An introduction to inflation after Planck: from theory to observations

These lecture notes have been written for a short introductory course on the status of inflation after Planck and BICEP2, given at the Xth Modave School of Mathematical Physics. The first objective is to give an overview of the theory of inflation: motivations, homogeneous scalar field dynamics, slow-roll approximation, linear theory of cosmological perturbations, classification of single field potentials and their observable predictions. This includes a pedagogical derivation of the primordial scalar and tensor power spectra for any effective single field potential. The second goal is to present the most recent results of Planck and BICEP2 and to discuss their implications for inflation. Bayesian statistical methods are introduced as a tool for model analysis and comparison. Based on the recent work of J. Martin et al., the best inflationary models after Planck and BICEP2 are presented. Finally a series of open questions and issues related to inflation are mentioned and briefly discussed.Comment: 39 pages, 9 figures, to be published in the proceedings of the Xth Modave School in Mathematical Physics. Important parts of those lecture notes draw from chapters 1 and 2 of the author's PhD thesis, arXiv:1109.557

Anamorphosis in hybrid inflation: How to avoid fine-tuning of initial conditions?

In order to generate more than 60 e-folds of accelerated expansion in original hybrid inflation, 2-fields trajectories are usually required to be initially fine-tuned in a very narrow band along the inflationary valley or in some isolated points outside it. From a more precise investigation of the dynamics, these points which can cover a non-negligible proportion of the space of sub-planckian initial field values, depending on the potential parameters, are shown to be organised in connected domains with fractal boundaries. They correspond to trajectories first falling towards the bottom of the potential, then climbing and slow-rolling back along the inflationary valley. The full parameter space, including initial velocities and all the potential parameters, is then explored by using Monte-Carlo-Markov-Chains (MCMC) methods. Results indicate that successful initial conditions (IC) outside the valley are not localized in the parameter space and are the dominant way to realise inflation, independently of initial field velocities. Natural bounds on parameters are deduced. The genericity of our results is confirmed in 5 other hybrid models from various framework.Comment: AIP Proceedings of the "Invisible Universe" conference, Palais de l'Unesco, Paris, 29 June - 4 July 200

Updated Constraints on Large Field Hybrid Inflation

We revisit the status of hybrid inflation in the light of Planck and recent BICEP2 results, taking care of possible transient violations of the slow-roll conditions as the field passes from the large field to the vacuum dominated phase. The usual regime where observable scales exit the Hubble radius in the vacuum dominated phase predicts a blue scalar spectrum, which is ruled out. But whereas assuming slow-roll one expects this regime to be generic, by solving the exact dynamics we identify the parameter space for which the small field phase is naturally avoided due to slow-roll violations at the end of the large field phase. When the number of e-folds generated at small field is negligible, the model predictions are degenerated with those of a quadratic potential. There exists also a transitory case for which the small field phase is sufficiently long to affect importantly the observable predictions. Interestingly, in this case the spectral index and the tensor to scalar ratio agree respectively with the best fit of Planck and BICEP2. This results in a \Delta \chi^2 \simeq 5.0 in favor of hybrid inflation for Planck+BICEP2 (\Delta \chi^2 \simeq 0.9 for Planck only). The last considered regime is when the critical point at which inflation ends is located in the large field phase. It is constrained to be lower than about ten times the reduced Planck mass. The analysis has been conducted with the use of Markov-Chain-Monte-Carlo bayesian method, in a reheating consistent way, and we present the posterior probability distributions for all the model parameters.Comment: 13 pages, 9 figures, comments welcom

Detecting the gravitational wave background from primordial black hole dark matter

The black hole merging rates inferred after the gravitational-wave detection by Advanced LIGO/VIRGO and the relatively high mass of the progenitors are consistent with models of dark matter made of massive primordial black holes (PBH). PBH binaries emit gravitational waves in a broad range of frequencies that will be probed by future space interferometers (LISA) and pulsar timing arrays (PTA). The amplitude of the stochastic gravitational-wave background expected for PBH dark matter is calculated taking into account various effects such as initial eccentricity of binaries, PBH velocities, mass distribution and clustering. It allows a detection by the LISA space interferometer, and possibly by the PTA of the SKA radio-telescope. Interestingly, one can distinguish this background from the one of non-primordial massive binaries through a specific frequency dependence, resulting from the maximal impact parameter of binaries formed by PBH capture, depending on the PBH velocity distribution and their clustering properties. Moreover, we find that the gravitational wave spectrum is boosted by the width of PBH mass distribution, compared with that of the monochromatic spectrum. The current PTA constraints already rule out broad-mass PBH models covering more than three decades of masses, but evading the microlensing and CMB constraints due to clustering.Comment: 12 pages, 4 figure

