A Framework for. Modified Gravity. Models of Cosmic Acceleration. Wayne Hu EFI, November 2008
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1 A Framework for Modified Gravity Models of Cosmic Acceleration Wayne Hu EFI, November 2008
2 Candidates for Acceleration Cosmological constant (cold dark matter) model ΛCDM is the standard model of cosmology Sucessful and highly predictive (falsifiable) ΛCDM
3 Candidates for Acceleration Yet need alternatives as foils to test the standard model Dvali-Gabadadze-Porrati (DGP) braneworld model: contender with fundamental physics aspirations, out of touch with the reality of observations ΛCDM DGP
4 Candidates for Acceleration Yet need alternatives as foils to test the standard model Modified action f(r) models: not serious contender from fundamental physics perspective, alternate propositions show how to sharpen tests of the standard model ΛCDM DGP f(r)
5 Charting Out the Expansion Standard candle: apparent brightness of objects with a fixed luminosity to judge distance Standard ruler: apparent (angular) separation of objects with a fixed physical separation to judge distance Supernovae 1998 Discovery Sound waves CMB+Galaxies
6 Predictive Power of Cosmological Constant Acceleration from a cosmological constant makes percent level predictions for growth of structure Mortonson, Hu, Huterer (2008)
7 Incomplete Geometry or Energy? General relativity says Gravity = Geometry And Geometry = Matter-Energy Could the missing energy required by acceleration be an incomplete description of how matter determines geometry?
8 Dynamics of Acceleration Dark energy or modified gravity? for same expansion different abundance of rare structures, gravitational lensing and redshift SPT DES WMAP SDSS
9 Parameterizing Acceleration Cosmic acceleration, like the cosmological constant, can either be viewed as arising from Missing, or dark energy, with w p/ ρ < 1/3 Modification of gravity on large scales G µν = 8πG ( ) Tµν M + Tµν DE F (g µν ) + G µν = 8πGT M µν Proof of principle models for both exist: quintessence, k-essence; DGP braneworld acceleration, f(r) modified action Compelling models for either explanation lacking Study models as illustrative toy models whose features can be generalized
10 Parameterized Post-Friedmann Description Smooth dark energy parameterized description on small scales: w(z) that completely defines expansion history, sound speed defines structure formation Parameterized description of modified gravity acceleration Many ad-hoc attempts violate energy-momentum conservation, Bianchi identities, gauge invariance; others incomplete Parameterize the degrees of freedom in the effective dark energy F (g µν ) = 8πGT DE µν retaining the metric structure of general relativity but with component non-minimally dependent on metric, coupled to matter Non-linear mechanism returns general relativity on small scales
11 Parameterized Post-Friedmann Description Smooth dark energy parameterized description on small scales: w(z) that completely defines expansion history, sound speed defines structure formation Parameterized description of modified gravity acceleration Many ad-hoc attempts violate energy-momentum conservation, Bianchi identities, gauge invariance; others incomplete Parameterize the degrees of freedom in the effective dark energy F (g µν ) = 8πGT DE µν retaining the metric structure of general relativity but with component non-minimally dependent on metric, coupled to matter Non-linear mechanism returns general relativity on small scales
12 Three Regimes Three regimes defined by γ= Φ/Ψ BUT with different dynamics Examples f(r) and DGP braneworld acceleration Parameterized Post-Friedmann description Non-linear regime follows a halo paradigm but a full parameterization still lacking and theoretical, workable, examples few General Relativistic Non-Linear Regime Scalar-Tensor Regime Conserved-Curvature Regime r r * c halos, galaxy large scale structure CMB Hu & Sawicki (2007) r
13 Outline Horizon scale (SNe, CMB): conserved curvature Large scale structure: quasi-static scalar-tensor regime Dark matter halos & the Galaxy: restoration of General Relativity Wenjuan Fang Marcos Lima Michael Mortonson Hiro Oyaizu Fabian Schmidt Hiranya Peiris Iggy Sawicki Yong-Seon Song Amol Upadhye Sheng (Wiley) Wang
