Application of cross correlations between CMB and large scale structure to constraints on the primordial non-Gaussianity

Application of cross correlations between CMB and large scale structure to constraints on the primordial non-Gaussianity

Yoshitaka Takeuchi Department of Physics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan    Kiyotomo Ichiki Department of Physics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan    Takahiko Matsubara Kobayashi-Maskawa Institute for the Origin of Particles and the Universe, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan
July 27, 2019
Abstract

The primordial non-Gaussianity of local type affects the clustering of dark matter halos, and the planned deep and wide photometric surveys are suitable for examining this class of non-Gaussianity. In our previous paper, we investigated the constraint from the cross correlation between CMB lensing potential and galaxy angular distribution on the primordial non-Gaussianity, without taking into account redshift slicing. To improve our previous analysis, in this paper, we add the galaxy lensing shear into our analysis and take into account redshift slicing to follow the redshift evolution of the clustering. By calculating 81 power spectra and using the Fisher matrix method, we find that the constraint on the primordial non-Gaussianity can be improved from to by including the galaxy-galaxy lensing shear cross correlations expected from the Hyper Suprime-Cam survey (HSC), in comparison with the constraint without any cross correlations. Moreover, the constraint can go down to by including the galaxy-CMB lensing cross correlations from the ACTPol and Planck experiments.

pacs:
98.80.-k, 98.62.Sb, 98.65.-r

I Introduction

Primordial non-Gaussianities have been intensively discussed, because any detection of them offers an important window into the early Universe. Standard single-field slow-roll inflation models predict small non-Gaussianity Mukhanov and Chibisov (1981); Allen et al. (1987); Gangui et al. (1994); Maldacena (2003); Acquaviva et al. (2003), so that the detection of large non-Gaussianity may turn down the standard model and suggest physics beyond the standard model. The most popular method to hunt for primordial non-Gaussianities is to measure higher order correlation functions of the cosmic microwave background (CMB) anisotropies such as the the three-point correlation function (bispectrum) and the four-point one (trispectrum). Non-vanishing signals of these higher order correlations may predict the presence of primordial non-Gaussianities. Komatsu et al. (2003); Spergel et al. (2007); Creminelli et al. (2007); Komatsu et al. (2009); Smith et al. (2009); Komatsu et al. (2011a).

Recently, many studies have revealed that the primordial non-Gaussianity of local type affects the large-scale structure (LSS) through the clustering of dark matter halos, and shown the modification of the halo mass function Lo Verde et al. (2008); Matarrese et al. (2000); Verde et al. (2001); D’Amico et al. (2011); Grossi et al. (2009); Yokoyama et al. (2011) and the halo bias Slosar et al. (2008); Dalal et al. (2008); Afshordi and Tolley (2008); Matarrese and Verde (2008); Yokoyama (2011); Gong and Yokoyama (2011); Desjacques et al. (2009), both by numerical simulations and by analytic calculations. Among several types of primordial non-Gaussianities, the local-type non-Gaussianity induces strong scale-dependence of the halo bias Slosar et al. (2008); Dalal et al. (2008); Afshordi and Tolley (2008); Matarrese and Verde (2008). This scale-dependent bias is also powerful tool to constrain the primordial non-Gaussianity from the observations of LSS independently of the method with CMB higher order correlation functions.

Using this scale-dependent feature in the halo bias, some measurements of the primordial non-Gaussianity have been done in Refs. Slosar et al. (2008); Afshordi and Tolley (2008); Xia et al. (2011). It is expected that future wide and deep surveys, such as Subaru Hyper Suprime-Cam (HSC) survey Miyazaki et al. (2006), Dark Energy Survey (DES) The Dark Energy Survey Collaboration (2005), Large Synoptic Survey Telescope (LSST) LSST Science Collaborations et al. (2009), and so on, will put tighter constraints on the primordial non-Gaussianity, which are expected to be comparable to those from the future CMB observations or more (e.g., Carbone et al. (2008, 2010); Oguri and Takada (2011); Cunha et al. (2010)).

To get a tighter constraint on the primordial non-Gaussianity, it is important to distinguish the signature of the primordial non-Gaussianity from the other effects more precisely, especially from the linear bias in the limited survey area. Thus, it is better for this purpose to combine unbiased observables which are sensitive to the matter power spectrum in the near Universe, such as weak gravitational lensing of CMB and galaxy lensing shear.

