Cosmological structure formation in a homogeneous dark energy background

There is now strong evidence that the current energy density of the Universe is dominated by dark energy with an equation of state w < -1/3, which is causing accelerated expansion. The build-up of structure within such Universes is subject to significant ongoing study, particularly through the spherical collapse model. This paper aims to review and consolidate progress for cosmologies in which the dark energy component remains homogeneous on the scales of the structures being modelled. The equations presented are designed to allow for dark energy with a general time-varying equation of state w(a). In addition to reviewing previous work, a number of new results are introduced: a new fitting formula for the linear growth factor in constant w cosmologies is given. A new formalism for determining the critical density for collapse is presented based on the initial curvature of the perturbation. The commonly used approximation to the critical density for collapse based on the linear growth factor is discussed for a general dark energy equation of state. Virialisation within such cosmologies is also considered, and the standard assumption that energy is conserved between turn-around and virialisation is questioned and limiting possiblities are presented.