Primordial Curvature Perturbation Research Articles

Enhanced curvature perturbation and primordial black hole formation in two-stage inflation with a break

We investigate a model of R2-gravity with a non-minimally coupled scalar field that gives rise to two-stage inflation with a break, that is, with an intermediate stage where inflation momentarily halts. We find that the power spectrum of the primordial curvature perturbation is significantly enhanced at the break scale, which can account for the primordial black hole (PBH) formation, without affecting the CMB constraint on large scales. The behavior of the curvature perturbation is carefully analyzed and we find a few notable new features in the spectrum. In particular, we find that the k3 growth of the spectrum of toward the end of the first stage of inflation. We argue that this is a universal feature common to all two-stage models where the field dominating the second stage is heavy during the first stage. By appropriately tuning the model parameters, we find that our model can realize the scenario of PBHs as the cold dark matter of the Universe. We also find that we can choose the parameters so that the spectrum of the induced gravitational waves from the enhanced curvature perturbation fits the NANOGrav-15yr data of pulsar timing array observation.

Constraints on the Primordial Curvature Power Spectrum and Reheating Temperature from the NANOGrav 15-Year Dataset

The stochastic signal observed by collaborations such as NANOGrav, PPTA, EPTA +InPTA, and CPTA may originate from gravitational waves induced by primordial curvature perturbations during inflation. This study investigates small-scale properties of inflation and reheating, assuming a log-normal form for the power spectrum of the primordial curvature and a reheating phase equation of state w=1/9. Inflation and reheating scenarios are thoroughly examined using Bayesian methods applied to the NANOGrav 15-year dataset. The analysis establishes constraints on the reheating temperature, suggesting Trh≳0.1Gev, consistent with Big Bang nucleosynthesis constraints. Additionally, the NANOGrav 15-year dataset requires the amplitude (A∼0.1) and width (Δ≲0.001) of the primordial curvature power spectrum to be within specific ranges. A notable turning point in the energy density of scalar-induced gravitational waves occurs due to a change in the equation of state w. This turning point signifies a transition from the reheating epoch to radiation domination. Further observations of scalar-induced gravitational waves could provide insights into the precise timing of this transition, enhancing our understanding of early Universe dynamics.

Constraints on primordial curvature power spectrum with kination era: Insights from NANOGrav 15-year data set

The stochastic signal detected by the NANOGrav, PPTA, EPTA, and CPTA collaborations can be attributed to gravitational waves induced by the primordial curvature perturbations generated during inflation. These scalar-induced gravitational waves provide valuable insights into the small-scale inflation and reheating epochs. In this paper, we assume an equation of state of w=1 during reheating and adopt a log-normal form for the primordial curvature power spectrum to elucidate the observed stochastic signal. The inflation and reheating scenarios are rigorously constrained utilizing Bayesian methods applied to the NANOGrav 15-year data set. The analysis yields constraints on the reheating temperature, indicating Trh≳0.01Gev, a result consistent with constraints derived from Big Bang nucleosynthesis. Furthermore, the NANOGrav 15-year data set necessitates the primordial curvature power spectrum's amplitude and width to satisfy A∼0.01 and Δ≲0.1, respectively. Due to the change in the equation of state w, there exists a turning point in the energy density spectrum of scalar-induced gravitational waves. This suggests that if more data about the scalar-induced gravitational waves is observed, it could potentially provide constraints on the time when the reheating epoch transitions to radiation domination.

Curvature perturbations from preheating with scale dependence

We extend the formalism to calculate non-Gaussianity of primordial curvature perturbations produced by preheating in the presence of a light scalar field. The calculation is carried out in the separate universe approximation using the non-perturbative delta N formalism and lattice field theory simulations. Initial conditions for simulations are drawn from a statistical ensemble determined by modes that left the horizon during inflation, with the time-dependence of Hubble rate during inflation taken into account. Our results show that cosmic variance, i.e., the contribution from modes with wavelength longer than the size of the observable universe today, plays a key role in determining the dominant contribution. We illustrate our formalism by applying it to an observationally-viable preheating model motivated by non-minimal coupling to gravity, and study its full parameter dependence.

Constraints on non-Gaussian primordial curvature perturbation from the LIGO-Virgo-KAGRA third observing run

The scalar-induced gravitational wave (SIGWs), arising from large amplitude primordial density fluctuations, provide a unique observational test for directly probing the epoch of inflation. In this work, we provide constraints on the SIGW background by taking into account the non-Gaussianity in the primordial density fluctuations, using the first three observing runs (O1-O3) data of the LIGO-Virgo-KAGRA collaboration. We find that the non-Gaussianity gives a non-negligible effect on the GW energy density spectrum and starts to affect the analysis of the O1-O3 data when the non-Gaussianity parameter is F NL > 3.55. Furthermore, the constraints exhibit asymptotic behavior given by F NL Ag = const. at large F NL limit, where Ag denotes the amplitude of the curvature perturbations. In the limit of large F NL, we placed a 95% credible level upper limit F NL Ag ≤ 0.115, 0.106, 0.112 at fixed scales of 1016, 1016.5, 1017 Mpc-1, respectively.