Cross correlations with complementary probes are expected to provide additional information on top of their respective auto correlations. The cross correlation between CMB and LSS is known to give information about the integrated Sachs-Wolfe (ISW) effect of CMB temperature anisotropy, which generates the secondary anisotropies due to the time variation of the gravitational potential Sachs and Wolfe (1967). Moreover, CMB lensing also has a correlation with LSS since the gravitational lensing of CMB is induced by the gravitational potential produced by LSS, and the galaxy lensing shear as well Jeong et al. (2009).

Since the effects of the CMB lensing are imprinted on small scales, the Wilkinson Microwave Anisotropy Probe (WMAP) satellite has not detected the CMB lensing directly. Recent ground based CMB experiments such as Atacama Cosmology Telescope (ACT), announced the detection of the CMB lensing signal, and an ongoing CMB observation by Planck The Planck Collaboration (2006) or various ground based experiments are expected to detect this signal more precisely. Therefore, we can be sure that the observation of CMB lensing will increase its accuracy and play an important role in observational cosmology in near future surveys.

In this paper we consider prospects for constraining the primordial non-Gaussianity through the scale-dependent bias from the near future surveys taking into account all auto and cross correlations among CMB, galaxy distribution and galaxy lensing shear. We shall show that galaxy-CMB lensing and galaxy-galaxy lensing cross correlations are particularly fruitful and allow us to improve the constraints on the primordial non-Gaussianity.

This paper is organized as follows. In section II we briefly review the effects of the primordial non-Gaussianity for large-scale structure and show the modifications to the mass function and bias of dark matter halos. In section III we summarize the observables employed in our analysis and describe the angular power spectra of the cross correlations between CMB and LSS. In section IV we mention the survey design and photometric redshift systematics employed in our analysis. In section V we explain the method and the setup of our analysis. In section VI we show the results particularly focusing on the constraints on the primordial non-Gaussianity. In section VII we discuss the effects of the photometric redshift systematics and the massive neutrino on the constraints. Then we assess the contribution of the cross correlations for the constraints. Moreover, we compare the constraints from some different surveys. Finally, in section VIII we summarize our conclusions. Through this paper we assume a spatially flat Universe for simplicity.

Ii Primordial non-Gaussianity

Deviations from Gaussian initial conditions are commonly parameterized in terms of the dimensionless parameter and the primordial non-Gaussianity of the local form is described as Komatsu and Spergel (2001); Verde et al. (2000); Gangui et al. (1994)

 Φ=ϕ+fNL(ϕ2−⟨ϕ2⟩), (1)

where is the curvature perturbation and is a Gaussian random field. On the subhorizon scale, is related to the Newtonian gravitational potential as .

The existence of the primordial non-Gaussianity, , indicates that the initial density field is positively or negatively skewed. Furthermore, the fact that the non-Gaussianity affects the clustering of dark matter halos or galaxies allows us to constrain on the non-Gaussianity from large-scale structure surveys. In particular, the local-type non-Gaussianity described by Eq. (1) induces a scale-dependent enhancement of the halo/galaxy power spectrum.

In the presence of the primordial non-Gaussianity, the mass function of clustering halos is modified and we adopt a non-Gaussian correction factor of the halo mass function based on the Edgeworth expansion Lo Verde et al. (2008):

 dn/dMdnG/dM=1+σMS36(ν2−3ν)−16d(σMS3)dlnν(ν−1ν), (2)

where is the skewness of the density field which is proportional to , is the rms of the linear density field smoothed on mass scale , is defined as and is the critical linear overdensity. is the mass function in the non-Gaussian case and is the one in the Gaussian case. For the Gaussian one, we adopt a model of Warren Warren et al. (2006).

Recent studies have shown that the local-type primordial non-Gaussianity produces a scale-dependent enhancement of the clustering of halo on large-scales,

 Pg(M,z,k)=b2G(M,z)P(k,z)→[bG+Δb(M,z,k)]2P(k,z), (3)

where is the galaxy power spectrum, is the matter power spectrum, is the bias in the Gaussian case and is the non-Gaussian correction of the halo bias described as Desjacques et al. (2009)

 Δb(M,z,k)=3Ωm,0H20k2T(k)D(z)fNLδc(bG(M,z)−1)−νδcddν(dn/dMdnG/dM). (4)

Here, and are the matter energy density normalized by the critical density and the Hubble parameter at present, is the linear growth rate normalized to the scale factor in the matter-dominant era and is the transfer function of linear matter density fluctuations. For the halo bias in the Gaussian case , we assumed the form presented by Sheth Sheth et al. (2001).

