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

2. Sawicki et al.astro-ph/0606285 Song et al.astro-ph/0606286 Summary New method for solving the evolution of linearised perturbations in DGP gravity using a scaling ansatz New method for solving the evolution of linearised perturbations in DGP gravity using a scaling ansatz Method valid and under control at horizon scales Method valid and under control at horizon scales Assuming HZ initial power spectrum, ISW-ISW too large to fit WMAP data 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 ISW-galaxy cross-correlation will be a robust discriminator in the future, independent of power spectrum

3. Sawicki et al.astro-ph/0606285 Song et al.astro-ph/0606286 What if we don’t know the real equations of gravity. Friedman equation could be 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 If we assumed GR were valid, we would interpret our observations as acceleration Doesn’t solve c.c. problem! Doesn’t solve c.c. problem! Why Modify Gravity? H 2 ¡ f ( H ) = 8 ¼ G N 3 ½ H 2 = 8 ¼ G N 3ý + 3 f ( H ) 8 ¼ G N |{z} ½ DE!

4. Sawicki et al.astro-ph/0606285 Song et al.astro-ph/0606286 DGP Gravity: Basic Setup Braneworld model with flat and infinite extra dimension Braneworld model with flat and infinite extra dimension Two competing gravities: Two competing gravities: Bulk 5D gravity Induced 4D gravity on brane 4D > 5D transition scale 4D > 5D transition scale Dvali et al. Phys. Lett. B485 (2000) T ¹º T ¹º R ( 4 ) h ¹º h ¹º r c = M2Pl M 3 5 D » H ¡ 1 0

5. Sawicki et al.astro-ph/0606285 Song et al.astro-ph/0606286 The Self-Accelerated Brane Project 5D equations to 4D to find effective e.o.m. for gravity on the 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 Plug in RW metric: Friedman equation is modified Two solution branches Two solution branches ‘—’ branch gives non-zero H for empty brane: asymptotic de Sitter phase ‘—’ branch gives non-zero H for empty brane: asymptotic de Sitter phase Choice determined by embedding of brane in bulk Choice determined by embedding of brane in bulk Maeda et al. Phys. Rev. D68:024033 (2003) Deffayet Phys. Lett. B502:199 (2001) G ¹º = 4 r c f ¹º(G ® ¯ ¡ 8 ¼ G N T ® ¯)¡ E ¹º H 2 ¨ H r c = 8 ¼ G N ½ 3

6. Sawicki et al.astro-ph/0606285 Song et al.astro-ph/0606286 Can linearise effective 4D equations in standard fashion Can linearise effective 4D equations in standard fashion Pretend E μν is a perfect fluid Pretend E μν is a perfect fluid Constrain E μν  through Bianchi identity Constrain E μν  through Bianchi identity E μν unavoidably generated by matter perturbations E μν unavoidably generated by matter perturbations Impossible to relate Δ E and π E from on-brane dynamics alone: no closure in 4D Impossible to relate Δ E and π E from on-brane dynamics alone: no closure in 4D k 2 a 2 © = ¹ 2 ½ 2 2 H r c 2 H r c ¡ 1 ¢ + ¹ 2 ½ 2 1 2 H r c ¡ 1 ¢ E © + ª = f ( H, _ H, ½ ) ¼ E Linear Perturbations Deffayet PRD 66 (2001) d s2= ¡ ( 1 + 2 ª ) d t2+ ( 1 + 2 © ) a2d x2 k 2 a 2 © = ¹ 2 ½ 2 ¢ © + ª = 0 r ¹ E ¹º = 4 r c r ¹ f ¹º _ q E + ± E 3 ¡ 2 9 k 2 ¼ E = 2 r c_H 3 H · ¢ + ¢ E 1 ¡ 2 H r c + g ( H, _ H ) k 2 ¼ E = 3 ¸

