Phantom dark energy and cosmological solutions without the Big Bang singularity

A. N. Baushev

Bogoliubov Laboratory of Theoretical Physics Joint Institute for Nuclear Research 141980 Dubna, Moscow Region, Russia

TAUP 2009, Italy, July, 2009. Introduction

Equations of state

p = αρ

Possible variants

1 α = 3 Relativistic matter α = 0 Non-relativistic matter 1 − 3 > α > −1 Quintessence α = −1 Cosmological constant α < −1 Phantom energy Figure: Caldwell, Kamionkowski, & Weinberg, 2003 Phantom energy models

∂ φ ∂ξφ L = − ξ − V (φ) 2

ρ = −φ˙2/2 + V (φ), p = −φ˙2/2 − V (φ)

³ ´ ˙2 p −φ /2 − V (φ) α ≡ = ³ ´ ρ −φ˙2/2 + V (φ) Phantom energy properties for various V (φ) I If the potential is not very steep (grows slower than V (φ) ∝ φ4), then α tends to −1, and the density becomes infinite only when t → ∞. I For steeper potentials a big rip singularity appears even if α → −1. I Even the parameter α can tend to −∞ for a very steep V (φ). I A very steep potential is necessary to provide a constant α < −1: for any polynomial potential, for instance, α tends to −1.

The case of V (φ) = m2φ2/2

φ¨ + 3Hφ˙ − m2φ = 0

r 2 m φ φ˙ ' mMp , H ' √ 3 Mp 6 The inevitability of the phantom field decay

Particle production in the cosmological gravitational field

Batista, Fabris, & Houndjo, 2007

I The universe is filled with a perfect fluid with α = const < −1 I The influence of the ’normal’ matter on the universe expansion was neglected. I the conformal time η is chosen so that η < 0, and the density becomes infinitive when η → −0

4 ρ = Cηβ, where β = norm 1 + 3α The system of cosmological equations for α = −4/3

We denote the phantom energy density by $ and its initial value by $0

− 2 $ ∝ η 3

µ ¶2 1 da 1 4 − 3 2 = 2 ($ + Cη ) (1) a dη 3Mp − 4 d($ + Cη 3 ) 1 da − 4 = ($ − 4Cη 3 ) (2) dη a dη dt = adη (3)

1 da 1 da H ≡ = a dt a2 dη Time dependence of the Hubble constant H Time dependence of the Hubble of α The universe properties after the phantom field decay

I It has just passed through the stage of very rapid (at least, exponential) expansion. I It is flat and homogeneous. I It is ’normal’ matter-dominated. I The Hubble constant is very large.

It is precisely these physical conditions that existed in our universe ∼ 13.7 milliard years ago, just after the inflation. Let us assume that the inflation was caused by the phantom field! Steady State on the average universe

I The universe leaves inflation being matter-dominated. I As it expands the density of matter decreases, while the phantom density grows: eventually the universe passes into the phantom-dominated stage. I The total density and the Hubble constant stop diminishing and begin to increase. Gradually the expansion becomes very fast, leading to an inflation-lake stage. I The phantom field decays into ’normal’ matter. I The cycle repeats. Thus, the universe eternally expands, while its density and other physical parameters oscillate over a wide range, never reaching the Plank values.