Evolution of Linear Perturbations under Time-Dependent Hubble Friction I: SR-USR-SR Inflation

Abstract

In this paper, we revisit the linear perturbation (including the comoving curvature perturbation and field perturbation) dynamics in the SR-USR-SR inflation with instantaneous transitions. Using the junction method and asymptotic expansions of Hankel functions, we derive accurate asymptotic expressions for the time evolution of mode functions and the resulting power spectrum, based on three systematic rules for identifying the dominant terms across transitions. Our results reveal that a finite dip of the final power spectrum arises from the cancellation between two growing modes within the linear perturbation theory, rather than between constant and growing terms as previously suggested. We also provide analytical descriptions of the amplitude enhancement and oscillatory features in the linear power spectrum, in agreement with numerical computations. These simple, tractable formulas not only facilitate theoretical calculations but also yield testable predictions for future CMB observations.

Figures (8)

• Figure 1: A sketch illustrating three representative $k$ regimes of the mode function $\chi_{k}(\tau)$, in relation to SR-USR-SR instantaneous transitions denoted by $\tau_{1}$ and $\tau_{2}$.
• Figure 2: The comparison between the exact solution \ref{['eq:chi_sol_AB']} (the blue solid curve) with coefficients \ref{['eq:A2B2']}, and the asymptotic solution \ref{['case1:usr_super']} (the red dashed curve), under the parameter choices: $k = 0.06$, $\tau_1 = -1$. The value of $\mathcal{P}_{\chi}^{\rm USR}(\tau, k)$ is normalized by the exact value of $\mathcal{P}_{\chi}^{\rm USR}(\tau_{1}, k)$ based on Eq. \ref{['eq:chi_sol_AB']}. The black dotted vertical line refers to the dip time $\tau_c$ given by Eq. \ref{['case1:tauc']}. The observed dip structure around $\tau_{c}$ emerges from the interplay between the positive $(-\tau)^{-6}$ term and negative $(-\tau)^{-3}$ term in Eq. \ref{['case1:usr_super']}.
• Figure 3: The comparisons between the exact solution \ref{['eq:chi_sol_AB']} (the blue solid curves) with the coefficients \ref{['eq:A3B3']}, and the asymptotic power spectrum \ref{['case1:sr2_super']} (the red dashed curves), for $k = 0.06$ and $\tau_1 = -1$ with $\tau_{2}$ scaled as $0.17 \tau_{1}$, $(2.2)^{1/3} \tau_{1}$, $(1.8)^{1/3} \tau_{1}$, $0.1 \tau_{1}$. All the results are normalized by the exact value of $\mathcal{P}_{\chi}^{\rm SR2}(\tau_{2}, k)$ based on Eq. \ref{['eq:chi_sol_AB']}. The dip time $\tau_{d}$ is given by Eq. \ref{['case1:taud']}, which only exists when $-\tau_{2} < -2^{1/3} \tau_{c}$.
• Figure 4: Time evolutions of the power spectrum $\mathcal{P}_{\chi}(\tau, k)$ across SR-USR-SR transitions, corresponding to Case 1 of Fig. \ref{['fig:k_regimes']} for $\tau_{2} = 0.8\tau_{1}, 2^{1/3}\tau_{c}, 1.8^{1/3}\tau_{c}, \tau_{c}, 0.9\tau_{c}, 0.1\tau_{c}$, respectively. The blue solid curves denote the exact solutions of power spectra based on Eqs. \ref{['eq:chi_sol_AB']}, \ref{['eq:A2B2']} and \ref{['eq:A3B3']}, while the red dashed curves represent the asymptotic solutions given by Eqs. \ref{['eq:sr1_full']}, \ref{['case1:usr_super']} and \ref{['case1:sr2_super']}. All results are normalized by the exact value of $\mathcal{P}_{\chi}^{\rm USR}(\tau_{1}, k)$ based on Eq. \ref{['eq:chi_sol_AB']}. The dip times $\tau_c$ and $\tau_{d}$ are defined in Eqs. \ref{['case1:tauc']} and \ref{['case1:taud']}, respectively.
• Figure 5: The final power spectrum $\mathcal{P}_{\chi}^{\rm SR2}(\tau_{\rm end}, k)$ in Eq. \ref{['case1:sr2_super_end']} with $\tau_{2} = 10^{-2} \tau_{1} = - 0.01$, which features a dip at $k_{\rm dip}$ expressed in Eq. \ref{['case1:sr2_kdip']} or equivalently Eq. \ref{['case1:sr2_kdip_2']}. The spectrum is nearly scale-invariant for $k \ll k_{\rm dip}/\sqrt{2}$, and transitions to a $k^4$ growth for $k>\sqrt{2}k_{\rm dip}$. The value of $\mathcal{P}_{\chi}^{\rm SR2}(\tau_{\rm end}, k)$ is normalized by $\mathcal{P}_{\chi}^{\rm USR}(\tau_{1}, k)$ based on Eq. \ref{['eq:chi_sol_AB']}.