Galaxy-CMB and galaxy-galaxy lensing on large scales: Sensitivity to primordial non-Gaussianity

A convincing detection of primordial non-Gaussianity in the local form of the bispectrum, whose amplitude is given by the $f_{\mathrm{NL}}$ parameter, offers a powerful test of inflation. In this paper, we calculate the modification of two-point cross-correlation statistics of weak lensing—galaxy-galaxy lensing and galaxy-cosmic microwave background (CMB) crosscorrelation—due to $f_{\mathrm{NL}}$. We derive and calculate the covariance matrix of galaxy-galaxy lensing, including cosmic variance terms. We focus on large scales ($l<100$) for which the shape noise of the shear measurement becomes irrelevant and cosmic variance dominates the error budget. For a modest degree of non-Gaussianity, $f_{\mathrm{NL}}=\pm 50$ modifications of the galaxy-galaxy-lensing signal at the 10% level are seen on scales $R\sim 300\mathrm{Mpc}$, and grow rapidly toward larger scales as $\propto R^2$. We also see a clear signature of the baryonic acoustic oscillation feature in the matter power spectrum at $\sim 150\mathrm{Mpc}$, which can be measured by next-generation lensing experiments. In addition, we can probe the local-form primordial non-Gaussianity in the galaxy-CMB lensing signal by correlating the lensing potential reconstructed from CMB with high-$z$ galaxies. For example, for $f_{\mathrm{NL}}=\pm 50$, we find that the galaxy-CMB lensing cross-power spectrum is modified by $\sim 10\%$ at $l\sim 40$, and by a factor of 2 at $l\sim 10$, for a population of galaxies at $z=2$ with a bias of 2. The effect is greater for more highly biased populations at larger $z$; thus, high-$z$ galaxy surveys cross correlated with CMB offer a yet another probe of primordial non-Gaussianity.

Figure 1

Coordinate system and ${\gamma }_1$ and ${\gamma }_2$. The shear along ${\mathit{e}}_1$ has ${\gamma }_1>0$ and ${\gamma }_2=0$, whereas the shear along ${\mathit{e}}_2$ has ${\gamma }_1<0$ and ${\gamma }_2=0$. The shear along ${\mathit{e}}_1+{\mathit{e}}_2$ has ${\gamma }_1=0$ and ${\gamma }_2>0$, whereas the shear along ${\mathit{e}}_1-{\mathit{e}}_2$ has ${\gamma }_1=0$ and ${\gamma }_2<0$.

Figure 2

Critical surface density ${\Sigma }_c(z_L;z_S)$ as a function of the source redshift $z_S$ for various lens redshifts that roughly correspond to the Two Degree Field Galaxy Redshift Survey (2dFGRS; $z_L=0.1$, solid), the main sample of the Sloan Digital Sky Survey (SDSS; $z_L=0.2$, dotted), the luminous red galaxies (LRGs) of SDSS ($z_L=0.3$, dashed), and the Large Synoptic Survey Telescope (LSST; $z_L=0.5$ and 0.8, dot-dashed and triple-dot-dashed, respectively).

Figure 3

The baryonic feature in the matter power spectrum, as seen in the galaxy-galaxy lensing $\Delta \Sigma (R)$ for several populations of lens galaxies with $b_1=2$ at $z_L=0.3$ (similar to SDSS LRGs, solid), $b_1=2$ at $z_L=0.5$ (higher-$z$ LRGs, dotted), $b_1=2$ at $z_L=0.8$ (galaxies that can be observed by LSST, dashed), and $b_1=5$ at $z_L=0.8$ (clusters of galaxies that can be observed by LSST, dot-dashed). The vertical line shows the location of the baryonic feature, $R_{\mathrm{BAO}}=106.9h^{-1}\mathrm{Mpc}$, calculated from the “$\mathrm{WMAP}+\mathrm{BAO}+\mathrm{SN}$ ML” parameters in Table 1 of 2. Note that we have used the linear matter power spectrum and the Gaussian initial condition ($f_{\mathrm{NL}}=0$) for this calculation.

Figure 4

Imprints of the local-type primordial non-Gaussianity in the galaxy-galaxy lensing $\Delta \Sigma (R)$ for the same populations of lens galaxies as in Fig. 3. The solid, dashed, and dotted lines show $f_{\mathrm{NL}}=0$, $\pm 50$, and $\pm 100$, respectively.

Figure 5

Fractional differences between $\Delta \Sigma (R)$ from non-Gaussian initial conditions and the Gaussian initial condition $|\Delta \Sigma (R;f_{\mathrm{NL}})/\Delta \Sigma (R;f_{\mathrm{NL}}=0)-1|$ calculated from the curves shown in Fig. 4. The dot-dashed, dashed, and dotted lines show $f_{\mathrm{NL}}=\pm 10$, $\pm 50$, and $\pm 100$, respectively, while the thin solid line shows $\propto R^2$ with an arbitrary normalization.