14 Illustrative Toy Models
15 Modified Action f(r) Model R: Ricci scalar or curvature f(r): modified action (Starobinsky 1980; Carroll et al 2004) S = d 4 x [ ] R + f(r) g 16πG + L m f R df/dr: additional propagating scalar degree of freedom (metric variation) f RR d 2 f/dr 2 : Compton wavelength of f R squared, inverse mass squared B: Compton wavelength of f R squared in units of the Hubble length B f RR 1 + f R R H H d/d ln a: scale factor as time coordinate
16 Modified Einstein Equation In the Jordan frame, gravity becomes 4th order but matter remains minimally coupled and separately conserved G αβ + f R R αβ ( ) f 2 f R g αβ α β f R = 8πGT αβ Trace can be interpreted as a scalar field equation for f R with a density-dependent effective potential (p = 0) 3 f R + f R R 2f = R 8πGρ For small deviations, f R 1 and f/r 1, f R 1 (R 8πGρ) 3 the field is sourced by the deviation from GR relation between curvature and density and has a mass m 2 f R 1 R = 1 3 f R 3f RR
17 DGP Braneworld Acceleration Braneworld acceleration (Dvali, Gabadadze & Porrati 2000) S = d 5 x [ (5) ( R (4) )] g 2κ + δ(χ) R 2 2µ + L 2 m with crossover scale r c = κ 2 /2µ 2 Influence of bulk through Weyl tensor anisotropy - solve master equation in bulk (Deffayet 2001) Matter still minimally coupled and conserved Exhibits the 3 regimes of modified gravity Weyl tensor anisotropy dominated conserved curvature regime r > r c (Sawicki, Song, Hu 2006; Cardoso et al 2007) Brane bending scalar tensor regime r < r < r c (Lue, Soccimarro, Starkman 2004; Koyama & Maartens 2006) Strong coupling General Relativistic regime r < r = (r 2 cr g ) 1/3 where r g = 2GM (Dvali 2006)
18 DGP field equations DGP Field Equations G µν = 4r 2 cf µν E µν where f µν is a tensor quadratic in the 4-dimensional Einstein and energy-momentum tensors ) (A αβ A αβ A2 f µν 1 12 AA µν 1 4 Aα µa να g µν 3 A µν G µν µ 2 T µν and E µν is the bulk Weyl tensor Background metric yields the modified Friedmann equation H 2 H r c = µ2 ρ 3 For perturbations, involves solving metric perturbations in the bulk through the master equation
19 Conserved Curvature Regime
20 Curvature Conservation On superhorizon scales, energy momentum conservation and expansion history constrain the evolution of metric fluctuations (Bertschinger 2006) For adiabatic perturbations in a flat universe, conservation of comoving curvature applies ζ = 0 where d/d ln a (Bardeen 1980) Gauge transformation to Newtonian gauge ds 2 = (1 + 2Ψ)dt 2 + a 2 (1 + 2Φ)dx 2 yields (Hu & Eisenstein 1999) Φ Ψ H H Φ ( H H H H ) Ψ = 0 Modified gravity theory supplies the closure relationship Φ = γ(ln a)ψ between and expansion history H = ȧ/a supplies rest.
21 Linear Theory for f(r) In f(r) model, superhorizon behavior persists until Compton wavelength smaller than fluctuation wavelength B 1/2 (k/ah) < 1 Once Compton wavelength becomes larger than fluctuation B 1/2 (k/ah) > 1 perturbations are in scalar-tensor regime described by γ = 1/2.
22 Linear Theory for f(r) In f(r) model, superhorizon behavior persists until Compton wavelength smaller than fluctuation wavelength B 1/2 (k/ah) < 1 Once Compton wavelength becomes larger than fluctuation B 1/2 (k/ah) > 1 perturbations are in scalar-tensor regime described by γ = 1/2. Small scale density growth enhanced and 8πGρ > R low curvature regime with order unity deviations from GR Transitions in the non-linear regime where the Compton wavelength can shrink via chameleon mechanism Given k NL /ah 1, even very small f R have scalar-tensor regime
23 Potential Growth On the stable B>0 branch, potential evolution reverses from decay to growth as wavelength becomes smaller than Compton scale Quasistatic equilibrium reached in linear theory with γ= Φ/Ψ=1/2 until non-linear effects restore γ=1 1.2 (Φ Ψ)/2 k/ah 0 =100 Φ- /Φ i w eff =-1, Ω DE =0.76, B 0 = ΛCDM a Hu, Song, Huterer && Sawicki Smith (2006) (2006)
