Dark energy from a quadratic equation of state

1. Dark energy from a quadratic equation of state Marco Bruni ICG, Portsmouth & Dipartimento di Fisica, Tor Vergata (Rome) & Kishore Ananda ICG, Portsmouth Marco Bruni, ICG, University of Portsmouth & Dipartimento di Fisica, Roma ``Tor Vergata” Paris 8/12/05

2. Outline • Motivations • Non-linear EoS and energy conservation • RW dynamics with a quadratic EoS • Conclusions Marco Bruni, ICG, University of Portsmouth & Dipartimento di Fisica, Roma ``Tor Vergata” Paris 8/12/05

3. Motivations • Acceleration (see Bean and other talks): • modified gravity; • cosmological constant ; • modified matter. • Why quadratic, P=Po +  +   /c ? • simplest non-linear EoS, introduces energy scale(s); • Mostly in general, energy scale -> effective cosmological constant ; • qualitative dynamics is representative of more general non-linear EoS’s; • truncated Taylor expansion of any P() (3 parameters); • explore singularities (brane inspired). “…my biggest blunder.” A. Einstein 2 Marco Bruni, ICG, University of Portsmouth & Dipartimento di Fisica, Roma ``Tor Vergata” Paris 8/12/05

4. Energy cons. & effective  • RW dynamics: • Friedman constraint: • Remarks: • If for a given EoS function P=P() there exists a such that P() = - , then  has the dynamical role of an effective cosmological constant. • A given non-linear EoS P() may admit more than one point . If these points exist, they are fixed points of energy conservation equation. Marco Bruni, ICG, University of Portsmouth & Dipartimento di Fisica, Roma ``Tor Vergata” Paris 8/12/05

5. Energy cons. & effective  • Further remarks: • From Raychaudhury eq., since , an accelerated phase is achieved whenever P() < -/3. • Remark 3 is only valid in GR. Remarks 1 and 2, however, are only based on conservation of energy. This is also valid (locally) in inhomogeneous models along flow lines. Thus Remarks 1 and 2 are valid in any gravity theory, as well as (locally) in inhomogeneous models. • Any point is a de Sitter attractor (repeller) of the evolution during expansion if +P()<0 (>0) for <  and +P()>0 (<0) for > . Marco Bruni, ICG, University of Portsmouth & Dipartimento di Fisica, Roma ``Tor Vergata” Paris 8/12/05

6. Energy cons. & effective  • For a given P(), assume a exists. • Taylor expand around : • Keep O(1) in  =  -  and integrate energy conservation to get: Marco Bruni, ICG, University of Portsmouth & Dipartimento di Fisica, Roma ``Tor Vergata” Paris 8/12/05

7. Energy cons. & effective  • Note that: , thus . • Assume and Taylor expand: • Then: • At O(1) in  and O(0) in , in any theory of gravity, any P() that admits an effective  behaves as -CDM; • For  > -1  -> , i.e.  is a de Sitter attractor. ¯ ¯ Marco Bruni, ICG, University of Portsmouth & Dipartimento di Fisica, Roma ``Tor Vergata” Paris 8/12/05 From energy cons. -> Cosmic No-Hair for non-linear EoS.

8. P=( +  /c) • Po =0,  = ± 1 • dimensionless variables: • Energy cons. and Raychaudhuri: • Friedman: Marco Bruni, ICG, University of Portsmouth & Dipartimento di Fisica, Roma ``Tor Vergata” Paris 8/12/05

9. P=( + /c) a b c • parabola: K=0; above K=+1, below K=-1 • dots: various fixed points; thick lines: separatrices • a:  > -1/3, no acc., qualitatively similar to linear EoS (different singularity) • b: -1<  <-1/3, acceleration and loitering below a threshold  • c:  < -1,  , de Sitter attractor, phantom for  <  Marco Bruni, ICG, University of Portsmouth & Dipartimento di Fisica, Roma ``Tor Vergata” Paris 8/12/05

10. P=( - /c) a b c • a:  < -1, all phantom, M in the past, singular in the future • b: -1<  <-1/3,  , de Sitter saddles, phantom for  >  • c:  >-1/3, similar to b, but with oscillating closed models • b and c: for  <  first acc., then deceleration Marco Bruni, ICG, University of Portsmouth & Dipartimento di Fisica, Roma ``Tor Vergata” Paris 8/12/05

11. P=Po+ • dimensionless variables: • Energy cons. and Raychaudhuri: • Friedman: Marco Bruni, ICG, University of Portsmouth & Dipartimento di Fisica, Roma ``Tor Vergata” Paris 8/12/05

12. P=Po+ a b c • a: Po>0, <-1: phantom for  > , recollapsing flat and oscillating closed models • b: Po>0, -1<<-1/3:similar to lower part of a • c: Po<0, -1/3<:phantom for  < , de Sitter attractor, closed loitering models. Marco Bruni, ICG, University of Portsmouth & Dipartimento di Fisica, Roma ``Tor Vergata” Paris 8/12/05

13. Full quadratic EoS • Left: =1, <-1, two , phantom in between • Right: =-1, >-1/3, two , phantom outside Marco Bruni, ICG, University of Portsmouth & Dipartimento di Fisica, Roma ``Tor Vergata” Paris 8/12/05

14. Conclusions • Non-linear EoS: • worth exploring as dark energy or UDM (but has other motivations); • dynamical, effective cosmological constant(s) mostly natural; • Cosmic No-Hair from energy conservation: evolutiona-la-CDM at O(0) in dP/d() and O(1) in  =  -  , in any theory gravity. • Quadratic EoS: • simplest choice beyond linear; • represents truncated Taylor expansion of any P() (3 parameters); • very reach dynamics: • allows for acceleration with and without  ; • Standard and phantom evolution, phantom -> de Sitter (no “Big Rip”); • Closed models with loitering, or oscillating with no singularity; • singularities are isotropic (as in brane models, in progress). • Constraints: high z, nucleosynthesis (>0), perturbations. Marco Bruni, ICG, University of Portsmouth & Dipartimento di Fisica, Roma ``Tor Vergata” Paris 8/12/05