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Probing non-Gaussianities with the high frequency tail of induced gravitational waves
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Abstract
We investigate in detail the spectrum of gravitational waves induced by a peaked primordial curvature power spectrum generated in single field inflationary models. We argue that the parameter can be inferred by measuring the high frequency spectral tilt of the induced gravitational waves. We also show that the intrinsically non-Gaussian impact of in is to broaden its peak, although at a negligible level in order not to overproduce primordial black holes. We discuss possible degeneracies in the high frequency spectral tilt between and a general equation of state of the universe w. Finally, we discuss the constraints on the amplitude, peak and slope (or equivalently, ) of the primordial power spectrum by combining current and future gravitational wave experiments with limits on distortions from the cosmic microwave background.
The in-in formalism provides a way to systematically organize the calculation of primordial correlation functions. Although its theoretical foundations are now firmly settled, the treatment of total time derivative interactions, incorrectly trivialized as “boundary terms”, has been the subject of intense discussions and conceptual mistakes. In this work, we demystify the use of total time derivatives — as well as terms proportional to the linear equations of motion — and show that they can lead to artificially large contributions cancelling at different orders of the in-in operator formalism. We discuss the treatment of total time derivative interactions in the Lagrangian path integral formulation of the in-in perturbation theory, and we showcase the importance of interaction terms proportional to linear equations of motion. We then provide a new route to the calculation of primordial correlation functions, which avoids the generation of total time derivatives, by working directly at the level of the full Hamiltonian in terms of phase-space variables. Instead of integrating by parts, we perform canonical transformations to simplify interactions. We explain how to retrieve correlation functions of the initial phase-space variables from the knowledge of the ones after canonical transformations. As an important first application, we find the explicit sizes of Hamiltonian cubic interactions in single-field inflation with canonical kinetic terms and for any background evolution, straight in terms of the primordial curvature perturbation and its canonical conjugate momentum, as well as the corresponding ones in the tensor sector, and the ones mixing scalars and tensors. We also briefly comment on quartic interactions. Our results are important for performing complete calculations of exchange diagrams in inflation, such as the (scalar and tensor) exchange trispectrum and the one-loop power spectrum. Being already written in a form amenable to characterize quantum properties of primordial fluctuations, they also promise to shed light on the non-linear dynamics of quantum states during inflation.
We present analytic integral solutions for the second-order induced gravitational waves (GWs). After presenting all the possible second-order source terms, we calculate explicitly the solutions for the GWs induced by the linear scalar and tensor perturbations during matter- and radiation-dominated epochs.
We compute the probability density distribution of maxima for a scalar random field in the presence of local non-gaussianities. The physics outcome of this analysis is the following. If we focus on maxima whose curvature is larger than a certain threshold for gravitational collapse, our calculations illustrate how the fraction of the Universe's mass in the form of primordial black holes (PBHs) changes in the presence of local non-gaussianities. We find that previous literature on the subject overestimates, by many orders of magnitude, the impact of local non-gaussianities on the PBH abundance. We explain the origin of this discrepancy, and conclude that, in realistic single-field inflationary models with ultra slow-roll, one can obtain the same abundance found with the gaussian approximation simply changing the peak amplitude of the curvature power spectrum by no more than a factor of two. We comment about the relevance of non-gaussianities for second-order gravitational waves.
We identify a characteristic pattern in the scalar-induced stochastic gravitational wave background from particle production during inflation. If particle production is sufficiently efficient, the scalar power spectrum exhibits 𝒪(1) oscillations periodic in k, characteristic of a sharp feature, with an exponentially enhanced envelope. We systematically study the properties of the induced spectrum of gravitational waves sourced after inflation and find that this inherits the periodic structure in k, resulting in a peak in the gravitational wave energy density spectrum with 𝒪(10 %) modulations. The frequency of the oscillation in the scalar power spectrum is determined by the scale of the feature during inflation and in turn sets the frequency of modulations in the gravitational wave signal. We present an explicit realisation of this phenomenon in the framework of multifield inflation, in the form of a strong sharp turn in the inflationary trajectory. The resulting stochastic background is potentially detectable in future gravitational wave observatories, and considerations of backreaction and perturbativity can be used to constrain the parameter space from the theoretical side. Our work motivates more extensive research linking primordial features to observable properties of the stochastic background of gravitational waves, and dedicated development in data analysis for their detection.
We give an explanation for the signal detected by NANOGrav as the stochastic gravitational wave background from binary mergers of primordial "Stupendously Large Black Holes"(SLABs) of mass M ∼ (1011-1012)M , and corresponding to roughly 0.1% of the dark matter. We show that the stringent bounds coming from μ distortions of the CMB can be surpassed if the perturbations resulting in these BHs arise from the non-Gaussian distribution of fluctuations expected in single field models of inflation generating a spike in the power spectrum. While the tail of the stochastic background coming from binaries with M ≲ 1011 M could both fit NANOGrav and respect μ distortions limits, they become excluded from large scale structure constraints.
