  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.
