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Initial Conditions for Supersymmetric Inflation

We perform a numerical investigation of the fields evolution in the supersymmetric inflationary model based on radiative corrections. Supergravity corrections are also included. We find that, out of all the examined initial data, only about 10% give an adequate amount of inflation and can be considered as ''natural''. Moreover, these successful initial conditions appear scattered and more or less isolated.

A couple of years ago Linde [1] has proposed, in the context of non-supersymmetric (SUSY) grand unified theories (GUTs), a clever inflationary scenario based on a coupled system of two real scalar fields one of which is a gauge non-singlet. This field remains displaced from its vacuum value during inflation and provides the vacuum energy which drives inflation while the other, which is a gauge singlet, is the slowly varying field during inflation. The main advantage of this hybrid inflationary scenario is that it can reproduce, in contrast to previous realizations of inflation, the observed temperature fluctuations in the cosmic background radiation (CBR) with natural values of the coupling constants. However, inflation terminates abruptly and is followed by a "waterfall" regime during which topological defects are easily produced if they are predicted by the symmetry breaking associated with inflation.
Recently [2,3], two variants of the above scenario, both sharing its naturalness, were proposed in the context of SUSY GUTs. One of them [2] relies on the lowest order nonrenormalizable contribution to the superpotential. This scenario has been called smooth hybrid inflation since the termination of the inflationary period is not as abrupt as in Linde's scheme. Rather, the entrance of the system into the oscillatory phase is quite smooth. Moreover, the system, already from the beginning of inflation, follows a particular valley of minima that leads to a particular point of the vacuum manifold and, thus, there is no production of topological defects in this case. This is certainly an advantage of this scenario since it makes it applicable even to GUT models where the symmetry breaking associated with inflation does imply the existence of (possibly) cosmologically disastrous topological defects like domain walls or magnetic monopoles. The main advantage though of smooth hybrid inflation is that it can be associated to the SUSY GUT scale M SG ≃ 2 × 10 16 GeV consistent with the unification of the minimal supersymmetric standard model (MSSM) gauge couplings. A possible shortcoming of this scenario is that the relevant part of inflation takes place at relatively high values [(1 − 3) × 10 17 GeV] of the inflaton field and, thus, supergravity and superstring corrections may not be so easily tamable.
The second SUSY variant [3] of the hybrid inflationary scenario utilizes the one loop radiative corrections to the inflationary potential. These corrections are non-trivial since SUSY is broken at the inflationary trajectory in field space. The termination of inflation is as in the non-SUSY Linde's scenario and topological defects can be copiously produced. It is then important to make sure that no cosmologically dangerous topological defects come into existence as a result of the symmetry breaking associated to inflation. Moreover, as it turns out, the scale at which this symmetry breaking takes place is about five times smaller than M SG . This can be considered as a disadvantage of the scheme since it forces us to abandon the successful unification of the MSSM gauge coupling constants. In this case, inflation may be chosen to take place during an intermediate symmetry breaking with no magnetic monopoles or domain walls associated with it. The important advantage of this scenario is that the relevant part of inflation takes place at values of the inflaton field of order 10 16 GeV which are low enough for supergravity and superstring corrections to be under relatively good control [4].
For an inflationary scenario to be considered fully successful, one has to show that it is obtainable for a wide class of "natural" initial values of the fields and their time derivatives.
In ref. [5] this issue was investigated for smooth hybrid inflation. The result was striking and unexpected. It was found that, for almost all the examined initial conditions (except for a narrow transition region), we get inflation with an adequate number of e-foldings. This is certainly an important advantage of this inflationary scenario. The purpose of this work is to discuss the question of initial conditions for the second SUSY hybrid inflationary scenario which is based on radiative corrections. In particular, we will try to see if there exists a wide class of "natural" initial conditions, i.e., a class of comparable initial values of the fields with vanishing time derivatives for which the system falls at the bottom of the valley of minima so that its subsequent evolution along this valley produces an adequate amount of inflation.
To this end, we solve numerically the evolution equations of the system for various sets of initial conditions. Our analysis also includes cases with all initial field values being much smaller than the Planck scale, where the results are expected to be less affected by replacing global by local SUSY. At the end, non-vanishing initial time derivatives of the fields as well as supergravity corrections were also taken into account.
