Power Spectrum Of Curvature Perturbations Research Articles

Solution for cosmological observables in the Starobinsky model of inflation

This paper focuses on the Starobinsky model of inflation in the Einstein frame. We derive solutions for various cosmological observables, such as the scalar spectral index ns\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n_s$$\\end{document}, the tensor-to-scalar ratio r and their runnings, as well as the number of e-folds of inflation, reheating, and radiation, with minimal assumptions. The impact of reheating on inflation is explored by constraining the equation of state parameter ωre\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega _{re}$$\\end{document} at the end of reheating. An equation linking inflation with reheating is established, which is solved for the spectral index ns\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n_s$$\\end{document}. Using consistency relations of the model, we determine the other observables while the number of e-folds during inflation Nk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_k$$\\end{document}, and the number of e-folds during reheating Nre\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_{re}$$\\end{document} are determined by their respective formulas involving ns\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n_s$$\\end{document}. We find remarkable agreement between the Starobinsky model and current measurements of the power spectrum of primordial curvature perturbations and the present bounds on the spectrum of primordial gravitational waves.

Non-Gaussianity in rapid-turn multi-field inflation

We show that theories of inflation with multiple, rapidly turning fields can generate large amounts of non-Gaussianity. We consider a general theory with two fields, an arbitrary field-space metric, and a potential that supports sustained, rapidly turning field trajectories. Our analysis accounts for non-zero field cross-correlation and does not fix the power spectra of curvature and isocurvature perturbations to be equal at horizon crossing. Using the δN formalism, we derive a novel, analytical formula for bispectrum generated from multi-field mixing on super-horizon scales. Rapid-turn inflation can produce a bispectrum with several potentially large contributions that are not necessarily of the local shape. We exemplify the applicability of our formula with a fully explicit model and show that the new contributions indeed can generate a large amplitude of local non-Gaussianity, f NL loc ∼ 𝒪(1). These results will be important when interpreting the outcomes of future observations.

Primordial black holes dark matter and secondary gravitational waves from warm Higgs-G inflation

We explore the role of dissipative effects during warm inflation leading to the small-scale enhancement of the power spectrum of curvature perturbations. In this paper, we specifically focus on non-canonical warm inflationary scenarios and study a model of warm Higgs-G inflation, in which the Standard Model Higgs boson drives inflation, with a Galileon-like non-linear kinetic term. We show that in the Galileon-dominated regime, the primordial power spectrum is strongly enhanced, leading to the formation of primordial black holes (PBH) with a wide range of the mass spectrum. Interestingly, PBHs in the asteroid mass window ∼ (1017–1023) g are generated in this model, which can explain the total abundance of dark matter in the Universe. In our analysis, we also calculate the secondary gravitational waves (GW) sourced by these small-scale overdense fluctuations and find that the induced GW spectrum can be detected in future GW detectors, such as LISA, BBO, DECIGO, etc. Our scenario thus provides a novel way of generating PBHs as dark matter and a detectable stochastic GW background from warm inflation. We also show that our scenario is consistent with the swampland and the trans-Planckian censorship conjectures and, thus, remains in the viable landscape of UV complete theories.

Cancellation of quantum corrections on the soft curvature perturbations

We study the cancellation of quantum corrections on the superhorizon curvature perturbations from subhorizon physics beyond the single-clock inflation from the viewpoint of the cosmological soft theorem. As an example, we focus on the transient ultra-slow-roll inflation scenario and compute the one-loop quantum corrections to the power spectrum of curvature perturbations taking into account nontrivial surface terms in the action. We find that Maldacena’s consistency relation is satisfied and guarantees the cancellation of contributions from the short-scale modes. As a corollary, primordial black hole production in single-field inflation scenarios is not excluded by perturbativity breakdown even for the sharp transition case in contrast to some recent claims in the literature. We also comment on the relation between the tadpole diagram in the in-in formalism and the shift of the elapsed time in the stochastic-δN formalism. We find our argument is not directly generalisable to the tensor perturbations.

