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Phase transitions in the early universe lead to a reduction in the equation of state of the primordial plasma. This exponentially enhances the formation rate of primordial black holes. However, this sensitivity to the equation of state is the same that primordial black hole abundances show to the primordial curvature power spectrum amplitude. In this paper, we investigate peaked power spectra and show the challenges associated with motivating populations of primordial black holes with standard model enhancements. The parametrisation of different power spectra plays an important role in this discussion. The allowed parameter space consistent with a large QCD phase transition impact on the primordial black hole abundance differs greatly. This is particularly evident for wider spectra., Comment: 18 pages plus appendices. 13 figures

A well-known route to form primordial black holes in the early universe relies on the existence of unusually large primordial curvature fluctuations, confined to a narrow range of wavelengths that would be too small to be constrained by Cosmic Microwave Background (CMB) anisotropies. This scenario would however boost the generation of $\mu$-type spectral distortions in the CMB due to an enhanced dissipation of acoustic waves. Previous studies of $\mu$-distortion bounds on the primordial spectrum were based on the assumptions of Gaussian primordial fluctuations. In this work, we push the calculation of $\mu$-distortions to one higher order in photon anisotropies. We discuss how to derive bounds on primordial spectrum peaks obeying non-Gaussian statistics under the assumption of local (perturbative or not) non-Gaussianity. We find that, depending on the value of the peak scale, the bounds may either remain stable or get tighter by several orders of magnitude, but only when the departure from Gaussian statistics is very strong. Our results are translated in terms of bounds on primordial supermassive black hole mass in a companion paper., Comment: 48 pages, 11 figures, minor typo is corrected

Explaining the origin of supermassive black holes via a primordial origin is severely challenged by the tight spectral distortion constraints on the amplitude of the primordial perturbations. Following the first calculation of how the $\mu$ constraints are modified by non-Gaussianity in a companion paper, we here make the first robust constraints on primordial black hole formation under large non-Gaussianity. Even the infinite $f_{\rm NL}$ limit is insufficiently non-Gaussian but much higher-order non-Gaussianity of the form ${\cal R}={\cal R}_{\rm G}^5$ may allow the formation of any mass primordial black hole without conflicting with distortion constraints. We caution that such extreme models face other challenges., Comment: 24 pages, 6 figures. v2: Minor typos corrected and references added to match the published version

All single-field inflationary models invoke varying degrees of tuning in order to account for cosmological observations. Mechanisms that generate primordial black holes (PBHs) from enhancement of primordial power at small scales posit inflationary potentials that transiently break scale invariance and possibly adiabaticity over a range of modes. This requires additional tuning on top of that required to account for observations at scales probed by cosmic microwave background (CMB) anisotropies. In this paper we study the parametric dependence of various single-field models of inflation that enhance power at small scales and quantify the degree to which coefficients in the model construction have to be tuned in order for certain observables to lie within specified ranges. We find significant tuning: changing the parameters of the potentials by between one part in a hundred and one part in $10^8$ (depending on the model) is enough to change the power spectrum peak amplitude by an order one factor. The fine-tuning of the PBH abundance is larger still by 1-2 orders of magnitude. We highlight the challenges imposed by this tuning on any given model construction. Furthermore, polynomial potentials appear to require significant additional fine-tuning to also match the CMB observations., Comment: 16 pages + appendices, 5 figures

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., Comment: 13 pages, 8 figures. Matches published version

These lecture notes are based on those presented at the Theoretical Aspects of Astroparticle Physics, Cosmology and Gravitation School at the Galileo Galilee Institute in Florence in 2021, https://agenda.infn.it/event/24368/. They aim to provide a pedagogical introduction and basic working knowledge of single-field inflation including the ultra-slow-roll regime, where the perturbations grow exponentially. This rapid growth is connected to the formation of primordial black holes (PBHs), a special dark matter candidate and probe of the initial conditions of the early universe. Although there are many textbooks and introductory texts about inflation, to the best of our knowledge there is no comparable introduction to ultra-slow-roll inflation. Furthermore, given their recent surge in popularity, there are numerous research articles and reviews on primordial black holes, however these notes aim to be more accessible for graduate students and those brand new to the topic. Some problems and solutions to primordial black hole-related calculations are also included. The reader of these lecture notes should come away being able to calculate the present-day abundance of primordial black holes produced from the density fluctuations left over at the end of single-field inflation with an ultra-slow-roll phase, and understand how this abundance compares with current observational constraints., Comment: 47 pages, 5 figures. Based on a short lecture course held at the GGI during March 2021

We examine the shape of the primordial black hole mass distribution arising from a peak in the primordial power spectrum. In light of improvements to the modelling, we revisit the claim that the effects of critical collapse produce a distribution that is not described by the commonly assumed lognormal, showing that this conclusion remains valid, particularly for narrow peaks where the shape of the mass distribution is insensitive to the peak properties and critical collapse determines a minimum width. We propose some alternative models that may better describe the shape, both for the narrow peak case and for much broader peaks where the effect of the peak shape is significant. We highlight the skew-lognormal and a generalised model motivated by the physics of critical collapse as the best of these possible alternatives. These models can be used as an accurate and fast approximation to the numerically calculated mass distribution, allowing for efficient implementation in an MCMC analysis. We advocate the use of one of these two models instead of the lognormal with sufficiently accurate data, such as future LIGO-Virgo observations, or when considering strongly mass dependent constraints on the PBH abundance., Comment: 15 pages + appendices, 5 figures; v3: additional discussion of mass distribution insensitivity to narrow peak shape, matches published version

