# Neutron mass fraction evolution: approximation

In Mukhanov's "Physical foundations of cosmology" on page 102 the author considers an equation for the evolution of the neutron mass fraction $$X_{n}\equiv n_{n}/(n_{p}+n_{n})$$: $$\tag 1 \dot{X}_{n}(t) = -\lambda_{n\to p}(t)(1+e^{-Q/T(t)})(X_{n} - X_{n,\text{eq}}), \quad X_{n}(t\to 0) = X_{n,\text{eq}}(t),$$ where

• $$\lambda_{n\to p}$$ the neutron-to-proton conversion rate (behaving as $$\lambda_{n\to p} \sim t^{-\alpha}$$, where $$\alpha > 0$$),
• $$Q = m_{n}-m_{p}$$,
• $$T(t)$$ is the temperature (behaving as $$T \sim 1/\sqrt{t}$$),
• $$X_{n,\text{eq}}(t)$$ is the neutron mass fraction in equilibrium, $$X_{n,\text{eq}}(t)= \frac{1}{1+e^{Q/T(t)}}, \quad X_{n,\text{eq}}(0) = \frac{1}{2}$$ The formal solution of $$(1)$$ is $$\tag 2 X_{n}(t) = X_{n,\text{eq}}(t) - \int \limits_{0}^{t}d\tilde{t} \dot{X}_{n,\text{eq}}(\tilde{t})\exp\left[-\int \limits_{\tilde{t}}^{t}d\bar{t}\lambda_{n\to p}(\bar{t})(1+e^{-Q/T(\bar{t})})\right]$$ The author says that for small enough $$t$$ the second term tends to zero and can be considered as a perturbation. Namely, he says that it is possible to write $$(2)$$ as the asymptotic series in increasing powers of $$\dot{X}_{n,\text{eq}}$$ using the integration by parts: $$\tag 3 X_{n}(t) = X_{n,\text{eq}}(t)\left[ 1-\frac{1}{\lambda_{n\to p}(t)(1+e^{-Q/T(t)})}\frac{\dot{X}_{n,\text{eq}}(t)}{X_{n,\text{eq}}(t)}+\dots\right]$$ I absolutely do not understand how to obtain $$(3)$$...