Figure 6

Same as Fig. 3, but with the expected $1\mathrm{\text{−}}\sigma$ uncertainties for full-sky lens surveys and a single lens redshift. Adjacent bins are highly correlated, with the correlation coefficients shown in Fig. 7. The open (filled) boxes show the binned uncertainties with (without) the cosmic variance term due to the cosmic shear field included. See Eqs. (28, 29) for the formulae giving open and filled boxes, respectively. We use the radial bin of size $\Delta R=5h^{-1}\mathrm{Mpc}$. For comparison, we also show $\Delta \Sigma (R)$ computed from the smooth power spectrum without the baryonic feature [56] (dashed lines). Note that the uncertainties are calculated for a single lens redshift slice, and thus they will go down as we add more lens redshift slices.

Figure 7

The cross-correlation-coefficient matrix $r_{ij}\equiv C_{ij}/\sqrt{C_{ii}{C}_{jj}}$, where $C_{ij}$ is the covariance matrix given in Eq. (27), for a radial bin of $\Delta R=5h^{-1}\mathrm{Mpc}$. We show $r_{ij}$ for the same populations of lens galaxies as shown in Figs. 3, 6. We use the same number of source galaxies and the same shape noise as in Fig. 6. The neighboring bins are highly correlated for $\Delta R<10h^{-1}\mathrm{Mpc}$.

Figure 8

Same as Fig. 4, but with the expected $1\mathrm{\text{−}}\sigma$ uncertainties for full-sky lens surveys and a single lens redshift. Adjacent bins are highly correlated. The open (filled) boxes show the binned uncertainties with (without) the cosmic variance term due to the cosmic shear field included. See Eqs. (28, 29) for the formulae giving open and filled boxes, respectively. We use logarithmic bins with $\Delta R=R/10$. Note that the uncertainties are calculated for a single lens redshift slice, and thus they will go down as we add more lens redshift slices.

Figure 9

Angular power spectrum of the galaxy-convergence cross correlation $C_l^{h\kappa }$ at various multipoles as a function of the lens redshift $z_L$ for two effective source redshifts $z_s=1$ (top) and 2 (bottom). We have divided $C_l^{h\kappa }$ by its maximum value. The solid, dotted, dashed, dot-dashed, and triple-dot-dashed lines show $l=10$, 50, 100, 350, and 1000, respectively.

Figure 10

Imprints of the local-type primordial non-Gaussianity in the galaxy-convergence cross-power spectrum, $l(l+1)C_l^{h\kappa }/(2\pi )$, for the same populations of lens galaxies as in Fig. 3. The solid, dashed, and dotted lines show $f_{\mathrm{NL}}=0$, $\pm 50$, and $\pm 100$, respectively.

Figure 11

Fractional differences between $C_l^{h\kappa }$ from non-Gaussian initial conditions and the Gaussian initial condition, calculated from the curves shown in Fig. 10. These differences are equal to $|\Delta b(l=k/d_A,z_L)|/b_1(z_L)$. The dashed and dotted lines show $f_{\mathrm{NL}}=\pm 50$ and $\pm 100$, respectively, while the thin solid lines show $l^{-2}$ with an arbitrary normalization.

Figure 12

Angular power spectra of the galaxy-galaxy correlation $C_l^h$ (thick dotted lines), the galaxy-convergence cross-correlation $C_l^{h\kappa }$ (thick solid lines), and the convergence-convergence correlation $C_l^{\kappa }$ (thick dashed lines) for the Gaussian initial condition ($f_{\mathrm{NL}}=0$). The four panels show the same populations of galaxies and clusters of galaxies as in Fig. 10. We also show the galaxy shot noise $1/n_L$ (thin dotted lines) as well as the source shape noise ${\sigma }_{\gamma }^2/n_S$ (thin dashed lines) for $N_L={10}^6$, ${\sigma }_{\gamma }=0.3$, and $n_S=3.5\times {10}^8{\mathrm{sr}}^{-1}$. We find $1/n_L\ll C_l^h$ and ${\sigma }_{\gamma }^2/n_S\ll C_l^{\kappa }$ for $l\lesssim 100$.

Figure 13

Same as Fig. 10, with the expected $1\mathrm{\text{−}}\sigma$ uncertainties for full-sky lens surveys and a single lens redshift. Adjacent bins are uncorrelated. The open (filled) boxes show the binned uncertainties with (without) the cosmic variance term due to the cosmic shear field included. We used Eq. (34) for the open boxes, and Eq. (34) with $C_l^{h\kappa }=0=C_l^{\kappa }$ for the filled boxes. We use logarithmic bins of $\Delta l=0.23l$. Note that the uncertainties are calculated for a single lens redshift slice, and thus they will go down as we add more lens redshift slices.

Figure 14

Angular power spectrum of the galaxy-CMB lensing $C_l^{h\kappa }$ at various multipoles as a function of the lens redshift $z_L$. We have divided $C_l^{h\kappa }$ by its maximum value. The solid, dotted, dashed, dot-dashed, and triple-dot-dashed lines show $l=10$, 50, 100, 350, and 1000, respectively.

