Curvature Power Spectrum Research Articles

Thermalization of large energy release in the early Universe

ABSTRACT Spectral distortions of the cosmic microwave background (CMB) provide a unique tool for learning about the early phases of cosmic history, reaching deep into the primordial Universe. At redshifts z ≲ 106, thermalization processes become inefficient and existing limits from COBE/FIRAS imply that no more than Δρ/ρ ≲ 6 × 10−5 ($95{{\ \rm per\ cent}}$ c.l.) of energy could have been injected into the CMB. However, at higher redshifts, when thermalization is efficient, the constraint weakens and Δρ/ρ ≃ 0.01−0.1 could in principle have occurred. Existing computations for the evolution of distortions commonly assume Δρ/ρ ≪ 1 and thus become inaccurate in this case. Similarly, relativistic temperature corrections become relevant for large energy release, but have previously not been modelled as carefully. Here, we study the evolution of distortions and the thermalization process after single large energy release at z ≳ 105. We show that for large distortions the thermalization efficiency is significantly reduced and that the distortion visibility is sizeable to much earlier times. This tightens spectral distortions constraints on low-mass primordial black holes with masses $M_{\rm PBH}\lesssim 2 \times 10^{11}\, {\rm g}$. Similarly, distortion limits on the amplitude of the small-scale curvature power spectrum at wavenumbers $k\gtrsim 10^4\, {\rm Mpc}^{-1}$ and short-lived decaying particles with lifetimes $t_X\lesssim 10^7\, {\rm s}$ are tightened, however, these still require a more detailed time-dependent treatment. We also briefly discuss the constraints from measurements of the effective number of relativistic degrees of freedom and light element abundances and how these complement spectral distortion limits.

Curvature perturbations from stochastic particle production during inflation

We calculate the curvature power spectrum sourced by spectator fields that are excited repeatedly and non-adiabatically during inflation. In the absence of detailed information of the nature of spectator field interactions, we consider an ensemble of models with intervals between the repeated interactions and interaction strengths drawn from simple probabilistic distributions. We show that the curvature power spectra of each member of the ensemble shows rich structure with many features, and there is a large variability between different realizations of the same ensemble. Such features can be probed by the cosmic microwave background (CMB) and large scale structure observations. They can also have implications for primordial black hole formation and CMB spectral distortions. The geometric random walk behavior of the spectator field allows us to calculate the ensemble-averaged power spectrum of curvature perturbations semi-analytically. For sufficiently large stochastic sourcing, the ensemble-averaged power spectrum shows a scale dependence arising from the time spent by modes outside the horizon during the period of particle production, in spite of there being no preferred scale in the underlying model. We find that the magnitude of the ensemble-averaged power spectrum overestimates the typical power spectra in the ensemble because the ensemble distribution of the power spectra is highly non-Gaussian with fat tails.

Generation of primordial black holes and gravitational waves from dilaton-gauge field dynamics

We study the observational signatures from particle production of a U(1) gauge field kinetically coupled to an inflaton. Regarding the form of gauge kinetic function, we consider the possibility that it becomes stabilized at a certain time, which makes the growing power of the gauge field evolve non-monotonically with a sharp transition. Remarkably, the copious production of the gauge field occurs on super-horizon scales at the late stage of inflation and perturbations are enhanced on the intermediate scales during inflation. We find that it can predict a bumpy shape of the curvature power spectrum which leads to the generation of primordial black holes as a dark matter after inflation. We also estimate two types of tensor modes sourced by the gauge field: the primordial gravitational waves generated during inflation and the induced gravitational waves provided by the enhanced curvature perturbation after inflation. We show that both of them are potentially testable with the future space-based gravitational wave interferometers.

