# How Does the Cosmological Constant Change with the Expansion of the Universe?

• TRB8985
In summary, the conversation discusses the incorrect assumption that the Hubble parameter (H) scales with temperature squared in a radiation-dominated universe. It also introduces the concept of dark energy in the form of a cosmological constant (Λ) and its relation to the critical energy density (ρ_critical). The question asks for the ratio of ρΛ / (3H2/8πG) at the time when the universe's temperature was 1019 GeV/kB. The solution involves multiplying 0.7 by the ratio of (H0 / H)2, which is derived from the assumption that the universe is radiation-dominated and H / H0 = a-2. This leads to (a2)2 = a
TRB8985

## Homework Statement

Suppose (incorrectly) that H scales as temperature squared all the way back until the time when the temperature of the universe was 1019 GeV/kB (i.e., suppose the universe was radiation dominated all the way back to the Planck time).

Also suppose that today the dark energy is in the form of a cosmological constant Λ, such that ρΛ today is equal to 0.7*ρcritical and ρΛ remains constant throughout the history of the universe. What was ρΛ / (3H2/8πG) back then?

(From Modern Cosmology by Dodelson, pg. 25)

## Homework Equations

ρ_critical = (3H02/8πG)

T = 1019 GeV/kB = 1.16045* 1032 K

T0 = 2.725 K

For a radiation-dominated universe, a ∝ t1/2.

## The Attempt at a Solution

I understand a part of the solution wherein ρΛ / ρcritical = 0.7, but I'm supposed to multiply this value by something.

In the answer key, Dodelson multiplies 0.7 by the ratio of (H0 / H)2. The text states:

"By assumption, the universe is forever radiation dominated (clearly not true today, but a good approximation early on), so H / H0 = a-2."

Given this, the inverse of H / H0 would result in a2, and since H scales as temperature squared, then (a2)2 = a4 which can then be applied to the ratio of the temperature. That latter part makes sense. However, I'm not quite understanding where Dodelson pulled the ratio of H0 / H from to get things started.

Could anyone provide any insight on this? Thank you very much for your help.

(This question is being attempted via an independent study and not a homework question. Additionally, there are no cosmology specialists at my university who could provide any useful feedback on how to attack this situation.)

TRB8985 said:

## Homework Statement

Suppose (incorrectly) that H scales as temperature squared ...

Multiply the expression in the exercise by one in the form ##1 = \rho_{cr} / \rho_{cr}##, then use equation (1.3) to substitute for ##\rho_{cr}## in the numerator (but not in the denominator).

TRB8985
George, your input was incredibly helpful and brought the entire picture together. Thank you so much for your help! I appreciate that.

## 1. What is the Cosmology Fine Tuning Problem?

The Cosmology Fine Tuning Problem is a question that arises from the observation that the fundamental constants and parameters of our universe seem to be precisely tuned to allow for the existence of life. This means that if these values were even slightly different, life as we know it would not be possible.

## 2. How does the Cosmology Fine Tuning Problem relate to the Big Bang Theory?

The Big Bang Theory is the prevailing scientific explanation for the origin of the universe. It states that the universe began as an incredibly hot and dense point and has been expanding and cooling ever since. The Cosmology Fine Tuning Problem is relevant to the Big Bang Theory because it raises the question of why the initial conditions of the universe were so precisely tuned to allow for the evolution of life.

## 3. What are some examples of fine tuning in the universe?

There are several examples of fine tuning in the universe, including the strength of the gravitational force, the mass of the proton, and the cosmological constant. If any of these values were even slightly different, the universe would not be able to support life.

## 4. Is there a scientific explanation for the fine tuning of the universe?

There is currently no widely accepted scientific explanation for the fine tuning of the universe. Some scientists propose the existence of multiple universes, with different values for fundamental constants, as a potential explanation. However, this theory is still speculative and has not been proven.

## 5. What are the implications of the Cosmology Fine Tuning Problem?

The implications of the Cosmology Fine Tuning Problem are still a topic of debate among scientists and philosophers. Some argue that it points to the existence of a higher power or intelligent designer, while others believe it is simply a coincidence or the result of natural processes. The debate surrounding this problem continues to drive research and exploration in the field of cosmology.

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