  We study the response of a quantitative trait to exponential directional
selection in a finite haploid population at the genetic and the phenotypic
level. We assume an infinite sites model, in which the number of new mutations
per generation in the population follows a Poisson distribution (with mean
$\Theta$) and each mutation occurs at a new, previously monomorphic site.
Mutation effects are beneficial and drawn from a distribution. Sites are
unlinked and contribute additively to the trait. Assuming that selection is
stronger than random genetic drift, we model the initial phase of the dynamics
by a supercritical Galton-Watson process. This enables us to obtain
time-dependent results. We show that the copy-number distribution of the mutant
in generation n, conditioned on non-extinction until n, is described accurately
by the deterministic increase from an initial distribution with mean 1. This
distribution is related to the absolutely continuous part $W^+$ of the random
variable, typically denoted $W$, that characterizes the stochasticity
accumulating during the mutant's sweep. On this basis, we derive explicitly the
(approximate) time dependence of the mutant frequency distribution, of the
expected mean and variance of the trait and of the expected number of
segregating sites. Unexpectedly, we obtain highly accurate approximations for
all times, even for the quasi-stationary phase where we refine classical
results. In addition, we find that $\Theta$ is the main determinant of the
pattern of adaptation at the genetic level, i.e., whether the initial
allele-frequency dynamics are best described by sweep-like patterns at few loci
or small allele-frequency shifts at many. The selection strength determines
primarily the rate of adaptation. The accuracy of our results is tested by
comprehensive simulations in a Wright-Fisher framework.