Iii Angular power spectrum

Here we briefly review the auto and cross correlations of various cosmological observables and their angular power spectra. In this paper we take into account CMB temperature () anisotropies, -mode polarization () and CMB lensing potential () for CMB observables and galaxy distribution (g) and weak lensing shear () for LSS observables.

iii.1 Integrated Sachs-Wolfe (ISW) effect

The decay of the Newtonian potential due to the presence of dark energy produces a differential gravitational redshift, and this effect is called the late-time integrated Sachs-Wolfe (ISW) effect. In a flat Universe the presence of ISW effect is a signature of dark energy, and induces a non-vanishing cross correlation between CMB temperature and large-scale structure measurements, such as galaxy distribution, weak lensing field and so on.

The contribution of the ISW effect to the CMB temperature field can be written as

 ΘISW(^n)=ΔTISW(^n)TCMB=−2TCMB∫χ∗0dχ˙Φ(χ^n,χ), (5)

where is the mean temperature of the CMB, is the direction to the line of sight, is the comoving distance and denotes the distance to the last scattering surface. is the gravitational potential and a dot denotes a derivative with respect to the conformal time.

The angular power spectrum of the cross correlation between CMB temperature through ISW effect and the other measurements can be written as

 CTXℓ=2π∫k2dkPΦ(k)ΔISWℓ(k)ΔXℓ(k), (6)

where

 ΔISWℓ(k)=3Ωm,0H20∫z∗0dzddz{D(z)a(z)}T(k)jℓ(kχ(z)), (7)

is the primordial power spectrum of as a function of the wave number and is the tilt of the primordial power spectrum. The functions and are the transfer function and the growth rate for linear matter density fluctuations, respectively, and is a spherical Bessel function. The kernel is for the other measurements, namely, CMB lensing potential, galaxy distribution and weak lensing shear in this paper; , respectively.

The noise spectra of CMB include detector noise and residual foreground contamination. Here, we assume the ideal condition that foreground contamination can be completely removable and include only Gaussian random detector noise of the form Knox (1995)

 NT,Pℓ=[∑ν{(θFWHMΔT,P)−2e−ℓ(ℓ+1)θFWHM/8ln2}]−1, (8)

where is the spatial resolution of the beam and represents the sensitivity to the temperature and polarization per pixel, respectively. These values are given for each of the frequency bands of channels and we show the values for two CMB experiments such as Planck The Planck Collaboration (2006) and ACT with new polarization sensitive receiver (ACTPol) Niemack et al. (2010) in Table 1.

iii.2 CMB lensing potential

CMB photons are deflected due to the gravitational potential produced by the large-scale structure on the way propagating to us (e.g., Ref. Lewis and Challinor (2006)). The relationship between the lensed temperature anisotropy and the unlensed one is given by the deflection angle as . The deflection is related to the line of sight projection of the gravitational potential as , where

 ψ(^n)=−2∫χ∗0dχχ∗−χχ∗χΨ(χ^n,χ). (9)

Here, is the (effective) lensing potential.

The angular power spectrum of the lensing potential can be written as

 Cψℓ=2π∫k2dkPΦ(k)[Δψℓ(k)]2, (10)

where

 Δψℓ(k)=3Ωm,0H20∫χ∗0dχχ∗−χχ∗χT(k)D(z(χ))a(χ)jℓ(kχ). (11)

The lensing potential can be reconstructed with quadratic statistics in the temperature and polarization data that are optimized to extract the lensing signal. To reconstruct the lensing potential, one needs to use the non-Gaussian nature imprinted into the lensed CMB statistics, and the noise of the lensing potential can be estimated as the reconstruction error Hu and Okamoto (2002); Okamoto and Hu (2003). In this paper, we estimate the noise spectrum of lensing potential following the technique developed in Okamoto and Hu (2003) optimally combining the temperature and polarization fields.

iii.3 Galaxy distribution

The luminous sources such as galaxies must be the most obvious tracers of the large-scale structure in the linear regime, and the projected density contrast of the galaxies can be written as

 δg,i(^n)=∫∞0dzbeff(z,k)ni(z)nAiδ(χ(z)^n,z), (12)

where the subscript represents the -th redshift bin. represents the matter density fluctuation, and and are the galaxy redshift distribution and the total number of galaxies per steradian in the -th redshift bin. The function is the weighted effective halo/galaxy bias defined as

 beff(z,k)=[∫∞MobsdMdndM]−1∫∞MobsdMb(M,z,k)dndM, (13)

where is the observable mass threshold, which is the minimum mass of the galaxy we can observe, and we take the value to be . The function and are the halo mass function and the halo bias in the case of the non-Gaussian initial condition defined in Eqs. (2) and (3), respectively. We show the dependence of on the redshift and the wave number with different mass thresholds and in Fig. 1.