24 PPF Correspondence On large scales, Bianchi identity requires covariant conservation of effective dark energy leaving: Metric ratio of Newtonian gravitational potentials g(a, k) = (Φ + Ψ)/(Φ Ψ) First relationship with the matter fluctuations f ζ (a) On linear scales, time-derivatives of metric can be dropped leading to a Poisson-like equation with: Metric ratio g(a, k) = (Φ + Ψ)/(Φ Ψ) Second relationship with the matter fluctuations f G (a) On non-linear scales associated with collapsed objects, acceptable modifications must return general relativity Transition between linear 2 halo and non-linear 1 halo regime
25 PPF f(r) Description Metric and matter evolution well-matched by PPF description Standard GR tools apply (CAMB), self-consistent, gauge invar Φ /Φ i k/h 0 = f(r) PPF Hu, Hu Huterer & Sawicki & Smith (2007); (2006) Hu (2008) a
26 Integrated Sachs-Wolfe Effect CMB photons transit gravitational potentials of large-scale structure If potential decays during transit, gravitational blueshift of infall not cancelled by gravitational redshift of exit Spatial curvature of gravitational potential leads to additional effect T/T = (Φ Ψ)
27 Integrated Sachs-Wolfe Effect CMB photons transit gravitational potentials of large-scale structure If potential decays during transit, gravitational blueshift of infall not cancelled by gravitational redshift of exit Spatial curvature of gravitational potential leads to additional effect T/T = (Φ Ψ)
28 ISW Quadrupole Reduction of potential decay can eliminate the ISW effect at the quadrupole for B 0 ~3/2 In conjunction with a change in the initial power spectrum can also bring the total quadrupole closer in ensemble average to the observed quadrupole 6C 2 /2π (µκ 2 ) ΛCDM total quadrupole ISW quadrupole Hu, Song, Huterer && Sawicki Smith (2006) (2006) B 0
29 ISW Quadrupole Reduction of large angle anisotropy for B 0 ~1 for same expansion history and distances as ΛCDM Well-tested small scale anisotropy unchanged l(l+1)c l /2π (µk 2 ) TT B 0 0 (ΛCDM) 1/2 3/2 Song, Hu & Sawicki (2006) multipole l
30 ISW-Galaxy Correlation Decaying potential: galaxy positions correlated with CMB Growing potential: galaxy positions anticorrelated with CMB Observations indicate correlation
31 Galaxy-ISW Anti-Correlation Large Compton wavelength B 1/2 creates potential growth which can anti-correlate galaxies and the CMB In tension with detections of positive correlations across a range of redshifts B 0 =0 B 0 =5 Hu, Song, Huterer Peiris & Smith Hu (2007) (2006)
32 PPF DGP Description Metric and matter evolution well-matched by PPF description Standard GR tools apply (CAMB), self-consistent, gauge invar. 1.0 Φ /Φ i DGP PPF k/h 0 =1 10 Hu, Hu Huterer & Sawicki & Smith (2007); (2006) Hu (2008) a
33 Extra dimension modify gravity on large scales 4D universe bending into extra dimension alters gravitational redshifts in cosmic microwave background 6000 fluctuation power Acceleration with Modified Gravity angular frequency 1000
34 CMB in DGP Adding cut off as an epicycle can fix distances, ISW problem Suppresses polarization in violation of EE data - cannot save DGP! Hu, Fang Huterer et al (2008) & Smith (2006)
35 CMB in DGP Adding cut off as an epicycle can fix distances, ISW problem Suppresses polarization in violation of EE data - cannot save DGP! Hu, Fang Huterer et al (2008) & Smith (2006)
36 Parameterized Post-Friedmann Parameterizing the degrees of freedom associated with metric modification of gravity that explain cosmic acceleration Simple models that add in only one extra scale to explain acceleration tend to predict substantial changes near horizon and hence ISW 1 Hu (2008) deviation from GR g 0
37 Linear Scalar Tensor Regime
38 Three Regimes Three regimes defined by γ= Φ/Ψ BUT with different dynamics Examples f(r) and DGP braneworld acceleration Parameterized Post-Friedmann description Non-linear regime follows a halo paradigm but a full parameterization still lacking and theoretical, workable, examples few General Relativistic Non-Linear Regime Scalar-Tensor Regime Conserved-Curvature Regime r r * c halos, galaxy large scale structure CMB Hu & Sawicki (2007) r
39 Linear Power Spectrum Linear real space power spectrum enhanced on small scales Degeneracy with galaxy bias and lack of non-linear predictions leave constraints from shape of power spectrum P L (k) (Mpc/h) B (ΛCDM) k (h/mpc)