We revisit the squeezed-limit non-Gaussianity in the single-field non-attractor inflation models from the viewpoint of the cosmological soft theorem. In the single-field attractor models, inflaton’s trajectories with different initial conditions effectively converge into a single trajectory in the phase space, and hence there is only one clock degree of freedom (DoF) in the scalar part. Its long-wavelength perturbations can be absorbed into the local coordinate renormalization and lead to the so-called consistency relation between n- and (n + 1)-point functions. On the other hand, if the inflaton dynamics deviates from the attractor behavior, its long-wavelength perturbations cannot necessarily be absorbed and the consistency relation is expected not to hold any longer. In this work, we derive a formula for the squeezed bispectrum including the explicit correction to the consistency relation, as a proof of its violation in the non-attractor cases. First one must recall that non-attractor inflation needs to be followed by attractor inflation in a realistic case. Then, even if a specific non-attractor phase is effectively governed by a single DoF of phase space (represented by the exact ultra-slow-roll limit) and followed by a single-DoF attractor phase, its transition phase necessarily involves two DoF in dynamics and hence its long-wavelength perturbations cannot be absorbed into the local coordinate renormalization. Thus, it can affect local physics, even taking account of the so-called local observer effect, as shown by the fact that the bispectrum in the squeezed limit can go beyond the consistency relation. More concretely, the observed squeezed bispectrum does not vanish in general for long-wavelength perturbations exiting the horizon during a non-attractor phase.
Ultra-light primordial black holes, with masses m PBH < 109g, evaporate before big-bang nucleosynthesis and can therefore not be directly constrained. They can however be so abundant that they dominate the universe content for a transient period (before reheating the universe via Hawking evaporation). If this happens, they support large cosmological fluctuations at small scales, which in turn induce the production of gravitational waves through second-order effects. Contrary to the primordial black holes, those gravitational waves survive after evaporation, and can therefore be used to constrain such scenarios. In this work, we show that for induced gravitational waves not to lead to a backreaction problem, the relative abundance of black holes at formation, denoted ΩPBH,f, should be such that ΩPBH,f < 10-4(m PBH/109g)-1/4. In particular, scenarios where primordial black holes dominate right upon their formation time are all excluded (given that m PBH > 10 g for inflation to proceed at ρ1/4 < 1016 GeV). This sets the first constraints on ultra-light primordial black holes.
The stochastic gravitational wave background (SGWB) offers a new opportunity to observe signals of primordial features from inflationary models. We study their detectability with future space-based gravitational waves experiments, focusing our analysis on the frequency range of the LISA mission. We compute gravitational wave spectra from primordial features by exploring the parameter space of a two-field inflation model capable of generating different classes of features. Fine-tuning in scales and amplitudes is necessary for these signals to fall in the observational windows. In some cases the scalar power spectrum can significantly exceed the ns=5 limit in single-field inflation and grow as fast as ns=9.1. Once they show up, several classes of frequency-dependent oscillatory signals, characteristic of different underlying inflationary physics, may be distinguished and the SGWB provides a window on dynamics of the primordial universe independent of cosmic microwave background and large-scale structure. To connect with future experimental data, we discuss two approaches of how the results may be applied to data analyses. First, we discuss the possibility of reconstructing the signal with LISA, which requires a high signal-to-noise ratio. The second more sensitive approach is to apply templates representing the spectra as estimators. For the latter purpose, we construct templates that can accurately capture the spectral features of several classes of feature signals and compare them with the SGWB produced by other physical mechanisms.
We compare primordial black hole (PBH) constraints on the power spectrum and mass distributions using the traditional Press Schechter formalism, peaks theory, and a recently developed version of peaks theory relevant to PBHs. We show that, provided the PBH formation criteria and the power spectrum smoothing are treated consistently, the constraints only vary by ∼ 10% between methods (a difference that will become increasingly important with better data). Our robust constraints from PBHs take into account the effects of critical collapse, the non-linear relation between ζ and δ, and the shift from the PBH mass to the power spectrum peak scale. We show that these constraints are remarkably similar to the pulsar timing array (PTA) constraints impacting the black hole masses detected by LIGO and Virgo, but that the μ-distortion constraints rule out supermassive black hole (SMBH) formation and potentially even the much lighter mass range of ∼(1-100) M that LIGO/Virgo probes.
In order to produce primordial black holes, or to enhance the spectrum of primordial gravitational waves at interferometer scales, cosmological inflation should include brief periods of nonattractor evolution, with departures from slow-roll conditions. To better understand these systems we develop a model independent, analytic approach to study cosmological fluctuations in single field inflationary models with short periods of slow-roll violation. We Taylor expand the equations for fluctuations in a quantity parametrizing the duration of slow-roll violation, and find solutions at leading order in this quantity. The resulting spectrum of perturbations is described by few parameters, and presents a pronounced dip followed by a rapid growth in amplitude, as typically found in numerical as well as analytical studies of similar scenarios. We determine new analytic relations between the position and depth of the dip, and the maximal enhancement of the spectrum toward small scales. For a single epoch of slow-roll violation we confirm previous studies finding that the steepest slope of the spectrum has spectral index ns−1=4. On the other hand, when considering multiple phases of slow-roll violation, the slope of the spectrum is generally enhanced. When two epochs of slow-roll violation occur, the spectral index can reach the value ns−1=8, indicating that the slope of the spectrum has memory of the history of non-slow-roll phases during inflation.
