Probing modified gravity theories with ISW and CMB lensing

We use the optimized skew-spectrum as well as the skew-spectra associated with the Minkowski functionals to test the possibility of using the cross-correlation of the integrated Sachs–Wolfe effect (ISW) and lensing of the cosmic microwave background (CMB) radiation to detect deviations in the theory of gravity away from General Relativity (GR). We find that the although both statistics can put constraints on modified gravity, the optimized skew-spectra are especially sensitive to the parameter B0 that denotes the Compton wavelength of the scalaron at the present epoch. We investigate three modified gravity theories, namely the post-parametrized Friedmann formalism; the Hu–Sawicki model and the Bertschinger–Zukin (BZ) formalism. Employing a likelihood analysis for an experimental setup similar to ESA's Planck mission, we find that, assuming GR to be the correct model, we expect the constraints from the first two skew-spectra, S(0)ℓ and S(1)ℓ⁠, to be the same: B0 < 0.45 at 95 per cent confidence level (CL) and B0 < 0.67 at 99 per cent CL in the BZ model. The third skew-spectrum does not give any meaningful constraint. We find that the optimal skew-spectrum provides much more powerful constraint, giving B0 < 0.071 at 95 per cent CL and B0 < 0.15 at 99 per cent CL, which is essentially identical to what can be achieved using the full bispectrum