Marco Bruni, ICG, University of Portsmouth & Dipartimento di Fisica, Roma ``Tor Vergata” Paris 8/12/05

Dark energy from a quadratic equation of state

Marco BruniICG, 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

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

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).

2

“…my biggest blunder.”A. Einstein

Marco Bruni, ICG, University of Portsmouth & Dipartimento di Fisica, Roma ``Tor Vergata” Paris 8/12/05

Energy cons. & effective

RW dynamics:

Friedman constraint: Remarks:

1. If for a given EoS function P=P() there exists a such that P() = - , then has the dynamical role of an effective cosmological constant.

2. 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

Energy cons. & effective

Further remarks:3. From Raychaudhury eq., since , an

accelerated phase is achieved whenever P() < -/3.

4. 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.

5. 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

Energy cons. & effective

1. For a given P(), assume a exists.

2. Taylor expand around :

3. 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

Energy cons. & effective

4. Note that: , thus .

5. Assume and Taylor expand:

6. Then:

a) At O(1) in and O(0) in , in any theory of gravity, any P() that admits an effective behaves as -CDM;

b) For > -1 -> , i.e. is a de Sitter attractor.¯

¯

From energy cons. -> Cosmic No-Hair for non-linear EoS.

Marco Bruni, ICG, University of Portsmouth & Dipartimento di Fisica, Roma ``Tor Vergata” Paris 8/12/05

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

P=( + /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 <

a b c

Marco Bruni, ICG, University of Portsmouth & Dipartimento di Fisica, Roma ``Tor Vergata” Paris 8/12/05

P=( - /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

ba c

Marco Bruni, ICG, University of Portsmouth & Dipartimento di Fisica, Roma ``Tor Vergata” Paris 8/12/05

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

P=Po+

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.

a b c

Marco Bruni, ICG, University of Portsmouth & Dipartimento di Fisica, Roma ``Tor Vergata” Paris 8/12/05

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

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: evolution a-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.

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