Future CMB integrated-Sachs-Wolfe-lensing bispectrum constraints on modified gravity in the parametrized post-Friedmann formalism

We forecast the constraints on both the Hu-Sawicki model and the Bertschinger-Zukin model of f(R) modified gravity within the parametrized post-Friedmann formalism for the Planck satellite experiment by performing the joint analysis of the integrated-Sachs-Wolfe-lensing bispectrum and cosmic microwave background power spectrum. We find that, even considering the temperature-temperature mode of cosmic microwave background power spectrum only, Planck data are expected to reduce the error bars on the modified gravity parameter B0 (related to the present value of the Compton wavelength of the extra scalar degree of freedom) at least one order magnitude compared with Wilkinson microwave anisotropy probe. The spectrum-bispectrum joint analysis can further improve the results by a factor ranging from 1.14 to 5.32 depending on the specific modified gravity model. One of our main results is that the cross correlation between the integrated-Sachs-Wolfe-lensing bispectrum and the power spectrum can be safely neglected when performing the joint analysis. For simplicity, we investigate only the likelihood of one parameter (B0) and fix all other cosmological parameters to their best-fit values in Wilkinson microwave anisotropy probe 7-yr results