The angular power spectrum of the galaxy distribution between -th and -th redshift bins can be written as

 Cgigjℓ=2π∫k2dkPΦ(k)Δgiℓ(k)Δgjℓ(k), (14)

where

 Δgiℓ(k)=∫dzbeff(z,k)ni(z)nAiT(k)D(z)jℓ(kχ). (15)

To estimate the signal-to-noise ratios and errors in parameter determination for each survey, we need to describe the noise contribution due to the finiteness in the number of sources associated with source samples. We can write the noise spectra from the shot noise as

 Ngigjℓ=δij1¯ni, (16)

where is the mean surface density of sources per steradian in the -th redshift bin.

iii.4 Weak lensing shear

The weak lensing shear (equivalently the convergence in the weak lensing limit) is a weighted integral of the density field of sources, which is directly related to the source galaxy redshift distribution. The average convergence of a light ray bundle from sources in the -th redshift bin is written as

 κi(^n)=∫∞0dχWi(χ)δ(χ^n,χ), (17)

where is the convergence weight function of kernel defined as

 Wi(χ(z))=3Ωm,0H202χ(z)a(z)∫∞zdz′ni(z′)nAiχ(z′)−χ(z)χ(z′). (18)

Here is the distance to the lens and is the one to the source.

The angular power spectra of the weak lensing shear between -th and -th redshift bins can be written as

 Cγiγjℓ=2π∫k2dkPΦ(k)Δγiℓ(k)Δγjℓ(k), (19)

where

 Δγiℓ(k)=∫dzWi(z)T(k)D(z)jℓ(kχ). (20)

The measurement of the shear from galaxy images has some uncertainties, one of which mainly comes from the intrinsic shape ellipticities of galaxies. The galaxy shapes can be treated statistically through a shape noise contribution and the noise spectra can be written as

 Nγiγjℓ=δijσ2γ¯ni, (21)

where is the intrinsic galaxy shear. In this paper, we assume the intrinsic galaxy shear to be as the empirically derived value Bernstein and Jarvis (2002) for all surveys.

iii.5 Cross correlation angular power spectrum

The effect of the primordial non-Gaussianity stands out the most in the galaxy-galaxy auto correlation (gg) through the bias parameter. As many previous works show, almost all the constraints on the primordial non-Gaussianity come from the galaxy-galaxy power spectrum Slosar et al. (2008); Afshordi and Tolley (2008); Xia et al. (2010a, b, 2011). The cross correlation power spectra, e.g., CMB Temperature-galaxy cross correlation (g), however, provide extra information and improve the errors of the parameters.

In our previous analysis Takeuchi et al. (2010), we paid a particular attention to the CMB lensing-galaxy cross correlation (g), and estimated the contribution to the constraint on the primordial non-Gaussianity. Here, we improve our analysis by adding the galaxy lensing shear, which may also correlated with galaxy distribution and CMB lensing, and estimate the errors of determining cosmological parameters including any possible auto and cross correlations.

Fig. 2 shows an example of the cross correlation angular power spectra, for galaxy-galaxy lensing (g), galaxy-CMB lensing (g) and CMB lensing-galaxy lensing (). The boxes around the curve show the expected measurement errors for HSC- survey (for an illustrating purpose, we take the logarithmic binning, as ). For the cross correlations with CMB lensing, we show the errors for two cases that one is for Planck and the other is Planck with ACTPol.

