Inflationary Trajectory Research Articles

Initial conditions for inflation — A short review

I give a brief review of the status of research on the nature of initial conditions required to obtain a period of cosmological inflation. It is shown that there is good evidence that in the case of large field models, the inflationary slow-roll trajectory is a local attractor in initial condition space, whereas it is not in the case of small field models.

Moduli backreaction on inflationary attractors

We investigate the interplay between moduli dynamics and inflation, focusing on the KKLT-scenario and cosmological $\alpha$-attractors. General couplings between these sectors can induce a significant backreaction and potentially destroy the inflationary regime; however, we demonstrate that this generically does not happen for $\alpha$-attractors. Depending on the details of the superpotential, the volume modulus can either be stable during the entire inflationary trajectory, or become tachyonic at some point and act as a waterfall field, resulting in a sudden end of inflation. In the latter case there is a universal supersymmetric minimum where the scalars end up, preventing the decompactification scenario. The gravitino mass is independent from the inflationary scale with no fine-tuning of the parameters. The observational predictions conform to the universal value of attractors, fully compatible with the Planck data, with possibly a capped number of e-folds due to the interplay with moduli.

Geometrical Destabilization of Inflation.

We show the existence of a general mechanism by which heavy scalar fields can be destabilized during inflation, relying on the fact that the curvature of the field space manifold can dominate the stabilizing force from the potential and destabilize inflationary trajectories. We describe a simple and rather universal setup in which higher-order operators suppressed by a large energy scale trigger this instability. This phenomenon can prematurely end inflation, thereby leading to important observational consequences and sometimes excluding models that would otherwise perfectly fit the data. More generally, it modifies the interpretation of cosmological constraints in terms of fundamental physics. We also explain how the geometrical destabilization can lead to powerful selection criteria on the field space curvature of inflationary models.

Trapping effects in inflation: Blue spectrum at small scales

We consider the inflationary model in which the inflaton $\phi$ couples to another scalar field $\chi$ via the interaction $g^2(\phi-\phi_0)^2\chi^2$ with a small coupling constant $g$ ($g^2 \sim 10^{-7}$). We assume that there is a sequence of "trapping points" $\phi_{0i}$ along the inflationary trajectory where particles of $\chi$-field become massless and are rather effectively produced. We calculate the power spectrum of inflaton field fluctuations originated from a backreaction of $\chi$-particles produced, using the Schwinger's "in-in" formalism. We show that the primary curvature power spectrum produced by these backreaction effects is blue, which leads to a strong overproduction of primordial black holes (PBHs) in subsequent radiation era.

CMB cold spot from inflationary feature scattering

We propose a “feature-scattering” mechanism to explain the cosmic microwave background cold spot seen from WMAP and Planck maps. If there are hidden features in the potential of multi-field inflation, the inflationary trajectory can be scattered by such features. The scattering is controlled by the amount of isocurvature fluctuations, and thus can be considered as a mechanism to convert isocurvature fluctuations into curvature fluctuations. This mechanism predicts localized cold spots (instead of hot ones) on the CMB. In addition, it may also bridge a connection between the cold spot and a dip on the CMB power spectrum at ℓ∼20.

Hyperbolic geometry of cosmological attractors

Cosmological alpha-attractors give a natural explanation for the spectral index n_s of inflation as measured by Planck while predicting a range for the tensor-to-scalar ratio r, consistent with all observations, to be measured more precisely in future detection of gravity waves. Their embedding into supergravity exploits the hyperbolic geometry of the Poincare disk or half-plane. These geometries are isometric under Mobius transformations, which include the shift symmetry of the inflaton field. We introduce a new Kahler potential frame that explicitly preserves this symmetry, enabling the inflaton to be light. Moreover, we include higher-order curvature deformations, which can stabilize a direction orthogonal to the inflationary trajectory. We illustrate this new framework by stabilizing the single superfield alpha-attractors.

