deletions | additions       diff --git a/results.tex b/results.tex  index 26b8e7f..660a553 100644  --- a/results.tex  +++ b/results.tex 
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To trace individual parameter uncertainties across the population we use the fractional uncertainties $\sigma_{\{\mathcal{M}_\mathrm{c},\,q\}}/\{\langle\mathcal{M}_\mathrm{c}\rangle,~\langle q\rangle\}$, where $\sigma_x$ and $\langle x\rangle$ are the standard deviation and mean of the distributions respectively. The mean (median) fractional uncertainties in chirp mass and mass ratio for the simulated population are $0.0675\%$ ($0.0640\%$) and $28.7\%$ ($28.4\%$).  The fractional uncertainties for both the chirp mass and the mass ratio both decrease as S/N increases as shown in Figure \ref{fig:Mc_q_std_snr}, which also shows results from the non-spinning analysis. As expected from Fisher matrix studies \citep[e.g.,]{FinnChernoff}, all except the $\sigma_q/\langle q\rangle$ from the spinning analysis appear to be inversely proportional to the S/N: the exception is better fit as $\propto \rho_\mathrm{net}^{-1/2}$. We do not suspect there is anything fundamental about the $\propto \rho_\mathrm{net}^{-1/2}$, rather it is a useful rule-of-thumb. The behaviour can still be understood from a Fisher matrix perspective, which predicts a Gaussian probability distribution (with width $\propto \rho_\mathrm{net}^{-1}$). Since the mass ratio is constrained to be $0 \leq q \leq 1$, if the width of a Gaussian is large, it is indistinguishable from a uniform distribution and the standard deviation tends to a constant $1/\sqrt{12} \simeq 0.289$. When the width of the Gaussian is small ($\lesssim 0.1$), the truncation of the distribution is negligible and the standard deviation behaves as expected, as is the case for the non-spinning results. The standard deviations obtained for the spinning runs lie in the intermediate regime, between being independent of S/N and scaling inversely with it; the mean (median) standard deviation $\sigma_q$ is $0.182$ ($0.183$).\footnote{The uncertainty for the symmetric mass ratio $eta $\eta  = m_1m_2/(m_1 + m_2)^2$, which is constrained to be $0 \leq \eta \leq 1/4$, does scale approximately as $\rho_\mathrm{net}^{-1}$. The mean (median) standard deviation $\sigma_\eta$ for the spinning runs is $2.00\times 10^{-2}$ ($1.95\times 10^{-2}$).} The mass--spin degeneracy broadens the posteriors for both the chirp mass and the mass ratio, a consequence of the latter is that the uncertainty does not decrease as rapidly with S/N (over the range considered here).   Projecting the tightly constrained chirp mass and poorly constrained mass ratio $90\%$ credible region from $\mathcal{M}_\mathrm{c}$--$q$ space into component-mass space makes it obvious how important mass-ratio uncertainties are for extracting astrophysical information. The credible regions in component-mass space are narrow bananas that lie along lines of constant chirp mass, bounded by the constraints on mass ratio (see Figure \ref{fig:comp_masses} for some examples posteriors).   
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