On the Necessity of Phantom Fields for Solving the Horizon Problem in Scalar Cosmologies

Abstract

We discuss the particle horizon problem in the framework of spatially homogeneous and isotropic scalar cosmologies.To this purpose we consider a Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime with possibly non-zero spatial sectional curvature (and arbitrary dimension), and assume that the content of the universe is a family of perfect fluids, plus a scalar field that can be a quintessence or a phantom (depending on the sign of the kinetic part in its action functional). We show that the occurrence of a particle horizon is unavoidable if the field is a quintessence, the spatial curvature is non-positive and the usual energy conditions are fulfilled by the perfect fluids. As a partial converse, we present three solvable models where a phantom is present in addition to a perfect fluid, and no particle horizon appears.

Integrable scalar cosmologies with matter and curvature

“…To our knowledge, the first examples of this inverse approach date back to 1980's and 1990's: we will mention, in particular, the papers by Lucchin and Matarrese [23], Barrow [2], Ellis and Madsen [12], Eashter [11]. More recently, nice "inverse" results have been obtained by Dimakis, Karagiorgos, Zampeli, Paliathanasis, Christodoulakis and Terzis [10], and by Barrow and Paliathanasis [3]; the same approach is also partly employed in [13], for the case of a phantom field. The feature specified in the cited papers to determine the scalar field potential is, for example, the dependence on cosmic time of one of the following observables: the scale factor, the Hubble parameter, the ratio between the pressure and the density produced by the scalar field alone, or jointly by scalar field and matter.…”

“…In the opposite case Θ(τ ) < δ k , we say that there is a particle horizon at time τ ; when k 0 the condition for a particle horizon reads Θ(τ ) < +∞, and this happens at some time τ if and only if it happens at all times τ after the Big Bang. Many FLRW cosmologies present particle horizons; it was shown in [13] that any FLRW cosmology with k 0, a (minimally coupled) scalar field and a matter fluid with equation of state p (m) = wρ (m) has a particle horizon (Θ(τ ) < +∞ for all τ after the Big Bang) if the matter fluid fulfills in the strict sense the strong energy condition (i.e., if the inequalities for ρ (m) , w in Eq. (2.1.13) hold strictly, with replaced by >).…”

“…Due to this, the hypothesis in [13, Eq. ( 34)] is not satisfied, which explains why the general conclusions of [13,Prop. 1] do not hold in this case.…”

“…(c) In most of the integrable cases presented in this work, a particle horizon appears; this fact can be checked by hand noting that the reciprocal of the scale factor, viewed as a function of cosmic time, diverges in a non-integrable way at the Big Bang. In the case of non-positive spatial curvature, the deep reason for this fact is explained in [13]; therein it is shown that a particle horizon occurs in all homogeneous and isotropic cosmologies with non positive spatial curvature, a self-interacting scalar field minimally coupled to gravity and a matter fluid with equation of state p (m) = w ρ (m) , fulfilling the strong energy condition. As shown in [13], the particle horizon is absent if, instead of a canonical scalar field, one considers a phantom field whose action functional contains an anomalous term corresponding to a negative kinetic energy.…”

“…In the case of non-positive spatial curvature, the deep reason for this fact is explained in [13]; therein it is shown that a particle horizon occurs in all homogeneous and isotropic cosmologies with non positive spatial curvature, a self-interacting scalar field minimally coupled to gravity and a matter fluid with equation of state p (m) = w ρ (m) , fulfilling the strong energy condition. As shown in [13], the particle horizon is absent if, instead of a canonical scalar field, one considers a phantom field whose action functional contains an anomalous term corresponding to a negative kinetic energy. It would be of some interest to search for FLRW integrable cosmologies with a phantom scalar field and matter; this subject is left to future investigations.…”