Cosmological gravity on all scales V: MCMC forecasts combining large scale structure and CMB lensing for binned phenomenological modified gravity

Abstract

As cosmology rapidly approaches the data-dominated phase of stage IV large scale structure surveys, the modelling of nonlinear scales has become a serious challenge that faces the community, particularly when analysing models beyond $w$CDM. In this work, we emulate the matter power spectrum in a phenomenological parameterisation of modified gravity in which a time-varying effective gravitational constant $μ$ and a gravitational slip $η$ are binned in redshift. We are able to achieve accuracy $<1\%$ in the modified gravity boost relative to COLA (COmoving Lagrangian Acceleration) simulations. We forecast the constraining power for each bin using a simulated $3\times 2$pt LSST Y10-like data vector and a $6\times 2$pt LSST Y10 x Simons Observatory cosmic microwave background (CMB) lensing data vector. We recover the characteristic degeneracy between $μ$ and $η$ previously identified in Fisher forecasts and demonstrate that the best-constrained direction corresponds to the combination $Σ=μ(1+η)/2$ which governs the lensing potential. We show that while large scale structure is sensitive to growth of structure at low redshift, CMB lensing extends the sensitivity to a higher redshift range. These results demonstrate that fast emulation of nonlinear modified-gravity effects enables full Bayesian analyses of model-agnostic gravity parameterisations with realistic survey data vectors and astrophysical systematics.

Figures (8)

• Figure 1: Posterior distributions for the cosmological and modified gravity parameters obtained from the $3\times2$pt analysis of the LSST Year 10 forecast for the first three redshift bins. The vertical and horizontal dashed lines indicate the fiducial $\Lambda$CDM values used to generate the mock data vector. As expected, the lowest redshift bin yields the strongest constraints due to the higher signal-to-noise of the clustering and lensing observables and the larger contribution from nonlinear structure formation within the scales that we consider $k_{\rm cut}\leq 0.5 h\,{\rm Mpc}^{-1}$.
• Figure 2: Comparison of the posterior constraints obtained from the $3\times2$pt (blue) and $6\times2$pt (red) data vectors for the highest redshift bin ($2.15<z<3$). The $6\times2$pt analysis additionally includes CMB lensing auto- and cross-correlations assuming Simons observatory-like survey specifications. The vertical and horizontal lines indicate the fiducial $\Lambda$CDM cosmology used to generate the mock data vector. The addition of CMB lensing tightens the constraint along the lensing-sensitive combination $\Sigma=\mu(1+\eta)/2$, reflecting the strong sensitivity of CMB lensing to the Weyl potential at high redshift.
• Figure 3: Identical as fig. \ref{['fig:6x2_bin5']} but this time focusing on bin 4. We note that the gain is relatively smaller compared to the highest redshift bin (bin 5) due to the reduced sensitivity of CMB lensing to this bin.
• Figure 4: Principal component analysis of the $\mu - \eta$ posterior distributions. The left panel shows the posterior samples for the lowest redshift bin obtained from the $3\times2$pt analysis (red points), with the corresponding PCA eigenmodes overplotted in yellow. The right panel shows the same analysis for the highest redshift bin, comparing the $3\times2$pt constraints (magenta points) with the $6\times2$pt constraints that include CMB lensing (cyan points). The principal component directions for the $3\times2$pt and $6\times2$pt analyses are indicated by the red and yellow lines respectively. The wide eigenmode corresponds to the well-known degeneracy between $\mu$ and $\eta$, while the tightly constrained direction closely aligns with the lensing-sensitive combination $\Sigma = \mu(1+\eta)/2$ which governs the amplitude of lensing observables. Note that the black dashed line indicates the line of constant $\Sigma$ in both panels. The inclusion of CMB lensing primarily tightens the constraint along this $\Sigma$ direction at high redshift.
• Figure 5: Forecast constraints obtained using an emulator trained on the halo model reaction formalism implemented in ReACT. The triangle plot shows posterior constraints on a subset of cosmological and modified gravity parameters for an LSST Y10 $3\times2$pt analysis. While the constraint on the modified gravity parameter $\mu$ is broadly consistent with that obtained using the Gaussian Process emulator, several $\Lambda$CDM parameters appear artificially overconstrained. This behaviour arises because the ReACT predictions fail for a non-negligible fraction of the prior volume at extreme values of $\sigma_8$, leading to gaps in the training data and biasing the emulator response near the edges of parameter space. As a result, the emulator spuriously suppresses parameter variations in regions where reliable training samples are unavailable, producing unrealistically tight posterior constraints.