Multifield Cosmological Perturbations at Third Order and the Ekpyrotic Trispectrum

TL;DR

The paper develops a third-order perturbation theory framework for two-field cosmologies using a covariant 1+3 formalism and applies it to ekpyrotic and cyclic models where density perturbations arise from the entropic mechanism. By deriving the full third-order evolution equations for adiabatic and entropy perturbations and defining the curvature perturbation $\zeta$ at third order, it computes the trispectrum characterized by $\tau_{NL}$ and $g_{NL}$ and relates them to $f_{NL}$ through $\zeta = \zeta_L + \frac{3}{5} f_{NL} \zeta_L^2 + \frac{9}{25} g_{NL} \zeta_L^3$. Analyzing two conversion channels—kinetic and ekpyrotic—the authors find distinct, regime-dependent predictions: ekpyrotic conversion yields negative $f_{NL}$ and positive $g_{NL}$ with large amplitudes, while kinetic conversion gives $f_{NL}$ of mixed sign and negative $g_{NL}$ of order $10^3$. These trispectrum signatures provide a strong observational discriminator between ekpyrotic/cyclic scenarios and inflation, highlighting the practical impact of higher-order perturbations for early-Universe cosmology.

Abstract

Using the covariant formalism, we derive the equations of motion for adiabatic and entropy perturbations at third order in perturbation theory for cosmological models involving two scalar fields. We use these equations to calculate the trispectrum of ekpyrotic and cyclic models in which the density perturbations are generated via the entropic mechanism. In these models, the conversion of entropy into curvature perturbations occurs just before the big bang, either during the ekpyrotic phase or during the subsequent kinetic energy dominated phase. In both cases, we find that the non-linearity parameters f_{NL} and g_{NL} combine to leave a very distinct observational imprint.

Figures (5)

• Figure 1: $f_{NL}$ as a function of the duration of conversion and for the values $\kappa_3=-5,0,5$ and $\epsilon=36$. In each case, we have plotted the results for four different reflection potentials, with the simplest potentials ($\phi_2^{-2},(\sinh{\phi_2})^{-2}$) indicated by solid lines, while the dashed ($(\sinh{\phi_2})^{-2}+(\sinh{\phi_2})^{-4}$) and dotted ($\phi_2^{-2}+\phi_2^{-6}$) lines give an indication of the range of values that can be expected. As the conversions become smoother, the predicted range of values narrows, and smooth conversions lead to a natural range of about $-50 \lesssim f_{NL} \lesssim + 60$ or so.
• Figure 2: $g_{NL}$ as a function of the duration of conversion, with $\kappa_3=\kappa_4=0$ and for four different reflection potentials, with the same line style assignments as in Fig. \ref{['FigurefNL']}. As the conversions become smoother, the predicted range of values narrows considerably, allowing us to make rather definite predictions.
• Figure 3: This figure shows $g_{NL}$ to be proportional to $\epsilon$, i.e. proportional to the equation of state $w_{ek}.$
• Figure 4: This figure indicates that $g_{NL}$ depends linearly in $\kappa_4,$ the parameter we are using to specify the fourth derivative of the ekpyrotic potential with respect to the entropic direction.
• Figure 5: $g_{NL}$ can be seen to depend approximately quadratically on $\kappa_3,$ the parameter determining the third derivative of the ekpyrotic potential with respect to the entropic direction.