Universe density in Hyperbolic Universe

• Quarlep
In summary, the equation for ##a(t)## is not solvable analytically, you can obtain this form by substituting ##\frac{\dot a}{a}## for ##H##, you get something of the form ## \int F(a)da=t-t_0 ##Then What should I do ? Is there's any way to escape this situation ?You can solve the equation for ##a(t)## numerically. However, it is not a simple task, and requires some knowledge of calculus.
Quarlep
I want to know Universe density according to this equation( ##k=-1##) ?

##H^2(t)-8πρG/3=-k/a^2(t)##
##ρ_U=ρ_m+p_r##
##ρ_U##=Universe density
##ρ_m##=Matter density

With three terms (##1/a^2, 1/a^3, 1/a^4##) the differential equation for ##a(t)## is not solvable analytically, you only have an implicit form in terms of an integral which cannot be expressed in terms of usual functions - you can obtain this form by substituting ##\frac{\dot a}{a}## for ##H##, you get something of the form ## \int F(a)da=t-t_0 ##

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Quarlep
Then What should I do ? Is there's any way to escape this situation ?

In ##k=-1## → ##Ω_0<1## Is this can help us ?

It depends what you're looking for. If you're brave enough I think you can express the integral in terms of standard elliptic integrals, which are available in most mathematical packages. But the easy route is just to integrate numerically, this works fine.

## \Omega_0<1 ## doesn't help - but the issue is not a hard one, the only thing is, the solution does not have a nice form. It is perfectly computable numerically though.

Quarlep
wabbit said:
If you're brave enough you can express tge integral in terms of standard elliptic integrals, wgich are available in most mathematical packaged. But the easy route is just to integrate numerically, tgis wirks fine.

I am in high school so I did not understand a single word what you're saying

wabbit said:
##Ω_0<1## doesn't help - but the issue is not a hard one, the only thing is, the solution does not have a nice form. It is perfectly computable numerically though.
I can't do that myself Should I open a new thread or somebody can do that ?

Sorry about that. Very impressive, when I was in high school in didn't know anything at all about the Friedman equation or about General Relativity !

To find ## a(t) ## (and therefore ## \rho(t) ## ), you need to program two things :
1. The numerical calculation of the integral ##G(a)=\int F(a)da##
2. The numerical solution of the equation ## G(a)=t ##

This is a rather tough with high school tools though, you definitely need calculus. And the answer is a computer program, not a formula.

What are you trying to achieve exactly, is your question about the qualitative behaviour of the solution ? There are some things you can tell without going through a complete solution.

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Quarlep
Quarlep said:
I want to know Universe density according to this equation( ##k=-1##) ?

##H^2(t)-8πρG/3=-k/a^2(t)##
##ρ_U=ρ_m+p_r##
##ρ_U##=Universe density
##ρ_m##=Matter density
Too many parameters, too few equations. Also, it depends a bit upon your conventions. There are two commonly-used conventions:
1. ##k = {-1, 0, 1}##. With this convention, the scale parameter ##a## becomes the radius of curvature.
2. ##a(now) = 1##. This is usually the easiest. With this convention, ##k## can take on any number, and represents the amount of spatial curvature today (when ##a = 1##).

One way to make things easier to deal with is to use the concept of density fractions. With density fractions, I can rewrite:

$$H^2(t) = {8\pi G \over 3} \left(\rho_m + \rho_r\right) - {k \over a^2}$$

as:

$$H^2(t) = const \left({\Omega_m \over a^3} + {\Omega_r \over a^4} + {\Omega_k \over a^2}\right)$$

Here I've introduced a constant on the left, and a series of ##\Omega## constants. The equation becomes simplest if we require than when ##a = 1##, ##\Omega_m + \Omega_r + \Omega_k = 1##. In this case, when ##a = 1##, we get:

$$H^2(now) = const$$

Where this is the same constant that goes out in front, so our constant is just ##H_0^2##, and the Friedmann equation becomes:

$$H^2(t) = H_0^2 \left({\Omega_m \over a^3} + {\Omega_r \over a^4} + {\Omega_k \over a^2}\right)$$

So now we have four unknowns, but only one equation. That's not enough to give an answer unless three of the unknowns are set. What's done in practice is to take some measurable quantity which is a function of ##H(t)##:

$$F(data) = f(H(t))$$

We then take lots of data points, and use computer simulations to figure out which choices for the constants ##H_0, \Omega_m, \Omega_r, \Omega_k,## and ##\Omega_\Lambda## fit the data most closely (that last one stems from the cosmological constant).

