## What is the difference between FRW and LCDM models

Noble P Abraham
Posts: 9
Joined: July 31 2008
Affiliation: School of Pure and Applied Physics, Mahatma Gandhi University, Kottayam

### What is the difference between FRW and LCDM models

Please help me, what is the difference between the following $d_L$ relations

1. FRW model
$d_L=5 \mbox{log} \left( (1+z) \frac{c}{H_0} |\Omega_k|^{-1/2}\ \mbox{Sinn} \left[ \ |\Omega_k|^{1/2} \int _{0}^{z} [(1+z^{\prime })^{2}(1+\Omega_{m}z^{\prime})-z^{\prime}(2+z^{\prime} )(\Omega _{\Lambda})]^{-1/2}\; dz^{\prime} \right] \right)$
where
$\Omega_k=1-\Omega_m-\Omega_\Lambda$
$\mbox{Sinn}(x) = \mbox{Sin}(x) \mbox{ for } \Omega_m + \Omega_\Lambda > 1$
$\mbox{Sinn}(x) = \mbox{Sinh}(x) \mbox{ for } \Omega_m + \Omega_\Lambda < 0$
and
$\mbox{Sinn}(x) = x \mbox{ for } \Omega_m + \Omega_\Lambda = 1$
(From Moncy & Narlikar, astro-ph/0111122, Drell et. al., astro-ph/9905027)

2. $\Lambda$CDM model
$d_L=5 \mbox{log} \left( (1+z) \frac{c}{H_0} |\Omega_k|^{-1/2}\ \mbox{Sinn} \left[ \ |\Omega_k|^{1/2} \int _{0}^{z} [\Omega_m (1+z^{\prime }) ^3+\Omega_k (1+z^{\prime })^2+\Omega_\Lambda]^{-1/2}\; dz^{\prime} \right] \right)$

where
$\Omega_k=\frac{k}{H_0 ^2}$
$\mbox{Sinn}(x) = \mbox{Sin}(x) \mbox{ for } k > 0$
$\mbox{Sinn}(x) = \mbox{Sinh}(x) \mbox{ for } k< 0$
and
$\mbox{Sinn}(x) = x \mbox{ for } k = 0$

(From Szydlowski and Godlowski, astro-ph/0509415)

Can I put $\Omega_k=1-\Omega_m-\Omega_\Lambda$ in the second case ($\Lambda$CDM) as well?

Kindly treat me as a beginner and correct me.

Ben Gold
Posts: 81
Joined: September 25 2004
Affiliation: University of Minnesota
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### What is the difference between FRW and LCDM models

Can I put &#937;k = 1 &#8722; &#937;m &#8722; &#937;&#923; in the second case (&#923;CDM) as well?
Yes, if you do that and perform some algebra you should find that they're exactly the same equation.