Introducing systematics for CMB-S4

 (Victor Buza), with inputs from John Kovac and Colin Bischoff

Proposal for including systematics in forecasts and sims for the CDT/S4

Additive Systematics

In the first section of this posting I introduce two types of additive systematics: a per-band irreducible residual (i.e. an unknown residual part that is effectively correlated across all redundant sub-maps of each single-freq survey, but not correlated across frequencies) and a common-mode irreducible residual that is present at the same level for all auto and cross spectra.

There are two questions I'm trying to answer, corresponding to bracketing extremes in our ability to describe the form of the systematic contamination in our analysis approach:

Per-band irreducible residual construction (using \(N_l\)) as template

For the per-band irreducible residual, for each auto-spectrum we inject a signal with a template that follows the \(N_l\) of that auto-spectrum times a power law in \(l\), with an amplitude at the level of X% of the \(N_l\). This signal \(S(X, \alpha)\) then has the shape: \[S(\nu_1,\nu_2,l) = X_i N_{l,i} \delta(\nu_1,\nu_2) \Big(\frac{l}{l_{pivot}}\Big)^{\alpha_i}\] where \(\delta\) is the usual Kronecker delta, \(l_{pivot}=80\), and \(i\) is the auto-spectrum index. We are adding these extra \(\alpha_i\) parameters to describe our effective uncertainty of the slope of this systematic signal, in order to avoid artificially good constraints that use information at higher ell to offer constraints at lower ell. The introduction of this signal changes the expectation values \(\mu(\theta)\), where \(\theta\) are all the usual model parameters, by: \[\tilde{\mu}(\theta, X_i, \alpha_i) = \mu(\theta) + S(X_i, \alpha_i)\] In a Fisher formalism one cares only about the derivatives of the expectation values with respect to the parameters (not the expectation values themselves), as seen from its definition here: \[ F_{ij} = \frac{\partial\mu^T}{\partial\theta_i} \Sigma^{-1} \frac{\partial\mu}{\partial\theta_j} + \frac{1}{2} Tr( \Sigma^{-1} \frac{\partial\Sigma}{\partial\theta_i} \Sigma^{-1} \frac{\partial\Sigma}{\partial\theta_i}) \] Where for us \(\Sigma(\theta) = \Sigma\) (i.e. the covariance matrix is fixed for a particular fiducial model), sending the second term to zero, as verified in previous studies of \(\sigma\) stastics, derived as the standard deviation of the recovered Maximum Likelihood values from an N-dimensional Maximum Likelihood search on sims. With that in mind, and with the introduction of these extra two parameters per frequency band (the full set of parameters is \(\rho = \{\theta, X_i, \alpha_i\}\)) we have: \[ F_{ij} = \frac{\partial\tilde{\mu}^T}{\partial\rho_i} \Sigma^{-1} \frac{\partial\tilde{\mu}}{\partial\rho_j} \] \[\frac{\partial\tilde{\mu}}{\partial\theta}=\frac{\partial\mu}{\partial\theta}\ \qquad \frac{\partial\tilde{\mu}}{\partial X_i}=\frac{\partial S(X_i, \alpha_i)}{\partial X_i}=N_{l,i} \delta(\nu_1,\nu_2)\Big(\frac{l}{l_{pivot}}\Big)^{\alpha_i} \qquad \frac{\partial\tilde{\mu}}{\partial \alpha_i}=\frac{\partial S(X_i, \alpha_i)}{\partial \alpha_i}=X_i N_{l,i} \delta(\nu_1,\nu_2)\Big(\frac{l}{l_{pivot}}\Big)^{\alpha_i} log\Big(\frac{l}{l_{pivot}}\Big)\] One can see that in this particular formulation, and with a fixed covariance matrix, the only dependence on \(X_i\) is indirectly through the expectation value derivatives with respect to \(\alpha\). Once the Fisher Matrix is constructed and inverted, we can assess the level of constraint degradation due to the introduction of this new signal described by 16 parameters (2 parameters x 8 frequencies).

Taking the baseline S4 scenario described in this posting -- 8 frequency bands, \(f_{sky}=0.03\), optimized over 1M det-yrs (resulting in the following \(N_l\)'s), 8 r+Foreground free parameters (turning dust/sync decorrelation off for now), and calculating a Fisher Matrix with and without the per-band irreducible residuals, we quantify its effects on \(\sigma_r\). Some notation: \(\sigma_{r,SystOff}\) -- no systematic residual, \(\sigma_{r,SystOn}\) -- with systematic residual.

Table 1:
Using the template described above for systematics modeling, we calculate the effects of a per-band irreducible systematic contamination on \(\sigma_r\) and constraints on this systematic.
\(\sigma_r, (\times 10^{-3})\)\(\Delta\)\(\epsilon\)
No Systematics (Fisher Ellipses)0.727----
{\(X_i=0.05\), free}, {\(\alpha_i=0\), fixed}0.7341.0%13.9%
{\(X_i=0.05\), free}, {\(\alpha_i=0\), free}0.7716.1%35.3%
{\(X_i=0.05\), free, \(P(X)=0.075\)}, {\(\alpha_i=0\), free}0.7422.1%20.4%
\[Where \qquad \epsilon = \sqrt{\frac{\sigma_{r,SystOn}^2-\sigma_{r,SystOff}^2}{\sigma_{r,SystOff}^2}}\times 100\% \qquad \Delta=\frac{\sigma_{r,SystOn}-\sigma_{r,SystOff}}{\sigma_{r,SystOff}}\times 100\% \]

Per-band irreducible residual (using \(r\) as template)

In this section I repeat the same procedure, except now I look at using a tensor signal with \(r=1\) as the systematic template. Our signal then has the following shape: \[S(\nu_1,\nu_2,l) = Y_i D^{tensor, r=1}_{l,i} \times\delta(\nu_1,\nu_2)\] Where we have one \(Y_i\) for each frequency. This means we now have 8 parameters instead of 16. It is clear that this case represents an evil scenario in which we have a systematic signal that looks exactly like \(r\) and our only saving grace is that we don't see it in the cross spectra. In this case, we would expect to see no constraining power from the auto spectra, and indeed we do see that!