Fractal initial conditions and natural parameter values in hybrid inflation

We show that the initial field values required to produce inflation in the two fields original hybrid model, and its supergravity F-term extension, do not suffer from any fine-tuning problem, even when the fields are restricted to be sub-planckian and for almost all potential parameter values. This is due to the existence of an initial slow-roll violating evolution which has been overlooked so far. Due to the attractor nature of the inflationary valley, these trajectories end up producing enough accelerated expansion of the universe. By numerically solving the full non-linear dynamics, we show that the set of such successful initial field values is connected, of dimension two and possesses a fractal boundary of infinite length exploring the whole field space. We then perform a Monte-Carlo-Markov-Chain analysis of the whole parameter space consisting of the initial field values, field velocities and potential parameters. We give the marginalised posterior probability distributions for each of these quantities such that the universe inflates long enough to solve the usual cosmological problems. Inflation in the original hybrid model and its supergravity version appears to be generic and more probable by starting outside of the inflationary valley. Finally, the implication of our findings in the context of the eternal inflationary scenario are discussed.Comment: 16 pages, 16 figures, uses RevTeX. Lyapunov exponents and references added, matches published versio

Probing Modified Gravity with Atom-Interferometry: a Numerical Approach

Refined constraints on chameleon theories are calculated for atom-interferometry experiments, using a numerical approach consisting in solving for a four-region model the static and spherically symmetric Klein-Gordon equation for the chameleon field. By modeling not only the test mass and the vacuum chamber but also its walls and the exterior environment, the method allows to probe new effects on the scalar field profile and the induced acceleration of atoms. In the case of a weakly perturbing test mass, the effect of the wall is to enhance the field profile and to lower the acceleration inside the chamber by up to one order of magnitude. In the thin-shell regime, results are found to be in good agreement with the analytical estimations, when measurements are realized in the immediate vicinity of the test mass. Close to the vacuum chamber wall, the acceleration becomes negative and potentially measurable. This prediction could be used to discriminate between fifth-force effects and systematic experimental uncertainties, by doing the experiment at several key positions inside the vacuum chamber. For the chameleon potential $V(\phi) = \Lambda^{4+\alpha} / \phi^\alpha$ and a coupling function $A(\phi) = \exp(\phi /M)$, one finds $M \gtrsim 7 \times 10^{16} \mathrm{GeV}$, independently of the power-law index. For $V(\phi) = \Lambda^4 (1+ \Lambda/ \phi)$, one finds $M \gtrsim 10^{14} \mathrm{GeV}$. A sensitivity of $a\sim 10^{-11} \mathrm{m/s^2}$ would probe the model up to the Planck scale. Finally, a proposal for a second experimental set-up, in a vacuum room, is presented. In this case, Planckian values of $M$ could be probed provided that $a \sim 10^{-10} \mathrm{m/s^2}$, a limit reachable by future experiments. Our method can easily be extended to constrain other models with a screening mechanism, such as symmetron, dilaton and f(R) theories.Comment: 13 pages, 12 figures, version accepted by PR

LIGO Lo(g)Normal MACHO: Primordial Black Holes survive SN lensing constraints

It has been claimed in Ref.[arXiv:1712.02240] that massive primordial black holes (PBH) cannot constitute all of the dark matter (DM), because their gravitational-lensing imprint on the Hubble diagram of type Ia supernovae (SN) would be incompatible with present observations. In this paper, we critically review those constraints and find several caveats on the analysis. First of all, the constraints on the fraction $\alpha$ of PBH in matter seem to be driven by a very restrictive choice of priors on the cosmological parameters. In particular, the degeneracy between $\Omega_{\rm M}$ and $\alpha$ is ignored and thus, by fixing $\Omega_{\rm M}$, transferred the constraining power of SN magnitudes to $\alpha$. Furthermore, by considering more realistic physical sizes for the type-Ia supernovae, we find an effect on the SN lensing magnification distribution that leads to significantly looser constraints. Moreover, considering a wide mass spectrum of PBH, such as a lognormal distribution, further softens the constraints from SN lensing. Finally, we find that the fraction of PBH that could constitute DM today is bounded by $f_{\rm PBH} < 1.09\ (1.38)$, for JLA (Union 2.1) catalogs, and thus it is perfectly compatible with an all-PBH dark matter scenario in the LIGO band.Comment: 7 pages, 7 figure

Primordial black holes from the QCD epoch: Linking dark matter, baryogenesis and anthropic selection

If primordial black holes (PBHs) formed at the quark-hadron epoch, their mass must be close to the Chandrasekhar limit, this also being the characteristic mass of stars. If they provide the dark matter (DM), the collapse fraction must be of order the cosmological baryon-to-photon ratio $\sim 10^{-9}$, which suggests a scenario in which a baryon asymmetry is produced efficiently in the outgoing shock around each PBH and then propagates to the rest of the Universe. We suggest that the temperature increase in the shock provides the ingredients for hot spot electroweak baryogenesis. This also explains why baryons and DM have comparable densities, the precise ratio depending on the size of the PBH relative to the cosmological horizon at formation. The observed value of the collapse fraction and baryon asymmetry depends on the amplitude of the curvature fluctuations which generate the PBHs and may be explained by an anthropic selection effect associated with the existence of galaxies. We propose a scenario in which the quantum fluctuations of a light stochastic spectator field during inflation generate large curvature fluctuations in some regions, with the stochasticity of this field providing the basis for the required selection. Finally, we identify several observational predictions of our scenario that should be testable within the next few years. In particular, the PBH mass function could extend to sufficiently high masses to explain the black hole coalescences observed by LIGO/Virgo.Comment: 37 pages, 3 figures, published in MNRA