PHYSICAL REVIEW D 103, 023536 (2021)
Editors' Suggestion
Decay of boson stars with application to glueballs and other real scalars
† ‡ Mark P. Hertzberg,* Fabrizio Rompineve, and Jessie Yang Institute of Cosmology, Department of Physics and Astronomy, Tufts University, Medford, Massachusetts 02155, USA
(Received 17 October 2020; accepted 10 December 2020; published 27 January 2021)
One of the most interesting candidates for dark matter is massive real scalar particles. A well-motivated example is from a pure Yang-Mills hidden sector, which locks up into glueballs in the early universe. The lightest glueball states are scalar particles and can act as a form of bosonic dark matter. If self-interactions are repulsive, this can potentially lead to very massive boson stars, where the inward gravitational force is balanced by the repulsive self-interaction. This can also arise from elementary real scalars with a regular potential. In the literature it has been claimed that this allows for astrophysically significant boson stars with high compactness, which could undergo binary mergers and generate detectable gravitational waves. Here we show that previous analyses did not take into proper account 3 → 2 and 4 → 2 quantum mechanical annihilation processes in the core of the star, while other work misinterpreted the classical 3 → 1 process. In this work, we compute the annihilation rates, finding that massive stars will rapidly decay from the 3 → 2 or 4 → 2 processes (while the 3 → 1 process is typically small). Using the Einstein-Klein- Gordon equations, we also estimate the binding energy of these stars, showing that even the densest stars do not have quite enough binding energy to prevent annihilations. For such boson stars to live for the current age of the universe and to be consistent with bounds on dark matter scattering in galaxies, we find the O 1 ≲ 10−18 3 → 2 following upper bound on their mass for ð Þ self-interaction couplings: M Msun when ≲ 10−11 4 → 2 processes are allowed and M Msun when only processes are allowed. We also estimate destabilization from parametric resonance which can considerably constrain the phase space further. Furthermore, such stars are required to have very small compactness to be long-lived.
DOI: 10.1103/PhysRevD.103.023536
I. INTRODUCTION plausible from the point of view of fundamental physics. They are also considered to be entirely natural, since they Perhaps the best motivation for physics beyond the do not appeal to any unnecessarily small parameters. In Standard Model is the presence of dark matter that comprises particular, they have no allowed elementary masses and are most of the mass of the universe. The most exciting aspect of fully described by only two quantities: the scale at which dark matter is that it may represent an entirely new sector of the theory becomes strong coupled Λ (which can be physics. Because of the current lack of discovery of any dark naturally small compared to any unification scale due to matter candidates that have direct couplings to the Standard the logarithmically slow running of coupling) and the size Model, including weakly interacting massive particles of the gauge group, such as SU N . It is also possible, of (WIMPs), it raises the possibility that dark matter may be ð Þ course, that physics beyond the Standard Model includes part of some hidden sector and/or associated with new very various other kinds of particles. Importantly, this may heavy particles (e.g., see Refs. [1–16]). include elementary spin 0 particles. Both the hidden spin Hidden sectors may include pure Yang-Mills inter- actions, i.e., new collections of massless spin 1 particles 1 and spin 0 particles are, of course, bosons, and as such with self-interactions. Such constructions are entirely they allow for a rich phenomenology; under some con- ditions they can organize into interesting states of high occupancy, as we will discuss in this paper. *[email protected] Let us address in more detail the case of hidden Yang- † [email protected] Mills. If there are indeed such sectors of interacting spin 1 ‡ [email protected] particles, it leads to various questions: What happens below the confinement scale? If the particles are thermally produced Published by the American Physical Society under the terms of in the early universe, what is their energy density today? Do the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the strong interactions lead to inconsistency with constraints the author(s) and the published article’s title, journal citation, on dark matter scattering in galaxies? And, very importantly, and DOI. Funded by SCOAP3. are there novel observational signatures of these sectors?