Primordial black holes (PBHs) can constitute the predominant fraction of dark matter (DM) if PBHs reside in the currently unconstrained “sublunar” mass range. PBHs originating from scalar perturbations generated during inflation can naturally appear with a broad spectrum in a class of models. The resulting stochastic gravitational wave (GW) background generated from such PBH production can account for the recently reported North American Nanohertz Observatory for Gravitational Waves (NANOGrav) pulsar timing array data signal, and will be testable in future GW observations by interferometer-type experiments such as Laser Interferometer Space Antenna (LISA). We show that the broad mass function of such PBH DM is already being probed by Subaru Hyper Suprime-Cam (HSC) microlensing data and is consistent with a detected candidate event. Upcoming observations of HSC will be able to provide an independent definitive test of the stochastic GW signals originating from such PBH DM production scenarios.
The NANOGrav collaboration for the pulsar timing array (PTA) observation recently announced evidence of an isotropic stochastic process, which may be the first detection of the stochastic gravitational-wave (GW) background. We discuss the possibility that the signal is caused by the second-order GWs associated with the formation of solar-mass primordial black holes (PBHs). This possibility can be tested by future interferometer-type GW observations targeting the stochastic GWs from merger events of solar-mass PBHs as well as by updates of PTA observations.
We discuss the effect of clustering for the determination of the merger rate of binary black holes in the LIGO/Virgo mass range. While for a Poissonian initial distribution, and assuming isolated binaries, the allowed fraction of Primordial Black Holes (PBHs) to dark matter (DM) is a few percent, we show that this bound can be relaxed if PBHs are clustered. More precisely we show that for large clustering the merger rate can drop with increasing fraction of PBHs, introducing a degeneracy in the parameters of the theory consistent with a given present merger rate, and allowing all the DM to be in the form of stellar mass PBHs. This degeneracy can however be broken by looking at the evolution of the merger rate with redshift. For a simple clustering model that we consider, we show that the LIGO/Virgo projected sensitivity can disentangle, through the observation of a stochastic background of gravitational waves, different clustered distributions having the same present merger rate.
The production rate of primordial black holes (PBHs) is often calculated by considering a nearly Gaussian distribution of cosmological perturbations, and assuming that black holes will form in regions where the amplitude of such perturbations exceeds a certain threshold. A threshold || for the curvature perturbation is somewhat inappropriate for this purpose, because it depends significantly on environmental effects, not essential to the local dynamics. By contrast, a threshold || for the density perturbation at horizon crossing seems to provide a more robust criterion. On the other hand, the density perturbation is known to be bounded above by a maximum limit || at the horizon entry and, given that || is comparable to ||, the density perturbation will be far from Gaussian near or above the threshold. In this paper, we provide a new plausible estimate for the primordial black hole abundance based on peak theory. In our approach, we assume that the curvature perturbation is given as a random Gaussian field with the power spectrum characterized by a single scale, while an optimized criterion for PBH formation is imposed, based on the locally averaged density perturbation around the nearly spherically symmetric high peaks. Both variables are related by the full non-linear expression derived in the long-wavelength approximation of general relativity. We do not introduce a window function, which is usually introduced to obtain the scale dependence of the spectrum. The scale of the inhomogeneity is introduced as a random variable in the peak theory, and the scale-dependent PBH fraction is automatically induced. We find that the mass spectrum is shifted to larger mass scales by one order of magnitude or so, compared to a conventional calculation. The abundance of PBHs becomes significantly larger than the conventional one, by many orders of magnitude, mainly due to the optimized criterion for PBH formation and the removal of the suppression associated with a window function.
We study the stochastic gravitational wave (GW) background induced by the primordial scalar perturbation with the spectrum having a lognormal peak of width Δ at k=k∗. We derive an analytical formula for the GW spectrum ΩGW for both narrow (Δ≪1) and broad (Δ≳ 1) peaks. In the narrow-peak case, the spectrum has a double peak feature with the sharper peak at k= 2k∗/3. On the infrared (IR) side of the spectrum, we find power-law behavior with a break at k=kb in the power-law index where it chages from k3 on the far IR side to k2 on the near IR side. We find the ratio of the break frequency to the peak frequency is determined by Δ as fb/fp≈3Δ, where fb and fp are the break and peak frequencies, respectively. In the broad-peak case, we find the GW spectrum also has a lognormal peak at k=k∗ but with a smaller width of Δ/2. Using these derived analytic formulae, we also present expressions for the maximum values of ΩGW for both narrow and broad cases. Our results will provide a useful tool in searching for the induced GW signals in the coming decades.
