Towards accurate modelling of the ISW effect, the non-linear contribution

In a universe with a cosmological constant, the large-scale gravitational potential varies in time and this is, in principle, observable. Using an N-body simulation of a $\Lambda$CDM universe, we show that linear theory is not sufficiently accurate to predict the power spectrum of the time derivative, $\dot{\Phi}$, needed to compute the imprint of large-scale structure on the cosmic microwave background (CMB). The linear part of the $\dot{\Phi}$ power spectrum (the integrated Sachs-Wolfe effect or ISW) drops quickly as the relative importance of $\Omega_{\Lambda}$ diminishes at high redshift, while the non-linear part (the Rees-Sciama effect or RS) evolves more slowly with redshift. Therefore, the deviation of the total power spectrum from linear theory occurs at larger scales at higher redshifts. The deviation occurs at $k\sim 0.1 $ $h$ Mpc$^{-1}$ at $z=0$. The cross-correlation power spectrum of the density $\delta$ with $\dot{\Phi}$ behaves differently to the power spectrum of $\dot{\Phi}$. Firstly, the deviation from linear theory occurs at smaller scales ($k\sim 1 $ $h$ Mpc$^{-1}$ at $z=0$). Secondly, the correlation becomes negative when the non-linear effect dominates. For the cross-correlation power spectrum of galaxy samples with the CMB, the non-linear effect becomes significant at $l\sim 500$ and rapidly makes the cross power spectrum negative. For high redshift samples, the cross-correlation is expected to be suppressed by $5-10%$ on arcminute scales. The RS effect makes a negligible contribution to the large-scale ISW cross-correlation measurement. However, on arc-minute scales it will contaminate the expected cross-correlation signal induced by the Sunyaev-Zel'dovich effect.