Primordial Non-Gaussianity and the Field-Level Cramer-Rao Bound

Abstract

Primordial non-Gaussianity is one of the most powerful probes of the inflationary epoch. The particle spectrum relevant to inflation, including masses and spins, is encoded in the precise form of statistical correlations of the adiabatic modes. Yet, in the presence of nonlinear structure formation, the optimal approach to measuring these signals remains unclear. Accurate modeling becomes crucial as late-time non-Gaussianty can become degenerate with primordial physics. Moreover, scale-dependent bias shows that information can move from non-Gaussian initial conditions to the amplitude of the Gaussian fluctuations. In this paper, we aim to clarify how primordial information is encoded in maps of galaxies. We use the field-level Cramer-Rao bound to investigate the ultimate limit of what can be extracted from realistic maps of the Universe. For local non-Gaussianity, we show that multi-tracer scale-dependent bias can exceed the sensitivity of conservative higher-point analyses. However, as expected, the multi-tracer analysis falls short of the optimal constraint when all the modes at the scale of the dark matter halos are included. We then forecast the potential reach of future surveys for equilateral and local non-Gaussianity. Equilateral in particular is highly sensitive to priors and modeling assumptions and can benefit dramatically from theoretical input such as the redshift evolution of the bias.

Figures (8)

• Figure 1: Bias parameters in the Eulerian frame, $b_1$ and $b_{\Phi}$, for halos with mass $M$, computed using Eqs. \ref{['eqn:b1_deriv']}-\ref{['eqn:bphi_deriv']} with exact expressions for $\sigma^2(M)$ in Eq. \ref{['eqn:sigma_def']} and Eq. \ref{['eqn:sigma2_exact']}. The numerical values of the cosmological parameters used are summarized in Table. \ref{['tab:parameters']}. Note that the naive analytic estimates in Eq. \ref{['eqn:b1_single']} approximate both $b_1$ and $b_{\Phi}$ as constants, which is only true when $M$ is sufficiently small.
• Figure 2: Comparison between the matter power spectrum, $P(k)$, and the halo shot noise, $N^2$, for a selection of different halo masses $M$. Here $P(k)$ is defined in Eq. \ref{['eqn:DM_density_field']} and Eq. \ref{['eqn:Pk_3']}, while the halo shot noise $N^2$ is defined in Eq. \ref{['eqn:poisson_noise_def']}, is evaluated numerically using the exact expression of $f(M)$ in Eqs. \ref{['eqn:hmf_convenient']}-\ref{['eqn:hmf_differential_exact']} and the Press-Schechter prediction of halo number density in Eq. \ref{['eqn:Press-Schechter']}. We also compute the mean halo number density in each mass bin, $\bar{n}_{\mathrm{bin}}\equiv N^{-2}$, for reference, assuming $h=0.7$Planck:2018vyg. Fiducial values of parameters chosen are listed in Table. \ref{['tab:parameters']}.
• Figure 3: Minimum and maximum wavenumber $k$ for different types of analysis. Here $k_{\min}\,\mathrm{noise}$ and $k_{\max}\,\mathrm{noise}$ are defined to be minimum and maximum $k$ such that $P(k)>N^2$ with $N^2$ being the halo shot noise. The expression of $P(k)$ is given in Eq. \ref{['eqn:DM_spectrum']} and Eq. \ref{['eqn:Pk_3']}, and $N^2$, defined in Eq. \ref{['eqn:poisson_noise_def']}, is evaluated numerically using the exact expression of $f(M)$ in Eqs. \ref{['eqn:hmf_convenient']}-\ref{['eqn:hmf_differential_exact']} and the Press-Schechter prediction of halo number density in Eq. \ref{['eqn:Press-Schechter']}. We also show the smallest scale of a halo analysis, $2\pi/R$, with $R$ being the size of a halo with mass $M$. Fiducial values of parameters chosen are listed in Table. \ref{['tab:parameters']}.
• Figure 4: Fisher information on $f_{\mathrm{NL}}^{\mathrm{loc}}$ for all analysis types considered in this section. The gradient-colored line denotes the Fisher information from a matter bispectrum analysis of the matter field, with $k_{\min}$ set by the minimum wavenumber for which $P(k)>N^2$ (i.e.$k_{\mathrm{noise},\,\min}$), where $N^2$ is the shot noise for halos of mass $M$, and $k_{\max}$ set by $2\pi/R$, with $R$ the size of a halo of mass $M$. This line illustrates the maximum amount of information contained in halo objects of mass $M$. The Fisher information is given by the first line of Eq. \ref{['eqn:SNR_expression_quote']}. The expression for $P(k)$ is given in Eqs. \ref{['eqn:DM_spectrum']} and \ref{['eqn:Pk_3']}, and $N$, defined in Eq. \ref{['eqn:poisson_noise_def']}, is evaluated numerically using the exact expression for $f(M)$ in Eqs. \ref{['eqn:hmf_convenient']}–\ref{['eqn:hmf_differential_exact']} together with the Press–Schechter prediction of the halo number density in Eq. \ref{['eqn:Press-Schechter']}. The color of each point along the line indicates the numerical value of $k_{\max}=2\pi/R$. The blue dashed line corresponds to a matter bispectrum analysis with the same $k_{\min}$ but with a fixed, conservative choice of $k_{\max}=0.1$ Mpc$^{-1}$. The orange line corresponds to a single-tracer analysis of halos of mass $M$, obtained by numerically integrating over $k$ using Eq. \ref{['eq:Ffinal_galaxy']}, with $k_{\min}$ and $k_{\max}$ set as above by $k_{\mathrm{noise},\,\min}$ and $2\pi/R$. Finally, the green line corresponds to a double-tracer analysis of two halo populations: heavier halos with mass $M_1=M$ and lighter halos with mass $M_2=1\,M_{\odot}$, obtained by numerically integrating over $k$ using Eq. \ref{['eqn:Fisher_final_multi']}, with the same choices of $k_{\min}$ and $k_{\max}$. We did not use any expressions that assume $P\gg N^2$ when computing the Fisher information. Fiducial parameter values are listed in Table \ref{['tab:parameters']}. The survey volume is taken to be $V_{\mathrm{survey}}=10^{10}$ Mpc$^3$.
• Figure 5: Forecasted marginalized uncertainties on $f_{\rm NL}$ for BOSS obtained using a greedy marginalization procedure. The horizontal axis shows the cumulative number of marginalized parameters added by the greedy algorithm. At each step, the parameter that maximally increases $\sigma(f_{\rm NL})$, conditional on the previously marginalized set, is added. The resulting greedy ordering differs between curves and is listed in Table. \ref{['tab:greedy-orderings']}. Solid curves assume bias parameters are shared across redshift bins, while dashed curves correspond to the conservative case in which bias parameters are treated independently in each bin. Blue curves show power-spectrum-only (PS) constraints, while orange curves include both the power spectrum and bispectrum (PS+BS). Shaded regions indicate broad parameter classes (bias, cosmological, and redshift-space nuisance parameters). Left: $\sigma(f_{\rm NL}^{\rm loc})$ for local primordial non-Gaussianity . Right: $\sigma(f_{\rm NL}^{\rm eq})$ for equilateral primordial non-Gaussianity .