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

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**March 05 2010**Please help me, what is the difference between the following [tex] d_L [/tex] relations

[tex]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)[/tex]

where

[tex]\Omega_k=1-\Omega_m-\Omega_\Lambda[/tex]

[tex]\mbox{Sinn}(x) = \mbox{Sin}(x) \mbox{ for } \Omega_m + \Omega_\Lambda > 1 [/tex]

[tex]\mbox{Sinn}(x) = \mbox{Sinh}(x) \mbox{ for } \Omega_m + \Omega_\Lambda < 0 [/tex]

and

[tex]\mbox{Sinn}(x) = x \mbox{ for } \Omega_m + \Omega_\Lambda = 1 [/tex]

(From Moncy & Narlikar, astro-ph/0111122, Drell et. al., astro-ph/9905027)

[tex]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)[/tex]

where

[tex]\Omega_k=\frac{k}{H_0 ^2}[/tex]

[tex]\mbox{Sinn}(x) = \mbox{Sin}(x) \mbox{ for } k > 0 [/tex]

[tex]\mbox{Sinn}(x) = \mbox{Sinh}(x) \mbox{ for } k< 0 [/tex]

and

[tex]\mbox{Sinn}(x) = x \mbox{ for } k = 0 [/tex]

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

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

Kindly treat me as a beginner and correct me.

**1. FRW model**[tex]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)[/tex]

where

[tex]\Omega_k=1-\Omega_m-\Omega_\Lambda[/tex]

[tex]\mbox{Sinn}(x) = \mbox{Sin}(x) \mbox{ for } \Omega_m + \Omega_\Lambda > 1 [/tex]

[tex]\mbox{Sinn}(x) = \mbox{Sinh}(x) \mbox{ for } \Omega_m + \Omega_\Lambda < 0 [/tex]

and

[tex]\mbox{Sinn}(x) = x \mbox{ for } \Omega_m + \Omega_\Lambda = 1 [/tex]

(From Moncy & Narlikar, astro-ph/0111122, Drell et. al., astro-ph/9905027)

**2. [tex]\Lambda[/tex]CDM model**[tex]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)[/tex]

where

[tex]\Omega_k=\frac{k}{H_0 ^2}[/tex]

[tex]\mbox{Sinn}(x) = \mbox{Sin}(x) \mbox{ for } k > 0 [/tex]

[tex]\mbox{Sinn}(x) = \mbox{Sinh}(x) \mbox{ for } k< 0 [/tex]

and

[tex]\mbox{Sinn}(x) = x \mbox{ for } k = 0 [/tex]

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

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

Kindly treat me as a beginner and correct me.