The SUSY inflationary scenario based on radiative corrections can be realized in the context of a SUSY GUT model with a (semisimple) gauge group G of rank ≥ 5 which breaks spontaneously directly to the standard model (SM) group G S at a scale M. The symmetry breaking of G to G S is obtained through a superpotential which includes the terms [6] W = κS(−M 2 +φφ). (1) Here,φ, φ is a conjugate pair of left-handed SM singlet superfields which belong to non-trivial representations of the gauge group G and reduce its rank through their vacuum expectation values (vevs) and S is a gauge singlet left-handed superfield. The superpotential in eq.(1) is in fact the only renormalizable superpotential consistent with a continuous U(1) R-symmetry under which W → e iθ W , S → e iθ S,φφ →φφ . The potential obtained from W in eq.(1), in the SUSY limit, is Vanishing of the D-terms is achieved along the D-flat directions where |φ| = |φ| . The SUSY vacuum lies on the particular D-flat directionφ * = φ. Restricting ourselves to this direction and performing appropriate gauge and R-transformations we can bring the complex S,φ,φ fields on the real axis, i.e., S ≡ σ √ 2 ,φ = φ ≡ 1 2 χ, where σ and χ are real scalar fields. The potential in eq.(2) then takes the form The vacua lie at < χ >= ±2M, < σ >= 0. We observe that the potential possesses an exact flat direction at χ = 0 with V (χ = 0, σ) = κ 2 M 4 . The mass squared of the field χ along this flat direction is given by m 2 χ = −κ 2 M 2 + 1 2 κ 2 σ 2 and remains non-negative for σ ≥ σ c = √ 2 M. This implies that, at χ = 0 and σ ≥ σ c , we obtain a flat-bottomed valley of minima which, as it stands, is not suitable for inflation. However, due to SUSY breaking along the bottom of this valley resulting from the non-zero value of V (χ = 0, σ), there exist non-trivial radiative corrections to the effective potential so that this valley acquires the negative slope which is necessary for inflation. The one loop corrected effective potential A region of the universe, where χ and σ happen to be almost uniform with negligible kinetic energies and with values close to the bottom of the valley of minima, follows this valley in its subsequent evolution and undergoes inflation. The temperature fluctuations of CBR produced during this inflation can be estimated to be [3] δT where GeV is the Planck mass, and N q ≃ 60 is the number of e-foldings of our present horizon scale during inflation. The Cosmic Background Explorer (COBE) result, δT /T ≃ 6.6 × 10 −6 , can then be reproduced with M = 5.82 × 10 15 GeV. Using eq.(6) one finds where σ q is the value of σ when the scale which evolved to our present horizon size crossed outside the de Sitter horizon during inflation. Inflation continues until σ = σ c where it terminates abruptly and is followed by a "waterfall", i.e., a sudden entrance into an oscillatory phase about a global minimum. This is checked by noting the magnitude of where the prime refers to derivatives with respect to σ . Since the system can fall into either of the two available global minima with equal probability, topological defects can be easily produced if they are predicted by the relevant symmetry breaking of the particular particle physics model one is employing.
We will now try to specify the initial conditions for the σ and χ fields which lead to the above described inflationary scenario. In other words, we will try to identify initial conditions for which the system falls at the bottom of the valley of minima of the effective potential so that its subsequent evolution along this valley produces an adequate amount of inflation.
We assume that, after "compactification" at some initial cosmic time, a region emerges in the universe where the scalar fields σ and χ and their time derivatives happen to be almost uniform and the energy density  5) and (6), though crucial for inflation, play no essential role in the issue of initial conditions and, thus, they will be ignored. In order to keep the formalism as clear as possible, we shall from now on use dimensionless variables for all relevant quantities as follows: where H is the Hubble parameter and ̺ the energy density. The evolution of the system in this region is governed by the following equations of motion where overdots denote derivatives with respect to the dimensionless timet, the rescaled Hubble parameterĤ is given bŷ andM = 2.39 × 10 −3 .