7. Sawicki et al.astro-ph/0606285 Song et al.astro-ph/0606286 Ä ­ ¡ 3 HF ( H ) _ ­ + µ F ( H ) k 2 a 2 + H K ( H ) r c + 2 H r c ¡ 1 r c RH ¶ ­ = 2 a 3 k 2 K ( H ) ¹ 2 ½ ¢ Dynamic Scaling (DS):Sawicki, Song and Hu astro-ph/0606285 Use Mukohyama’s equation for master variable for scalar bulk perturbations Use Mukohyama’s equation for master variable for scalar bulk perturbations Wave equation in the bulk Wave equation in the bulk Components of E μν are functions of Ω evaluated at the brane Components of E μν are functions of Ω evaluated at the brane Re-express Bianchi identity in terms of Ω Re-express Bianchi identity in terms of Ω If can obtain R, we can close equations! If can obtain R, we can close equations! Bulk Parameterisation ¡ Ã _ ­ n b 3 !. + @ @ y µ n b 3 @­ @ y ¶ ¡ n k 2 b 5 ­ = 0 R ´ 1 ­ @­ @ y Mukohyama PRD 62 (2000) b ´ a(1 + Hjyj)n ´ 1 + Ã _ H H + H ! j y j Quasi-Static (QS):Koyama and Maartens JCAP 0601 (2006) _ q E + ± E 3 ¡ 2 9 k 2 ¼ E = 2 r c_H 3 H · ¢ + ¢ E 1 ¡ 2 H r c + g ( H, _ H ) k 2 ¼ E = 3 ¸

8. Sawicki et al.astro-ph/0606285 Song et al.astro-ph/0606286 Scaling Ansatz Assume Ω exhibits a scaling behaviour to reduce master PDE to an ODE Assume Ω exhibits a scaling behaviour to reduce master PDE to an ODE Require that Ω 0 at the causal horizon, yH = ξ hor Require that Ω 0 at the causal horizon, yH = ξ hor Use numerical integration and iteration to find value of R(a) for each mode k 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 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 Find that iteration converges quickly and is stable to variation of initial guess ­ = A(p)a p G(y H=» h or) G ( y H = » h or ) y H=» h or

9. Sawicki et al.astro-ph/0606285 Song et al.astro-ph/0606286 Caveat Emptor Linearised, pure de Sitter solution of DGP has ghost Linearised, pure de Sitter solution of DGP has ghost Matter appears to stabilise system at the classical level: no uncontrolled growth observed here Matter appears to stabilise system at the classical level: no uncontrolled growth observed here Potentials do grow as  a 1+ε on approach to de Sitter (future!), but not exponential Potentials do grow as  a 1+ε on approach to de Sitter (future!), but not exponential At small distances, DGP exhibits strong coupling At small distances, DGP exhibits strong coupling Linearisation not appropriate for Linearisation not appropriate for For dark matter haloes, R ~ r *, so linear theory OK to describe interactions of haloes — not their structure 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 r < r ¤ ´ ( r 2 c r g ) 1 = 3

10. Sawicki et al.astro-ph/0606285 Song et al.astro-ph/0606286 Does the Geometry Fit? Tension between SNe and CMB data excludes a flat DGP cosmology 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 BAO would exclude the model at 4.5σ, but potentially affected by strong-coupling regime not a robust test of DGP­m = 0. 22 h = 0. 66 ­m = 0. 20 ­ k = 0. 039 h = 0. 80 ­m = 0. 21 ­ k = 0. 032 h = 0. 76 ­m = 0. 24 h = 0. 66 ¢ Â 2 ( DGP ¡ LCDM )

11. Sawicki et al.astro-ph/0606285 Song et al.astro-ph/0606286 Evolution of Potentials ( © ¡ ª )= 2 k = 0. 01 M pc ¡1 ( © ¡ ª ) = 2 k = 0. 001 M pc ¡ 1

12. Sawicki et al.astro-ph/0606285 Song et al.astro-ph/0606286 ISW Effect in DGP Faster decay of potential strengthens ISW effect Faster decay of potential strengthens ISW effect Signal at low multipoles 4x ΛCDM Signal at low multipoles 4x ΛCDM Excluded if primordial spectrum scale invariant Extremely hard to reduce significantly by modifying off-brane gradient ` ( ` + 1 ) C ` = 2 ¼ `

13. Sawicki et al.astro-ph/0606285 Song et al.astro-ph/0606286 Final stages of decay of Φ similar to ΛCDM Final stages of decay of Φ similar to ΛCDM No significant modification to signal Φ decays much earlier in DGP Φ decays much earlier in DGP Large ISW cross- correlation with high- z galaxies ISW-Galaxy Cross-Correlation S/N = 2.5 S/N = 5.5

14. Sawicki et al.astro-ph/0606285 Song et al.astro-ph/0606286 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 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 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 Decay of Newtonian potential is much faster and occurs earlier than in GR Results in a much stronger ISW effect 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 Galaxy-ISW cross-correlation differs significantly for high-z SNe and offers a robust test of DGP gravity, independent of perturbation power spectrum