  We investigate the impact of nonlinear evolution of the gravitational
potentials in the LCDM model on the Integrated Sachs-Wolfe (ISW) contribution
to the CMB temperature power spectrum, and on the cross-power spectrum of the
CMB and a set of biased tracers of the mass. We use an ensemble of N-body
simulations to directly follow the potentials and compare results to
perturbation theory (PT). The predictions from PT match the results to high
precision for k<0.2 h/Mpc. We compute the nonlinear corrections to the angular
power spectrum and find them to be <10% of linear theory for l<100. These
corrections are swamped by cosmic variance. On scales l>100 the departures are
more significant, however the CMB signal is more than a factor 10^3 larger at
this scale. Nonlinear ISW effects therefore play no role in shaping the CMB
power spectrum for l<1500. We analyze the CMB--density tracer cross-spectrum
using simulations and renormalized bias PT, and find good agreement. The usual
assumption is that nonlinear evolution enhances the growth of structure and
counteracts linear ISW on small scales, leading to a change in sign of the
CMB-LSS cross-spectrum at small scales. However, PT analysis suggests that this
trend reverses at late times when the logarithmic growth rate
f(a)=dlnD/dlna<0.5 or om_m(a)<0.3. Numerical results confirm these expectations
and we find no sign change in ISW-LSS cross-power for low redshifts.
Corrections due to nonlinearity and scale dependence of the bias are found to
be <10% for l<100, therefore below the S/N of the current and future
measurements. Finally, we estimate the CMB--halo cross-correlation coefficient
and show that it can be made to match that for CMB--dark matter to within 5%
for thin redshift shells, mitigating the need to model bias evolution.
