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 Lambda cold dark matter universe, we show that linear theory is not sufficiently accurate to predict the power spectrum of the time derivative, Phi, needed to compute the imprint of large-scale structure on the cosmic microwave background (CMB). The linear part of the. Phi power spectrum [the integrated Sachs-Wolfe effect (ISW)] drops quickly as the relative importance of Omega(Lambda) 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 similar to 0.1 h Mpc(-1) at z = 0. The cross-correlation power spectrum of the density delta with Phi behaves differently from the power spectrum of. Phi. First, the deviation from linear theory occurs at smaller scales (k similar to 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 similar to 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.