We conduct a thorough Bayesian analysis of the possibility that the black hole merger events seen in gravitational waves are primordial black hole (PBH) mergers. Using the latest merger rate models for PBH binaries drawn from a lognormal mass function we compute posterior parameter constraints and Bayesian evidences using data from the first two observing runs of LIGO-Virgo. We account for theoretical uncertainty due to possible disruption of the binary by surrounding PBHs, which can suppress the merger rate significantly. We also consider simple astrophysically motivated models and find that these are favoured decisively over the PBH scenario, quantified by the Bayesian evidence ratio. Paying careful attention to the influence of the parameter priors and the quality of the model fits, we show that the evidence ratios can be understood by comparing the predicted chirp mass distribution to that of the data. We identify the posterior predictive distribution of chirp mass as a vital tool for discriminating between models. A model in which all mergers are PBH binaries is strongly disfavoured compared with astrophysical models, in part due to the over-prediction of heavy systems having $\mathcal{M}_{{\rm chirp}} \gtrsim 40 \, M_\odot$ and positive skewness over the range of observed masses which does not match the observations. We find that the fit is not significantly improved by adding a maximum mass cut-off, a bimodal mass function, or imposing that PBH binaries form at late times. We argue that a successful PBH model must either modify the lognormal shape of the initial mass function significantly or abandon the hypothesis that all observed merging binaries are primordial. We develop and apply techniques for analysing PBH models with gravitational wave data which will be necessary for robust statistical inference as the gravitational wave source sample size increases., Comment: 29+8 pages, 18 figures. Minor edits to match version published in Physical Review D

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 $\sim$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 $\zeta$ and $\delta$, 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 the LIGO and Virgo, but that the $\mu$-distortion constraints rule out supermassive black hole (SMBH) formation and potentially even the much lighter mass range of $\sim$(1-100) $\mathrm{M}_\odot$ that LIGO/Virgo probes., Comment: 24 pages + Appendices, 8 figures; Matches published version

We consider a disformal coupling between Standard Model matter and a cubic Galileon scalar sector, assumed to be a relict of some other physics that solves the cosmological constant problem rather than a solution in its own right. This allows the energy density carried by the Galileon scalar to be sufficiently small that it evades stringent constraints from the integrated Sachs-Wolfe effect, which otherwise rules out the cubic Galileon theory. Although the model with disformal coupling does not exhibit screening, we show there is a `screening-like' phenomenon in which the energy density carried by the Galileon scalar is suppressed during matter domination when the quadratic and cubic Galileon operators are both relevant and the quadratic sector has a stable kinetic term. We obtain the explicit 3+1 form of Maxwell's equations in the presence of the disformal coupling, and the wave equations that govern electromagnetic waves. The disformal coupling is known to generate a small mass that modifies their velocity of propagation. We use the WKB approximation to study electromagnetic waves in this theory and show that, despite remarkable recent constraints from the LIGO/Virgo observatories that restrict the difference in propagation velocity between electromagnetic and gravitational radiation to roughly 1 part in $10^{15}$, the disformal coupling is too weak to be constrained by events such as GW170817 or by the dispersion of electromagnetic radiation at different wavelengths., Comment: 18 pages, 1 figure. v2: Stability analysis added. This is an author-created, un-copyedited version of an article published in JCAP. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at 10.1088/1475-7516/2021/10/085

It is commonly assumed that the energy density of the Universe was dominated by radiation between reheating after inflation and the onset of matter domination 54,000 years later. While the abundance of light elements indicates that the Universe was radiation dominated during Big Bang Nucleosynthesis (BBN), there is scant evidence that the Universe was radiation dominated prior to BBN. It is therefore possible that the cosmological history was more complicated, with deviations from the standard radiation domination during the earliest epochs. Indeed, several interesting proposals regarding various topics such as the generation of dark matter, matter-antimatter asymmetry, gravitational waves, primordial black holes, or microhalos during a nonstandard expansion phase have been recently made. In this paper, we review various possible causes and consequences of deviations from radiation domination in the early Universe - taking place either before or after BBN - and the constraints on them, as they have been discussed in the literature during the recent years., Comment: 67 pages, 18 figures. v2: Discussion and references added. Accepted for publication in The Open Journal of Astrophysics

We have calculated the detectable merger rate of primordial black holes, as a function of the redshift, as well as the binary's mass ratio, total mass and chirp mass (observables that have not previously been explored in great detail for PBHs). We consider both the current and design sensitivity of LIGO and five different primordial black hole mass functions, as well as showing a comparison to a predicted astrophysical black hole merger rate. We show that the empirical preference for nearly equal-mass binaries in current LIGO/Virgo data can be consistent with a PBH hypothesis once observational selection effects are taken into account. However, current data do exclude some PBH mass distributions, and future data may be able to rule out the possibility that all observed BH mergers had a primordial origin., Comment: 23 pages + appendices, 17 figures; v2: Minor changes, clarification of probabilities in table 1 supersedes published version

If the primordial curvature perturbation followed a Gaussian distribution, primordial black holes (PBHs) will be Poisson distributed with no additional clustering. We consider local non-Gaussianity and its impact on the initial PBH clustering and mass function due to mode coupling between long and short wavelength modes. We show that even a small amount of non-Gaussianity results in a significant enhancement on the PBH initial clustering and subsequent merger rate and that the PBH mass function shifts to higher mass PBHs. However, as the clustering becomes strong, the local number density of PBHs becomes large, leading to a large theoretical uncertainty in the merger rate., Comment: 22 pages, 6 figures

