I'm not sure when I'm going to get round to writing up the conclusions of this. Here's a summary.

Using the lensed [tex]C_l[/tex] seems to work fine for parameter estimation as long as you compute the theoretical result accurately. In particular the 2nd-order perturbative result (e.g.

astro-ph/0001303) is

*not* accurate enough. Here's a comparison with the correct result

Red is the 2nd order result, and biases parameters for Planck. The blue result is that of

astro-ph/9505109, which is actually pretty much accurate enough for Planck. The very accurate 'correct' result was generated using the Nov 2004 version of

CAMB using a new full-sky generalization of the result in

astro-ph/9505109 (work with Anthony Challinor).

Using the accurate [tex]C_l[/tex] calculation, for a typical idealized Planck-like simulation I get parameter constraints for a vanilla model like this:

Here black is analysing the lensed sky with the lensed [tex]C_l[/tex], red is analysing the unlensed sky with the unlensed [tex]C_l[/tex], and blue is the wrong result if you analyse the lensed sky but neglect lensing in the [tex]C_l[/tex] calculation. The lensed results neglects non-gaussianity of the lensed field.

[The simulation input model had [tex]n_s = 0.99[/tex], [tex]A_s = 2.5 \times 10^{-9}[/tex], [tex]\Omega_b h^2 = 0.022[/tex], [tex]\Omega_b h^2 = 0.12[/tex], [tex]\tau=0.15[/tex] and [tex]h=0.72[/tex]].