### 3.7 Multi-state boson stars

It turns out that excited BSs, as dark matter halo candidates, provide for flatter, and hence more
realistic, galactic rotation curves than ground state BSs. The problem is that they are generally unstable to
decay to their ground state. Combining excited states with the ground states in what are called multi-state
BSs is one way around this.
Although bosons in the same state are indistinguishable, it is possible to construct non-trivial
configurations with bosons in different excited states. A system of bosons in different states that only
interact with each other gravitationally can be described by the following Lagrangian density

where is the particular complex scalar field representing the bosons in the -state with
nodes. The equations of motion are very similar to the standard ones described in Section 2.2, with two
peculiarities: (i) there are independent KG equations (i.e., one for each state) and (ii) the stress-energy
tensor is now the sum of contributions from each mode. Equilibrium configurations for this system were
found in [25].
In the simplest case of a multi-state boson, one has the ground state and the first excited state. Such
configurations are stable if the number of particles in the ground state is larger than the number of particles
in the excited state [25, 7]

This result can be understood as the ground state deepens the gravitational potential of the
excited state, and thereby stabilizing it. Unstable configurations migrate to a stable one via
a flip-flop of the modes; the excited state decays, while the ground jumps to the first exited
state, so that the condition (78) is satisfied. An example of this behavior can be observed in
Figure 8.
Similar results were found in the Newtonian limit [215], however, with a slightly higher stability
limit . This work stresses that combining several excited states makes it
possible to obtain flatter rotation curves than only with ground state, producing better models for
galactic dark matter halos (see also discussion of boson stars as an explanation of dark matter in
Section 5.3).

Ref. [110] considers two scalar fields describing boson stars that are phase shifted in time with respect
to each other, studying the dynamics numerically. In particular, one can consider multiple scalar fields with
an explicit interaction (beyond just gravity) between them, say . Refs. [37, 38] construct
such solutions, considering the individual particle-like configurations for each complex field as interacting
with each other.