53. Eternal inflation: stochastic approach 1 (Inflationary perturbations 7)

Posted by Dmitry
June 3, 2008

This post is the next in the series devoted to the discussion of physics of cosmological perturbations. Today I will start the physical regime which is ofthe most interest for me - regime where super-Hubble cosmological perturbations are of the order 1.

As we have found, inflation typically predicts slightly red power spectrum of primordial perturbations due to the necessity to have graceful exit from inflation. “Red” tilt of the power spectrum means that the amplitude of primordial perturbations is larger at smaller k or longer length scales. Why the tilt of inflationary perturbations should be read, can also be easily understood if we recall that longer length scales correspond to earlier moments of time in the evilution of the Unverse.

Let us take for simplicity a chaotic inflationary model with a monotonic potential V(\phi) having a single minimum \phi=0. Earlier moments of time correspond to larger values of \phi, because inflaton field slowly rolls down towards its minimum \phi=0 during inflation. At the moment of time t=t_{0} the modes with k=a(t_{0})H(t_{0}) cross the horizon and their amplitude freezes at the level

\sqrt{\langle\delta\phi_{k}^{2}\rangle}\approx\frac{H(t_{0})}{2\pi}. (1)

Clearly, the earlier is the the moment t_{0}, the larger is H(t_{0}), since, according to the Friedmann equation

H^{2}(t_{0})=\frac{8\pi}{3M_{P}^{2}}V(\phi(t_{0})). (2)

Therefore, the amplitude of generated primordial perturbations is larger at earlier times than at later times.

If we follow this logic, we can conclude that there may exist wavelength

\lambda_{EI}=\frac{2\pi}{k_{EI}} (3)

such that the relative amplitude of primordial perturbations \delta_{k} will become of order of 1, and our perturbative treatment of primordial perturbations (i.e., description in terms of background field \phi and small fluctuations \delta\phi near it) may break down. This is indeed the case for many inflationary models and another typical prediction of inflation. Again, considering chaotic inflatioary model with monotonic potential V(\phi) let us find the moment of time t_{EI} when the inflationary perturbation theory breaks down. During one Hubble time \Delta t\sim H^{-1} the value of the inflaton field decreases by

\Delta\phi\sim\dot{\phi}\Delta t\sim\frac{1}{3H^{2}}\frac{\partial V}{\partial\phi}\sim\frac{M_{P}^{2}}{8\pi V}\frac{\partial V}{\partial\phi} (4)

due to the action of the classical force (derivative of the potential w.r.t. the scalar field) and, as we said, there will be fluctuations \delta\phi which for l\sim k^{-1}\sim H^{-1} have the amplitude

|\delta\phi|\sim\frac{H}{2\pi}\sim\sqrt{\frac{2V}{3\pi M_{P}^{2}}}. (5)

When

(M_{P}^{2}/V)(\partial V/\partial\phi)\lesssim\sqrt{V/M_{P}^{2}} (6)

or, in other words, when V\gtrsim\epsilon M_{P}^{4}, where \epsilon is the usual slow roll parameter, \Delta\phi\sim\delta\phi, and at the corresponding time scale inflationary perturbation theory breaks down. Typically, at a similar time scale (and associated wave length) non-gaussianity parameter NG defined at the previous lecture also becomes of the order of 1.

Let us find the critical value of field \phi_{EI} for the chaotic inflationary model with the potential \lambda\phi^{4}. Using our estimations we find that

\phi_{EI}\sim\lambda^{-1/6}M_{P}. (7)

Now, if we set initial condition for inflation at planckian energy density which corresponds to \phi_{I}\sim\lambda^{-1/4}M_{P}, we see that the inflation can proceed in the non-perturbative regime for quite a long time if \lambda is small enough.

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Cosmological perturbations and large scale structure, Cosmology, Fellow Craft, Lectures on de Sitter spacetime, Quantum field theory

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