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 P different states that only interact with each other gravitationally can be described by the following Lagrangian density

P --1--- ∑ 1[ ab ¯(n) (n) (|| (n)||2)] ℒ = 16πG R − 2 g ∂aϕ ∂bϕ + V ϕ , (77 ) n=1
where ϕ(n) is the particular complex scalar field representing the bosons in the n-state with n − 1 nodes. The equations of motion are very similar to the standard ones described in Section 2.2, with two peculiarities: (i) there are n 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]

(1) (2) N ≥ N . (78 )
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 (78View Equation) is satisfied. An example of this behavior can be observed in Figure 8View Image.
View Image

Figure 8: Left: The maximum of the central value of each of the two scalar fields constituting the multi-state BS for the fraction η = 3, where η ≡ N (2)∕N (1) defines the relative “amount” of each state. Right: The frequencies associated with each of the two states of the multi-state BS. At t = 2000, there is a flip in which the excited state (black solid) decays and the scalar field in the ground state (red dashed) becomes excited. Discussed in Section 3.7. Reprinted with permission from [25]; copyright by APS.

Similar results were found in the Newtonian limit [215], however, with a slightly higher stability limit N (1) ≥ 1.13N (2). 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 ( ) V |ϕ (1)||ϕ(2)|. Refs. [37, 38] construct such solutions, considering the individual particle-like configurations for each complex field as interacting with each other.

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