Friedmann's equation for a^3 with Λ, ρm

In summary: The conclusion is that the proposed equation to prove is wrong as it stands.In summary, the proposed equation a3(t)= ρo/2Λ [cosh(sqrt(24πGΛ)*t) -1] cannot be proven by changing into a variable of u, where u=2Λa3/ρo, as the relationship between Λ and ρm is arbitrary and the equation does not take into account the changing nature of ρm as the universe expands.
  • #1
QuarkDecay
47
2
Homework Statement
We have a flat Universe with a positive Λ, density matter ρ[SUB]m[/SUB]. Need to prove the formula for a[SUP]3[/SUP](t) below from the first Friedmann equation and changing the variable from the u=(...) given below
Relevant Equations
- Friedmann's first equation; H[SUP]2[/SUP]= 8πGρ[SUB]total[/SUB]/3 - k/a[SUP]2[/SUP]
- New variable; u=2Λa[SUP]3[/SUP]/ρ[SUB]o[/SUB]
We need to prove that a3(t)= ρo/2Λ [cosh(sqrt(24πGΛ)*t) -1] by changing into a variable of u, where
u=2Λa3o

From Friedmann's second equation we know that Λ= ρm/ 2
Also ρm= ρo/ a3

[First attempt]
I begin from Friedmann's equation where (for here), ρtotal= ρm + Λ and k=0;

a'2/a2 = 8πG(ρm + Λ)/3 , (a'=da/dt)

⇒ a'/a = sqrt ((8πG/3) * (3ρo/2a3) )

⇒∫ da/a = ∫sqrt((8πG/3) * (3ρo/2a3))dt

⇒ lna = sqrt((8πG) * (ρo/2a3)) t

⇒ a= exp [sqrt((8πG) * (ρo/2a3))t ]

which is not close with what we're trying to prove, and I didn't use the variable u I was supposed to. The solution is the correct for a flat universe, but when we're looking for the a(t). Seems like I have to change the a into u completely inside friedmann's equation.

So I tried this too; (second attempt)

a'2/a2 = 8πG(ρm + Λ)/3

⇒ a'/a = sqrt ((8πG/3) * (3ρo/2a3)) = sqrt(4πGρo/a3) ⇒

⇒ da/dt = a*sqrt(4πGρo/a3) = sqrt(a24πGρo/a3)=

= sqrt(4πGρo/a) (a)

and now changing the variable from a to u;

u=2Λa3o ⇒ a3 = uρo/2Λ ⇒

a= [uρo/2Λ] 3/2 (1)

So,
da= (ρo/2Λ) 3/2 u1/2du (2) (a),(1),(2) → ... du/u1/2= sqrt(8πGΛ)dt = sqrt(8πGΛ)t

Where the cosh() is still missing...
In the first attempt I had the exact same problem too. The first integral was always da/sqrt(a) that doesn't give a cosh for a solution
 
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  • #2
QuarkDecay said:
From Friedmann's second equation we know that Λ= ρm/ 2
This is not true. You have not been given the relation between ##\Lambda## and ##\rho_m## and the proportion between them is arbitrary. Furthermore, ##\rho_m## is going to change as the universe expands, which ##\Lambda## will not.

QuarkDecay said:
8πG(ρm + Λ)/3 , (a'=da/dt)

⇒ a'/a = sqrt ((8πG/3) * (3ρo/2a3) )
Which means that this is wrong. As stated above ##\Lambda## will depend on ##a## in a different way from ##\rho_m##.
 

Related to Friedmann's equation for a^3 with Λ, ρm

1. What is Friedmann's equation for a^3 with Λ, ρm?

Friedmann's equation for a^3 with Λ, ρm is a mathematical equation used in cosmology to describe the expansion of the universe. It relates the rate of expansion (Hubble parameter) to the density of matter (ρm) and the cosmological constant (Λ).

2. How is Friedmann's equation derived?

Friedmann's equation is derived from the Einstein field equations, which are a set of equations that describe the relationship between the curvature of spacetime and the distribution of matter and energy in the universe. It is a key component of the Friedmann-Lemaître-Robertson-Walker (FLRW) metric, which is used to model the universe.

3. What does the a^3 term represent in Friedmann's equation?

The a^3 term in Friedmann's equation represents the scale factor, which is a measure of the expansion of the universe. It is a function of time and describes how the distances between objects in the universe change over time.

4. How does the cosmological constant (Λ) affect Friedmann's equation?

The cosmological constant (Λ) is a term in Friedmann's equation that represents the energy density of empty space. It has a repulsive effect on the expansion of the universe, counteracting the attractive force of gravity. This leads to an accelerating expansion of the universe.

5. Can Friedmann's equation be used to predict the future of the universe?

Yes, Friedmann's equation, along with other cosmological models and data, can be used to make predictions about the future of the universe. It suggests that the expansion of the universe will continue to accelerate due to the influence of the cosmological constant, leading to a "heat death" scenario where all matter and energy are evenly distributed and the universe reaches a state of maximum entropy.

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