Expanding Present Time Universe at Omega 0 = 1: Help Needed

In summary, the present time of a matter-dominated universe with \Omega_{0}>1 can be calculated using the formula t_{0}=H_{0}^{-1}\frac{\Omega_{0}} {2(\Omega_{0}-1)^{3/2}}[cos^{-1}(2\Omega_{0}^{-1}-1)-\frac{2}{\Omega_{0}}(\Omega_{0}-1)^{1/2}], which can be simplified by making the substitution x = \Omega_0 - 1. However, expanding the formula at \Omega_{0}=1 is difficult and requires using l'Hôpital's Rule. Another method suggested is to make use of trigonometric identities,
  • #1
knightq
23
0
the present time of matter dominated universe for [tex]\Omega_{0}>1[/tex] is :
[tex]t_{0}=H_{0}^{-1}\frac{\Omega_{0}} {2(\Omega_{0}-1)^{3/2}}[cos^{-1}(2\Omega_{0}^{-1}-1)-\frac{2}{\Omega_{0}}(\Omega_{0}-1)^{1/2}][/tex]
how to expand this at[tex]\Omega_{o}=1[/tex]??help me
 
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  • #2
This seems like a much more complicated formula than it should be for a matter-only universe. How did you come by it?
 
  • #3
you can find this fomula in many books,say Kolb and Turner P53
 
  • #4
knightq said:
you can find this fomula in many books,say Kolb and Turner P53
Hmm, okay. I'll trust that you have it correct, then.

A first trick is just to make a simple substitution:

[tex]x = \Omega_0 - 1[/tex]

This doesn't change the problem, just makes it easier to work with.

Now, the derivatives are obviously going to get a little bit messy, but whenever you find yourself running into the problem where you have a fraction with both numerator and denominator equal to zero, you merely make use of l'Hôpital's Rule to find the result.
 
  • #5
yes,let x=[tex]\((Omega_{0}-1)^{1/2}[/tex]will be more simple.the difficulty is we can't expand cos-1(x) at x=1,the caculation through l'hospital is too difficult ,we may differentiate 3 times
 
  • #6
Hint:

Make:

[tex]
2 \, \Omega^{-1}_{0} - 1 = \cos{2 \, p} \Leftrightarrow \Omega_{0} = \frac{2}{1 + \cos{2 \, p}} = \frac{1}{\cos^{2}{p}}
[/tex]

and

[tex]
\Omega_{0} - 1 = \frac{1}{\cos^{2}{p}} - 1 = \tan^{2}{p}
[/tex]

The limit [itex]\Omega_{0} \rightarrow 1 + 0[/itex] is equivalent to the limit [itex]p \rightarrow 0[/itex] (from any direction).
 
  • #7
Be careful: arccos(x) is not analytic around x=1. To first order it behaves like √(1-x), so that it is real for x < 1 and imaginary for x > 1.
 
  • #8
I can get the right answer through Dickfore's method ,thank you!
 
  • #9
cool. post it here for future generations to learn from your example.
 

Related to Expanding Present Time Universe at Omega 0 = 1: Help Needed

1. What is the concept of "Expanding Present Time Universe at Omega 0 = 1"?

"Expanding Present Time Universe at Omega 0 = 1" is a theoretical concept in cosmology that suggests the rate of expansion of the universe is directly related to the amount of matter and energy present in the universe. Omega 0 represents the critical density of the universe, and when it is equal to 1, it means that the universe will continue to expand forever.

2. What is the significance of this concept in understanding the universe?

The concept of "Expanding Present Time Universe at Omega 0 = 1" is significant because it helps us understand the fate of the universe. If Omega 0 is less than 1, it suggests that the expansion of the universe will eventually slow down and stop, leading to a "Big Crunch" where the universe collapses back in on itself. If Omega 0 is greater than 1, it indicates that the universe will continue to expand forever.

3. How do scientists measure the value of Omega 0?

Scientists measure the value of Omega 0 by studying the distribution of matter and energy in the universe. This can be done through observations of the cosmic microwave background radiation, galaxy surveys, and other astronomical measurements.

4. Is the value of Omega 0 constant or does it change over time?

The value of Omega 0 is believed to be constant, meaning that it does not change over time. However, it is important to note that this value is still a subject of ongoing research and may be refined as we continue to learn more about the universe.

5. Can this concept be tested or proven?

The concept of "Expanding Present Time Universe at Omega 0 = 1" is a theoretical framework that is supported by observational evidence. However, it is still a subject of ongoing research and cannot be definitively proven at this time. As our understanding of the universe continues to advance, new evidence may emerge that either supports or challenges this concept.

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