Non-linear Equation of State:
Dark Energy Parametrization,
Dynamics
 and Singularities
Marco Bruni
ICG, Portsmouth & Dipartimento di Fisica, Tor Vergata (Rome)

Credits
Work with:
 Kishore Ananda, ICG, Portsmouth:
Ananda & Bruni, Physical Review D, 74, 023523 (2006).   astro-ph/0512224
Ananda & Bruni, Physical Review D, 74, 023524 (2006).        gr-qc/0603131
In progress:
 Amedeo Balbi & Claudia Quercellini, Tor Vergata (Roma 2)
Preliminary attempts:
Alan Coley and Woei Chet, Dalhousie University, Canada
Valerio Bozza, Salerno

Outline
Standard cosmology
Motivations
Non-linear EoS and energy conservation
RW dynamics with a quadratic EoS
Bianchi I and V dynamics with a quadratic EoS
Large scale perturbations of the past attractor
Work in progress: affine EoS as unified DM and cosmological tests
Summary

Standard cosmology
Receipt for modeling based on 3 main ingredients:
Homogeneous isotropic background, FRW models
Perturbations (e.g. CMB)
Numerical simulations (structure formation)

Standard FRW models in GR
FRW dynamics:
energy conservation
Raychaudhuri eq.
Friedman constraint
EoS, typically linear, p=wr
Early universe: inflation based on scalar field(s)

Main dynamical components
Inflation provides initial conditions, almost scale invariant perturbations
Radiation follows
Dark Matter
Dark Energy (L)

Observational Constraints
Singularities in GR
FRW models have a matter dominated singularity (spatial curvature negligible)
This is non-generic: for p=wr (w<1) matter is negligible, singularities are velocity dominated, Weyl curvature dominates over Ricci curvature
Kasner (vacuum Bianchi I) is paradigmatic
BLK conjecture: Bianchi IX ÒmixmasterÓ (a chaotic sequence of Kasner phases) is generic
No mixmaster for w=1 and massless scalar field
Isotropic singularity for w>1

Motivations for a non-linear EoS
Acceleration and dark energy:
cosmological constant L (standard LCDM);
modified 4-d gravity;
Higher dimensions effects (branes);
Back reaction;
modified matter sector:
Quintessence;
Chaplygin gas;
Phenomenological approach (this talk).

"Isotropy and singularities:"
Isotropy and singularities:
In GR, the observed isotropic universe cannot emerge out of generic initial conditions
Penrose conjecture: initial state isotropic=low gravitational entropy (Weyl tensor)
Inflation great for perturbations, but does need fairly homogeneous isotropic initial conditions to start (low energy anyway)
In the brane scenario, the effective energy momentum tensor gives r    terms in the equations, making the singularity isotropic, as shown by many for homogeneus models, and by a perturbative analysis [Dunsby, Goheer, MB, Coley, PRD 69, 101303 (2004); Goheer, Dunsby, Coley, MB, PRD 70, 123517 (2004)]
w>1 needed in ekpyrotic/cyclic and pre-big bang for isotropic singularity [Erickson, Wesley, Stenhardt, Turok, PRD 69, 063514 (2004)]

Motivations for a quadratic EoS
Why quadratic, P=Po + ar + e r /rc ?
simplest non-linear EoS, introduces energy scale(s);
Mostly in general, energy scale -> effective cosmological constant rL;
qualitative dynamics is representative of more general non-linear EoSÕs;
Parametrizes a vast class of DE models: truncated Taylor expansion of any  P(r) (3 parameters);
explore singularities and bounces or cyclic solutions (brane and pre-big bang inspired).

Energy cons. & effective rL
Covariant energy conservation:
Note: phantom if r+P(r)<0
Remarks:
If for a given EoS function P=P(r) there exists a rL such that P(rL) = - rL, then rL has the dynamical role of an effective cosmological constant.
A given non-linear EoS P(r) may admit more than one point rL. If these points exist,  they  are fixed points of the energy conservation equation.

Energy cons. & effective rL
Further remarks:
From  Raychaudhury eq., since                                   , an accelerated phase is achieved whenever P(r) < -r/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 rL is a de Sitter attractor (repeller) of the evolution during expansion if r+P(r)<0 (>0) for r< rL and r+P(r)>0 (<0) for r> rL.

Energy cons. & effective rL
For a given P(r), assume a rL exists.
Taylor expand around rL:
Keep O(1) in Dr = r - rL and integrate energy conservation to get:

Energy cons. & effective rL
Note that:                  , thus                               .
Assume                and Taylor expand:
Then:
At O(1) in Dr and O(0) in a, in any theory of gravity, any P(r) that admits an effective rL behaves as        L-CDM (Dr>0);
For a > -1 r -> rL, i.e. rL is a de Sitter attractor.