The formation and abundance of primordial black holes (PBHs) arising from the curvature perturbation $\zeta$ is studied. The non-linear relation between $\zeta$ and the density contrast $\delta$ means that, even when $\zeta$ has an exactly Gaussian distribution, significant non-Gaussianities affecting PBH formation must be considered. Numerical simulations are used to investigate the critical value and the mass of PBHs which form, and peaks theory is used to calculate the mass fraction of the universe collapsing to form PBHs at the time of formation. A formalism to calculate the total present day PBH abundance and mass function is also derived. It is found that the abundance of PBHs is very sensitive to the non-linear effects, and that the power spectrum $\mathcal{P}_\zeta$ must be a factor of $\sim2$ larger to produce the same number of PBHs as the linear model (where the exact value depends on the critical value for a region to collapse and form a PBH). This also means that the derived constraints on the small-scale power spectrum from constraints on the abundance of PBHs are weaker by the same factor., Comment: 29 pages, 7 figures. V2: updated to match published version

The expansion of the Universe is understood to have accelerated during two epochs: in its very first moments during a period of Inflation and much more recently, at $z < 1$, when Dark Energy is hypothesized to drive cosmic acceleration. The undiscovered mechanisms behind these two epochs represent some of the most important open problems in fundamental physics. The large cosmological volume at $2 < z < 5$, together with the ability to efficiently target high-$z$ galaxies with known techniques, enables large gains in the study of Inflation and Dark Energy. A future spectroscopic survey can test the Gaussianity of the initial conditions up to a factor of ~50 better than our current bounds, crossing the crucial theoretical threshold of $\sigma(f_{NL}^{\rm local})$ of order unity that separates single field and multi-field models. Simultaneously, it can measure the fraction of Dark Energy at the percent level up to $z = 5$, thus serving as an unprecedented test of the standard model and opening up a tremendous discovery space., Comment: Science white paper submitted to the Astro2020 Decadal Survey

We recently showed that postulated ultracompact minihalos with a steep density profile do not form in realistic simulations with enhanced initial perturbations. In this paper we assume that a small fraction of the dark matter consists of primordial black holes (PBHs) and simulate the formation of structures around them. We find that in this scenario halos with steep density profiles do form, consistent with theoretical predictions. If the rest of the dark matter consists of weakly interacting massive particles (WIMPs), we also show that WIMPs in the dense innermost part of these halos would annihilate and produce a detectable gamma-ray signal. The non-detection of this signal implies that PBHs make up at most one billionth of the dark matter, provided that their mass is greater than one millionth of the mass of the Sun. Similarly, a detection of PBHs would imply that the remaining dark matter could not be WIMPs., Comment: 10 pages, 8 figures. v2: Minor changes to match the published version

We derive analytic bounds on the shape of the primordial power spectrum in the context of single-field inflation. In particular, the steepest possible growth has a spectral index of $n_s - 1 = 4$ once transients have died down. Its primary implication is that any constraint on the power spectrum at a particular scale can be extrapolated to an upper bound over an extended range of scales. This is important for models which generate relics due to an enhanced amplitude of the primordial scalar perturbations, such as primordial black holes. In order to generate them, the power spectrum needs to grow many orders of magnitude larger than its observed value on CMB scales - typically achieved through a phase of ultra slow-roll inflation - and is thus subject to additional constraints at small scales. We plot all relevant constraints including CMB spectral distortions and gravitational waves sourced by scalar perturbations at second order. We show how this limits the allowed mass of PBHs, especially for the large masses of interest following recent detections by LIGO and prospects for constraining them further with future observations. We show that any transition from approximately constant $\epsilon$ 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 $\epsilon$, offering a complementary perspective on how ultra slow-roll can be achieved., Comment: 18 pages + appendices, 12 figures. v2: typos corrected and references added. v3: minor changes, matches published version

The grand challenges of contemporary fundamental physics---dark matter, dark energy, vacuum energy, inflation and early universe cosmology, singularities and the hierarchy problem---all involve gravity as a key component. And of all gravitational phenomena, black holes stand out in their elegant simplicity, while harbouring some of the most remarkable predictions of General Relativity: event horizons, singularities and ergoregions. The hitherto invisible landscape of the gravitational Universe is being unveiled before our eyes: the historical direct detection of gravitational waves by the LIGO-Virgo collaboration marks the dawn of a new era of scientific exploration. Gravitational-wave astronomy will allow us to test models of black hole formation, growth and evolution, as well as models of gravitational-wave generation and propagation. It will provide evidence for event horizons and ergoregions, test the theory of General Relativity itself, and may reveal the existence of new fundamental fields. The synthesis of these results has the potential to radically reshape our understanding of the cosmos and of the laws of Nature. The purpose of this work is to present a concise, yet comprehensive overview of the state of the art in the relevant fields of research, summarize important open problems, and lay out a roadmap for future progress., Comment: White Paper for the COST action "Gravitational Waves, Black Holes, and Fundamental Physics", 272 pages, 12 figures; v4: updated references and author list. Overall improvements and corrections. To appear in Classical and Quantum Gravity

Making use of definitive new lattice computations of the Standard Model thermodynamics during the quantum chromodynamic (QCD) phase transition, we calculate the enhancement in the mass distribution of primordial black holes (PBHs) due to the softening of the equation of state. We find that the enhancement peaks at approximately $0.7M_\odot$, with the formation rate increasing by at least two orders of magnitude due to the softening of the equation of state at this time, with a range of approximately $0.3M_\odot

Simple models of single-field inflation in the very early universe can generate the observed amplitude and scale dependence of the primordial density perturbation, but models with multiple fields can provide an equally good fit to current data. We show how future observations will be able to distinguish between currently favoured models. If a curvaton field generates the primordial perturbations after inflation, we show how the total duration of inflation can be measured., Comment: 7 pages, 3 figures -- matches published version

The spectral index of scalar perturbations is an important observable that allows us to learn about inflationary physics. In particular, a detection of a significant deviation from a constant spectral index could enable us to rule out the simplest class of inflation models. We investigate whether future observations could rule out canonical single-field slow-roll inflation given the parameters allowed by current observational constraints. We find that future measurements of a constant running (or running of the running) of the spectral index over currently available scales are unlikely to achieve this. However, there remains a large region of parameter space (especially when considering the running of the running) for falsifying the assumed class of slow-roll models if future observations accurately constrain a much wider range of scales.

