55. Eternal inflation: stochastic approach 3 (Inflationary perturbations 7)

Posted by Dmitry
June 5, 2008

Last time we have found that dynamics of the inflaton field (more precisely, its expectation value w.r.t. to the distribution among different Hubble patches) is determined by the Langevin equation.

As we know, there are two descriptions of the Brownian motion: in terms of the Langevin equation and in terms of the Fokker-Planck equation describing diffusion of the probability distribution to find a randomly moving particle at given \Phi in a given moment of time t. This Fokker-Planck equation has the form

\frac{\partial\rho(\phi,t)}{\partial t}=\frac{H_{0}^{3}}{8\pi^{2}}\frac{\partial^{2}\rho}{\partial\Phi^{2}}+\frac{1}{3H_{0}}\frac{\partial}{\partial\Phi}\left(\frac{\partial V}{\partial\Phi}\rho\right). (1)

The probability distribution \rho(\phi,t) describes how the values of \phi are distributed among different Hubble patches in the multiverse and how they are correlated with each other in different Hubble patches.

The general solution to the Fokker-Planck equation (1) is given by

\rho=\exp\left(-\frac{4\pi^{2}\delta V(\Phi)}{3H_{0}^{4}}\right)\sum_{n}c_{n}\psi_{n}(\Phi)\exp\left(-\frac{E_{n}H_{0}^{3}(t-t_{0})}{4\pi^{2}}\right), (2)

where \psi_{n} and E_{n} are respectively the eigenfunctions and the eigenvalues of the effective Hamiltonian

\hat{H}=-\frac{1}{2}\frac{\partial^{2}}{\partial\Phi^{2}}+W(\Phi). (3)

Here

W(\Phi)=\frac{8\pi^{4}}{9H_{0}^{8}}\left(\frac{\partial\delta V}{\partial\Phi}\right)^{2}-\frac{2\pi^{2}}{3H_{0}^{4}}\frac{\partial^{2}\delta V}{\partial\Phi^{2}} (4)

is a functional of the scalar field potential V(\Phi). It is often denoted as the superpotential due to its “supersymmetric” form: the Hamiltonian (3) can be rewritten as

\hat{H}=\hat{Q}^{\dagger}\hat{Q}, (5)

where

\hat{Q}=-\partial/\partial\Phi+v'(\Phi) (6)

with

v(\phi)=4\pi^{2}V(\Phi)/(3H_{0}^{4}). (7)

The eigenfunctions and eigenstates of the Hamiltonian (3) satisfy the Schrodinger equation

\frac{1}{2}\frac{\partial^{2}\psi_{n}}{\partial\Phi^{2}}+(E_{n}-W(\Phi))\psi_{n}=0. (8)

Its solutions have the following features:

1. The eigenvalues of the Hamiltonian (3) are all positive definite due to the supersymmetric form of W(\Phi). Assuming normalizable wavefunctions \psi_{n}(\Phi), the ground state \psi_{0}(\Phi) corresponds to the zero eigenvalue and defines the steady state solution of the Fokker-Planck equation. One can easily check that the ground state has the form

\psi_{0}\sim{\rm Const.}e^{-v(\Phi)}, (9)

where {\rm Const} is defined from the normalization condition.

2. The contributions from eigenfunctions of excited states \psi_{n>0}(\Phi) to the solution (2) become exponentially quickly damped with time.

Arbitrary correlation functions of the inflaton field \Phi can be easily found given the probability distribution \rho(\Phi,t). Namely,

\langle\Phi^{n}(t)\rangle=({\rm Norm})^{-1}\int d\Phi\Phi^{n}\rho(\Phi,t), (10)

where {\rm Norm}=\int d\Phi\rho(\Phi,t).

In the limit t\to\infty only the ground state in the solution (2) survives, and one can write

\langle\Phi^{n}(t\to\infty)\rangle=({\rm Norm})^{-1}\int d\Phi\Phi^{n}\exp(-2v(\Phi)). (11)

This expression in principle completely defines the asymptotic structure of the distribution of \Phi in spacetime (and the structure of the spacetime itself). If we are interested in finite t behavior, the situation becomes more complicated because the constants c_{n} in the general solution (2) are defined as convolutions of \psi_{n} with the distribution function \rho_{0}(\Phi,t=t_{0}) determining the structure of the spacetime in the initial moment of time, and we can hardly determine such a thing from observations.

Finally, let me make some concluding remarks. First of all, we notice that the stochastic formalism works well exactly in the regime when quantum fluctuations of the inflaton field become stronger than the effect of the classical force \frac{\partial V}{\partial\phi} - i.e., in the regime where stochastic force in the Langevin equation is stronger than the lassical force. Therefore, stochastic approach is suitable for the description of physics where the standard inflationary perturbation theory breaks down. It also describes the structure of spacetime at superlarge scales. This structure according to the expression (11) is rather special - the hypersurface of constant \Phi in the spacetime is strongly fluctuating at very long scales and the distribution of these fluctuations is very far from being Gaussian (compare it to what happens at smaller scales where quantum fluctuations of the inflaton are suppressed with respect to the effect of the classical force - at the present horizon scale fluctuations are almost exactly gaussian!). In fact, one can show that the hypersurface of the constant value of the inflaton field is fractal, but this is beyond the scope of our present discussion.

Finally, let me introduce a couple of exercises which will be a good warm-up if you are willing to learn the physics of eternal inflation better.

Problem: chaotic inflation with quadratic potential

Consider a chaotic inflationary model with potential V(\phi)=\frac{1}{2}m^{2}\phi^{2}, derive the corresponding Lanfevin and Fokker-Planck equations. Hint: instead of the world time t use the number of efolds N=\log(a).

Problem: average length of inflationary stage

Calculate the average number of efoldings \langle N\rangle for the model with potential V(\phi)=V_{0}+\frac{1}{3}\mu\phi^{3}. Note that inflation ends in the regime when stochastic random force f has negligible effect on the evoluton of the inflaton field.

Problem: non-gaussianities from superlarge scales

According to the \delta N formalism, the curvature perturbation \zeta can be defined as \zeta=N-\langle N\rangle. Show how to calculate the arbitrary one-point correlation function of the form \langle\zeta^{n}\rangle using the stochastic approach.

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Cosmological perturbations and large scale structure, Cosmology, Lectures on de Sitter spacetime, Master Mason

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