Imprints of Oscillatory Bispectra on Galaxy Clustering

TL;DR

The paper analyzes how oscillatory squeezed bispectra from inflation imprint on galaxy clustering, by computing the scale-dependent halo bias in resonant non-Gaussianity and cosmological collider models using Conformal Fermi Coordinates to isolate physical long-short couplings. It shows that the halo bias exhibits scale-oscillations with an envelope akin to equilateral NG and a halo-mass modulation, but the overall signal is too small for upcoming surveys to detect. For cosmological colliders, the analysis confirms that gauge artifacts from the consistency relation are removed in CFC, leaving non-analytic log-oscillations in the squeezed limit and corresponding mass-dependent signatures. The work connects primordial non-Gaussianity to late-time large-scale structure observables and lays out forecasted constraints and avenues for future exploration, including extensions to tidal fields and anisotropic long modes.

Abstract

Long-short mode coupling during inflation, encoded in the squeezed bispectrum of curvature perturbations, induces a dependence of the local, small-scale power spectrum on long-wavelength perturbations, leading to a scale-dependent halo bias. While this scale dependence is absent in the large-scale limit for single-field inflation models that satisfy the consistency relation, certain models such as resonant non-Gaussianity show a peculiar behavior on intermediate scales. We reconsider the predictions for the halo bias in this model by working in Conformal Fermi Coordinates, which isolate the physical effects of long-wavelength perturbations on short-scale physics. We find that the bias oscillates with scale with an envelope similar to that of equilateral non-Gaussianity. Moreover, the bias shows a peculiar modulation with the halo mass. Unfortunately, we find that upcoming surveys will be unable to detect the signal because of its very small amplitude. We also discuss non-Gaussianity due to interactions between the inflaton and massive fields: our results for the bias agree with those in the literature.

Figures (6)

• Figure 1: Constraints in the $\alpha$ -- $\delta n_\text{s}$ plane from Planck temperature and polarization angular spectra. We see that the whole $95\%\,\mathrm{CL}$ allowed region is consistent with having a monotonicity parameter $b_\ast < 1$. Therefore, we can safely take $\delta n_\text{s} = 2.0d-3\times\alpha^{0.63}$ (orange line) as a rough estimate for the maximum value of $\delta n_\text{s}$ allowed by Planck.
• Figure 2: The squeezed bispectrum of Eq. \ref{['eq:squeezed_CFC_bispectrum']} for $\alpha=50$ and $k_\ast = 5e-2\mathrm{Mpc}^{-1}$, integrated over $\cos\theta = \hat{\bm{k}}_s\cdot\hat{\bm{k}}_\ell$, for fixed $k_s$ (top panel) and as function of $(k_\ell,k_s)$ (bottom panel). As we can see from Eqs. \ref{['eq:resonant_bispectrum']}, \ref{['eq:enhancement']}, the bispectrum is periodic under $k_\ast\to k_\ast\exp(2\pi n/\alpha)$, so we do not vary it in the plots. The two plots in the top panel have $k_s = e-2\mathrm{Mpc}^{-1}$ (dashed lines represent negative values): we see that in the intermediate squeezing regime the bispectrum in CFC (orange line) is similar to that in global coordinates (blue line). More precisely, it oscillates linearly with $k_\ell$ with a frequency proportional to $\alpha$, and has an envelope (shown as the grey line in the second plot of the top panel) different from a simple power law. For $k_\ell\lesssim k_s/\alpha$ (the red dotted lines show $k_\ell=k_s/\alpha$) we start to see a difference between the two bispectra, while for $k_\ell\ll k_s/\alpha$ the CFC one becomes $\propto k^2_\ell/k^2_s$ without oscillations in $k_\ell$, as shown in Eq. \ref{['eq:enhancement']}. The contour plot in the bottom panel shows the full angle-averaged bispectrum for varying $k_\ell$ and $k_s$ (also here the red dotted line shows $k_\ell=k_s/\alpha$): we again see that in the ultra-squeezed regime $k_\ell\ll k_s/\alpha$ (upper-left corner of the plot) the response of the small-scale power spectrum to variations in $k_\ell$ goes quickly to zero, and the only oscillations are of the form $\cos(\alpha\log k_s)$.
• Figure 3: Top panel: non-Gaussian halo bias in global coordinates (blue line) and CFC (red line) for $\alpha=10$ as a function of $k$ at $M = e16{h^{-1}\,M_\odot}$, together with that for equilateral non-Gaussianity (green line) for $f_\text{NL}^\text{equil}=1$Schmidt:2010gwDesjacques:2011mqDesjacques:2016bnm. The latter can be converted into the prediction of single-field slow-roll models, for which $f_\text{NL}^\text{equil}\sim d-2$Cabass:2016cgp, or that of $P(\phi,X)$ theories with non-canonical speed of sound $c^2_\text{s}\neq 1$, that have $f_\text{NL}^\text{equil}\sim(1-c^2_\text{s})/c^2_\text{s}$Creminelli:2013cgaCabass:2016cgp. Dashed lines indicate negative values of $\Delta b_1$. Bottom panel: $\Delta b_1(k)$ in CFC for $\alpha=20$ and $M = e16{h^{-1}\,M_\odot}$, together with the fit of Eq. \ref{['eq:fit']} (black dotted line). The vertical line denotes the point at which the transfer function $T(k)$, entering in the definition of $\Delta b_1$ through $\mathcal{M}(k)$ in Eq. \ref{['eq:bias_exp-P']}, starts to affect its scale dependence: from the bottom panel, we see that for $k/k_\text{eq}\approx 10$ this causes the scaling of the non-Gaussian bias to slightly deviate from $\sim k^2$.
• Figure 4: Non-Gaussian halo bias correction in CFC as a function of halo mass for $k= h\,e-3\mathrm{Mpc}^{-1}$ (top panel) and $k=h\,e-2\mathrm{Mpc}^{-1}$ (bottom panel): we see that taking $k$ near to $k_\text{eq}\approx e-2\mathrm{Mpc}^{-1}$ substantially diminishes the amplitude of the oscillations. However, we see that in both cases the envelope of the oscillations is an increasing function of the halo mass: comparing with the green lines, we see that the bias increases with $M$ faster than equilateral non-Gaussianity. Notice that in these plots we normalize $f_\mathrm{NL}^\mathrm{equil}$ to match $\Delta b_1$ at $M = e11{h^{-1}\,M_\odot}$, in order to better compare the dependence on the halo mass. A more precise discussion is presented in Section \ref{['sec:observations']}.
• Figure 5: Dependence of $\Delta b_1$ on $k$ at $M=e13{h^{-1}\,M_\odot}$ (top panel) and $M$ at $k = h\,e-3\mathrm{Mpc}^{-1}$ (bottom panel). We see that, as expected, $\Delta b_1\sim1/\sqrt{k}$ on large scales: due to the transfer function, departures from this scaling are seen at $k\sim k_\text{eq}$. While for $\mu = 1$ the dependence on the halo mass is very mild, for $\mu = 5$ oscillations with $M$ can be seen at large halo masses $M\gtrsim e14{h^{-1}\,M_\odot}$.