Table 2:
Effects of a per-band irreducible systematic contamination on \(\sigma_r\), and constraints on this systematic.
\(\sigma_r, (\times 10^{-3})\)\(\Delta\)\(\epsilon\)
No Systematics (Fisher Ellipses)0.727----
{\(Y_i=0.05\), free}0.7989.8%45.3%
{\(Y_i=0.05\), free, \(P(Y)=7.5\times 10^{-4}\)}0.7432.2%21.1%
No Systematics (special case: No Auto-spectra)0.80010.0%45.9%

Common-mode irreducible residual construction

In comparison to the per-band residual, which only introduces a residual for the auto spectra, a common-mode residual is present in all auto and cross spectra, at the same amplitude, making it harder to separate from a signal like the CMB. This means only one parameter, and no Kronecker delta for the residual signal: \[S(\nu_1,\nu_2,l) = X N^{raw}_l \Big(\frac{l}{l_{pivot}}\Big)^\alpha\] where, for now, for a common template we use the raw noise of the experiment (combined across all bands, as plotted here): \[N^{raw}_l=\frac{1}{\sqrt{\sum^{n_{expt}}_i{1/N_{l,i}^2}}}\] The rest follows the same path. Calculating a Fisher Matrix with and without this common-mode residual yields:

Table 3:
Effects of a common-mode irreducible systematic contamination on \(\sigma_r\), and constraints on this systematic.
\(\sigma_r, (\times 10^{-3})\)\(\Delta\)\(\epsilon\)\(\sigma_X\)\(\sigma_\alpha\)
No Systematics (Fisher Ellipses)0.727--------
{\(X=0.05\), free}, {\(\alpha=0\), fixed}0.7695.8%34.5%0.0423--
{\(X=0.05\), free}, {\(\alpha=0\), free}1.582117.6%193.3.6%0.27604.346
{\(X=0.05\), free, \(P(X)=0.0290\)}, {\(\alpha=0\), free}0.7422.1%20.4%0.02900.804

Common-mode irreducible residual (using \(r\) as template)

Similarly to the per-band residual, we also look at a common-mode residual that follows a tensor signal template. \[S(\nu_1,\nu_2,l) = Y D^{tensor, r=1}_{l,i}\] Where now we only have one parameter Y that tells us the amplitude of the injected signal in all auto and cross spectra. Such a signal is completely indistinguishable from \(r\) (i.e. fully degenerate with \(r\)), and therefore the degradation on \(\sigma_r\) is fully determined by the prior on \(Y\), as seen below.

Table 4:
Effects of a common-mode irreducible systematic contamination on \(\sigma_r\), and constraints on this systematic.
\(\sigma_r, (\times 10^{-3})\)\(\Delta\)\(\epsilon\)\(\sigma_X\)
No Systematics (Fisher Ellipses)0.727------
{\(Y=0.05\), free, \(P(Y)=1.5\times 10^{-4}\)}0.7422.1%20.4%0.00015

Biases on \(r\) due to an analysis blind to the form of residual systematics

With the above four cases in mind, we can set off to calculate the level of biases on \(r\) one would get if one were to inject these systematic signals into the data, and be completely agnostic to them in the model. To do this, I run a global Maximum Likelihood search over the default 8 parameters (r+foregrounds), before, and after the injection of systematics. The difference in the recovered \(r\) values between these cases tells us about the level of bias (i.e. what fraction of \(\sigma_r\)). I do this iteratively in order to figure what what signals will yield a bias on \(r\) that is close to 20% of \(\sigma_r\).

Figure 1:

Recovered ML values before and after injection of per-band and common-mode systematics with the two templates described above. The \(X\) and \(Y\) values have been iteratively found to give us a bias equivalent to 20% of \(\sigma_r\).

It is easy to note that the \(X\) and \(Y\) values here are lower than the values from obtained by including these parameters in the Fisher Model, and marginalizing over them. This is not surprising. The level of systematics we can tolerate will be lower when we have no information about the systematics in our maps than when we can model and marginalize over them.

To derive measurement requirements from this analysis, we can look at the following plot, and read-off the power at \(l=80\) from the common-mode and per-band residuals. On this plot are the four level of systematics in the table above, that for each of the four cases yield a bias on \(r\) at the level of 20% of \(\sigma_r\), as well as the template shapes used for the systematics.

Figure 2:

CM -- stands for Common-Mode; PB -- stands for Per-Band.

"Takeaways" Bullets:

Bandpass Systematics

This work is in progress: Similarly to the additive systematics, we are proposing a simplified model of bandpass systematics, a first path through which is decribed by adding nuissance parameters that describe uncertainties in the band-centers, specified as \(Z\)% of the band-center. Here we are trying to answer similar questions as before: what levels of correlated common-mode band-center errors, and uncorrelated level of rms deviation, offer a bias that is equivalent to 20% of \(\sigma_r\). This will also yield a recipe for including band-center uncertainties into DC3.0