2470-0010=2021=103(2)=023536(21) 023536-1 Published by the American Physical Society HERTZBERG, ROMPINEVE, and YANG PHYS. REV. D 103, 023536 (2021)
To address these issues, let us start in the very early “boson stars” (for a review see Refs. [30–33]). For both universe. We know that at high temperatures of the early glueballs and for elementary scalars with a regular poten- 4 universe, the particles exhibit asymptotic freedom, and so tial, one anticipates self-interactions, including ∼λ4ϕ , etc. we have a gas of almost free particles. When the hidden For repulsive self-interactions this can give rise to very sector temperature is on order of the strong coupling scale dense boson stars. This was carefully studied originally in ∼Λ, the particles are expected to lock up into color neutral Ref. [34] for a complex scalar field with a global Uð1Þ states, known as “glueballs” (for a review, see Ref. [17]). symmetry (recent work includes Ref. [35]). They solved the The lightest glueball states are expected to be spin 0 full Einstein-Klein-Gordon equations of motion for spheri- particles, whose mass mϕ is of the order the strong coupling cally symmetric time independent solutions, finding that scale Λ. As the temperature of the dark sector Td lowers the maximum star mass is further, these glueballs become nonrelativistic, and can act ffiffiffiffiffi as a form of dark matter in the late universe. If one just p 3 λ4M ∼ Pl focuses on freeze-out, the relic abundance can be estimated Mmax 2 ð1Þ 2 3 mϕ to be roughly [7] Ωϕ ∼ N ξ ðΛ=ð100 eVÞÞ where ξ ≡ Td=T is the ratio of temperatures of the dark to visible ξ ∼ 1 pffiffiffiffi sectors. If , then one evidently needs rather small (M ≡ 1= G ≈ 1.2 × 1019 GeV). At this maximum mass, Λ Pl values of , no larger than a few eV, to avoid over- the physical radius is only a factor of ∼2 larger than the production. However, there are important regimes in which corresponding Schwarzschild radius. For masses above the abundance can be modified due to self-interactions M , boson star solutions do not exist; the density is so 3 → 2 max leading to processes (this will be relevant to the rates high that such configurations can collapse to a black hole. we compute in stars later). For aspects of glueball and self- Interestingly, if λ4 ¼ Oð1Þ, then this scaling is similar to the – interacting dark matter relic abundances, see Refs. [18 25]. Chandrasekhar mass for white dwarf stars M ∼ M3 =m2; On the other hand, if the temperature of the hidden sector wd Pl p one is effectively replacing a repulsion from Pauli exclu- is small (ξ ≪ 1), then one can have larger strong coupling sion of fermions by a self-interaction repulsion of bosons. scales without overclosing the universe. As some of us Then if mϕ ≲ GeV, this can be of the order of a solar mass showed in Ref. [26], there is a very reasonable scenario in ∼ 1057 which the inflaton φ decays predominantly to the Standard Msun GeV, or larger. Model through the dimension three coupling to the Higgs In the literature, this result has been applied to boson stars φH†H, while its decays to hidden sector Yang-Mills is from real scalars, including glueballs and elementary sca- suppressed as it would occur through the dimension five lars. It is not clear that glueballs have the required repulsive μν interaction [and in the case of the SUð2Þ gauge group, some coupling φGμνG . This leads to the expectation ξ ≪ 1 recent lattice calculations suggest it may, in fact, be attractive (such as ξ ≈ 0.006 for reasonable parameters considered [36,37]]. Since we do not know the sign for a generic gauge in Ref. [26]). group, we will simply work under the assumption that it may Furthermore, glueballs exhibit scattering in galaxies. be repulsive for some cases, and this is, of course, certainly The scattering cross section to mass ratio is expected to 3 4 possible for scalars in other kinds of theories. To apply the be on the order σϕ=mϕ ∼ few=ðΛ N Þ [27]. While con- 2 result to glueballs, one estimates the quartic coupling as straints from the bullet cluster imply σϕ=mϕ ≲ cm =g ≈ 2 2 λ4 ∼ ð4πÞ =N and mϕ ∼ fewΛ. Then using the above 1=ð60 MeVÞ3 [28,29], to satisfy this bound one needs bound on Λ ≳ Λ in order to satisfy bullet bluster con- Λ ≳ Λ Λ ∼ 100 =N4=3 bc bc where bc MeV . This in turn implies straints, one finds that the maximum mass can be as large that Ωϕ is large unless ξ ≪ 1, which is compatible with the as M ∼ M3 N5=3=ð100 MeVÞ2 ∼ 100M N5=3. reasoning in Ref. [26] (although for extremely high N, one max Pl sun Hence one can readily have boson star masses that are would need to compensate with extremely small ξ, which larger than a solar mass by choosing the strong coupling seems less plausible). Or alternatively, if 3 → 2 processes scale accordingly. Alternatively, if one is simply studying are significant, this can reduce the abundance further. elementary scalars, one can just impose the appropriate If we move beyond the glueball motivation, we can, in values of mϕ and λ4 to obtain such massive stars and fact, consider any dark matter candidate that organizes into satisfying bullet cluster constraints. massive scalars with self-interactions. As an example, we Very interestingly, Refs. [27,38–40] pointed out that if could just study an elementary scalar with a renormalizable one has M ≳ Oð10ÞM and if these boson stars potential. Generally, these are all interesting candidates that max sun undergo a merger, they can emit gravitational waves with are the focus of this study. a frequency and amplitude that could possibly be detectable at LIGO, and even heavier stars at LISA, with a signal that A. Novel signature could be potentially distinguished from that of black hole An interesting possibility is that the dark matter scalars mergers (for a focus on complex scalars, see Refs. [41,42]). organize into gravitationally bound systems, so-called This presents a novel signature of these hidden sectors.
