Falsifying paradigms for cosmic acceleration

Consistency relations between growth of structure and expansion history observables exist for any physical explanation of cosmic acceleration, be it a cosmological constant, scalar field quintessence, or a general component of dark energy that is smooth relative to dark matter on small scales. The high-quality supernova sample anticipated from an experiment like the SuperNova/Acceleration Probe and cosmic microwave background data expected from Planck thus make strong predictions for growth and expansion observables that additional observations can test and potentially falsify. We perform a Markov chain Monte Carlo likelihood exploration of the strength of these consistency relations based on a complete parametrization of dark energy behavior by principal components. For $\Lambda \mathrm{CDM}$, future supernova and cosmic microwave background data make percent level predictions for growth and expansion observables. For quintessence, many of the predictions are still at a level of a few percent with most of the additional freedom coming from curvature and early dark energy. While such freedom is limited for quintessence where phantom equations of state are forbidden, it is larger in the smooth dark energy class. Nevertheless, even in this general class predictions relating growth measurements at different redshifts remain robust, although predictions for the instantaneous growth rate do not. Finally, if observations falsify the whole smooth dark energy class, new paradigms for cosmic acceleration such as modified gravity or interacting dark matter and dark energy would be required.

Figure 1

The first 15 PCs of $w(z)$ (increasing variance from bottom to top), with 500 redshift bins between $z=0$ and $z_{\max }=1.7$. The vertical dashed line shows the minimum redshift of the data assumed for computing the PCs, $z_{\min }^{\mathrm{SN}}=0.03$. The PCs are offset vertically from each other for clarity with dotted lines showing the zero point for each component.

Figure 2

Redshift-dependent quantities for the fiducial flat $\Lambda \mathrm{CDM}$ cosmology with ${\Omega }_{\mathrm{m}}=0.24$ and $h=0.73$ as assumed for PC construction and for the default data sets for MCMC, including the fractions of the total density in matter and ark energy, ${\Omega }_{\mathrm{m}}(z)$ and ${\Omega }_{\mathrm{DE}}(z)$ (top: solid blue and dashed red lines, respectively), comoving angular diameter distance $D(z)$ (middle: solid blue line), inverse of the expansion rate $H^{-1}(z)$ (middle: dashed red line), growth function relative to early times (bottom: solid blue line), and growth rate $f=1+d\ln G/d\ln a$ (bottom: dashed red line). The vertical dotted line is plotted at $z_{\max }=1.7$, the maximum redshift for the PCs and for SNe in the likelihood analysis.

Figure 3

Illustration of the model classes (large circles) and subclasses (represented by points within different regions of the circles) that we consider in this paper. Open points mark the initial (simplest) type of model within each class, and arrows indicate paths to the more complex models in the class. The four quadrants separate flat and curved models and those with and without early dark energy. The $\Lambda \mathrm{CDM}$ class does not contain models with significant early dark energy by definition.

Figure 4

Forecasted predictions from SNAP SN and Planck CMB data for growth and expansion observables, showing the influence of curvature on predictions for $\Lambda \mathrm{CDM}$ models. Shaded regions enclose 68% C.L. regions and curves without shading are upper and lower 95% C.L. limits, plotted as fractional differences from the fiducial flat $\Lambda \mathrm{CDM}$ cosmology. The model classes are flat $\Lambda \mathrm{CDM}$ (light gray areas) and $\Lambda \mathrm{CDM}$ with nonzero spatial curvature (dark blue areas). An example model with nonzero curvature is also shown (dashed red curve). Figures 5, 6, 7, 8, 9, 10, 11, 14 all follow the same format.

Figure 5

Effects of generalization of flat $\Lambda \mathrm{CDM}$ (light gray areas) to quintessence (dark blue areas) (example model: dashed red lines). Quintessence is defined to have $-1\le w\le 1$ with $w(z)$ parametrized by 15 PCs. Here $w_{\infty }=-1$ and ${\Omega }_{\mathrm{K}}=0$ to eliminate early dark energy and curvature.

Figure 6

Effects of early dark energy (dark blue areas) (example model: dashed red lines) on quintessence models (light gray areas). Quintessence models from Fig. 5 are generalized to have $w_{\infty }$ vary, with ${\Omega }_{\mathrm{K}}=0$ to eliminate spatial curvature.

Figure 7

Effects of curvature (dark blue areas) (example model: dashed red lines) on quintessence models (light gray areas). Quintessence models from Fig. 5 are generalized to have ${\Omega }_{\mathrm{K}}$ vary, with $w_{\infty }=-1$ to eliminate early dark energy.

Figure 8

Effects of curvature and early dark energy (dark blue areas) (example model: dashed red lines) on quintessence models (light gray areas). Quintessence models from Fig. 5 are generalized to have both ${\Omega }_{\mathrm{K}}$ and $w_{\infty }$ vary.

Figure 9

Effects of generalizing $-1\le w\le 1$ quintessence models (light gray areas) to smooth dark energy with $-5\le w\le 3$ (dark blue areas) (example model: dashed red lines). The smooth dark energy model class generalizes the quintessence models from Fig. 5 using the same 15 PCs for $w(z<z_{\max })$ with $w_{\infty }=-1$ and ${\Omega }_{\mathrm{K}}=0$ to eliminate early dark energy and spatial curvature.

Figure 10

Effects of early dark energy (dark blue areas) (example model: dashed red lines) on smooth dark energy models (light gray areas). Smooth dark energy models from Fig. 9 are generalized to have $w_{\infty }$ vary, with ${\Omega }_{\mathrm{K}}=0$ to eliminate spatial curvature.

