A model of accelerating dark energy in decelerating gravity Matts Roos University 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: , [astro-ph] TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAA

Contents I.Introduction II.The DGP model III.The Chaplygin gas model IV.A combined model V.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 T   mass 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 M Pl, the corresponding scale in the 5-dim. bulk is M 5. R and R 5 are the corresponding Ricci scalars. Matter fields act on the brane only, gravity is felt throughout the bulk. Below a cross-over length scale r c, 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 << r c, the H /r c term vanishes ) the usual FL equation with no acceleration. When H ~ r c or H > r c the H /r c 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=H 0 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/

The basic DGP model. (Davis & al., arXiv:astro- ph/ ).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/ ).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 r c, both have the same asymptotic behavior:  for a / r c  constant  like  CDM)  for a / r c >> 1,  1 / a 3 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. * Magnitudes: Magnitudes: Luminosity distance: Luminosity distance: Hubble expansion: Hubble expansion: * arXiv:astro-ph/ which includes the ”passed” set in Wood-Vasey & al., * arXiv:astro-ph/ which includes the ”passed” set in Wood-Vasey & al., arXiv: astro-ph/ and the ”Gold” set in Riess & al., Ap.J. 659 (2007)98. arXiv: astro-ph/ 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:  m 0 = Thus the –function isWe also use a weak constraint from CMB data:  m 0 = Thus the   –function is The best fit hasThe best fit has   = for 190 degrees of freedom, (  CDM scores   = ). The parameter values are The 1  errors correspond to   best

Best fit (at +) and 1  contour in 3-dim. space. The lines correspond to the flat-space condition at  A values +1  central (2), and -1  (3)

Best fit (at +) and 1  contour 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  w eff is singular at the points  eff = 0. This shows that the definition of w eff is not very useful The region of singularities in the  A,  rc  space is indicated by a straight line in the previous figure.

An example:  m,  A at best values,  rc at its -1  value 0.6 (upper curve), and at its highest value 1.06 allowed by requiring  eff to be non-negative (lower curve).

w eff (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 1  contour in the  rc,  A  -plane

V. Conclusions 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 became temporarily negative in the past, violating the weak energy condition. 7. w eff changed from super-acceleration to acceleration sometime in the range 0 < z < 1. In the future it approaches w eff = The ”coincidence problem” is a consequence of the time-independent value of r c, a braneworld property.