Cédric Deffayet(APC & IAP, Paris)

1/ DGP model (in 5D) or « brane induced gravity »

2/ Cosmology and phenomenology

Q2C Warrenton 2006

DGP gravityTheory and

Phenomenology

From quantum to CosmosFundamental Physics

Research in Space

Why being interested in this model ?

One way to modify gravity at « large distances »… and get rid of dark energy ?

Dark matter or dark energy ?

Changing the dynamics of gravity ?

Historical example the success/failure of both approaches: Le Verrier and

• The discovery of Neptune• The non discovery of Vulcan… but that of General Relativity

Standard

model

5D Minkowski bulk space-time

1. The DGP model (or brane-induced gravity).

Dvali, Gabadadze, Porrati, 2000 A « brane world » model: our 4D space-time is a surface embedded in a large space-time

Special equations of motion for gravity

If equal to zero, standard, 4D, Einstein equations (~ = c =1)

5D Einstein tensor Brane localization

Standard

model

Action principle for this

Peculiar to DGP model

Usual 5D brane world action

• Brane localized kinetic term for the graviton

• Will generically be induced by quantum corrections

A special hierarchy between M(5) and MP is required to render the model phenomenologically interesting

Phenomenological interestA new way to modify gravity at large distance, with a new type of phenomenology … (Important to have such models, if only to disentangle what does and does not depend on the large distance dynamics of gravity in what we know about the Universe)

Theoretical interestConsistent (?) non linear massive gravity …

DGP model

How does that work ? A scalar toy model for DGP

5D D’Alembertian 4D D’Alembertian Source for

For a localized static source, one find the following response

5D potential at large distances

4D potential at small distances

Transition

• Newtonian potential on the brane behaves as

V(r) / r1

V(r) /r21

4D behavior at small distances

5D behavior at large distances

• The crossover distance between the two regimes is given by

This enables to get a “4D looking” theory of gravity out of one which is not, without having to assume a compact (Kaluza-Klein) or “curved” (Randall-Sundrum) bulk.

• But the tensorial structure of the graviton propagator is that of a massive graviton (gravity is mediated by a continuum of massive modes)

Leads to the van Dam-Veltman-Zakharov discontinuity on Minkowski background!

Back to the DGP model :

This coefficient equals +1 in Schwarschild solution

ds2 = à e÷(ú)dt2+eõ(ú)dú2+eö(ú)ú2dÒ22

÷(r) = àrrS(1

õ(r) = + 21

rrS(1

Wrong light bending!

+ …

+ …

+ 327 ï + :::

à821ï + :::

with ï = m4r5rS

Vainshtein ‘72

Introduces a new length scale r in the problem below which the perturbation theory diverges!

V

with rv = (rSmà4)1=5For the sun: bigger than solar system!

The vDVZ discontinuity as seen in Schwarzschild-type solution of « massive gravity » (DGP model, see thereafter!)

So, what is going on at smaller distances?

Vainshtein’s answer (1972):

There exists an other perturbative expansion at smaller distances, reading:

÷(r) = àrrS 1+O r5=2=r5=2v

ð ñn o

õ(r) = + rrS 1+O r5=2=r5=2v

ð ñn o

with rà1v / m4=5

This goes smoothly toward Schwarschild as m goes to zero

No warranty that this solution can be matched with the other for large r! Boulware, Deser ‘72

C.D. ‘01

2. Phenomenology of DGP model2.1 homogeneous cosmology

a(t0) a(t)

The dynamics of the scale factor a(t) of our 4D Universe (the brane) is governed by the modified Friedmann equation

With

Analogous to standard (4D) Friedmann equations

for small Hubble radii

Early cosmology as usual

Late time cosmology

Brane cosmology in 5D Minkowski bulk with no R term on the brane (i.e.: solution to 5D Einstein-Hilbert Action)

Depending on the sign of

Late time deviation from standard cosmology Self accelerating solution (asympotes de Sitter space even with zero matter energy density)

One virtue of DGP model: can get accelerated universe by large distance modification of gravity (C.D (‘01); C.D., Dvali, Gabadaze (‘02)).

