Variation of Density Parameters with Redshift

In summary: My post and my question deals with how these parameters evolve with redshift. I was struck by the fact that the matter density parameter itself tends towards unity as z tends to infinity for almost any reasonable value of said parameter at the present epoch. For the flatness condition to be true, this must mean that the dark energy density parameter tends to zero as z tends to infinity. I set about trying to show that, and I managed to, but I am wondering whether my analysis is correct. The problem of fine tuning in cosmology arises when trying to explain why the universe has to have been extremely close to flat initially in order to match present-day observations. The matter density parameter tends towards unity as z approaches
  • #1
cepheid
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I was looking at section 7.6 of Longair's Galaxy Formation, and in it, he is talking about the flatness problem, or fine-tuning problem. In it, he shows that if you define a general (i.e. varies with cosmic time and hence with redshift) density parameter [itex] \Omega_m [/itex] for matter by analogy with [itex] \Omega_{m,0} [/itex] (the matter density parameter at the present epoch), it would be defined as follows:

[tex] \Omega_m = \frac{8 \pi G \rho_m}{3H^2} [/tex]​

Where H(z) is the Hubble parameter (so that H0 = H(0))

Since H = [itex] \dot{a}/a [/itex], and a = 1/(1+z), we can get directly from the Friedman equation:

[tex] H = H_0 [(1+z)^2(\Omega_{m,0} z +1) - \Omega_{\Lambda ,0}z(z+2)]^{1/2} [/tex]​

I've used a slightly non-standard notation of including a zero in the subscript of the dark energy density parameter to explicitly indicate that I'm talking about the value of this parameter at the present epoch. The reason for doing so is that, later on, I'll be talking about the value of this parameter not at the present epoch. Anyway, from this, Longair shows that

[tex] \Omega_m = \frac{\Omega_{m,0}}{\left(\frac{\Omega_{m,0} z + 1}{1+z}\right) - \Omega_{\Lambda ,0} \left(\frac{1}{1+z} - \frac{1}{(1+z)^3}\right)} [/tex]​

From this, it can be shown that if [itex] \Omega_m [/itex] had a value that differed even slightly from 1 initially, then the current value [itex] \Omega_{m,0} [/itex] would be drastically different from 1 (it would be either huge or miniscule...small variations initially would cause it to diverge rapidly as z approached 0). Another way of looking at this is that there is a very wide range of value of the parameter [itex] \Omega_{m,0} [/itex] that all basically converge to 1 at high redshifts. It's certainly true for any value between 0 and 1. This is the flatness, or "fine-tuning" problem. Why is the universe seemingly required to have been exceedingly close to flat initially in order to accord with present-day observations?

This result got me thinking. It is shown (elsewhere in the book) that the present condition for flatness is that the total density parameter is 1:

[tex] \Omega_{\textrm{tot},0} = \Omega_{m,0} + \Omega_{\Lambda ,0} =1 [/tex]​

We think that this is probably true (e.g. they are ~0.3 and ~0.7, speaking very loosely). But, I was bothered by the fact that if the universe is flat now, then it must always have been flat, suggesting that:

[tex] \Omega_{\textrm{tot}} = \Omega_{m} + \Omega_{\Lambda} =1 [/tex]​

is true generally (where these are now functions of z). But I have just shown that [itex] \Omega_m \rightarrow 1 [/itex] all by itself as z approaches infinity! Therefore, the only way that this condition can be satisfied is if [itex] \Omega_{\Lambda} \rightarrow 0 [/itex] as z approaches infinity. Longair makes no mention of this, so I set about to figure out if this was true on my own. I came up with what I thought was a reasonable defnition for the total density parameter (again, by analogy with the existing ones):

[tex] \Omega_{\textrm{tot}} = \frac{8 \pi G \rho_{\textrm{tot}}}{3H^2} [/tex]

[tex] \Omega_{\textrm{tot}} = \frac{8 \pi G (\rho_m + \rho_v)}{3H^2} [/tex]​

where the 'm' and 'v' stand for 'matter' and 'vacuum energy' respectively:

[tex] \Omega_{\textrm{tot}} = \frac{8 \pi G \rho_m }{3H^2} + \frac{8 \pi G \rho_v }{3H^2} [/tex]​

The first term is just the matter density parameter from before, therefore the second term must be the dark energy density parameter:

[tex] \Omega_{\textrm{tot}} = \Omega_m + \frac{8 \pi G \rho_v }{3H^2} [/tex]

[tex] \Rightarrow \frac{8 \pi G \rho_v }{3H^2} \equiv \Omega_{\Lambda} [/tex]​

The question is, does it go to zero in the limit of infinite z? Using the definition of H from before:

[tex] \Omega_{\Lambda} = \frac{8 \pi G \rho_v }{3H_0^2 [(1+z)^2(\Omega_{m,0} z +1) - \Omega_{\Lambda ,0}z(z+2)]} [/tex]​

Everything that is not in square brackets is just, by definition, [itex] \Omega_{\Lambda, 0} [/itex]

