Testing the dark energy consistency with geometry and growth

We perform parametric tests of the consistency of the standard $w\mathrm{CDM}$ model in the framework of general relativity by carefully separating information between the geometry and growth of structure. We replace each late-Universe parameter that describes the behavior of dark energy with two parameters: one describing geometrical information in cosmological probes, and the other controlling the growth of structure. We use data from all principal cosmological probes; of these, Type Ia supernovae, baryon acoustic oscillations, and the peak locations in the cosmic microwave background angular power spectrum constrain the geometry, while the redshift space distortions, weak gravitational lensing, and abundance of galaxy clusters constrain both geometry and growth. Both geometry and growth separately favor the $\mathrm{\Lambda }\mathrm{CDM}$ cosmology with the matter density relative to critical ${\mathrm{\Omega }}_M\simeq 0.3$. When the equation of state is allowed to vary separately for probes of growth and geometry, we find again a good agreement with the $\mathrm{\Lambda }\mathrm{CDM}$ value ($w\simeq -1$), with the major exception of redshift-space distortions which favor less growth than in $\mathrm{\Lambda }\mathrm{CDM}$ at $3\text{−}\sigma$ confidence, favoring the equation of state $w^{\text{grow}}\simeq -0.8$. The anomalous growth favored by redshift space distortions has been noted earlier, and is common to all Redshift space distortions data sets, but may well be caused by systematics, or be explained by the sum of the neutrino masses higher than that expected from the simplest mass hierarchies, $m_{\nu }\simeq 0.45\mathrm{eV}$. On the whole, the constraints are tight even in the new, larger parameter space due to impressive complementarity of different cosmological probes.

Figure 1

Plot showing the set of 472 supernovae used in this work. Error bars are from diagonal-only statistical errors. The black line shows the best-fit $\mathrm{\Lambda }\mathrm{CDM}$ model with parameter values from column 2 of Table 7.

Figure 2

RSD data used in our analysis, shown in the $f{\sigma }_8\text{−}F$ plane; more details can be found in Table 4. The black line shows the best-fit $\mathrm{\Lambda }\mathrm{CDM}$ model with our best-fit parameter values given in the second column of Table 7. The low-redshift 6dFGS measurement does not have an associated value for $F(z)$, and we, therefore, only show its horizontal error bar. The BOSS constraint on $F$ is obtained from the covariance of $H(z)$ and $D_A(z)$; see Appendix pp2 for details. The dashed error ellipse corresponds to an alternative RSD measurement at $z=0.57$ from Samushia et al. [54]; for details, see Sec. 6.

Figure 3

Fiducial constraints from cosmological probes before the geometry-growth parameter split. We show the 68% and 95% confidence constraints in the ${\mathrm{\Omega }}_M\text{−}{\sigma }_8$ plane assuming $w=-1$ held constant (left panel) and in the ${\mathrm{\Omega }}_M\text{−}w$ plane (right panel). In the labels, “EU” refers to our early-Universe prior, while “SH” refers to the sound horizon prior; see Table 6 for relevant details.

Figure 4

68% and 95% confidence constraints in the split ${\mathrm{\Omega }}_M$ plane with the equation of state held constant at the $\mathrm{\Lambda }\mathrm{CDM}$ value ($w^{\text{geom}}=w^{\text{grow}}=-1$). As in Fig. 3, “EU” refers to our early-Universe prior, while “SH” refers to the sound horizon prior.

Figure 5

68% and 95% confidence constraints in the split $w$ plane. Note that the combined $2\text{−}\sigma$ contour does not pass through the $w^{\text{geom}}=w^{\text{grow}}$ line. As before, “EU” refers to our early-Universe prior, while “SH” refers to the sound horizon prior. Individual CMB results have been omitted due to the poor constraints they provide in this plane, but they are included in the combined constraint. See text for details.

Figure 6

Dependence of our results on the RSD data and their analyses. Left panel: Combined constraints for the case where we replace the $z=0.57$ RSD measurement from [56] with the alternative BOSS measurement that uses the same raw data but a different analysis [54]; see Fig. 2. The combined constraints are now only slightly less discrepant with the $w^{\text{geom}}=w^{\text{grow}}$ line. Right panel: Combined constraints, but without the RSD data employed. The combined contour is now larger in the growth direction; however it is still somewhat discrepant with the $w^{\text{geom}}=w^{\text{grow}}$ line, though less so than with the RSD data included. See text for details.

Figure 7

Left panel: The effects on the combined constraints when the sum of the neutrino masses $m_{\nu }$ is allowed to vary, compared to our fiducial assumption of holding it fixed at 0.06 eV. The constraints are now fully consistent with the $w^{\text{geom}}=w^{\text{grow}}$ line. Right panel: Posterior likelihood on $m_{\nu }$ for when ${\mathrm{\Omega }}_M$ and $w$ are split (wider curve), and when $\text{growth}=\text{geometry}$ correspondence (${\mathrm{\Omega }}_M^{\text{geom}}={\mathrm{\Omega }}_M^{\text{grow}}$ and $w^{\text{geom}}=w^{\text{grow}}$) is enforced (narrower curve). In both cases a value of $m_{\nu }\simeq 0.45\mathrm{eV}$ is preferred; see text for details.

Figure 8

Top: Number of galaxy clusters within a given richness bin in the MaxBCG data set. Errors shown are the diagonal parts of the covariance matrix. The step function shown uses the parameter values from the best-fit $\mathrm{\Lambda }\mathrm{CDM}$ model (column 2 of Table 7). The data are summarized in Table 8. Bottom: Mean mass of galaxy clusters within the given richness bin in the MaxBCG data set. The step function uses the same parameter values as the top figure. The data are summarized in Table 9.

Figure 9

Same as the left panel of Fig. 3, but the various probes have been separated for easier viewing. The smaller, dark set of contours corresponds to all probes combined.

Figure 10

Same as the right panel of Fig. 3, but the various probes have been separated for easier viewing. The smaller, dark set of contours corresponds to all probes combined.

Figure 11

Same as Fig. 4, but the various probes have been separated for easier viewing. The smaller, dark set of contours corresponds to all probes combined.

Figure 12

Same as Fig. 5, but the various probes have been separated for easier viewing. The smaller, dark set of contours corresponds to all probes combined.