Rapidly-Varying Speed of Sound, Scale Invariance and Non-Gaussian Signatures

TL;DR

The paper shows that a scale-invariant curvature perturbation can arise for any constant equation-of-state parameter when the sound speed $c_s(t)$ varies in time, yielding two branches: Case I inflationary expansion with decreasing $c_s$ and Case II ekpyrotic contraction with increasing $c_s$. While the two-point function remains scale-invariant in both cases, the three-point function (non-Gaussianity) exhibits strong running and distinct shapes—equilateral in Case I and local in Case II—providing a clear discriminant between the histories. Gravitational waves are not scale-invariant in general: tensor modes are red-tilted for the expanding branch and strongly blue-tilted for the contracting branch, limiting their CMB detectability in the latter. The work highlights a broader, single-field framework (including DBI-like models) that can address the standard cosmological problems while making testable predictions for non-Gaussianities and gravitational waves, thus offering new avenues for observational discrimination of early-universe scenarios.

Abstract

We show that curvature perturbations acquire a scale invariant spectrum for any constant equation of state, provided the fluid has a suitably time-dependent sound speed. In order for modes to exit the physical horizon, and in order to solve the usual problems of standard big bang cosmology, we argue that the only allowed possibilities are inflationary (albeit not necessarily slow-roll) expansion or ekpyrotic contraction. Non-Gaussianities offer many distinguish features. As usual with a small sound speed, non-Gaussianity can be relatively large, around current sensitivity levels. For DBI-like lagrangians, the amplitude is negative in the inflationary branch, and can be either negative or positive in the ekpyrotic branch. Unlike the power spectrum, the three-point amplitude displays a large tilt that, in the expanding case, peaks on smallest scales. While the shape is predominantly of the equilateral type in the inflationary branch, as in DBI inflation, it is of the local form in the ekpyrotic branch. The tensor spectrum is also generically far from scale invariant. In the contracting case, for instance, tensors are strongly blue tilted, resulting in an unmeasurably small gravity wave amplitude on cosmic microwave background scales.

Figures (5)

• Figure 1: Comparison between the exact formula \ref{['FNLI']} and its linear fitting \ref{['fNLfit1']} in the expanding case, for the parameter region of interest: $-1<\alpha<0$. Values of $f_{NL}^{({\rm I})}$ are evaluated as a function of $\alpha$ for different $c_s$ and $f_X$: from top to bottom, $({\bar{c}}_s, f_X) = (0.25, 1);( 0.15, 0.1); (0.12, 0.9); (0.1, 0.1)$.
• Figure 2: In the� $(c_{s\;{\rm end}}, \epsilon)$ plane we plot the region corresponding to $f_{\rm NL} < 100$ with the choice $f_X=1$. The upper curve corresponds to the upper bound $f_{\rm NL} = 100$.
• Figure 3: In the top two panels we plot the non-Gaussian ampitude in the inflationary branch, $-{\cal A}^{\rm (I)}(1,x_2,x_3)/(x_2 x_3)$ for ${\bar{c}}_{s} = 0.1$, $f_X=0.9$, $\alpha = -0.3$ ($\epsilon = 0.08$, top left) and $\alpha = -0.9$ ($\epsilon = 0.3$, top right). Thus the dominant contribution peaks for equilateral configurations ($x_2 \approx x_3 \approx 1$). To see how this differs from DBI inflation, in the bottom two panels we subtract the dominant DBI-like (equilateral) contribution and plot $|{\cal A}^{\rm (I)} - A_{\rm equi}| /(x_2 x_3)$, where ${\cal A}^{\rm (I)}$ and $A_{\rm equi}$ have each been normalized to one in the equilateral limit $x_2 = x_3 = 1$. Thus the subdominant contribution peaks for squashed configurations ($x_2 \approx x_3 \approx 0.5$).
• Figure 4: The angle $\theta$ defined in eq. \ref{['theta']} is plotted as a function of $\alpha$ in the expanding case. In the left panel ${\bar{c}}_s=0.05$ and $f_X =0.5$, in the right panel we choose instead ${\bar{c}}_s=0.5$ and $f_X =0.5$. The distinguishability between our shape and DBI's increases with $|\alpha |$ and with ${\bar{c}}_s$.
• Figure 5: We plot ${\cal A}^{\rm (II)}(1,x_2,x_3)/(x_2 x_3)$ for ${c}_{s\, \rm end} = .95$, $f_X=1$, $\beta = 0.1$ (left) and ${c}_{s\, \rm end} = .9$, $\beta = 0.01$ (right).