Model-independent reconstruction of the primordial curvature power spectrum from PTA data

Recently released data from pulsar timing array (PTA) collaborations provide strong evidence for a stochastic signal consistent with a gravitational-wave background, potentially originating from scalar-induced gravitational waves (SIGWs). However, in order to determine whether the SIGWs with a specific power spectrum of curvature perturbations can account for the PTA signal, one needs to estimate the energy density of the SIGWs, which can be computationally expensive.In this paper, we use a model-independent approach to reconstruct the primordial curvature power spectrum using a free spectrum cross over from 101 Mpc-1 to 1020 Mpc-1 with NANOGrav 15-year data set.Our results can simplify the task of assessing whether a given primordial curvature power spectrum can adequately explain the observed PTA signal without calculating the energy density of SIGWs.

Primordial black hole mass function with mass gap

ABSTRACT Primordial black holes (PBHs) are mainly characterized by their mass function, in which there may be some huge suppression for certain mass spans. If this is the case, the absence of these PBHs will form mass gaps. In this paper, we investigate the PBH mass function with the mass gap. First, to obtain a data-supported PBH mass function with the mass gap for subsolar masses PBHs, we fine-tune the coefficients of a model-independent power spectrum of primordial curvature perturbations. Then we take this unique PBH mass function into consideration and calculate the energy-density spectrum of the stochastic gravitational-wave background from PBH mergers. We find the location of its first peak almost has no relationship with the mass gap and is only determined by the probability distribution of frequencies at which PBH binaries merge. Apart from the first peak, there must be an accompanying smaller trough at a higher frequency resulting from the mass gap. Therefore, the detection of this smaller trough will provide more information about inflation and PBH formation.

Constraints on primordial curvature power spectrum with pulsar timing arrays

The stochastic signal detected by NANOGrav, PPTA, EPTA, and CPTA can be explained by the scalar-induced gravitational waves. In order to determine the scalar-induced gravitational waves model that best fits the stochastic signal, we employ both single- and double-peak parameterizations for the power spectrum of the primordial curvature perturbations, where the single-peak scenarios include the δ-function, box, lognormal, and broken power law model, and the double-peak scenario is described by the double lognormal form.Using Bayesian inference, we find that there is no significant evidence for or against the single-peak scenario over the double-peak model, with log (Bayes factors) among these models ln ℬ < 1.Therefore, we cannot distinguish the different shapes of the power spectrum of the primordial curvature perturbation with the current sensitivity of pulsar timing arrays.

One-loop corrections in power spectrum in single field inflation

We revisit the one-loop correction in curvature perturbation power spectrum in models of single field inflation which undergo a phase of ultra slow-roll (USR) inflation. We include the contributions from both the cubic and quartic interaction Hamiltonians and calculate the one-loop corrections on the spectrum of the CMB scale modes from the small scale modes which leave the horizon during the USR phase. It is shown that the amplitude of one-loop corrections depends on the sharpness of the transition from the USR phase to the final slow-roll phase. For an arbitrarily sharp transition, the one-loop correction becomes arbitrarily large, invalidating the perturbative treatment of the analysis. We speculate that for a mild transition, the large one-loop corrections are washed out during the subsequent evolution after the USR phase. The implications for primordial black holes formation are briefly reviewed.

Inflation from Multiple Pseudo-scalar Fields: Primordial Black Hole Dark Matter and Gravitational Waves

We study a model of inflation with multiple pseudo-scalar fields coupled to a U(1) gauge field through Chern–Simons interactions. Because of parity-violating interactions, one polarization of the gauge field is amplified, yielding to an enhanced curvature perturbation power spectrum. Inflation proceeds in multiple stages, as each pseudo-scalar field rolls toward its minimum, yielding to distinct multiple peaks in the curvature perturbation power spectra at various scales during inflation. The localized peaks in the power spectra generate primordial black holes that can furnish a large fraction of dark matter abundance. In addition, gravitational waves with nontrivial spectra are generated that are in the sensitivity ranges of various forthcoming GW observatories.

