40. Inflation: field-theoretic description (Inflationary perturbations 4)

Posted by Dmitry

May 9, 2008

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This is the next post in the series based on my lectures on cosmological perturbations. Last time I discussed how inflation can be described only in terms of effective equation of state (with negative pressure). Today I am going to show how this equation of state can be realized at the quasi-classical level of QFT.

In order to describe the physics of inflation, a QFT model should have a distinctive feature: its hydrodynamic modes (i.e., such modes that their relaxation time goes to infinity while the wavelength goes to infinity) have to be described by the effective equation of state $p\approx-\rho$ . As a simple working example, let us consider a self-interacting scalar field $\varphi$ with potential $V(\varphi)$ ; at the level of phenomenology it can be a fundamental or a composite field (condensate of some kind).

The energy density stored in the hydrodynamic modes of \varphi is given by

$\rho=\frac{1}{2}(\dot{\varphi})^{2}+V(\phi),$ (1)

while the corresponding pressure is

$p=\frac{1}{2}(\dot{\varphi})^{2}-V(\phi).$ (2)

The realization of the de Sitter stage is possible if the kinetic energy of the scalar field is negligible compared to its potential energy. More precisely, expansion of the Universe will accelerate if

$\ddot{\varphi}\ll H\dot{\varphi},$ $\dot{\varphi}\ll H\varphi.$ (3)

Indeed, as follows from the Eqs. (1) and (2), $\rho\approx V(\varphi)$ and $p\approx-V(\phi)\approx-\rho$ in this case, so the Universe is de Sitter-like.

Dynamics of inflationary stage is the determined by the equation of motion for the scalar field

$\ddot{\varphi}+3H\dot{\varphi}+\frac{\partial V}{\partial\varphi}=0,$ (4)

where the friction term is defined by the Friedmann equation

$H^{2}=\frac{8\pi}{3M_{P}^{2}}\left(\frac{1}{2}(\dot{\varphi})^{2}+V(\varphi)\right).$ (5)

When the slow roll conditions (3) are valid, Hubble friction in the Eq. (5) dominates over the kinetic term and scalar field starts to slowly roll down towards the minimum of its potential. In this regime, one effectively has

$\frac{8\pi}{M_{P}^{2}}V(\varphi)\frac{d\varphi}{dN}=-\frac{\partial V}{\partial\varphi}$ (6)

(where is again the number of e-folds) with the solution

$N(\varphi)=\frac{M_{P}^{2}}{8\pi}\int_{\varphi}^{\varphi_{{\rm max}}}\frac{V(\phi)d\varphi}{\partial V/\partial\varphi},$

determining the number of e-folds of accelerated expansion $a(N)=a_{i}e^{N}$ as a function of $\varphi$ (please note that the number of e-folds turns out to be a more appropriate variable than the physical time during accelerated expansion stage; there is a deep physics in this statement, as we will see later when will discuss stochastic approach to eternal inflation). De Sitter stage can start at some

$\varphi_{{\rm max}}\lesssim\varphi_{P}$

such that $V(\varphi_{P})\sim M_{P}^{4}$ and continue until the conditions (3) break down at $\varphi=\varphi_{{\rm SR}}.$ The value of the Hubble parameter

$H\sim\frac{\sqrt{V(\varphi)}}{M_{P}^{2}}$

will slowly ( $|\dot{H}|\ll H^{2}$ ) decrease from $H_{i}=H(\varphi_{{\rm max}})$ to $H_{f}=H(\varphi_{{\rm SR}})$ , while the value of scale factor will quasiexponentially grow.

Let us show what happens explicitly taking the simplest possible model with potential

$V(\varphi)=\frac{1}{2}m^{2}\varphi^{2}$ .

Slow roll conditions (3) are satisfied when

$M_{P}\lesssim\varphi\lesssim\frac{M_{P}^{2}}{m},$

i.e., in the very wide range of possible values of $\varphi$ if $m\ll M_{P}$ . From the Eq. (6) we find

$\varphi(t)\approx\varphi_{{\rm max}}-\frac{mM_{P}}{\sqrt{12\pi}}(t-t_{i}).$

Therefore, the Hubble parameter decreases quadratically with time, while the scale factor grows as

$a(t)=a_{0}e^{\int Hdt}=a_{f}e^{-\frac{m^{2}}{6}(t-t_{f})^{2}},$

where $a_{f}$ is its value in the end of inflation. The overall length of the de Sitter stage is given by

$\delta t=t_{f}-t_{i}\approx\sqrt{12\pi}\frac{\varphi_{{\rm max}}}{mM_{P}}\lesssim2\sqrt{6\pi}\frac{M_{P}}{m^{2}},$

while the total number of e-folds accumulated during inflation is

$N\lesssim\exp\left(\frac{M_{P}^{2}}{m^{2}}\right)\sim\exp\left(10^{10}\right),$

where we took $m=10^{-5}\, M_{P}$ consistent with COBE normalization. As we see, the overall de Sitter stage could be extremely long, and the the size of homogeneous isotropic region by many orders of magnitude may exceed the present horizon size. Only last 60 or so e-folds of inflation give rise to the structure of the gravitational potential seen at near-horizon scale in the present universe.

The last thing remained to be explained in this Section is the Hamilton-Jacobi formalism for inflation. Often, it is more convenient to represent the Hubble parameter H=H(t) as a function of field $\varphi$ itself (of course, this can be done only if the field $\varphi$ changes monothonically with time). Second order differential equation (4) is equivalent to a pair of equations for the field and the Hubble parameter

$\dot{\varphi}=-\frac{M_{P}^{2}}{4\pi}H'(\varphi),$

$(H'(\varphi))^{2}-\frac{12\pi}{M_{P}^{2}}H^{2}(\phi)=-\frac{32\pi^{2}}{m_{P}^{2}}V(\varphi).$ (7)

The Eq. (7) is known as the Hamilton-Jacobi equation for inflation. Defining the slow roll parameter

$\epsilon=\frac{M_{P}^{2}}{4\pi}\left(\frac{H'}{H}\right)^{2},$

one can rewrite it as

$H^{2}(\varphi)\left(1-\frac{1}{3}\epsilon(\varphi)\right)=\frac{8\pi}{3M_{P}^{2}}V(\varphi).$

The meaning for the slow roll parameter $\epsilon$ is clear from the Friedmann equation $\frac{\ddot{a}}{a}=H^{2}(1-\epsilon)$ – it shows how rapidly effective cosmological constant changes with time. The reason why we mention the Gamilton-Jacobi equation here is that inflationary observables are typically represented as functions of slow roll parameter(s) and the Hubble scale at a given scale, not at a given time.

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

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44. Cosmological perturbations in the presence of scalar field (Inflationary perturbations 5)

40. Inflation: field-theoretic description (Inflationary perturbations 4)

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