Near-Horizon Solution to Near-Horizon Solution to DGP PerturbationsDGP Perturbations

Ignacy SawickiIgnacy Sawicki, Yong-Seon Song, Wayne Hu, Yong-Seon Song, Wayne HuUniversity of ChicagoUniversity of Chicago

astro-ph/0606285astro-ph/0606285astro-ph/0606286astro-ph/0606286

Sawicki Sawicki et al.et al. astro-astro-ph/0606285ph/0606285Song Song et al.et al.astro-ph/0606286astro-ph/0606286

SummarySummary New method for solving the evolution of linearised New method for solving the evolution of linearised

perturbations in DGP gravity using a scaling perturbations in DGP gravity using a scaling ansatzansatz

Method valid and under control at horizon scalesMethod valid and under control at horizon scales

Assuming HZ initial power spectrum, ISW-ISW too Assuming HZ initial power spectrum, ISW-ISW too large to fit WMAP datalarge to fit WMAP data

ISW-galaxy cross-correlation will be a robust ISW-galaxy cross-correlation will be a robust discriminator in the future, independent of power discriminator in the future, independent of power spectrumspectrum

Sawicki Sawicki et al.et al. astro-astro-ph/0606285ph/0606285Song Song et al.et al.astro-ph/0606286astro-ph/0606286

What if we don’t know the What if we don’t know the realreal equations equations of gravity. Friedman equation could beof gravity. Friedman equation could be

If we assumed GR were validIf we assumed GR were valid,, we would we would interpret our observations as accelerationinterpret our observations as acceleration

Doesn’tDoesn’t solve c.c. problem! solve c.c. problem!

Why Modify Gravity?Why Modify Gravity?

HH22 ¡¡ ff ((HH)) ==88¼¼GGNN

33½½

HH22 ==88¼¼GGNN

33

ÃÃ

½½++33ff ((HH))88¼¼GGNN|| {{zz }}

½½DDEE

!!

Sawicki Sawicki et al.et al. astro-astro-ph/0606285ph/0606285Song Song et al.et al.astro-ph/0606286astro-ph/0606286

DGP Gravity: Basic SetupDGP Gravity: Basic Setup

Braneworld model Braneworld model with flat and infinite with flat and infinite extra dimensionextra dimension

Two competing Two competing gravities:gravities: Bulk 5D gravityBulk 5D gravity Induced 4D gravity on Induced 4D gravity on

branebrane 4D > 5D transition 4D > 5D transition

scalescale

Dvali Dvali et al.et al. Phys. Lett. Phys. Lett. B485B485 (2000) (2000)

T¹ º

T¹ º

R(4)

h¹ º

h¹ º

rrcc ==MM 22

PPllMM 33

55DD»» HH ¡¡ 11

00

Sawicki Sawicki et al.et al. astro-astro-ph/0606285ph/0606285Song Song et al.et al.astro-ph/0606286astro-ph/0606286

The Self-Accelerated BraneThe Self-Accelerated Brane Project 5D equations to 4D to find effective Project 5D equations to 4D to find effective

e.o.m. for gravity on the branee.o.m. for gravity on the brane

Plug in RW metric: Friedman equation is Plug in RW metric: Friedman equation is modifiedmodified

Two solution branchesTwo solution branches ‘—’ ‘—’ branch gives non-zero branch gives non-zero HH for empty brane: for empty brane:

asymptotic de Sitter phaseasymptotic de Sitter phase Choice determined by embedding of brane in bulkChoice determined by embedding of brane in bulk

Maeda Maeda et al.et al. Phys. Rev. Phys. Rev. D68:D68:024033 024033 (2003) (2003)

Deffayet Phys. Lett. Deffayet Phys. Lett. B502:B502:199 (2001) 199 (2001)

GG¹¹ ºº == 44rrccff¹¹ ºº ((GG®®̄̄ ¡¡ 88¼¼GGNNTT®®̄̄ )) ¡¡ EE¹¹ ºº

HH22 ¨̈HHrrcc

==88¼¼GGNN½½

33

Sawicki Sawicki et al.et al. astro-astro-ph/0606285ph/0606285Song Song et al.et al.astro-ph/0606286astro-ph/0606286

Can linearise effective 4D equations in standard fashionCan linearise effective 4D equations in standard fashion

