Density reconstruction from biased tracers and its application to primordial non-Gaussianity

Large-scale Fourier modes of the cosmic density field are of great value for learning about cosmology because of their well-understood relationship to fluctuations in the early universe. However, cosmic variance generally limits the statistical precision that can be achieved when constraining model parameters using these modes as measured in galaxy surveys, and moreover, these modes are sometimes inaccessible due to observational systematics or foregrounds. For some applications, both limitations can be circumvented by reconstructing large-scale modes using the correlations they induce between smaller-scale modes of an observed tracer (such as galaxy positions). In this paper, we further develop a formalism for this reconstruction, using a quadratic estimator similar to the one used for lensing of the cosmic microwave background. We incorporate nonlinearities from gravity, nonlinear biasing, and local-type primordial non-Gaussianity, and verify that the estimator gives the expected results when applied to $N$-body simulations. We then carry out forecasts for several upcoming surveys, demonstrating that, when reconstructed modes are included alongside directly observed tracer density modes, constraints on local primordial non-Gaussianity are generically tightened by tens of percents compared to standard single-tracer analyses. In certain cases, these improvements arise from cosmic variance cancellation, with reconstructed modes taking the place of modes of a separate tracer, thus enabling an effective “multitracer” approach with single-tracer observations.

Figure 1

Reconstruction noise power spectra for estimators that use each of the quadratic mode-couplings discussed in Sec. 2b. We omit curves for the $c_{01}$ and $c_{02}$ estimators, which are greater than the upper limit of the plot. The G (“growth”) estimator has the lowest noise by far. These curves are computed for a DESI-like survey, but the hierarchy between them is unchanged for the other surveys we consider. The signal to noise on reconstructed modes (not shown) is likewise much higher for the G estimator than for S or T, justifying our use of the G estimator for our main results.

Figure 2

Left: contamination in the expectation value of G estimator, corresponding to separate multiplicative biases on the amplitude of a reconstructed mode, computed for a DESI-like survey. Blue solid line is the estimator growth bias shown for comparison. Dashed lines indicate negative values. Several of these curves inherit the $k^{-2}$ scaling of the scale-dependent bias in ${\delta }_{\mathrm{g}}$ arising from nonzero $f_{\mathrm{NL}}$, implying that reconstructed modes can be used to constrain $f_{\mathrm{NL}}$ in the same way. Right: ratio of scale-dependent bias from $f_{\mathrm{NL}}$ (for a fiducial value of $f_{\mathrm{NL}}=1$) to total bias for ${\delta }_{\mathrm{g}}$ (solid) and ${\delta }_{\mathrm{r}}$ (dot-dashed). Local primordial non-Gaussianity has roughly the same relative contribution to the bias of ${\delta }_{\mathrm{g}}$ or reconstructed modes.

Figure 3

Cross correlations of estimators ${\widehat{\mathrm{\Delta }}}_{\alpha }$ corresponding to the growth, shift, and tidal mode-couplings with the linear density field ${\delta }_1$. We compare theory predictions (lines) with simulations (points) for three different smoothing scales, $R=20h^{-1}\mathrm{Mpc}$, $R=10h^{-1}\mathrm{Mpc}$ and $R=4h^{-1}\mathrm{Mpc}$, corresponding to maximum wave numbers $k_{\max }=0.05h{\mathrm{Mpc}}^{-1}$, $0.1h{\mathrm{Mpc}}^{-1}$, and $0.25h{\mathrm{Mpc}}^{-1}$ respectively. In this figure, we plot $⟨{\widehat{\mathrm{\Delta }}}_{\alpha }{\delta }_1⟩/N_{\alpha \alpha }$ (in contrast to what is defined in Eq. (16), in simulations we define the estimators ${\widehat{\mathrm{\Delta }}}_{\alpha }$ without a prefactor $N_{\alpha \alpha }$). We find very good agreement for the growth estimator for all smoothing scales, and also reasonably good agreement for the other estimators.

Figure 4

Autocorrelations of the quadratic estimators ${\widehat{\mathrm{\Delta }}}_{\alpha }$, for the same smoothing scales shown in Fig. 3. The predictions for the growth estimator agree with simulations for all smoothing scales. However, for other estimators predictions agree with simulations for large smoothing scales but for the low smoothing scales, the predictions slightly disagree with simulation results as higher-order terms become more important.

Figure 5

Comparison of the auto power spectrum of the growth estimator ${\widehat{\mathrm{\Delta }}}_{\mathrm{G}}$, normalized by $N_{\mathrm{GG}}$ computed from theory (in red), with $(⟨{\widehat{\mathrm{\Delta }}}_{\mathrm{G}}{\delta }_1⟩{)}^2/P_{\mathrm{lin}}$ (in blue). We compare simulation results (points) with theory predictions (lines) for the same smoothing scales as Figs. 3 and 4. We again find excellent agreement between simulations and theory. In the bottom right panel, we plot the cross-correlation coefficient $r_{\mathrm{G}{\delta }_1}$ between the growth estimator and the linear density field for three smoothing scales. We see that $r_{\mathrm{G}{\delta }_1}>0.9$ for $R=4h^{-1}\mathrm{Mpc}$ which is why in the bottom left panel, $⟨{\widehat{\mathrm{\Delta }}}_{\mathrm{G}}{\widehat{\mathrm{\Delta }}}_{\mathrm{G}}⟩$ is signal dominated.

