# Update on Neff Forecasts

Dan writing (input from Erminia, Joel and Alex)

I will present updates on the forecasting for N_{eff} and how it will impact our ability to reach the target of σ(N_{eff}) = 0.027.

Unless otherwise stated, our forecasts use lensed spectra on 40 percent of the sky with l_{min}=30 and l_{max}=5000, with 2 arcmin beams. Y_{p} is varied with N_{eff} to be consistent with BBN for the same value of N_{eff}. We have also included the lensing power spectrum using iterative delensing. Planck has been included on an additional 20 percent of the sky and at low l in TT +tau prior.

## Temperature versus Polarization

As shown in the figure below, TE drives the constraints for N_{eff}. Reaching the target is particularly sensitive to TE with l > 2000.

A consequence of this statement is that our N_{eff} forecasts are not particularly sensitive to component separation. Specifically, we can use Planck TT without losing much/any constraining power. Foregrounds in EE are sufficiently low that they do not impact constraints. Detailed studies of component separation will be presented in a separate post, but this serves to motivate why we have ignored it here. Measurement requirements therefore depend more on the resolution and overall sensitivity rather than sensitivities in individual frequencies.

## Point Sources and Atmosphere

Point sources act much like an additional source of noise (modulo the correction from the beam). Point sources in TT therefore affect the signal to noise of both TT and TE at high-l. We will estimate the contribute to TT/EE from unresolved point source as

D_{TT,ps}(l=3000) = 6 ( μK)²

D_{EE,ps}(l=3000) = 3× 10^{-3} × 6 ( μK)²

where we used the measured few percent polarization fraction of point sources in Planck/SPT to estimate D_{EE,ps}(l=3000). In practice, the polarization fraction would have to be order 1 for the EE point sources to have any effect on our forecasts.

We will determine the level of atmospheric noise from the model presented by Matthew Hasselfield at the SLAC meeting:

N_{l}^{TT}= N_{0}^{TT}(1+ (l /3400)^{-4.7})

N_{l}^{EE}= N_{0}^{EE}(1+ (l /340)^{-4.7})

where the factor of 10 reduction in l_{knee}^{EE} was estimated from the polarization fraction.

The solid/dashed lines in the left figure represent 2'/1' beams respectively (we use 2' in the rest of the figures). The impact of the point sources is more important at low noise, as the point sources act like an irreducible noise source in TT. At higher noise, the atmosphere is more important as it scales with the white noise amplitude. We notice that these effects are generally small. We have used lensed spectra because it was shown in Green, Meyers and van Engelen (2016) that the forecasts with delensed spectra are well approximated by the lensed spectra when Y_{p} does not vary independently. Further improvements in delensing, particularly at high-l are possible but can at most reach the Green curve (in the absence of point sources or atmospheric noise).

We can also consider the impact of the atmosphere in polarization by changing the model to

N_{l}^{EE}= N_{0}^{EE}(1+ (l /l_{knee})^{α=-4.7, -4})

We notice that the atmosphere has a relatively small impact on N_{eff}, presumably because the information is coming from smaller angular scales. However, this is not a universal property of the cosmological parameters, as we can see from n_{s}. The solid/dashed line correspond to α=-4.7/-4.0 in EE holding the atmospheric noise in TT fixed in all cases.

**Summary:** Reasonable expectations for point sources and atmosphere have a relatively small impact on our ability to reach the N_{eff} target. 10 percent changes do occur, but this likely lies within the accuracy of the forecasts themselves.

## Beam / Pointing Calibration

Our ability to calibrate the power spectra at high-l is important for N_{eff}. A bias in the damping tail of the spectra would look like a bias in N_{eff}. Furthermore, marginalizing over these uncertainties reduces the sensitivity to N_{eff}. We will give a simple model of the beam uncertainty to as

B_{l} = exp[θ_{1 arcmin}² l(l +1) /(2 log 8) (b_1 + b_2 (l/3000) + b_3 (l /3000)^2 ) ]

where the b_{i} are in units of arcmin² and b_{1} is the uncertainty in the width of a Gaussian beam. We have added the additional terms to model more complicated beam shapes. This parametrization can be expanded to include many other terms, but with little change. At the level of Fisher matrices, this parameterization is equivalent to many others. We have focused on these terms as they are most degenerate with N_{eff}. We have chosen this particular model as these terms of the most degenerate with the measurement of N_{eff} from the damping tail.

With no external prior, we see that there our constraints weaken significantly. We recover the case without the beam uncertainties when we take the 1σ prior to be less that 0.003 arcmin². The beam uncertainty for these various priors in shown below. Since N_{eff} is driven by l > 2000, we should interpret this as the statement that we need 0.1 percent calibration. This is not a surprising requirement, as cosmic variance is order 0.1 percent on those scales. Therefore, we are requiring that the error in the power spectra from the beam uncertainties are below cosmic variance.

This requirement is compared to the performance of ACT as reported in Hasselfield et al. (2013). It is unclear is out 0.1 percent requires significant improvement on the ACT performance. ACT is within a factor of a few of this requirement on the reverent scales, but also does not show the same l-dependence as our model. The l-dependence of our model is of particular concern of N_{eff} because it is generate with a change to the damping tail which scales as exp[-l²/l²_{d}]. A more l-indendent beam uncertainty might weaken constraints on other parameters, like A_{s}, but does not impact N_{eff}.

**Summary:** Calibrating the beam to order 10^{-3} for the modes 2000 < l < 4000 is required to reach our target for N_{eff}. Specifically, l-dependent uncertainties in the beam on these scales are degenerate with physics in the damping tail and can reduce the sensitivity to N_{eff} and/or bias the measurement. Larger beam uncertainties are tolerable if the l-dependence is significantly different from a change to damping tail.