We perform three-dimensional simulations of structure formation in the early Universe, when boosting the primordial power spectrum on approximately kpc scales. We demonstrate that our simulations are capable of producing power-law profiles close to the steep $\rho\propto r^{-9/4}$ halo profiles that are commonly assumed to be a good approximation to ultracompact minihalos (UCMHs). However, we show that for more realistic initial conditions in which halos are neither perfectly symmetric nor isolated, the steep power-law profile is disrupted and we find that the Navarro-Frenk-White profile is a better fit to most halos. In the presence of background fluctuations even extreme, nearly spherical initial conditions do not remain exceptional. Nonetheless, boosting the amplitude of initial fluctuations causes all structures to form earlier and thus at larger densities. With sufficiently large amplitude of fluctuations we find that values for the concentration of typical halos in our simulations can become very large. However, despite the signal coming from dark matter annihilation inside the cores of these halos being enhanced, it is still orders-of-magnitude smaller compared to the usually assumed UCMH profile. The upper bound on the primordial power spectrum from the non-observation of UCMHs should therefore be re-evaluated., Comment: 19 pages, 20 figures. v2: Minor changes to match the published version

Observational constraints on the abundance of primordial black holes (PBHs) constrain the allowed amplitude of the primordial power spectrum on both the smallest and the largest ranges of scales, covering over 20 decades from $1-10^{20}/ \rm{Mpc}$. Despite tight constraints on the allowed fraction of PBHs at their time of formation near horizon entry in the early universe, the corresponding constraints on the primordial power spectrum are quite weak, typically ${\cal P}_\mathcal{R}\lesssim 10^{-2}$ assuming Gaussian perturbations. Motivated by recent claims that the evaporation of just one PBH would destabilise the Higgs vacuum and collapse the universe, we calculate the constraints which follow from assuming there are zero PBHs within the observable universe. This extends the constraints right down to the horizon scale at the end of inflation, but does not significantly tighten the existing power spectrum constraints, even though the constraint on PBH abundance can decrease by up to 46 orders of magnitude. This shows that no future improvement in observational constraints can ever lead to a significant tightening in constraints on inflation (via the power spectrum amplitude). The power spectrum constraints are weak because an order unity perturbation is required in order to overcome pressure forces. We therefore consider an early matter dominated era, during which exponentially more PBHs form for the same initial conditions. We show this leads to far tighter constraints, which approach ${\cal P}_\mathcal{R}\lesssim10^{-9}$, albeit over a smaller range of scales and are very sensitive to when the early matter dominated era ends. Finally, we show that an extended early matter era is incompatible with the argument that an evaporating PBH would destroy the universe, unless the power spectrum amplitude decreases by up to ten orders of magnitude., Comment: 16 pages, 5 figures. Minor clarifications and references added, matches published version

We study the stochastic distribution of spectator fields predicted in different slow-roll inflation backgrounds. Spectator fields have a negligible energy density during inflation but may play an important dynamical role later, even giving rise to primordial density perturbations within our observational horizon today. During de-Sitter expansion there is an equilibrium solution for the spectator field which is often used to estimate the stochastic distribution during slow-roll inflation. However slow roll only requires that the Hubble rate varies slowly compared to the Hubble time, while the time taken for the stochastic distribution to evolve to the de-Sitter equilibrium solution can be much longer than a Hubble time. We study both chaotic (monomial) and plateau inflaton potentials, with quadratic, quartic and axionic spectator fields. We give an adiabaticity condition for the spectator field distribution to relax to the de-Sitter equilibrium, and find that the de-Sitter approximation is never a reliable estimate for the typical distribution at the end of inflation for a quadratic spectator during monomial inflation. The existence of an adiabatic regime at early times can erase the dependence on initial conditions of the final distribution of field values. In these cases, spectator fields acquire sub-Planckian expectation values. Otherwise spectator fields may acquire much larger field displacements than suggested by the de-Sitter equilibrium solution. We quantify the information about initial conditions that can be obtained from the final field distribution. Our results may have important consequences for the viability of spectator models for the origin of structure, such as the simplest curvaton models., Comment: 27 pages without appendices (total 35 pages), 6 figures, 1 table, matches published version in JCAP

We consider the possible decay of the inflaton into curvaton particles during reheating and analyse its effect on curvaton scenarios. Typical decay curvatons are initially relativistic then become non-relativistic, changing the background history of the Universe. We show that this change to the background is the only way in which observational predictions of the scenario are modified. Moreover, once the required amplitude of perturbations is fixed by observation there are no signatures of such decays in other cosmological observables. The decay curvatons can prevent the Universe from becoming dominated by the curvaton condensate, making it impossible to match observations in parts of parameter space. This constrains the branching ratio of the inflaton to curvaton to be less than of order $0.1$ typically. If the branching ratio is below about $10^{-4}$ it has negligible impact on the model parameter space and can be ignored., Comment: 9 pages PDFLatex with 6 incorporated figures. Minor updates to match PRD published version

