Effects of Scale-Dependent Non-Gaussianity on Cosmological Structures

TL;DR

The paper investigates scale-dependent primordial non-Gaussianity arising in single-field models with changing sound speed, notably DBI inflation, and its imprint on cosmological structures. It parameterizes the running of non-Gaussianity with a simple ansatz $f^{eff}_{NL}(k)=f^{eff}_{NL}(k_{CMB})(k/k_{CMB})^{-2\kappa}$ and develops an analytic framework using Edgeworth expansions to connect primordial skewness to the non-Gaussian halo mass function and cluster counts. It forecasts constraints from upcoming cluster surveys and CMB priors, showing that even modest running can noticeably alter cluster abundances and that the evolved galaxy bispectrum can help distinguish between local and equilateral shapes and their running. The work underscores the value of multi-scale observations (clusters, galaxies, and CMB) to probe inflationary physics beyond scale-invariant predictions and highlights the DBI scenario as a concrete bridge between extra-dimensional geometry and observable non-Gaussian signatures.

Abstract

The detection of primordial non-Gaussianity could provide a powerful means to test various inflationary scenarios. Although scale-invariant non-Gaussianity (often described by the $f_{NL}$ formalism) is currently best constrained by the CMB, single-field models with changing sound speed can have strongly scale-dependent non-Gaussianity. Such models could evade the CMB constraints but still have important effects at scales responsible for the formation of cosmological objects such as clusters and galaxies. We compute the effect of scale-dependent primordial non-Gaussianity on cluster number counts as a function of redshift, using a simple ansatz to model scale-dependent features. We forecast constraints on these models achievable with forthcoming data sets. We also examine consequences for the galaxy bispectrum. Our results are relevant for the Dirac-Born-Infeld model of brane inflation, where the scale-dependence of the non-Gaussianity is directly related to the geometry of the extra dimensions.

Figures (9)

• Figure 1: The quantity $|f_{NL}(k)|$ for several different values of the running of the non-Gaussianity $n_{NG}-1\equiv d\ln f_{NL}/d\ln k$. The solid line has $n_{NG}-1=0$, the dashed $n_{NG}-1=0.2$, and the dot-dashed $n_{NG}-1=0.6$. The shaded region in the upper left hand corner shows the range that is excluded at $95\%$ confidence by current CMB data Creminelli:2006rz for equilateral shape non-Gaussianity (plotted is the more conservative lower bound on $f_{NL}$) . The shaded regions on the right shows the range of scales probed by the galaxy bispectrum and by clusters. The range of scales probed by the bispectrum depends (among other things) on the redshift of the survey, survey volume and the number density of galaxies, the above plot assumes $V\sim 10 h^{-3}Gpc^3$, $z\sim 1$ and the maximum $k$ is determined by the nonlinear scale Sefusatti:2007ih.
• Figure 2: (a) The shape of the primordial bispectrum for the local model, $\mathcal{A}_{local}(1,k_2,k_3)/(k_2k_3)/f_{NL}$. The domain of the plot is restricted to ${\bf k}_1+{\bf k}_2+{\bf k}_3=0$. (b) Contour plot of the fractional difference between the local form of non-Gaussianity and the DBI shape. Shaded regions show contours of (beginning from the upper left-hand corner) $(\mathcal{A}_{local}-\mathcal{A}_c)/\mathcal{A}_c=$$0$, $0.05$, $0.1$, $0. 5$, $1$, $2$, $10$. (c) The dominant shape in the primordial bispectrum for the DBI model, plotted is $\mathcal{A}_c(1,k_2,k_3)/(k_2k_3)/f_{NL}^c$.(d) Contour plot of the fractional difference between the equilateral form of non-Gaussianity and the DBI shape. Shaded regions show contours of (beginning from the upper left-hand corner) $(\mathcal{A}_{equil}-\mathcal{A}_c)/\mathcal{A}_c=$$0$, $0.01$, $0.02$, $0.05$, $0. 1$, $0.25$.
• Figure 3: (a) The smoothed variance. (b) The smoothed skewness for the local, equilateral and DBI type models all with $f^{eff}_{NL}=-256$ and $\kappa=0$. (c) The fractional difference between the smoothed skewness for the DBI-type model and that for the scale-dependent equilateral model with several values of the parameter $\kappa$. (d) The scale-dependence of the non-Gaussianity is visible in $S_3\sigma={\langle \delta^3 \rangle}/{\langle \delta^2 \rangle}^{3/2}$ for the DBI model (just the $\mathcal{A}_c$ term).
• Figure 4: The ratio of the non-Gaussian mass function to the Gaussian mass function for DBI inflation (green curves, just the $c$ term) with a sound speed that saturates the bound on non-Gaussianity at the CMB scales, and for the equilateral shape of the bispectrum with $f_{NL}^{eq}(k_{CMB})=332$ (magenta upper curves) and $f_{NL}^{eq}(k_{CMB})=-256$ (blue lower curves, showing smaller deviation from Gaussian than the $c$-term). The solid horizontal line is the Gaussian prediction, the dotted curves have no running of the non-Gaussianity ($\kappa=0$), the dashed curves have non-Gaussianity that increases on small scales $\kappa=-0.1$ and the dot-dashed curves have $\kappa=-0.3$. The shaded regions show the regimes in which corrections to the non-Gaussian mass function from the $(S_3\sigma)^2$ term reach $5\%$ -- that is, in the shaded regions the validity of truncating the Edgeworth expansion at the first term is uncertain (see Appendix B). The validity of the expansion depends on the magnitude of the skewness: the left hand boundary is where the mass function for DBI-type with $\kappa=-0.1$ becomes invalid, the right cross-hatched region is where the mass function for $f_{NL}^{eq}(k)=332$ and $\kappa=0$ breaks down. All other curves become invalid somewhere between the two boundaries, except the $\kappa=-0.3$ cases, which becomes invalid at lower mass but where the deviation from Gaussianity is larger than shown in the range of the plot. For example, for the equilateral model with $f_{NL}=-256$, $\kappa=-0.3$, and $z=0$, the expansion is valid for $M<1.55\times10^{15}h^{-1}M_{sun}$; at $z=0.9$ the same curve is unreliable above $M=2.25\times10^{14}h^{-1}M_{sun}$.
• Figure 5: (a) The number of clusters with $M> M_{lim}=1.75\times 10^{14}h^{-1}M_{sun}$ per redshift interval per $deg.^2$ shown for $f_{NL}^{eq}(k_{CMB})=+332$ (magenta), $f_{NL}^{eq}(k_{CMB})=-256$ (blue) and for DBI-type inflation (green curves, just the $c$-term) with a sound speed that saturates the CMB bounds. The solid curve is the Gaussian prediction, the dotted curves have $\kappa=0$, the dashed curve has $\kappa=-0.1$ and the dot-dashed have $\kappa=-0.3$. (b) The ratio of the quantities in (a) to the Gaussian $dN/dz$. Figures (c) and (d) reproduce (a) and (b) but also show the change in $dN/dz$ for a Gaussian cosmology if the mass threshold $M_{lim}$ or $\sigma_8$ were changed. Clearly precise knowledge of these parameters is necessary to use cluster number counts to constrain primordial non-Gaussianity. The hatched and cross-hatched regions are the same as in the previous figure.