written in a more convenient form, eliminating $\gamma$ : $$\ln \left( \frac{T_c}{T_{c0}} \right) = 2\, T_c\, \sum_{\omega_n\,\geq\,0}\left[\frac{v_\text{F}}{2}\,\int_{-\infty}^{+\infty}\mathrm{Re}\left(\frac{F^{\dagger}_{\downarrow\uparrow}}{\Delta_{0}}\right)\,\mathrm{d}p -  \frac{\pi}{\omega_n} \right]\;,
\label{eq. SC}$$ where $v_\text{F}$ is the Fermi velocity in the absence of SOC, $T_c$ is the critical temperature of the wire in presence of Zeeman and Rashba interactions, while $T_{c0}$ is the critical temperature in the absence of these fields, i.e. for $h_{x}=h_{z}=0$ and $\alpha=0$.

A simple way of seeing the influence of spin-orbit coupling on the emergence of the modulated phase is to expand Eq. (\[eq. SC\]) in series with respect to the wave-vector $q$ as : $$\begin{aligned}
\ln\left(\frac{T_c}{T_{c0}}\right) &=& 2\,\pi\,T_c\,\sum_{\omega_n \geq 0}\left[-\frac{h_x^2 + h_z^2}{\omega_n^3} + \frac{7\,p_\text{F}^2}{\omega_n^7}\,h_x^2\,h_z\,\alpha^3\,q\right.\notag\\ 
&&\left.- \frac{v_\text{F}^2}{4\,\omega_n^3}\,q^2\right]\;.
\label{SCE-DL}\end{aligned}$$ This expansion is justified by the fact that the wave-vector $q$ is small near the transition towards the nonuniform phase. We derived Eq. (\[SCE-DL\]) assuming a weak Zeeman field ($h_x,\, h_z \ll T_c$) and weak spin-orbit coupling ($\alpha\,p_\text{F} \ll T_c$). The chemical potential is larger than all the other energies: $\mu \gg h_x,\,h_z,\, E_\text{so}$, where $E_{so} = \frac{1}{2}\,m\,\alpha^2$. Therefore the Green’s function (\[eq. Fdu-single\]) may be expanded in series with respect to the small energies $h_x$, $h_z$ and $\alpha\,p$. Because of the $p$ integration in Eq. (\[eq. SC\]), only 