Falsifying Paradigms for Cosmic Acceleration

Michael J. Mortonson, Wayne Hu, & Dragan Huterer

Abstract

Future measurements of cosmic distances and growth can test and potentially falsify classes of dark energy models including the cosmological constant and quintessence. The distance-redshift relation measured by future supernova surveys will strongly constrain the expansion history of the universe. Under the assumption of a particular dark energy scenario, limits on the expansion rate place bounds on the evolution of the growth of large-scale structure. We present the anticipated predictions for growth and expansion observables from a future SNAP supernova sample combined with CMB data from Planck. Although simple models like flat LCDM are easiest to falsify, strong consistency tests exist even for general dark energy models that include dynamical dark energy at low redshift, early dark energy, and nonzero spatial curvature.

- DarkEnergyModel Classes
- Equation of state at z < 1.7
- Cosmological constant (L) w = -1
- Quintessence -1 < w(z) < 1
- Smooth dark energy -5 < w(z) < 3
- Early dark energy: constant w∞

Spatial curvature:WK Expansion and Growth Predictions from MCMC

We use Markov Chain Monte Carlo (MCMC) likelihood analysis to find dark energy models in each class that fit the fiducial SNAP + Planck distance data well. By computing the expansion and growth observables for each of these distance-matched models, we find the ranges in which the observables are predicted to lie for each model class. A model class can be falsified if follow-up observations are inconsistent with these predictions. The full set of parameters used for MCMC is:

LCDM

The predicted ranges of the observables H, D, G, and G0 relative to the fiducial flat LCDM cosmology are shown here for two dark energy model classes: flat LCDM (grey) and LCDM with nonzero curvature (blue). The inner shaded areas are 68% CL regions and the outer curves enclose 95% CL.

Predictions from SNAP + Planck for all observables in the context of LCDM, with or without curvature, are quite strong [especially H(z = 1) for WK = 0].

Quintessence

Smooth Dark Energy

Without the quintessence prior on w(z), predictions are no longer one-sided relative to flat LCDM. Due to the large amount of freedom in general smooth dark energy models with both early dark energy and curvature, predictions for an observable at any single redshift are weak and prior-dependent.

Tests of this model class require observations at multiple redshifts. For example, the redshift dependence of the growth function depends differently on early dark energy and curvature.

Flat quintessence models with no early dark energy still make strong predictions for the expansion and growth histories when matched to SN and CMB distances, even though the PC param-etrization allows these models to span the space of -1 < w(z) < 1.

Principal Components

Complete basis for dark energy equation of state w(z), ordered by how well each component is measured by the fiducial data (SNAP + Planck).

Data

We assume the following future data will be available for dark energy model predictions:

With curvature and early dark energy, pre-dictions weaken, but only in one direction since the equation of state at z < 1.7 can only be larger than in the fiducial w = -1 model. To match the CMB di-stance, this restriction forbids closed uni-verses and requires the amount of early dark energy to be small.

Given future SN and CMB data consistent with flat LCDM, observed deviations of more than a few percent from the expected values of these observables would falsify the cosmo-logical constant paradigm for acceleration.

Certain comb-inations of G and G0that are correlated with WK must yield the same number at different red-shifts, serving as one consistency test for these general dark energy models.

- SNAP Type Ia supernovae (SNe) at z < 1.7, measuring with magnitude errors
- Planck CMB constraints on angular diameter distance to z* = 1089 and Wmh2.

- PC shapes are eigenfunctions of the Fisher matrix of the data.
- Uncertainties are eigenvalues.
- Need ~15 PCs for complete-ness in observables.

PC amplitudes {ai} are the model parameters for quintessence and smooth dark energy.

- Additional weak priors based on current data (11% H0, 4% distance to z = 0.35, 2.5% fraction of dark energy at z*)

Observables

H (expansion rate)

D (comoving angular diameter distance)

G (linear growth function normalized to z=1000)

G0 (linear growth function normalized to z=0)

f (growth rate)

Growth Index

The growth index g has been proposed as another test of consistency between the expansion and growth rates, since with g = 0.55 is a good approximation for models close to flat LCDM (below, left). However, for general dark energy models matched to distances this approximation can break down. Strong time dependence of the equation of state can widen the allowed range of growth indices, and for models beyond quintessence, the standard parametrization of g has a singularity when Wm(z) crosses unity in closed universe models.

Reference

e-print:

Mortonson, Hu, & Huterer (2008)

arXiv:0810.1744

Fiducial Cosmology

Flat LCDM with:

Wm = 0.24

WK = 0

h = 0.73

Quintessence (left) and smooth DE (right) [flat, no EDE; curved, EDE]

LCDM [flat; curved]