How Does Energy Density Change with Scale Factor in Cosmology?

[cosmoshadow]

Homework Statement

The fluid equation in cosmology is given as:

$$\dot{\epsilon}$$ + 3*($$\dot{a}$$/a)*($$\epsilon$$+P) = 0

Where $$\epsilon$$ is the energy density and a(t) is a scale factor.

Using the equation of state, P = w*$$\epsilon$$, show how $$\epsilon$$ change with a(t).

Homework Equations

$$\dot{\epsilon}$$ + 3*($$\dot{a}$$/a)*($$\epsilon$$+P) = 0
P = w*$$\epsilon$$

The Attempt at a Solution

I can solve for the equation to the point where I re-arrange it to look like this:

$$\dot{\epsilon}$$/$$\epsilon$$ = -3*(1+w)*($$\dot{a}$$/a)

I do not know how to proceed from here. I know that this equation is supposed to end up like this,

[tex]\epsilon_w(a)[/tex] = [tex]\epsilon_w,0[/tex]*a^-3*(1+w)

but I do not know how to get to this point. Can someone assist me please?

 

[cosmoshadow]

can someone take a look at this? I'm pretty sure its a simple operation that I'm failing to realize.

 

[cosmoshadow]

bump?

 

[scottie_000]

You have your equation
$$\frac{\dot\epsilon}{\epsilon} = -3(w+1)\frac{\dot a}{a}$$
From here you can eliminate the time-dependence
$$\frac{d\epsilon}{\epsilon} = -3(w+1)\frac{da}{a}$$
and this is a differential equation involving just $$\epsilon$$ and $$a$$ you can solve by integrating both sides

 

[paranormal]

I would suggest considering the physical meaning behind the fluid equation in cosmology. This equation is a manifestation of the conservation of energy, specifically for the energy density of the universe. The left side represents the change in energy density over time, while the right side represents the expansion of the universe and its effect on the energy density.

To understand how the energy density changes with the scale factor, we can use the equation of state, P = w*epsilon, which relates the pressure (P) to the energy density (epsilon) through a constant parameter w. This equation tells us that as the universe expands (a increases), the pressure will decrease, leading to a decrease in the energy density.

To show this mathematically, we can substitute the equation of state into the fluid equation, giving us:

\dot{\epsilon} + 3*(\dot{a}/a)*(\epsilon+w*\epsilon) = 0

We can then factor out epsilon to get:

\dot{\epsilon} + 3*(\dot{a}/a)*\epsilon*(1+w) = 0

We can then divide both sides by epsilon to get:

\dot{\epsilon}/\epsilon + 3*(\dot{a}/a)*(1+w) = 0

This can be rearranged to get:

\dot{\epsilon}/\epsilon = -3*(1+w)*(\dot{a}/a)

Now we can integrate both sides with respect to time to get:

ln(\epsilon) = -3*(1+w)*ln(a) + C

Where C is a constant of integration. We can exponentiate both sides to get:

\epsilon = e^C * a^(-3*(1+w))

We can then use the initial conditions of the universe (e.g. the energy density at a certain time) to determine the value of the constant C. This will give us the full equation for how the energy density changes with the scale factor:

\epsilon(a) = \epsilon(0) * a^(-3*(1+w))

This shows that as the scale factor increases, the energy density decreases due to the expansion of the universe. We can also see that the value of w, which represents the type of energy present in the universe, will affect the rate at which the energy density decreases with the scale factor.