Suppose you have a universe where [itex] Ω_{Λ}=1[/itex] and the rest are zero. Then the Friedmann equations are:(adsbygoogle = window.adsbygoogle || []).push({});

[itex] (\frac{H}{H_{0}})^{2} + \frac{k}{(aH_{0})^2} = \Omega_{\Lambda}=1 [/itex]

since [itex] \Omega= \Omega_{\Lambda}=1 [/itex] we have a flat universe and so [itex]k=0[/itex]

This leaves us with:

[itex] (\frac{H}{H_{0}})^{2}=1 [/itex]

or

[itex]\frac{H}{H_{0}}=1 \rightarrow H=H_{0}[/itex]

If I try to write find the age of this universe today, it will give me:

[itex] \frac{1}{a} \frac{da}{dt} = H = H_{0}[/itex]

If I integrate from [itex]a=0[/itex] to [itex]a=1[/itex] and so time from 0 to T (today):

[itex] \int_{0}^{1} \frac{da}{a} = H_{0} T [/itex]

I am getting negative age T.... because the integral is logarithmically divergent at 0.

An additional problem also appears by solving for [itex]a[/itex] as a differential equation and get:

[itex] a(t) = a_{0} e^{H_{0} t} [/itex]

from which you see that to solve [itex]a(t)=0 [/itex] (so to find the time connecting [itex]a_{0}[/itex] to [itex]a=0[/itex]) you need to get [itex]H_{0} t= -∞[/itex]

Both these cases seem unphysical to me...? Any help for clarification?

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# FRW cosmology [itex]Ω_{Λ}[/itex] domination

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