Spherical collapse in modified gravity with the Birkhoff theorem

We study structure formation in a phenomenological model of modified gravity which interpolates between Lambda cold dark fritter (Lambda CDM) and Dvali-Gabadadze-Porrati (DGP) gravity. In our model, the Friedmann equation assumes the form H-2(a) = (1 - Omega(m))H-alpha(a) + Omega(m)/a(3) with the interpolation parameter a. Generalization of spherical collapse by using the Birkhoff theorem along with the modified growth equation shows that the overdensity for spherical collapse delta(c) in these models is significantly lowered compared to Lambda CDM, leading to enhanced number densities of massive clusters and enhanced cluster merging rates. We find that delta(c)(z) is well fitted by a function of the form delta(c)(z) = a - bexp(-cz). We examine the sensitivity of Planck's and SPT's Sunyaev-Zel'dovich survey to constrain the modified gravity parametrization, and find that these experiments can easily distinguish between models with a cosmological constant and modified gravity by including prior constraints from cosmic microwave background (CMB) temperature and polarization anisotropies. Applying a Fisher matrix formalism yields an expected accuracy of Delta alpha similar or equal to 0.04 (SPT), ..., 0.07(Planck) for a combined measurement of the cluster redshift distribution and the CMB temperature and polarization power spectrum, assurning Lambda CDM as the fiducial cosmology.