A model of accelerating dark energy in decelerating gravity

Matts RoosUniversity of Helsinki

Department of Physical Sciences and

Department of Astronomy

Tuorla and Tartu Observatories’ Autumn Meeting in Cosmology and Large Scale Structure, 4-5 October 2007

arXiv:0704.0882, 0707.1086 [astro-ph]

Contents

I. IntroductionII. The DGP modelIII. The Chaplygin gas modelIV. A combined modelV. Conclusions

I. Introduction

The Universe is expanding, as is well known from:

the luminosity distances of type SNIa supernovae,

the location of the first acoustic peak in the CMB TT power spectrum, or the shift parameter

baryonic acoustic oscillations (BAO).

Supernovae of type SNIa exhibit accelerating expansion since z ~ 0.5 .

I. Introduction

The Einstein equation

describes the most general kind of interaction that a massless spin-two particle can have (the graviton), and is consistent with general principles: special relativity, the covariance principle, the equivalence principle, unitarity, and stability.

R is the symmetric Ricci tensor,

R is the spatial curvature (the Ricci scalar), both containing derivatives of the metric tensor g to lowest order.

I. Introduction

The energy densities (mass, pressure, stress) in the Universe form components in the Stress-Energy Tensor T The geometry of the Universe depends on its energy content Tmass causes curvature.

Let’s choose the Robertson-Walker metric

From Einstein’s equation one then derives the Friedmann-Lemaître equation (FL)

where Einstein’s equation describes (i) an open Universe in expansion, k =-1, or (ii) a closed Universe in contraction, k = +1,

but never a Universe in acceleration.

I. Introduction

The accelerated expansion must be explained

1) either by changes to the spacetime geometry on the lefthand side of Einstein’s equation

2) or by the introduction of some new energy density on the righthand side, in the energy-momentum tensor T

(Other viable explanations are not explored here.)

Let’s study one model of each kind.

II The DGP* model A braneworld model of modified gravity is the DGP model. The action of gravity can be written

The mass scale on our 4-dim. brane is MPl , the corresponding scale in the 5-dim. bulk is M5 . R and R5 are the corresponding Ricci scalars.

• Matter fields act on the brane only, gravity is felt throughout the bulk. Below a cross-over length

scale rc , gravity acts also on the brane.

* Dvali-Gabadadze-Porrati

II The DGP model

• The DGP model has a self-accelerating branch, on which gravity leaks out from the brane to the bulk, thus getting weaker on the brane (at late time, i.e. now).

This branch has a ghost.

• On the self-decelerating branch gravity leaks in from the bulk onto the brane, thus getting stronger on the brane. This branch has no ghosts.

• The scale has to be tuned to give the right cosmic accceleration

at the right time, but this fine-tuning is much less extreme than for the cosmological constant.

II The DGP model

• The FL equation takes the form (here G

We shall only consider flat spacetime, k = 0, when the FL equation simplifies to

When H << rc, the H /rc term vanishes ) the usual FL equation with no acceleration.

• When H ~ rc or H > rc the H /rc term causes acceleration.

• At late times when the total energy density goes like this becomes a de Sitter acceleration.

In the FL equation corresponds to self-acceleration, corresponds to self-deceleration. m is the baryonic and DM energy density, is any other energy density component.

Let us replace the densities by the dimensionless density parameters , and

The FL equation can be solved for H:

Today when H=H0 and a=1, the basic DGP (without in flat space is

II The DGP model

The basic DGP model has only 2 parameters (m, rc

Generalized DGP models include a third parameter n,

corresponding to n + 4 dimensional bulk spacetime.

The flatness condition in the self-accelerated branch does not fit data as well as does the corresponding

condition in CDM with 2 parameters, m +

The self-decelerated branch does not yield an

accelerated expansion.

Basic DGP (n = 1) is a poor fit, n = 2 CDM) and n = 3 are good fits when constrained by SNLS, ESSENCE, BAO, and WMAP3 data.

n>3 fits are poor, need more parameters (k or other).

Below is plotted rc versus m (the solid line corresponds to k=0).

Rydbeck, Fairbairn and Goobar, astro-ph/0701495

The basic DGP model. (Davis & al., arXiv:astro-ph/0701510).The dashed line shows the flat

version.

III The Chaplygin gas model

In higher dimensional space-times, tachyons may

move in the bulk, into our brane, and out of it.