Negative absolute temperatures (NAT) are an exotic thermodynamical consequence of quantum physics which has been known since the 1950's (having been achieved in the lab on a number of occasions). Recently, the work of Braun et al (2013) has rekindled interest in negative temperatures and hinted at a possibility of using NAT systems in the lab as dark energy analogues. This paper goes one step further, looking into the cosmological consequences of the existence of a NAT component in the Universe. NAT-dominated expanding Universes experience a borderline phantom expansion ($w<-1$) with no Big Rip, and their contracting counterparts are forced to bounce after the energy density becomes sufficiently large. Both scenarios might be used to solve horizon and flatness problems analogously to standard inflation and bouncing cosmologies. We discuss the difficulties in obtaining and ending a NAT-dominated epoch, and possible ways of obtaining density perturbations with an acceptable spectrum., Comment: 14 pages, 3 figures

If the Standard Model (SM) Higgs is weakly coupled to the inflationary sector, the Higgs is expected to be universally in the form of a condensate towards the end of inflation. The Higgs decays rapidly after inflation - via non-perturbative effects - into an out-of-equilibrium distribution of SM species, which thermalize soon afterwards. If the post-inflationary equation of state of the universe is stiff, $w \simeq +1$, the SM species eventually dominate the total energy budget. This provides a natural origin for the relativistic thermal plasma of SM species, required for the onset of the `hot Big Bang' era. The viability of this scenario requires the inflationary Hubble scale $H_*$ to be lower than the instability scale for Higgs vacuum decay, the Higgs not to generate too large curvature perturbations at cosmological scales, and the SM dominance to occur before Big Bang Nucleosynthesis. We show that successful reheating into the SM can only be obtained in the presence of a non-minimal coupling to gravity $\xi \gtrsim 1$, with a reheating temperature of $T_{\rm RH} \gtrsim \mathcal{O}(10^{10})\xi^{3/2}(H_*/10^{14}{\rm GeV})^2~{\rm GeV}$., Comment: 6 pages, 2 figures, minor changes with new figures to match published version in PLB

Observations of the microwave background fluctuations suggest a scale-dependent amplitude asymmetry of roughly 2.5 sigma significance. Inflationary explanations for this 'anomaly' require non-Gaussian fluctuations which couple observable modes to those on much larger scales. In this Letter we describe an analysis of such scenarios which significantly extends previous treatments. We identify the non-Gaussian 'response function' which characterizes the asymmetry, and show that it is non-trivial to construct a model which yields a sufficient amplitude: many independent fine tunings are required, often making such models appear less likely than the anomaly they seek to explain. We present an explicit model satisfying observational constraints and determine for the first time how large its bispectrum would appear to a Planck-like experiment. Although this model is merely illustrative, we expect it is a good proxy for the bispectrum in a sizeable class of models which generate a scale-dependent response using a large eta parameter., Comment: 5 pages. v2: Minor changes to match version published in Phys. Rev. D

Primordial black holes represent a unique probe to constrain the early universe on small scales - providing the only constraints on the primordial power spectrum on the majority of scales. However, these constraints are strongly dependent on even small amounts of non-Gaussianity, which is unconstrained on scales significantly smaller than those visible in the CMB. This paper goes beyond previous considerations to consider the effects of a bispectrum of the equilateral, orthogonal and local shapes with arbitrary magnitude upon the abundance of primordial black holes. Non-Gaussian density maps of the early universe are generated from a given bispectrum and used to place constraints on the small scale power spectrum. When small, we show that the skewness provides an accurate estimate for how the constraint depends on non-Gaussianity, independently of the shape of the bispectrum. We show that the orthogonal template of non-Gaussianity has an order of magnitude weaker effect on the constraints than the local and equilateral templates., Comment: 11 pages, 4 figures, updated to match published version in JCAP02(2016)029, Journal of Cosmology and Astroparticle Physics, Volume 2016, February 2016

If the primordial bispectrum is sufficiently large then the CMB hemispherical asymmetry may be explained by a large-scale mode of exceptional amplitude which perturbs the zeta two-point function. We extend previous calculations, which were restricted to one- or two-source scenarios, by providing a method to compute the response of the two-point function in any model yielding a 'local-like' bispectrum. In general, this shows that it is not the reduced bispectrum fNL which sources the amplitude and scale-dependence of the mode coupling but rather a combination of 'response functions'. We discuss why it is difficult to construct successful scenarios and enumerate the fine-tunings which seem to be required. Finally, we exhibit a concrete model which can be contrived to match the observational constraints and show that to a Planck-like experiment it would appear to have |fNL-local| ~ |fNL-equi| ~ |fNL-ortho| ~ 1. Therefore, contrary to previous analyses, we conclude that it is possible to generate the asymmetry while respecting observational constraints on the bispectrum and low-ell multipoles even without tuning our location on the long-wavelength mode., Comment: 38 pages plus references, 7 figures. v2: Minor changes to match version published in JCAP. This is an author-created, un-copyedited version of an article published in JCAP. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at http://dx.doi.org/10.1088/1475-7516/2016/06/025

Primordial black holes (PBHs) are black holes which may have formed very early on during the radiation dominated era in the early universe. We present here a method by which the large scale perturbations in the density of primordial black holes may be used to place tight constraints on non-gaussianity if PBHs account for dark matter (DM). The presence of local-type non-gaussianity is known to have a significant effect on the abundance of primordial black holes, and modal coupling from the observed CMB scale modes can significantly alter the number density of PBHs that form within different regions of the universe, which appear as DM isocurvature modes. Using the recent \emph{Planck} constraints on isocurvature perturbations, we show that PBHs are excluded as DM candidates for even very small local-type non-gaussianity, $|f_{NL}|\approx0.001$ and remarkably the constraint on $g_{NL}$ is almost as strong. Even small non-gaussianity is excluded if DM is composed of PBHs. If local non-Gaussianity is ever detected on CMB scales, the constraints on the fraction of the universe collapsing into PBHs (which are massive enough to have not yet evaporated) will become much tighter., Comment: 23 pages, 11 figures. V2: minor corrections and changes, matches published version