Iv Survey design

The errors in determining cosmological parameters from observations of LSS depend a great deal on the survey design, such as the survey region, the mean redshift and photometric redshift errors. Here we summarize the survey design employed in our analysis.

iv.1 Redshift distribution of galaxies

We assume the redshift distribution of galaxy samples with a function of the form:

 n(z)=¯ngβΓ[(α+1)/β]zαzα+10exp[−(zz0)β], (22)

where is the mean number density of source galaxies at all redshifts, , and are free parameters. We adopt , , and the parameter is related to the mean redshift as

 zm=∫dzzn(z)¯ng=z0Γ[(α+2)/β]Γ[(α+1)/β], (23)

and is determined in such a way that the mean redshift fits the value in Table 2 for each survey. The normalization of the redshift distribution function is fixed by the total number density of galaxies:

 nA=∫∞0dzn(z). (24)

iv.2 Photometric redshift systematics

In the galaxy imaging survey whose observables are galaxy distribution and galaxy shear, we measure a large number of galaxies so that the systematic uncertainties play a correspondingly important role with statistical errors. Although systematics may include many effects and these effects depend on the details of the observations Bernstein (2009); Das et al. (2011), we include only the effect of photometric redshift errors.

Imperfect calibration of photometric redshifts induces a residual scatter and bias with respect to the true redshifts, and uncertainties in the redshifts distort the volume element. Following the photometric redshift model as described in Ma et al. (2006), the probability distribution of photometric redshift given the true redshift , is modeled as a Gaussian distribution

 p(zph|z)=1√2πσzexp[−(z−zph−zbias)22σ2z], (25)

where and are the scatter and bias, respectively, which are arbitrary function of redshift .

The best-estimated distribution for objects in the -th photometric redshift bin with , , can be written as

 ni(z) = ∫z(i+1)phz(i)phdzphn(z)p(zph|z), (26) = 12n(z)[erf(xi+1)−erf(xi)],

where is defined as

 xi≡(z(i)ph−z+zbias(z))/√2σz(z). (27)

In the same way as Eq. (24), the total number density of galaxies in the -th bin becomes

 nAi=∫∞0dzni(z). (28)

We assume a fiducial redshift scatter and bias following Das et al. (2011) as

 σz(z)=σ(i)z(1+z), (29) zbias(z)=z(i)bias(1+z), (30)

where and are defined for each redshift bin. We set the fiducial values as , for each redshift bin and all surveys.

V Forecasts

We estimate the parameter errors using the Fisher matrix formalism. We summarize the Fisher matrix formalism taking into account the cross correlation between CMB and LSS, and we describe the setup for the CMB and LSS observables.

v.1 Fisher matrix formalism

Under the assumptions that the observables follow the Gaussian statistics, we can obtain the information on a set of parameters from the Fisher matrix written as Tegmark et al. (1997)

 Fℓ,ij=fskyℓmax∑ℓmin(2ℓ+1)2Tr[Cℓ,iC−1ℓCℓ,jC−1ℓ], (31)

where is the sky coverage of the experiment, is the signal plus noise covariance matrix, and is the derivative of with respect to the parameter .

For CMB experiments, we consider two cases: that one is the constraint from Planck and the other is from Planck with ACTPol. It is well known that combining the results of the satellite CMB experiment with the ground based one can improve the constraints, e.g., combining WMAP with ACTDunkley et al. (2010), ACBARReichardt et al. (2009) or so on. For these two cases, we define the total Fisher matrix as below, respectively, under the assumption for the size of survey area as . Here , and are the sky coverage of the ACTPol, the LSS and the Planck surveys, respectively, and we define the overlap region between CMB experiments and LSS surveys as .

• LSS + Planck:

 Fℓ,ij=fCrossskyℓLSSmax∑ℓmin=2FCrossℓ,ij+fPlanckskyℓCMBmax∑ℓLSSmax+1FCMBℓ,ij+(fPlancksky−fCrosssky)ℓLSSmax∑ℓmin=2FCMBℓ,ij, (32)
• LSS + Planck + ACTPol:

 Fℓ,ij=fCrossskyℓLSSmax∑ℓmin=2FCrossℓ,ij∣∣ACTPol+fACTPolskyℓCMBmax∑ℓLSSmax+1FCMBℓ,ij∣∣ACTPol+(fLSSsky−fCrosssky)ℓLSSmax∑ℓmin=2FCrossℓ,ij∣∣Planck+(fPlancksky−fLSSsky)ℓLSSmax∑ℓmin=2FCMBℓ,ij∣∣Planck+(fPlancksky−fACTPolsky)ℓCMBmax∑ℓLSSmax+1FCMBℓ,ij∣∣Planck, (33)

where

 FCrossℓ,ij = (2ℓ+1)2Tr[CCrossℓ,i(CCrossℓ)−1CCrossℓ,j(CCrossℓ)−1], (34) FCMBℓ,ij = (2ℓ+1)2Tr[CCMBℓ,i(CCMBℓ)−1CCMBℓ,j(CCMBℓ)−1],

and different superscripts for and represent different observations or CMB experiments, e.g., and are the maximum multipole we can take for the CMB experiment and the LSS survey, respectively. Here is the minimum multipole we can take and we use in all cases. We summarize these values in Table 4.