40 Power Spectrum Data Linear power spectrum enhancement fits SDSS LRG data better than ΛCDM but Shape expected to be altered by non-linearities 30 P(k) (10 4 Mpc 3 /h 3 ) 10 3 f(r) ΛCDM (non-linear fit) Hu, Song, Huterer Peiris & Smith Hu (2007) (2006) 0.01 k (h/mpc) 0.1
41 Redshift Space Distortion Relationship between velocity and density field given by continuity with modified growth rate (f v = dlnd/dlna) Redshift space power spectrum further distorted by Kaiser effect 0.55 B 0 =1 f v (ΛCDM) k (h/mpc)
42 Non-Linear GR Regime
43 Three Regimes Three regimes defined by γ= Φ/Ψ BUT with different dynamics Examples f(r) and DGP braneworld acceleration Parameterized Post-Friedmann description Non-linear regime follows a halo paradigm but a full parameterization still lacking and theoretical, examples few: f(r) now fully worked General Relativistic Non-Linear Regime Scalar-Tensor Regime Conserved-Curvature Regime r r * c halos, galaxy large scale structure CMB r
44 Non-Linear Chameleon For f(r) the field equation 2 f R 1 3 (δr(f R) 8πGδρ) is the non-linear equation that returns general relativity High curvature implies short Compton wavelength and suppressed deviations but requires a change in the field from the background value δr(f R ) Change in field is generated by density perturbations just like gravitational potential so that the chameleon appears only if f R 2 3 Φ, else required field gradients too large despite δr = 8πGδρ being the local minimum of effective potential
45 Non-Linear Dynamics Supplement that with the modified Poisson equation 2 Ψ = 16πG δρ δr(f R) Matter evolution given metric unchanged: usual motion of matter in a gravitational potential Ψ Prescription for N-body code Particle Mesh (PM) for the Poisson equation Field equation is a non-linear Poisson equation: relaxation method for f R Initial conditions set to GR at high redshift
46 Environment Dependent Force Chameleon suppresses extra force (scalar field) in high density, deep potential regions Oyaizu, Hu, Lima, Huterer Hu &(2008) Smith (2006)
47 Environment Dependent Force For large background field, gradients in the scalar prevent the chameleon from appearing Oyaizu, Hu, Lima, Huterer Hu &(2008) Smith (2006)
48 N-body Power Spectrum PM-relaxation code resolves the chameleon transition to GR: greatly reduced non-linear effect P(k)/P GR (k) Oyaizu, Hu, Lima, Huterer Hu &(2008) Smith (2006) k (h/mpc)
49 N-body Power Spectrum Artificially turning off the chameleon mechanism restores much of enhancement P(k)/P GR (k) Oyaizu, Hu, Lima, Huterer Hu &(2008) Smith (2006) k (h/mpc)
50 N-body Power Spectrum Models where the chameleon absent today (large field models) show residual effects from a high redshift chameleon P(k)/P GR (k) Oyaizu, Hu, Lima, Huterer Hu &(2008) Smith (2006) k (h/mpc)
51 Distance Predicts Growth With smooth dark energy, distance predicts scale-invariant growth to a few percent - a falsifiable prediction Mortonson, Hu, Huterer (2008)
52 Mass Function Enhanced abundance of rare dark matter halos (clusters) with extra force Rel. deviation dn/dlogm Lima, Hu, Schmidt, Huterer Oyaizu, & Smith Hu (2006) (2008) M 300 (h -1 M )
53 Halo Model Power spectrum trends also consistent with halos and modified collapse f R0 =10-4 P(k)/P GR (k) halo model + modified collapse k (h/mpc) Lima, Hu, Schmidt, Huterer Oyaizu, & Smith Hu (2006) (2008)
54 f (R) Solar System Tests
55 Solar System Constraint Cassini constraint on PPN γ-1 <2.3x10-5 Easily satisfied if galactic field is at potential minimum f Rg <4.9x10-11 Allows even order unity cosmological fields γ-1 (x ) f R0 =0.1 f R0 =0.05 f R0 =0.01 f R0 =0.001 f R0 =0 n=4 Hu & Sawicki (2007) r/r 10 4
56 Summary Lessons from f(r) and DGP braneworld examples 3 regimes: large scales: conservation determined intermediate scales: scalar-tensor small scales: non-linear or GR Large and intermediate scales parameterized by metric ratio γ or g = (Φ + Ψ)/(Φ Ψ) as in PPN but with different dynamics Order unity modifications of Friedmann eqn suggest problematic order unity deviations in horizon scale physics (e.g. DGP) unless a second scale exists (e.g. f(r)) Small scales: non-linear mechanism and modified halo model N-body (PM-relaxation) simulations with non-linear chameleon mechanism show strongest deviations at intermediate scales
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