Analytic approximations for the primordial power spectrum with Israel junction conditions

This work compares cosmological matching conditions used in approximating generic pre-inflationary phases of the Universe. We show that the joining conditions for primordial scalar perturbations assumed by Contaldi et al. [J. Cosmol. Astropart. Phys. 07 (2003) 002] are inconsistent with the physically motivated Israel junction conditions; however, performing general relativistic matching with the aforementioned constraints results in unrealistic primordial power spectra. Eliminating the need for ambiguous matching, we look at an alternative semianalytic model for producing the primordial power spectrum allowing for finite duration cosmological phase transitions.

Figure 1

Instantaneous transition in the comoving horizon between a period of kinetic dominance and de Sitter inflation as used in the Contaldi approximation. We have set $a=0$ at the Planck epoch; however, the convention that $a=1$ at the present epoch is not used but instead denotes the time of the instantaneous phase transition. Based on Fig. 3 in [14].

Figure 2

Analytic primordial power spectra generated from the Contaldi approximation for BD, RHM, HD, and RSET vacuum initial conditions. ${ϵ}_{\mathrm{I}}$ is set to 0.0127 in order to normalize the power spectrum to 1. Based on Fig. 4 from [14].

Figure 3

Numerical primordial power spectrum for a Starobinsky potential, $V(\varphi )={\mathrm{\Lambda }}^4(1-e^{-\sqrt{2/3}\varphi }{)}^2$, with ${\mathrm{\Lambda }}^4=0.01$, where initial conditions for $\varphi$, $\dot{\varphi }$ have resulted in a preinflationary phase of kinetic domination. The spectra have been normalized to 1 for comparison with the spectra produced from the Contaldi approximation.

Figure 4

Comparison of analytic primordial power spectrum from the Contaldi approximation and numerical primordial power spectrum from a Starobinsky potential for BD initial perturbation conditions. $k_{\mathrm{t}}$ is set to 1 in the Contaldi approximation.

Figure 5

Analytic primordial power spectra produced from the Contaldi approximation using cosmological matching conditions for a hypersurface defined by $\mathrm{\Sigma }:\overline{\rho }+\delta \rho =\text{const}$. BD, RHM, HD, and RSET initial conditions are shown. ${ϵ}_{\mathrm{I}}$ has been set to 0.0127 for comparison with Contaldi matching.

Figure 6

Analytic primordial power spectra produced from the Contaldi approximation using cosmological matching conditions for a hypersurface defined by $\mathrm{\Sigma }:\overline{\varphi }+\delta \varphi =\text{const}$. BD, RHM, HD, and RSET initial conditions are shown. ${ϵ}_{\mathrm{I}}$ has been set to 0.0127 for comparison with Contaldi matching.

Figure 7

Numerical primordial power spectra for increasingly sharp cosmological phase transitions generated by the semianalytic model presented in this section. ${ϵ}_{\mathrm{I}}=0.0001$ and $\alpha =2$, 10, 60 with BD initial conditions for perturbation modes set at the maximum of the analytic comoving horizon. The duration of the transitions are 3.25, 0.65, and $0.12e$-folds, respectively.

Figure 8

The upper plot compares primordial power spectra produced from an analytic background with a sufficiently sudden duration phase transition defined by Eq. (83) taking $\alpha =60$ and that from cosmological matching conditions with hypersurface of constant $\varphi$ applied to the instantaneous phase transition in the Contaldi approximation. ${ϵ}_{\mathrm{I}}$ is set to 0.0001 and $k_t$ is set to 0.98945 in the analytic power spectrum to align with the background of the numerical solution. RSET initial conditions have been used. The lower plot shows the percent error which remains small for scales obeying the condition expressed by (89).

Figure 9

Characteristic convex inflationary potential from parametric reconstruction for a smooth comoving horizon defined by Eq. (83) with a slow transition corresponding to ${ϵ}_{\mathrm{I}}\ll 1$ and $\alpha <3$. This example has $\alpha =1$ and ${ϵ}_{\mathrm{I}}=0.0001$. The regions of the potential corresponding to kinetic dominance and inflation specified using Eqs. (87) and (88).

Figure 10

Characteristic steplike inflationary potential from parametric reconstruction for a smooth comoving horizon defined by Eq. (83) with a sudden transition corresponding to ${ϵ}_{\mathrm{I}}\ll 1$ and $\alpha >3$. This example has $\alpha =10$ and ${ϵ}_{\mathrm{I}}=0.0001$. The regions of the potential corresponding to kinetic dominance and inflation are specified using Eqs. (87) and (88).