Neff and Beam Calibration
Dan writing (input from Tom Crawford, Matthew Hasselfield, Christian Reichardt and Alex van Engelen )
This is an extension of the beam calibration discussion in an earlier post: Update on Neff Forecasts. The main point of this new post is to use the ACT beam eigenmodes directly, rather than the simpler parameterization of the beam uncertainty given there. Using the beam eigenmodes will give an unambiguous comparison between the calibration requirements needed to reach σ(Neff) = 0.027 and the calibration levels that have been achieved in ACT (for example).
The forecasts here use the same parameters as those in the earlier post, including point-sources and atmosphere.
ACT Beam Eigenmodes
The details for how the ACT beams are calibrated can be found in Hasselfield et al. (2013), and the beams themselves are available at https://lambda.gsfc.nasa.gov/product/act/actpol_beams_get.cfm.
For our purposes here, the measured sky is a convolution of the true sky with the beam shape, Bl, such that
Clobs= Bl2 Cl
By measuring Bl, we can deconvolve it to determine Cl. Of course, there is some error on our measurement of Bl so we can't do this perfectly. This uncertainty is defined using some basis of function that diagonalize the covariance matrix of that calibration measurement : the beam eigenmodes. The above link gives these functions with a normalization such that their amplitude is given by a gaussian distribution with zero mean and unit variance.
Here they are as a function of l:
The dashed grey line of this plot is (2l+1)-1/2. This line is shown as a rough estimate of where the beam uncertainty is larger than the cosmic variance (for the ACT level of calibration). Clearly the largest beam eigenmode is larger than this line for l > 3000.
Marginalizing over the ACT Beam Eigenmodes
Given the beam eigenmodes, it is easy to marginalize over their amplitude in a forecast. The derivative of the amplitude of a given eigenmode is just
2( δBl / Bl) Cl
and we include these in the Fisher matrix. For the ACT-level of calibration, we would put a prior on the beam parameters by adding a diagonal fisher matrix with 1s in the position of the beam amplitudes and zeros everywhere else. We will test the dependence on the calibration level by multiplying this matrix by a constant.
We can see the impact on σ(Neff) below:
We clearly see that the ACT calibration is insufficient to achieve the sensitivity predicted for a fixed beam. It looks like a factor of 10 better than ACT is needed to be roughly equivalent to a fixed beam and to be close to σ(Neff) = 0.03.
The need for beam calibration at this level is consistent with the observation that most of the improvement in Neff comes from l = 2000-4000 in TE. From the figure comparing the beam eigenmodes to cosmic variance, we see that a significant fraction of these modes are going to be lost due to marginalization over the amplitude of the largest beam eigenmodes. The factor of 10 required in our forecasts is somewhat larger than one would estimate by requiring the beams to lie below the CV curve but not dramatically so.
A quick look at the SPT beams (https://lambda.gsfc.nasa.gov/product/spt/spt_beam_real_2013_info.cfm) suggests comparable calibration levels for SPT and would likely lead to the same conclusion.