1. DGP gravity Theory and Phenomenology
Cédric Deffayet
(APC & IAP, Paris)
DGP gravity
Theory and
Phenomenology
From quantum to Cosmos
Fundamental Physics
Research in Space
Q2C Warrenton 2006
1/ DGP model (in 5D) or « brane induced gravity »
2/ Cosmology and phenomenology

2. Why being interested in this model ?
One way to modify gravity at « large distances »
… and get rid of dark energy ?
Changing the dynamics of gravity ?
Dark matter or dark energy ?
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

3. 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
Standard model
5D Minkowski
bulk space-time
Special equations of motion for gravity
5D Einstein tensor
Brane localization
If equal to zero, standard, 4D, Einstein equations (~ = c =1)

4. Action principle for this
Standard model
Action principle for this
Usual 5D brane world action
Peculiar to
DGP model
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

5. DGP model
Phenomenological interest
A 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 interest
Consistent (?) non linear massive gravity …

6. How does that work ? A scalar toy model for DGP
5D D’Alembertian
Source for 
4D D’Alembertian
For a localized static source, one find the following response
5D potential at large distances
4D potential at small distances
Transition

7. Newtonian potential on the brane behaves as
Back to the DGP model :
Newtonian potential on the brane behaves as
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!

8. This coefficient equals +1 in Schwarschild solution + …
The vDVZ discontinuity as seen in Schwarzschild-type solution of « massive gravity » (DGP model, see thereafter!)
+ …
This coefficient equals +1 in Schwarschild solution
+ …
Vainshtein ‘72
Wrong light bending!
Introduces a new length scale r in the problem below which the perturbation theory diverges! V
For the sun: bigger than solar system!

9. So, what is going on at smaller distances?
Vainshtein’s answer (1972):
There exists an other perturbative expansion at smaller distances, reading:
with
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

10. 2. Phenomenology of DGP model 2.1 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
C.D. ‘01
Analogous to standard (4D) Friedmann equations
for small Hubble radii
Early cosmology as usual

11. Late time deviation from standard cosmology
Late time cosmology
Depending on the sign of
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)).
Brane cosmology in 5D Minkowski bulk with no R term on the brane (i.e.: solution to 5D Einstein-Hilbert Action)
Late time deviation from standard cosmology

12. DGP self accelerating phase
The brane (first) Friedmann equation
Can be rewritten as
Phase diagram
with  = +1
Maartens, Majerotto
Acts as a cosmological constant if  = +1
with
Same number of parameter as CDM

13. Strictly speaking, only SN observations are depending solely on the background evolutions
Vs. CDM
Maartens, Majerotto ‘06
DGP
CMB and more importantly Baryon oscillations should be re-computed taking into account the peculiarities of DGP gravity

14. 2. 2 Back to the van Dam-Veltman-Zakharov discontinuity…
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)
large rc
small rc
Solution of 4D GR with cosmic fluid
Solution of 5D GR with a brane source
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…..

15. Related to strong self interaction of the brane bending sector
Perturbative study of Schwarzschild type solutions of DGP model on a flat background space-time:
Gruzinov, Porrati, Lue, Lue & Starkman, Tanaka
Potential: D D D
Tensor D D D
structure:
Tensorial structure of massive gravity
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.

16. This has been generalized to cosmological backgrounds
Lue, Starkman ’02
(see also Dvali, Gruzinov,
Zaldarriaga ‘02)
GR terms
Correction
depending on the cosmological phase
For the Earth
Universal perihelion precession
Best prospect to detect this effect: lunar ranging experiments (BEPPI COLUMBO mission ?)

17. One can get effective (4D) equations of motion
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)

18. 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

19. 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 !

20. 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
Interesting playground to investigate « massive gravity » (a candidate for a consistent theory of « massive gravity »)
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 !