Trajectories with suppressed tensor-to-scalar ratio in Aligned Natural Inflation

In Aligned Natural Inflation, an alignment between different potential terms produces an inflaton excursion greater than the axion scales in the potential. We show that, starting from a general potential of two axions with two aligned potential terms, the effective theory for the resulting light direction is characterized by four parameters: an effective potential scale, an effective axion constant, and two extra parameters (related to ratios of the axion scales and the potential scales in the 2−field theory). For all choices of these extra parameters, the model can support inflation along valleys (in the 2−field space) that end in minima of the potential. This leads to a phenomenology similar to that of single field Natural Inflation. For a significant range of the extra two parameters, the model possesses also higher altitude inflationary trajectories passing through saddle points of the 2−field potential, and disconnected from any minimum. These plateaus end when the heavier direction becomes unstable, and therefore all of inflation takes place close to the saddle point, where—due to the higher altitude—the potential is flatter (smaller ϵ parameter). As a consequence, a tensor-to-scalar ratio r = O ( 10−4 − 10−2 ) can be easily achieved in the allowed ns region, well within the latest 1 σ CMB contours.

Nonuniqueness of classical inflationary trajectories on a high-dimensional landscape

Motivated by the string landscape, the inflationary potential may be high dimensional and complicated. We propose a new way to construct some high-dimensional random potentials, and simulate inflationary dynamics on top of that, for up to 50 dimensions in field space. Especially, random bifurcations of classical inflationary trajectory are studied. It is shown that the bifurcation probability increases as a function of the number of dimensions. Those random bifurcations are not consistent with observations and dramatically limit the parameter space of inflation on a complicated landscape. For example, in 10 dimensions, only $1{0}^{\ensuremath{-}3}--1{0}^{\ensuremath{-}6}$ of the parameter space volume leads to unique classical trajectories. The rest is ruled out by random bifurcations.

Untangling features in the primordial spectra

We discuss the possible existence of features in both the primordial power spectrum and bispectrum generated during a stage of single field cosmic inflation. We argue that there are two main classes of features: those produced by a sudden time variation of the sound speed of curvature perturbations, and those produced by a sudden change in the expansion rate during inflation. The former are known to be produced by heavy fields, when the inflationary background trajectory in field space undergoes a bend, whereas the latter are known to be produced by features in the inflaton potential encountered as the inflaton field descends its slope. In general, features are expected to be the result of these two sources combined, however, it is possible that one source dominated over the other, resulting in a distinctive pattern that may be observationally tested. We deduce a relation that gives us the shape of features in the bispectrum provided that we know the shape of features in the power spectrum, and show that each one of these two classes of features leaves a particular footprint in the distribution of perturbations that could be uncovered by a joint analysis of the primordial power spectrum and bispectrum.

Supersymmetry breaking and inflation from higher curvature supergravity

The generic embedding of the $R+R^2$ higher curvature theory into old-minimal supergravity leads to models with rich vacuum structure in addition to its well-known inflationary properties. When the model enjoys an exact R-symmetry, there is an inflationary phase with a single supersymmetric Minkowski vacuum. This appears to be a special case of a more generic set-up, which in principle may include R-symmetry violating terms which are still of pure supergravity origin. By including the latter terms, we find new supersymmetry breaking vacua compatible with single-field inflationary trajectories. We discuss explicitly two such models and we illustrate how the inflaton is driven towards the supersymmetry breaking vacuum after the inflationary phase. In these models the gravitino mass is of the same order as the inflaton mass. Therefore, pure higher curvature supergravity may not only accommodate the proper inflaton field, but it may also provide the appropriate hidden sector for supersymmetry breaking after inflation has ended.

Geodesic completeness and homogeneity condition for cosmic inflation

There are two disjointed problems in cosmology within General Relativity (GR), which can be addressed simultaneously by studying the nature of geodesics around $t\rightarrow 0$, where $t$ is the physical time. One is related to the past geodesic completeness of the inflationary trajectory due to the presence of a cosmological singularity, and the other one is related to the homogeneity condition required to inflate a local space-time patch of the universe. We will show that both the problems have a common origin, arising from how the causal structure of null and timelike geodesics are structured within GR. In particular, we will show how a non-local extension of GR can address both problems, while satisfying the null energy condition for the matter sources.