The inflationary origin of primordial black holes (PBHs) relies on a large enhancement of the power spectrum Δζ of the curvature fluctuation ζ at wavelengths much shorter than those of the cosmic microwave background anisotropies. This is typically achieved in models where ζ evolves without interacting significantly with additional (isocurvature) scalar degrees of freedom. However, quantum gravity inspired models are characterized by moduli spaces with highly curved geometries and a large number of scalar fields that could vigorously interact with ζ (as in the cosmological collider picture). Here we show that isocurvature fluctuations can mix with ζ inducing large enhancements of its amplitude. This occurs whenever the inflationary trajectory experiences rapid turns in the field space of the model leading to amplifications that are exponentially sensitive to the total angle swept by the turn, which induce characteristic observable signatures on Δζ. We derive accurate analytical predictions and show that the large enhancements required for PBHs demand noncanonical kinetic terms in the action of the multifield system.
The scalar perturbation induced gravitational waves are a probe of the primordial density perturbation spectrum on small scales. In this paper, we show that they can also probe the thermal history of the universe. We assume the universe underwent a stage with a constant equation of state parameter w, followed by the radiation-dominated stage of the conventional big bang universe. We find that the infrared slope of the power spectrum of the induced stochastic gravitational wave background for decelerating cosmologies is related to the equation of state of the universe. Furthermore, the induced gravitational wave spectrum has in general a broken power-law shape around the scale of reheating. Interestingly, below the threshold 0w= of the equation of state parameter, the broken power-law presents a peak for a Dirac delta peak in the scalar spectrum. For a finite width peak, the threshold changes to w=-1/15 depending on the value of the width. In some cases, such a broken power-law gravitational wave spectrum may degenerate to the spectrum from other sources like phase transitions or global cosmic strings.
Primordial black holes (PBHs) generated by gravitational collapse of large primordial overdensities can be a fraction of the observed dark matter. In this paper, we introduce a mechanism to produce a large peak in the primordial power spectrum (PPS) in two-field inflationary models characterized by two stages of inflation based on a large non-canonical kinetic coupling. This mechanism is generic to several two-field inflationary models, due to a temporary tachyonic instability of the isocurvature perturbations at the transition between the two stages of inflation. We numerically compute the primordial perturbations from largest scales to the small scales corresponding to that of PBHs using an extension of BINGO (BI-spectra and Non-Gaussianity Operator). Moreover we numerically compute the stochastic background of gravitational waves (SBGW) produced by second order scalar perturbations within frequencies ranging from nano-Hz to KHz that covers the observational scales corresponding to Pulsar Timing Arrays, Square Kilometer Array to that of Einstein telescope. We discuss the prospect of its detection by these proposed and upcoming gravitational waves experiments.
We consider the possibility that the majority of dark matter in our Universe consists of black holes of primordial origin. We determine the conditions under which such black holes may have originated from a single-field model of inflation characterized by a quartic polynomial potential. We also explore the effect of higher-dimensional operators. The large power spectrum of curvature perturbations that is needed for a large black hole abundance sources sizable second order tensor perturbations. The resulting stochastic background of primordial gravitational waves could be detected by the future space-based observatories LISA and DECIGO or-A s long as we give up on the dark matter connection-by the ground-based Advanced LIGO-Virgo detector network.
We review potential low-frequency gravitational-wave sources, which are expected to be detected by Taiji, a Chinese space-based gravitational-wave detector, estimate the detection rates of these gravitational-wave sources and present the parameter estimation of massive black hole binaries.
Primordial black holes (PBHs) may result from high peaks in a random field of cosmological perturbations. In single field inflationary models, such perturbations can be seeded as the inflaton overshoots a small barrier on its way down the potential. PBHs are then produced through two distinct mechanisms, during the radiation era. The first one is the familiar collapse of large adiabatic overdensities. The second one is the collapse induced by relic bubbles where the inflaton field is trapped in a false vacuum. The latter are due to rare backward fluctuations of the inflaton which prevented it from overshooting the barrier in horizon sized regions. We consider (numerically and analytically) the effect of non-Gaussianities on the threshold for overdensities to collapse into a PBH. Since typical high peaks have some dispersion in their shape or profile, we also consider the effect of such dispersion on the corresponding threshold for collapse. With these results we estimate the most likely channel for PBH production as a function of the non-Gaussianity parameter fNL. We also compare the threshold for collapse coming from the perturbative versus the non perturbative template for the non-Gaussianity arising in this model. We show that i) for fNL 3.5, the population of PBH coming from false vacuum regions dominates over that which comes from the collapse of large adiabatic overdensities, ii) the non-perturbative template of the non-Gaussianities is important to get accurate results. iii) the effect of the dispersion is small in determining the threshold for the compaction function, although it can be appreciable in determining the threshold amplitude for the curvature perturbation at low fNL. We also confirm that the volume averaged compaction function provides a very accurate universal estimator for the threshold.