023536-2 DECAY OF BOSON STARS WITH APPLICATION TO … PHYS. REV. D 103, 023536 (2021)
This is such an exciting possibility, it acted as a motivation II. BOSON STARS for the present work. When glueballs form, they organize into spin 0 scalar particles. In the effective field theory formalism, we can B. Outline of this work organize this into a scalar field ϕ. These scalars interact In this work, we critically examine whether such stars directly with one another in the effective theory, which is a are, in fact, long-lived. While a complex scalar with an consequence of the microscopic hidden gluon interactions. internal Uð1Þ symmetry, of the sort studied in Ref. [34], In this work, the most important form of the interactions carries a conserved particle number, the same is not true for will be the leading order operators as organized into a scalar a real scalar. In this case there is no distinction between the potential V. We can expand this as particle and its antiparticle. Such real scalars can undergo λ quantum annihilation processes in the core of the star, 1 2 2 3mϕ 3 λ4 4 λ5 5 VðϕÞ¼ mϕϕ þ ϕ þ ϕ þ ϕ þ ; ð2Þ including 3ϕ → 2ϕ and 4ϕ → 2ϕ processes. Such proc- 2 3! 4! 5!mϕ esses will cause the star to evaporate away. In this paper we compute these annihilation rates for the boson stars, finding where λn are dimensionless couplings. There could be a that while the processes are negligible for very low mass tower of higher dimension operators, including higher stars, they are extremely important for high mass stars order derivative operators in the effective theory, but these including those above a solar mass. We find that this leads leading terms will be sufficient to describe the most to short lifetimes, unless the couplings are very small relevant effects in this work. This formalism is obviously (associated with huge gauge groups for glueballs). also applicable to other scalars, including elementary However, even for small couplings, the stars whose mass scalars with a renormalizable potential. So our analysis would be relevant to LIGO/LISA typically have low will be quite general. In the case of glueballs formed from compactness and so the gravitational wave emission would the group SUðNÞ, the couplings are expected to scale as n−2 be suppressed (unless one takes extreme parameters). λn ∼ ð4π=NÞ , with n ¼ 3; 4; …. For a generic real scalar Furthermore, we estimate possible decays from parametric the same hierarchy of couplings may occur too resonance, finding that this cuts down the available parameter space considerably further. ðn−2Þ=2 λn ∼ λ ðλ ≡ λ4Þ: ð3Þ One might hope that the star carries enough binding energy to prevent these number changing processes. We But we do not suppose that the prefactors are necessarily find that even though the most massive stars have appreci- close to 1. In the case of a real scalar, one can also imagine able binding energy, it is not sufficient to prevent the that it is endowed with some discrete Z2 symmetry radiation. Finally, for completeness, we also compute 3ϕ → ϕ → −ϕ, forbidding all the odd powers. We shall consider 1ϕ processes, which had previously been claimed to be the this case in this work also. most important process in the work of Ref. [43] (in the Of course, the field is also coupled to gravity with action context of axions). We show that the calculations provided (we use units c ¼ ℏ ¼ 1 and signature − þþþ) in that work, while appearing as a quantum mechanical Z process, are, in fact, properly captured by classical field 4 pffiffiffiffiffiffi R 1 μν S ¼ d x −g − g ∂μϕ∂νϕ − VðϕÞ ; ð4Þ theory when the rescaling to the final decay rate of the 16πG 2 condensate is obtained, and we discuss the prefactor. For the massive stars of interest, these 3ϕ → 1ϕ classical where R is the Ricci scalar, G is Newton’s gravitational 4ϕ → processes are found to typically be small, while the constant, and gμν is the metric of spacetime. 