RW dynamics
We now consider the dynamics in RW models with quadratic EoS
P=Po + ar + e r /rc
First Òhigh energyÓ EoS, neglecting Po
Then Òlow energyÓ affine EoS, e=0
Finally, most significant cases with full EoS

P=r(a + e r/rc)
Po =0, e = ± 1
dimensionless variables:
Energy cons. and Raychaudhuri:
Friedman:

P=r(a + r/rc)
a: a > -1/3, no acc., qualitatively similar to linear EoS (different singularity)
b: -1< a <-1/3, acceleration and loitering below a threshold r
c: a < -1, rL , de Sitter attractor, phantom for r < rL
b and c show Einstein saddle  point

P=r(a - r/rc)
a: a < -1, all phantom, M in the past, singular in the future
b: -1< a <-1/3, rL , de Sitter saddles, phantom for r > rL
c: a >-1/3, similar to b, with cyclic closed models around Einstein centre
c: for r < rL first accelerating, then deceleration

Affine EoS: P=Po+ar
dimensionless variables:
Energy cons. and Raychaudhuri:
Friedman:

P=Po+ar
a: Po>0, a<-1: phantom for r > rL; for  r < rL recollapsing flat and cyclic closed models (around Einstein fixed point)
b: Po>0, -1<a<-1/3: similar to lower part of a
c: Po<0, -1/3<a: phantom for r < rL,  de Sitter attractor, closed loitering models (close to Einstein fixed point).

Full quadratic EoS
Left: e=1, a<-1, two rL , phantom in between
Right: e=-1, a>-1/3, two rL , phantom  outside
Both show Einstein and cyclic models, and bounces

Anisotropic dynamics and singularities
We now focus on P= ar + r /rc   and look at singularities, considering in order:
Bianchi I models
Bianchi V models
Perturbations of the isotropic past attractor of the Bianchi models

 Bianchi I dynamics
left: a>-1, Minkowski future attractor
right: a<-1,de Sitter future attractor, phantom below separatrix   r = rL
non phantom: isotropic past attractor in both cases, Kasner is a saddle point (past attractor for phantom)

Bianchi V dynamics
X and Z as before, C spatial negative curvature variable
left: a>-1, Minkowski future attractor
right: a<-1,de Sitter future attractor, phantom below r = rL
non phantom: isotropic past attractor in both cases, Kasner and CS are a saddle points (Kasner past attractor for phantom)

The isotropic past attractor IS
IS is an isotropic FRW state, the past attractor of singular FRW models with same parameters (Z=0).
In general, the scale factor tends to a finite value as  at the singularity
The singularity is matter dominated, with shear effects negligible

Perturbations of IS
We consider large scales perturbations:
all models considered are non-accelerating at early times (high energies);
This implies that for any l there is a time t* such that for t<t*  l is larger than horizon
Hence in the t->0 limit we are interested, large scale analysis is enough

Perturbations of IS
we use gauge-invariant covariant variables, in particular the shear, vorticity, electric and magnetic Weyl tensors
we use expansion normalised variables, and look at asymptotic behaviour as the singularity is approached, a->as

Perturbations of IS
For standard linear EoS  P = w r, these variables diverge at the singularity, signaling anisotropy.
For P= ar + r /rc , after splitting in scalar vector and tensor modes, we find that all the expansion normalised perturbations vanish at the singularity, as we found in the brane case.
As in the brane case, the r term makes the singularity isotropic, dominating over shear.

Work in progress
Program: quadratic EoS as a way of parametrizing DE, or even Unified Dark Matter, comparing with observations (Balbi and Quercellini + MB)
First attempt: affine EoS as UDM and DE+DM, flat models
Note that an exact a=e=0 model is equivalent to LCDM for r> rL
Use: age, supernovae, baryon oscillations, CMB peaks.

 Flat model,  UDM with Affine EoS
Simplest model: k=0, P=Po+ar;               2 parameters, like LCDM with k
Reach dynamics: 4 cases, 2 standard and 2 phantom, independently of field eqs. (gravity theory).
Falsifiable, tight constraints
Admits scalar field version

Likelihood: Age of the Universe
Use

Likelihood: Supernovae
SNIa data (Reiss golden sample)

Likelihood: Baryon oscillations
SDSS luminous red galaxies
Parameter

Likelihood: CMB first peak
WMAP, CBI, ACBAR
Shift parameter R

Likelihood: Total
Scalar field for affine UDM
From P=Po+ar we can derive a scalar field potential
Start from

Potential
Summary
Non-linear EoS:
worth exploring as dark energy or UDM, has further motivations;
dynamical, effective cosmological constant(s) mostly natural;
Cosmic No-Hair from energy conservation: evolution a-la L-CDM at O(0) in dP/dr(rL) and O(1) in Dr = r - rL , in any theory of gravity.
Quadratic EoS:
simplest choice beyond linear;
represents truncated Taylor expansion of any  P(r) (3 parameters);
very reach dynamics:
allows for acceleration with and without rL ;
Standard and phantom evolution, phantom -> de Sitter (no ÒBig RipÓ);
Closed models with loitering, or oscillating with no singularity;
singularities are isotropic, as in brane models.
Affine EoS as UDM pass tests so far and has scalar field equivalent (background).
Constraints: high z, nucleosynthesis (b>0), perturbations.

Some open questions
Is the singularity locally or globally isotropic? (issue similar for de Sitter, cf. Starobinski)
Is it extendable (geodesic completeness)?
Causality?
Bounces in closed FRW models: are these stable? Can they be used to model pre-big-bang and perturbations? (currently exploring this idea with Valerio Bozza in Salerno; cf. Brandenberger et al. vs. Kaloper et al. 2006).
Does isotropy survive in strong non-linear regime? A first attempt at Bianchi IX seems to confirm it (in progress with Coley and Woei Chet).