Modified initial power spectrum and too big to fail problem

ABSTRACT The galactic scale challenges of dark matter such as ‘missing satellite’ problem and ‘too big to fail’ problem are the main caveats of standard model of cosmology. These challenges could be solved either by implementing the complicated baryonic physics or it could be considered as an indication to a new physics beyond the standard model of cosmology. The modification of collisionless dark matter models or the standard initial conditions are two promising venues for study. In this work, we investigate the effects of the deviations from scale invariant initial curvature power spectrum on number density of dark matter haloes. We develop the non-Markov extension of the excursion set theory to calculate the number density of dark matter substructures and dark matter halo progenitor mass distribution. We show that the plausible solution to ‘too big to fail’ problem could be obtained by a Gaussian excess in initial power in the scales of k* ∼ 3 h Mpc−1 that is related to the mass scale of M* ∼ 1011 M⊙. We show that this deviation leads to the decrement of dark matter subhaloes in galactic scale, which is consistent with the current status of the non-linear power spectrum. Our proposal also has a prediction that the number density of Milky Way-type galaxies must be higher than the standard case.

The 21-cm signals from ultracompact minihaloes as a probe of primordial small-scale fluctuations

ABSTRACT Ultracompact minihaloes (UCMHs) can form after the epoch of matter–radiation equality, if the density fluctuations of dark matter have significantly large amplitude on small scales. The constraint on the UCMH abundance allows us to access such small-scale fluctuations. In this paper, we present that, through the measurement of 21-cm fluctuations before the epoch of reionization, we can obtain a constraint on the UCMH abundance. We calculate the 21-cm signal from UCMHs and show that UCMHs provide the enhancement of the 21-cm fluctuations. We also investigate the constraint on the UCMH abundance and small-scale curvature perturbations. Our results indicate that the upcoming 21-cm observation, the Square Kilometre Array (SKA), provides the constraint on amplitude of primordial curvature power spectrum, ${\cal A}_{\zeta } \lesssim 10^{-6}$ on 100 ≲ k ≲ 1000 Mpc−1. Although it is not stronger than the one from the non-detection of gamma-rays induced by dark matter annihilation in UCMHs, the constraint by the SKA will be important because this constraint is independent of the dark matter particle model.

Primordial black holes from a tiny bump/dip in the inflaton potential

Scalar perturbations during inflation can be substantially amplified by tiny features in the inflaton potential. A bump-like feature behaves like a local speed-breaker and lowers the speed of the scalar field, thereby locally enhancing the scalar power spectrum. A bump-like feature emerges naturally if the base inflaton potential Vb(ϕ) contains a local correction term such as Vb(ϕ)[1+ε(ϕ)] at ϕ=ϕ0. The presence of such a localised correction term at ϕ0 leads to a large peak in the curvature power spectrum and to an enhanced probability of black hole formation. Remarkably this does not significantly affect the scalar spectral index nS and tensor to scalar ratio r on CMB scales. Consequently such models can produce higher mass primordial black holes (MPBH⩾ 1 M⊙) in contrast to models with `near inflection-point potentials' in which generating higher mass black holes severely affects nS and r. With a suitable choice of the base potential—such as the string theory based (KKLT) inflation or the α-attractor models—the amplification of primordial scalar power spectrum can be as large as 107 which leads to a significant contribution of primordial black holes (PBHs) to the dark matter density today, fPBH = Ω0, PBH/Ω0,DM ∼ O(1). Interestingly, our results remain valid if the bump is replaced by a dip. In this case the base inflaton potential Vb(ϕ) contains a negative local correction term such as Vb(ϕ)[1−ε(ϕ)] at ϕ=ϕ0 which leads to an enhanced probability of PBH formation. We conclude that primordial black holes in the mass range 10−17 M⊙ ⩽ MPBH ⩽ 100 M⊙ can easily form in single field inflation in the presence of small bump-like and dip-like features in the inflaton potential.

