Dark energy from a quadratic equation of state
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Transcript of Dark energy from a quadratic equation of state
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).
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“…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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