What values of a satisfy Ωm≈1 in the matter density parameter equation?

[ryanwilk]

Homework Statement

Starting with the equation below, I need to:
- Show that there is a range of values for a for which Ω_m≈1
- Derive expressions for the values of a at the endpoints of this range.

Homework Equations

Ω_m(a) = Ω_m0/[Ω_m0+Ω_r0/a+Ω_v0a³].

(0 signifies present day values, m=matter, r=radiation, v=vacuum)

The Attempt at a Solution

Just setting Ω_m=1 leads to a⁴ = -Ω_r0/Ω_v0, which gives imaginary values for a, which is obviously not right. However, I can't see any other way of solving this problem to get real values...Thanks in advance,
Ryan.

 

[George Jones]

The question is unclear to me. What does [itex]\Omega_m \approx 1[/itex] mean? Clearly, the expression gives [itex]\Omega_m \left( a \right) < 1[/itex] always. To get a handle on things, I had my computer plot [itex]\Omega_m \left( a \right)[/itex]. Try finding the values of [itex]a[/itex] for which [itex]\Omega_m \left( a \right) > 0.95[/itex]. Instead of 0.95 you could use some other number nearer to or farther from 1.