2ϕ or 3ϕ → 2ϕ quantum processes that we focus on can be very important. Our paper is organized as follows: In Sec. II we present the A. Spherical symmetry basic effective field theory, show the corresponding classical Under some conditions, the scalars can condense into equations of motion for a spherically symmetric star, and states of high occupancy, which is usually well described recap the properties of stars in a single harmonic approxi- by classical field theory (we shall return to quantum mation. In Sec. III we compute the quantum annihilation corrections in the next section). A condensate for a system rates in the core of the stars. In Sec. IV we use these results to with gravity is a localized system: either a gravitationally derive bounds on the mass and compactness. In Sec. V we bound “boson star” or a potentially bound (for attractive also consider parametric resonance. In Sec. VI we estimate interactions) “oscillon.” This work will focus on repulsive whether the star’s binding energy can prevent decays. In interactions mainly (though some of our reasoning will be Sec. VII we compute the classical radiation and compare to relevant for attractive interactions, too), so the only relevant existing claims in the literature. In Sec. VIII we briefly solution is the gravitationally bound boson star. The lowest discuss decays in boson stars supported by quantum pres- energy configurations are expected to be spherically sure. Finally, in Sec. IX we conclude. symmetric.
023536-3 HERTZBERG, ROMPINEVE, and YANG PHYS. REV. D 103, 023536 (2021)
In this case, it is useful to go to spherical coordinates Substituting in the above metric and carrying out the ðr; θ; φÞ, where quantities only depend on radius r (as well derivatives leads to as time t). Any spherically symmetric spacetime metric can be written in the Schwarzschild coordinates 2 B0 A0 A A_ B_ ϕ00 þ þ − ϕ0 − ϕ̈þ − ϕ_ ¼AV0: ð12Þ r 2B 2A B 2A 2B ds2 ¼ −Bðr;tÞdt2 þ Aðr;tÞdr2 þ r2ðdθ2 þ sin2 θdφ2Þ; ð5Þ There is also a third Einstein equation from the time-space _ where grr ¼ A and gtt ¼ B are functions that can depend on components, which arises from Gtr ¼ A=ðrAÞ and radius and time that we need to solve for; we also need to _ 0 Ttr ¼ ϕϕ . However, this is redundant with the above set solve for the scalar field ϕðr; tÞ. of equations and so is not needed.
B. Einstein-Klein-Gordon equations C. Single harmonic approximation The Einstein equations Gμν ¼ 8πGTμν follow from In general a solution for a real scalar field will have a ϕ varying the action S above with respect to gμν and extrem- complicated time dependence. A boson star or oscillon izing. Here Gμν is the Einstein tensor and Tμν is the energy- is a periodic solution of the classical equations of motion. It momentum tensor provided by the scalar field ϕ. Because can therefore be expanded in a harmonic expansion of spherical symmetry, we only need to report on the time X ϕ Φ ω and radius components of the Einstein equations. The ðr; tÞ¼ nðrÞ cosð ntÞð13Þ Einstein tensor can be shown to be n¼1 A0 1 1 (this solution exists up to exponentially small corrections, G ¼ B þ 1 − ; ð6Þ which we will return to in Sec. VII). For small amplitude tt rA2 r2 A stars one can be sure that this expansion is dominated by the first harmonic with ω close to, but slightly less than, the B0 1 1 − 1 − mass mϕ. For very large amplitude stars, especially those Grr ¼ A 2 : ð7Þ rAB r A that are at masses approaching the maximum mass, higher order terms can become more important. By inserting this Here we use a notation in which a prime means a derivative into the above equations of motion, along with a similar with respect to radius r and a dot means a derivative