Inflation and supersymmetry

Volume 131B, number 1,2,3 PHYSICS LEqTERS 10 November 1983

INFLATION AND SUPERSYMMETRY

Andreas A L B R E C H T and Paul S T E I N H A R D T Department of Physics, University of Pennsylvania, Philadelphia, PA 19104, USA

Received 2 May 1983

The constraints on grand unified models required to obtain an inflationary universe scenario are outlined, especially a new assessment of the reheating that follows inflation. The implications for various supersymmetric models are discussed.

The new inflationary universe scenario based upon the Coleman-Weinberg grand unified theories (CW-GUTs) appears to be a simple approach to solving the cosmological homo- geneity, isotropy, entropy, flatness/oldness, monopole and domain wall problems of the standard hot big bang model [1-4]. The cosmological problems are solved because in CW- GUTs the universe as it cools undergoes an epoch of t remendous exponential (de Sitter) expansion followed by a return to a Fr iedmann- Rober tson-Walker (FRW) expansion. The new inflationary universe scenario also has two liabilities: (1) CW-GUTs require the unnatural fine-tuning of the Higgs field mass to zero; and (2) recent, calculations of the energy density perturbations in the new inflationary universe scenario have shown that their amplitude in CW-GUTs is too big to be consistent with astrophysical observations [5,6].

The general expression for the fractional fluctuation in the energy density, ~p/pIH, on a given scale when the physical scale (with comoving scale = l) equals the Hubble radius (H -1) in the FRW phase is [5,6]:

~P/DIH(I) = 0(1)H2/4;1,=,,~,) (1)

where 4~ is the time derivative of the expec- tation value of the Higgs field, ~b, and h(l) is the time at which the physical scale expanded bey- ond the Hubble radius during the de Sitter phase. Because ~b and 4~ are nearly constant during the inflationary epoch, the spectrum of

density fluctuations is nearly scale-invariant- qualitatively just the spectrum of fluctuations Zel 'dovieh and collaborators [7] proposed is necessary to explain the origin and organization of galaxies. Quantitatively, 6p/plH ~ 10 -4 is large enough to produce galaxies but not so large as to be inconsistent with the observed isotropy of the cosmic microwave background. For CW- GUTs, ~ ( t f )~ < H e (basically because H is the only dimensionful scale) [8], so the perturbation amplitude is O ( 1 ) - m u c h too big.

We believe that the proper perception of the situation is that the inflationary cosmology is, by itself, a totally self-consistent and powerful model for resolving many cosmological problems, including the problem of generating a scale-invariant spectrum of density fluctuations; but the phase transition necessary for inflation must derive from a G U T model different from CW-GUTs. In short, the problem is one of par- ticle physics, not cosmology.

In light of a deepening understanding of the inflationary transition, one can give a "prescrip- t ion" for the necessary form of the effective potential, V(~b), as a function of the order parameter, &, usually a Higgs field which measures the degree of spontaneous symmetry breaking (SSB). By V(~b), we mean the effective potential with the couplings set at values ap- propriate for the de Sitter phase of the transition [8]. The prescription is: (1) The symmetric phase, say ~ = 0, must be unstable (no barrier) or metastable (a barrier exists) with respect to

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Volume 131B, number 1,2,3 PHYSICS LETTERS 10 November 1983

the SSB phase. (2) There must be a "flat" portion (on the stable phase side of any barrier) followed by a steep portion decreasing to the SSB minimum. The flat portion must be such that V"(qb) ~ H 2, where H ~ M 2 / M p , M

v~/a(~b) along the flat portion of the potential and Mp = Planck mass = 1.2 × 1019 GeV; this condition insures that the evolution of ~b over the flat portion of the potential is slow compared to the expansion rate of the universe [2,3]. (3) The steep portion of the potential must be such that V"(~b)>> H 2, so that the time variation of • is rapid compared to the expansion rate and the Higgs field radiates particles to reheat the universe [4]. (4) The flat portion of V(tb) must have a width, A~b > H, or else de Sitter fluctuations speed up the evolution towards the stable minimum too much to obtain sufficient inflation [9, 10]. (5) There may be some ad- ditional constraints, if, as commonly occurs in (SUSY) models, the Higgs field responsible for reheating decouples from light fields.