Bayesian Implications for the Primordial Black Holes from NANOGrav’s Pulsar-Timing Data Using the Scalar-Induced Gravitational Waves

Assuming that the common-spectrum process in the NANOGrav 12.5-year dataset has an origin of scalar-induced gravitational waves, we study the enhancement of primordial curvature perturbations and the mass function of primordial black holes, by performing the Bayesian parameter inference for the first time. We obtain lower limits on the spectral amplitude, i.e., A≳10−2 at 95% confidence level, when assuming the power spectrum of primordial curvature perturbations to follow a log-normal distribution function with width σ. In the case of σ→0, we find that the primordial black holes with 2×10−4−10−2 solar mass are allowed to compose at least a fraction 10−6 of dark matter. Such a mass range is shifted to more massive regimes for larger values of σ, e.g., to a regime of 4×10−3−0.2 solar mass in the case of σ=1. We expect the planned gravitational-wave experiments to have their best sensitivity to A in the range of 10−4 to 10−7, depending on the experimental setups. With this level of sensitivity, we can search for primordial black holes throughout the entire parameter space, especially in the mass range of 10−16 to 10−11 solar masses, where they could account for all dark matter. In addition, the importance of multi-band detector networks is emphasized to accomplish our theoretical expectation.

Reconstruction of power spectrum of primordial curvature perturbations on small scales from primordial black hole binaries scenario of LIGO/VIRGO detection

As a candidate bound for the Binary Black Hole (BBH) merger events detected by LIGO/Virgo, Primordial Black Holes (PBHs) provide a useful tool to investigate the primordial curvature perturbations on small scales. Using the GWTC-1 to GWTC-3 catalogs, under the scenario that PBHs originate from large primordial curvature perturbations on small scales during inflationary epoch, we for the first time reconstruct the power spectrum of primordial curvature perturbations on small scales. It is found that the value of the amplitude of the primordial power spectrum is enhanced to $\mathcal{O}(10^{-2})$ on scales $\mathcal{O}(1)$ pc. This may imply the validity of PBH as a possible BBH merger candidate.

Primordial black holes from Higgs inflation with a Gauss-Bonnet coupling

Primordial black holes (PBHs) can be the source for all or a part of today's dark matter density. Inflation provides a mechanism for generating the seeds of PBHs in the presence of a temporal period where the velocity of an inflaton field $\phi$ rapidly decreases toward 0. We compute the primordial power spectra of curvature perturbations generated during Gauss-Bonnet (GB) corrected Higgs inflation in which the inflaton field has not only a nonminimal coupling to gravity but also a GB coupling. For a scalar-GB coupling exhibiting a rapid change during inflation, we show that curvature perturbations are sufficiently enhanced by the appearance of an effective potential $V_{\rm eff}(\phi)$ containing the structures of plateau-type, bump-type, and their intermediate type. We find that there are parameter spaces in which PBHs can constitute all dark matter for these three types of $V_{\rm eff}(\phi)$. In particular, models with bump- and intermediate-types give rise to the primordial scalar and tensor power spectra consistent with the recent Planck data on scales relevant to the observations of cosmic microwave background. This property is attributed to the fact that the number of e-foldings $\Delta N_c$ acquired around the bump region of $V_{\rm eff}(\phi)$ can be as small as a few, in contrast to the plateau-type where $\Delta N_c$ typically exceeds the order of 10.

Measuring the primordial curvature perturbations from the scalar induced gravitational waves