The covariance matrix and each of its matrix elements is defined as

 [CCross,CMBℓ]XY = CXYℓ+NXYℓδXY, (36) where X,Y = {T,E,ψ,gi,γi(for Cross)T,E,ψ(for CMB),

is the angular power spectrum and is the noise spectrum for each auto or cross correlation. We defined and as and in Eq. (8), and in Eqs. (16) and (21), respectively, and is the noise spectrum of CMB lensing potential we estimate following the technique developed in Okamoto and Hu (2003). Then we assume that there is no correlation of noises between the different observables and different redshift bins. Moreover, we assume that -mode polarization () has correlation only with temperature anisotropies () because -mode polarization can be generated through the Thomson scattering dominantly on the last scattering surface, .

Incidentally the assumption that is not strictly correct Lewis et al. (2011) because the large-scale polarization -mode generated by scattering at reionization correlates with sources of CMB lensing potential over long distance. Moreover, if the galaxies are at high-redshift, the assumption that is also not correct Challinor and Lewis (2011). However the influence of these assumptions for our results, especially constraints on the primordial non-Gaussianity, can be small because sample galaxies in our analysis are at low redshift.

Here we explain the detail of our Fisher matrices given in Eqs. (32) and (LABEL:eq:Fisher_II). In Eq. (32) the first term is responsible for the cross correlation between LSS and CMB, the second one is for CMB observables at the higher multipoles than the LSS survey can probe, and the last one is for CMB observables in the remaining survey area of CMB which does not overlap with the LSS survey. In Eq. (LABEL:eq:Fisher_II) the first term is responsible for the cross correlation between LSS and CMB with ACTPol, the second one is for CMB observables with ACTPol at the higher multipoles than the LSS survey can probe, the third one is for the cross correlation between LSS and CMB with Planck in the overlapping region which does not overlap with ACTPol, the forth one is for CMB observables with Planck in the region which does not overlap with the LSS survey, and the last one is for CMB observables with Planck at the higher multipoles than the LSS survey can probe in the region which does not overlap with ACTPol.

In our analysis, the total number of non-zero angular power spectra is 81 for the full cross correlation case (), while it is 5 for the only CMB case ().

v.2 Setup

Given the measurement design of the surveys, we can estimate the errors in determining the cosmological parameters using the Fisher matrix formalism. The formalism tells us how well the given surveys can measure the cosmological parameters around fiducial cosmological model. Therefore, the parameter forecasts by Fisher matrix formalism depend on the choice of the fiducial model and the number of free parameters.

As for our fiducial cosmological model, we assume a flat CDM model and we include the following 9 cosmological parameters, which is based on the WMAP seven-year results Komatsu et al. (2011b). The density parameters of cold dark matter, baryon and dark energy are , and , respectively; dark energy equation of state parameter is ; the optical depth at reionization is ; and the primordial power spectrum parameters are the spectrum tilt and the amplitude of the primordial power spectrum which is normalized at . We also include the primordial non-Gaussianity of local type modeled by the non-linear parameter . For specifying the galaxy bias, we include the minimum mass of halo hosting the galaxies, for which we can observe as a free parameter. The fiducial values are

 {100Ωbh2, Ωch2, ΩΛ, τ, nS, Δ2R, w, fNL, logMobs}={0.20, 0.1109, 0.72, 0.086, 0.96, 2.4×10−9, −1, 0, 11.7}. (37)

Throughout this paper we assume a spatially flat Universe, and the Hubble parameter is adjusted to keep our Universe flat when we vary .

To calculate the angular power spectra including non-linear effects on the angular power spectrum of the galaxy distribution, cosmic shear and lensing potential, we use the CAMB code Lewis et al. (2000) and the HALOFIT code Smith et al. (2003). We, however, want to remove the burden of the uncertainty of the non-linear calculation as much as possible, so our estimation preferably is based on the information from large-scale for LSS where the linear prediction is reliable.