Figure 11

Effects of curvature (dark blue areas) (example model: dashed red lines) on smooth dark energy models (light gray areas). Smooth dark energy models from Fig. 9 are generalized to have ${\Omega }_{\mathrm{K}}$ vary, with $w_{\infty }=-1$ to eliminate early dark energy.

Figure 12

Effects of priors on the degeneracy between curvature and $w(z)$ PCs for the smooth dark energy models with curvature (but not early dark energy) in Fig. 11. Top: 1D marginalized posterior probability $P({\Omega }_{\mathrm{K}})$ for the default priors that are flat in PC amplitudes ${\alpha }_i$ (solid, dark blue line), and for alternate priors that are flat in the density of each PC at $z_{\max }$ relative to $z=0$, ${\rho }_i(z_{\max })/{\rho }_i(0)$ as defined in Eq. (13) (dot-dashed, red line). The dotted curve is the mean likelihood distribution for flat ${\alpha }_i$ priors. Bottom: Probability contours of ${\Omega }_{\mathrm{K}}$ vs ${\alpha }_2$ at 68% and 95% C.L. for the same priors on PCs as in the top panel. Dashed lines mark the fiducial values, ${\Omega }_{\mathrm{K}}=0$ and ${\alpha }_2=0$.

Figure 13

Effects of priors on smooth dark energy models with nonzero curvature. Dark blue lines: flat top-hat priors on ${\alpha }_i$ as in Fig. 11. Light gray areas: flat priors on the density of individual PCs, ${\rho }_i(z_{\max })/{\rho }_i(0)$.

Figure 14

Effects of curvature and early dark energy (dark blue areas; example model: dashed red lines) on smooth dark energy models (light gray areas). Smooth dark energy models from Fig. 9 are generalized to have both $w_{\infty }$ and ${\Omega }_{\mathrm{K}}$ vary.

Figure 15

Effects of priors on smooth dark energy models with curvature and early dark energy. Same as Fig. 13, but comparing priors for the dark blue model in Fig. 14.

Figure 16

Growth functions of MCMC samples in the smooth dark energy model class ($-5\le w\le 3$) that include either early dark energy (EDE) at $z>z_{\max }$ ($w_{\infty }\ne -1$: top panels), curvature (${\Omega }_{\mathrm{K}}\ne 0$: middle panels), or both (bottom panels). We plot growth relative to early times in the left column of panels, and growth relative to the present on the right. Dashed red curves show growth in the fiducial flat $\Lambda \mathrm{CDM}$ model. Samples are selected randomly from those with likelihoods satisfying $\Delta {\chi }^2\le 4$, but for visual clarity we plot samples that are approximately evenly spaced in $G(z=0)$ (left panels) or $G_0(z=4)$ (right panels). The dotted vertical line in each panel marks $z=z_{\max }$.

Figure 17

Comparison of ${\Omega }_{\mathrm{K}}$ and linear combinations of the growth function relative to flat $\Lambda \mathrm{CDM}$, $\Delta G_0(z)$, and $\Delta G(z)$, at $z=z_{\max }$ and $z=1$ for randomly selected models (with $\Delta {\chi }^2\le 4$) in the smooth dark energy class with early dark energy and curvature. The coefficients used here are $K_1(z=1)=0.24$, $K_2(z=1)=0.10$, $K_1(z=z_{\max })=0.17$, and $K_2(z=z_{\max })=0.09$. Note that these specific values may not produce accurate estimates of ${\Omega }_{\mathrm{K}}$ for fiducial cosmologies other than the one assumed here.

Figure 18

Effects of curvature on the $\Lambda \mathrm{CDM}$ growth rate. Predictions from future SN and CMB data for the growth rate $f$, plotted relative to the fiducial model, and for the growth index $\gamma =\ln f/\ln [{\Omega }_{\mathrm{m}}(z)]$. The model classes here are $\Lambda \mathrm{CDM}$ either assuming flat geometry (light gray areas) or allowing nonzero curvature (dark blue areas) (example model: dashed red lines). For flat $\Lambda \mathrm{CDM}$ the predictions for $\gamma$ are so tight that the 68% region is obscured by the thickness of the 95% line. The three example models plotted here and in Figs. 19, 20 are the same as the ones in Figs. 4, 8, 14, respectively.

Figure 19

Effects of curvature and early dark energy on the quintessence growth rate. Same as Fig. 18 but for flat quintessence models with $w_{\infty }=-1$, i.e. no early dark energy (light gray areas), and quintessence with both nonzero curvature and early dark energy (dark blue areas) (example model: dashed red lines).

Figure 20

Effects of curvature and early dark energy on the smooth dark energy growth rate. Same as Fig. 19 but for smooth dark energy models ($-5\le w\le 3$).

Figure 21

Index for dark energy model classes compared in figures in Sec. 3. The number of parameters varied in the MCMC likelihood analysis increases from left to right. For extensions to the baseline model within each class, the additional parameters (${\Omega }_{\mathrm{K}}$, $w_{\infty }$, or both) are listed. Red numbers along lines connecting two models indicate the number of the figure in this paper in which we plot growth and expansion predictions for that pair of models.

Figure 22

Examples of equations of state constructed from the 15 PCs in Fig. 1, where the PC amplitudes satisfy the priors given by Eqs. (a10, a13) for $-1\le w\le 1$ (solid black lines) and $-5\le w\le 3$ (dashed red lines).

Figure 23

Test of PC completeness for quintessence models. Predictions for growth and expansion observables from MCMC with 10 (light gray lines) and 15 (dark blue lines) PCs for flat quintessence models ($-1\le w\le 1$) with $w_{\infty }=-1$ (no early dark energy).

Figure 24

Same as Fig. 23 for nonflat quintessence models with early dark energy.

Figure 25

Same as Fig. 23 for smooth dark energy models ($-5\le w\le 3$) including curvature and early dark energy.