We outline the experimental concept and key scientific capabilities of AION (Atom Interferometer Observatory and Network), a proposed experimental programme using cold strontium atoms to search for ultra-light dark matter, to explore gravitational waves in the mid-frequency range between the peak sensitivities of the LISA and LIGO/Virgo/ KAGRA/INDIGO/Einstein Telescope/Cosmic Explorer experiments, and to probe other frontiers in fundamental physics. AION would complement other planned searches for dark matter, as well as probe mergers involving intermediate-mass black holes and explore early-universe cosmology. AION would share many technical features with the MAGIS experimental programme, and synergies would flow from operating AION in a network with this experiment, as well as with other atom interferometer experiments such as MIGA, ZAIGA and ELGAR. Operating AION in a network with other gravitational wave detectors such as LIGO, Virgo and LISA would also offer many synergies.
The issue of the gauge invariance of gravitational waves arises if they are produced in the early universe at second-order in perturbation theory. We address it by dividing the discussion in three parts: the production of gravitational waves, their propagation in the real universe, and their measurement.
We study the induced primordial gravitational waves (GWs) coming from the effect of scalar perturbation on the tensor perturbation at the second order of cosmological perturbation theory. We use the evolution of the standard model degrees of freedom with respect to temperature in the early Universe to compute the induced gravitational waves background. Our result shows that the spectrum of the induced GWs is affected differently by the standard model degrees of freedom than the GWs coming from the first-order tensor perturbation. This phenomenon is due to the presence of scalar perturbations as a source for tensor perturbations, and it is effective around the quark gluon deconfinement and electroweak transition. In case of considering a scalar spectral index larger than 1 at small scales or a non-Gaussian curvature power spectrum, this effect can be observed by gravitational wave observatories.
We revisit the effects of an early matter-dominated era on gravitational waves induced by scalar perturbations. We carefully take into account the evolution of the gravitational potential, the source of these induced gravitational waves, during a gradual transition from an early matter-dominated era to the radiation-dominated era, where the transition timescale is comparable to the Hubble time at that time. Realizations of such a gradual transition include the standard perturbative reheating with a constant decay rate. Contrary to previous works, we find that the presence of an early matter-dominated era does not necessarily enhance the induced gravitational waves due to the decay of the gravitational potential around the transition from an early matter-dominated era to the radiation-dominated era.
If the black holes detected by LIGO/VIRGO are primordial black holes (PBHs) sourcing from a large primordial curvature perturbation on small scales, the corresponding induced gravitational waves (GWs) would peak at nanohertz that is detectable by the current and future observations of pulsar timing array (PTA). In this paper we show that with the mass function estimated from the merger rate of LIGO O1 and O2 events, the induced GWs from such a curvature perturbation with a Gaussian narrow peak at some small scale would be in a seemingly mild tension with current constraints from PTA. However, if the curvature perturbation is of local-type non-Gaussianity with a non-linear parameter fNL≳(10), the tension could be relieved. Nevertheless, such an induced GWs must be detectable by the Square Kilometer Array in a decade or less.
In the context of transient constant-roll inflation near a local maximum, we derive the non-perturbative field redefinition that relates a Gaussian random field with the true non-Gaussian curvature perturbation. Our analysis shows the emergence of a new critical amplitude ζ*, corresponding to perturbations that prevent the inflaton from overshooting the local maximum, thus becoming trapped in the false minimum of the potential. For potentials with a mild curvature at the local maximum (and thus small non-Gaussianity), we recover the known perturbative field redefinition. We apply these results to the formation of primordial black holes, and discuss the cases for which ζ* is smaller or of the same order than the critical value for collapse of spherically symmetric overdensities. In the latter case, we present a simple potential for which the power spectrum needs an amplitude 10 times smaller that in the Gaussian case for producing a sizeable amount of primordial black holes.
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 ns - 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 ϵ 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 ϵ, offering a complementary perspective on how ultra slow-roll can be achieved.
Gravitational waves (GWs) have the potential to probe the entirety of cosmological history due to their nearly perfect decoupling from the thermal bath and any intervening matter after emission. In recent years, GW cosmology has evolved from merely being an exciting prospect to an actively pursued avenue for discovery, and the early results are very promising. As we highlight in this paper, spectral distortions (SDs) of the cosmic microwave background (CMB) uniquely probe GWs over six decades in frequency, bridging the gap between astrophysical high- and cosmological low-frequency measurements. This means SDs will not only complement other GW observations, but will be the sole probe of physical processes at certain scales. To illustrate this point, we explore the constraining power of various proposed SD missions on a number of phenomenological scenarios: early-universe phase transitions (PTs), GW production via the dynamics of SU(2) and ultra-light U(1) axions, and cosmic string (CS) network collapse. We highlight how some regions of parameter space were already excluded with data from COBE/FIRAS , taken over two decades ago. To facilitate the implementation of SD constraints in arbitrary models we provide GW2SD. This tool calculates the window function, which easily maps a GW spectrum to a SD amplitude, thus opening another portal for GW cosmology with SDs, with wide reaching implications for particle physics phenomenology.