Combined features in the primordial spectra induced by a sudden turn in two-field DBI inflation

We investigate the features generated by a sharp turn along the inflationary trajectory in a two-field model of Dirac-Born-Infeld inflation, where one of the fields is heavy. Distinct features are generated by two different effects: the mixing of the light and heavy modes during the turn, on the one hand, and the resonance between the oscillations along the heavy direction after the turn, on the other hand. Contrary to models with standard kinetic terms, the resonance effect is not strongly suppressed because the action contains derivative interactions. Working in the potential basis, we study the oscillations after the turn and compute the amplitude of the mixing and resonance features in the power spectrum, as well as in the bispectrum for the latter effect. We find that the amplitudes and positions of these combined features obey specific consistency relations, which could be confronted with cosmological data.

BICEP's acceleration

The recent Bicep2 [1] detection of, what is claimed to be primordial B-modes, opens up the possibility of constraining not only the energy scale of inflation but also the detailed acceleration history that occurred during inflation. In turn this can be used to determine the shape of the inflaton potential V(ϕ) for the first time — if a single, scalar inflaton is assumed to be driving the acceleration. We carry out a Monte Carlo exploration of inflationary trajectories given the current data. Using this method we obtain a posterior distribution of possible acceleration profiles ϵ(N) as a function of e-fold N and derived posterior distributions of the primordial power spectrum P(k) and potential V(ϕ). We find that the Bicep2 result, in combination with Planck measurements of total intensity Cosmic Microwave Background (CMB) anisotropies, induces a significant feature in the scalar primordial spectrum at scales k∼ 10-3 Mpc -1. This is in agreement with a previous detection of a suppression in the scalar power [2].

Non-gaussian signatures of general inflationary trajectories

We carry out a numerical calculation of the bispectrum in generalised trajectories of canonical, single-field inflation. The trajectories are generated in the Hamilton-Jacobi (HJ) formalism based on Hubble Slow Roll (HSR) parameters. The calculation allows generally shape and scale dependent bispectra, or dimensionless fNL, in the out-of-slow-roll regime. The distributions of fNL for various shapes and HSR proposals are shown as an example of how this procedure can be used within the context of Monte Carlo exploration of inflationary trajectories. We also show how allowing out-of-slow-roll behaviour can lead to a bispectrum that is relatively large for equilateral shapes.

D7-brane chaotic inflation

We analyze string-theoretic large-field inflation in the regime of spontaneously-broken supergravity with conventional moduli stabilization by fluxes and non-perturbative effects. The main ingredient is a shift-symmetric Kähler potential, supplemented by flux-induced shift symmetry breaking in the superpotential. The central technical observation is that all these features are present for D7-brane position moduli in Type IIB orientifolds, potentially allowing for a realization of the axion monodromy proposal in a string theory compactification. Furthermore, our model is explicit enough to address issues of control and moduli stabilization quantitatively. On the one hand, in the large complex structure regime the D7-brane position moduli inherit a shift symmetry from their mirror-dual Type IIA Wilson lines. On the other hand, the Type IIB flux superpotential generically breaks this shift symmetry and allows, by appealing to the large flux discretuum, to tune the relevant coefficients to be small. The shift-symmetric direction in D7-brane moduli space can then play the role of the inflaton: While the D7-brane circles a certain trajectory on the Calabi–Yau many times, the corresponding F-term energy density grows only very slowly, thanks to the above-mentioned tuning of the flux. To be successful our model requires that the dilaton, all complex structure moduli and all D7-brane moduli except the inflaton are fixed at leading order by fluxes. Then the large-field inflationary trajectory can be realized in a regime where Kähler, complex structure and other brane moduli are stabilized in a conventional manner, as we demonstrate using the example of the Large Volume Scenario.

PLANCK and WMAP constraints on generalised Hubble flow inflationary trajectories

We use the Hamilton-Jacobi formalism to constrain the space of possible single field, inflationary Hubble flow trajectories when compared to the WMAP and PLANK satellites Cosmic Microwave Background (CMB) results. This method yields posteriors on the space of Hubble Slow Roll (HSR) parameters that uniquely determine the history of the Hubble parameter during the inflating epoch. The trajectories are used to numerically determine the observable primordial power spectrum and bispectra that can then be compared to observations. Our analysis is used to infer the most likely shape of the inflaton potential V(ϕ) and also yields a prediction for, ℬ, the dimensionless amplitude of the non-Gaussian bispectrum.