Smooth vs instant inflationary transitions: steepest growth re-examined and primordial black holes

Primordial black holes (PBHs) can be produced by a range of mechanisms in the early universe. A particular formation channel that connects PBHs with inflationary phenomenology invokes enhanced primordial curvature perturbations at small scales. In this paper, we examine how rapidly the background can transition between different values of the parameters of the Hubble hierarchy in the context of single-clock inflation, which must ultimately derive from a consistent derivative expansion for the background inflaton field. We discuss artefacts associated with instant or very rapid transitions, and consider the impact on the steepest power spectrum growth and the formation of PBHs. In particular, we highlight the robustness of the k 4 steepest growth previously found for single-field inflation with conservatively smoothed transitions and limits on how much the amplitude of the power spectrum can grow, and demonstrate that the PBH mass distribution is sensitive to the artefacts, which go away when the transitions are smoothed. We also show that the mass distribution is relatively insensitive to the steepness of the growth and subsequent decay of the power spectrum, depending primarily on the peak amplitude and the presence of any plateaus that last more than an e-fold. The shape of the power spectrum can of course be constrained by other tracers, and so understanding the physical limitations on its shape remains a pertinent question.

Fully non-Gaussian Scalar-Induced Gravitational Waves

Scalar-Induced Gravitational Waves (SIGWs) represent a particular class of primordial signals which are sourced at second-order in perturbation theory whenever a scalar fluctuation of the metric is present. They form a guaranteed Stochastic Gravitational Wave Background (SGWB) that, depending on the amplification of primordial scalar fluctuations, can be detected by GW detectors. The amplitude and the frequency shape of the scalar-induced SGWB can be influenced by the statistical properties of the scalar density perturbations. In this work we study the intuitive physics behind SIGWs and we analyze the imprints of local non-Gaussianity of the primordial curvature perturbation on the GW spectrum. We consider all the relevant non-Gaussian contributions up to fifth-order in the scalar seeds without any hierarchy, and we derive the related GW energy density ΩGW(f). We perform a Fisher matrix analysis to understand to which accuracy non-Gaussianity can be constrained with the LISA detector, which will be sensitive in the milli-Hertz frequency band. We find that LISA, neglecting the impact of astrophysical foregrounds, will be able to measure the amplitude, the width and the peak of the spectrum with an accuracy up to 𝒪(10-4), while non-Gaussianity can be measured up to 𝒪(10-3). Finally, we discuss the implications of our non-Gaussianity expansion on the fraction of Primordial Black Holes.

Supermassive primordial black holes from inflation

There is controversy surrounding the origin and evolution of our universe's largest supermassive black holes (SMBHs). In this study, we consider the possibility that some of these black holes formed from the direct collapse of primordial density perturbations. Since the mass of a primordial black hole is limited by the size of the cosmological horizon at the time of collapse, these SMBHs must form rather late, and are naively in conflict with constraints from CMB spectral distortions. These limits can be avoided, however, if the distribution of primordial curvature perturbations is highly non-Gaussian. After quantifying the departure from Gaussianity needed to evade these bounds, we explore a model of multi-field inflation — a non-minimal, self-interacting curvaton model — which has all the necessary ingredients to yield such dramatic non-Gaussianities. We leave the detailed model building and numerics to a future study, however, as our goal is to highlight the challenges associated with forming SMBHs from direct collapse and to identify features that a successful model would need to have. This study is particularly timely in light of recent observations of high-redshift massive galaxy candidates by the James Webb Space Telescope as well as evidence from the NANOGrav experiment for a stochastic gravitational wave background consistent with SMBH mergers.

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.

