Statistics question in cosmology The following are the approximate parameters for the Benchmark model in cosmology:
$H_0 = 73.8 km s^{-1} Mpc^{-1})$, $\Omega_{m,0} = 0.266$, and $\Omega_{\Lambda,0} = 0.734$.
Given that we know $H_0$ very precisely to this value in a flat universe (meaning neglecting the negligible contribution from radiation, $\Omega_{m,0} + \Omega_{\Lambda,0} = 1$) with $\Omega_{\Lambda,0}$ being an unknown value very close to the given number, if we want to make a measurement of $\Omega_{\Lambda,0}$ with an accuracy of 5%, what is the maximum percent error we can tolerate on our measurement of luminosity distance at z = 0.5, 1.0, and 1.5?
Extra note: luminosity distance can be approximated as
$$d_L \approx \frac{c}{H_0} z (1 + \frac{1 - q_0}{2} z)$$
Where
$$q_0 = \frac{1}{2} \Omega_{m,0} - \Omega_{\Lambda,0}$$
Due to covid, I've been unable to take the statistical physics course that was part of my plan, so it has been since the mid 2000's since I've had a stats course. There are a bunch of follow-up questions, but I'm hoping once I grasp the approach to this one I will be able to do them myself.
 A: Please see the comment, however it might be best to make $\Omega_{m,0}$ the subject of $\Omega_{m,0} + \Omega_{\Lambda,0} = 1$ and replace it and $q_0$ in your luminosity distance formula, so it only depends on the cosmological constant density parameter.  Redshifts are usually known really accurately, so you could then vary  the cosmological constant density parameter by 5% in each case and see what percentage variation occurs for the luminosity distance, for different redshifts.
There is probably a more mathematical way with differentials to do it, but it finds the answer.
It must be said that most of the assumptions being made in cosmology at the moment are likely to be wrong, theories such as this, which claims that the Hubble parameter should be halved and a different scale factor- redshift relation used, might be just as viable.
https://www.researchgate.net/publication/342040580_A_New_Solution_of_the_Friedman_Equations
P.S the speed of light should be in km/s
A: Write your equation as $d_L(z) = A(z)+B(z) \Omega_{\Lambda,0}$
Then,
$ \Omega_{\Lambda,0} =  {d_L - A \over B}$
and an error $\epsilon $ in $d_L$ translates into an error $\delta$ in $\Omega$:
$ \Omega_{\Lambda,0}+\delta =  {d_L - A \over B}+{\epsilon \over B} $.
The relative error in $\Omega$ is ${\delta \over \Omega} = {\epsilon \over d_L -A} = {{\epsilon \over d_L} \over 1-A/d_L}$ compared to the relative error in the luminosity $\epsilon \over d_L$.
You are left with the task of calculating  $A,B$.
PS: A course in statistical physics does not deal with these issues. I believe it is discussed in lab courses. See experimental error.