[tex] \Omega_{\Lambda} = \frac{\Omega_{\Lambda ,0}}{[(1+z)^2(\Omega_{m,0} z +1) - \Omega_{\Lambda ,0}z(z+2)]} [/tex]​

Now, the nice thing about this is that IF [itex] \Omega_{m,0} + \Omega_{\Lambda ,0} =1 [/itex], then you can show using a little bit of algebra that the denominator (the part in square brackets) becomes:

[tex] \Omega_{\Lambda} = \frac{\Omega_{\Lambda ,0}}{[\Omega_{m,0}(1+z)^3 +\Omega_{\Lambda ,0}]} [/tex]​

I'm confident that that does indeed approach zero as z approaches infinity. My question is simple: have I attempted to reconcile the apparent problem in the proper way, or is my attempt to define a dark energy density parameter that varies with cosmic epoch totally out to lunch?
 
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  • #2
For a while, most theorists just assumed that [tex]\Omega_\Lambda = 0[/tex]. The correct statement is that [tex]\Omega_{tot}[/tex] must be absurdly close to one in the distant past for it to be anywhere near one now.
 
  • #3
Thanks for your response Chalnoth,

Chalnoth said:
For a while, most theorists just assumed that [tex]\Omega_\Lambda = 0[/tex].

Yes, I understand that the observational evidence for a non-zero cosmological constant is relatively recent. However, when most people talk about [itex] \Omega_\Lambda [/itex] or any density parameter for that matter, they are usually talking about that parameter at the present epoch. My post and my question deals with how these parameters evolve with redshift. I was struck by the fact that the matter density parameter itself tends towards unity as z tends to infinity for almost any reasonable value of said parameter at the present epoch. For the flatness condition to be true, this must mean that the dark energy density parameter tends to zero as z tends to infinity. I set about trying to show that, and I managed to, but I am wondering whether my analysis is correct.
 
  • #4
Fine tuning is definitely a problem in cosmology, and it is probably true Omega had to be almost exactly 1 over the entire history of the universe for reasons you noted. Why is the mystery, expansion dilutes the matter density of the universe and dark energy compensates almost exactly. Weird.
 
  • #5
cepheid said:
Thanks for your response Chalnoth,
Yes, I understand that the observational evidence for a non-zero cosmological constant is relatively recent. However, when most people talk about [itex] \Omega_\Lambda [/itex] or any density parameter for that matter, they are usually talking about that parameter at the present epoch. My post and my question deals with how these parameters evolve with redshift. I was struck by the fact that the matter density parameter itself tends towards unity as z tends to infinity for almost any reasonable value of said parameter at the present epoch. For the flatness condition to be true, this must mean that the dark energy density parameter tends to zero as z tends to infinity. I set about trying to show that, and I managed to, but I am wondering whether my analysis is correct.
In a spatially-flat universe, [tex]\Omega[/tex] is just the fraction of the total density of any given matter or energy type. So whatever type of matter/energy dilutes the slowest will always tend toward [tex]\Omega = 1[/tex].

If you have no dark energy, and the universe is perfectly flat, then that is matter. But if you have dark energy, which is required to dilute more slowly than normal matter in order for it to explain the current acceleration, then it is the dark energy instead that will approach [tex]\Omega = 1[/tex], not the matter. Dark energy has the added feature that it also dilutes more slowly than the effect of the curvature, such that you no longer need a perfectly-flat universe for this to be true.

(In a matter-dominated universe with some curvature, either [tex]\Omega_m \rightarrow 0[/tex] and [tex]\Omega_k \rightarrow 1[/tex] (open case), or [tex]\Omega_m \rightarrow \infty[/tex] and [tex]\Omega_k \rightarrow -\infty[/tex] such that [tex]\Omega_m + \Omega_k = 1[/tex] (closed case).)
 

Related to Variation of Density Parameters with Redshift

What is the concept of "Variation of Density Parameters with Redshift"?

The variation of density parameters with redshift refers to the change in the density of matter and energy in the universe as we look back in time, or towards higher redshifts. This is due to the expansion of the universe, which causes the density of matter and energy to decrease over time.

What are density parameters in cosmology?

Density parameters in cosmology are dimensionless quantities that represent the ratio of the density of a particular component (such as matter or dark energy) to the critical density of the universe. They are important in understanding the overall composition of the universe and how it evolves over time.

How does the density of matter change with redshift?

The density of matter decreases with increasing redshift. This is because as the universe expands, the matter is spread out over a larger volume, resulting in a lower density. This can be seen in the decreasing matter density parameter as we look towards higher redshifts.

What is the impact of dark energy on density parameters with redshift?

Dark energy, which is believed to be the dominant component of the universe, has a significant impact on the variation of density parameters with redshift. It is thought to be responsible for the accelerating expansion of the universe, leading to a decrease in the matter density parameter and an increase in the dark energy density parameter over time.

How is the variation of density parameters with redshift measured?

Various methods can be used to measure the variation of density parameters with redshift, including observations of the cosmic microwave background radiation, the large-scale distribution of galaxies, and the brightness of supernovae. These measurements provide important insights into the composition and evolution of the universe.

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