Is Modified Gravity Required by Observations? An Empirical Consistency Test of Dark Energy Models

TL;DR

The paper tests whether observations require modifications to gravity by enforcing a geometry-growth consistency check through parameter-splitting, introducing meta-parameters $\Omega_\Lambda^{\rm (geom)}$ vs $\Omega_\Lambda^{\rm (grow)}$ and $w^{\rm (geom)}$ vs $w^{\rm (grow)}$. By fitting SN, CMB, galaxy clustering, and weak lensing data with MCMC, the authors assess whether the split parameters agree as expected under GR; ΛCDM is consistent, while current data place tight limits on deviations, with $\Delta\Omega_\Lambda$ around $-0.0044$ (68% CL) and $\Delta w$ around $0.37$ (68% CL). The analysis shows that theories like DGP are increasingly disfavored, as growth and geometry constraints strongly align, and the method provides a crude but useful parametrization of the space of modified gravity theories. The work demonstrates the potential of future surveys to drastically tighten these consistency checks, enhancing our ability to distinguish dark energy dynamics from genuine modifications of gravity. The approach can be extended to more complex dark energy models and alternative gravity theories as data quality improves.

Abstract

We apply the technique of parameter-splitting to existing cosmological data sets, to check for a generic failure of dark energy models. Given a dark energy parameter, such as the energy density Omega_Lambda or equation of state w, we split it into two meta-parameters with one controlling geometrical distances, and the other controlling the growth of structure. Observational data spanning Type Ia Supernovae, the cosmic microwave background (CMB), galaxy clustering, and weak gravitational lensing statistics are fit without requiring the two meta-parameters to be equal. This technique checks for inconsistency between different data sets, as well as for internal inconsistency within any one data set (e.g., CMB or lensing statistics) that is sensitive to both geometry and growth. We find that the cosmological constant model is consistent with current data. Theories of modified gravity generally predict a relation between growth and geometry that is different from that of general relativity. Parameter-splitting can be viewed as a crude way to parametrize the space of such theories. Our analysis of current data already appears to put sharp limits on these theories: assuming a flat universe, current data constrain the difference Omega_Lambda(geom) - Omega_Lambda(grow) to be -0.0044 +/- 0.0058 (68% C.L.); allowing the equation of state w to vary, the difference w(geom) - w(grow) is constrained to be 0.37 +/- 0.37 (68% C.L.). Interestingly, the region w(grow) > w(geom), which should be generically favored by theories that slow structure formation relative to general relativity, is quite restricted by data already. We find w(grow) < -0.80 at 2 sigma. As an example, the best-fit flat Dvali-Gabadadze-Porrati (DGP) model approximated by our parametrization lies beyond the 3 sigma contour for constraints from all the data sets.

Figures (3)

• Figure 1: Joint constraints on $\Omega_{\Lambda}^{\rm (geom)}$ and $\Omega_{\Lambda}^{\rm (grow)}$ in a $\Lambda$CDM model (upper panel) and the normalized likelihood distribution of $\Delta\Omega_\Lambda\equiv\Omega_\Lambda^{\rm (geom)}-\Omega_\Lambda^{\rm (grow)}$ (lower panel). Here the equation of state parameters are fixed as $w^{\rm (geom)}=w^{\rm (grow)}=-1$. The contours and curves show the $68\%$ confidence limits from the marginalized distributions. The thick gray lines show $\Omega_{\Lambda}^{\rm (geom)}=\Omega_{\Lambda}^{\rm (grow)}$. The data sets used are described in the text. Different contours and curves represent constraints from different combinations of the data sets. The smallest contour and the most narrow curve (black solid line) represent constraints from all the data. No significant difference is found and deviations are constrained to $\Delta\Omega_\Lambda=-0.0044^{+0.0058+0.0108}_{-0.0057-0.0119}$ ($68\%$ and $95\%$ C.L.).
• Figure 2: Variations of CMB temperature power spectra due to different changes of $\Omega_\Lambda^{\rm (geom)}$ and $\Omega_\Lambda^{\rm (grow)}$ (with all the other cosmological parameters fixed) as illustrated in the inset on the $\Omega_\Lambda^{\rm (geom)}$vs.$\Omega_\Lambda^{\rm (grow)}$ plane. The black solid curve corresponds to the black square symbol, which is our best--fit $\Lambda$CDM model with $\Omega_\Lambda^{\rm (geom)}=\Omega_\Lambda^{\rm (grow)}=0.744$. The blue dashed curve corresponds to the blue circular symbol, which is obtained from the best--fit model by fixing $\Omega_\Lambda^{\rm (geom)}=\Omega_\Lambda^{\rm (grow)}$ and increasing both parameters by 0.03. The red dot--dashed curve corresponds to the red triangular symbol, which is obtained by fixing $\overline{\Omega}_\Lambda$ and increasing $\Omega_\Lambda^{\rm (grow)}$ by 0.03 while decreasing $\Omega_\Lambda^{\rm (geom)}$ by 0.03.
• Figure 3: Joint constraints on $w^{\rm (geom)}$ and $w^{\rm (grow)}$ in a QCDM model (upper panel) and the normalized likelihood distribution of $\Delta w\equiv w^{\rm (geom)}-w^{\rm (grow)}$ (lower panel). Here the energy density parameters are fixed as $\Omega_{\rm DE}^{\rm (geom)} =\Omega_{\rm DE}^{\rm (grow)}$. The contours and curves show the $68\%$ confidence limits from the marginalized distributions. The thick gray lines show $w^{\rm (geom)}=w^{\rm (grow)}$. The data sets used are described in the text. Different contours and curves represent constraints from different combinations of the data sets (see legend in Fig. \ref{['fig:LCDM']}). The smallest contour and the most narrow curve (black solid line) represent constraints from all the data. No significant difference is found and deviations are constrained to $\Delta w=0.37^{+0.37+1.09}_{-0.36-0.53}$ ($68\%$ and $95\%$ C.L.). The star--shaped symbol corresponds to the effective $w^{\rm (geom)}$ and $w^{\rm (grow)}$, which approximately match the expansion history and the growth history, respectively, of a flat DGP model with our best--fit $\Omega_m$.