Cosmic scalar fields with flat potential ∗

The dynamics of cosmic scalar fields with flat potential is studied. Their contribution to the expansion rate of the universe is analyzed, and their behaviour in a simple model of phase transitions is discussed. ∗Work supported by the research program FP52 of the Foundation for Fundamental Research of Matter (FOM). ∗∗e-mail: [email protected] The universe is observed to expand [1], and this expansion is suspected to have accelerated at a late stage of its evolution [2, 3]. Furthermore, observation of the cosmic microwave background indicates the universe is spatially flat [4]. The energy density of scalar fields has been recognized to contribute to the expansion of the universe [5, 6, 7], and has been proposed to explain inflation [8], as well as the presently observed expansion [9, 10, 11, 12, 13]; in ref. [14] an attempt has been made to confront the data with the predictions for a minimally coupled scalar field with an a priori unknown potential. An overview of models with many references has been given in [15]. As suggested by the standard model of the electroweak interactions and its many proposed small-distance extensions with or without supersymmetry, the dynamics of the universe involves a large number of scalar fields interacting with gravity, gauge fields and fermionic matter. The observed isotropy and homogeneity of the universe do not allow for the existence of long-range electric and magnetic fields, but neutral scalar fields can have non-trivial dynamics in an expanding FRW-type universe. For a number of minimally coupled scalar fields φi with a potential V [φ] the relevant dynamical equations for a flat universe (k = 0) are φ̈i + 3Hφ̇i + V,i = 0, 1 2 ∑ i φ̇i + V [φ] + ρ = 3H2 8πG , (1) where H is the Hubble parameter, and ρ is the contribution of matter to the density of the universe. It should be noted, that we absorb any cosmological constant in a constant contribution to the potential V [φ]. Constant scalar fields. As observed in [5, 6, 7] simple non-trivial solutions to the coupled equations (1) are provided by constant fields at stationary points of the potential: V,i = 0, Vm + ρ = 3H2 8πG . (2) Here Vm is the value of the potential at the stationary point (presumably a minimum), and ρ represents the density of ordinary relativistic and cold nonrelativistic matter, with present densities ρr,0 and ρnr,0. Then the last equation becomes in terms of the scale factor σ(t) = a(t)/a0: σ̇ = hσ + 8πG 3 ( ρr,0 σ2 + ρnr,0 σ ) , h = 8πGVm 3 , (3) provided Vm > 0. Hence for large universes the late-time behaviour of the scale factor is described by a(t) ∼ a0e0. (4) This is consistent with the observed present state of the universe if the Hubble parameter is h ≃ 70 km/sec/Mpc, of the order of the inverse life time of the universe. As is well-known, the corresponding value of Vm is extremely small: Vm = ρc ≃ 10−122 in Planck units. In contrast, for Vm = 0, we obtain the standard radiation/matter dominated solutions with a(t) ∼ tκ, where κ = 1/2 or κ = 2/3. If Vm = −3ω2/(8πG) < 0, 1 there are oscillating solutions with maximal amplitude ā = σ̄a0 given by the solution of ωσ̄ − 8πG 3 (ρnr,0 σ̄ + ρr,0) = 0. (5) The last two cases can be consistent with a flat universe only if the density of matter and radiation equals or exceeds the critical density ρc, which is definitely contradicted by the best current estimates of the densities of luminous and cold dark matter. Flat potentials. A different evolution of the Hubble parameter H(t) is obtained by allowing one or more scalar fields to depend on time. We consider the situation where the first field is dynamical: φ1 = φ(t), whilst all other fields are constant: φi = vi (i = 2, 3, ..., N). Then V (φ) = V [φ1 = φ;φi = vi]. Thus we have φ̈+ 3Hφ̇+ V ′ = 0, 1 2 φ̇ + V (φ) + ρ = 3H2 8πG . (6) In general any solution depends sensitively on the initial value φinit, which is difficult to control. However, this problem is absent if the potential is invariant under field translations φ̃ = φ+ ǫ. Therefore it is of some interest to study flat potentials V (φ) = V0. Such potentials are characteristic for Goldstone bosons, and they occur in many supersymmetric models. From the flatness of V one derives a conservation law for the field momentum per co-moving volume: φ̈+ 3Hφ̇ = 0 ⇒ σφ̇ = γ = constant. (7) After multiplication by σ6 the other equation then becomes γ2 2 + ρσ + V0σ 6 = 3 8πG ( σH )2 = 1 24πG ( dσ3 dt )2 . (8) The solutions can be characterized qualitatively as in the case of constant scalar fields: a. For V0 > 0, the potential provides the leading term, and for large a we find as before: a(t) ≃ a0 e0, h = √ 8πGV0 3 . (9) b. For V0 = 0, the leading term is provided by the matter density ρ, and the evolution of the scale factor is again of the form a(t) ∼ tκ, where κ = 1/2 for relativistic matter, and κ = 2/3 for cold non-relativistic matter. c. Finally, for V0 < 0 there are oscillating solutions, with the maximal scale ā = σ̄a0 a solution of γ2 2 + ρ̄ σ̄ + V0 σ̄ 6 = 0, (10) which is a direct generalization of eq.(5). Matter-dominated regime. The precise evolution of the scale factor and the scalar field can be solved analytically in the matter-dominated regime, in which ρa3 = constant. In the following we describe the solutions in some detail.