Near-Horizon Solution to DGP Perturbations

1. astro-ph/0606285astro-ph/0606286 Near-Horizon Solution to DGP Perturbations Ignacy Sawicki, Yong-Seon Song, Wayne Hu University of Chicago

2. Summary • New method for solving the evolution of linearised perturbations in DGP gravity using a scaling ansatz • Method valid and under control at horizon scales • Assuming HZ initial power spectrum, ISW-ISW too large to fit WMAP data • ISW-galaxy cross-correlation will be a robust discriminator in the future, independent of power spectrum

3. Ã ! G 8 ¼ ( ) N G f H 8 3 2 ( ) ¼ H f H N ¡ 2 ½ = H + ½ = 3 G 3 8 ¼ N | { z } ½ D E Why Modify Gravity? • What if we don’t know the real equations of gravity. Friedman equation could be • If we assumed GR were valid, we would interpret our observations as acceleration • Doesn’t solve c.c. problem!

4. ( ) 4 R T T h h ¹ ¹ ¹ ¹ º º º º 2 M 1 ¡ H P l r = » 3 0 c M D 5 DGP Gravity: Basic Setup • Braneworld model with flat and infinite extra dimension • Two competing gravities: • Bulk 5D gravity • Induced 4D gravity on brane • 4D > 5D transition scale Dvali et al. Phys. Lett. B485 (2000)

5. H G 8 ¼ ½ N 2 H ¨ = ( ) G f G G T E 3 4 8 ¡ ¡ r r ¼ = ¯ N ¯ c c ¹ º ¹ º ® ® ¹ º The Self-Accelerated Brane • Project 5D equations to 4D to find effective e.o.m. for gravity on the brane • Plug in RW metric: Friedman equation is modified • Two solution branches • ‘—’ branch gives non-zero H for empty brane: asymptotic de Sitter phase • Choice determined by embedding of brane in bulk Maeda et al. Phys. Rev. D68:024033 (2003) Deffayet Phys. Lett. B502:199 (2001)

6. 2 2 2 2 2 k k H 2 _ 1 · ¸ r ¹ ½ ¹ ½ ¹ ½ ± H ¢ ¢ 2 2 + ¹ ¹ E f r r 4 c r E E © © ¢ ¢ ¢ r _ 2 2 2 2 2 + 2 = c ( ) ( ( ) = ) k H H k d ª d © d _ = = c ¹ º ¹ º 3 1 2 E 1 2 + ¡ + ¡ + + + t q g ¼ ¼ x s a 2 2 = E E E H H = 2 2 1 2 2 2 1 ¡ ¡ , a a r r H H 3 9 3 1 2 ¡ c c r c © _ ª 0 + ( ) © ª f H H = + ½ ¼ = E , , Linear Perturbations • Can linearise effective 4D equations in standard fashion • Pretend Eμν is a perfect fluid • Constrain Eμν through Bianchi identity • Eμνunavoidably generated by matter perturbations • Impossible to relate ΔEand πE from on-brane dynamics alone: no closure in 4D Deffayet PRD 66 (2001)

7. @ ­ 1 R ( j j ) b H ´ 1 + Ã ! ´ a y _ . ­ @ 2 µ ¶ ­ @ @ ­ k y 2 3 µ ¶ n n k H H 2 1 2 ¡ r a _ Ã ! ­ Ä _ 0 2 · ¸ _ c ¡ + ¡ ± H ¢ ¢ ( ) ( 2 ) 2 ­ H F H ­ F H R H ­ ¢ + 3 H = ¡ + + + r ¹ ½ E E = _ 3 3 5 2 2 c b @ b @ b ( ) = j j ( ) ( ) k 2 H H H k 2 _ 1 3 K H k K H + + + ¡ n y y + ´ n y q g ¼ ¼ a r r = E E E H c c , H H 3 9 3 1 2 ¡ r c Bulk Parameterisation • Use Mukohyama’s equation for master variable for scalar bulk perturbations • Wave equation in the bulk • Components of Eμνare functions of Ω evaluated at the brane • Re-express Bianchi identity in terms of Ω • If can obtain R , we can close equations! Mukohyama PRD 62 (2000) Quasi-Static (QS): Koyama and Maartens JCAP 0601 (2006) Dynamic Scaling (DS): Sawicki, Song and Hu astro-ph/0606285

8. ( = ) G H » y ( ) ( = ) p h ­ A G H » = H » o r p a y y = h h o r o r Scaling Ansatz • Assume Ωexhibits a scaling behaviour to reduce master PDE to an ODE • Require that Ω0 at the causal horizon, yH = ξhor • Use numerical integration and iteration to find value of R(a) for each mode k • This allows p to vary as fn of scale factor and k • Find that iteration converges quickly and is stable to variation of initial guess

9. = 2 1 3 ( ) < ´ r r r r ¤ g c Caveat Emptor • Linearised, pure de Sitter solution of DGP has ghost • Matter appears to stabilise system at the classical level: no uncontrolled growth observed here • Potentials do grow as  a1+ε on approach to de Sitter (future!), but not exponential • At small distances, DGP exhibits strong coupling • Linearisation not appropriate for • For dark matter haloes, R ~ r*, so linear theory OK to describe interactions of haloes — not their structure e.g. Charmousis et al. hep-th/0604086

10. 2 ( ) ¢ D G P L C D M ¡ Â ­ ­ ­ ­ 0 0 0 0 2 2 2 2 1 0 4 2 = = = = m m m m . . . . ­ ­ h h 0 0 0 0 6 6 0 0 6 6 3 3 2 9 = = = = k k . . . . h h 0 0 7 8 6 0 = = . . Does the Geometry Fit? • Tension between SNe and CMB data excludes a flat DGP cosmology • a slightly open universe fits the bill • BAO would exclude the model at 4.5σ, but potentially affected by strong-coupling regime • not a robust test of DGP

11. ( ( ) ) = = © © ª ª 2 2 ¡ ¡ 1 1 ¡ ¡ k k M M 0 0 0 0 1 0 1 p p c c = = . . Evolution of Potentials

12. ( ) = ` ` C 1 2 + ¼ ` ` ISW Effect in DGP • Faster decay of potential strengthens ISW effect • Signal at low multipoles 4x ΛCDM • Excluded if primordial spectrum scale invariant • Extremely hard to reduce significantly by modifying off-brane gradient

13. ISW-Galaxy Cross-Correlation • Final stages of decay of Φsimilar to ΛCDM • No significant modification to signal • Φdecays much earlier in DGP • Large ISW cross-correlation with high-z galaxies S/N = 2.5 S/N = 5.5

14. Concluding Remarks • Presented a new method of solving for linear perturbations in DGP theory, opening the study of cosmology at scales where gravity is modified • DGP cosmology fits the geometry of universe provided that a small positive curvature is added • Decay of Newtonian potential is much faster and occurs earlier than in GR • Results in a much stronger ISW effect • Galaxy-ISW cross-correlation differs significantly for high-z SNe and offers a robust test of DGP gravity, independent of perturbation power spectrum