Intuitive understanding of non-gaussianity in ekpyrotic and cyclic models

TL;DR

This paper presents a physically intuitive, semi-analytic framework to estimate the bispectrum in ekpyrotic and cyclic cosmologies, showing that the intrinsic non-Gaussianity scales as the geometric mean of the equation-of-state parameters during the ekpyrotic phase and the conversion epoch, $f_{NL}^{intrinsic}=O(\sqrt{\epsilon_{ek}\epsilon_c})$, typically far larger than inflationary expectations. It analyzes multiple entropy-to-curvature conversion channels—during kinetic-energy domination, during the ekpyrotic phase, and after the crunch/bang transition via modulated reheating—and derives how each scenario shapes the total $f_{NL}$, including sign and magnitude. A key result is the strong correlation between $f_{NL}$ and the scalar spectral tilt $n_s$, such that smaller $|f_{NL}|$ tends to accompany a bluer spectrum, providing a testable diagnostic. The authors argue that Planck and LSS measurements should reveal non-Gaussianity if the ekpyrotic/cyclic picture is realized, and that the $f_{NL}$–$n_s$ plane constitutes a powerful discriminator between these models and simple inflationary scenarios.

Abstract

It has been pointed out by several groups that ekpyrotic and cyclic models generate significant non-gaussianity. In this paper, we present a physically intuitive, semi-analytic estimate of the bispectrum. We show that, in all such models, there is an intrinsic contribution to the non-gaussianity parameter f_{NL} that is determined by the geometric mean of the equation of state w_{ek} during the ekpyrotic phase and w_{c} during the phase that curvature perturbations are generated and whose value is O(100) or more times the intrinsic value predicted by simple slow-roll inflationary models, f_{NL}^{intrinsic} = O(0.1). Other contributions to f_{NL}, which we also estimate, can increase |f_{NL}| but are unlikely to decrease it significantly, making non-gaussianity a useful test of these models. Furthermore, we discuss a predicted correlation between the non-gaussianity and scalar spectral index that sharpens the test.

Figures (4)

• Figure 1: The trajectory in field space reflects off a boundary at $\phi_2=0.$ The entropy perturbation, denoted $\delta s$, is orthogonal to the trajectory. The bending causes the conversion of entropy modes into adiabatic modes $\delta \sigma$, which are perturbations tangential to the trajectory.
• Figure 2: The evolution of the linear entropy perturbation during conversion: the solid (blue) line shows the actual evolution calculated numerically, while the dashed (purple) line shows the approximate solution (\ref{['entropys1conversion']}) with $\omega=3,$$\dot{\theta}=-1/t_{ref}$ and $t_{ref} =-400 M_{Pl}^{-1}$. For the purposes of illustration, $\delta s^{(1)}(t_{ref})$ has been normalized to $1$.
• Figure 3: A comparison of the results from numerical calculations with the fitting formula given in Eq. (\ref{['fNLapproximation']}) and indicated by the thick black line. Here we have fixed the value of $\epsilon_{ek} = 36.$ The plot confirms that $f_{NL}$ then grows linearly with $\kappa_3,$ in good agreement with (\ref{['fNLapproximation']}). The sample models (dashed and dotted lines) are representative of the range of models and parameters shown in Lehners:2007wc. We have similarly checked the dependence on $\sqrt{\epsilon}$ when the value of $\kappa_3$ is kept fixed.
• Figure 4: A plot for characterizing the correlation between $|f_{NL}|$ and scalar spectral tilt, $n_s -1$, here illustrated for the case of the cyclic model in which the conversion from entropic to curvature perturbations occurs during the kinetic energy dominated phase just before the big crunch/big bang transition. Different curves correspond to different fixed amounts of skewness $\kappa_3$ in the potential (the central curve corresponds to $\kappa_3=4\sqrt{2/3}$), while we vary the steepness of the potential $\epsilon_{ek}.$ The curves show the general trend that $|f_{NL}|$ increases as the spectrum becomes redder. Simple inflationary models correspond to the narrow horizontal hashed (red) strip with $|f_{NL}| \lesssim 1$. The shaded rectangle represents the current observational constraints on $f_{NL}$ and tilt (95% confidence) from WMAP5 Komatsu:2008hk.