1. J. Tang, J. Weller, A. Zablocki
Probing Modified Gravity with Supernovae and Galaxy Cluster Counts 2 for 1
O. Mena, J. Santiago and JW
PRL, 96, , 2006
J. Tang, J. Weller, A. Zablocki
astro-ph/

2. New Gravitational Action
Einstein Gravity has not been tested on
large scales (Hubble radius)
But in general:
Simple approach: F(R) = R+mRn

3. Well known for n>1  early de Sitter e.g. Starobinsky (1980)
Interest here: Late time modification  n<0 (inverse curvature)
modification becomes important at low curvature and can lead to accelerated expansion [Capozziello, Carloni, Troisy (’03), Carroll, Duvvuri, Trodden, Turner (’03), Carroll, De Felice, Duvvuri, Easson, Trodden, Turner (’04)]
purely gravitational alternative to dark energy

4. 1/R model accelerated attractor: [CDDETT] vacuum solutions:H
dH/dt
accelerated attractor: [CDDETT]
vacuum solutions:
de Sitter (unstable)
Future Singularity
power law acceleration a(t) ~ t2
For  = eV corrections only important today
Observational consequences similar to dark energy with w = -2/3

5. General f(R) actions
e.g. 2(n+1)/Rn , with n>1 have late-time acceleration
Can satisfy observational constraints form Supernovae

6. Non-Cosmological Constraints on f(R) Theories
General Brans-Dicke theories:
f(R) models in Einstein frame ( = 0):
Simplest model (1/Rn) ruled out by observations of distant Quasars and the deflection of their light by the sun with VLBI: >35000 [Chiva (‘03), Soussa, Woodard (‘03),...]
Can we really do this transformation ? V. Faraoni 2006

7. The New Model Unstable de Sitter solution
[CDDTT’04]
Unstable de Sitter solution
Corrections negligible in the past (large curvature), but dominant for R  2; acceleration today for   H0 (Again why now problem and small parameter)
Late time accelerated attractor [CDDTT’04]

8. Example 1/RR Model Unstable de Sitter Point (H=const)
Two late time attractors:
acceleration with p=3.22
deceleration with p=0.77 H

9. Afraid of Ghosts ?
In the presence of ghosts: negative energy states, hence background unstable towards the generation of small scale inhomogeneities
If one chooses: c = -4b in action, there are NO GHOSTS: I. Navarro and K. van Acoleyen 2005
In general F(R,Q-4P) with Q=RR and P=R R has no ghosts, however...

10. We are still afraid of tachyons
Q=4P is necessary, but not sufficient condition for positive energy eigenstates (vanishing of 4th order terms is guarantied so)
Also have to check 2nd order derivatives for finite propagation speeds (De Felice et al. astro-ph/ )
some parameter combination are still allowed !
For higher inverse powers 1/(aR2+bP+cQ)n there is hope !

11. Solar Systems Tests Linear expansion around Schwarzschild metric
Navarro et al. 2005

12. Non-Cosmological Tests
with critical radius
for solar system: 10pc !
for galaxies: 102kpc
for clusters: 1Mpc !

13. Modified Friedman Equation
Stiff, 2nd order non-linear differential equation, solution is hard numerical problem - initial conditions in radiation dominated era are close to singular point.
Source term is matter and radiation: NO DARK ENERGY
Effectively dependent on 3 extra parameters:

14. Dynamical Analysis  is fixed by the dynamical behavior of the system
Four special values of 
For   1: both values of  are acceptable
For 1    2: =+1 hits singularity in past
For 2    4 : =-1 hits singularity in past
For 2    3 : stable attractor that is decelerated for <32/21 and accelerated for larger .
For 3    4 : no longer stable attractor and singularity is reached in the future through an accelerated phase. For small this appears in the past.