On the slope of the curvature power spectrum in non-attractor inflation

The possibility that primordial black holes constitute a fraction of dark matter motivates a detailed study of possible mechanisms for their production. Black holes can form by the collapse of primordial curvature fluctuations, if the amplitude of their small scale spectrum gets amplified by several orders of magnitude with respect to CMB scales. Such enhancement can for example occur in single-field inflation that exhibit a transient non-attractor phase: in this work, we make a detailed investigation of the shape of the curvature spectrum in this scenario. We make use of an analytical approach based on a gradient expansion of curvature perturbations, which allows us to follow the changes in slope of the spectrum during its path from large to small scales. After encountering a dip in its amplitude, the spectrum can acquire steep slopes with a spectral index up to ns−1 = 8, to then relax to a more gentle growth with ns −1 ≲ 3 towards its peak, in agreement with the results found in previous literature. For scales following the peak associated with the presence of non-attractor phase, the spectrum amplitude then mildly decays, during a transitional stage from non-attractor back to attractor evolution. Our analysis indicates that this gradient approach offers a transparent understanding of the contributions controlling the slope of the curvature spectrum. As an application of our findings, we characterise the slope in frequency of a stochastic gravitational wave background generated at second order from curvature fluctuations, using the more accurate information we gained on the shape of curvature power spectrum.

Constant roll and primordial black holes

The constant-roll inflation with small positive value of the constant-roll parameter β≡ (ϕ̇)/(Hϕ̇)=const. has been known to produce a slightly red-tilted curvature power spectrum compatible with the current observational constraints. In this work, we shed light on the constant-roll inflation with negative β and investigate how a stage of constant-roll inflation may realize the growth in the primordial curvature power spectrum necessary to produce a peaked spectrum of primordial black hole abundance. We first review the behavior of constant-roll models in the range of parameters −32<β<0, which allows for a constant-roll attractor stage generating a blue-tilted curvature power spectrum without superhorizon growth. As a concrete realization, we consider a potential with two slow-roll stages, separated by the constant-roll stage, in a way that satisfies the current constraints on the power spectrum and the primordial black hole abundance. The model can produce primordial black holes as all dark matter, LIGO-Virgo events, or OGLE microlensing events. Due to the range of different scalar tilts allowed by the constant-roll potential, this construction is particularly robust and testable by future observations.

Thermal history of the early Universe and primordial gravitational waves from induced scalar perturbations

We study the induced primordial gravitational waves (GW) 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 bakcground. Our result shows that the spectrum of the induced GW is affected differently by the standard model degrees of freedom than the GW coming from 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 one at small scales or a non-Gaussian curvature power spectrum this effect can be observed by gravitational wave observatories.

Primordial black hole formation in inflationary α -attractor models

The formation of primordial black holes is studied using superconformal inflationary $\alpha$-attractors. An inflaton potential is constructed with a plateau-like region which brings about the onset of an ultra slow-roll region where the required enhancement in the curvature power spectrum takes place. This is accomplished by carefully fine-tuning the parameters in the potential. For the parameter sets, the amount of observable inflation is such that the curvature perturbation $\mathcal{P}_{\mathcal{R}}$ peaks at $\sim 10^{14}\text{Mpc}^{-1}$, producing PBHs in the mass window $10^{16}\text{g}\leq M \leq 10^{18}\text{g}$. The reheating period after inflation is taken into account to determine whether or not PBH formation takes place in such a phase. Finally, the spectrum of second order gravitational waves, that can arise due to the curvature perturbation enhancement, is calculated.

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

In the model where Primordial Black Holes (PBHs) form from large primordial curvature (C) perturbations, i.e., CPBHs, constraints on PBH abundance provide in principle constraints on the primordial curvature power spectrum. This connection however depends necessarily on the details of PBH formation mechanism. In this paper we provide, for the first time, constraints on the primordial curvature power spectrum from the latest limits on PBH abundance, taking into account all the steps from gravitational collapse in real space to PBH formation. In particular, we use results from numerical relativity simulations and peak theory to study the conditions for PBH formation for a range of perturbation shapes, including non-linearities, perturbation profile and a careful treatment of smoothing and filtering scales. We then obtain updated PBH formation conditions and translate that into primordial spectrum constraints for a wide range of shapes and abundances. These updated constraints cover a range of scales not probed by other cosmological observables. Our results show that the correct and accurate modelling of non-linearities, filtering and typical perturbation profile, is crucial for deriving meaningful cosmological implications.

