “SN ”, we obtain $1.12$ with $z_{\rm max}=2$ (improved by a factor of
$\sim 8.7$ from $z_{\rm max}=1.1$), and $0.728$ with $z_{\rm max}=3$ (improved
by a factor of $\sim 1.5$ from $z_{\rm max}=2$); while for the dark energy
independent constraints from “SN + $g\kappa^{\rm g}$ \+ $g\kappa^{\rm CMB}$”,
we get $0.00926$ with $z_{\rm max}=2$ (improved by a factor of $\sim 4.3$ from
$z_{\rm max}=1.1$), and $0.00541$ with $z_{\rm max}=3$ (improved by a factor
of $\sim 1.7$ from $z_{\rm max}=2$). Even if we can increase the supernovae’s
redshift distribution only to $z_{\rm max}=2$, the efforts are rewarding
enough judging from the improvements on curvature constraints from “SN” and
its combination with galaxy-lensing cross-correlations. Again, we find as
before that combining galaxy-lensing cross-correlations and “SN” resulting in
significant improvement, even for the case with $g\kappa^{\rm SN}$, which has
a relative lower $S/N$.

However, for the combination of all five probes “SN + $g\kappa^{\rm g}$ \+
$g\kappa^{\rm CMB}$ \+ BAO + CMB”, the improvement from increasing $z_{\rm
max}$ is much milder, especially when increasing from $z_{\rm max}=2$ to
$z_{\rm max}=3$, reflecting the subdominant role of “SN” in the combined
probes (probably because BAO already provides the high-$z$ information up to
$z=3$), and the rapid decrease with redshift of galaxy distribution at high
$z$. We find the constraints from the combination of the five probes is still
$\sim 10^{-3}$. However, significant improvement on curvature constraint can
be possible with standard candles whose redshift distribution has a larger
fraction at high redshift, say $z>3$, which we postpone for future study.

Figure 4:  Forecast 1$\sigma$ constraints on curvature as a function of the
maximal redshift of supernovae for “SN”(black) , “SN + $g\kappa^{\rm
SN}$”(red), “SN + $g\kappa^{\rm g}$ \+ $g\kappa^{\rm CMB}$”(green) and “SN +
$g\kappa^{\rm g}$ \+ $g\kappa^{\rm CMB}$ \+ BAO + CMB”(blue, denoted as
“All”), respectively. The total number of supernovae is kept fixed at $4\times
10^{5}$. The dark energy equation of state parameter $w$ is assumed to be a
binned function parametrized by its values in $50$ bins from $a=0$ to $a=1$.

##  IV discussion

Our calculations above have adopted several simplifications. We have neglected
a few systematic effects such as photometric redshift errors, shear
calibration errors, galaxy intrinsic alignments etc. For photometric redshift
errors, the LSST galaxies’ photometry will have high enough quality to provide
a rms accuracy $\sigma/(1+z)$ of 0.02 Ivezić  _et al._ (2019). This is in
general much smaller compared to our bin width of $\Delta z=0.1$. Therefore,
we expect photometric redshift errors would not change at least the order of
magnitude of our results. For shear calibration errors, recent analysis by
Prat  _et al._ (2019) found that if the multiplicative shear bias $m$ from
LSST can be calibrated to $\sigma(m)=0.001$, which is the requirement set in
the LSST science book LSST Science Collaboration  _et al._ (2009), its effect
on cosmological constraints would be negligible. We then assume LSST shear
calibration will be accurate enough not to change much of the cosmological
constraints we have obtained. For intrinsic alignment, which probably will
weaken our constraints on curvature to some extent, the effect can be
minimized if the redshift separation between the lens and source galaxies are
increased to be large enough. Considering the ultimate constraints on
curvature we obtain is at the level of $10^{-3}$, one order of magnitude
larger than the desired level of $10^{-4}$, we do not analyze the effect of
intrinsic alignment together with other systematic effects (including those we
have not mentioned in the above, such as the narrow lens bin approximation,
lensing dilution and galaxy lensing boost factors) quantitatively here, which
will not change our primary finding.

