Designing an Inflation Galaxy Survey: how to measure $σ(f_{\rm NL}) \sim 1$ using scale-dependent galaxy bias

TL;DR

This paper investigates the design requirements for galaxy surveys to measure primordial non-Gaussianity at the level σ(f_NL) ∼ 1 by exploiting scale-dependent galaxy bias in the power spectrum. Using a Fisher-matrix framework with multiple tracers and a Halo Occupation Distribution based on the SHMR to map halos to observable galaxies, the authors quantify how survey depth, volume, redshift accuracy, and mass-proxy scatter affect the f_NL constraint. They find that a very large-volume, deep, imaging survey—ideally full-sky and leveraging multitracer information—can reach the desired precision, with photometric or low-resolution redshifts being sufficient when the halo-mass proxy scatter is small. The work also contrasts imaging versus spectroscopic surveys for this purpose and discusses real-world surveys (EUCLID, DESI, LSST, SPHEREx, J-PAS, SKA) as to their potential to achieve σ(f_NL) ∼ 1, while highlighting that controlling large-scale systematics and potentially incorporating the bispectrum could further improve constraints.

Abstract

The most promising method for measuring primordial non-Gaussianity in the post-Planck era is to detect large-scale, scale-dependent galaxy bias. Considering the information in the galaxy power spectrum, we here derive the properties of a galaxy clustering survey that would optimize constraints on primordial non-Gaussianity using this technique. Specifically, we ask the question what survey design is needed to reach a precision $σ(f_{\rm NL}^{\rm loc}) \approx 1$. To answer this question, we calculate the sensitivity to $f_{\rm NL}^{\rm loc}$ as a function of galaxy number density, redshift accuracy and sky coverage. We include the multitracer technique, which helps minimize cosmic variance noise, by considering the possibility of dividing the galaxy sample into stellar mass bins. We show that the ideal survey for $f_{\rm NL}^{\rm loc}$ looks very different than most galaxy redshift surveys scheduled for the near future. Since those are more or less optimized for measuring the BAO scale, they typically require spectroscopic redshifts. On the contrary, to optimize the $f_{\rm NL}^{\rm loc}$ measurement, a deep, wide, multi-band imaging survey is preferred. An uncertainty $σ(f_{\rm NL}^{\rm loc}) = 1$ can be reached with a full-sky survey that is complete to an $i$-band AB magnitude $i \approx 23$ and has a number density $\sim 8$ arcmin$^{-2}$. Requirements on the multi-band photometry are set by a modest photo-$z$ accuracy $σ(z)/(1+z) < 0.1$ and the ability to measure stellar mass with a precision $\sim 0.2$ dex or better (or another proxy for halo mass with equivalent scatter). While here we focus on the information in the power spectrum, even stronger constraints can potentially be obtained with the galaxy bispectrum.

Figures (10)

• Figure 1: The number density and bias of galaxies above a stellar mass cut $M_*$ at various redshifts. Different curves are for different assumed scatter in stellar mass, $\sigma_{\log M_*}$ (see text), relative to a deterministic stellar mass-halo mass relation, Eq. (\ref{['eq:stel2hal']}). The red curves correspond to the fiducial model for $\sigma_{\log M_*}$, based on simulations. The number density in the top panel is the number densities of central galaxies, or, equivalently, of different halos and the bias is the mean bias of those halos. Brighter objects typically live in more massive halos, which are rarer, and therefore have a larger bias.
• Figure 2: The differential Fisher information in $f_{\rm NL}$ per unit redshift as a function of minimum stellar mass, assuming the galaxy sample consists of all galaxies above the stellar mass cut. We assume a sky coverage $f_{\rm sky} = 0.75$. Dashed lines show the single-tracer case and solid lines show the optimal multitracer case. Different colors correspond to different assumed scatters in $\log_{10}M_*$ at fixed host halo mass, where $M_*$ is the stellar mass of the halo's central galaxy. The number density on the upper horizontal axis is the total galaxy number density (central plus satellite) corresponding to the minimum stellar mass $M_*$ on the lower axis, in the case of the default $\sigma(\log M_*) \approx 0.25$. The information on $f_{\rm NL}$ goes through zero when the mean halo bias of the sample equals unity. For instance, at $z = 0.5$, this occurs at a stellar mass cut $M_{*,{\rm min}} \approx 4 \cdot 10^{9} M_\odot$ (cf. the bias in Figure \ref{['fig:num and bias vs mstar']}), with the exact value depending on the assumed stellar mass scatter.
• Figure 3: Fisher information per unit redshift at $z = 1$ as a function of the minimum wave vector $k_{\rm min}$ included in the Fisher matrix (sky coverage $f_{\rm sky} = 0.75$). We fix $k_{\rm max} = 0.1 h/$Mpc. We show results for a moderate density survey (black), $M_{*,{\rm min}} = 10^{11} M_\odot$, for which the multitracer technique (solid) improves the $f_{\rm NL}$ constraint little over the single-tracer case (dashed), but also for a very dense survey (red), $M_{*,{\rm min}} = 10^{10} M_\odot$, for which using multiple tracers improves the signal-to-noise squared by an order of magnitude. In all cases, the constraining power strongly improves as $k_{\rm min}$ is lowered.
• Figure 4: Uncertainty in $f_{\rm nl}$ as a function of comoving survey volume $V$ based on the Fisher information per unit volume calculated at $z=1$. Thick curves take into account the variation with survey volume of the minimum wave vector that can be used in the analysis $k_{\rm min} = \pi/V^{1/3}$. Thin curves fix $k_{\rm min}$ to the value corresponding to the smallest volume shown, $V = 0.1 (h^{-1}$Gpc$)^3$. Left: moderate density survey, $M_{*,{\rm min}} = 10^{11} M_\odot$. Right: high density survey, $M_{*,{\rm min}} = 10^{10} M_\odot$. An important advantage of large volume surveys is the ability to measure very large modes (small $k_{\rm min}$). In order to constrain primordial non-Gaussianity at the level $\sigma(f_{\rm NL}) \sim 1$, it is crucial to use a very large volume $V > 100 (h^{-1}$Gpc$)^{-3}$.
• Figure 5: Left: Fisher information on $f_{\rm NL}$ per unit redshift at $z = 1$ as a function of maximum wave vector $k_{\rm max}$. We fix $k_{\rm min} = 10^{-3} h/$Mpc. A large amount of the information on $f_{\rm NL}$ comes from the very largest scales, $k < 0.01 - 0.02h/$Mpc. This relaxes the requirements on redshift accuracy and modeling of nonlinearities in the galaxy power spectrum. Right: The ratio of the Fisher information per unit redshift with a redshift uncertainty $\sigma(z) = \sigma_{z,0} \, (1 + z)$ to the Fisher information with spectroscopic redshifts (modeled as $\sigma(z) = 0$). We here assume the default $k_{\rm min} = 0.001 h/$Mpc, $k_{\rm max} = 0.1 h/$Mpc and $z=1$. The plot shows that even large redshift uncertainties, $\sigma_{z,0} \lesssim 0.1$, are tolerable. The multitracer constraints are a bit more sensitive to redshift accuracy as the information content on $f_{\rm NL}$ is skewed toward smaller scales relative to the single-tracer case.