Constraints on the primordial curvature power spectrum from primordial black holes

Large-amplitude density perturbations may have collapsed during the radiation dominated epoch of the Universe to form primordial black holes (PBHs). There are several constraints to the abundance of PBHs that stem from evaporation or gravitational effects. Due to the connection between primordial perturbations and the formation of PBHs, constraints on the present-day abundance of PBHs can be translated into limits on the primordial curvature power spectrum. We introduce several new observational and forecasted constraints to the amplitude of the primordial power spectrum and incorporate in our analysis uncertainties in the critical overdensity for collapse and considerations of ellipsoidal collapse. Our results provide the most stringent limits from PBHs on the primordial curvature power spectrum on small scales.

Constraints on the curvature power spectrum from primordial black hole evaporation

We estimate the maximum allowed amplitude for the power spectrum of the primordial curvature perturbations, \U0001d4abR(k), on all scales from the absence of any detection signal of sub-solar mass black holes. In particular we analyze the constraints on the PBHs and we focus on the low mass limit where the Hawking radiation is expected to significantly influence the big bang observables, considering also different early cosmic histories. We derive the upper bounds for the variance of density perturbations, σ(M), for any possible reheating temperature as well as for the cosmological scenario of a scalar condensate domination. We expect our results to have considerable implications for models designed to generate PBHs, especially in the low mass range, and provide additional constraints to a large class of inflationary models.

Quintessential isocurvature in separate universe simulations

In a universe with quintessence isocurvature, or perturbations in dark energy that are independent from the usual curvature perturbations, structure formation is changed qualitatively. The existence of two independent fields, curvature and isocurvature, causes the growth rate of matter perturbations to depend on their initial conditions. The quintessence perturbations cause their growth to depend on scale. We perform the first separate universe simulations for this cosmology. We demonstrate that the power spectrum response and the halo bias depend on scale and initial conditions and that the presence of the isocurvature mode changes the mapping from these quantities to the halo auto- and cross-power spectra, and the squeezed-limit bispectrum. We compare the bias to several models, finding reasonable agreement with both a power-spectrum-response model with one free parameter and a model that fits two independent bias parameters for curvature and isocurvature sourced fluctuations. We also verify that simulation responses to pure isocurvature and pure curvature modes can be linearly combined to reproduce responses with different ratios of isocurvature and curvature. This allows our results to be used to predict the halo power spectrum and stochasticity with arbitrary large-scale curvature and isocurvature power spectra. In an appendix, we study the generation of quintessence isocurvature during inflation and show that a modified kinetic term is typically required to produce observable isocurvature modes in a field with $w_Q\approx -1$.

Trans-Planckian effects in warm inflation

We study the effect of a non-trivial vacuum prescription on warm inflation observables, namely the power spectrum of the comoving curvature perturbation. Non-trivial choice of vacuum, can provide information about trans-Planckian physics. Traditionally, the initial condition for inflation is chosen to be the usual Bunch-Davies vacuum. Another more reasonable choice of vacuum is the so-called α-vacua. Because the duration of inflation is not infinite, as it is assumed in the Bunch-Davies case, imposing the initial condition at infinite past is not sensible and one must utilize another vacuum prescription. In this paper, working in the slow-roll regime during warm inflation, the initial condition for inflaton fluctuations is imposed at finite past, i.e. the α-vacua. We show that this non-trivial vacuum prescription results in oscillatory correction to the comoving curvature power spectrum, which is scale dependent both in amplitude and frequency. Having obtained this scale dependent power spectrum, we consider its late time footprints and compare our results with observational data and other proposed models for the comoving curvature power spectrum.

Observational constraints on two-field warm inflation

We study the two-field warm inflation models with a double quadratic potential and a linear temperature dependent dissipative coefficient. We derived the evolution equation of all kinds of perturbations without assuming slow-roll approximation, and obtained the curvature power spectrum at the end of inflation with a fully numerical method. Then we compute the scalar spectral index $n_s$, tensor-to-scalar ratio $r$ for several representative potentials, and compare our results with observational data. At last, we use Planck data to constrain the parameters in our models. This work is a natural extension of single-field warm inflation, and the aim of this work is to present some features of multi-field warm inflation using a simple two-field model.