We introduce an alternative parametrisation for the scale dependence of the non-linearity parameter $f_{\rm NL}$ in quasi-local models of non-Gaussianity. Our parametrisation remains valid when $f_{\rm NL}$ changes sign, unlike the commonly adopted power law ansatz $f_{\rm NL}(k) \propto k^{ n_{f_{\rm NL}} }$. We motivate our alternative parametrisation by appealing to the self-interacting curvaton scenario, and as an application, we apply it to the CMB power asymmetry. Explaining the power asymmetry requires a strongly scale dependent non-Gaussianity. We show that regimes of model parameter space where $f_{\rm NL}$ is strongly scale dependent are typically associated with a large $g_{\rm NL}$ and quadrupolar power asymmetry, which can be ruled out by existing observational constraints., Comment: 21 pages. Comments welcome. v2: Minor changes and corrections to match the published version

We perform a Bayesian model comparison for scenarios within the quadratic curvaton model, determining the degree to which both are disfavoured with respect to the $\Lambda$CDM concordance model and single-field quadratic inflation, using the recent \emph{Planck} data release. Despite having three additional model parameters, the simplest curvaton scenario is not disfavoured relative to single-field quadratic inflation, and it becomes favoured against this single-field model when we include the joint BICEP/Keck/\emph{Planck} analysis. In all cases we assume an instantaneous inflaton decay and no surviving isocurvature perturbations. Despite the success of \emph{Planck} reaching its forecast measurement accuracy, we show that the current constraints on local non-Gaussianity are insufficiently precise to have any significant impact on the evidence ratios so far. We also determine the precision $\sigma(f_{\mathrm{NL}})$ required by future measurements assuming a fiducial value of $f_{\mathrm{NL}}=-5/4$ or $10.8$ to no longer disfavour the curvaton against the $\Lambda$CDM parametrisation, and we discuss the effect that the predicted increase in precision from future measurements on $f_{\mathrm{NL}}$ may have. We show that our results are not very sensitive to our choice of priors., Comment: 14 pages, 1 figure, 5 tables, Version 2

We discuss how primordial non-Gaussianity of the curvature perturbation helps to constrain models of the early universe. Observations are consistent with Gaussian initial conditions, compatible with the predictions of the simplest models of inflation. Deviations are constrained to be at the sub percent level, constraining alternative models such as those with multiple fields, non-canonical kinetic terms or breaking the slow-roll conditions. We introduce some of the most important models of inflation which generate non-Gaussian perturbations and provide practical tools on how to calculate the three-point correlation function for a popular class of non-Gaussian models. The current state of the field is summarised and an outlook is given., Comment: Lectures aimed at an audience with little or no previous knowledge of the subject. Lectures held at the II JPBCosmo school in Espirito Santo, Brasil

The effects of non-gaussianity on the constraints on the primordial curvature perturbation power spectrum from primordial black holes (PBHs) are considered. We extend previous analyses to include the effects of coupling between the modes of the horizon scale at the time the PBH forms and super-horizon modes. We consider terms of up to third order in the Gaussian perturbation. For the weakest constraints on the abundance of PBHs in the early universe (corresponding to a fractional energy density of PBHs of $10^{-5}$ at the time of formation), in the case of gaussian perturbations, constraints on the power spectrum are $\mathcal{P}_{\zeta}<0.05$ but can significantly tighter when even a small amount of non-gaussianity is considered, to $\mathcal{P}_{\zeta}<0.01$, and become approximately $\mathcal{P}_{\zeta}<0.003$ in more special cases. Surprisingly, even when there is negative skew (which naively would suggest fewer areas of high density, leading to weaker constraints), we find that the constraints on the power spectrum become tighter than the purely gaussian case - in strong contrast with previous results. We find that the constraints are highly sensitive to both the non-gaussianity parameters as well as the amplitude of super-horizon perturbations., Comment: 15 pages, 9 figures. V2: minor corrections and additions, closely matches published version

We reinspect the calculation for the mass fraction of primordial black holes (PBHs) which are formed from primordial perturbations, finding that performing the calculation using the comoving curvature perturbation $\mathcal{R}_{c}$ in the standard way vastly overestimates the number of PBHs, by many orders of magnitude. This is because PBHs form shortly after horizon entry, meaning modes significantly larger than the PBH are unobservable and should not affect whether a PBH forms or not - this important effect is not taken into account by smoothing the distribution in the standard fashion. We discuss alternative methods and argue that the density contrast, $\Delta$, should be used instead as super-horizon modes are damped by a factor $k^{2}$. We make a comparison between using a Press-Schechter approach and peaks theory, finding that the two are in close agreement in the region of interest. We also investigate the effect of varying the spectral index, and the running of the spectral index, on the abundance of primordial black holes., Comment: 16 pages, 7 figures. Version 2: updated to match published version and include journal reference. Version 3 and 4: minor corrections, conclusions unchanged

We carry out a comprehensive analysis of the simplest curvaton model, which is based on two non-interacting massive fields. Our analysis encompasses cases where the inflaton and curvaton both contribute to observable perturbations, and where the curvaton itself drives a second period of inflation. We consider both power spectrum and non-Gaussianity observables, and focus on presenting constraints in model parameter space. The fully curvaton-dominated regime is in some tension with observational data, while an admixture of inflaton-generated perturbations improves the fit. The inflating curvaton regime mimics the predictions of Nflation. Some parts of parameter space permitted by power spectrum data are excluded by non-Gaussianity constraints. The recent BICEP2 results [1] require that the inflaton perturbations provide a significant fraction of the total perturbation, ruling out the usual curvaton scenario in which the inflaton perturbations are negligible, though not the admixture regime where both inflaton and curvaton contribute to the spectrum., Comment: 7 pages, 9 figures; v2 matches journal accepted version