In this work, we obtain our dark energy-independent constraints on curvature
by assuming the equation of state of dark energy is parametrized by its values
in $50$ $a$ bins from $a=0$ to $a=1$ and marginalizing over all these
parameters. Compared to previous work on this topic by Knox (2006) and
Mortonson (2009), this approach is more model-independent: the method proposed
by Knox (2006) depends on the assumption that dark energy is completely
negligible in “matter-dominated” regime, while ours allows dark energy to be
non-negligible even at early times (early dark energy); Mortonson (2009)
parametrizes $w$ of dark energy at low redshift ($z\;\hbox to0.0pt{\lower
2.5pt\hbox{$\sim$}\hss}\raise 1.5pt\hbox{$<$}\;1.7$) with $15$ principle
components (PCs), but calculations of the PCs typically depend on the
specifics of both the data set and cosmological model used to obtain the
Fisher matrix from which they are derived.

However, compared to previous work by Bernstein (2006), our approach is not as
model-independent. Bernstein (2006) probes curvature purely from the
relationship between $r_{L}$, $r_{LS}$ and $r_{S}$. By marginalizing over the
distances of $r_{L}$, $r_{S}$ which are integrals of functions of the Hubble
expansion rate, the obtained constraints do not depend on any energy component
of the Universe or dynamics that governs its expansion, but only on the
validity of the FRW metric. Our approach is less general in the sense that we
in addition assume energy-momentum conservation and the validity of the
Friedmann equation if cosmic acceleration is due to dark energy, or if cosmic
acceleration is due to modified gravity, its effect on the Universe’s
expansion can be viewed equivalently as an effective dark energy, which holds
for most interesting modified gravity models (see e.g., Hu and Sawicki (2007);
Fang  _et al._ (2008a)). Therefore, though in this paper we use the term of
“dark energy”-independent constraints on curvature, our constraints are
actually independent on the unknown mechanism for cosmic acceleration
including both dark energy and modified gravity.

Since we obtain our curvature constraints by marginalizing over the
contribution of dark energy or “effective” dark energy to the Universe’s
expansion, while Bernstein (2006) obtain theirs by marginalizing over the
distances, our constraints will be tighter than theirs. We conclude that
Bernstein (2006) provide a pure metric probe of curvature which does not
depend on how the Universe expands, while we probe the curvature in a way that
is independent on how (“effective”) dark energy affects the Universe’s
expansion.

There are also many works in the literature that utilize measurements of the
angular diameter distances and Hubble expansion rates to obtain model-
independent constraints on curvature, e.g. Clarkson  _et al._ (2007); Takada
and Doré (2015); Cai  _et al._ (2016); Yu and Wang (2016); Wei and Wu (2017);
Wang  _et al._ (2020). These works typically need to estimate derivatives of
the angular diameter distances or to reconstruct the Hubble expansion history
using some model-independent smoothing techniques such as the Gaussian
process. Thus, it may be hard to achieve an accuracy as tight as
$\sigma_{\Omega_{K}}\sim 10^{-4}$ robustly using these methods.

In this work, we have focused on constraining curvature using probes of the
Universe’s geometry. One can surely add in probes of the Universe’s growth of
structure to tighten the constraints. Actually, Mortonson (2009) has studied
model independent constraints on curvature from combining geometry probes with
growth probes through measuring the abundance of X-ray clusters. However,
their work is done within the “smooth” dark energy paradigm Fang  _et al._
(2008b) and assumes the validity of general relativity, hence it does not
apply to dark energy with nontrivial clustering properties or modified
gravity. Also, probes of structure growth are usually subject to systematics
from theoretical modeling of baryonic physics and growth of structure on
nonlinear scales. Future work on using probes of structure growth to constrain
curvature in a dark energy-independent way should take into account of all
these problems, which may be challenging.

