© CAMBRIDGE UNIVERSITY PRESS 1999

1.6.7 Inflation Summary

The key features of all inflation scenarios are a period of superluminal expansion, followed by (``re-'')heating which converts the energy stored in the inflaton field (for example) into the thermal energy of the hot big bang.

Inflation is generic: it fits into many versions of particle physics, and it can even be made rather natural in modern supersymmetric theories as we have seen. The simplest models have inflated away all relics of any pre-inflationary era and result in a flat universe after inflation, i.e., = 1 (or more generally ₀ + = 1). Inflation also produces scalar (density) fluctuations that have a primordial spectrum

(1.12)

where V is the inflaton potential and n_p is the primordial spectral index, which is expected to be near unity (near-Zel'dovich spectrum). Inflation also produces tensor (gravity wave) fluctuations, with spectrum

(1.13)

where the tensor spectral index n_t (1 - n_p) in many models.

The quantity (1 - n_p) is often called the ``tilt'' of the spectrum; the larger the tilt, the more fluctuations on small spatial scales (corresponding to large k) are suppressed compared to those on larger scales. The scalar and tensor waves are generated by independent quantum fluctuations during inflation, and so their contributions to the CMB temperature fluctuations add in quadrature. The ratio of these contributions to the quadrupole anisotropy amplitude Q is often called T/S Q_t² / Q_s²; thus the primordial scalar fluctuation power is decreased by the ratio 1 / (1 + T/S) for the same COBE normalization, compared to the situation with no gravity waves (T = 0). In power-law inflation, T/S = 7(1 - n_p). This is an approximate equality in other popular inflation models such as chaotic inflation with V() = m^{2 2} or ⁴. But note that the tensor wave amplitude is just the inflaton potential during inflation divided by the Planck mass, so the gravity wave contribution is negligible in theories like the supersymmetric model discussed above in which inflation occurs at an energy scale far below m_Pl. Because gravity waves just redshift after they come inside the horizon, the tensor contributions to CMB anisotropies corresponding to angular wavenumbers >> 20, which came inside the horizon long ago, are strongly suppressed compared to those of scalar fluctuations. The indications from presently available data (Netterfield et al. 1997; cf. Tegmark 1996, and Silk's article in this volume) are that the CMB amplitude is rather high for 200, approximately in agreement with the predictions of standard CDM with h 0.5, _b 0.1, and scalar spectral index n_p = 1. This suggests that there is little room for gravity-wave contributions to the low- CMB anisotropies, i.e., that T/S << 1. Thus tests of inflation involving the gravity-wave spectrum will be very difficult. Fortunately, inflation can be tested with the data expected soon from the next generation of CMB experiments, since it makes very specific and discriminatory predictions regarding the relative locations of the acoustic peaks in the spectrum, for example the ratio of the first peak location to the spacing between the peaks ₁ / 0.7-0.9 (Hu & White 1996, Hu et al. 1997).

On the other hand, inflation is also Alice's restaurant where, according to the Arlo Guthrie song, ``...you can get anything you want ... excepting Alice''. It's not even clear what ``Alice'' you can't get from inflation. It was initially believed that inflation predicts a flat universe. But now we know that you

• can get ₀ < 1 (with = 0), as discussed in the previous section.
• can make models consistent with supergravity and the sort of four-dimensional physics expected from superstrings, in which case one may expect that inflation occurs at a relatively low energy scale, which implies T/S 0, a low reheat temperature implying no production of topological defects and presumably requiring that baryosynthesis occur at the electroweak phase transition, and plenty of inflation implying that ₀ 1.
• can alternatively get strings or other topological defects such as textures during or at the end of inflation (e.g. Hodges & Primack 1991) - which however probably requires tuning of the inflation and/or string model, for example to avoid a fractal pattern of structure-forming defects, which would conflict with the observed homogeneity of structure on very large scales.

And in many versions of inflation, the most reasonable answer to the question ``what happened before inflation'' appears to be eternal inflation, which implies that in most of the meta-universe, exponentially far beyond our horizon, inflation never stopped.