Magnification effect on the detection of primordial non-Gaussianity from photometric surveys

We present forecast results for constraining the primordial non-Gaussianity from photometric surveys through a large-scale enhancement of the galaxy clustering amplitude. In photometric surveys, the distribution of observed galaxies at high redshifts suffers from the gravitational-lensing magnification, which systematically alters the number density for magnitude-limited galaxy samples. We estimate size of the systematic bias in the best-fit cosmological parameters caused by the magnification effect, particularly focusing on the primordial non-Gaussianity. For upcoming deep and/or wide photometric surveys like the Hyper Suprime-Cam, the Dark Energy Survey and the Large Synaptic Survey Telescope, the best-fit value of the non-Gaussian parameter, $f_{\mathrm{NL}}$, obtained from the galaxy count data is highly biased, and the true values of $f_{\mathrm{NL}}$ would typically go outside the $3\mathrm{\text{−}}\sigma$ error of the biased confidence region, if we ignore the magnification effect in the theoretical template of angular power spectrum. The additional information from cosmic shear data helps not only to improve the constraint, but also to reduce the systematic bias. As a result, the size of systematic bias on $f_{\mathrm{NL}}$ would become small enough compared to the expected $1\mathrm{\text{−}}\sigma$ error for the Hyper Suprime-Cam and the Dark Energy Survey, but it would still be serious for deep surveys with $z_{\mathrm{m}}\gtrsim 1.5$, like the Large Synaptic Survey Telescope. Tomographic technique improves the constraint on $f_{\mathrm{NL}}$ by a factor of 2–3 compared to the one without tomography, but the systematic bias would increase.

Figure 1

The galaxy autopower spectrum $C_{\ell }^{nn}$ (top) and the shear-galaxy cross-power spectrum $C_{\ell }^{\gamma n}$ (bottom). We plot the angular power spectra in the presence/absence of the magnification effect (black solid/red dotted) for the Gaussian initial condition, $f_{\mathrm{NL}}=0$, and also show the angular power spectra in the absence of the magnification effect in the non-Gaussian case, $f_{\mathrm{NL}}=\pm 10$ (magenta solid). For comparison, we plot the contribution of the magnification effect ($2g_3{\mu }_3$, and ${\mu }_3{\mu }_3$ in the top and ${\gamma }_3{\mu }_3$ in the bottom panel).

Figure 2

The dependence of galaxy autopower spectra on the parameter $f_{\mathrm{NL}}$ and the magnification effect for each tomographic bin. We define the fractional difference of galaxy autopower spectra, $\delta C_{\ell }/C_{\ell }$ which is given in two cases: $C_{\ell }^{g_i{g}_i}(f_{\mathrm{NL}}=10)/C_{\ell }^{g_i{g}_i}(f_{\mathrm{NL}}=0)-1$ (magenta lines) and $C_{\ell }^{n_i{n}_i}(f_{\mathrm{NL}}=0)/C_{\ell }^{g_i{g}_i}(f_{\mathrm{NL}}=0)-1$ (black lines). The solid, dashed and dotted lines represent $(i,j)=(1,1)$, (2,2) and (3,3) bin, respectively.

Figure 3

The signal-to-noise ratio of each power spectrum in the presence/absence of the magnification effect (red/green) for a survey with the mean redshift $z_m=1.0$ and the number density of galaxy $N_g=35{\mathrm{arcmin}}^{-2}$. We show the signal-to-noise ratio for all power spectra normalized by $\sqrt{f_{\mathrm{sky}}}$ since the signal-to-noise ratio is proportional to $\sqrt{f_{\mathrm{sky}}}$.

Figure 4

The $1\mathrm{\text{−}}\sigma$ error contours on $b_0-f_{\mathrm{NL}}$ (top left), $b_z-f_{\mathrm{NL}}$ (bottom left) and $b_z-b_0$ (bottom right) planes for HSC. In each panel, we show the constraints in the case with (solid line) and without (dashed line) the magnification effect in theoretical template. In each case, we show the results obtained from galaxy number counts alone, and further including cosmic shear of galaxies. Note that CMB prior information from Planck is included in all cases. Information from the primary CMB is summed up to $\ell =3000$, and other signals are included up to ${\ell }_{\max }=1000$. We also note that the fiducial values of the galaxy bias parameters and slope indices are chosen as $b_0=1.5$, $b_z=2.0$, $s_1=0.5$, $s_2=1.0$ and $s_3=1.5$.

Figure 5

Same as Fig. 4 but for DES.

Figure 6

Same as Fig. 4 but for LSST. The inset shows the zoomed error contours around the fiducial values.

Figure 7

The systematic bias on $f_{\mathrm{NL}}$ as a function of ${\ell }_{\max }$ for $+g$ (red line) and $+g+\gamma$ (green line). The shaded regions denote the $1\mathrm{\text{−}}\sigma$ constraint on $f_{\mathrm{NL}}$ for $+n$ (thin line) and $+n+\gamma$ (thick line). Each panel show the case with $z_{\mathrm{m}}=0.5$, 1.0, and 2.0. The galaxy subsamples are divided into three redshift bins ($N_{\mathrm{bin}}=3$). We assume $N_{\mathrm{g}}=100{\mathrm{arcmin}}^{-2}$ and $f_{\mathrm{sky}}=0.5$.

Figure 8

Same as Fig. 7 but for $N_{\mathrm{bin}}=1$. In this case, we choose the slope as $s_1=1.0$. Note that the range of $y$-axis is three times larger than that in Fig. 7.

Figure 9

Dependence of the systematic bias and $1\mathrm{\text{−}}\sigma$ constraint on the slope parameter. We vary only the one slope parameter ($s_1$, $s_2$ or $s_3$) and parametrized by a parameter $s$ as $s_1=0.5+s$, $s_2=1.0+s$, or $s_3=1.5+s$. The parameter $s$ is varied from $-0.5$ to 0.5. The red, green and blue lines show the systematic bias with varying $s_1$, $s_2$ and $s_3$, respectively. The gray and black shaded regions represent the $1\mathrm{\text{−}}\sigma$ constraint in the case of $+g$ and $+g+\gamma$, respectively, with varying the slope parameter in third redshift bin, $s_3$. We assume LSST to calculate the noise spectrum.