Cosmological parameters of modified gravity Levon Pogosian Simon Fraser University Burnaby, BC, Canada In collaborations with R. Crittenden, A. Hojjati, K. Koyama, A. Silvestri, G.-B. Zhao
Two questions 1. Is data consistent with equations of GR? apply conventional measures of goodness of fit apply additional consistency tests ( triggers ): E G, growth index γ, (consistency with GR need not imply inconsistency with modified gravity) 2. What are constraints on Modified Gravity? test specific modified gravity models or classes of models develop a framework that a) maps onto parameters of reasonable modified gravity models b) includes the answer to Question 1
Popular triggers The growth index Wang and Steinhardt (1998) Linder & Cahn (2007) Simple (if one number). OK as a consistency test on a subset of data Allows for unphysical dynamics, not a good fit parameter E G = (Galaxy X Lensing) / (Galaxy X Velocity) Zhang, Liguori, Bean and Dodelson (2007) Directly observable A way of storing data (in some idealized limit) Consistency of triggers with GR need not rule out modifications of gravity
Building a framework for linear perturbations in modified gravity Things we can agree to keep Metric theory, FRW background initial conditions, particle content pick frame in which matter is conserved ignore radiation, use Newtonian gauge In GR, Poisson and anisotropy equations are algebraic in Fourier space: In modified gravity, Poisson and anisotropy equations can be dynamical, e.g.
Approaches to building parameterized frameworks (1) Parameterize new terms in the equations of motion ( new PPF of Baker et al) uniquely maps onto parameters of specific theories (2) Parameterize terms in the effective action (Gubitosi et al; Bloomfield et al) 6 functions of time for a general single field (3) Parameterize the effective dark sector (Pearson & Battye) (4) Use algebraic relations in Fourier space (Hu & Sawicki, MGCAMB and this talk) parameterization of solutions maps onto parameters of specific theories in the quasi-static limit Hu & Sawicki s PPF: separate super- and sub-horizon evolution, then link the two
Fitting unknown functions to data Pick a specific functional form, e.g. if motivated by theory o different forms lead to different bounds Discretize into bins o works well for forecasts o use Principal Component Analysis (PCA) to gain insights
PCA forecast for µ(a,k) and γ(a,k) for DES and LSST discretize µ and γ on a (z,k) grid treat each pixel, µ ij and γ ij, as a free parameter discretize w(z) on the same z-grid and treat w i as free parameters, along with Ω c, Ω b, h, n s, A s, τ and bias calculate the Fisher Matrix for 800+ parameters diagonalize to find principal components of µ and γ PCA provides variances of uncorrelated parameters α m Zhao, LP, Silvestri, Zylberberg (2009); Hojjati, Zhao, et al (2011)
Eigenmodes and uncertainty in their amplitudes for LSST+ Zhao, LP, Silvestri, Zylberberg (2009); Hojjati, Zhao, et al (2011)
PCA is a useful forecast tool Tells us the features of functions that can be measured best Tells us the regions in (k,z) that are most constraining Offers a clean way to compare different observational probes Allows to project forecasts onto parameters of other models Provides insight into degeneracies with other parameters Questions PCA cannot answer Which parameters should we fit to data? Fitting only the best constrained modes assume the amplitudes of poorly constrained modes are known to be exactly zero! Which eigenmodes are physically meaningful? E.g. the k-dependence cannot be arbitrary
Theoretical priors are subjective but unavoidable make them explicit o let theory fix the k-dependence o use explicit smoothness priors on time-dependence
The form of equations for scalar perturbations Poisson: Anisotropy: EOM for extra fields: Quasi-Static Approximation same Even powers of k for scalar degrees of freedom on isotropic backgrounds Second order in k if only one scalar DOF and 2 nd order EOM Silvestri, LP, Buniy, arxiv:1302.1193
The Quasi-Static Approximation Under QSA, a key conceptual difference (i.e. theories vs solutions) between the parametrized frameworks disappears QSA includes two separate assumptions: 1) Time derivatives are much smaller than space derivatives 2) The sub-horizon limit: 2 implies 1 in LCDM, but not in theories with extra degrees of freedom How detectable are non-quasi-static departures from GR? Rapid time variations are generic, but (so far) unobservable in viable models (e.g. f(r), chameleon, symmetron, dilaton, ) Near horizon scale departures from GR are observable, in principle, but (so far) too small in viable models
A way to proceed Locality constrains µ and γ to be ratios of polynomials in k in the quasi-static limit Silvestri, LP, Buniy, arxiv:1302.1193 Restrict to single scalar field models with second order equations of motion Functions p i (a) are expected to be slowly varying Super-horizon modifications are allowed Fit bins of p i (a) with an added explicit prior that correlates neighboring bins Crittenden, Zhao, LP, Samushia, Zhang, arxiv:1112.1693, arxiv:1207.3804 (Can use the same idea to fit bins of smooth functions of time in other frameworks)
true w(z) MCMC using many w-bins
true w(z) no prior MCMC using many w-bins reconstructed w(z) o MCMC will not converge o large variance o zero bias
true w(z) Excessively strong prior MCMC using many w-bins reconstructed w(z) o tiny error bars (small variance) o large bias
true w(z) reasonable prior MCMC using many w-bins reconstructed w(z) o moderate variance o insignificant bias
What is reasonable? Smooth features (well constrained by data) are not biased by the prior Noisy features (poorly constrained by data) are determined by the prior Procedure: a) Decide on a desired smoothness scale b) Build a prior covariance matrix c) Use PCA to compare data with data+prior d) Tune the prior so that eigenmodes with periods larger than the smoothness scale are not affected 10 4 1/σ 2 10 2 Data Prior Data + Prior 10 0 10-2 0 5 10 15 20 mode number Crittenden, Zhao, LP, Samushia, Zhang, PRD (1112.1693); PRL (1207.3804)
Summary Consistency of some of the triggers (e.g. E G ) with GR need not automatically rule out all modifications of gravity We need parametrized frameworks to determine the allowed parameter space for alternative gravity models preferably in a model-independent way PCA is a useful forecast tool, but does not tell you what to fit to data General frameworks must be supplemented by priors to eliminate unphysical solutions and make it possible to fit functions to data