Figure 15

Imprints of the local-type primordial non-Gaussianity in the galaxy-CMB lensing power spectrum $l(l+1)C_l^{h\kappa }/(2\pi )$ for the same populations of lens galaxies as in Fig. 3. The solid, dashed, and dotted lines show $f_{\mathrm{NL}}=0$, $\pm 50$, and $\pm 100$, respectively.

Figure 16

Same as Fig. 15, but for high-$z$ lens galaxies with $b_1=2$ at $z_L=2$ (top-left), $b_1=2.5$ at $z_L=3$ (top-right), $b_1=3$ at $z_L=4$ (bottom-left), and $b_1=3.5$ at $z_L=5$ (bottom-right).

Figure 17

Same as Fig. 11, but for high-$z$ lens galaxies with $b_1=2$ at $z_L=2$ (top-left), $b_1=2.5$ at $z_L=3$ (top-right), $b_1=3$ at $z_L=4$ (bottom-left), and $b_1=3.5$ at $z_L=5$ (bottom-right).

Figure 18

Angular power spectra of the galaxy-galaxy correlation $C_l^h$ (thick dotted lines), the galaxy-convergence cross correlation $C_l^{h\kappa }$ (thick solid lines), and the convergence-convergence correlation $C_l^{\kappa }$ (thick dashed lines) for the Gaussian initial condition ($f_{\mathrm{NL}}=0$). The four panels show the same populations of galaxies and clusters of galaxies as in Fig. 15. We also show the galaxy shot noise $1/n_L$ (thin dotted lines) as well as the lens reconstruction noise $N_l^{\kappa }$ (think dashed lines) for $N_L={10}^6$ and $N_l^{\kappa }\simeq 6\times {10}^{-8}{\mathrm{sr}}^{-1}$ (for multipoles much smaller than that corresponds to the beam size of $4^{\prime }$). We find $1/n_L\ll C_l^h$ and $N_l^{\kappa }\ll C_l^{\kappa }$ for $l\lesssim 100$.

Figure 19

Same as Fig. 15, but with 1-sigma uncertainty due to the shape noise of source galaxies [filled box, Eq. (29)] and full error budget [empty box, diagonal of Eq. (27)] including the cosmic variance. We use the multipole bins of size $\Delta l=0.23l$. For uncertainty of CMB lensing reconstruction, we assume the nearly-perfect reference experiment of Hu and Okamoto [58]: white detector noise ${\Delta }_T={\Delta }_P/\sqrt{2}=1\mu \mathrm{K}\mathrm{arcmin}$, and FWHM of the beam $\sigma =4^{\prime }$.

Figure 20

Same as Fig. 18, but for the high redshift lens galaxies shown in Fig. 16. For these populations (and with $N_L={10}^6$), the shot noise is about the same as the galaxy power spectrum, i.e., $C_l^h\simeq 1/n_L$.

Figure 21

Same as Fig. 19, but for the high redshift lens galaxies shown in Fig. 16.

Figure 22

Top: Convergence-convergence angular power spectrum from two different methods: the exact calculation [Eq. (c14), symbols] and Limber’s approximation [Eq. (35), solid lines]. Bottom: Fractional differences between Limber’s approximation and the exact integration. Symbols are the same as the top panel. Grey symbols show the absolute values of negative values.

Figure 23

Same as Fig. 22, but for the galaxy-convergence cross angular power spectrum with $f_{\mathrm{NL}}=0$ and $b_1=1$.

Figure 24

Same as Fig. 22, but for the non-Gaussian correction (i.e., the term proportional to $\Delta b(k)$) to the galaxy-convergence cross angular power spectrum. We show the corrections with $f_{\mathrm{NL}}=1$ and $b_1=2$.

Figure 25

Same as Fig. 23, but for the galaxy-CMB lensing.

Figure 26

Same as Fig. 24, but for the galaxy-CMB lensing.

Figure 27

Top: Same as Fig. 3, but also showing the exact result [Eq. (c13), thick lines] on top of the result from Limber’s approximation [Eq. (32), thin lines]. Bottom: Fractional difference of Limber’s approximation relative to the exact result.

Figure 28

Same as Fig. 27, but for larger $R$. Thick lines are the results of the exact integration, while the thin lines are Limber’s approximation. The Limber approximation over predicts $\Delta \Sigma (R)$ for large $R$, but the error is at most 5% for $R<500h^{-1}\mathrm{Mpc}$. The error is the largest for the lowest $z_L$, as a physical separation $R$ at a lower redshift corresponds to a larger angular separation on the sky.

Figure 29

Fractional differences in the non-Gaussian correction terms, $\Delta {\Sigma }_{\mathrm{nG}}$, from Limber’s approximation and the exact integration. Using Limber’s approximation, we overpredict the non-Gaussian correction by $\sim 20\%$ at $R=300h^{-1}\mathrm{Mpc}$ for $z_L=0.3$.

Figure 30

Same as Fig. 4, but with the exact integration instead of Limber’s approximation.