Superhorizon curvature perturbation in ultraslow-roll inflation

Abstract

We study the growth of superhorizon modes in the curvature perturbation during an ultra-slowroll or a large-η phase in single-field inflation. In a simple toy model, we derive the two-point correlation function of the curvature perturbation and show that the requirement for causality restricts the growth rate and hence puts a lower limit on the value of η. The toy model is then realized by considering an inflation potential with an inflection point. Our study is useful to assessing the growth of the curvature perturbation that seeds the formation of primordial black holes.

Constant-rate inflation: primordial black holes from conformal weight transitions

“…with very sharp transitions across different rolling phases and δ ≈ 0 in each phase [9,[12][13][14]18]. First of all, the power spectrum P R of the gauge invariant curvature perturbation R on superhorizon scales exhibits a (k/k * ) 4 scaling for k k * , where k * ≈ −1/η * is the horizon-crossing scale at which the inflation begins with a negative constant rate δ < −3/2 (Phase 2 in figure 1).…”

“…That enhanced primordial density fluctuations from inflationary models with large curvature perturbations on small scales is the mainstream scenario for PBH formation. PBHs from models of inflation, with targeting masses spanning in the range of 10 −17 -10 9 M , can be realized either in the single-field [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] or multi-field [22][23][24][25][26][27][28][29][30][31][32][33][34][35] framework.…”

“…In this work, we focus on the class of PBH models of inflation in the single-field framework with an approximately constant rate of rolling [36][37][38][39][40]: δ ≡ φ/(H φ) ≈ constant, where φ is the inflaton field and H is the Hubble parameter during inflation. The analytic structure of the power spectrum generated by a constant-rate inflation has been investigated in [11,12,18,19,41], showing an essential departure from the standard slow-roll predictions [9] due to the significant violation of slow-roll conditions in the background dynamics [8]. Quantum diffusion induced by the off-attractor nature of the constant-rate rolling can introduce non-negligible corrections on the large-scale spectrum via the stochastic effect [13,[42][43][44][45][46].…”

On the Constant-Roll Inflation with Large and Small ηH

“…In Figure 1, we also see that the analytical power spectrum (12) for the scalar perturbation approximates the exact result very well even when η H > 3 and the curvature perturbations grow on superhorizon scales. Due to the growth of the curvature perturbations on superhorizon scales for the constant-roll model (18) with η H > 3/2, the scalar power spectrum (12) should be evaluated at the end of inflation instead of the horizon crossing [18,[46][47][48][49]. This brings a new problem on the evaluation of the power spectrum, because inflation does not end naturally and some unknown mechanisms need to be introduced to end inflation in the model (18), but this problem does not exist in the model (22).…”

Quantum loop effects to the power spectrum of primordial perturbations during ultra slow-roll inflation

“…Generally speaking, for having sufficiently large fluctuations, the inflationary dynamics has to deviate from slow-roll (SR) [12,13]. * dslee@gms.ndhu.edu.tw † nkw@phys.sinica.edu.tw Ultra-slow-roll (USR) inflation has been proposed as a transient phase of single-field inflation to generate large small-scale perturbations [14][15][16][17]. The idea of the USR is to consider a very flat potential of the inflaton field when its equation of motion, given by the Klein-Gordon equation, in a Friedmann-Robertson-Walker (FRW) cosmological space time, is dominated by the cosmological friction term rather than the slope of the potential in the SR inflation case.…”