We have argued previously that SUSY models in general and, in particular, SUSY models which employ the Wit ten-O'Raifear- taigh symmetry breaking scheme (WOSUSY models) are ideal candidates for meeting all these conditions [11]. We would like to present a reassessment of the reheating process in the WOSUSY models, which was claimed to be a serious problem. We will then consider other attempts made to employ SUSY to implement inflation and point out some difficulties.

The attraction of WOSUSY models is that all of the conditions listed above for a successful inflationary cosmology can be met without the fine-tuning of parameters. Furthermore, unlike other SUSY models, it is not difficult to arrange an effective potential where the SU(3) × SU(2) × U(1) phase is the stable minimum at zero temperature [12]. The effective potential in WOSUSY models as a function of some scalar field, ~b, can be expressed for ~b > M (where M is a fundamental scale that appears in the model) as [11]:

V((~ ) = C l M 4 ln(~b2/M 2)

+ c2M4(qb2/M 2 - ½c3) 2 + c4, (2)

where the first term represents the one loop corrected SUSY potential, the second term represents the form of the gravitational corrections (taking as our guide the form of the lowest order gravity corrections if the model derives from an N = 1 supergravity potential) and the constant term which must be adjusted to insure that V(~bM) = 0 at the minimum ibM (c3/2)t/2Mp (corresponding to setting the cosmological constant to zero in the SSB phase). The coefficients, ci, are functions of the scalar and gauge couplings which in turn obey renor- realization group relations. In particular cl > 0 for ~b ~> M and, depending upon the model, may change sign as the scale of ~b increases; Witten [13] originally depended upon this possibility in his inverse hierarchy scheme to dynamically generate a minimum for ~b >> M. Here, by in- troducing Mp (gravity) by hand into the model, the gravitational corrections guarantee a minimum at tb -Mp(c3 ~> 1). This avoids various technical problems with the inverse hierarchy scenario, while sacrificing all of its phenomenological advantages. Nevertheless, for our pur- poses it is sufficient to suppose that eq. (2) represents the generic form of an effective low energy theory derived from an O'Raifeartaigh SUSY model coupled to gravity.

The potential for ~b ~< M depends upon fur- ther details in the theory, but in general 4~ = 0 may be unstable (no barrier) or metastable (with barrier) with respect to the SSB minimum. As the temperature of the universe decreases, a fluctuation drives a region of radius O ( H -1) (a causal horizon volume) coherently towards an SSB minimum and at some point ~b evolves to a value O(M). The cosmology only depends upon the subsequent evolution of tb which can be determined from eq. (2) and is independent of the fine-tuning of parameters.

The Hubble parameter during the evolution is H 2 ~ 2 7 r c 2 c 2 M 4 / 3 M 2. For ~b < Mp, V"(~b) ~- ClM4/gb 2 ~ H 2 for Cl ~ c2c 2 and the potential is flat enough for inflation to occur; since A~b > M-> H during the inflation de Sitter fluctuations do not play a role in the evolution of ~b. At ~b = ~bM, V"(~b) ~ 4 c 2 c 3 M 4 / M 2 > H 2, ~ > M 2 > H 2, and reheating is at least feasible. Fur-

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ther, because q~ > H 2 during inflation, one even finds that the amplitude of density fluctuations, eq. (1), is small in these models. Thus, criteria (1)-(4) and small density fluctuations are obtained naturally in these m o d e l s - and without fine-tuning of parameters (to obtain t~p/PlH 10 -4 generally requires an adjustment, but not a fine-tuning, of parameters) [6].

