In an Einstein-de Sitter (matter-dominated) universe, the scale factor is proportional to time to the power of:
* 3/2
* 2/3
* 1/2
* -1/3
To determine the relationship between the scale factor and time in an Einstein-de Sitter universe, we need to consider the Friedmann equation. The Friedmann equation relates the expansion rate of the universe (Hubble parameter) to the matter density of the universe.
In an Einstein-de Sitter universe, the dominant component of the universe is non-relativistic matter. The Friedmann equation in this case is:
H² = (8πGρ) / 3,
where H is the Hubble parameter, G is the gravitational constant, and ρ is the matter density. Since we are considering a matter-dominated universe, the matter density ρ scales as a^(-3), where a is the scale factor.
Differentiating the Friedmann equation with respect to time, we get:
2H(dH/dt) = (-8πG/3)(dρ/dt) = (-8πG/3)(-3d(a^(-3))/dt).
Noting that d(a^(-3))/dt = -3(a^(-3))(da/dt), we can rewrite the equation as:
2H(dH/dt) = 8πG(a^(-3))(3(a^(-3))(da/dt)) = 24πG(a^(-6))(da/dt).
Now, let's evaluate da/dt. Differentiating the scale factor with respect to time, we have:
da/dt = (da/dθ)(dθ/dt),
where θ is the conformal time. In an Einstein-de Sitter universe, the conformal time is related to time by dθ = dt/a(t), or dθ/dt = 1/a(t). Therefore, da/dt = (da/dθ)(dθ/dt) = (da/dθ)(1/a).
Plugging this expression for da/dt into our equation, we have:
2H(dH/dt) = (24πG)(a^(-6))(da/dθ)(1/a) = (24πG)(a^(-7))(da/dθ).
Now, we can rewrite the equation once again in terms of the conformal time θ:
2H(dH/dθ) = 24πG(a^(-7))(da/dθ).
In an Einstein-de Sitter universe, the Hubble parameter is given by H = (2/3)(1/θ). Plugging this into the equation, we get:
2(2/3)(1/θ)(d(2/3)(1/θ)/dθ) = 24πG(a^(-7))(da/dθ),
which simplifies to:
2(2/3)(1/θ)((-2/3)(1/θ^2)) = 24πG(a^(-7))(da/dθ).
Simplifying further, we have:
(4/9θ^2)(1/θ^2) = 24πG(a^(-7))(da/dθ),
(4/9)(1/θ^4) = 24πG(a^(-7))(da/dθ).
Finally, by rearranging the terms, we find:
(da/dθ) = (4/9)(24πG)(θ^4)(a^(-7)).
Now, notice that the left-hand side is the derivative of the scale factor with respect to the conformal time, and the right-hand side consists of constants multiplied by powers of θ and a. Integrating both sides, we obtain:
a = C(θ^((4/9) - 7)),
where C is a constant of integration. Since the scale factor a is proportional to time t, we have θ ∝ t in an Einstein-de Sitter universe. Therefore, the relationship between the scale factor a and time t is given by:
a ∝ t^((4/9) - 7) = t^(-(29/9)).
Comparing this with the options given, we find that the scale factor is proportional to time to the power of -29/9.