Some questions related to the Cosmological Constant

In summary, Barbara Ryden's introduction to cosmology book states that introducing ##\Lambda## into Poisson's equation can allow for a static universe, if ##\Lambda = 4\pi G\rho##. However, the book's definition of ##\epsilon_{\Lambda}## as ##\frac{c^2\Lambda}{8\pi G}## does not make sense, as it should be ##\frac{c^4\Lambda}{8\pi G}## according to other sources. This suggests that the book's information may be incorrect or that the definition of ##\Lambda## for a static universe is different from the general definition. Additionally, the equation ##\Lambda = \
  • #1
Arman777
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In Barbara Ryden's introduction to cosmology book its written that
"Introducing ##\Lambda## into the Poisson's equation allows the universe to be static, if you set ##\Lambda = 4\pi G\rho##"

Then later on, in the book energy density of the ##\Lambda## defined as ##\epsilon_{\Lambda} = \frac{c^2\Lambda}{8\pi G}##

But this not make much sense. Since in this case
$$\epsilon_{\Lambda} = \frac{c^2 4\pi G\rho}{8\pi G} $$
$$\epsilon_{\Lambda} = \rho/2 c^2 $$
Which is not correct.

From other sources I see that actually

##\Lambda = 8\pi G\rho## and
##\epsilon_{\Lambda} = \frac{c^4\Lambda}{8\pi G}##

And it seems that other sources definiton is the correct one. So the information on the book is wrong ? Or the ##\Lambda## for the static universe case is different then the general ##\Lambda## ?

Also,
##\epsilon_{\Lambda} = \frac{c^4\Lambda}{8\pi G}## and we know that ##\Omega_{\Lambda} = \frac{\epsilon_{\Lambda}}{\epsilon_c}##
and ##\epsilon_c = \frac{3H^2c^2}{8 \pi G}##

so we get

$$\Lambda = \frac{3H^2\Omega_{\Lambda}} {c^2}$$

So is this means that ##\Lambda## is actually a time dependent thing since it involves ##H(t)## and ##\Omega_{\Lambda}## ?
 
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  • #2
Arman777 said:
$$\epsilon_{\Lambda} = \rho/2 c^2 $$
Which is not correct.
Its correct with ##c^2## in the numerator.
Arman777 said:
$$\Lambda = \frac{3H^2\Omega_{\Lambda}} {c^2}$$

So is this means that ##\Lambda## is actually a time dependent thing since it involves ##H(t)## and ##\Omega_{\Lambda}## ?
It only appears to be. With ##\Omega_\Lambda=\frac{8\pi{G}\rho_\Lambda}{3H^2}## you obtain ##\Lambda=8\pi{G}\rho_\Lambda##.
 
  • #3
timmdeeg said:
Its correct with c2c2c^2 in the numerator.
Well 2 is dimensionless so it has not much effect on the equation itself. But I don't think its normal to have a 2.

timmdeeg said:
##\Lambda = 8 \pi G\rho_{\Lambda}##
So the equations on the book is wrong ?

timmdeeg said:
It only appears to be.
Hmm I see. Its also interesting. Then ##H^2(t)\Omega_{\Lambda}(t) = Constant## for all times
 
  • #4
Arman777 said:
Well 2 is dimensionless so it has not much effect on the equation itself. But I don't think its normal to have a 2.
If you combine the first 2 equations in your OP you get 2 in the denominator. Nothing wrong.
Your first equation follows from the 2. Friedmann Equation with the 2. derivative of the scale factor set zero (because static).
 
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  • #5
Hmm okay than
 

FAQ: Some questions related to the Cosmological Constant

1. What is the Cosmological Constant?

The Cosmological Constant, denoted by the Greek letter lambda (Λ), is a term in Einstein's theory of general relativity that represents the energy density of the vacuum of space. It is essentially a measure of the overall energy of the universe and plays a crucial role in understanding the expansion of the universe.

2. How does the Cosmological Constant affect the expansion of the universe?

The Cosmological Constant has a repulsive effect on the expansion of the universe, meaning it counteracts the gravitational pull of matter and causes the universe to expand at an accelerating rate. This effect becomes more significant as the universe expands, leading to the theory of "dark energy" which is thought to be responsible for this acceleration.

3. What is the current value of the Cosmological Constant?

The current accepted value of the Cosmological Constant is approximately 10^-52 m^-2, which is incredibly small. This value was determined through observations of the cosmic microwave background radiation and the expansion rate of the universe.

4. Why is the Cosmological Constant important in understanding the universe?

The Cosmological Constant is important because it helps us understand the overall energy and expansion of the universe. It also plays a crucial role in the formation of galaxies and the evolution of the universe. Without it, our understanding of the universe would be incomplete.

5. Can the Cosmological Constant change over time?

According to current theories, the Cosmological Constant is considered to be a fundamental constant of the universe and does not change over time. However, some theories suggest that it may vary over extremely long periods of time, which could have significant implications for the fate of the universe.

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