Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials

We investigate the construction of diffusions consisting of infinitely numerous Brownian particles moving in $\mathbb{R}^d$ and interacting via logarithmic functions (two-dimensional Coulomb potentials). These potentials are very strong and act over a long range in nature. The associated equilibrium states are no longer Gibbs measures. We present general results for the construction of such diffusions and, as applications thereof, construct two typical interacting Brownian motions with logarithmic interaction potentials, namely the Dyson model in infinite dimensions and Ginibre interacting Brownian motions. The former is a particle system in $\mathbb{R}$, while the latter is in $\mathbb{R}^2$. Both models are translation and rotation invariant in space, and as such, are prototypes of dimensions $d=1,2$, respectively. The equilibrium states of the former diffusion model are determinantal or Pfaffian random point fields with sine kernels. They appear in the thermodynamical limits of the spectrum of the ensembles of Gaussian random matrices such as GOE, GUE and GSE. The equilibrium states of the latter diffusion model are the thermodynamical limits of the spectrum of the ensemble of complex non-Hermitian Gaussian random matrices known as the Ginibre ensemble.Comment: Published in at http://dx.doi.org/10.1214/11-AOP736 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

The CMB and the measure of the multiverse

In the context of eternal inflation, cosmological predictions depend on the choice of measure to regulate the diverging spacetime volume. The spectrum of inflationary perturbations is no exception, as we demonstrate by comparing the predictions of the fat geodesic and causal patch measures. To highlight the effect of the measure---as opposed to any effects related to a possible landscape of vacua---we take the cosmological model, including the model of inflation, to be fixed. We also condition on the average CMB temperature accompanying the measurement. Both measures predict a 1-point expectation value for the gauge-invariant Newtonian potential, which takes the form of a (scale-dependent) monopole, in addition to a related contribution to the 3-point correlation function, with the detailed form of these quantities differing between the measures. However, for both measures both effects are well within cosmic variance. Our results make clear the theoretical relevance of the measure, and at the same time validate the standard inflationary predictions in the context of eternal inflation.Comment: 28 pages; v2: reference added, some clarification

Is Our Universe Natural?

It goes without saying that we are stuck with the universe we have. Nevertheless, we would like to go beyond simply describing our observed universe, and try to understand why it is that way rather than some other way. Physicists and cosmologists have been exploring increasingly ambitious ideas that attempt to explain why certain features of our universe aren't as surprising as they might first appear.Comment: Invited review for Nature, 11 page

Universality of the Volume Bound in Slow-Roll Eternal Inflation

It has recently been shown that in single field slow-roll inflation the total volume cannot grow by a factor larger than e^(S_dS/2) without becoming infinite. The bound is saturated exactly at the phase transition to eternal inflation where the probability to produce infinite volume becomes non zero. We show that the bound holds sharply also in any space-time dimensions, when arbitrary higher-dimensional operators are included and in the multi-field inflationary case. The relation with the entropy of de Sitter and the universality of the bound strengthen the case for a deeper holographic interpretation. As a spin-off we provide the formalism to compute the probability distribution of the volume after inflation for generic multi-field models, which might help to address questions about the population of vacua of the landscape during slow-roll inflation.Comment: 24 pages, 5 figure

A geometric bound on F-term inflation

We discuss a general bound on the possibility to realise inflation in any minimal supergravity with F-terms. The derivation crucially depends on the sGoldstini, the scalar field directions that are singled out by spontaneous supersymmetry breaking. The resulting bound involves both slow-roll parameters and the geometry of the K\"ahler manifold of the chiral scalars. We analyse the inflationary implications of this bound, and in particular discuss to what extent the requirements of single field and slow-roll can both be met in F-term inflation.Comment: 14 pages, improved analysis, references added, matches published versio

On Inflation with Non-minimal Coupling

A simple realization of inflation consists of adding the following operators to the Einstein-Hilbert action: (partial phi)^2, lambda phi^4, and xi phi^2 R, with xi a large non-minimal coupling. Recently there has been much discussion as to whether such theories make sense quantum mechanically and if the inflaton phi can also be the Standard Model Higgs. In this note we answer these questions. Firstly, for a single scalar phi, we show that the quantum field theory is well behaved in the pure gravity and kinetic sectors, since the quantum generated corrections are small. However, the theory likely breaks down at ~ m_pl / xi due to scattering provided by the self-interacting potential lambda phi^4. Secondly, we show that the theory changes for multiple scalars phi with non-minimal coupling xi phi dot phi R, since this introduces qualitatively new interactions which manifestly generate large quantum corrections even in the gravity and kinetic sectors, spoiling the theory for energies > m_pl / xi. Since the Higgs doublet of the Standard Model includes the Higgs boson and 3 Goldstone bosons, it falls into the latter category and therefore its validity is manifestly spoiled. We show that these conclusions hold in both the Jordan and Einstein frames and describe an intuitive analogy in the form of the pion Lagrangian. We also examine the recent claim that curvature-squared inflation models fail quantum mechanically. Our work appears to go beyond the recent discussions.Comment: 14 pages, 2 figures. Version 2: Clarified findings and improved wording. Elaborated important sections and removed an unnecessary section. Added references. Version 3: Updated towards JHEP version. Version 4: Final JHEP versio

Inflation on the Brane with Vanishing Gravity

Many existing models of brane inflation suffer from a steep irreducible gravitational potential between the branes that causes inflation to end too early. Inspired by the fact that point masses in 2+1 D exert no gravitational force, we propose a novel unwarped and non-supersymmetric setup for inflation, consisting of 3-branes in two extra dimensions compactified on a sphere. The size of the sphere is stabilized by a combination of a bulk cosmological constant and a magnetic flux. Computing the 4D effective potential between probe branes in this background, we find a non-zero contribution only from exchange of level-1 KK modes of the graviton and radion. Identifying antipodal points on the 2-sphere projects out these modes, eliminating entirely the troublesome gravitational contribution to the inflationary potential.Comment: 19 pages, 11 figures, JHEP forma

Scalar Three-point Functions in a CDL Background

Motivated by the FRW-CFT proposal by Freivogel, Sekino, Susskind and Yeh, we compute the three-point function of a scalar field in a Coleman-De Luccia instanton background. We first compute the three-point function of the scalar field making only very mild assumptions about the scalar potential and the instanton background. We obtain the three-point function for points in the FRW patch of the CDL instanton and take two interesting limits; the limit where the three points are near the boundary of the hyperbolic slices of the FRW patch, and the limit where the three points lie on the past lightcone of the FRW patch. We expand the past lightcone three-point function in spherical harmonics. We show that the near boundary limit expansion of the three-point function of a massless scalar field exhibits conformal structure compatible with FRW-CFT when the FRW patch is flat. We also compute the three-point function when the scalar is massive, and explain the obstacles to generalizing the conjectured field-operator correspondence of massless fields to massive fields.Comment: 42 pages + appendices, 10 figures; v2, v3: minor correction

Spinning Conformal Correlators

We develop the embedding formalism for conformal field theories, aimed at doing computations with symmetric traceless operators of arbitrary spin. We use an index-free notation where tensors are encoded by polynomials in auxiliary polarization vectors. The efficiency of the formalism is demonstrated by computing the tensor structures allowed in n-point conformal correlation functions of tensors operators. Constraints due to tensor conservation also take a simple form in this formalism. Finally, we obtain a perfect match between the number of independent tensor structures of conformal correlators in d dimensions and the number of independent structures in scattering amplitudes of spinning particles in (d+1)-dimensional Minkowski space.Comment: 46 pages, 3 figures; V2: references added; V3: tiny misprint corrected in (A.9