We compare the spectrum of the stochastic gravitational wave background produced in several models of cosmic strings with the common-spectrum process recently reported by NANOGrav. We discuss theoretical uncertainties in computing such a background, and show that despite such uncertainties, cosmic strings remain a good explanation for the potential signal, but the consequences for cosmic string parameters depend on the model. Superstrings could also explain the signal, but only in a restricted parameter space where their network behavior is effectively identical to that of ordinary cosmic strings.
Gravitational waves (GWs) induced by scalar curvature fluctuations are an important source of the cosmological GW background and a crucial counterpart of the primordial black hole scenario. However, doubts have been cast on the theoretically predicted induced GW spectrum due to its seeming gauge dependence. In this paper, we shed light on the gauge dependence issue of the induced GW spectrum in general cosmological backgrounds. First, inspired by the Hamiltonian formalism we provide very simple formulas for the tensor modes at second order in cosmological perturbation theory. We also emphasize the difference between observable and gauge invariant variables. Second, we argue that the Newton (or shear-free) gauge is suitable for both the calculation of induced GWs and the physical interpretation. We then show that, most notably, the induced GW spectrum is invariant under a set of reasonable gauge transformations, i.e., physically well behaved on small scales, once the source term has become inactive. This includes the commonly used flat, constant Hubble and synchronous gauges but excludes the comoving slicing gauge. We also show that a particular solution of the GW equation in a dust dominated universe while the source term is active can be gauged away by a small change of gauge.
We show that the recent NANOGrav result can be interpreted as a stochastic gravitational wave signal associated to formation of primordial black holes from high-amplitude curvature perturbations. The indicated amplitude and power of the gravitational wave spectrum agrees well with formation of primordial seeds for supermassive black holes.
Primordial black holes (PBHs) might be formed in the early Universe and could comprise at least a fraction of the dark matter. Using the recently released GWTC-2 dataset from the third observing run of the LIGO-Virgo Collaboration, we investigate whether current observations are compatible with the hypothesis that all black hole mergers detected so far are of primordial origin. We constrain PBH formation models within a hierarchical Bayesian inference framework based on deep learning techniques, finding best-fit values for distinctive features of these models, including the PBH initial mass function, the fraction of PBHs in dark matter, and the accretion efficiency. The presence of several spinning binaries in the GWTC-2 dataset favors a scenario in which PBHs accrete and spin up. Our results indicate that PBHs may comprise only a fraction smaller than 0.3% of the total dark matter, and that the predicted PBH abundance is still compatible with other constraints.
We study the general infrared behavior of the power spectrum of a stochastic gravitational wave background produced by stress tensor in the form bilinear in certain dynamical degrees of freedom. We find ΩGW∝k3 for a very wide class of the sources which satisfy a set of reasonable conditions. Namely, the k3 scaling is universally valid when the source term is bounded in both frequency and time, is effective in a radiation-dominated stage, and for k smaller than all the physical scales associated with the source, like the peak frequency, peak width, and time duration, etc. We also discuss possible violations of these conditions and their physical implications.
For primordial black holes (PBHs) to form a considerable fraction of cold dark matter, the required amplitude of primordial scalar perturbations is quite large (Pζ(k)∼10−2) if PBH is formed in radiation epoch. In alternate cosmological histories, where additional epoch of arbitrary equation of state precede radiation epoch, the dynamics of PBH formation and relevant mass ranges can be different leading to lower requirement of primordial power at smaller scales of inflation. Moreover, this alternate history can modify the predictions for the gravitational wave (GW) spectrum, which can be probed by upcoming GW observations. In this paper we show that an early kination epoch can lead to percent level abundance of PBH for a lower amplitude of Pζ(k) as compared to PBH formation in a standard radiation epoch. Moreover, we calculate the effect of early kination epoch on the GW spectrum for first and second orders in perturbation theory which show enhancement in the amplitude of the GW spectrum in a kination epoch with respect to that in a standard radiation epoch.
The detection of binary black hole coalescences by LIGO and Virgo has aroused the interest in primordial black holes (PBHs), because they could be both the progenitors of these black holes and a compelling candidate of dark matter (DM). PBHs are formed soon after the enhanced scalar perturbations reenter horizon during the radiation dominated era, which would inevitably induce gravitational waves as well. Searching for such scalar induced gravitational waves (SIGWs) provides an elegant way to probe PBHs. We perform the first direct search for the signals of SIGWs accompanying the formation of PBHs in the North American Nanohertz Observatory for Gravitational waves (NANOGrav) 11-year dataset. No statistically significant detection has been made, and hence we place a stringent upper limit on the abundance of PBHs at 95% confidence level. In particular, less than one part in a million of the total DM mass could come from PBHs in the mass range of [2×10−3,7×10−1] M⊙.