Figure 6

2D slices of the 3D linear density field (left panel) and the growth estimator ${\widehat{\mathrm{\Delta }}}_{\mathrm{G}}$ (right panel). For the growth estimator we used $R=4h^{-1}\mathrm{Mpc}$ smoothing which corresponds to $k_{\max }=0.25h{\mathrm{Mpc}}^{-1}$. We apply an external smoothing of $R=20h^{-1}\mathrm{Mpc}$ to both the linear and reconstructed fields. As expected, we find that the reconstruction reproduces many of the large-scale features in the linear density field.

Figure 7

Probability distribution functions (histograms) of the linear density field and the reconstructed field from the halo density field of mass bin I. As in Fig. 6, we use $k_{\max }=0.25h{\mathrm{Mpc}}^{-1}$ for the reconstruction and apply an external smoothing scale of $R=20h^{-1}\mathrm{Mpc}$ to both the linear field and the reconstructed field. The PDFs of the reconstructed field are scaled to have the same variance as the linear field, and shifted to have mean 0. We find that the PDF of the reconstructed field is very close to Gaussian. Note that here we have applied a low-$k$ cutoff to the modes used for reconstruction of $k_{\min }=0.05h{\mathrm{Mpc}}^{-1}$ in order to match the approach in our forecast section below.

Figure 8

Expected error bars on the reconstructed power spectrum $P_{\mathrm{rr}}$ for the surveys and redshift bins we consider (blue), along with cosmic-variance-limited error bars (orange), computed for bandpowers with $\mathrm{\Delta }K=0.002h{\mathrm{Mpc}}^{-1}$. Downward arrows indicate error bars whose lower limits fall outside of the $y$ axis range. High-precision measurements of the power spectrum of reconstructed modes will be possible in several cases, even in the presence of a 21 cm foreground wedge for PUMA.

Figure 9

Forecasts for a DESI-like survey. Left: signal and noise power spectra involved in the forecast. The galaxy auto spectrum is well above the shot noise, while the auto spectrum of reconstructed modes ($P_{\mathrm{rr}}$) is roughly an order of magnitude below both the reconstruction noise ($N_{\mathrm{GG}}$) in the quadratic estimator and the shot noise contribution to the estimator variance. Center: expected constraints on $f_{\mathrm{NL}}$ when only ${\delta }_{\mathrm{g}}$ is used (solid), or when ${\delta }_{\mathrm{r}}$ is also used (dotted). We assume that ${\delta }_{\mathrm{g}}(\mathit{K})$ cannot be directly measured for $K<K_{\min }$, and marginalize over the $b_1$, $b_2$, and $b_{s^2}$ bias parameters. Right: ratio of ${\delta }_{\mathrm{g}}+{\delta }_{\mathrm{r}}$ and ${\delta }_{\mathrm{g}}$-only cases from the center panel. We only notice an improvement for higher values of $K_{\min }$, corresponding to using ${\delta }_{\mathrm{r}}$ but not ${\delta }_{\mathrm{g}}$ at $K<K_{\min }$.

Figure 10

The analog of the right panel of Fig. 9, with a variety of (mostly artificial) modifications to the forecasts. There is no improvement in $\sigma (f_{\mathrm{NL}})$ when ${\delta }_{\mathrm{r}}$ is neglected at $K<K_{\min }$, indicating that the inclusion of ${\delta }_{\mathrm{r}}$ at $K<K_{\min }$ drives the improvement. Greater improvements are achieved for higher galaxy number density or if $k_{\max }$ can be increased by a factor of 2, with milder changes if the fiducial $b_2$ value is set to zero or shot noise on $P_{\mathrm{rr}}$ and $P_{\mathrm{gr}}$ is neglected.

Figure 11

As Fig. 9, for low-redshift (top panels) and high-redshift (bottom panels) bins of a MegaMapper-like survey. The former has greater signal to noise on reconstructed modes than DESI, leading to a greater improvement in $\sigma (f_{\mathrm{NL}})$ when these modes are included in the forecast. For the latter, the shot noise contributions to $P_{\mathrm{rr}}$ and $P_{\mathrm{gr}}$ are comparatively much larger, leading to a different scale-dependence for the improvement in $\sigma (f_{\mathrm{NL}})$.

Figure 12

Modifications to the base MegaMapper forecasts. For the low-redshift bin (left panel), as for DESI, the improvement in $\sigma (f_{\mathrm{NL}})$ is driven mostly by modes of ${\delta }_{\mathrm{r}}$ with $K<K_{\min }$, with further improvement possible for higher galaxy number density. For the high-redshift bin (right panel), neglecting ${\delta }_{\mathrm{r}}$ at $K<K_{\min }$ makes no difference, indicating that for lower $K_{\min }$ values, cosmic variance cancellation between ${\delta }_{\mathrm{g}}$ and ${\delta }_{\mathrm{r}}$ at the same $K$ is driving the improvement in $\sigma (f_{\mathrm{NL}})$. There are several ways to obtain greater improvements, as discussed in the main text.