When theχ field starts oscillating,Ĥ(t) satisfies the equation as one deduces from eq.(19). Eqs. (14) and (18) then givė Substituting into eq.(20) we, furthermore, get Note that eqs. Upon substituting eq.(19) into eq.(13) we geẗ Forσ constant, this is a differential equation of the Emden type whose solution satisfying the (approximate) boundary conditions att 0 is given bŷ Eq.(11), fort ≫t 0 , averaged over one oscillation ofχ then giveŝ This equation implies where x(t) ≡tσ(t),ẋ(t) > 0.We see that x(t) is a periodic function of time corresponding to a bounded motion of x in a logarithmic potential. The turning point x m (> 0) is defined through the relation and eq.(26) becomes (x(t) > 0) where Φ(x) is the probability integral. The turning point x m can be estimated from eq.(27) and the fact thatẋ forσ(0) ≫ 1. This implies that the system,att =t 0 , starts from a position far away from the turning point which could subsequently be reached after the lapse of time δt ≈ whereχ m is the amplitude ofχ. The decreasing "frequency" ofσ which isχ m /2 ≤M 2 /σ soon becomes much smaller thanĤ and eq.(29) reduces to 3Ĥσ + 1 4χ The variation ofσ within one expansion time is then estimated to be |∆σ/σ| ∼χ 2 m /12 ≤ M 4 /3σ 2 ≪ 1. This justifies the assumption thatσ does not change much. The overall conclusion is that, at times much larger thant d ≈ 2/ √ 3M 2 , theχ field falls into the valley of minima of the potential in eq.( 12 ) and relaxes at the bottom of this valley whereas theσ field still remains more or less unchanged. After that, the radiative corrections in eq.(5) come into play and the system follows the valley of minima towards the supersymmetric vacuum and, therefore, the hybrid inflationary scenario is realized for initial values of the fields satisfying the inequalityσ ≫ 1,χ. These initial conditions, however, cannot be considered as totally satisfying because there is a considerable discrepancy between the initial values of the fields. Also, the inclusion of supergravity is expected to invalidate the above discussion of initial conditions which involve values of the fieldσ ≥ 1.
For SUSY inflation based on radiative corrections to be considered as a fully successful inflationary scenario, one must show that it is obtained for a wide class of initial conditions which are more "natural" than the ones just discussed. This can be checked only numerically.
3. 1 ≥χ ≥ 0.1, 1 ≥σ ≥ 0.1 at the same points as in (2) introducing equal positive initial velocities so that the initial kinetic energy equals the corresponding potential energy.
The last two cases, although corresponding to pretty low values of the initial energy density, were included since our results there are expected to remain essentially unaffected if we replace global by local SUSY with canonical Kaehler potential. On the contrary, the results of case (1) should be strongly affected by supergravity corrections. To see this we reexamined this case by including supergravity corrections with minimal Kaehler potential. This is done by replacing the potential energy density in eq.(12) and its derivatives with respect toχ and σ in eqs.(10) and (11) byV and its respective derivatives.
The integration of the two coupled equations was performed by implementing a variant of the Bulrish-Stoer [8] variable step method in a Fortran program. As a general rule, the initial step for the dimensionless time variablet was chosen to be 2 while the sought accuracy was put to 10 −15 . This choice was found to ensure reasonable stability in all cases.
The results of our search are shown in Figs Fig.1. This area, however, although leading to successful inflation cannot be considered as being "natural" since it requires relatively large differences between the initial values of the fields and is destroyed by supergravity as we shall soon see. Points following evolution type (c) are significantly better since, for many of them, the initial values of the fields are more or less of the same order of magnitude. These points, however, are scattered and do not appear to constitute an extended and well bounded continuous area as in the case of the smooth hybrid inflationary model. We have checked in more detail the area around most of the filled circle points and found out that almost all the points in their immediate neighborhood correspond to the evolution type (a) suggesting that evolution of type (c) occurs only for more or less isolated points. We conclude then that, in the model analyzed here, only about 10% of the examined points lead to successful inflation and lie in the area which can be considered "natural". Finally, the results with supergravity corrections included (Fig.4) show that the filled triangle area disappears and we are left with evolution of the types (a) and (b). This is to be expected, since the very steep walls of the potential in theσ direction do not allow theσ field to linger at large values for long. It is remarkable to realize that here also only about 10% of the examined points lead to adequate inflation. The initial energy density ρ can be calculated from eqs. (9),(12) for global SUSY or (9),(31) in the supergravity case with κ given by eq. (8) where σ Q /σ c is taken close to 2 [4]. It turns out that, for many initial conditions corresponding to type (c) evolution, we get adequately large initial energy densities. In summary, only a scattered set of "natural" but more or less isolated initial conditions, constituting about 10% of the examined cases, do lead to the SUSY inflationary model based on radiative corrections. It appears that this scenario is less satisfactory than smooth hybrid inflation which is obtainable for almost all initial conditions.

ACKNOWLEDGMENTS
This work was supported in part by E.U. grant ERBFMRXCT 960090 FIGURE CAPTIONS Fig. 1: Evolution patterns for theσ-χ system with vanishing initial velocities. Open circles represent points that do not lead to inflation. Filled triangles give adequate inflation having only theχ field oscillating.
Filled circles give adequate inflation with both fields oscillating initially.