In our estimation, we include the information from temperature anisotropies (), -mode polarization () and reconstructed lensing potential () for CMB. The range of multipoles are and the survey area is taken as for Planck and for ACTPol, respectively. On the other hand, for the galaxy survey () and galaxy weak lensing survey (), the ranges of multipole are and the survey areas are for both of them with HSC. Furthermore we take into account all the cross correlation we can consider and we assume the optimal condition that there is no correlation between different patches, so that the area having a correlation between CMB and LSS corresponds to the overlapped survey area. We summarize the values we use in the following calculation in TABLE 4.

Vi Result

vi.1 Signal-to-Noise ratio

To see the errors in the cross correlation power spectra more quantitatively, we estimate the signal-to-noise ratio () for each -bin and the cumulative (). We define the for a cross correlation between X and Y, following Peiris and Spergel (2000), as

for each -bin:

 (SN)2(ℓ)≡fskyℓ(i)max∑ℓ(i)min(2ℓ+1)×(CXYℓ)2(CXXℓ+NXXℓ)(CYYℓ+NYYℓ)+(CXYℓ)2, (38)

stacked :

 (SN)2(≤ℓmax)≡fskyℓmax∑ℓ=2(2ℓ+1)×(CXYℓ)2(CXXℓ+NXXℓ)(CYYℓ+NYYℓ)+(CXYℓ)2, (39)

where and are the maximum and minimum multipoles of the -th multipole bin, respectively, and , are the fiducial and noise spectra, respectively. The index represents .

Figure  3 shows an example of for each given by Eq. (38) and cumulative by Eq. (39), for the galaxy-galaxy lensing (g), galaxy-CMB lensing (g) and CMB lensing-galaxy lensing () cross correlations. For the galaxy-CMB lensing and CMB lensing-galaxy lensing with HSC-Planck experiments, the amplitude of decreases at high- bins and the stacked saturates due to the noise of CMB lensing at small scales. On the other hand, we can expect the lager at high- bins for the case with HSC-ACTPol. The amplitude of at low- bins where the signature of the primordial non-Gaussianity shines out is very low in all the cases. However, comparing galaxy-CMB lensing with galaxy-galaxy lensing signals, the amplitude of the former is larger than the latter in high-redshift bins and the difference becomes even greater for the with HSC-ACTPol.

vi.2 Constraints on the primordial non-Gaussianity

We estimate the errors in the determination of cosmological parameters for the HSC- survey using the Fisher matrix formalism, and show the main results, namely, the 1 confidence limit constraints in Fig. 4 and Table 5, in which we consider 10 cosmological parameters: the fraction of the massive neutrino density parameter to the matter density parameter in addition to the nine cosmological parameters defined in Sec. V.2.

The simplest case, which is from only galaxy clustering, puts the constraint on as , and the constraint can be improved to by combining the CMB-only result though CMB has no information about primordial non-Gaussianity on the level of the 2-point function without cross correlations with galaxy clustering. This is because the information from CMB helps to determine the galaxy bias by breaking degeneracies with other cosmological parameters, and then the degeneracy between and can be broken effectively. Hence the constraint on is also improved dramatically after combining CMB information, which can be seen in Table 5, and these tendencies have been seen in our previous analysis Takeuchi et al. (2010).

We show four cases to see the improvement by adding the different observables in Fig. 4. In the first case we use CMB and galaxy distribution (CMB + g); in the second case we add CMB lensing to the first case (CMB + g + ); in the third case we add shear to the first case (CMB + g + ); and in the last case we use all the information available. We take into account all the available power spectra. The selection of auto and cross correlations in Fig. 4 and Table 5 corresponds to each other in each case. We obtain the constraints on as with Planck and HSC. Moreover, we can improve the constraint by including ACTPol as , thanks to the galaxy-CMB lensing cross correlations.

Vii Discussion

vii.1 Selection of the fiducial model

We assume imaging surveys for the LSS galaxy surveys, and we have to take into account the influence of the various systematics, e.g.,, magnification effects due to gravitational lensing Namikawa et al. (2011), photometric redshift(photo-) errors Ma et al. (2006) and so on (see, for example, Bernstein (2009); Das et al. (2011)). Moreover, as mentioned in the previous section, the parameter forecasts by Fisher matrix formalism depend on the choice of the fiducial model and free parameters. In Fig. 5, we summarize the effects of selecting fiducial models.