We study cosmological tensor perturbations induced by second-order scalar perturbations in the presence of anisotropic non-Gaussianity. This class of induced tensor modes arises on superhorizon scales through the anisotropic coupling between long modes and short modes. We show that scalar perturbations on the inflationary Hubble scale to the Silk damping scale at recombination contribute to the induced tensor powerspectrum at the cosmic microwave background (CMB) scale, and that the induced tensor spectrum becomes almost scale-invariant. The former property suggests that measurements of the CMB offer a test of tiny scale physics. However, the latter implies the secondary effect may contaminate the primordial tensor spectrum, which tells us the energy scale of inflation. We derive the induced tensor modes originated from two concrete examples of anisotropic non-Gaussianity; statistically anisotropic scalar non-Gaussianity and scalar-scalar-tensor non-Gaussianity, and discuss observational consequences of extremely short scale physics. Also, we comment on various possibilities of enhancing the induced spectrum with nonstandard early Universe physics.
The ultraslow-roll (USR) inflationary models predict large-amplitude scalar perturbations at small scales which can lead to the primordial black hole production and scalar-induced gravitational waves. In general, scalar perturbations in the USR models can only be obtained using a numerical method because the usual slow-roll approximation breaks. In this work, we propose an analytical approach to estimate the scalar spectrum which is consistent with the numerical result. We find that the USR inflationary models predict a peak with power-law slopes in the scalar spectrum and energy spectrum of gravitational waves, and we derive the expression of the spectral indexes in terms of the inflationary potential. In turn, the inflationary potential near the USR regime can be reconstructed from the negative spectral index of the gravitational wave energy spectrum.
We revisit and clarify the gauge dependence of gravitational waves generated at second order from scalar perturbations. In a universe dominated by a perfect fluid with a constant equation-of-state parameter w, we compute the energy density of such induced gravitational waves in the Newtonian, comoving, and uniform curvature gauges. Huge differences are found between the Newtonian and comoving gauge results for any w(≥0). This is always caused by the perturbation of the shift vector. Interestingly, the Newtonian and uniform curvature gauge calculations give the same energy density for w>0. In the case of w=0, the uniform curvature gauge result differs only by a factor from that of the comoving gauge, but it deviates significantly from that of the Newtonian gauge. Our calculation is done analytically for w=0 and w=1/3, and our result is consistent with the previous numerical one.
The Einstein Telescope (ET), a proposed European ground-based gravitational-wave detector of third-generation, is an evolution of second-generation detectors such as Advanced LIGO, Advanced Virgo, and KAGRA which could be operating in the mid 2030s. ET will explore the universe with gravitational waves up to cosmological distances. We discuss its main scientific objectives and its potential for discoveries in astrophysics, cosmology and fundamental physics.
In this paper we calculate the scalar induced gravitational waves (SIGWs) accompanying the formation of primordial black holes during the radiation dominated era in three different gauges, i.e., the synchronous gauge, Newton gauge, and uniform curvature gauge, and we find that the energy density spectra of SIGWs, ΩGW(k), are identical in these three different gauges.
We analytically calculate the scalar induced gravitational waves (SIGWs) and find a log-dependent slope of SIGW in the infrared regions (f<fc), namely nGW(f)=3−2/ln(fc/f), and nGW(f)=2−2/ln(fc/f) near the peak if the power spectrum of scalar curvature perturbation is quite narrow, where fc is roughly the peak frequency. Such a log-dependent slope can be taken as a new template for distinguishing SIGW from other sources.
Gravitational waves (GWs) are inevitably produced by second-order terms in cosmological perturbation theory. Most notably, the so-called induced (GWs) are a window to the small scales part of the primordial spectrum of fluctuations and a key counterpart to the primordial black hole (PBH) scenario. However, semi-analytical solutions are only known for matter and radiation domination eras. In this paper, we present new analytic integral formulas for the induced GWs on subhorizon scales in a general cosmological background with a constant equation-of-state. We also discuss applications to a peaked primordial scalar power spectrum and the PBH scenario.
We study gauge (in)dependence of the gravitational waves (GWs) induced from curvature perturbations. For the GWs produced in a radiation-dominated era, we find that the observable (late-time) GWs in the transverse-traceless (synchronous) gauge and in the Newtonian gauge are the same in contrast to a claim in the literature. We also mention the interpretation of the gauge dependence of the tensor perturbations which appears in the context of the induced GWs.
We study the production of gravitational waves from primordial scalar perturbations in double-inflection-point inflation, in which one of the inflection points leads to a primordial power spectrum consistent with CMB observations at large scales and the other generates a large peak in the power spectrum of scalar perturbations at small scales. We calculate the energy spectrum of the reduced gravitational waves and find that the gravitational-wave signal can be detected by future space-based laser interferometers.
We consider the steepest rate at which the power spectrum from single field inflation can grow, with the aim of providing a simple explanation for the k4 growth found recently. With this explanation in hand we show that a slightly steeper k5(log k)2 growth is, in fact, possible. Moreover, we argue that the power spectrum after a steep growth cannot immediately decay, but must remain large for the k modes that exit during a ∼2e-fold period. We also briefly consider how a strong growth can affect the spectral index of longer wavelengths preceding the growth and show that even the conversion of isocurvature modes likely cannot lead to a stronger growth. These results have implications for the formation of primordial black holes and other phenomena that require a large amplitude of power spectrum at short scales.
