Bias from gas inhomogeneities in the pressure profiles as measured from
  X-ray and SZ observations
  X-ray observations of galaxy clusters provide emission measure weighted
spectra, arising from a range of density and temperature fluctuations in the
intra-cluster medium (ICM). This is fitted to a single temperature plasma
emission model to provide an estimate of the gas density and temperature, which
are sensitive to the gas inhomogeneities. Therefore, X-ray observations yield a
potentially biased estimate of the thermal gas pressure, P_X. At the same time
Sunyaev-Zeldovich (SZ) observations directly measure the integrated gas
pressure, P_SZ. If the X-ray pressure profiles are strongly biased with respect
to the SZ, then one has the possibility to probe the gas inhomogeneities, even
at scales unresolved by the current generation of telescopes. At the same time,
a weak bias has implications for the interchangeable use of mass proxies like
Y_SZ and Y_X as cosmological probes. In this paper we investigate the
dependence of the bias, defined as b_P(r)=P_X(r)/P_SZ(r)-1, on the
characteristics of fluctuations in the ICM taking into account the correlation
between temperature and density fluctuations. We made a simple prediction of
the irreducible bias in idealised X-ray vs SZ observations using
multi-temperature plasma emission model. We also provide a simple fitting form
to estimate the bias given the distribution of fluctuations. Analysing a sample
of 16 clusters extracted from hydrodynamical simulations, we find that the
median value of bias is within +/-3% within R_500, it decreases to -5% at
R_500<r<1.5R_500 and then rises back to ~0% at r >~ 2R_500. The scatter of
b_P(r) between individual relaxed clusters is small -- at the level of <0.03
within R_500, but turns significantly larger (0.25) and highly skewed at r>~
1.5R_500. For any relaxed cluster we find |b_P(r)| < 15% within R_500, across
different implementations of input physics in the simulations. [abridged]