vii.1.1 Photometric redshift error

For the constraint on , any significant difference is not found between with and without the photometric redshift scatter systematics. This is because the redshift scatter changes the overall amplitude of the galaxy distribution (g) and cosmic shear () as shown in Fig. 2. On the other hand, the effect of non-Gaussianity, , emerges only at large angular scales, and it can be distinguishable. However, the effects of the photometric redshift error are significant for the parameters which are determined through the information about the redshift evolution such as and . Although the effect of the parameter is to change the bias parameter and is similar to the effect of the redshift scatter, a significant difference is not found in the figure.

vii.1.2 Massive neutrino

Various experiments, such as ground based experiments of neutrino oscillation, predict the non-zero neutrino mass Schwetz et al. (2008); Maltoni et al. (2004). Therefore, we need to include the mass of neutrinos for a more accurate estimation of the cosmological parameters. There are, however, some debatable subtle arguments about taking the mass of neutrinos into account in models for the mass function and the halo bias. These models have been tested by N-body simulations without massive neutrinos and are not guaranteed enough for cosmological models with massive neutrinos. The analytic formula for the scale-dependent bias presented in Eq. (4) is also for models without massive neutrinos.

For the mass function, it is found that the influence of massive neutrinos is very small Ichiki and Takada (2011). Thus, we just assume that the halo mass function and halo bias models can be applied even if the massive neutrinos are included.

Comparing the results of and models in Fig. 5, a significant difference can be seen in the constraint on . The existence of massive neutrinos affects the large-scale structure formation through its large velocity dispersion and alter the amplitude of the matter power spectrum. This effect is similar to the effect from and these two parameters are strongly degenerated with each other as shown in Fig. 4. Note that the degrade of the constraint on in the case of the model comes from the strong degeneracy between them and we have confirmed that there is little change in the constraint if we fix the value of (and do not take it as a free parameter). For the constraints on itself, however, the difference of the fiducial models whether or not is not so significant.

In summary, focusing on the constraint of , the selection of the fiducial model makes little impact on the result. On the other hand, the constraints on some other parameters depend on the fiducial model, especially on the existence of neutrino mass and photometric redshift error.

vii.2 Contribution of cross correlations

Now let us discuss the contribution from ACTPol experiment which will extract the information of CMB lensing more efficiently than Planck.

As shown Fig. 4 and Table 5, we find that almost all the constraints on each plane are determined through galaxy lensing information except for and . However, CMB lensing also gives important contributions if the ACTPol experiment is included. Focusing on the constraints on the - plane, both the CMB lensing and galaxy lensing signals improve the constrains slightly by including their cross correlations, though the most of the constraints are determined by the galaxy auto correlation.

We show how the cross correlations improve the constraints in Fig. 6 and Table 6. We find that most of the constraints are mainly determined through the auto correlations on each plane but we can still gain a little benefit from the cross correlations. By including all cross correlations among CMB, galaxy distribution and galaxy lensing shear, we can expect improvement for the constraint on with HSC and Planck, and improvement with HSC, Planck and ACTPol. In particular, the impact of the cross correlation between galaxy and galaxy lensing (g) is significant, especially in the - plane. This result is consistent with that obtained by Namikawa et al. (2011); Oguri and Takada (2011) and g will play a very important role in the constraint on dark energy parameters.

Finally, to see the impact of cross correlations for the constraint on the primordial non-Gaussianity, let us focus on the constraints on the - plane in Fig. 6. We find that the cross correlation between galaxy and CMB lensing (g) will improve the constraint on when the Planck and ACTPol experiments are combined. However we have little benefit from the cross correlations if we use only Planck for the CMB experiment.

vii.3 Dependence on mass threshold Mobs

The parameter reflects the mass threshold we can observe as galaxy and relates to the galaxy bias through Eq. (13). We can interpret from Fig. 1 that the larger value of predicts the larger bias. We show the constraints for the different fiducial value of in Fig. 7. Only in this figure, we plot instead of for the purpose of illustrating the different fiducial models on the same planes.

The difference which comes from the different fiducial values of appears especially in the constraints on and , while the constraints on the other parameters are not altered so much. This is because the constraints on and come mainly from the galaxy distribution (g), while the constraints of the others come mainly from the other observables. As for the constraints on , the larger value of can lead to a tighter constraint on . The larger value of means picking out the higher mass objects. Because high mass objects exhibit a large bias, the non-Gaussian correction to the bias becomes large. Furthermore, the correction to the mass function also becomes large for high mass objects. For these reasons, the effect of the non-Gaussianity through the effective bias