Spatial part of Robertson-Walker metric

The spatial part of the FRW metric can be written as $$d\Sigma^2=d\rho^2+f^2(\rho)(d\theta^2+{sin}^2\theta d\phi^2)$$ where $$f(\rho)$$ satisfies $$\frac{df}{d\rho}=\frac{f(2\rho)}{2f(\rho)}.$$ I am trying to derive the form of $$f(\rho)$$ by using a power series expansion $$f(\rho)=\sum a_n \rho^n$$ and show that $$f(\rho)$$ can be $$\rho$$, $$R\sin(\rho/R)$$ or $$R\sinh(\rho/R)$$. I am getting stuck.

What should be further steps?

Should be:

$$f(\rho) = \sum a_n \rho^n$$
$$df/d\rho = \sum n a_n \rho^{n-1}$$
$$f(2 \rho) = \sum a_n (2 \rho)^n = \sum 2^n a_n \rho^n$$

You plug what above in the equation, you multiply L.H.S. and R.H.S. times the expression for $$f(\rho)$$ and move all the terms to the L.H.S. Then order the terms as per powers of $$\rho$$, e.g. $$\rho^0$$, $$\rho^1$$, $$\rho^2$$, etc. For each power you equate the coefficient to zero. You should get a recurring formula relating $$a_n$$ to $$a_{n-1}$$.

DERIVATION
$$df/d\rho = f(2 \rho) / (2 f(\rho))$$
$$(df/d\rho) (2 f(\rho)) = f(2 \rho)$$
$$(\sum_n n a_n \rho^{n-1}) 2 (\sum_m a_m \rho^m) - \sum_l 2^l a_l \rho^l = 0$$
$$\sum_n \sum_m 2 n a_n a_m \rho^{n-1+m} - \sum_l 2^l a_l \rho^l = 0$$
Equating the coefficients of power $$l$$ requires $$n - 1 + m = l$$, that is $$m = l + 1 - n$$, hence
$$a_l = 2^{-(l - 1)} \sum_{n = 0}^{l + 1} n a_n a_{l + 1 - n}$$

Solution I
$$l = 0, n = 0, 1$$
$$a_0 = 2 a_1 a_0$$
$$a_0 (2 a_1 - 1) = 0$$
Let us consider $$a_0 = 0$$ and $$a_1$$ undefined

$$l = 1, n = 0, 1, 2$$
$$a_1 = a_1^2$$
$$a_1 (a_1 - 1) = 0$$
$$a_1 = 1$$ defined

$$l = 2, n = 0, 1, 2, 3$$
$$a_2 = 2^{-1} (a_2 + 2 a_2)$$
$$a_2 = 0$$

$$l = 3, n = 0, 1, 2, 3, 4$$
$$a_3 = 2^{-2} (a_3 + 3 a_3)$$
$$a_3$$ undefined

If we choose $$a_3 = 0$$, then the equation for $$a_l$$ with $$l \ge 4$$ provides a zero value if the previous coefficients up to $$a_{l - 1}$$ are zero, except for $$a_1$$.

To summarize we got the following solution:
$$a_1 = 1$$
$$a_l = 0$$ for $$l \ne 1$$
That is $$f (\rho) = \rho$$
It is the zero curvature geometry of FRW metric, i.e. a flat universe.

Note: To look for the positive/negative curvature solutions you have to exploit the fact that $$a_3$$ is undefined. You assume it is non zero and then proceed. However it is more laborious.

• I used the same method. I am getting stuck. Could you please include a small part of the derivation? – Tejas P Nov 15 '18 at 11:15
• I edited my answer adding the section DERIVATION. – Michele Grosso Nov 17 '18 at 17:05