Higher-order modified Starobinsky inflation

An extension of the Starobinsky model is proposed. Besides the usual Starobinsky Lagrangian, a term proportional to the derivative of the scalar curvature, ∇μR∇μR, is considered. The analyzis is done in the Einstein frame with the introduction of a scalar field and a vector field. We show that inflation is attainable in our model, allowing for a graceful exit. We also build the cosmological perturbations and obtain the leading-order curvature power spectrum, scalar and tensor tilts and tensor-to-scalar ratio. The tensor and curvature power spectrums are compared to the most recent observations from BICEP2/Keck collaboration. We verify that the scalar-to-tensor rate r can be expected to be up to three times the values predicted by Starobinsky model.

Primordial black holes from α-attractors

We consider primordial black hole (PBH) production in inflationary α-attractors. We discuss two classes of models, namely models with a minimal polynomial superpotential as well as modulated chaotic ones that admit PBHs. We find that a significant amplification of the curvature power spectrum \U0001d4abR can be realized in this class of models with a moderate tuning of the potential parameters. We consistently examine the PBH formation during radiation and additionally during reheating eras where the background pressure is negligible. It is shown that basic features of the curvature power spectrum are explicitly related with the postinflationary cosmic evolution and that the PBH mass and abundance expressions are accordingly modified. PBHs in the mass range 10−16−10−14 M⊙ can form with a cosmologically relevant abundance for a power spectrum peak \U0001d4abR ∼ 10−2 and large reheating temperature and, furthermore, for a moderate peak \U0001d4abR ∼ 10−5 and reheating temperature Trh∼ 107 GeV, characteristic of the position of the power spectrum peak. Regarding the CMB observables, the α-attractor models utilized here to generate PBH in the low-mass region predict in general a smaller ns and larger r and αs parameter values compared to the conventional inflationary α-attractor models.

N-body simulations of structure formation in thermal inflation cosmologies

Thermal inflation models (which feature two inflationary stages) can display damped primordial curvature power spectra on small scales which generate damped matter fluctuations. For a reasonable choice of parameters, thermal inflation models naturally predict a suppression of the matter power spectrum on galactic and sub-galactic scales, mimicking the effect of warm or interacting dark matter. Matter power spectra in these models are also characterised by an excess of power (with respect to the standard ΛCDM power spectrum) just below the suppression scale. By running a suite of N-body simulations we investigate the non-linear growth of structure in models of thermal inflation. We measure the non-linear matter power spectrum and extract halo statistics, such as the halo mass function, and compare these quantities with those predicted in the standard ΛCDM model and in other models with damped matter fluctuations. We find that the thermal inflation models considered here produce measurable differences in the matter power spectrum from ΛCDM at redshifts z>5 for wavenumbers k∊[2,64] h Mpc−1, while the halo mass functions are appreciably different at all redshifts in the halo mass range Mhalo∊ [109,1012] h−1 M⊙ resolved by our simulations. The halo mass function at z=0 for thermal inflation displays an enhancement of around ∼ 20% with respect to ΛCDM and a damping at lower halo masses, with the position of the enhancement depending on the value of the free parameter in the model. The enhancement in the halo mass function (with respect to ΛCDM) increases with redshift, reaching ∼ 40% at z=5. We also study the accuracy of the analytical Press-Schechter approach, using different filters to smooth the density field, to predict halo statistics for thermal inflation. We find that the predictions with the smooth-k filter we proposed in a separate paper agree with the simulation results over a wider range of halo masses than is the case with other filters commonly used in the literature.

Primordial black holes and second order gravitational waves from ultra-slow-roll inflation

The next generation of space-borne gravitational wave detectors may detect gravitational waves from extreme mass-ratio inspirals with primordial black holes. To produce primordial black holes which contribute a non-negligible abundance of dark matter and are consistent with the observations, a large enhancement in the primordial curvature power spectrum is needed. For a single field slow-roll inflation, the enhancement requires a very flat potential for the inflaton, and this will increase the number of e-folds. To avoid the problem, an ultra-slow-roll inflation at the near inflection point is required. We elaborate the conditions to successfully produce primordial black hole dark matter from single field inflation and propose a toy model with polynomial potential to realize the big enhancement of the curvature power spectrum at small scales while maintaining the consistency with the observations at large scales. The power spectrum for the second order gravitational waves generated by the large density perturbations at small scales is consistent with the current pulsar timing array observations.