© CAMBRIDGE UNIVERSITY PRESS 1999

1.6.5 A Supersymmetric Inflation Model

We have already considered, in connection with cold dark matter candidates, why supersymmetry is likely to be a feature of the fundamental theory of the particle interactions, of which the present ``Standard Model'' is presumably just a low-energy approximation. If the higher-energy regime within which cosmological inflation occurs is described by a supersymmetric theory, there are new cosmological problems that initially seemed insuperable. But recent work has suggested that these problems can plausibly be overcome, and that supersymmetric inflation might also avoid the fine-tuning otherwise required to explain the small inflaton coupling corresponding to the COBE fluctuation amplitude. Here the problems will be briefly summarized, and an explanation will be given of how one such model, due to Ross & Sarkar (1996; hereafter RS96) overcomes them. (An interesting alternative supersymmetric approach to inflation is sketched in Dine et al. 1996.)

When Pagels and I (1982) first suggested that the lightest supersymmetric partner particle (LSP), stable because of R-parity, might be the dark matter particle, that particle was the gravitino in the early version of supersymmetry then in fashion. Weinberg (1982) immediately pointed out that if the gravitino were not the LSP, it could be a source or real trouble because of its long lifetime ~ M_Pl² / m_3/2³ ~ (m_3/2 / TeV)^-3 10³ s, a consequence of its gravitational-strength coupling to other fields. Subsequently, it was realized that supersymmetric theories can naturally solve the gauge hierarchy problem, explaining why the electroweak scale M_EW ~ 10² GeV is so much smaller than the GUT or Planck scales. In this version of supersymmetry, which has now become the standard one, the gravitino mass will typically be m_3/2 ~ TeV; and the late decay of even a relatively small number of such massive particles can wreck BBN and/or the thermal spectrum of the CBR. The only way to prevent this is to make sure that the reheating temperature after inflation is sufficiently low: T_RH 2 x 10⁹ GeV (for m_3/2 = TeV) (Ellis, Kim, & Nanopoulos 1984, Ellis et al. 1992).

This can be realized in supergravity theories rather naturally (RS96). Define M M_Pl / (8 )^1/2 = 2.4 x 10¹⁸ Gev. Break GUT by the Higgs field with vacuum expectation value (vev) <> ~ 10¹⁶ GeV. Break supersymmetry by a gaugino condensate < > ~ (10¹³ GeV)³; then the gravitino mass is m_3/2 ~ <> / M² ~ TeV. Inflation is expected to inhibit such breaking, so it must occur afterward. The inflaton superpotential has the form I = ² M f( / M), with the corresponding potential

(1.9)

with minimum at ₀. Demanding that at this minimum the potential actually vanishes V(₀) = 0, i.e., that the cosmological constant vanishes, implies that I(₀) = (ðI / ð)₀ = 0. The simplest possibility is I = ² ( - ₀)² / M. Requiring that ðV / ð|₀ = 0 for a sufficiently flat potential, implies that ₀ = M and that the second derivative also vanishes at the origin; thus

(1.10)

(Holman, Raymond, & Ross 1984). This particular inflaton potential is of the ``new inflation'' type, and corresponds to tilt n_p = 0.92 and a number of e-folds during inflation

(1.11)

assuming that the starting value of the inflaton field _in is sufficiently close to the origin (which has relatively small but nonvanishing probability - the field presumably has a broad initial distribution). Matching The COBE fluctuation amplitude requires that / M = 1.4 x 10^-4, which in turn implies that N ~ 10³, m ~ ² / M ~ 10¹¹ GeV, T_RH ~ 10⁵ GeV (parametric resonance reheating does not occur). Such a low reheat temperature insures that there will be no gravitino problem, and requires that the baryon asymmetry be generated by electroweak baryogenesis - which appears to be viable as long as the theory contains adequately large CP violation.

Note the following features of the above scenario: inflation occurs at an energy scale far below the GUT scale, so there is essentially no gravity wave contribution to the large-angle CMB fluctuations (i.e., T / S 0) even through there is significant tilt (n_p = 0.92 for the particular potential above); there is a low reheat temperature, so electroweak baryogenesis is required; and the universe is predicted to be very flat since there are many more e-folds than required to solve the flatness problem.