Splitting the matter force spectrum Several of the observables we consider depend on the matter force spectrum – namely galaxy clustering and lensing, the RSD and the CMB power spectrum. Because we use "growth" to describe the evolution of perturbations in the late Universe, we assume that the early shape of the power spectrum is determined by geometrical parameters. As arguments for the HaloFitfit function, we use the mixed linear power spectrum from Eq.(1) and we use the growth versions of the cosmological parameters.
The main effects of changing Ωgeom are a scaling of the normalization of the power spectrum and a change in the wavenumber where it peaks. We use zi¼3.5 as our confidence value for the redshift at which growth parameters begin to control the evolution of the matter power spectrum. The geometric information mainly consists of the distance to the last scattering surface and the sound horizon size at recombination.
The probability of a DES supernova is a multivariate Gaussian in the difference between the predicted and measured values of the distance modulus μ. Pantheon[73] module, modified to use the DES measurements instead of the original Pantheon supernova sample.
ANALYSIS CHOICES AND PROCEDURE We use the same parameters and parameter priors as
We confirm that this compressed probability is an accurate representation of the Planck constraints in this five-dimensional parameter space—in other words, that the Planck probability is approximately Gaussian—by checking that the chain samples for the full Planck probability follow aχ2 distribution when it is evaluated. relative to the mean and covariance used in the compressed likelihood. Combining the post-reconstruction BAO and pre-reconstruction full-form fits tightens constraints on DMðzÞ by about 10%. The analysis presented in this paper was blinded in the sense that all analysis choices were fixed and we ensured that the pipeline passed a number of predefined validation tests before looking at the true cosmological results.
When performing parameter estimation on the real data, we masked the cosmology results using the following strategies:. We postprocessed all chains so that the mean of the posterior distributions lay on our fiducial cosmology. We maintained these constraints until we confirmed that the assay passed several sets of validation tests:.
We confirmed that our results cannot be significantly biased by any of the sampling schemes used in our validation tests. Upon further investigation, as described in Appendix D, we found that a similar posterior shift manifests itself in the analysis of synthetic data, so we believe it is due to a parameter-space projection effect rather than a property of the real DESs -data. After passing another step of internal review, we completed the analysis by updating the plots to show non-offset posteriors, computer stress, and model comparison statistics and writing descriptions of the results.
This choice was motivated by the fact that the sampling error in the maximum posterior estimate means that the compressed likelihood is more accurate when centered on the mean. Suspiciousness is a quantity constructed from the Bayesian R evidence ratio designed to remove the dependence of the voltage metric on the first choice. To translate this into a more quantitative measure of consistency, we use the fact that the quantity d−2logS follows approximately the probability distribution aχ2d, where is the number of parameters constrained by both data sets.
The Bayesian ratio of evidence and suspicion defined in equations (21)–(24) can be interpreted as testing the hypothesis that data sets A and B are described by a common set of cosmological parameters as opposed to two independent sets.
RESULTS: SPLIT PARAMETERS
The fact that the confidence regions intersect the Ωgrowm ¼Ωgeom line but are asymmetrically distributed around it is reflected in the Bayesian measure of 1.5σtension with ΛCDM. In the middle panel of Fig. 3, we show the combination of DES data with external geometric measurements from the CMB and BAO (Ext-geo). To understand the appearance of the lower bound, note that the measurement of a given late-time density fluctuation amplitude allows for arbitrarily small values of Ωgrowm because little or no structure growth over time can be compensated for by the large initial amplitude of As.
We see that when we split Ωm, the 68% confidence intervals for DES and DESþExt-geo intersect the Ωgrowm ¼Ωgeom line, while that of DESþExt-all just touches the ΛCDM line, favoring Ωgrowm >Ωgeom. When we evaluate the fraction of posterior volume above and below 0, we find that the fraction of posterior volume. Marginalized posterior of the differenceΩgrowm −Ωgeom , from fitting the split-Ωm model to the DES, DESþExt-geo, and DESþExt-all data combinations.
First, due to the difference in the constraining power on Ωgrow and Ωge, some asymmetry is expected in these marginalized posteriors, even if the data are consistent with ΛCDM. 12 of Appendix C showing versions of this plot for synthetic data generated with Ωgrowm ¼Ωgeom. This means that in cases where the shape of the posterior is influenced by the anterior boundary, e.g. Ωgrowm, this necessarily affected the shape of the marginalized posterior for Ωgrowm −Ωgeom.
Taking these caveats into account and comparing with the simulated results in Appendix C, we see that the DESþExt-all probability distribution is shifted to higher Ωgrowth −Ωgeom than found in simulated analyses. In the table, for the splitΩm model, we show one-dimensional marginalized constraints on Ωgrowth and Ωgeom. Because we expect the one-dimensional marginalized posteriors to be subject to significant projection effects for DES-only constraints on the splitΩm model and for the DESþExt-all constraints when splitting both Ωmandw, as discussed in Sect.IV and Appendix B, we report not parameter bounds for those cases.
As an alternative model comparison statistic for the split-Ωm model, we additionally report pðΩgrowm >Ωgeom Þ, the proportion of posterior volume with Ωgrowm >Ωgeom.
RESULTS: IMPACT OF GROWTH-GEOMETRY SPLIT ON OTHER PARAMETERS
This preference, combined with the DES upper limit onΩgrowm, likely drives the 2σ tension between Ext-all and DES, and appears to be responsible for pulling the combined DESþExt-all constraints away from the ΛCDMΩgrowm ¼Ωgeom line. It is instructive to examine how the constituent Planck and BOSS probabilities combine to produce the Ext-all contours. Constraints from DES and Ext-all external data sets, which include the compressed Planck probability, BOSS DR12 BAO and BOSS DR12 RSD.
Only the DES results are shown in blue, the Ext-all results are shown in pink, and their combination is shown in unshaded purple outlines. Putting all this together, we see that the shape of the Ext-all posterior strongly depends on the ratio of the Planck measurement of As to the BOSS measurement of σ8, as well as the extent to which the posterior degrees of freedom affect how deterministically the Planck As constraint maps into σ8. For example, lowering the Planck As constraints slightly or raising the BOSS σ8 constraints would shift the Ext-all constraints towards lower P .
Our Ext-all constraints are weaker than this because using a compressed Planck probability causes us to lose information about a degeneracy between P. To explore how our results would be affected by stronger Pmν constraints, in Figs. 8 we show the DESþExt-geo and DESþExt-all constraints on Ωgrowm andΩgeom when the sum of the neutrino masses is fixed to its minimum value, 0.06 eV. We find that the minimum neutrino mass assumption allows us to constrain Ωgrowm either with DESþExt-geo alone or with Ext-all data alone, and that the DESþ Ext-all fixed neutrino mass constraints are dominated by information from the external data.
Constraints on the parameters most relevant to the description. late-time growth shown for data sets comprising Ext-all. We can see in Figure 6 that there is a significant negative degeneracy between Ωgrow and AIA present in the DES posterior. When we partition Ωm, the Ωgrowm −AIA degeneracy causes the DES-only constraints to expand significantly, with most of the posterior volume residing in the region of small Ωgrowm and high AIA.
Outermost (bottom) constraints as second and third panels in Fig. 3, but with the sum of neutrino masses fixed at 0.06 eV.
DISCUSSION
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