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). |