Pretend Pretend EEμνμν is a perfect fluid is a perfect fluid

Constrain Constrain EEμνμν through Bianchi identity through Bianchi identity

EEμνμν unavoidably unavoidably generated by matter perturbationsgenerated by matter perturbations ImpossibleImpossible to relate to relate ΔΔEE and and ππE E from on-brane dynamics from on-brane dynamics

alone: alone: no closure no closure in 4Din 4D

k2

a2©=

¹ 2½2

2Hrc2Hrc ¡ 1

¢ +¹ 2½2

12Hrc ¡ 1

¢ E

©+ ª = f (H, _H,½)¼E

Linear PerturbationsLinear Perturbations

Deffayet PRD Deffayet PRD 6666 (2001) (2001)

ddss22 == ¡¡ ((11++ 22ªª ))ddtt22 ++ ((11++ 22©©))aa22ddxx22

kk22

aa22©©==

¹¹ 22½½22

¢¢

©©++ ªª == 00

rr ¹¹ EE¹¹ ºº == 44rrccrr ¹¹ ff¹¹ ºº__qqEE ++

±±EE

33¡¡

2299kk22¼¼EE ==

22rrcc__HH

33HH

··¢¢ ++ ¢¢ EE

11¡¡ 22HHrrcc++ gg((HH,, __HH))kk22¼¼EE ==33

¸̧

Sawicki Sawicki et al.et al. astro-astro-ph/0606285ph/0606285Song Song et al.et al.astro-ph/0606286astro-ph/0606286

ÄÄ ¡¡ 33HHFF ((HH)) __ ++µµ

FF ((HH))kk22

aa22++

HHKK ((HH))rrcc

++22HHrrcc ¡¡ 11

rrccRRHH

¶¶

==22aa33

kk22KK ((HH))¹¹ 22½½¢¢

Dynamic Scaling (DS)Dynamic Scaling (DS):: Sawicki, Song and Hu astro-Sawicki, Song and Hu astro-ph/0606285ph/0606285

Use Mukohyama’s equation for master variable Use Mukohyama’s equation for master variable for scalar bulk perturbationsfor scalar bulk perturbations Wave equation in the bulkWave equation in the bulk

Components of Components of EEμνμν are functions of are functions of ΩΩ evaluated at the evaluated at the branebrane

Re-express Bianchi identity in terms of Re-express Bianchi identity in terms of ΩΩ

If can obtain If can obtain RR , we can close equations! , we can close equations!

Bulk ParameterisationBulk Parameterisation

¡¡

ÃÃ__

nnbb33

!! ..

++@@@@yy

µµnnbb33

@@ @@yy

¶¶¡¡

nnkk22

bb55 == 00

RR ´́11

@@ @@yy

Mukohyama PRD Mukohyama PRD 6262 (2000) (2000)

bb ´́ aa((11++ HH jjyyjj))

nn ´́ 11++

ÃÃ__HHHH

++ HH

!!

jjyyjj

Quasi-Static (QS)Quasi-Static (QS):: Koyama and Maartens JCAP Koyama and Maartens JCAP 06010601 (2006)(2006)

__qqEE ++±±EE

33¡¡

2299kk22¼¼EE ==

22rrcc__HH

33HH

··¢¢ ++ ¢¢ EE

11¡¡ 22HHrrcc++ gg((HH,, __HH))kk22¼¼EE ==33

¸̧

Sawicki Sawicki et al.et al. astro-astro-ph/0606285ph/0606285Song Song et al.et al.astro-ph/0606286astro-ph/0606286

Scaling AnsatzScaling Ansatz Assume Assume ΩΩ exhibits a scaling exhibits a scaling

behaviour to reduce master behaviour to reduce master PDE to an ODEPDE to an ODE

Require that Require that ΩΩ 0 at the 0 at the causal horizon, causal horizon, yHyH = = ξξhorhor

Use numerical integration Use numerical integration and iteration to find value of and iteration to find value of RR((aa) for each mode ) for each mode kk This allows This allows pp to vary as fn of to vary as fn of

scale factor and scale factor and kk Find that iteration converges Find that iteration converges

quickly and is stable to quickly and is stable to variation of initial guessvariation of initial guess

== AA((pp))aappGG((yyHH==»»hhoorr))

-6 -4 -2 0 2-10

-5

0

log a

R

0 0.5 10

0.5

1

k/aH=1

k/aH=10

k/aH=30

GG(( yy

HH== »»

hh oo rr))

yyHH==»»hhoorr

Sawicki Sawicki et al.et al. astro-astro-ph/0606285ph/0606285Song Song et al.et al.astro-ph/0606286astro-ph/0606286

Caveat EmptorCaveat Emptor

Linearised, Linearised, pure pure de Sitter solution of DGP has de Sitter solution of DGP has ghostghost

Matter appears to stabilise system at the classical Matter appears to stabilise system at the classical level: level: no no uncontrolled growth observed hereuncontrolled growth observed here

Potentials do grow as Potentials do grow as aa 1+1+εε on approach to de Sitter on approach to de Sitter ((future!future!), but ), but notnot exponential exponential