15. Solving the Friedman Equation for n=1
Numerical codes can not solve this from initial conditions in radiation dominated era or matter domination
Approximate analytic solution in distant past

16. Perturbative Solution for =1
good approximation in the past

17. Solution and Conditions

18. Specific Conditions For example with =-4 at a=0.2:
In general all 3 conditions break down at a >

19. Dynamics of best fit model

20. Approximation and Numerical Solution
Very accurate for z ≥ few (7), better than 0.1% with HE2=8G/ the standard Einstein gravity solution at early times.
Use approximate solution as initial condition at z=few (7) for numerical solution (approximation very accurate and numerical codes can cope)

21. Fit to Supernovae Data
Include intrinsic magnitude of Supernovae as free parameter: Degenerate with value of H0 or better absolute scale of H(z). Measure all dimensionful quantities in units of
Remaining parameters:  and
 1 leads to very bad fits of the SNe data; remaining regions
low
high

22. very good fits, similar to CDM (2 = 183.3)
Fit to Riess et al (2004) gold sample; a compilation of 157 high confidence Type Ia SNe data.
very good fits, similar to CDM (2 = 183.3)
Universe hits
singularity in the
past; but at lower
redshift then closest
SNe

23. Combining Datasets
In order to set scale use prior from Hubble Key Project: H0 = 728 km/sec/Mpc [Freedman et al. ‘01]
Prior on age of the Universe: t0 > 11.2 Gyrs [Krauss, Chaboyer ‘03]
low
high
marginalized 0.07 < m < 0.21 (95% c.l.); require dark matter

24. CMB for the Brave Small scale CMB anisotropies are mainly affected by
the physical cold dark matter and baryon densities and
the angular diameter distance to last scattering

25. Angular Diameter Distance to Last Scattering
For the brave:
Angular diameter distance to last
last scattering with WMAP data -
might as well be bogus ! Need full perturbation
analysis

26. But Lesson from Dark Energy
(Weller & Lewis 2003)
On large scales: INTEGRATED SACHS WOLFE EFFECT
change in potential along line of sight
Respecting Einstein !
w=-1
w=-2
w=-0.6

27. The Importance of Perturbations in Dark Energy
without perturbations
with perturbations
Linear Perturbation Theory might be important for the
modified gravity models as well !

28. Conclusions
Inverse curvature gravity models can lead to accelerated expansion of the Universe and explain SNe data without violation of solar system tests. No need for dark energy !
Use of other data sets like CMB, LSS, Baryon Oscillations and clusters require careful analysis of perturbation regime and post - Newtonian limit on cluster scales (in progress)
So far No alternative for dark matter ! But only studied one functional form (n=1) ! Some ideas by Navarro et al. with logarithmic functions ?

29. The Model vs Dark Energy
Require also small parameter: 
Larger n for truly physical models
Ghost free version has only scalar degree of freedom: is there a simple scalar-tensor theory ?
Is there any motivation for this model ?
“If at first an idea is not absurd, there is no hope for it”

30. Can we possibly distinguish dark energy from modified gravity in future observations ?

31. Another Example for Modified Gravity Model - DGP
Brane-world inspired scenario
large extra dimension
Standard model confined to the brane
Gravity can leak of the brane into 5th dimension - cross over scale rc
Modification of Friedman equations
Dvali, Gabadadze, Porrati 2000

32. Modified Friedman Equation in DGP Model
modified equation
accelerated branch as a solution
For flat Universe, condition:

33. Effective Equation of State of DGP Model
Comparison to dark energy component
parameterization: w(a)= (1-a)

34. DGP Model and Supernovae Observations
magnitude - redshift relation in DGP model
SNAP data, 0.3<z<1.7; # 2000; M = 0.15
Supernovae can not distinguish DGP from Dark Energy!
fiducial model: m = 0.3
H0 = 72 km/s/Mpc, DGP

35. Galaxy Cluster Counts
Distribution of clusters measured in Nbody simulations (modern day Press-Schechter)
Predict mass per redshift bin
very sensitive to growth factor

36. The Growth Factor From 5D perturbations (Maartens & Koyama 2006)
For : std gravity mimic DE model
significant difference

37. Cluster Counts in DGP Model
DGP number counts for 8 = 0.75, n=1, Mlim=1.71014h-1M(from ‘SPT’)
mock data assuming Poisson errors
mimic DE model
different rc
Error’s from SNAP w0=0.05; wa=0.2; m=0.03; 8=0.03 (WMAP3+SDSS)
8=0.01 (Planck+LSS)
CDM
w=-0.8
fixed mass limit !
significant difference between
mimic DE and DGP: >1

38. Conclusions
Purely geometrical probes can not distinguish DGP from dark energy
Possibly true for general modified gravity theories
Different dynamics of growth factor for same background evolution in dark energy and DGP models
Cluster number counts significantly different for DGP and mimic DE
So are weak lensing and possibly BAO measuremens
Parameter degeneracy weakens this result, but still significant: cluster counts can distinguish modified gravity from dark energy
Result holds most likely for more detailed analysis