Annu. Rev. Astron. Astrophys. 1991. 29: 325-362
Copyright © 1991 by . All rights reserved

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4.1 The Scale Invariant Spectrum

The most attractive feature of inflation, from the point of view of an astronomer, is the possibility that inflation may provide the seed perturbations that grow to form the structures we see today. In this section we provide an overview of how this is achieved and what difficulties arise.

We have discussed in Section 3.1 how inflation may lead to seed perturbations. To construct a working model from these ideas, we must (a) have a scenario that produces the observed structures from seed - perturbations and decide on the form of the seed perturbations needed, and (b) compute explicitly the nature of perturbations produced by inflation. A comparison of the outcomes of (a) and (b) will decide the measure of success achieved by inflation.

The first task can be achieved in principle by determining rho (x, t0) today observationally and extrapolating it backwards theoretically. In practice, of course, this is an impossibly difficult task! The structures seen today are the result of very complicated nonlinear evolution in the ``recent'' past (say 0 < z < 50), and we do not have a sufficiently well-defined theoretical formalism to allow us to extrapolate back in time. (Neither do we know rho (x, t0) to sufficient accuracy.) An indirect approach to this problem is, therefore, necessary.

We know from observations of MBR that the density perturbations in the universe must have been quite small (<< 1) at the time of decoupling (z approx 103), for all astrophysically relevant scales (27, 96, 128). We also notice from Equation 16 that all these relevant scales ``entered the Hubble radius'' in the radiation-dominated era, i.e. before decoupling. It follows that the linear approximation is valid when the density perturbations enter the Hubble radius. One can, therefore, characterize the density perturbations completely by giving the amplitude (<< 1) of each perturbation when it enters the Hubble radius. In other words, we need to specify the function

Equation 31 31.

where tenter (k) is the time at which the perturbation labeled by k enters the Hubble radius decided by the equation 2 pi k-1 a (tenter) = cH-1 (tenter). [Here most physical quantities depend only on the magnitude k = |k|; in such a situation, we simplify notation by writing, say delta (k, t), instead of delta (k, t), etc. We may also use deltak to denote the same quantity as was done earlier.]

The study of the linear perturbation theory allows us to evolve further a perturbation that enters the Hubble radius (see e.g. 98, 102). Perturbations with lambda < lambdaeq appeq 13 Mpc (Omega h2)-1 theta 2, carrying mass Meq = 3.19 x 1014 Msun (Omega h2)-2 theta 6, enter the Hubble radius in the radiation-dominated era; they grow by a small factor S - about O(10) or so - until t = teq and grow in proportion with a (t) afterwards. Those perturbations with lambda > lambdaeq enter the Hubble radius when the universe is matter-dominated and will grow as a (t) if the wavelength is bigger than the Jeans length. (For the scales in which we are interested, this condition is usually satisfied). Taking F (k) = alpha kn, one can easily work out delta (k, t) for all k at t > teq. We obtain

Equation 32a
Equation 32b 32.

where M is the mass carried by the particular perturbation. Scales with lambda > cH-1 (t) are still outside the Hubble radius at this time; it can be shown that they grow as a2(t) in the radiation phase and as a (t) in the matter dominated era; they also satisfy the scalings given above for lambda > lambdaeq (see e.g. 98).

The quantity k3 |deltak|2 is directly related to the r.m.s. fluctuation in the mass M (R) contained in a size R. For a power-law spectrum |deltak|2 propto kn, it can be easily seen that the average value of <(deltaM / M)2) in a region of size R is proportional to k3 |deltak|2 at k = R-1. Therefore (deltaM / M)2 propto M-(n+3)/3 for M < Meq and (deltaM / M)2 propto M-(n+7)/3 for M > Meq.

Some constraints on the form and amplitude of this spectrum can be obtained from the bounds on the temperature anisotropies of MBR. A perturbation of size lambda will produce anisotropies in the MBR at angular scale theta appeq 0.55' Omega h (lambda / 1 Mpc) (see e.g. 98). A scale of lambdaH = 65 (Omega h2)-1/2 Mpc, subtending an angle of thetaH appeq 0.870 Omega 1/2 (zdec / 1100)-1/2, will be entering the Hubble radius at the time of decoupling t = tdec. Temperature anistropies in MBR at larger angles (theta > thetaH approx 10) arise from perturbations that are still outside the Hubble radius at t = tdec and are due to Sachs-Wolfe effect: (DeltaT / T) approx 0.5 delta (k, tdec) (106). Fluctuations at smaller scales are due to baryons that are coupled to photons: (DeltaT / T) approx 0.33 deltabaryon (k, tdec) (see e.g. 66 and references cited therein). Bounds on (DeltaT / T) from (1°-30°) imply that delta ltapprox 10-4 at scales corresponding to (65-3000)h-1 Mpc today. Such a uniformity suggests that (n + 3) gtapprox -0.1. One can also show that if delta approx O(1)at M approx 1015 g appeq 3 x 10-19 Msun, there will be far too many compact black holes today, thus suggesting (n + 3) ltapprox 0.2 (21, 22). These considerations therefore favor (n + 3) = 0, i.e. a n = - 3 spectrum with the amplitude of about 10-6-10-4 when each perturbation enters the Hubble radius.

Zeldovich and, independently, Harrison had argued, based on theoretical considerations, that at the time of entering the Hubble radius, the perturbations should have the index n = -3 (51, 134). In other words, F (k) ident |delta (k, tenter (k))|2 propto k-3 or K3 F (k) should be a constant. In that case <(deltaM / M)2> will be independent of the scale R at t = tenter (k), giving equal power at all scales at the instant of entering the horizon.

Three points need to be stressed regarding the above discussion. First, note that k3 |delta (k, t)|2 is not expected to be scale-invariant (in fact, it will not be); it is only the quantity k3 |delta [k, tenter (k)]|2 that is expected to be independent of k. That is, each scale enters the Hubble radius with an amplitude which is independent of k; but, of course, each scale enters at a different time tenter (k). Second, the excess mass at scale R is in general related to the integrated power spectrum and depends on |delta (k)|2 at all 0 < k < R-1 rather than just to power at k = R-1. Finally, theoretical calculations are expected to fix only |delta (k, t)|2; the phase of delta (k, t) is not known. This is equivalent to supposing that [rho (x, t) - rhobar (t)] is a random variable with zero mean and some specified dispersion. What is determined by the theory is the power spectrum of. this random variable, which is the Fourier transform of the two-point correlation function <[rho (x + y, t) [rho (y, t)> where the brackets denote statistical averaging over the random phases of delta (k, t).

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