An analytical approximation of the evolution of the primordial curvature perturbation in the ultraslow-roll inflation: an extended study

Cosmic inflation can enter an ultraslow-roll (USR) stage, if there is a plateau on the inflaton potential. During this stage, the primordial curvature perturbation Rk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathscr {R}}}_k$$\\end{document} and its power spectrum PR\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathscr {P}}}_{{\\mathscr {R}}}$$\\end{document} can be remarkably enhanced on small scales. In this work, an analytical approximation is provided to systematically study the evolution of Rk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathscr {R}}}_k$$\\end{document} in the USR inflation. We first discuss the asymptotic solutions of the moduli and arguments of Rk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathscr {R}}}_k$$\\end{document} and its time derivative Rk,N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathscr {R}}}_{k,N}$$\\end{document} on the sub- and super-horizon scales separately and find that all these solutions have simple exponential forms. Then, Rk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathscr {R}}}_k$$\\end{document} on five typical scales are investigated in order. Our analytical approximation predicts that Rk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathscr {R}}}_k$$\\end{document} first revolves around the origin in the complex plane, but if it crosses the horizon around the start of the USR stage, there appears a subsequent linear evolution towards or away from the origin. This behavior naturally explains the shape of PR\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathscr {P}}}_{{\\mathscr {R}}}$$\\end{document} from the sharp dip to the peak and matches the numerical results perfectly. Moreover, the minimum of PR\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathscr {P}}}_{{\\mathscr {R}}}$$\\end{document} is exactly proved to be nonvanishing. Our analytical approximation will help the model building in primordial black hole and gravitational wave physics.

Cosmic topology. Part IIa. Eigenmodes, correlation matrices, and detectability of orientable Euclidean manifolds

If the Universe has non-trivial spatial topology, observables depend on both the parameters of the spatial manifold and the position and orientation of the observer. In infinite Euclidean space, most cosmological observables arise from the amplitudes of Fourier modes of primordial scalar curvature perturbations. Topological boundary conditions replace the full set of Fourier modes with specific linear combinations of selected Fourier modes as the eigenmodes of the scalar Laplacian. We present formulas for eigenmodes in orientable Euclidean manifolds with the topologies E 1–E 6, E 11, E 12, E 16, and E 18 that encompass the full range of manifold parameters and observer positions, generalizing previous treatments. Under the assumption that the amplitudes of primordial scalar curvature eigenmodes are independent random variables, for each topology we obtain the correlation matrices of Fourier-mode amplitudes (of scalar fields linearly related to the scalar curvature) and the correlation matrices of spherical-harmonic coefficients of such fields sampled on a sphere, such as the temperature of the cosmic microwave background (CMB). We evaluate the detectability of these correlations given the cosmic variance of the observed CMB sky. We find that topologies where the distance to our nearest clone is less than about 1.2 times the diameter of the last scattering surface of the CMB give a correlation signal that is larger than cosmic variance noise in the CMB. This implies that if cosmic topology is the explanation of large-angle anomalies in the CMB, then the distance to our nearest clone is not much larger than the diameter of the last scattering surface. We argue that the topological information is likely to be better preserved in three-dimensional data, such as will eventually be available from large-scale structure surveys.

Exploring Primordial Curvature Perturbation on Small Scales with the Lensing Effect of Fast Radio Bursts

Cosmological observations, e.g., cosmic microwave background, have precisely measured the spectrum of primordial curvature perturbation on larger scales, but smaller scales are still poorly constrained. Since primordial black holes (PBHs) could form in the very early Universe through the gravitational collapse of primordial density perturbations, constraints on the PBH could encode much information on primordial fluctuations. In this work, we first derive a simple formula for the lensing effect to apply PBH constraints with the monochromatic mass distribution to an extended mass distribution. Then, we investigate the latest fast radio burst observations with this relationship to constrain two kinds of primordial curvature perturbation models on small scales. This suggests that, from the null search result of lensed fast radio bursts in currently available observations, the amplitude of primordial curvature perturbation should be less than 8 × 10−2 at the scale region of 105–106 Mpc−1. This corresponds to an interesting mass range relating to binary black holes detected by LIGO–Virgo–KAGRA and future Einstein Telescope or Cosmic Explorer.

Constraints on the primordial curvature power spectrum by pulsar timing array data: a polynomial parameterization approach

The recent stochastic signal observed jointly by NANOGrav, parkes pulsar timing array, European pulsar timing array, and Chinese pulsar timing array can be accounted for by scalar-induced gravitational waves (SIGWs). The source of the SIGWs is from the primordial curvature perturbations, and the main contribution to the SIGWs is from the peak of the primordial curvature power spectrum. To effectively model this peak, we apply the Taylor expansion to parameterize it. With the Taylor expansion parameterization, we apply Bayesian methods to constrain the primordial curvature power spectrum based on the NANOGrav 15 year data set. The constraint on the primordial curvature power spectrum possesses a degree of generality, as the Taylor expansion can effectively approximate a wide range of function profiles.

