# S4 Inflation Chapter Plot Suggestions

## 2015-06-16  (Victor Buza) Updated 2016-06-22: Updated Section 3 to include No Delensing, No Foregrounds, and No Decorrelation cases.

This is a simple posting in which I put up for discussion some plots for Chapter 2 of the S4 Science Book. There are two new types of plots: $$r-n_s$$ and $$r-n_t$$, and two old types of plots: "$$\sigma_r$$ vs Effort" and "$$\sigma_r$$ vs $$f_{sky}$$," which have been updated to include decorrelation, and to have an extra panel depicting the "fraction rms of the lensing residual" and "the fraction of total effort going towards delensing."

## 1. $$r-ns$$

For this plot, I assume that the $$n_s$$ constraint comes from $$TT, TE, EE$$ and the $$r$$ constraint comes from $$BB$$. Under this assumption, I can take the $$\sigma_{n_s}$$ achieved by the large survey, and the $$\sigma_r$$ achieved by the small survey, and form a perfectly non-degenerate ellipse. For the large survey, I was guided by this posting, and picked $$n_s=0.9655$$ and $$\sigma_{n_s}=0.002$$. For the small survey, I was guided by the decorrelation section of this posting, and picked $$r=[0,0.01]$$ and the minima $$\sigma_r=[0.00075, 0.00159]$$ for $$f_{sky}=[0.01, 0.10]$$.

## 2. $$r-n_t$$

In this section, I extend our model to include $$n_t$$ as a parameter. I run CAMB with the same Cosmology as before (w/ $$n_t=0$$), but now pick the pivot scale -- $$k_t$$ to break the $$r-n_t$$ degeneracy, and calculate $$\frac{\partial C_l^{BB}}{\partial n_t}$$ for the extra dimension in the Fisher Matrix. The two cases I consider are $$r=0.01$$ and $$r=0.10$$, though the second one is mostly out of curiosity. Both cases are using the $$r=0.01$$ optimized distribution, and an $$f_{sky}=0.1$$, meaning that the $$r=0.10$$ is quite non-optimal.

Some literature I found making $$n_t$$ forecasts:

The literature finds smaller $$\sigma_{n_t}$$, but they also have different assumptions on what the instrument characteristics are, in some cases vastly so; if I change to a Knox formulation of the BPCM (as described in this posting) based on the instrument characteristics for some of the literature case, I can confirm that I recover similar constraints.