Following C08, we transform the covariance matrices to a more physical parameter space, parameterized by the variables \(t_c\), \(b^2\), \(\tau_0^2\), \(r\), and \(f_0\), given by the inverse mapping
\[r = \left( \frac{\delta}{f_0} \right)^{1/2} \label{eqn:r}\]
\[b^2 = 1 - \frac{r T}{\tau} \label{eqn:bsq}\]
\[\tau_0^2 = \frac{T \tau}{4 r} \label{eqn:tau0sq}\]
The covariance matrix of the physical parameters is then found by the transformation
\[\text{Cov}'(...) = J^T \text{Cov}(...) J\]
with \(J\) the Jacobian matrix
\[J = \frac{\partial (t_c, b^2, \tau_0^2, r, f_0)}{\partial (t_c, \tau, T, \delta, f_0)} = \left( \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 0 & \frac{T r}{\tau ^2} & \frac{T}{4 r} & 0 & 0 \\ 0 & -\frac{r}{\tau } & \frac{\tau }{4 r} & 0 & 0 \\ 0 & -\frac{T}{2 f_0 r \tau } & -\frac{T \tau }{8 f_0 r^3} & \frac{1}{2 f_0 r} & 0 \\ 0 & \frac{T r}{2 f_0 \tau } & \frac{T \tau }{8 f_0 r} & -\frac{r}{2 f_0} & 1 \\ \end{array} \right).\]
| Symbol | Expression |
| \(\Delta\) | \(T^2 \left(\delta ^2 \xi +24 f_0^2 (2 \alpha \nu -\beta \nu +4 \mu )+48 \delta f_0 \eta (\beta -\alpha )\right) / \left( \tau^2 f_0^2 \right)\) |
| \(\Theta\) | \(\left( 24 f_0^2 (6 \beta \kappa \tau +\nu T (\alpha -\beta ))-12 \delta f_0 (\theta \tau +2 \eta T (\alpha -2 \beta ))+\delta ^2 \xi T \right) / \left( \tau f_0^2 \right)\) |
| \(\Lambda\) | \(\left( 24 f_0^2 (\nu T (\alpha -\beta )-6 \beta \kappa \tau )+12 \delta f_0 (\theta \tau -2 \eta T (\alpha -2 \beta ))+\delta ^2 \xi T \right) / \left( \tau f_0^2 \right)\) |
| \(\Pi\) | \(\left( \delta \xi T-12 f_0 (\theta \tau +2 \eta T (\alpha -\beta )) \right) / \left( \tau f_0 \right)\) |
| \(\Sigma\) | \(\left( 12 f_0 (\theta \tau +2 \eta T (\beta -\alpha ))+\delta \xi T \right) / \left( \tau f_0 \right)\) |
| \(\Upsilon\) | \(\left( 288 f_0^2 \kappa \tau T (\beta -\alpha )-24 \delta f_0 \theta \tau T \right) / \left( \tau^2 f_0^2 \right)\) |
| \(\Phi\) | \(24 \upsilon\) |
| \(\Psi\) | \(\left( \delta \xi +24 \beta f_0 \eta \right) / f_0\) |
| \(\Omega\) | \(\left( \delta ^2 \xi -24 \beta f_0 (f_0 \nu -2 \delta \eta ) \right) / f_0^2\) |
\label{tab:cov3vars}
As before, we define several variables so that we can write the covariance matrix compactly. For \(\tau > t_{exp}\), they are given in Table \ref{tab:cov3vars}. With these definitions, the transformed covariance matrix in the \(\tau > t_{exp}\) case becomes
\[\text{Cov}(\{t_c,b^2,\tau_0^2,r,f_0\},\{t_c,b^2,\tau_0^2,r,f_0\};~\tau > t_{exp}) = \\ \frac{\sigma ^2}{\Gamma} \left( \begin{array}{ccccc} -\frac{3 \tau }{f_0^2 r^4 \phi } & 0 & 0 & 0 & 0 \\ 0 & \frac{\Delta +\Upsilon +\Phi }{4 f_0^2 r^2 \zeta \tau } & \frac{\tau (\Delta -\Phi )}{16 f_0^2 r^4 \zeta } & -\frac{\Theta }{4 f_0^2 r^2 \zeta \tau } & \frac{\Pi }{2 f_0 r \zeta \tau } \\ 0 & \frac{\tau (\Delta -\Phi )}{16 f_0^2 r^4 \zeta } & \frac{\tau ^3 (\Delta -\Upsilon +\Phi )}{64 f_0^2 r^6 \zeta } & -\frac{\tau \Lambda }{16 f_0^2 r^4 \zeta } & \frac{\tau \Sigma }{8 f_0 r^3 \zeta } \\ 0 & -\frac{\Theta }{4 f_0^2 r^2 \zeta \tau } & -\frac{\tau \Lambda }{16 f_0^2 r^4 \zeta } & \frac{\Omega }{4 f_0^2 r^2 \zeta \tau } & -\frac{\Psi }{2 f_0 r \zeta \tau } \\ 0 & \frac{\Pi }{2 f_0 r \zeta \tau } & \frac{\tau \Sigma }{8 f_0 r^3 \zeta } & -\frac{\Psi }{2 f_0 r \zeta \tau } & \frac{\xi }{\zeta \tau } \\ \end{array} \right). \label{eqn:cov3}\]