Log-periodic gravitational-wave background beyond Einstein gravity

Periodic patterns of logarithmic oscillations can arise in primordial curvature perturbation spectra and in the associated gravitational-wave background via different mechanisms. We show that, in the presence of log oscillations, the spectral shape of the stochastic background has a unique parametrization independent of its physical origin. We also show that this log-periodic modulation can be generated in any scenario beyond Einstein gravity endowed with an approximate discrete scale invariance, a symmetry typical of deterministic fractal spacetimes that could emerge in quantum gravity under certain conditions. We discuss how a log-oscillatory spectral shape arises from concrete inflationary models beyond Einstein gravity and the prospects for detection in Einstein Telescope and other next-generation gravitational-wave observatories. We find that these instruments will be sensitive to log-periodic features if the detection is made with a high signal-to-noise ratio (SNR) and that the error scales as .

ON-GAUSSIANITY AND LOOP CORRECTIONS IN A QUADRATIC TWO-FIELD SLOW-ROLL MODEL OF INFLATION. PART I

I am showing in this paper that it is possible to attain very high, including observable, values for the leve! of non-gaussianity f N L in a particular quadratic two-field slow-roll model of inflation with canonical kinetic terms. This is done by taking care of loop corrections both in the spectrum Pζ and the bispectrum Bζ; of the primordial curvature perturbation ζ. Sizable values for f N L arise even if ζ is generated during inflation. Five issues are considered when constraining the available parameter space: 1. we must ensure that we are in a perturbative regime so that the ζ series expansion, and its truncation, are valid. 2. we must apply the correct condition for the (possible) loop dominance in Bζ and/or Pζ. 3 we must satisfy the spectrum normalisation condition. 4. we must satisfy the spectral tilt constraint. 5. we must have enough inflation to sol ve the horizon problem.

NON-GAUSSIANITY AND LOOP CORRECTIONS IN A QUADRATIC TWO-FIELD SLOW-ROLL MODEL OF INFLATION. PART 11

We calculate the trispectrum Tζ (k1 , k2 , k3, k4) of the primordial curvature perturbation ζ, gener­ated during a slow-roll inflationary epoch and considering a quadratic two-component scalar potential and canonical kinetic terms. We consider one-loop and tree leve] contlibutions, and show that it is pos­sible to attain observable values for the leve! of non-gaussianity T N L, if Tζ; is dominated by the one-loop contlibution. This work is performed by taking into account that there exists sorne physical restrictions that constrain the available parameter window. Such conditions are: thc existence of sorne coupling constants that guarantee the calculation in a perturbative regime, the relative weight of the one-loop and tree leve! contributions, the normalisation of the spectrum, the observed spectral index, and the mini mal amount of inflation required to solve the horizon problem.

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.

The evolution of the primordial curvature perturbation in the ultraslow-roll inflation

A plateau on the background inflaton potential can lead inflation into the ultraslow-roll (USR) stage, during which the primordial curvature perturbation {{{mathscr {R}}}} can still remarkably evolve outside the horizon. In this paper, the evolution of the Fourier mode {{{mathscr {R}}}}_k and the resulting power spectrum {{mathscr {P}}}_{{{mathscr {R}}}}(k) are systematically investigated. It is found that, when a scale crosses the horizon well before or after the USR stage, {{{mathscr {R}}}}_k only revolves around the origin in the complex plane. In contrast, if a scale crosses the horizon around the start of the USR stage, the argument of {{{mathscr {R}}}}_k will be nearly frozen and {{{mathscr {R}}}}_k will have a subsequent linear evolution accordingly, in which it may stop rather near or extremely far away from the origin, naturally explaining the entire growing stage of {{mathscr {P}}}_{{{mathscr {R}}}}(k) on small scales from the minimum to the peak. Moreover, there is a sharp dip in {{mathscr {P}}}_{{{mathscr {R}}}}(k), but the lowest point cannot reach 0 exactly. Our work provides a whole picture for {{{mathscr {R}}}}_k and {{mathscr {P}}}_{{{mathscr {R}}}}(k) in the USR inflation and will help our understanding of the relevant primordial black hole and gravitational wave physics.

Confronting sound speed resonance with pulsar timing arrays

The stochastic signal detected by pulsar timing arrays (PTAs) has raised great interest in understanding its physical origin. Assuming the signal is a cosmological gravitational-wave background produced by overly large primordial curvature perturbations, we investigate the sound speed resonance effect with an oscillatory behavior using the combined PTA data from NANOGrav 15-yr data set, PPTA DR3, and EPTA DR2.We find that the stochastic signal can be explained by the induced gravitational waves sourced by the sound speed resonance mechanism, with the oscillation frequency f * ∈ [1.51, 4.90] × 10-7Hz and the start time of oscillation |τ 0| ∈ [2.05, 106] × 107s.