##  V summary

Accurate constraints on curvature provide a powerful probe of inflation.
However, current accurate constraints on curvature are almost all derived upon
simple assumptions of dark energy such as assuming it is the cosmological
constant. Considering the large uncertainties in our theoretical understanding
about dark energy, constraints with these assumptions may lead to unreliable
conclusions when they are used to test inflation models. Hence, for a robust
test of inflation models, it is important to obtain constraints on curvature
that are independent on uncertainties in our knowledge about dark energy. In
this paper, we have investigated such constraints on curvature from the
geometrical probe constructed from galaxy-lensing cross-correlations and its
combination with other common geometrical probes.

We study the galaxy-magnification, galaxy-shear, and galaxy-CMB lensing cross-
correlations, with magnification measured from the type Ia supernovae’s
brightnesses. We find for the Stage IV dark energy survey of LSST and the
Stage IV CMB survey, the galaxy-magnification cross-correlation
(“$g\kappa^{\rm SN}$”) can be detected with signal-to-noise ratio $S/N=104$,
the galaxy-shear cross-correlation (“$g\kappa^{\rm g}$”) with $S/N=2291$, and
the galaxy-CMB lensing cross-correlation (“$g\kappa^{\rm CMB}$”) with
$S/N=1842$. We include the supernovae Hubble diagram (“SN”) to break parameter
degeneracy, which is available with the same supernovae data set used to
measure “$g\kappa^{\rm SN}$”. We obtain dark energy independent constraints on
$\Omega_{K}$ of $0.723$ from “SN + $g\kappa^{\rm SN}$”, $0.0417$ from “SN +
$g\kappa^{\rm g}$”, and $0.04$ from “SN + $g\kappa^{\rm g}$ \+ $g\kappa^{\rm
CMB}$” for the LSST and Stage IV CMB experiment. We find that the galaxy-
lensing cross-correlation plays a significant role in tightening the curvature
constraint by breaking the degeneracy between curvature and the dark energy
parameters, especially when dark energy is completely unknown. We find the
constraint from “SN + $g\kappa^{\rm g}$ \+ $g\kappa^{\rm CMB}$” is better than
that from a Stage IV BAO experiment, but not as good when BAO is combined with
the Planck measurement for the acoustic scale in the CMB. Adding the galaxy-
lensing cross-correlations to the combined probe of “SN + BAO + CMB” results
in a factor of 7 improvement in the dark energy independent constraints on
curvature, but much milder improvement when dark energy is known to be the
cosmological constant. We obtain our ultimate constraint on $\Omega_{K}$ of
$0.0013$ from “SN + $g\kappa^{\rm g}$ \+ $g\kappa^{\rm CMB}$ \+ BAO + CMB”.
Our analysis also shows that tighter constraints can be obtained with better
knowledge about dark energy.

We investigate the possibility of tightening the curvature constraints further
by increasing the redshift extension of supernovae or standard candles in
general, while keeping the total number fixed at the same value. We find
though the “SN” alone and its combination with galaxy-lensing cross
correlations have significant improvements on curvature constraints, the
combined probes of “SN + $g\kappa^{\rm g}$ \+ $g\kappa^{\rm CMB}$ \+ BAO +
CMB” does not. However, improvements can still be achievable with a larger
fraction of standard candles at high redshift, larger than that for the LSST
galaxies which we have assumed for the supernovae in our analysis. While this
is hard to realize with supernovae, it can be easier to achieve with other
types of standard candles such as quasars Risaliti and Lusso (2019). We plan
to investigate more about this in a future paper.

## Acknowledgements

We thank Neal Dalal and Lam Hui, discussions with whom initiated this work,
and thank Pengjie Zhang, Jun Zhang, Bhuvnesh Jain for useful conversations.
This work is supported by the National Natural Science Foundation of China
Grants No.11773024, No.11653002, No.11421303, by the China Manned Space
Program through its Space Application System, by the Fundamental Research
Funds for Central Universities, and by the CAS Interdisciplinary Innovation
Team.

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