The problem with these models, as reported in ref. [11], is that even though the curvature of the potential at ~b = ~bu is greater than H 2, 4) decouples from light fields. The evolution of ~b and the radiation energy density, Pr, can be expressed as [4,11]

q~ + 3H4; + 8/4~ + V'(~b) = 0, (3)

/gr + 4Hot = 8, (4)

where

t~ = o t ~ 2 ( M / 6 ) a ~ Id, ( ~ ) ~ 2 (5)

represents the energy density per unit time converted from the scalar field energy density, p~ = ~4) 2 + V(~b), to pr. Here 8 has been expressed in the most general form Consistent with its dimensions. For CW-GUTs, where no M is present, a = 0 and p~ is found to rapidly be converted to pr as 4> oscillates about 4) = ~bu. In ref. [11], it was shown that a = 6 in the Dimo- poulos-Raby [12] WOSUS~( model, because of the extreme decoupling of 4) from light fields in this model. As a result, reheating is difficult.

One way around the reheating problem is to find a model with less decoupling. We find that the necessary criterion is that once the 8-term in eq. (3) is greater than or equal to the 3H4~ term, then the remaining p~ is converted to p~ efficiently. Thus, one can show analytically ttrat sufficient decoupling such that a ~< 2 in eq. (5) still does not induce a reheating problem. However, we have found it difficult to find a model that maintains the desirable features of these models and which does not yield a >> 2 or very strong decoupling.

An important new result is that, even with strong decoupling and Pr = 0 initially, significant reheating of light particles is possible. Since t = th/4~ ~ H -1 as 4) oscillates around the SSB minimum, eqs. (3)-(5) can be solved for all 8 to

show that, independent of the degree of decoupling, there is always a time (~/.£-1) at which p, = Pr, after which the universe is radiation dominated [14]. This result can be crucial both for the inflationary universe and for the cosmology of invisible axions. The reheating temperature (i.e. when p~ = Or) for the WOSUSY model is given by Tr ~ plr/4 = O(1)a 1/2

× (gp/c~M)(a-1)/2(M/Mp)(a-2)/2g. (6)

For M = 1012 GeV, and ~bu = 1019 GeV, as in the Dimopoulos-Raby WOSUSY model [12], Tr 10 MeV, and ref. [11] was correct in concluding that the reheating was insufficient to generate baryon asymmetry after the transition. However, we also see the way out of this dilemma: If M - O (1016-17 GeV) the reheating temperature is Tr = 10 l°-13 GeV, large enough to be compatible with the observed entropy of the universe and potentially large enough that baryon asymmetry can be generated after fur- ther expansion in the reheated FRW phase (of course, it remains to be shown in a detailed model that the light particles produced during reheating are the baryons and other particles we observe in the universe.) The amount of inflation is rather insensitive to M and M - O (1017 GeV) even generates 8 p / p [ n ~ 10 -4 in eq. (1)!

Recently, there have been attempts to obtain inflation from other classes of SUSY models. In these models, the effective potential is a poly- nomial in th with quadratic, cubic, quartic [15] and, in some cases, higher power nonrenor- malizable interactions [16]. A careful study of the conditions necessary for inflation shows that without extreme fine-tuning and inclusion of non-renormalizable terms, A~b is less than H (unless ~bM, the SSB minimum, exceeds Mp) [14]. If h~b < H, de Sitter fluctuations cause tb to evolve too quickly to obtain sufficient inflation. The suggestion that the fundamental mass scales in these theories should be set at M~ (so-called primordial inflation [16] makes matters even worse since than H ~> Mp and tb < H for all ~b < ~bM -- Mp. The de Sitter fluctuations again spoil inflation.

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Thus, only the WOSUSY models at present can obtain inflation and small density fluctuations, without fine-tuning of parameters. Naturally such a model must be reconciled with phenomenology, which was not our goal here, but at least we have shown that there already exists a generic class of models that can lead to a successful inflationary cosmology.

This work would not have been possible without the previous collaborative work with J. Bardeen, S. Dimopoulos, E. Kolb, W. Fischler, S. Raby and M. Turner. We especially thank M. Gell-Mann, who urged us to reconsider the reheating problem independent of other phenomenological considerations. This work was supported in part by D O E contract No. EY-76-C-02-3071 and a D O E Outstanding Junior Investigator Grant.

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Inflation and supersymmetry