Bispectrum signatures of a modified vacuum in single field inflation with a small speed of sound

TL;DR

The paper investigates how deviations from the Bunch-Davies vacuum in single-field inflation with a small speed of sound modify the primordial bispectrum, focusing on general P(X,φ) models and DBI inflation. By introducing a Bogolyubov parameter $\beta$ with phase $\delta$, the authors derive vacuum-induced corrections to the three-point function and analyze their projection onto standard templates (local, equilateral, orthogonal) to extract observational constraints. They find that the boosted bispectrum in the small-$c_s$ regime yields strong bounds on $|\beta|$, often tighter than power-spectrum constraints, with the exact limits depending on $\delta$ and whether the leading term vanishes (as in DBI) where subleading contributions dominate. The work emphasizes the bispectrum as a powerful probe of high-energy physics during inflation and highlights the need for oscillatory templates to capture the full signature of initial-state modifications. Overall, the results constrain vacuum-state deviations in a way that is highly complementary to, and often stronger than, existing power-spectrum limits.

Abstract

Deviations from the Bunch-Davies vacuum during an inflationary period can leave a testable imprint on the higher-order correlations of the CMB and large scale structures in the Universe. The effect is particularly pronounced if the statistical non-Gaussianity is inherently large, such as in models of inflation with a small speed of sound, e.g. DBI. First reviewing the motivations for a modified vacuum, we calculate the non-Gaussianity for a general action with a small speed of sound. The shape of its bispectrum is found to most resemble the 'orthogonal' or 'local' templates depending on the phase of the Bogolyubov parameter. In particular, for DBI models of inflation the bispectrum can have a profound 'local' template feature, in contrast to previous results. Determining the projection into the observational templates allows us to derive constraints on the absolute value of the Bogolyubov parameter. In the small sound speed limit, the derived constraints are generally stronger than the existing constraint derived from the power spectrum. The bound on the absolute value of the Bogolyubov parameter ranges from the 10^-6 to the 10^-3 level for H/Λ_c = 10^-3, depending on the specific details of the model, the sound speed and the phase of the Bogolyubov parameter.

Figures (8)

• Figure 1: DBI inflation; IR and UV branches resulting in different evolution for $c_s$, leading to different motivations for changes in the Initial Conditions (I.C.)
• Figure 2: Here we plotted the 'density' of the different non-Gaussian shapes. As can be seen from the color (shading), large density corresponds to lighter color (shading). It is clear from inspection that the equilateral template has zero density on the collinear line (spanned by $x_{2}+x_{3}=1$), while the leading order term from initial state modifications maximizes in the vicinity of this line, resulting in virtually zero overlap (which is already small due to the oscillatory nature of the leading term).
• Figure 3: Constraining contours for different values of $p=c_{s}k_{1}\eta_{0}$ and two values of $\delta$. The constraints for intermediate values of $\delta$ should lie somewhere in between. For values $\pi/2<\delta<3\pi/2$ one has to consider the opposite (in sign) constraints on the various $f_{\mathrm{NL}}$. The constraint for $p=1000$ and $\delta=0$ is too tight to be visible in this graph. For $\delta=\pi/2$ and $p=1000$ the constraint is still weaker than the constraint for $\delta=0$ and $p=100$.
• Figure 4: The shape of the first subleading contribution (eq.\ref{['eq:Fsl_1']}) for different values of $\delta$. For $\delta=0$ the shape of this function is very similar to the leading order term. Once $\delta\neq0$ the shape picks up a significant local feature which leads us to expect a significant overlap with the local non-Gaussian template. Since the oscillations are still present, we also expect a comparable overlap with the orthonormal template. As for the second subleading contribution (eq. \ref{['eq:Fsl_2']}, not shown) we find similar behavior, but opposite in sign, resulting in a small but non-negligible cancellation.
• Figure 5: The dot product between the subleading ($F_{\mathrm{sl}}=F_{\mathrm{sl(1)}}+F_{\mathrm{sl(2)}}$) and local template (multiplied with $p$) for two values of $p$. Although the dot product is much smaller for $\delta=0$ it is non-zero for both $p$. However, there exist values of the phase for which the dot product (and consequently the leakage or fudge factor) is exactly zero. For such $\delta$ the local template would not be suitable (or at least limited) for detecting any deviation from a BD state (in a DBI model of inflation) and we will have to use the much less capable equilateral template.