The scalar induced gravitational waves are produced from primordial curvature perturbations in the second order of perturbations. We constrain the fractional energy density of scalar induced gravitational waves from gravitational waves observations. If there is no detection of the scalar induced gravitational waves, the fractional energy density of scalar induced gravitational waves is constrained by some upper limits. Depends on these upper limits, we can obtain the constraints on the power spectrum of the primordial curvature perturbations. For a power-law scalar power spectrum, the constraints on the power spectrum are affected by adding the upper limit of scalar induced gravitational waves from Square Kilometer Array (SKA). In the standard model, the mean values of the scalar amplitude and the spectral index shift to lower values when SKA is added to the combination of Cosmic Microwave Background (CMB) and Baryon Acoustic Oscillation (BAO) datasets, namely $\ln(10^{10}A_s)=3.038\pm0.013$ and $n_s=0.9589^{+0.0021}_{-0.0011}$ at $68\%$ confidence level. We also consider the effects of the existing ground-based gravitational-wave detectors, the existing Pulsar Timing Arrays (PTAs) and Five-hundred-meter Aperture Spherical radio Telescope (FAST), while the constraints from CMB+BAO datasets are totally within their upper limits of scalar induced gravitational waves. Furthermore, we characterize the scalar fluctuation spectrum in terms of the spectral index $n_s$ and its first two derivatives. We calculate corresponding power spectrum of scalar induced gravitational waves theoretically and give the constraints on the running of the spectral index and the running of the running of the spectral index.

Constraints on ultraslow-roll inflation from the third LIGO-Virgo observing run

The nonattractor evolution in ultra-slow-roll (USR) inflation results in the amplification of superhorizon curvature perturbations and then induces a strong and detectable stochastic gravitational wave background. In this letter, we search for such a stochastic gravitational wave background in data from the third LIGO-Virgo observing run and place constraints on the USR inflationary models. The $e$-folding number of the USR phase are constrained to be $\Delta N \lesssim 2.9$ at the 95% confidence level and the power spectrum of curvature perturbations amplified during the USR phase is constrained to be $\log_{10}P_{R\mathrm{p}}<-1.7$ at the scales $2.9\times10^5 ~\mathrm{pc^{-1}} \lesssim k \lesssim 1.7\times10^{11}~\mathrm{pc^{-1}}$. Besides, we forecast the ability of future experiments to constrain USR inflation, and find $P_{R\mathrm{p}}\lesssim 10^{-3.6}$ for LISA and Taiji, $P_{R\mathrm{p}}\lesssim 10^{-3.3}$ for Cosmic Explore and Einstein Telescope, $P_{R\mathrm{p}}\lesssim 10^{-5.5}$ for DECIGO and Big Bang Observer and $P_{R\mathrm{p}}\lesssim 10^{-5.2}$ for Square Kilometre Array.

Towards a reliable reconstruction of the power spectrum of primordial curvature perturbation on small scales from GWTC-3

Primordial black holes (PBHs) can be both candidates of dark matter and progenitors of binary black holes (BBHs) detected by the LIGO-Virgo-KAGRA collaboration. Since PBHs could form in the very early Universe through the gravitational collapse of primordial density perturbations, the population of BBHs detected by gravitational waves encodes much information on primordial curvature perturbation. In this work, we take a reliable and systematic approach to reconstruct the power spectrum of the primordial curvature perturbation from GWTC-3, under the hierarchical Bayesian inference framework, by accounting for the measurement uncertainties and selection effects. In addition to just considering the single PBH population model, we also report the results considering the multi-population model, i.e., the mixed PBH and astrophysical black hole binaries model. We find that the maximum amplitude of the reconstructed power spectrum of primordial curvature perturbation can be ∼2.5×10−2 at O(105)Mpc−1 scales, which is consistent with the PBH formation scenario from inflation at small scales.

Primordial black hole dark matter from inflation: The reverse engineering approach

Constraining the inflationary epoch is one of the aims of modern cosmology. In order to fully exploit current and future small-scale observations, it is necessary to devise tools to directly relate them to the early Universe's dynamics. We present here a novel reverse engineering approach able to connect fundamental late-time observables to consistent inflationary dynamics and, eventually, to the inflaton potential. Employing this procedure, we are able to describe which conditions can give rise to a raised plateau in the power spectrum of curvature perturbations at small scales, which are not constrained by CMB observations. Within this new phenomenologically driven approach, we find that inflation can generate a raised plateau in the spectrum of curvature perturbations that potentially connects three fundamental observables; a dominant component of the dark matter in the form of asteroid-mass/atomic-size primordial black holes, detectable signals in stochastic gravitational waves, and a subdominant fraction of stellar-mass primordial black holes mergers.