Primordial black holes (PBHs) can form in the early Universe from the collapse of rare, large density fluctuations. They have never been observed, but this fact is enough to constrain the amplitude of fluctuations on very small scales which cannot be otherwise probed. Because PBHs form only in very rare large fluctuations, the number of PBHs formed is extremely sensitive to changes in the shape of the tail of the fluctuation distribution - which depends on the amount of non-Gaussianity present. We first study how local non-Gaussianity of arbitrary size up to fifth order affects the abundance and constraints from PBHs, finding that they depend strongly on even small amounts of non-Gaussianity and the upper bound on the allowed amplitude of the power spectrum can vary by several orders of magnitude. The sign of the non-linearity parameters (f_{NL}, g_{NL}, etc) are particularly important. We also study the abundance and constraints from PBHs in the curvaton scenario, in which case the complete non-linear probability distribution is known, and find that truncating to any given order (i.e. to order f_{NL} or g_{NL}, etc) does not give accurate results., Comment: 17 pages, 9 figures. V2. Minor changes included to match the published version. Lower resolution graphics included to aid printing of the document

The new Planck constraints on the local bispectrum parameter fnl are about 10^5 times tighter than the current constraints on the trispectrum parameter gnl, which means that the allowed numerical values of the second and third order terms in the perturbative expansion of the curvature perturbation are comparable. We show that a consequence of this is that if gnl is large enough to be detectable, then it will induce a large variation between the observable value of fnl and its value in a larger inflated volume. Even if there were only a few extra efoldings between the beginning of inflation and horizon crossing of our Hubble horizon, an observably large gnl means that fnl is unlikely to be as small as its current constraint, regardless of its true background value. This result is very general, it holds regardless of how many fields contributed to the curvature perturbation. We also generalise this result to other shapes of non-Gaussianity, beyond the local model. We show that the variance of the 3-point function in the squeezed limit is bounded from below by the square of the squeezed limit of the 4-point function., Comment: 8 pages, 2 figures

We study the evolution of the non-linear curvature perturbation during perturbative reheating, and hence how observables evolve to their final values which we may compare against observations. Our study includes the evolution of the two trispectrum parameters, $\gnl$ and $\taunl$, as well as the scale dependence of both $\fnl$ and $\taunl$. In general the evolution is significant and must be taken into account, which means that models of multifield inflation cannot be compared to observations without specifying how the subsequent reheating takes place. If the trispectrum is large at the end of inflation, it normally remains large at the end of reheating. In the classes of models we study, it is very hard to generate $\taunl\gg\fnl^2$, regardless of the decay rates of the fields. Similarly, for the classes of models in which $\gnl\simeq\taunl$ during slow--roll inflation, we find the relation typically remains valid during reheating. Therefore it is possible to observationally test such classes of models without specifying the parameters of reheating, even though the individual observables are sensitive to the details of reheating. It is hard to generate an observably large $\gnl$ however. The runnings, $\nfnl$ and $\ntaunl$, tend to satisfy a consistency relation $\ntaunl=(3/2)\nfnl$, but are in general too small to be observed for the class of models considered regardless of reheating timescale., Comment: 17 pages, 8 figures. v2: minor changes to match the published JCAP version

Primordial perturbations with wavelengths greater than the observable universe shift the effective background fields in our observable patch from their global averages over the inflating space. This leads to a landscape picture where the properties of our observable patch depend on its location and may significantly differ from the expectation values predicted by the underlying fundamental inflationary model. We show that if multiple fields are present during inflation, this may happen even if our horizon exit would be preceded by only a few e-foldings of inflation. Non-Gaussian statistics are especially affected: for example models of local non-Gaussianity predicting |f_NL|>> 10 over the entire inflating volume can have a probability up to a few tens of percent to generate a non-detectable bispectrum in our observable patch |fNL^{obs.}|<10. In this work we establish systematic connections between the observable local properties of primordial perturbations and the global properties of the inflating space which reflect the underlying high energy physics. We study in detail the implications of both a detection and non-detection of primordial non-Gaussianity by Planck, and discover novel ways of characterising the naturalness of different observational configurations., Comment: 21 pages, 6 figures

We compute the scale dependence of fNL for models of multi-field inflation, allowing for an arbitrary field space metric. We show that, in addition to multi-field effects and self interactions, the curved field space metric provides another source of scale dependence, which arises from the field-space Riemann curvature tensor and its derivatives. The scale dependence may be detectable within the near future if the amplitude of fNL is not too far from the current observational bounds., Comment: (v1) 9 pages, 1 figure; (v2) 10 pages, minor updates including the title, to appear in Physics Letters B

Inflation is the leading paradigm for explaining the origin of primordial density perturbations and the observed temperature fluctuations of the cosmic microwave background. However many open questions remain, in particular whether one or more scalar fields were present during inflation and how they contributed to the primordial density perturbation. We propose a new observational test of whether multiple fields, or only one (not necessarily the inflaton) generated the perturbations. We show that our test, relating the bispectrum and trispectrum, is protected against loop corrections at all orders, unlike previous relations., Comment: 6 pages; v2: references added, minor changes; v3: Extended discussion, including the role of gravitational corrections. References added. Version accepted for publication in PRD

