On Primordial Density Perturbation and Decaying Speed of Sound

Abstract

The decaying speed of sound can lead to the emergence of the primordial density perturbation in any expanding phase, even if the expansion is decelerated. Recently, some proposals have been given to implement this mechanism, in which it was found that the primordial spectrum of scalar perturbation can be scale invariant. In this note, we will give more insights for the details of this seeding mechanism.

Figures (2)

• Figure 1: The figure of $\ln{({1\over ah})}$ with respect to $\ln{a}$, see black solid lines. The red solid lines are the perturbation modes with wave number $k$. The black dashed line is that of $\ln{({c_s\over ah})}$ with respect to $\ln{a}$. The blue region below the $\ln{a}$ axis denotes the decelerated expanding phase in which $0<n<1$. In principle the primordial perturbation can not be generated in such a phase. However, when a decaying $c_s$ is introduced, the effective wave number $c_sk$ of perturbation mode will not unchange any more, which will decrease with the time. This under certain condition will make the evolution of their physical wavelengths able to be faster than that of $1/h$, see the red dashed lines, thus the corresponding mode will leave the horizon, see Ref. Piao0609 for details. This may be also equally explained as that the perturbation modes with wave number $k$, see the red solid lines in the left side of reheating point, leave the sound horizon $c_s/h$. The upper right region of inflation line is actually the superinflationary region.
• Figure 2: The figure of the $\log ({m_p\over T_e})$ with respect to $n$ showing how enough efolding number is obtained. The solid line is for the case that the energy scale $\sim m_p$ at the beginning time of phase generating the primordial perturbations and the dashed line is that of the energy scale $\sim m_p/10^4$. The region above the corresponding line is that with enough efolding number.