At small distances, DGP exhibits strong couplingAt small distances, DGP exhibits strong coupling Linearisation not appropriate forLinearisation not appropriate for For dark matter haloes, For dark matter haloes, RR ~ ~ rr**, so linear theory OK to , so linear theory OK to

describe interactions of haloes — not their structuredescribe interactions of haloes — not their structure

e.g. Charmousis e.g. Charmousis et al.et al. hep-th/0604086 hep-th/0604086

rr << rr¤¤ ´́ ((rr22cc rrgg))11==33

Sawicki Sawicki et al.et al. astro-astro-ph/0606285ph/0606285Song Song et al.et al.astro-ph/0606286astro-ph/0606286

Does the Geometry Fit?Does the Geometry Fit?

0

5

10

WMAP3+SNLS WMAP3+SNLS+KP

fDGPoDGP

Tension between SNe and CMB data excludes a flat DGP Tension between SNe and CMB data excludes a flat DGP cosmologycosmology a slightly open universe fits the billa slightly open universe fits the bill

BAO would exclude the model at 4.5BAO would exclude the model at 4.5σσ, but potentially , but potentially affected by strong-coupling regime affected by strong-coupling regime not a robust test of DGPnot a robust test of DGP

mm == 00..2222

hh== 00..6666

mm == 00..2200

kk == 00..003399

hh== 00..8800

mm == 00..2211

kk == 00..003322

hh== 00..7766

mm == 00..2244

hh== 00..6666

¢¢ÂÂ

22 ((DD

GGPP

¡¡LL CC

DDMM

))

Sawicki Sawicki et al.et al. astro-astro-ph/0606285ph/0606285Song Song et al.et al.astro-ph/0606286astro-ph/0606286

Evolution of PotentialsEvolution of Potentials

0.5

0.6

0.7

0.8

0.9

1

-2 -1 0

log a

(( ©©¡¡

ªª)) ==

22

kk == 00..0011MMppcc¡¡ 11

0.5

0.6

0.7

0.8

0.9

1

-2 -1 0

log a

LCDM QS DS

(( ©©¡¡

ªª)) ==

22

kk == 00..000011MMppcc¡¡ 11

Sawicki Sawicki et al.et al. astro-astro-ph/0606285ph/0606285Song Song et al.et al.astro-ph/0606286astro-ph/0606286

ISW Effect in DGPISW Effect in DGP

Faster decay of Faster decay of potential strengthens potential strengthens ISW effectISW effect

Signal at low Signal at low multipoles 4x multipoles 4x ΛΛCDMCDM Excluded if primordial Excluded if primordial

spectrum scale spectrum scale invariantinvariant

Extremely hard to Extremely hard to reduce significantly by reduce significantly by modifying off-brane modifying off-brane gradientgradient

`̀((̀̀++

11 ))CC̀

`==22 ¼¼

`̀2 10 10010

-11

10-10

10-9

2 10 10010

-11

10-10

10-9

2 10 10010

-11

10-10

10-9

2 10 10010

-11

10-10

10-9

Sawicki Sawicki et al.et al. astro-astro-ph/0606285ph/0606285Song Song et al.et al.astro-ph/0606286astro-ph/0606286

Final stages of Final stages of decay of decay of ΦΦ similar to similar to ΛΛCDMCDM No significant No significant

modification to modification to signalsignal

ΦΦ decays much decays much earlier in DGPearlier in DGP Large ISW cross-Large ISW cross-

correlation with high-correlation with high-zz galaxies galaxies

ISW-Galaxy Cross-ISW-Galaxy Cross-CorrelationCorrelation

S/N = 2.5S/N = 2.5

S/N = 5.5S/N = 5.5

Sawicki Sawicki et al.et al. astro-astro-ph/0606285ph/0606285Song Song et al.et al.astro-ph/0606286astro-ph/0606286

Concluding RemarksConcluding Remarks Presented a new method of solving for linear Presented a new method of solving for linear

perturbations in DGP theory, opening the study of perturbations in DGP theory, opening the study of cosmology at scales where gravity is modifiedcosmology at scales where gravity is modified

DGP cosmology fits the geometry of universe DGP cosmology fits the geometry of universe provided provided that a small positive curvature is addedthat a small positive curvature is added

Decay of Newtonian potential is much faster and Decay of Newtonian potential is much faster and occurs earlier than in GRoccurs earlier than in GR Results in a much stronger ISW effectResults in a much stronger ISW effect

Galaxy-ISW cross-correlation differs significantly Galaxy-ISW cross-correlation differs significantly for high-for high-zz SNe and offers a robust test of DGP SNe and offers a robust test of DGP gravity, independent of perturbation power gravity, independent of perturbation power spectrumspectrum