Let's say that we have the sum of the utility of a social planner
$$\int_{0}^{\infty}U\left(C\right)e^{-\rho t}dt$$
where $C$ is the total consumption. If we want to write this by a per capita variable $c=\frac{C}{L}$ where $L=e^{nt}$ is the total number of population that grows exogenously at rate $n$. We can reformulate this such as
$$\int_{0}^{\infty}U\left(c\right)e^{nt}e^{-\rho t}dt$$
Until now, everything is straightforward. However, if we have started by a specified functional form such as a CRRA utility
$$\int_{0}^{\infty}\frac{C^{1-\sigma}}{1-\sigma}e^{-\rho t}dt$$
We would not have reached the result above and should have some exponent term with $\sigma$. What is the logic behind? There is something I miss.