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Specifically, I need to find the cosmological parameters $Ω_i$ from a certain data set (with error bars) of redshift $z$ and luminosity distance $d_L$ using a Fisher Information matrix. I know that my data goes accordingly to $$ d_L(z)= \begin{cases} \frac{d_H}{\sqrt{|\Omega_k|}}\sinh(\sqrt{\Omega_k}d_c(z)/d_H) & \quad \text{if } \Omega_k>0\\ d_c(z) & \quad \text{if } \Omega_k=0\\ \frac{d_H}{\sqrt{|\Omega_k|}}\sin(\sqrt{-\Omega_k}d_c(z)/d_H) & \quad \text{if } \Omega_k<0\\ \end{cases} $$

and that $$ d_c(z)=d_H\int\limits^z_0\frac{dz'}{H(z')}\ \ ;\ \ H(z)=\left[\Omega_m(1+z)^3+\Omega_k(1+z)^2+\Omega_{\Lambda}\right]^{1/2} $$

where $d_H$ is a constant. I'm a little lost on how to determine the $\Omega_i$ from there, since I have no prior information nor I know the PDF. I'm suspecting that it should be $d_L$ normalized, is this correct?

Thanks!

PS: I wasn't sure if this should be posted here or in a statistics forum, please let me know!

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  • $\begingroup$ This is really a question about line-fitting. All of the functions and parameters are fairly smooth and well behaved, and the parameters you are looking for are (likely) bounded. This is a perfect situation for an MCMC type approach, but even doing a grid search shouldn't be too bad for so few parameters. Because this question involves so many different topics, it's difficult to answer on SE; perhaps you can find a collaborator who uses the same language (e.g. python, C, etc) and has experience with line-fitting. $\endgroup$ – DilithiumMatrix Oct 2 '16 at 16:57
  • $\begingroup$ I'm aware that it's a better job for MCMC, but my advisor required me to use Fisher information. Also, there are two more parameters a didn't include for simplicity, the equation of state $w$ for the Dark Energy and the $\Omega_r$, the parameter for radiation, but it would be fairly easy to generalize for these cases. $\endgroup$ – Renan Nobuyuki Hirayama Oct 2 '16 at 18:13
  • $\begingroup$ This has been cross-posted from Astronomy. Renan, please don't do this; post a question on one site. If you then feel it would be better elsewhere, flag the post and ask a moderator to migrate it elsewhere. Cross-posting splits answers and potential information up, which is inefficient. $\endgroup$ – HDE 226868 Oct 2 '16 at 18:25
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    $\begingroup$ Sorry about that, I'm new to SE. I'll delet the from Astronomy then, thanks. $\endgroup$ – Renan Nobuyuki Hirayama Oct 2 '16 at 18:43

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