conformal time in Lambda-driven expansion

What’s the relation between conformal time and cosmic time (or equivalently between comoving and proper distance) during epochs of exponential expansion? Can you write the relation as a power-law?

physics bonus question: What’s the reason why magnetic fields play such an important role in astrophysics, and electrical fields so little?

maths bonus question: Can you determine the derivative of \(\exp(-1/(1-x^\alpha))\) both at \(x=0\) and \(x=1\) for arbitrary \(\alpha>0\)?