The possibility that primordial black holes (PBHs) represent all of the dark matter (DM) in the Universe and explain the coalescences of binary black holes detected by LIGO/Virgo has attracted a lot of attention. PBHs are generated by the enhancement of scalar perturbations which inevitably produce the induced gravitational waves (GWs). We calculate the induced GWs up to the third-order correction which not only enhances the amplitude of induced GWs, but also extends the cutoff frequency from 2k* to 3k*. Such effects of the third-order correction lead to an around 10% increase of the signal-to-noise ratio (SNR) for both LISA and pulsar timing array (PTA) observations, and significantly widen the mass range of PBHs in the stellar mass window accompanying detectable induced GWs for PTA observations including IPTA, FAST and SKA. On the other hand, the null detections of the induced GWs by LISA and PTA experiments will exclude the possibility that all of the DM is comprised of PBHs and the GW events detected by LIGO/Virgo are generated by PBHs.
We study gravitational waves induced from the primordial scalar perturbations at second order around the reheating of the Universe. We consider reheating scenarios in which a transition from an early matter-dominated era to the radiation-dominated era completes within a timescale much shorter than the Hubble time at that time. We find that an enhanced production of induced gravitational waves occurs just after the reheating transition because of fast oscillations of scalar modes well inside the Hubble horizon. This enhancement mechanism just after an early matter-dominated era is much more efficient than a previously known enhancement mechanism during an early matter era, and we show that the induced gravitational waves could be detectable by future observations if the reheating temperature TR is in the range TR≲7×10−2 GeV or 20 GeV≲TR≲2×107 GeV. This is the case even if the scalar perturbations on small scales are not enhanced relative to those on large scales, probed by the observations of the cosmic microwave background.
As potential candidates of dark matter, primordial black holes (PBHs) are within the core scopes of various astronomical observations. In light of the explosive development of gravitational wave (GW) and radio astronomy, we thoroughly analyze a stochastic background of cosmological GWs, induced by overly large primordial density perturbations, with several spikes that was inspired by the sound speed resonance effect and can predict a particular pattern on the mass spectrum of PBHs. With a specific mechanism for PBH formation, we for the first time perform the study of such induced GWs that originate from both the inflationary era and the radiation-dominated phase. We report that, besides the traditional process of generating GWs during the radiation-dominated phase, the contribution of the induced GWs in the sub-Hubble regime during inflation can become significant at the critical frequency band because of a narrow resonance effect. All contributions sum together to yield a specific profile of the energy spectrum of GWs that can be of observable interest in forthcoming astronomical experiments. Our study sheds light on the possible joint probe of PBHs via various observational windows of multimessenger astronomy, including the search for electromagnetic effects with astronomical telescopes and the stochastic background of relic GWs with GW instruments.
The idea that primordial black holes (PBHs) can comprise most of the dark matter of the Universe has recently reacquired a lot of momentum. Observational constraints, however, rule out this possibility for most of the PBH masses, with a notable exception around 10−12 M⊙. These light PBHs may be originated when a sizable comoving curvature perturbation generated during inflation reenters the horizon during the radiation phase. During such a stage, it is unavoidable that gravitational waves (GWs) are generated. Since their source is quadratic in the curvature perturbations, these GWs are generated fully non-Gaussian. Their frequency today is about a millihertz, which is exactly the range where the LISA mission has the maximum of its sensitivity. This is certainly an impressive coincidence. We show that this scenario of PBHs as dark matter can be tested by LISA by measuring the GW two-point correlator. On the other hand, we show that the short observation time (as compared to the age of the Universe) and propagation effects of the GWs across the perturbed Universe from the production point to the LISA detector suppress the bispectrum to an unobservable level. This suppression is completely general and not specific to our model.
We study gravitational waves (GWs) induced by non-Gaussian curvature perturbations. We calculate the density parameter per logarithmic frequency interval, ΩGW(k), given that the power spectrum of the curvature perturbation PR(k) has a narrow peak at some small scale k*, with a local-type non-Gaussianity, and constrain the nonlinear parameter fNL with the future LISA sensitivity curve as well as with constraints from the abundance of the primordial black holes (PBHs). We find that the non-Gaussian contribution to ΩGW increases as k3, peaks at k/k*=4/3, and has a sharp cutoff at k=4k*. The non-Gaussian part can exceed the Gaussian part if PR(k)fNL2≳1. If both a slope ΩGW(k)∝kβ with β∼3 and the multiple-peak structure around a cutoff are observed, it can be recognized as a smoking gun of the primordial non-Gaussianity. We also find that if PBHs with masses of 1020 to 1022 g are identified as cold dark matter of the Universe, the corresponding GWs must be detectable by LISA-like detectors, irrespective of the value of PR or fNL.