Figure 13

As Fig. 9, for low-redshift (top panels) and high-redshift (bottom panels) bins of a PUMA-like survey, treating the 21 cm brightness temperature in the same way as ${\delta }_{\mathrm{g}}$ in our other forecasts, and translating thermal noise on the brightness temperature into an effective tracer number density for computing shot noise. We show $\sigma (f_{\mathrm{NL}})$ either neglecting or incorporating the effects of the 21 cm foreground wedge; at high $z$, the benefit to $\sigma (f_{\mathrm{NL}})$ from including reconstructed modes is greater in the presence of the wedge, since there are fewer ${\delta }_{\mathrm{g}}$ modes that can be directly measured in that case. The results for the low-redshift bin are similar to those for MegaMapper, while larger improvements in $\sigma (f_{\mathrm{NL}})$ are possible at higher redshift.

Figure 14

Modifications to the base PUMA forecasts, neglecting the foreground wedge. In the low-$z$ bin, modes of ${\delta }_{\mathrm{r}}$ with $K<K_{\min }$ are entirely responsible for the improvement in $\sigma (f_{\mathrm{NL}})$, with most other modifications having little effect. In the high-$z$ bin, the blue dotted curve demonstrates that the $\sigma (f_{\mathrm{NL}})$ improvement comes from a combination of low-$K$ modes of ${\delta }_{\mathrm{r}}$ and cosmic variance cancellation at higher $K$. The improvement would get better if the thermal noise could be reduced (which maps onto a higher $\overline{n}$ in these forecasts).

Figure 15

The ratio of $\sigma (f_{\mathrm{NL}})$ for the ${\delta }_{\mathrm{g}}+{\delta }_{\mathrm{r}}$ and ${\delta }_{\mathrm{g}}$-only forecasts for the high-$z$ PUMA bin, where $\overline{n}$ is taken to infinity and $k_{\max }$ is artificially increased, assuming that our quadratic bias model is valid to arbitrarily high $k$. The left and right panels correspond to two representatives values of $K_{\min }$. We show forecasts after marginalizing over bias parameters (black points) and without any marginalization (red points), along with the expected $k_{\max }^{-3/2}$ scaling (dashed lines, each normalized to the highest-$k_{\max }$ point plotted). We find the unmarginalized curves quickly approach the ideal scaling, while the marginalized forecasts show small deviations from this scaling. This shows that large increases in constraining power are possible in principle for surveys with very high number density and a large allowed value of $k_{\max }$.

Figure 16

Cross-correlation coefficients between G, S, and T estimators, as given by Eq. (c12), for our DESI-like survey forecast (the other surveys give similar results) We see that all three estimators are highly correlated, implying, through Eq. (c14), that applying bias-hardening will strongly increase the noise of the resulting estimator.

Figure 17

We plot the contamination curves from numerical and analytic approximation. It can be seen that for the $\phi \phi$, $b_{11}$ and $b_{02}$ terms we have a $\frac{1}{K^2}$ behavior. For the analytical calculation of the $b_{02}$ term we use an approximation $({\int }_{q_{\min }}^{q_{\max }}dq{q}^2\frac{P_{\mathrm{lin}}^2(\mathit{q})}{P_{\mathrm{NL}}^2(\mathit{q})}\frac{1}{M(\mathit{q})}/{\int }_{q_{\min }}^{q_{\max }}dq{q}^2\frac{P_{\mathrm{lin}}^2(\mathit{q})}{P_{\mathrm{NL}}^2(\mathit{q})})\approx \frac{1}{2M(q_{\min })}$. We also show the absolute value of the numerical $b_{01}$ curve, with an approximate constant value on large scales.

Figure 18

The fraction of small-scale modes lying outside the foreground wedge, for the low-$z$ (solid) and high-$z$ (dashed) PUMA redshift bins we use in our forecasts. To a good approximation, the reconstruction noise integrals $N_{\alpha \beta }$ will simply be scaled by the inverse of these fractions in the presence of the wedge.

Figure 19

Forecasts for a PUMA-like survey, analogous to Fig. 13 except for using a cutoff on the line-of-sight component of accessible ${\delta }_{\mathrm{g}}$ modes instead of an isotropic $K$ cutoff. This is motivated by the fact that 21 cm foregrounds will preferentially obscure modes with low wave number components along the line of sight. The absolute values of $\sigma (f_{\mathrm{NL}})$ are slightly higher when using $K_{\parallel ,\min }$ instead of $K_{\min }$, since more modes are eliminated with a $K_{\parallel }$ cut, but the improvement in $\sigma (f_{\mathrm{NL}})$ from including reconstructed modes is qualitatively similar to the case with $K_{\min }$.