Take a potential of the form

so that the field has a ground state at 1 . In a flat FRW background, and V define

from which

III The Chaplygin gas model

A special case of a tachyon field with a constant potential V 2() = A > 0, is Chaplygin gas, a dark energy fluid with density and pressure p

and an Equation of State • The continuity equation is then

which can be integrated to give

where B is an integration constant.

• Thus this model has two parameters, A and B, in addition to m . It has no ghosts.

III The Chaplygin gas model

• At early times this gas behaves like pressureless dust

• at late times like CDM, causing acceleration:

• Chaplygin gas then has a cross-over length scale

• This model does not fit data well, unless one modifies it and dilutes it with extra parameters.

Standard Chaplygin gas fit (Davis & al., arXiv:astro-ph/0701510).The dashed line shows the flat

version.

IV A combined Chaplygin-DGP model

• Both the Chaplygin gas model and the DGP model are characterized by a bulk/brane crossover scale rc,

• both have the same asymptotic behavior: for a / rc constantlike CDM)

for a / rc >> 1 , 1 / a3

• Both models have problems explaining dark energy.• Consider then a model combining Chaplygin gas

acceleration with DGP self-deceleration, in which the cross-over lengths are assumed identical

IV A combined model

• The new energy density is then

where• The FL equation becomes

• For the self-decelerating branch one chooses The flat-space condition at the present time (a=1) is then

• This does not reduce to CDM for any choice of parameters.

IV A combined model

• We fit supernova data, redshifts and magnitudes, to H(z) using the 192 SNeIa in the compilation of Davis & al.using the 192 SNeIa in the compilation of Davis & al.**

Magnitudes:Magnitudes:

Luminosity distance:Luminosity distance:

Hubble expansion:Hubble expansion:

**arXiv:astro-ph/ 0701510 which includes the ”passed” set in Wood-Vasey & al.,arXiv:astro-ph/ 0701510 which includes the ”passed” set in Wood-Vasey & al.,

arXiv: astro-ph/ 0701041 and the ”Gold” set in Riess & al., Ap.J. 659 (2007)98.arXiv: astro-ph/ 0701041 and the ”Gold” set in Riess & al., Ap.J. 659 (2007)98.

From A. Goobar talk at Cosmo07. Credit: M.Sullivan

• “Third year” SNLS

Hubble Diagram (preliminary)• 3/5 years of SNLS• ~240 distant SNe Ia• rms ~ 0.17mag• The curve is a standard

model fit.

IV A combined model

• We also use a weak constraint from CMB data: We also use a weak constraint from CMB data: mm

00 = 0.24 +- 0.09. Thus the = 0.24 +- 0.09. Thus the –function is–function is

• The best fit has The best fit has = 195.5 for 190 degrees of freedom, (CDM scores = 195.6 ).

• The parameter values are

The 1 errors correspond to best + 3.54.

Best fit (at +) and 1contour in 3-dim. space.The lines correspond to the flat-space condition at A

values +1central (2), and -1 (3)

Best fit (at +) and 1contour in 3-dim. space.

• One may define an effective dynamics by

• Note, however, that eff can be negative for some z in some part of the parameter space. Then

the Universe undergoes an anti-deSitter evolution the weak energy condition is violated weff is singular at the points eff = 0.

This shows that the definition of weff is not very useful

• The region of singularities in the A ,rcspace is

indicated by a straight line in the previous figure.

An example: m ,A at best values, rc at its -1value 0.6 (upper curve), and at its highest value 1.06 allowed by

requiring eff to be non-negative (lower curve).

weff (z) for a selection of points along the 1 contour in the rc ,A-plane

The deceleration parameter q (z) for a selection of points along the 1contour in the rc ,A-plane

V. Conclusions - 11. Chaplygin gas embedded in self-decelerated DGP geometry with the condition of equal cross-over scales

fits supernova data as well as does CDM.

2. The model has only 3 parameters.

3. It has no ghosts.

4. The model cannot be reduced to CDM, it is unique.

5. It needs no extreme fine-tuning, as does CDM.

V. Conclusions - 2

6. became temporarily negative in the past, violating the weak energy condition.

7. weff changed from super-acceleration to acceleration sometime in the range 0 < z < 1. In the future it approaches weff = -1.

8. The ”coincidence problem” is a consequence of

the time-independent value of rc , a braneworld property.