Passing through the bounce in the ekpyrotic models

TL;DR

The paper critically assesses perturbation transfer through the ekpyrotic bounce, showing that the commonly used singular-bounce prescriptions yield spectra that depend on unphysical normalizations and do not reproduce exact transitions in simple limits. By analyzing exact toy models for scalar-field and hydrodynamical bounces, it demonstrates that quantities like $ icefrac{ ho+p}{ ho}$ and $m{ abla} ext{ perturbations}$ do not behave simply through the bounce, and that the post-bounce spectrum is highly sensitive to the detailed bounce dynamics. It further shows that a conserved quantity such as $oldsymbol{}$ is not guaranteed to remain constant during the bounce, and that a test scalar field or gravitational-wave perturbation reveals spectra that depend on the bounce shape, undermining a universal ekpyrotic prediction. The work argues for a non-singular, higher-dimensional, theory-wide treatment of the bounce to obtain robust cosmological predictions and to assess ekpyrotic scenarios as viable alternatives to inflation.

Abstract

By considering a simplified but exact model for realizing the ekpyrotic scenario, we clarify various assumptions that have been used in the literature. In particular, we discuss the new ekpyrotic prescription for passing the perturbations through the singularity which we show to provide a spectrum depending on a non physical normalization function. We also show that this prescription does not reproduce the exact result for a sharp transition. Then, more generally, we demonstrate that, in the only case where a bounce can be obtained in Einstein General Relativity without facing singularities and/or violation of the standard energy conditions, the bounce cannot be made arbitrarily short. This contrasts with the standard (inflationary) situation where the transition between two eras with different values of the equation of state can be considered as instantaneous. We then argue that the usually conserved quantities are not constant on a typical bounce time scale. Finally, we also examine the case of a test scalar field (or gravitational waves) where similar results are obtained. We conclude that the full dynamical equations of the underlying theory should be solved in a non singular case before any conclusion can be drawn.

Figures (10)

• Figure 1: Schematic representation of the old ekpyrotic model as a bulk -- boundary branes in an effective five dimensional theory. Our Universe is to be identified with the visible brane, and a bulk brane is spontaneously nucleated near the hidden brane, moving towards our universe to produce the Big-Bang singularity and primordial perturbations. In the new ekpyrotic scenario, the bulk brane is absent and it is the hidden brane that collides with the visible one, generating the hot Big Bang singularity.
• Figure 2: Scale factor in the new ekpyrotic scenario. The Universe starts its evolution with a slow contraction phase $a\propto (-\eta)^{1+\beta}$ with $\beta=-0.9$ on the figure. The bounce itself is explicitly associated with a singularity which is approached by the scalar field kinetic term domination phase, and the expansion then connects to the standard Big-Bang radiation dominated phase.
• Figure 3: Characteristic functions of the new ekpyrotic model [See Eqs. (\ref{['wexp']}), (\ref{['Hekp']}), (\ref{['rhoekp']}) and (\ref{['cs2ekp']})]. The parameters are chosen as $\ell _0=1$ and $\omega _1=-1$ (the ekpyrotic scenario requires $\omega_1$ to be negative).
• Figure 4: The divergence in the Bardeen potential and the gauge invariant energy density perturbation near the bounce in the new ekpyrotic scenario. Parameters are chosen as $\ell _0=1$ and $\omega _1=-1$. The Bardeen potential $\Phi$ is from Eq. (\ref{['Phidiv']}) with $s B_1=B_2=1$, while $k^2s B_1=k^2B_2=1$ for $\epsilon_{_{\rm m}}$ in Eq. (\ref{['solepsm']}).
• Figure 5: Time evolution of the Bardeen potential during the radiation to matter domination transition. The full line shows the exact solution (\ref{['potradmat']}), the dashed line represents the usual approximation (\ref{['PhiUsual']}), and the dotted line is obtained using the new proposal (\ref{['PhiRuth']}) for the junction conditions. This last approximation is not in fact constant but, as discussed below Eq. (\ref{['PhiRuth']}), only hardly varying at all on the scale shown. In order to ensure that initial conditions are identical for the three curves, we have set numerically $\Phi_{\rm i} = \Phi_{\rm i}' = 1$ [see above Eq. (\ref{['phii']})], and used $B_1 \simeq -b^2 a^2_{\rm eq} \eta_{\rm i}^4 (b \Phi_{\rm i} + 8 \eta_{\rm eq} \Phi_{\rm i}')/9\eta_{\rm eq}^2$ and $B_2\simeq 3/2 (\Phi_{\rm i} + \eta_{\rm i} \Phi_{\rm i}'/3)$ in Eq. (\ref{['potradmat']}) valid in the limit $\eta_{\rm i}\ll \eta_{\rm eq}$.