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3.3. Shape of the fluctuation spectrum

A common assumption to be found in the literature is that the amplitude of a fluctuation when it comes within the horizon varies as some power of the mass contained within the horizon,

Equation 3.20 (3.20)

with the power-law index alpha, and the amplitude (fixed by M*) as free parameters.

This hypothesis is made purely for simplicity rather than from any compelling physical reason (a hypothesis of maximal ignorance). Zel'dovich (1972) has suggested an adiabatic spectrum with alpha = 0, i.e. (delta rho / rho)H = K where K is some constant. The motivation behind this choice of spectrum (often referred to as constant curvature fluctuations) is that it presents no preferred scales. Zel'dovich argues that the constant K might be related to the total entropy per baryon, the entropy being generated by the damping of short wavelength perturbations (though see Barrow and Matzner, 1977). If we choose alpha < 0, it implies that the Universe is non-Friedmannian with divergent curvature perturbations on large scales. If alpha > 0, it implies that the Universe was non-Friedmannian at early times. In the latter case it is likely that order-unity fluctuations on scales of the horizon would lead to a prolific production of black holes with masses approx M* (Carr, 1975; Barrow and Carr, 1978) (and a cosmological density in black holes Omegablack hole >> 1) and would disrupt the standard picture of nucleosynthesis. This is unattractive, but in order to avoid these problems, some physical process is required which cuts off the spectrum (3.20) such that delta rho / rho << 1 for M ltapprox M* (Press and Vishniac, 1980b).

Gott and Rees (1975) suggested an initial spectrum of isothermal perturbations obeying the Zel'dovich constant curvature hypothesis. If we let tequ be the epoch at which baryon and radiation densities are equal, rhoB(tequ) = rhoR(tequ), and Mequ is the total mass in baryons with the horizon at tequ, then from Eq. (3.20),

Equation 3.21 (3.21)

where MB(tH) is the total mass in baryons within the horizon at time tH. Gott and Rees' constant curvature hypothesis then yields delta rhoB / rhoB alpha MB-1/3. As in the case of adiabatic perturbations, any spectrum of the form (3.20) with alpha > 0 leads to non-Friedmannian behaviour on small scales. In the case of isothermal perturbations this occurs when perturbations are matter-dominated on horizon scales at early times. The evolution of non-linear isothermal perturbations, and their effect on primordial nucleosynthesis has been considered in some detail by Hogan (1978).

Finally we note that the observed entropy per baryon ratio may have an explanation in terms of baryon non-conserving and CP violating processes acting at very early times when T approx 1028 K. Several authors (Weinberg, 1982; Press and Vishniac, 1980b) have made the important point that if the entropy per baryon is fixed in terms of purely microscopic parameters, then any initial isothermal perturbations would be erased, and only adiabatic fluctuations could survive. Ways out of this possible difficulty are discussed in Section 9.

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