Gravitational Phase Transition

Rotating self-gravitating fluid bodies embedded in an external medium can collapse in one of two ways, according to studies of analytical approximations. They can collapse dynamically by becoming gravitationally unstable, or they can collapse via a phase transition, before dynamical instability sets in. The dual possibility arises when the equilibrium solutions have a triple-value region, as shown in the plot at left. Since the phase transition mass is often much less than the dynamical collapse mass, structure formation via phase transition represents an interesting possibility in cosmological scenarios.

A common calculation used to estimate the typical mass scale of condensations in a cosmological scenario goes as follows. Start by considering a sphere with an internal temperature and an external pressure prescribed by the scenario. An equilibrium geometry exists in which the force of gravity is balanced by the body's internal pressure. Now consider increasing the body's mass. There comes a point at which no equilibrium state exists: the thermal energy of the body is insufficient to counteract the pull of gravity and the body collapses. This critical mass is known as the Jeans collapse mass. It is used as a mass yardstick not just in cosmology but also in studies of star formation.

Now consider the same situation, but imagine the body possesses some angular momentum and spheroidal symmetry. There may now be two critical collapse masses [1]. The first is a dynamical collapse mass, similar to the Jeans mass, above which the body must collapse to a rotationally supported condensed state, since a diffuse pressure supported state no longer exists [2]. The second critical collapse mass is the phase transition mass, which is often much less than the Jeans mass [1]. At this mass the body has a choice between two stable equilibria. It "prefers" to be in the condensed state when its mass is larger than the critical mass and to be in the diffuse state otherwise, in much the same way a van der Waals gas prefers to be a gas when its temperature is above a critical temperature and to be a liquid otherwise.

Structure formation via phase transition could solve some long-standing problems in star formation [1]. It also represents an interesting possibility for cosmology. For example, a simple cosmological scenario featuring spheroids with body temperature that stays constant over time, embedded in a medium of decreasing external pressure, finds the multi-equilibria regime begins at redshift z <~ 100. Subsequently, the dynamical collapse mass grows rapidly, while the phase transition mass remains approximately the same at about a galaxy mass [3].

One standard analytic approximation of an isothermal structure in detailed pressure balance is a Weber spheroid: a constant density spheroid rotating as a solid body [2]. This model may be made somewhat more general with little effort. Namely, spheroids with nested density distribution, differential rotation, and a central mass "seed" (useful in cosmological scenarios) have the same equations of equilibrium as Weber spheroids, differing only in the scaling coefficients [3].

One can also add another cosmologically useful accessory: an embedding infinite isothermal dark matter halo. When this is done, one finds there exists a critical halo temperature (equivalent to a critical halo mass) above which multiple equilbria vanish and only rotationally supported condensed states are allowed [4].

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