Hybrid inflation in the complex plane

Supersymmetric hybrid inflation is an exquisite framework to connect inflationary cosmology to particle physics at the scale of grand unification. Ending in a phase transition associated with spontaneous symmetry breaking, it can naturally explain the generation of entropy, matter and dark matter. Coupling F-term hybrid inflation to soft supersymmetry breaking distorts the rotational invariance in the complex inflaton plane — an important fact, which has been neglected in all previous studies. Based on the δ N formalism, we analyze the cosmological perturbations for the first time in the full two-field model, also taking into account the fast-roll dynamics at and after the end of inflation. As a consequence of the two-field nature of hybrid inflation, the predictions for the primordial fluctuations depend not only on the parameters of the Lagrangian, but are eventually fixed by the choice of the inflationary trajectory. Recognizing hybrid inflation as a two-field model resolves two shortcomings often times attributed to it: the fine-tuning problem of the initial conditions is greatly relaxed and a spectral index in accordance with the PLANCK data can be achieved in a large part of the parameter space without the aid of supergravity corrections. Our analysis can be easily generalized to other (including large-field) scenarios of inflation in which soft supersymmetry breaking transforms an initially single-field model into a multi-field model.

Upper bound on the tensor-to-scalar ratio in GUT-scale supersymmetric hybrid inflation

We explore the upper bound on the tensor-to-scalar ratio r in supersymmetric (F-term) hybrid inflation models with the gauge symmetry breaking scale set equal to the value 2.86⋅1016 GeV, as dictated by the unification of the MSSM gauge couplings. We employ a unique renormalizable superpotential and a quasi-canonical Kähler potential, and the scalar spectral index ns is required to lie within the two-sigma interval from the central value found by the Planck satellite. In a sizable region of the parameter space the potential along the inflationary trajectory is a monotonically increasing function of the inflaton, and for this case, r≲2.9⋅10−4, while the spectral index running, |dns/dln⁡k|, can be as large as 0.01. Ignoring higher order terms which ensure the boundedness of the potential for large values of the inflaton, the upper bound on r is significantly larger, of order 0.01, for subplanckian values of the inflaton, and |dns/dln⁡k|≃0.006.

No-scale D-term inflation with stabilized moduli

We study the consistency of hybrid inflation and moduli stabilization, using the Kallosh–Linde model as an example for the latter. We find that F-term hybrid inflation is not viable since inflationary trajectories are destabilized by tachyonic modes. On the other hand, D-term hybrid inflation is naturally compatible with moduli stabilization due to the absence of a large superpotential term during the inflationary phase. Our model turns out to be equivalent to superconformal D-term inflation and it therefore successfully accounts for the CMB data in the large-field regime. Supersymmetry breaking can be incorporated via an O'Raifeartaigh model. For GUT-scale inflation one obtains stringent bounds on the gravitino mass. A rough estimate yields 105 GeV≲m3/2≲1010 GeV, contrary to naive expectation.

Cosmic inflation in a landscape of heavy-fields

Heavy isocurvature fields may have a strong influence on the low energy dynamics of curvature perturbations during inflation, as long as the inflationary trajectory becomes non-geodesic in the multi-field target space (the landscape). If fields orthogonal to the inflationary trajectory are sufficiently heavy, one expects a reliable effective field theory describing the low energy dynamics of curvature perturbations, with self-interactions determined by the shape of the inflationary trajectory. Previous work analyzing the role of heavy-fields during inflation have mostly focused in the effects on curvature perturbations due to a single heavy-field. In this article we extend the results of these works by studying models of inflation in which curvature perturbations interact with two heavy-fields. We show that the second heavy-field (orthogonal to both tangent and normal directions of the inflationary trajectory) may significantly affect the evolution of curvature modes. We compute the effective field theory for the low energy curvature perturbations obtained by integrating out the two heavy-fields and show that the presence of the second heavy-field implies the existence of additional self-interactions not accounted for in the single heavy-field case. We conclude that future observations will be able to constrain the number of heavy fields interacting with curvature perturbations.