We study the effects of perturbative reheating on the evolution of the curvature perturbation \zeta, in two-field inflation models. We use numerical methods to explore the sensitivity of f_NL, n_s and r to the reheating process, and present simple qualitative arguments to explain our results. In general, if a large non-Gaussian signal exists at the start of reheating, it will remain non zero at the end of reheating. Unless all isocurvature modes have completely decayed before the start of reheating, we find that the non-linearity parameter, f_NL, can be sensitive to the reheating timescale, and that this dependence is most appreciable for `runaway' inflationary potentials that only have a minimum in one direction. For potentials with a minimum in both directions, f_NL can also be sensitive to reheating if a mild hierarchy exists between the decay rates of each field. Within the class of models studied, we find that the spectral index n_s, is fairly insensitive to large changes in the field decay rates, indicating that n_s is a more robust inflationary observable, unlike the non-linearity parameter f_NL. Our results imply that the statistics of \zeta, especially f_NL, can only be reliably used to discriminate between models of two-field inflation if the physics of reheating are properly accounted for., Comment: 25 pages, 8 figures. Comments welcome

Primordial Black Holes (PBH's) can form in the early Universe from the collapse of large density fluctuations. Tight observational limits on their abundance constrain the amplitude of the primordial fluctuations on very small scales which can not otherwise be constrained, with PBH's only forming from the extremely rare large fluctuations. The number of PBH's formed is therefore sensitive to small changes in the shape of the tail of the fluctuation distribution, which itself depends on the amount of non-Gaussianity present. We study, for the first time, how quadratic and cubic local non-Gaussianity of arbitrary size (parameterised by f_nl and g_nl respectively) affects the PBH abundance and the resulting constraints on the amplitude of the fluctuations on very small scales. Intriguingly we find that even non-linearity parameters of order unity have a significant impact on the PBH abundance. The sign of the non-Gaussianity is particularly important, with the constraint on the allowed fluctuation amplitude tightening by an order of magnitude as f_nl changes from just -0.5 to 0.5. We find that if PBH's are observed in the future, then regardless of the amplitude of the fluctuations, non-negligible negative f_nl would be ruled out. Finally we show that g_nl can have an even larger effect on the number of PBH's formed than f_nl., Comment: 9 pages, 5 figures, v2: version to appear in Phys. Rev. D with minor changes, v3: typos corrected (including factor of 1/2 in erfc prefactor), no changes to results

We propose a method to probe higher-order correlators of the primordial density field through the inhomogeneity of local non-Gaussian parameters, such as f_NL, measured within smaller patches of the sky. Correlators between n-point functions measured in one patch of the sky and k-point functions measured in another patch depend upon the (n+k)-point functions over the entire sky. The inhomogeneity of non-Gaussian parameters may be a feasible way to detect or constrain higher-order correlators in local models of non-Gaussianity, as well as to distinguish between single and multiple-source scenarios for generating the primordial density perturbation, and more generally to probe the details of inflationary physics., Comment: 16 pages, 2 figures; v2: Minor changes and references added. Matches the published version

We propose a novel scenario to generate primordial magnetic fields during inflation induced by an oscillating coupling of the electromagnetic field to the inflaton. This resonant mechanism has two key advantages over previous proposals. First of all, it generates a narrow band of magnetic fields at any required wavelength, thereby allaying the usual problem of a strongly blue spectrum and its associated backreaction. Secondly, it avoids the need for a strong coupling as the coupling is oscillating rather than growing or decaying exponentially. Despite these major advantages, we find that the backreaction is still far too large during inflation if the generated magnetic fields are required to have a strength of order 10^{-15} Gauss today on observationally interesting scales. We provide a more general no-go argument, proving that this problem will apply to any model in which the magnetic fields are generated on subhorizon scales and freeze after horizon crossing., Comment: 13 pages + 7 pages of appendices and references, 3 figures, matches published version

We study the scale dependence of the non-linearity parameters f_NL and g_NL in curvaton models with self-interactions. We show that the spectral indices n_fNL=d ln|f_NL|/(d ln k) and n_gNL=d ln |g_NL|/(d ln k) can take values much greater than the slow--roll parameters and the spectral index of the power spectrum. This means that the scale--dependence of the bi and trispectrum could be easily observable in this scenario with Planck, which would lead to tight additional constraints on the model. Inspite of the highly non-trivial behaviour of f_NL and g_NL in the curvaton models with self-interactions, we find that the model can be falsified if g_NL(k) is also observed., Comment: 19 pages, many figures. v2: Figure 4 replaced with a corrected normalisation, conclusions unchanged. Matches version published in JCAP

We investigate the signature of primordial non-Gaussianities in the weak lensing bispectrum, in particular the signals generated by local, orthogonal and equilateral non-Gaussianities. The questions we address include the signal-to-noise ratio generated in the Euclid weak lensing survey (we find the 1sigma-errors for fNL are 200, 575 and 1628 for local, orthogonal and equilateral non-Gaussianities, respectively), misestimations of fNL if one chooses the wrong non-Gaussianity model (misestimations by up to a factor of +/-3 in fNL are possible, depending on the choice of the model), the probability of noticing such a mistake (improbably large values for the chi^2-functional occur from fNL 200 on), degeneracies of the primordial bispectrum with other cosmological parameters (only the matter density Omega_m plays a significant role), and the subtraction of the much larger, structure-formation generated bispectrum. If a prior on a standard wCDM-parameter set is available from Euclid and Planck, the structure formation bispectrum can be predicted accurately enough for subtraction, and any residual structure formation bispectrum would influence the estimation of fNL to a minor degree. Configuration-space integrations which appear in the evaluation of chi^2-functionals and related quantities can be carried out very efficiently with Monte-Carlo techniques, which reduce the complexity by a factor of O(10^4) while delivering sub-percent accuracies. Weak lensing probes smaller scales than the CMB and hence provide an additional constraint on non-Gaussianities, even though they are not as sensitive to primordial non-Gaussianities as the CMB., Comment: 10 pages, 9 figures