Steepest growth of the power spectrum and primordial black holes

Abstract

We derive analytic bounds on the shape of the primordial power spectrum in the context of single-field inflation. In particular, the steepest possible growth has a spectral index of n s − 1 = 4 once transients have died down. Its primary implication is that any constraint on the power spectrum at a particular scale can be extrapolated to an upper bound over an extended range of scales. This is important for models which generate relics due to an enhanced amplitude of the primordial scalar perturbations, such as primordial black holes. In order to generate them, the power spectrum needs to grow many orders of magnitude larger than its observed value on CMB scales -typically achieved through a phase of ultra slow-roll inflation -and is thus subject to additional constraints at small scales. We plot all relevant constraints including CMB spectral distortions and gravitational waves sourced by scalar perturbations at second order. We show how this limits the allowed mass of PBHs, especially for the large masses of interest following recent detections by LIGO and prospects for constraining them further with future observations. We show that any transition from approximately constant slow-roll inflation to a phase where the power spectrum rapidly rises necessarily implies an intervening dip in power. We also show how to reconstruct a potential that can reproduce an arbitrary time-varying , offering a complementary perspective on how ultra slow-roll can be achieved. * C.Byrnes@sussex.ac.uk

Gauge independence of induced gravitational waves

“…Also, uv 1 is possible only at the corners of the integration region, so the net effect of gauge dependence after integration over u and v is suppressed. 7 A corollary of this discussion is that the large enhancement of the induced GWs found in Ref. [19], which one may suspect a gauge artifact, is not a gauge artifact.…”

“…Gravitational waves (GWs) are a useful probe of small-scale curvature perturbations [3][4][5][6][7][8] since if the latter is enhanced, a sizable amount of GWs is produced at the second order of the cosmological perturbation. We call it the induced GWs 1 (see early works [9][10][11][12] and recent developments [13][14][15][16][17][18][19][20][21][22]).…”

From primordial black holes abundance to primordial curvature power spectrum (and back)

“…We refer the interested reader to Ref. [108], where the change in the constraints assuming different power spectra shapes is shown. Since many of these constraints are forecast and the specifics of the instruments are unknown, in this work we use the curves found in Ref.…”

Initial clustering and the primordial black hole merger rate

“…The term In order to derive a mass function, we will, again, make the simplifying assumption that all PBH formation occurs at a single epoch, with a Dirac-delta form for the power spectrum, as in equation (17). It is noted that such a form for the power spectrum is not physical, and the fastest growth for the power spectrum as a function of the wavevector k has been shown to be ∼ k 4 [64], or slightly steeper in the case of a more contrived scenario [65] (see also [66]), which means that PBH formation at a single scale is not physical, but rather that it occurs over a range of scales.…”