Modified Gravity David Parkinson University of Queensland Dark Fluids: Failure of GR?

• Einsteinian gravity appears to break down on scales much larger than the solar system (rotation curves of galaxies, acceleration of Hubble expansion) • This requires us to introduce some Dark Fluids

• Instead of Dark Energy, maybe the law of gravity is wrong c.f. perihelion of Mercury - Vulcan vs. Einstein

• But new theories must match Earth/Solar system tests, and so require cross-over scale when new physics takes over Modified Gravity

• General relativity (Einsteinian gravity) is a theory that relates the curvature of space-time to the energy-density of matter and energy • It can be understood as the theory of a non-trivially interacting massless helicity 2 particle. • Therefore any modification can be considered as changing its degrees of freedom • But these modifications may not recover GR in the limiting case • The vDVZ discontinuity: in massive gravity where mgraviton goes to zero, extra scalar degrees of freedom exist, breaking solar system tests Models • Consider three such modifications in the literature:

• f(R) gravity • DGP gravity • Galileon gravity 1. f(R) gravity

• Extra scalar degree of freedom function of Ricci scalar f(R)

• If so if f(R)->0, we recover ordinary GR. • If we vary this action with respect to the metric, we produce the modified Einstein equations

• Constant f(R) is just the Λ term (the cosmological constant) • For f(R) linear with R, Newton’s constant G is rescaled Matching ΛCDM

• Using Ricci scalar from the homogeneous FRW metric into the Einstein equation, we modify the Hubble law

• Such models have the same expansion history as ordinary DE, but make different structure formation predictions.

• We match to a form of H(z) given by

(Note that ρDE is not the Dark Energy density, merely an effective density induced in the Hubble law by the modification of gravity) • We can now reconstruct f(R) from the expansion history using Reconstructed f(R)

All curves have same expansion history, but different growth history, determined by B0

Brane 2. DGP gravity • In the Dvali-Gabadadze-Porrati braneworld gravity (or DGP), gravity leaks from the 4-dimensional Minkowski brane into the 5- dimensional “bulk” at large scales. Brane • Extra-scalar is ‘brane-bending field’ - giving warping of extra dimension • On small scales, gravity is effectively bound to the brane, and the usual 4D Newtonian gravity is recovered • The cross-over scale is fixed by the apparent ‘acceleration’ at late-times to be of order the Hubble constant Bulk today - fixing size of extra scalar DGP Theory

• In the DGP model, the Friedmann equation is • So in the flat case, when ρm→0,

H→1/rc.

• This has an effective ‘equation of state’ of w≅-0.7 at late time (in the flat case, assuming

Ωm), and there is already pressure on this model from SN 3. Galileon Gravity • A generalisation of the 4D DGP model • New scalar degree of freedom ᴨ emerges when considering decoupling limit of DGP (M5, MPl →∞)

• Action of ᴨ field invariant under Galilean shift- symmetry - Galileon

• Contains same self-accelerating solution as DGP, but free from ghost instabilities that plagued it

• Does not get renormalised to any loop in Testing gravity through Structure Formation

• Perturbations away from a homogeneous density are parameterised by δ

• These small perturbations in matter density grow by attracting and accumulating material under gravity • The rate of growth on large scales (linear physics) is set by the theory of gravity

• We find rate of growth is a power law of the density of matter f(R) growth DGP Structure formation

• In the DGP model, the linear growth is modified

• Here Geff

• In the flat case this is solved to give γ=11/16, but can take other values in a curved universe Screening mechanisms

• If the gravity is different, we can test it on lab or solar system system scales

• e.g. fifth force effect, or scale-dependent GNewton • Three “screening mechanisms” save the theories 2 2 2 L⊃ -½Z(ɸ0)(∂δɸ) -½m (ɸ0)δɸ +(β(ɸ)/mP)δɸδT

• Vainshtein mechanism: higher-order corrections (cubic and above) recover GR on scales smaller than Vainshtein radius (DGP, Galileon) • Chameleon mechanism: mass of field large enough to suppress range of fifth force (f(R) theories) • Symmteron mechnism: direct coupling to stress-energy tensor (T) is small Cosmological tests

Turnover Non-linear structure Primordial modes BAO scales formation k=10-4 k=10-3 k=10-2 k=0.1 k=1 k=10 k=100

Scale- Changed growth Screened regime

dependent/stochastic (e.g. Geff=4/3 GN) (e.g. fifth force) bias Clusters CMB, Redshift-space Lyman-α or low All-sky surveys distortions mass galaxies Thank you Extra Slides Successes of GR

• General Relativity is one of the most successful physical theories of the 20th century. • It recovers Newton’s law of Gravity in the weak field limit • It predicts: • The precession of the Perihelion of Mercury (detected by Le Verrier, 1859) • The bending of light around massive objects (Gravitational lensing, detected by Arthur Eddington, 1919) • Gravitational time dilation (Pound-Rebka experiment, 1959) • Gravitational time delay (Shapiro delay, 1966) • Frame-dragging (Gravity Probe B, 2011) • Gravitational waves (not yet detected) • Black holes (not yet detected) • And the Hubble expansion, of course! WiggleZ Survey

• WiggleZ is a spectroscopic galaxy redshift survey conducted on the AAT • It covers 1000 square degrees over the southern sky, and has measured the redshift of 250,000 galaxies • It targets bright, star- forming galaxies at high redshift by using the GALEX satellite to generate a source catalogue • It started in 2006 and finished in January 2011 (this year!) The WiggleZ Team • University of Queensland: Michael Drinkwater, Tamara Davis, David Parkinson, Signe Reimer-Sorensen • Swinburne: Chris Blake, Carlos Contreras, Warrick Couch, Darren Croton, Karl Glazebrook, Tornado Li, Felipe Marin, Greg Poole, Emily Wisniowski • AAO: Sarah Brough, Matthew Colless, Mike Pracy, Rob Sharp • Scott Croom (USyd), Ben Jelliffe (USyd), David Woods (UBC), Kevin Pimblet Redshift-space distortions

• The motions of galaxies are perturbed by the local gravitational field • The Power spectrum/correlation function in the line of sight is distorted relative to the transverse direction • Assuming these motions are generated by matter perturbations, we can measure the growth of structure 2D Power Spectra Growth of structure Conclusions

• We can use the WiggleZ redshift-space distortion measurements of the growth of structure to probe different models of modified gravity: • f(R) - which uses only geometric quantities similar to GR • DGP - where the potential leaks away into an extra dimension • Both models make very different predictions for the formation of structure, but f(R) predicts a growth index which varies with redshift and scale • Combing WiggleZ growth data with WMAP measurements of the amplitude of perturbations at early times, gives strong constraints on these models

-3 • the Compton Wavelength of the f(R) theory B0<10 (at 95% conf.) • the DGP theory is under significant tension, and may be ruled out