In the description of linear perturbation theory given above, we assumed that some small perturbations existed in the early universe which are amplified through gravitational instability. To provide a complete picture we need a mechanism for generation of these initial perturbations. One such mechanism is provided by inflationary scenario which allows for the quantum fluctuations in the field driving the inflation to provide classical energy density perturbations at a late epoch. (Originally inflationary scenarios were suggested as pseudo-solutions to certain pseudo-problems; that is only of historical interest today and the only reason to take the possibility of an inflationary phase in the early universe seriously is because it provides a mechanism for generating these perturbations.) We shall now discuss how this can come about.
The basic assumption in inflationary scenario is that the universe underwent a rapid - nearly exponential - expansion for a brief period of time in very early universe. The simplest way of realizing such a phase is to postulate the existence of a scalar field with a nearly flat potential. The dynamics of the universe, driven by a scalar field source, is described by:
where Mpl = (8 G)-1/2. If the potential is nearly flat for certain range of , we can introduce the " slow roll-over" approximation, under which these equations become:
For this slow roll-over to last for reasonable length of time, we need to assume that the terms ignored in the Eq. (151) are indeed small. This can be quantified in terms of the parameters:
which are taken to be small. Typically the inflation ends when this assumption breaks down. If such an inflationary phase lasts up to some time tend then the universe would have undergone an expansion by a factor expN(t) during the interval (t, tend) where
One usually takes N 65 or so.
Before proceeding further, we would like to make couple of comments regarding such an inflationary phase. To begin with, it is not difficult to obtain exact solutions for a(t) with rapid expansion by tailoring the potential for the scalar field. In fact, given any a(t) and thus a H(t) = ( / a), one can determine a potential V() for a scalar field such that Eq. (151) are satisfied (see the first reference in ). One can verify that, this is done by the choice:
Given any H(t), these equations give ((t), V(t)) and thus implicitly determine the necessary V(). As an example, note that a power law inflation, a(t) = a0 t p (with p >> 1) is generated by:
while an exponential of power law
can arise from
Thus generating a rapid expansion in the early universe is trivial if we are willing to postulate scalar fields with tailor made potentials. This is often done in the literature.
The second point to note regarding any inflationary scenarios is that the modes with reasonable size today originated from sub-Planck length scales early on. A scale 0 today will be
at the end of inflation (if inflation took place at GUT scales) and
at the beginning of inflation if the inflation changed the scale factor by A 1030. Note that begin < Lp for 0 <3 Mpc!! Most structures in the universe today correspond to transplanckian scales at the start of the inflation. It is not clear whether we can trust standard physics at early stages of inflation or whether transplanckian effects will lead to observable effects [20, 21].
Let us get back to conventional wisdom and consider the evolution of perturbations in a universe which underwent exponential inflation. During the inflationary phase the a(t) grows exponentially and hence the wavelength of any perturbation will also grow with it. The Hubble radius, on the other hand, will remain constant. It follows that, one can have situation in which a given mode has wavelength smaller than the Hubble radius at the beginning of the inflation but grows and becomes bigger than the Hubble radius as inflation proceeds. It is conventional to say that a perturbation of comoving wavelength 0 "leaves the Hubble radius" when 0 a = dH at some time t = texit(0). For t > texit the wavelength of the perturbation is bigger than the Hubble radius. Eventually the inflation ends and the universe becomes radiation dominated. Then the wavelength will grow ( t1/2) slower than the Hubble radius ( t) and will enter the Hubble radius again during t = tenter(0). Our first task is to relate the amplitude of the perturbation at t = texit(0) with the perturbation at t = tenter(0).
We know that for modes bigger than Hubble radius, we have the conserved quantity (see Eq. (97)
At the time of re-entry, the universe is radiation dominated and entry (2/3) . On the other hand, during inflation, we can write the scalar field as a dominant homogeneous part plus a small, spatially varying fluctuation: (t, x) = 0(t) + f(t, x). Perturbing the equation in Eq. (151) for the scalar field, we find that the homogeneous mode 0 satisfies Eq. (151) while the perturbation, in Fourier space satisfies:
Further, the energy momentum tensor for the scalar field gives [with the "dot" denoting (d / d) = a(d / dt)]:
It is easy to see that is negligible at t = texit since
Using the conservation law exit = entry, we get
Thus, given a perturbation of the scalar field fk during inflation, we can compute its value at the time of re-entry, which - in turn - can be used to compare with observations.
