Modified gravity constraints with Planck ISW-lensing bispectrum

TL;DR

This work tests late-time modifications of gravity within the Horndeski class using the EFT of dark energy, focusing on the ISW effect via CMB lensing cross-correlations and Planck PR4 data. It models perturbations with $oldsymbol{rac{rac{oldsymbol{rac{alpha_i(a)=c_i olinebreak mod}}}}}$ on a background $w(a)=w_0+w_a(1-a)$ and varies $c_B$ and $c_M$, applying stability conditions and MCMC inference. The key result is that including the ISW-lensing bispectrum tightens MG constraints by ~20%, reduces the MG parameter-space volume by 40–80%, and yields $c_B<1.2$ at 95% CL, while allowing a phantom crossing and showing MG is mildly preferred over ΛCDM depending on SN data; Planck PR4 data reduce deviations from GR compared to Planck PR3. Overall, the ISW-lensing probe proves to be a powerful constraint on late-time gravity modifications and highlights the importance of high-quality Planck data for MG tests.

Abstract

We present updated constraints on modified gravity by including the Integrated Sachs-Wolfe (ISW) effect from CMB lensing-CMB temperature cross-correlations, based on the latest Planck PR4 maps. Utilizing the Effective Field Theory of dark energy approach and adopting the $w_0w_a$CDM background cosmological model, we find that including the CMB ISW lensing cross-correlations tighten constraints on the modified gravity parameters by approximately $20\%$, reducing the viable parameter space by $40-80\%$. We derive constraints from Planck CMB, Planck and ACT CMB lensing, DESI DR1 BAO, CMB ISW-lensing, and type Ia supernovae (SN Ia) data. The constraints on the EFT parameters controlling the kinetic braiding and non-minimal coupling are consistent with General Relativity (GR) at the $95\%$ CL. In particular, we obtain a bound on the kinetic braiding parameter, $c_B < 1.2$ at $95\%$ CL. In the $w_0$-$w_a$ parameter space, our results imply a crossing of the phantom divide, $w=-1$. The modified gravity model shows a mild preference over $Λ$CDM at the $1.8σ$, $2.6σ$ and $3.2σ$ levels for the combinations with Pantheon+, Union3 and DESY5 supernova datasets. We find that using the latest $\texttt{HiLLiPoP}+\texttt{LoLLiPoP}$ likelihoods alleviates the departure of modified gravity parameters from the GR-values compared to results using {\it Planck} 2018 data. This paper underlines the importance of the ISW lensing probe in constraining late-time modifications of gravity.

Figures (5)

• Figure 1: Linear theory predictions for $C_\ell^{T\phi}$ in the MG model while fixing the standard 6 cosmological parameters and $(w_0,w_a)$ to their best-fit values obtained in the CMB (PR3)+DESI+PantheonPlus analysis of ref. Chudaykin:2024gol. Black line represents the $w_0w_a{\rm CDM}$ prediction for the same best-fit cosmology, where the dark energy perturbations are computed using the parametrized post-Friedmann approach. The blue points with $1\sigma$ error bars represent the ISW lensing bispectrum estimate in the range $2<\ell<84$ from the Planck PR4 maps Carron:2022eum.
• Figure 2: Theoretical predictions for the ISW lensing bispectrum $C_\ell^{T\phi}$ ( left panel) and the temperature power spectrum $C_\ell^{TT}$ ( right panel) in the MG model for different parameter choices selected from the MCMC chains of the CMB (PR3)+DESI+PantheonPlus MG analysis in ref. Chudaykin:2024gol (for details on the selection algorithm, see the main text). For clarity, we report only the EFT parameters: $(c_M,c_B)=(-0.55,1.02)$ (in red), $(0.14,0.61)$ (in gray, best-fit model), $(0.43,0.69)$ (in purple), $(0.77,0.36)$ (in green), and $(1.20,0.31)$ (in brown). The $\Lambda$CDM prediction (dashed line) corresponds to the best-fit model from the same dataset. The blue points with $1\sigma$ error bars represent the $C_\ell^{T\phi}$ estimate from the Planck PR4 NPIPE maps Carron:2022eum and the $C_\ell^{TT}$ measurement from the Commander likelihood Planck:2018vyg.
• Figure 3: 68% and 95% marginalized constraints in the $c_B-c_M$ plane for various data combinations in the MG model. Adding the ISW lensing data shrinks the credible-interval contours, excluding scenarios with $c_B\gtrsim1$.
• Figure 4: 68% and 95% marginalized constraints on $w_0$ and $w_a$ parameters in the MG (solid) and $w_0w_a{\rm CDM}$ (dashed) models. The constraints are obtained from the combinations of $\rm CMB+DESI+ISWL$ with PantheonPlus (in blue), Union3 (in orange), and DESY5 (in green).
• Figure 5: Posterior distributions of the cosmological parameters in the MG model from the combinations of CMB+DESI+ISWL with PantheonPlus (blue), Union3 (orange) and DESY5 (green) datasets.