# S4-Lensing

Neelima and Blake writing

We were asked within SO to provide ballpark S4 measurement requirements for lensing.

**For neutrino mass:**

S4 science target:

sigma(\summnu) = 30 meV (lensing) with sigma(tau) = 0.01 and DESI

sigma(\summnu) = 15 meV (lensing) with sigma(tau) = 0.002 (CV-limit) and DESI

S4 measurement target analysis:

The plot below assumes 1uK-arcmin noise for 10% of the sky (or other noise levels as indicated), and scales the noise as the sky fraction changes to keep observation time fixed. The beam assumed is 2' at 150 GHz, which number we are converging on for lensing, primarily due to halo lensing considerations, however, for removing a large amount of lensing for r, we do not want much worse resolution at 150 GHz. We find that 10% of the sky at 1uk-arcmin achieves the neutrino mass targets. The second plot (with the same noise/beam assumptions as above) shows that going to slightly higher sky fractions gains more lensing S/N (which is useful for cross correlations), even though it doesn't help much with neutrino mass.

(plot by Byeonghee Yu using code from Mathew Madhavacheril)

(plot by Byeonghee Yu using code from Mathew Madhavacheril)

At this point, we cannot argue that Galactic synchrotron or dust will not impact lensing measurements. To foreground-clean, having a 40 GHz and 270 GHz channel are then necessary to prevent the noise from blowing up. We explored the configuration of frequencies of 40,90,150,220,270 GHz with angular resolutions of 6.6,2.8,1.8,1.2,1.0 arcminutes at noise levels of 5, 1, 1, 5, 9 uk-arcmin, respectively. This resulted in the plot below, which gives a bit higher effective noise than the 1uk-arcmin we were aiming for. We also explored lowering the noise in the 40 GHz channel, and adding a 30 GHz channel, which gets close to1uk-arcmin effective noise over some ell range.

(plot by Colin Hill)

**S4 measurement requirement for neutrino mass:**
Measure I, Q, and U at frequencies of 30,40,90,150,220,270 GHz with angular resolutions of 9, 6.6, 2.8, 1.8, 1.2, 1.0 arcminutes. Can observe either 10% of sky with map noise levels (statistical + systematic) of 5, 3, 1, 1, 5, 9 uK-arcmin (so 1uk-arcmin effective noise in foreground cleaned map), or 20% of sky with map noise levels (statistical + systematic) of 7, 4.2, 1.4, 1.4, 7, 12.7 uK-arcmin (so 1.4uk-arcmin effective noise in foreground cleaned map). Have polarization atmospheric ell_knee be below 3000.

**For r:**

S4 science target:

sigma(r) = 5e-4 for fiducial r=0

S4 measurement target analysis:

There are multiple paths to reach a given sigma(r) target. Either going deeper in a low-resolution survey and requiring less delensing, or the reverse. We see an example in the plot below which has AL as a free parameter. However, some of these configurations require more detectors than others.

(plot by Josquin Errard and David Alonso)

Analytical approach to optimization between high-res and low-res surveys:

While a detailed forecasting approach as is being pursued by this group is required, a simple analytical optimization can be very useful to give cross checks and additional insight.

To make progress on this, we rewrite the residual lensing B-mode power as a function of the noise on the high-resolution survey. To find such a function, we fit a polynomial to the delensing simulation results of Hirata/Seljak 2003, as can be seen in the following plot (red curve fit by black curve).

(plot by Blake Sherwin)

Plugging in, this expression below is a good fit to within a few percent for noise below 5uK-arcmin:

where, to simplify things, we define map noise in uk-arcmin, N.

This implies

Now that we have a relation between the tradeoffs between high and low-resolution noise, we can optimize. A simple starting point would be to assume that the noise on the high-resolution experiment is N/sqrt(x), and on the low-resolution experiment is N/sqrt(1-x), where x is the fraction of detectors at high-resolution. We can then optimize to find the ideal noise level relationship:

(plot by Blake Sherwin)

(This is justified with more detail in this short writeup. File:PreliminaryWriteup.pdf) So a good rule of thumb (from simple optimization) for noise levels of order 1-2 uK-arcmin is to have roughly equal effective, foreground-cleaned, noise levels in the low-res and high-res surveys for the same overlapping patch of sky.

We note that a simple extension is to investigate the errors as a function not just of the ratio of noises on high and low resolution surveys, but also on the sky fraction. This is very much work in progress. (Note that in the equation below, we are letting noise tradeoff with sky fraction.)

From the plot above (which has AL as a free parameter), extrapolating to 1.75uK-arcmin low-res noise requires about 80% of the lensing power to be removed, which will leave about 2.3 uk-arcmin lensing noise in the low-res map. We need about 2uk-arcmin noise in the high-res survey to achieve 2.3uk-arcmin residual lensing noise in the low-res map (see second plot below).

Some advantages of observing 10% of the sky as opposed to 5% are that the level of delensing required is more relaxed, one can have multiple patches with different foreground levels to cross check results, and for the same 10% of sky, one can reach the neutrino mass target in a cost effective way without loosing too many detectors to a wider survey. However, separate deep and wide surveys can also work if we are not limited too much by number of detectors.

**S4 measurement requirement for delensing for r:**
Measure I, Q, and U at frequencies of 30,40,90,150,220,270 GHz with angular resolutions of 9, 6.6, 2.8, 1.8, 1.2, 1.0 arcminutes. Observe 10% of sky with map noise levels (statistical + systematic) of 10, 6, 2, 2, 10, 18 uK-arcmin (so 2uk-arcmin effective noise in foreground cleaned map). Have polarization atmospheric ell_knee be below XXX.

Other work on delensing:

- Foreground estimates converging. Next steps are simple estimates of delensing biases from template cross-correlations.

- Simulations are being shared and pushed through maximum likelihood delensing pipe (at Sussex, one being developed at Berkeley...)

- Ideas for joint optimiziation of delensing and r measurement are being pursued (should coordinate with Victor++)