Equation (166) connects a classical energy density perturbation fk at the time of exit with the corresponding quantity k at the time of re-entry. The next important - and conceptually difficult - question is how we can obtain a c-number field fk from a quantum scalar field. There is no simple answer to this question and one possible way of doing it is as follows: Let us start with the quantum operator for a scalar field decomposed into the Fourier modes with qk(t) denoting an infinite set of operators:
We choose a quantum state | > such the expectation value of k(t) vanishes for all non-zero k so that the expectation value of (t, x) gives the homogeneous mode that drives the inflation. The quantum fluctuation around this homogeneous part in a quantum state | > is given by
It is easy to verify that this fluctuation is just the Fourier transform of the two-point function in this state:
Since k characterises the quantum fluctuations, it seems reasonable to introduce a c-number field f(t, x) by the definition:
This c-number field will have same c-number power spectrum as the quantum fluctuations. Hence we may take this as our definition of an equivalent classical perturbation. (There are more sophisticated ways of getting this result but none of them are fundamentally more sound that the elementary definition given above. There is a large literature on the subject of quantum to classical transition, especially in the context of gravity; see e.g. ) We now have all the ingredients in place. Given the quantum state | >, one can explicitly compute k and then - using Eq. (166) with fk = k - obtain the density perturbations at the time of re-entry.
The next question we need to address is what is | >. The free quantum field theory in the Friedmann background is identical to the quantum mechanics of a bunch of time dependent harmonic oscillators, each labelled by a wave vector k. The action for a free scalar field in the Friedmann background
can be thought of as the sum over the actions for an infinite set of harmonic oscillators with mass m = a3 and frequency k2 = k2 / a2. (To be precise, one needs to treat the real and imaginary parts of the Fourier transform as independent oscillators and restrict the range of k; just pretending that qk is real amounts the same thing.) The quantum state of the field is just an infinite product of the quantum state k[qk, t] for each of the harmonic oscillators and satisfies the Schrodinger equation
If the quantum state k[qk, t'] of any given oscillator, labelled by k, is given at some initial time, t', we can evolve it to final time:
where K is known in terms of the solutions to the classical equations of motion and [qk', t'] is the initial state. There is nothing non-trivial in the mathematics, but the physics is completely unknown. The real problem is that unfortunately - in spite of confident assertions in the literature occasionally - we have no clue what [qk', t'] is. So we need to make more assumptions to proceed further.
One natural choice is the following: It turns out that, Gaussian states of the form
preserve their form under evolution governed by the Schrodinger equation in Eq. (172). Substituting Eq. (174) in Eq. (172) we can determine the ordinary differential equation which governs Bk (t). (The Ak (t) is trivially fixed by normalization.) Simple algebra shows that Bk(t) can be expressed in the form
where fk is the solution to the classical equation of motion:
For the quantum state in Eq. (174), the fluctuations are characterized by
Since one can take different choices for the solutions of Eq. (176) one get different values for k and different spectra for perturbations. Any prediction one makes depends on the choice of mode functions. One possibility is to choose the modes so that k represents the instantaneous vacuum state of the oscillators at some time t = ti. (That is Re Bk(ti) = (1/2)k2(ti), say). The final result will then depend on the choice for ti. One can further make an assumption that we are interested in the limit of ti -; that is the quantum state is an instantaneous ground state in the infinite past. It is easy to show that this corresponds to choosing the following solution to Eq. (176):
which is usually called the Bunch-Davies vacuum. For this choice,
where the second result is at t = texit which is what we need to use in Eq. (166), (Numerical factors of order unity cannot be trusted in this computation). We can now determine the amplitude of the perturbation when it re-enters the Hubble radius. Eq. (166) gives:
One sees that the result is scale invariant in the sense that k3|k|entry2 is independent of k.
It is sometimes claimed in the literature that scale invariant spectrum is a prediction of inflation. This is simply wrong. One has to make several other assumptions including an all important choice for the quantum state (about which we know nothing) to obtain scale invariant spectrum. In fact, one can prove that, given any power spectrum (k), one can find a quantum state such that this power spectrum is generated (for an explicit construction, see the last reference in ). So whatever results are obtained by observations can be reconciled with inflationary generation of perturbations.
To conclude the discussion, let us work out the perturbations for one specific case. Let us consider the case of the 4 model for which
we can write
so that the result in Eq. (180) becomes:
We do get scale invariant spectrum but the amplitude has a serious problem. If we take N 50 and note that observations require k3/2 k ~ 10-4 we need to take 10-15 for getting consistent values. Such a fine tuning of a dimensionless coupling constant is fairly ridiculous; but over years inflationists have learnt to successfully forget this embarrassment.
Our formalism can also be used to estimate the deviation of the power spectrum from the scale invariant form. To the lowest order we have
Let us define the deviation from the scale invariant index by (n - 1) = (d ln2 / d lnk). Using
one finds that
Thus, as long as and are small we do have n 1; what is more, given a potential one can estimate and and thus the deviation (n-1).
Finally, note that the same process can also generate spin-2 perturbations. If we take the normalised gravity wave amplitude as hab = (16 G)1/2 eab , the mode function behaves like a scalar field. (The normalisation is dictated by the fact that the action for the perturbation should reduce to that of a spin-2 field.) The corresponding power spectrum of gravity waves is
Comparing the two results
we get (tensor / scalar)2 16 << 1. Further, if (1 - n) 4 (see Eq. (187) with ~ ) we have the relation (tensor / scalar)2 (3) (1 - n) which connects three quantities, all of which are independently observable in principle. If these are actually measured in future it could act as a consistency check of the inflationary paradigm.