  Primordial isocurvature fluctuations between photons and either neutrinos or
non-relativistic species such as baryons or dark matter are known to be
sub-dominant to adiabatic fluctuations. Perturbations in the relative densities
of baryons and dark matter (known as compensated isocurvature perturbations, or
CIPs), however, are surprisingly poorly constrained. CIPs leave no imprint in
the cosmic microwave background (CMB) on observable scales, at least at linear
order in their amplitude and zeroth order in the amplitude of adiabatic
perturbations. It is thus not yet empirically known if baryons trace dark
matter at the surface of last scattering. If CIPs exist, they would spatially
modulate the Silk damping scale and acoustic horizon, causing distinct
fluctuations in the CMB temperature/polarization power spectra across the sky:
this effect is first order in both the CIP and adiabatic mode amplitudes. Here,
temperature data from the Wilkinson Microwave Anisotropy Probe (WMAP) are used
to conduct the first CMB-based observational search for CIPs, using
off-diagonal correlations and the CMB trispectrum. Reconstruction noise from
weak lensing and point sources is shown to be negligible for this data set. No
evidence for CIPs is observed, and a 95%-confidence upper limit of $1.1\times
10^{-2}$ is imposed to the amplitude of a scale-invariant CIP power spectrum.
This limit agrees with CIP sensitivity forecasts for WMAP, and is competitive
with smaller scale constraints from measurements of the baryon fraction in
galaxy clusters. It is shown that the root-mean-squared CIP amplitude on 5-100
degree scales is smaller than 0.07-0.17 (depending on the scale) at the
95%-confidence level. Temperature data from the Planck satellite will provide
an even more sensitive probe for the existence of CIPs, as will the upcoming
ACTPol and SPTPol experiments on smaller angular scales.
