1. a viable candidate for dark energy?
Is higher-order gravity
a viable candidate for dark energy?
Luca Amendola
INAF/Osservatorio Astronomico di Roma
Tema: models that can be ruled out/cant be ruled out
Collaboration: D. Polarski, S. Tsujikawa, R. Gannouji, C. Charmousis, S. Davis, E. Magliaro
Firenze GGI 2006

2. Observations are converging…
…to an unexpected universe

3. What do we know about cosmic expansion ?
Nucleosynthesis (z~109)
CMB (z~1000)
Standard candles (z~1)
Perturbations (z~0-1000)

4. Four hypotheses on dark energy
A) Lambda & friends
B) scalar field
C) modified gravity
D) non-linear effect
What dark energy can do for us

5. Two classes of models A) Lambda & friends
B) scalar field
C) modified gravity
D) non-linear effect
1) Models which are almost impossible
to rule out
2) Models which are easy to rule out
For some mysterious reason DE models are always either too easy or too difficult to rule out…

6. Lambda & friends A) Lambda & friends B) scalar field
C) modified gravity
D) non-linear effect
Perche’ I fluidi perfetti generalmente tendono a Lambda . Nota che sotto w>-1 non c’e’ (quasi mai) big rip !!

7. Scalar field A) Lambda & friends B) scalar field C) modified gravity
D) non-linear effect
What dark energy can do for us

8. An ultra-light scalar field
It is more general
Scalars are predicted by fundamental theories
Observational requirements:
Slow evolution
Light mass
V()
What dark energy can do for us 
Compton wavelength = Hubble length

9. The coupling
But beside the potential there can be also a coupling…

10. Dark energy as scalar gravity
T(m)= CT(m)
T= -CT(m)
coupled conservation laws :
First basic
property:
C2/G = scalar-to-tensor ratio

11. Dark energy as scalar gravity
Jordan frame
Einstein frame

12. An extra gravity
Newtonian limit: the scalar interaction generates an attractive extra-gravity
in real space
Give limits to beta on baryons
Yukawa term

13. How about local gravity constraints ?α
Give limits to beta on baryons λ

14. The fourfold way out NO DE uninteresting Make mφ very large
Chameleon models
Give limits to beta on baryons
Coupled DE....
Make mφ very large
Make βvery small
Make them density dependent
Make them species dependent

15. A species-dependent interaction
T(cdm)= CT(cdm)
T= -CT(cdm)
T(bar)= 0
T(rad)= 0

16. Two qualitatively different cases: weak coupling  strong coupling 

17. Weak coupling: density trends
today
No coupling
mat
rad
 = 0
a ~ tp
p = 2/3
MDE:
field
coupling
rad
mat
 = /9
a ~ tp
p = 6/(42+9)
MDE:
field
MDE
come possiamo testare il valore di beta ?
kinetic phase, indep. of potential!

18. Deceleration and acceleration
Assume
V(f) = f-a
today
mat
rad
Dominated by
potential energy α
Dominated by
kinetic energy β
field
The equation of state w=p/r
depends on b during fMDE and on a during tracking:
we = 4b2/9 : past value (decelerated)
wf = -2/(a+2) : present value (accelerated)

19. cl)
WMAP and the coupling 
cl)
Even with boomerang data!
Planck: 
Scalar force 100 times weaker than gravity

20. Modified 3D gravity Higher order gravity ! Simplest case:
A) Lambda & friends
B) scalar field
C) modified gravity
D) non-linear effect
Higher order gravity !
Simplest case:
What dark energy can do for us
Capozziello,Turner,
Carroll, Odintsov…
L.A., S. Capozziello, F. Occhionero, 1992

21. Modified N-dim gravity
Simplest case:
What dark energy can do for us

22. Faces of the same physics
Extra-dim.
degrees of freedom
Higher order
gravity
Coupled scalar field
What dark energy can do for us
Scalar-tensor gravity

23. The simplest higher-order gravity
is in fact a scalar-tensor model in the Jordan frame
and a coupled dark energy model
in the Einstein Frame
What dark energy can do for us

24. General higher order gravity
In Einstein Frame
What dark energy can do for us
Coupled dark energy with a strong coupling !

25. Is this already ruled out by local gravity?
is a scalar-tensor theory with Brans-Dicke
parameter ω=0 or β=1/2
What dark energy can do for us
Therefore if
and f(R) is not yet ruled out
see eg. Nojiri & Odintsov 2003; Brookfield et al. 2006

26. The simplest case
In Einstein Frame
What dark energy can do for us

27. R+1/R model : the φMDE today mat rad field rad mat field MDE
What dark energy can do for us
field
MDE
Jordan F:
Caution:
Plots in the
Einstein frame!

28. R+1/R model : the φMDE today More exactly mat rad During this phase,
field
rad
mat
and therefore
What dark energy can do for us
field
MDE
Caution:
Plots in the
Einstein frame!

29. R+Rn model L.A., S. Tsujikawa, D. Polarski 2006
What dark energy can do for us
L.A., S. Tsujikawa, D. Polarski 2006

30. Distance to last scattering in R+Rn model
What dark energy can do for us

31. A view from the Jordan Frame
What dark energy can do for us
Plots by R. Gannouji, U. of Montpellier

32. How far can we extend this?
An autonomous dynamical system
What dark energy can do for us

33. Analytical results: critical points
For all f(R) theories:
The wrong matter era
(the t1/2 behavior) exists always
The good matter era
(the t2/3 behavior) exists
only if m(-1)=0
What dark energy can do for us
This immediately rules out many cases:

34. Gallery of Failed Cosmologies
What dark energy can do for us
Plots by R. Gannouji, U. of Montpellier

35. Numerical results a) Dark energy domination is always preceded by
the wrong matter era (if any), except...
b) …when the acceleration is driven
by a Lambda term as in
What dark energy can do for us

36. General f(R, Ricci, Riemann)
we find that an exact matter era exists only
for very special combination of parameters
What dark energy can do for us
so probably most of these models are ruled out.

37. Back to second order gravity
Is this the most general second-order scalar-tensor theory ?
No!
This is a Coupled Gauss Bonnet model. Can we put
constraints on it without specifying the potential/couplings?
What dark energy can do for us
Yes, if the scalar field is a DE field

38. Coupled Gauss-Bonnet Everything depends on Observable deviations
from linearized
Newtonian gravity
What dark energy can do for us

39. Coupled Gauss-Bonnet Observable deviations from linearized
Newtonian gravity
in the slow-rolling and
small-coupling limit
What dark energy can do for us

40. Observing a Gauss-Bonnet term
Direct observables
Growth of matter fluctuations
ISW effect
What dark energy can do for us
L.A., C. Charmousis,S. Davis,
astro-ph/

41. Searching for a perfect dark energy model
Quanto a figure perfette o nobili,
credo che per murare le quadre
sieno più perfette che le sferiche,
ma per ruzzolare o condurre i carri
stimo più perfette le tonde che le
triangolari. (Il Saggiatore, 1623)
As to the perfection or nobility
of geometric figures, I think that
for works in masonry square figures
are more perfect than spherical ones,
but to roll or drive a carriage I consider
the circular figures more perfect
than the triangular ones.
(Il Saggiatore, 1623)
What dark energy can do for us

42. An ultra-light scalar field
Hubble size
Galactic size
Adopting a PGB
potential
Abundance
What dark energy can do for us
Mass
L.A. & R. Barbieri 2005