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Probing Modified Gravity with Supernovae and Galaxy Cluster Counts 2 for 1

Probing Modified Gravity with Supernovae and Galaxy Cluster Counts 2 for 1. O. Mena, J. Santiago and JW PRL, 96, 041103, 2006 J. Tang, J. Weller, A. Zablocki astro-ph/0609028. But in general:. New Gravitational Action. Einstein Gravity has not been tested on large scales (Hubble radius).

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Probing Modified Gravity with Supernovae and Galaxy Cluster Counts 2 for 1

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  1. Probing Modified Gravity with Supernovae and Galaxy Cluster Counts2 for 1 O. Mena, J. Santiago and JW PRL, 96, 041103, 2006 J. Tang, J. Weller, A. Zablocki astro-ph/0609028

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

  3. Well known for n>1  early de Sittere.g. Starobinsky (1980) • Interest here: Late time modificationn<0 (inverse curvature) • modification becomes important at low curvatureand 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. dH/dt H 1/R model • accelerated attractor: [CDDETT] • vacuum solutions: • de Sitter (unstable) • Future Singularity • power law acceleration a(t) ~ t2 • For  = 10-33 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-timeacceleration • 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. [CDDTT’04] The New Model • 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/0604154) • some parameter combination are still allowed ! • For higher inverse powers 1/(aR2+bP+cQ)n there is hope !

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

  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 deceleratedfor <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 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 > 0.1-0.2

  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. low high Fit to Supernovae Data • Include intrinsic magnitude of Supernovae as free parameter:Degenerate with value of H0or 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

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

  27. without perturbations with perturbations The Importance of Perturbations in Dark Energy 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. “If at first an idea is not absurd, there is no hope for it” 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 ?

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

  33. Effective Equation of State of DGP Model • Comparison to dark energy component • parameterization:w(a)=-0.77+0.27(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 gravitymimic DE model • significant difference

  37. Cluster Counts in DGP Model fixed mass limit ! • 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 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

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