Quarlep
Quarlep said:
I can't do that myself Should I open a new thread or somebody can do that ?
I don't think this is exactly what you've asked for, but this website calculates a bunch of different quantities based upon a variety of different inputs:
http://www.astro.ucla.edu/~wright/CosmoCalc.html

Quarlep
Chalnoth said:
##H_0, \Omega_m, \Omega_r, \Omega_k,## and ##\Omega_\Lambda## fit the data most closely (that last one stems from the cosmological constant).

In our universe ##H_0## is known 70 and ##Ω_m=0.05## ... so we can change this things isn't it.I mean this quantites describes known universe.So Can I put ##H_0=65## to fit the data ?

Quarlep said:
In our universe ##H_0## is known 70 and ##Ω_m=0.05## ... so we can change this things isn't it.I mean this quantites describes known universe.So Can I put ##H_0=65## to fit the data ?
##\Omega_m## is not 0.05. This parameter includes both normal matter and dark matter.

Planck's 2015 results report the best-fit estimates of these parameters are:
##H_0 = 68##km/sec/Mpc
##\Omega_m = 0.31##

Anyway, if you tried to fit the data with these parameter values, but ##\Omega_\Lambda = 0##, you'd find pretty quickly that the data doesn't fit at all.

I am trying to find density of universe ) If I put matter density 0.31 then what's the meaning to find density.We have already decided it.Is there any calculator which modified by 2015 plank results.

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You can't just compute it... you have to make experiments in order to decide its value and try to fit the data... It's a free parameter.

Chalnoth said:
In this site How can I put data ?

Ok,I understand the idea.But I want to ask something.My question is simple so I thought that there's no need to open a new thread and its related this question.
##H^2(t)-8ρπG/3=-k/a^2(t)## (Lets suppose a(t)=1) then
##H^2(t)-8ρπG/3=-k## In equations ##k## can be -1,0,1.
Then If ##k<0## then ##-k>0## then
##H^2(t)-8ρπG/3>0 ##→Hyperbolic Universe. This means ##Ω_k>0## I mean If ##k## negative ##Ω_k## must be positive isn't it ? I am confused here.
In cosmology calculator says ##Ω_k=1-Ω_m-Ω_Λ##

(Its look like hyperbolic universe I am trying to understand)
Chalnoth said:
Here I've introduced a constant on the left, and a series of ##\Omega## constants. The equation becomes simplest if we require than when ##a = 1##, ##\Omega_m + \Omega_r + \Omega_k = 1##.
Thanks

1. What is a hyperbolic universe?

A hyperbolic universe is a type of non-Euclidean geometry where the angles of a triangle add up to less than 180 degrees. This type of universe is characterized by negative curvature, meaning that parallel lines will eventually diverge from each other.

2. How is the universe density calculated in a hyperbolic universe?

In a hyperbolic universe, the density is calculated using the equation ρ = 3H^2 /8πG, where ρ is the density, H is the Hubble constant, and G is the gravitational constant. This equation takes into account the negative curvature of the universe and the expansion rate.

3. How does the universe density in a hyperbolic universe differ from that of a flat or positively curved universe?

In a flat universe, the density is equal to the critical density, meaning that the universe will eventually stop expanding. In a positively curved universe, the density is greater than the critical density and the universe will eventually collapse. In a hyperbolic universe, the density is less than the critical density and the universe will continue to expand forever.

4. Are there any observable effects of the universe density in a hyperbolic universe?

Yes, the density of the universe affects the overall curvature of space-time, which in turn affects the paths of light and other celestial objects. This can be observed through the bending of light from distant galaxies and the redshift of light from objects moving away from us.

5. How does the universe density in a hyperbolic universe impact the fate of the universe?

The density of the universe plays a crucial role in determining the fate of the universe. In a hyperbolic universe, the density is less than the critical density, meaning that the universe will continue to expand forever. This also means that the universe will eventually become more and more sparse, as matter and energy are spread out over increasingly larger distances.

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