Towards accurate modelling of the integrated Sachs-Wolfe 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 Λ cold dark matter universe, we show that linear theory is not sufficiently accurate to predict the power spectrum of the time derivative, needed to compute the imprint of large-scale structure on the cosmic microwave background (CMB). The linear part of the power spectrum [the integrated Sachs-Wolfe effect (ISW)] drops quickly as the relative importance of Ω _Λ diminishes at high redshift, while the non-linear part [the Rees-Sciama effect (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 ∼ 0.1 h Mpc ^-1 at z = 0. The cross-correlation power spectrum of the density δ with behaves differently from the power spectrum of. First, the deviation from linear theory occurs at smaller scales (k ∼ 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 ∼ 500 and rapidly makes the cross-power spectrum negative. For high-redshift samples, the cross-correlation is expected to be suppressed by 5-10 per cent on arcminute scales. The RS effect makes a negligible contribution to the large-scale ISW cross-correlation measurement. However, on arcminute scales it will contaminate the expected cross-correlation signal induced by the Sunyaev-Zel'dovich effect.