Primordial black holes and gravitational waves in hybrid inflation with chaotic potentials

We study the formation of primordial black hole (PBH) dark matter and the generation of scalar induced secondary gravitational waves (SIGWs) in a non-supersymmetric model of hybrid inflation with chaotic (polynomial-like) potential, including one-loop radiative corrections. A radiatively corrected version of these models is entirely consistent with Planck's data. By adding non-canonical kinetic energy term in the lagrangian, the inflation experiences a period of ultra-slow-roll, and the amplitude of primordial power spectrum is enhanced to O(10−2). The enhanced power spectra of primordial curvature perturbations can have both sharp and broad peaks. A wide mass range of PBHs is realized in our model with the frequencies of scalar induced gravitational waves ranged from nHz to kHz. We present several benchmark points where the PBH mass generated during inflation is around (1−100)M⊙, (10−9−10−7)M⊙ and (10−16−10−11)M⊙. The PBHs can make up most of the dark matter with masses around (10−16−10−11)M⊙ and (1−100)M⊙, and their associated SIGWs can be probed by the upcoming ground and space-based gravitational wave (GW) observatories. The evidence of stochastic process recently reported by NANOGrav may be interpreted as SIGWs associated with the formation of PBHs. These SIGWs may also be tested by future interferometer-type GW observations of SKA, DECIGO, LISA, BBO, TaiJi, TianQin, CE and ET.

Smooth coarse-graining and colored noise dynamics in stochastic inflation

We consider stochastic inflation coarse-grained using a general class of exponential filters. Such a coarse-graining prescription gives rise to inflaton-Langevin equations sourced by colored noise that is correlated in e-fold time. The dynamics are studied first in slow-roll for simple potentials using first-order perturbative, semi-analytical calculations which are later compared to numerical simulations. Subsequent calculations are performed using an exponentially correlated noise which appears as a leading order correction to the full slow-roll noise correlation functions of the type 〈ξ(N)ξ(N')〉(n) ∼ (cosh[n(N-N'+1])-1. We find that the power spectrum of curvature perturbations \U0001d4ab ζ is suppressed at small e-folds, with the suppression controlled by n. Furthermore, we use the leading order, exponentially correlated noise and perform a first passage time analysis to compute the statistics of the stochastic e-fold distribution \U0001d4a9 and derive an approximate expression for the mean number of e-folds 〈\U0001d4a9〉. Comparing analytical results with numerical simulations of the inflaton dynamics, we show that the leading order noise correlation function can be used as a very good approximation of the exact noise, the latter being more difficult to simulate.

Model-independent approach to effective sound speed in multi-field inflation

For any physical system satisfying the Einstein’s equations, the comoving curvature perturbations satisfy an equation involving the momentum-dependent effective sound speed, valid for any system with a well defined energy-stress tensor, including multi-fields models of inflation. We derive a general model-independent formula for the effective sound speed of comoving adiabatic perturbations, valid for a generic field-space metric, without assuming any approximation to integrate out entropy perturbations, but expressing the momentum-dependent effective sound speed in terms of the components of the total energy-stress tensor. As an application, we study a number of two-field models with a kinetic coupling between the fields, identifying the single curvature mode of the effective theory and showing that momentum-dependent effective sound speed fully accounts for the predictions for the power spectrum of curvature perturbations. Our results show that the momentum-dependent effective sound speed is a convenient scheme for describing all inflationary models that admit a single-field effective theory, including the effects of entropy pertubations present in multi-fields systems.

Formation of primordial black holes after Starobinsky inflation

In this paper, we adapted the Appleby–Battye–Starobinsky model of [Formula: see text] gravity towards describing double cosmological inflation and formation of primordial black holes with masses up to [Formula: see text] g in the single-field model. We found that it is possible to get an enhancement of the power spectrum of scalar curvature perturbations to the level beyond the Hawking (black hole evaporation) limit of [Formula: see text] g, so that the primordial black holes resulting from gravitational collapse of those large primordial perturbations can survive in the present universe and form part of cold dark matter. Our results agree with the current measurements of cosmic microwave background radiation within [Formula: see text] but require fine-tuning of the parameters.