Phase diagramwith = +1

Maartens, Majerotto

DGP self accelerating phase

with

The brane (first) Friedmann equation

Can be rewritten as

Acts as a cosmological constant if = +1

Same number of parameter as CDM

DGP

Vs. CDMMaartens, Majerotto ‘06

Strictly speaking, only SN observations are depending solely on the background evolutions

CMB and more importantly Baryon oscillations should be re-computed taking into account the peculiarities of DGP gravity

• Exact cosmological solutions provide an explicit example of interpolation between theories with different tensor structure for the graviton propagator.

C.D.,Gabadadze, Dvali, Vainshtein (2002)

Solution of 4D GR with cosmic fluid

Solution of 5D GR with a brane source

large rc

small rc

Comes in support of a « Vainshtein mechanism » [non perturbative

recovery of the « massless » solutions] at work in DGP…… Recently an other exact solution found by Kaloper for localized relativistic source showing the same recovery…..

2. 2 Back to the van Dam-Veltman-Zakharov discontinuity…

• Perturbative study of Schwarzschild type solutions of DGP model on a flat background space-time:

Gruzinov, Porrati, Lue, Lue & Starkman, Tanaka

Potential: 4D 4D 5D

Tensor 4D 5D 5Dstructure:

Vainshtein radius for DGP model

Related to strong self interaction of the brane bending sector

C.D.,Gabadadze, Dvali, Vainshtein; Arkani-Hamed, Georgi Schwartz; Rubakov; Luty, Porrati, Rattazzi.

Tensorial structure of massive gravity

This has been generalized to cosmological backgrounds

GR terms Correction depending on the cosmological phase

Lue, Starkman ’02(see also Dvali, Gruzinov, Zaldarriaga ‘02)

For the Earth

Universal perihelion precession Best prospect to detect this effect: lunar ranging experiments (BEPPI COLUMBO mission ?)

2.3 Cosmological perturbations (linearized theory on a cosmological background)

One can get effective (4D) equations of motion which have the form (e.g. for matter on the brane with vanishing anisotropic stress) C.D. ‘02

Gravitationnal potentials

Usual 4D Einstein equations

Bulk « Weyl fluid » anistropic stress…

Has no local evolution equation

This has been put to zero by various authors for no good reasons (equivalent to « declare » that the model has no vDVZ discontinuity !)

Correct analysis done by Lue, Scoccimarro, Starkman; Koyama, Maartens ) non standard growth of LSS, yields 8 < 0.8 (at two sigma level)

2. 4 The dark side of DGP gravity…

and

Dvali, Gabadadze, Kolanovic, Nitti; Kiritsis, Tetradis, Tomaras; Antoniadis, Minasian, Vanhove; Kohlprath; Kohlprath, Vanhove

Need for a good underlying quantum gravity construction

Meaning of this strong coupling scale, UV completion at a scale even lower than Luty, Porrati, Rattazzi; Dvali;

Gabadadze; Nicolis, Rattazi; Rubakov

Interesting issues related to comparison between linearized solution and spherically symmetric perturbative solutions

Gabadadze, Iglesias

A Ghost in the self accelerating phase Luty, Porrati, Rattazi; Nicolis, Rattazzi; Koyama; Gorbunov, Koyama

But appears at the cutoff of the scalar part of the theory, also issues with the choice of boundary conditions C.D. Gabadadze, Iglesias in preparation

Sibiryakov; Charmousis, Kaloper,Gregory, Padilla.

Recent claim: no possible UV completion in a well-behaved theory ?

Adams, Arkani-Hamed, Dubovsky, Nicolis, Rattazzi

Not in DGP model, but at best in some limit where gravity has been decoupled !

Conclusions DGP gravity

• Modifies gravity at large distances

• Has a well defined action principle

• Accelerates universe expansion with no c.c. and the same # of parameters as CDM

• Can be distinguished from CDM

• Exciting observables linked to the « Vainshtein mechanism »: gravity is (also) modified at distances smaller than cosmological

More work needed to enlightened the dark side!

IN PARTICULAR, ONE SHOULD KEEP IN MIND THE LOW CUTOFF OF THE SCALAR PART OF THE THEORY… AS A CONSEQUENCE, COMPARISONS WITH PRECISION DATA ARE TO BE CONSIDERED WITH SOME CAUTION !

• Interesting playground to investigate « massive gravity » (a candidate for a consistent theory of « massive gravity »)