I have 400 supernovas type Ia's distance modulus and its error and redshift. how can I obtain an elliptic curve like the one in image for my datas? where did that elliptic came from?
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$\begingroup$ try to look confidence intervals as areas.. $\endgroup$– seVenVo1dCommented Dec 31, 2020 at 11:27
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$\begingroup$ I,m not familiar with these experimental concepts,can you explain it more? $\endgroup$– Ali RayatCommented Dec 31, 2020 at 21:06
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$\begingroup$ I might help a bit but you need to give me more equations about the data you are using and what you are deriving...like from the Distance modulus and redshift you can find the $q$ and $j$ right ? and that $q$ and $j$ depends on $\Omega_m$ and $\Omega_{\Lambda}$. $\endgroup$– seVenVo1dCommented Dec 31, 2020 at 22:43
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$\begingroup$ However you ll find $q$ and $j$ with some errors which means you ll also find $\Omega_m$ and $\Omega_{\Lambda}$ with some error. When you plot 2 axis with both error bars you get an error as area. I am not sure this is the correct way but that is what I would try at least. Or, as I said earlier, if you can provide more equations that would be more helpful $\endgroup$– seVenVo1dCommented Dec 31, 2020 at 22:44
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This type of plot is known as a contour plot. It represents the 95% confidence interval of the joint posterior distribution of the two parameters $\Omega_m$ and $\Omega_{\Lambda}$ after marginalisation over other parameters present in the model (if using supernova data, these will likely be parameters relating to the standardisation of the light curves, such as the colour and stretch).
The posterior distribution (the likelihood multiplied by the prior, see Bayes' theorem) is typically obtained using Markov chain Monte Carlo sampling, as for high dimensions (i.e. a large number of parameters) it is extremely computationally expensive to compute directly.
For this method, you need to write a likelihood for your data (essentially calculate the $\chi^2$ statistic for your cosmological model of choice given the data) and consider what your prior is (for example, you can impose the physical conditions that $0 \leq \Omega_m \leq 1$ and $0 \leq \Omega_{\Lambda} \leq 1$).
There are a number of tools available to perform MCMC sampling for cosmological parameters, the most popular being emcee (written in Python) and CosmoMC (written in Fortran and designed for use with the Boltzmann code CAMB).
