# CMB-S4 wiki:Update on beam uncertainties and science goals

Update on beam uncertainties, 1/26/2021:

Dan Grin/Francis-Yan Cyr Racine writing (input from Neil Gockner-Wald, Christian Reichardt, Srini Raghunathan). Dan implemented the first round of changes to the DRAFT code, obtaining results consistent with these. Francis-Yan improved the code to be pythonic (e.g. use dictionaries and classes), obtaining consistent results.

This is an extension of the beam uncertainty discussion in https://cmb-s4.org/wiki/index.php/Update_on_Neff_Forecasts and https://cmb-s4.org/wiki/index.php/Neff_and_Beam_Calibration. We wished to explore the impact of beam uncertainty in realistic experiments. To this end, we have adapted the DRAFT Fisher forecasting tool (https://github.com/sriniraghunathan/cmbs4_fisher_forecasting) created by Srini (+ Joel Meyers and others) to suit our purposes, and the working version is available at (https://github.com/dgrin1/s4beams/tree/master/cmbs4_fisher_forecasting). Please use the fcyr branch.

ACTPol beams are obtained from the 2020 papers, based on data products publicly available at https://lambda.gsfc.nasa.gov/product/act/act_dr4_anc_prod_info.cfm and described in https://phy-act1.princeton.edu/public/saiola/act_dr4_A20.pdf, and https://phy-act1.princeton.edu/public/saiola/act_dr4_C20.pdf

SPT3G beams are based on a combination of planetary and field-source calibration, as partially described in https://arxiv.org/abs/2101.01684.

Beams are modeled as a multiplicative power-spectrum contribution
with an unknown amplitude, and derivative (in the usual Fisher formula) of
dC_{l}/da_{i}=2( δB_{l,i} / B_{l,i}) C_{l}, where i is a label for beam eigenmodes (eigenvectors of the covariance matrix in some basis), see for further explanation. The beam is expanded as B_{l,i}=Σ _{i} a_{i} e_{i,l}.

Fisher-level uncertainties for N_{eff} and n_{s} are obtained by applying the Fisher matrix information with a prior F_{jk} → F_{ jk}+δ_{ j=k=i}1/σ_{k,}^{2}, where the index i denotes the amplitude of the beam eigenmode in question. Eigenvectors are provided to us with a normalization such that a prior of σ_{k,}^{2}=1 corresponds to reproducing the current uncertainty of this experiment.

Below, we obtain uncertainty in N_{eff} relative to the target uncertainty as a function of an overall rescaling of all the beam eigenmodes.

We see that uncertainty in N_{eff} scales monotonically, flattening for sufficiently low σ a_{eigen} to a plateau with properties set by the overall noise level, f_{sky}, and the overall mean beam width (a property of each experiment). Some variation is evident between seasons, detector sets, and sky patches, as seen in the top set of figures. Beam measurements for different seasons (S13,14,15)/detector sets (PA1, PA2, PA3)/ sky regions (D1,D5,D6,D8, AA,BN) are provided and separately modeled (for ACTPol so far, soon with SPTPol, too). We see that beam characterization uncertainty sets how efficiently the desired σ_{Neff } is approached, but the amplitude of the plateau is controlled by overall noise level and beam, which must also be sufficiently low/narrow.

We show results as forecast for an ACTPol-type and an SPT-3G type experiment. We see that uncertainty in N_{eff} scales monotonically, flattening for sufficiently low σ _{aeigen} to a plateau with properties set by f_{sky}, and the overall mean beam width. Achieving the desired σ N_{eff} requires a high sky fraction and eigenmode amplitudes (beam noise levels) a factor of ~3 better than current experiments.

We explore the joint dependence on the overall noise level and beam uncertainty below, finding that very good noise levels and low beam uncertainties are required to truly hit the desired science-driving sensitivity to N_{eff}. The below plot is for fsky=0.6.

We must complete the analysis for SPT, and rescale beam eigen-modes to narrower overall beam widths, and explore the effect of calibration on non-thermal sources, as well as temperature-polarization leakage. We also have preliminary results for n_{s} as well as N_{eff}, shown below.

We see that beam characterization uncertainty sets how efficiently the desired σ_{ns } is approached, but the amplitude of the plateau is controlled by overall noise level and beam, which must also be sufficiently low/narrow.

In the bottom set of figures, we show results as forecast for an ACTPol-type and an SPT-3G type experiment. We see that uncertainty in n_{s} scales monotonically, flattening for sufficiently low σ _{aeigen} to a plateau with properties set by f_{sky}, and the overall mean beam width. Achieving the desired σ n_{s} requires a high sky fraction and eigenmode amplitudes (beam noise levels) a factor of ~3 better than current experiments.

The below plot is for fsky=0.6, showing the impact of noise.