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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.

ObiDonKenobi
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  • There is a Physics Stack Exchange that may be more suitable for your question. Also the initial assumption that we know $H_0$ very precisely is very questionable, there is currently enormous controversy about it's value (see Hubble tension) – John Hunter Mar 07 '21 at 18:40
  • I unfortunately get in this loop where I ask a question at Physics stack exchange and they say "go to the math stack exchange," so now when I post here, it's suggested I go to the Physics stack exchange. The two are so inter-related that I don't know what to do about that. – ObiDonKenobi Mar 07 '21 at 19:04
  • Also good point about the Hubble parameter for today; but I interpreted the instructor's question to mean "let's assume that we do know it accurately" when saying "given that..." – ObiDonKenobi Mar 07 '21 at 19:05

2 Answers2

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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

John Hunter
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    Thank you, I will try this, then. I do have some trouble where if I have a question, I'll ask Physics Stack Exchange and they'll say "you should post this in math stack exchange." Then when I post here, I'm told "you should post this in physics stack exchange." Just goes to show how interrelated they are, I guess.

    As for the Hubble parameter not being known precisely, the professor is at least saying this. I'm interpreting the problem saying "given that" to just mean "for the purposes of this problem, assume that we know it precisely." Thanks for the interesting link, also.

     – ObiDonKenobi Mar 07 '21 at 19:10
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    @ObiDonKenobi I've left a comment here but I'll say that I appreciate well the situation you describe. I've asked a lot of questions in several sites and the strategy I use now is to adjust the "flavor" of the question post to match that of other questions in the target site. In Math SE for example I'd begin the question with "math words" so it feels more mathematical in the beginning, then move the physics stuff to the bottom and label it "background information". – uhoh Mar 07 '21 at 23:41
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    @ObiDonKenobi Physics SE is a royal pain sometimes, the site is too big and everybody has an opinion... But if you are lucky you can get excellent answers there sometimes. You can sometimes head off the "better asked on" comments by ensuring your tags already include mathematics and mathematical-physics tags AND you acknowledge that your question is not "on simplification of a mathematical expression"... – uhoh Mar 07 '21 at 23:42
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    @ObiDonKenobi because those tag definitions say explicitly "please ask it at math.stackexchange.com". Every SE site is like a different country, it takes a while to get used to each site's traditions, peculiar personalities and culture. – uhoh Mar 07 '21 at 23:42
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    @uhoh Noted, thanks for the tip. I'll try this if I have more questions down the line, which I'm sure I will with this cosmology course, lol. Though, I'm proud to have done a lot of it on my own. There are occasional things that just trip me and my study group up. I'm a night worker so I can't make the professor's office hours for questions :( – ObiDonKenobi Mar 08 '21 at 21:55
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    @ObiDonKenobi for me SE is a really useful way to interact intellectually/scientifically with others asynchronously (I live on the opposite side of Earth from where I originated, so I'm "shifted" as well) Sometimes the act of writing up the question carefully and thoughtfully produces an Aha! before it even gets posted, but I always continue to post them because answers can sometimes be surprising. Good luck with the cosmology course + night shift working! – uhoh Mar 08 '21 at 22:02
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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.

user619894
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  • Thank you for the excellent response. I took the early physics courses/lab courses in the mid/early 2000's as well, unfortunately I don't still have my notes from then. This is one of those unfortunate problems of being a nontraditional student trying to come back after a long break. – ObiDonKenobi Mar 08 '21 at 21:57