# Detection significance for r=0.003

*Colin Bischoff, 2018-11-12*

A science target for the small area survey is to detect r=0.003 at five sigma. While our Fisher forecast can quickly provide sigma(r) for specified experiment configuration, foregrounds, and cosmological parameters, detection significance is not accurately determined from a Gaussian approximation to the r likelihood. In particular, for small sky areas the likelihood is significantly skewed and cuts off sharply for r→0, which improves our confidence to rule out r=0.

To obtain some estimates of detection significance, I used CosmoMC to run a full BICEP/Keck-style likelihood for four experiment configurations. The A_{L} values correspond to either four or seven years of delensing effort in each case, with numbers drawn from John+Victor's spreadsheet.

- Chile sky mask (data challenge 04b), residual A
_{L}=0.270 - Chile sky mask (data challenge 04b), residual A
_{L}=0.337 - Pole sky mask (data challenge 04c), residual A
_{L}=0.081 - Pole sky mask (data challenge 04c), residual A
_{L}=0.106

For all experiment configurations, I assumed r=0.003 and foreground model 00 with A_{dust}=4.25 μK^{2} at 353 GHz (β_{dust}=1.6, T_{dust}=19.6 K, α_{dust}=-0.4) and A_{sync}=3.8 μK^{2} at 23 GHz (β_{sync}=-3.1, α_{sync}=-0.6). I constructed a bandpower covariance matrix from the set of power spectra described here. The covariance matrix construction rescales signal levels to match the cosmology+foregrounds model described here, and it observes a distinction between signal and noise degrees of freedom.

In place of simulated bandpowers, I used the model expectation values for each of my four configurations. This is meant to represent an average experimental outcome for each case, but it would be better to repeat this analysis on an ensemble of simulations.

With the bandpowers and their covariance in hand, I ran each configuration through CosmoMC to obtain the marginalized likelihood for r. The full likelihood includes the following eight dimensions:

- r
- A
_{dust} - β
_{dust}, with Gaussian prior [1.6, 0.11] - α
_{dust}, with flat prior on the interval [-1, 0] - A
_{sync} - β
_{sync}, with Gaussian prior [-3.1, 0.3] - α
_{sync}, with flat prior on the interval [-1, 0] - Δ
_{dust}(decorrelation parameter), assuming that decorrelation scales linearly with ℓ

Note that A_{L} was not a free parameter in the likelihood -- it was fixed to appropriate value for each experiment configuration.

Figure 1 shows the marginalized likelihoods for r from this analysis (peak normalized). The likelihood curves for Pole cut off more sharply for r→0 than the corresponding curves for Chile, because the sky area observed from Pole is much smaller. I think it is a coincidence that the high-side tails match so well for “Chile, A_{L}=0.270” and “Pole, A_{L}=0.106”. Summary statistics, including detection significance, are provided here. I haven't tried to figure out the cause of the low bias in the likelihood peak -- because I used model expectation values instead of bandpowers, I think that the global maximum likelihood point (before marginalization) must correspond to the model input parameters (unless there are small disagreements in the shape of the lensing and/or tensor theory spectra).

Site | A_{L} |
peak(r) | sigma(r) | L(0) / L_{peak} |
detection significance |
---|---|---|---|---|---|

Chile | 0.270 | 0.00249 | 0.00102 | 5.23e-2 | 2.4 σ |

Chile | 0.337 | 0.00257 | 0.00111 | 7.01e-2 | 2.3 σ |

Pole | 0.081 | 0.00261 | 0.00080 | 4.30e-4 | 3.9 σ |

Pole | 0.106 | 0.00273 | 0.00089 | 2.81e-3 | 3.4 σ |

Below I have included an alternate version of the likelihood figure, where each curve has been centered at zero and rescaled by sigma(r). These curves might be useful as templates to estimate detection significance for other values of r, if they are rescaled according to a Fisher estimate of sigma(r).