# Scale factor negative

Can the scale factor in a Robertson-Walker metric with $$k=0$$ (flat universe, with only matter, no cosmological constant or quintessence) depend on a polynomial function?

For example:

$$a(t)=-t^3+4t^2-t+3.$$

Because the strange thing is that this type of scale factor become negative on the infinite. So here we have that $$a(t)$$ become 0 at max 3 times.

• When $k=0$, the solution is $$a(t)=a_0t^{\frac{2}{3(w+1)}}$$, which is certainly not a polynomial like the one you describe. Oct 4, 2015 at 19:02
• Yes, but it depends on the behaviour of the matter. For now it doesn't count much the real solution, but the meaning of a (possible)solution as the one I wrote. Oct 4, 2015 at 19:09
• I'm not sure I follow. That's the solution for $k=0$. In a homogeneous FLRW universe with $k=0$, that's the solution that exists. Oct 4, 2015 at 19:13
• Ok, if I ask you can, in general, the scale factor be negative or be as the polynomial that I wrote? Oct 4, 2015 at 19:23
• It can be negative, but I don't know if a polynomial form like the one you wrote can arise. Oct 4, 2015 at 19:28

The only approach I know relating $$H(t)$$ and $$a(t)$$ is from
$$a(t) = A t^{2/3}$$.
The usual convention is that $$a = 1$$ for the corresponding value of t being the current age of the universe, say $$t=t_{now}$$. Then
$$A = t_{now}^{-2/3}$$.