with harmonic expansion for metric components Aðr; tÞ and respect to time t. Bðr; tÞ, one obtains an infinite tower of coupled nonlinear The energy momentum tensor arises from varying the ordinary differential equations for the prefactors ΦnðrÞ; scalar field contribution of S with respect to the metric AnðrÞ;BnðrÞ. In principle, one can numerically solve this complicated system. Interesting work on this was done in 4 Tμν ¼ ∂μϕ∂νϕ þ gμνLϕ: ð8Þ Refs. [44,45] for the ϕ potential, where the tower of harmonics was constructed and numerical investigations The time and radius components are readily found to be were done. However, we leave a full treatment of all this for future work. 1 2 B 2 A useful approximation is to assume that the system T ¼ ϕ_ þ ðϕ0Þ þ BV; ð9Þ tt 2 2A is dominated by only one harmonic of frequency ω.We write A 1 ϕ_ 2 ϕ0 2 − pffiffiffi Trr ¼ 2 þ 2 ð Þ AV: ð10Þ ϕ ≈ 2Φ ω B ðr; tÞ ðrÞ cosð tÞð14Þ pffiffiffi Combining Eqs. (6) and (7) with Eqs. (9) and (10) we have (the factor of 2 is for convenience). This is accurate at a pair of Einstein equations for A and B. Note that these small amplitudes, as it is dominated by the fundamental in equations only involve spatial derivatives for A and B, that case anyhow, and provides a rough estimate of the which reflects the fact that these are constrained variables; behavior at high amplitudes, too. Of course, by inserting there are no dynamical (tensor) components of the metric this into the equations of motion it will not satisfy them due to spherical symmetry. precisely as it will generate higher harmonics. So the idea Finally, we need the equation of motion for the scalar of the single harmonic approximation is to time average the ϕ field . This comes from varying the action S with respect equations of motion over a period of oscillation T ¼ 2π=ω. ϕ to , giving Self-consistently then, we need to approximate the ffiffiffiffiffiffi ffiffiffiffiffiffi metric as time independent, writing Aðr; tÞ ≈ A¯ ðrÞ and ∂ p− μν∂ ϕ p− 0 μð gg μ Þ¼ gV : ð11Þ Bðr; tÞ ≈ B¯ ðrÞ. On the right-hand side of the Einstein
023536-4 DECAY OF BOSON STARS WITH APPLICATION TO … PHYS. REV. D 103, 023536 (2021) equations we need the time average of the energy momen- A¯ → 1; B¯ → 1; Φ → 0 as r → ∞ and Φ0 → 0 as r → 0. The tum tensor, which is ground state solution is identified by having no nodes. One can readily integrate up the hGtti¼8πGhTtti 1 B ¯ ω2Φ2 Φ0 2 equation to solve for A in terms of the enclosed mass hTtti¼2 þ 2 ð Þ þ BhVi; ð15Þ A MencðrÞ as 1 A 2 2 2 2 −1 T ω Φ Φ0 − A V : 16 GMencðrÞ h rri¼2 þ 2 ð Þ h i ð Þ A¯ ðrÞ¼ 1 − : ð21Þ B r In addition, a useful way to handle the Klein-Gordon equation pffiffiffi The enclosed mass is just defined as the (weighted) integral in Eq. (12) is to multiply throughout by 2 cosðωtÞ and then of the energy density time average. This gives Z r 2 0 0 pffiffiffi 4π 0 02 0 ¯ 0 00 B A 0 A 2 0 MencðrÞ¼ dr r hTttðr Þi=Bðr Þ: ð22Þ Φ þ þ − Φ þ ω Φ ¼ 2AhcosðωtÞV i: ð17Þ 0 r 2B 2A B The corresponding mass of the star, or total energy, is The time average of the potential V is simple if the potential has only even powers of ϕ. If there are odd powers ∞ 3 M ¼ Mencð Þ: ð23Þ of ϕ, such as ∼λ3mϕϕ , then the averaging is more subtle. Naively one just time averages those terms to zero. But, in Note that A¯ → 1 as r → 0. However, the value of B¯ or Φ fact, the presence of those terms means that the field as r → 0 is not specified uniquely. Let us call them ¯ ¯ undergoes asymmetric oscillations; it spends more time on B0 ≡ Bð0Þ and Φ0 ≡ Φð0Þ, respectively. Their values are one side of the potential than the other. A proper treatment related to the value of ω. It is useful to trade in B¯ for another ϕ of this would include a time independent term in , dimensionless function as providing a nonzero mean hϕi ≠ 0 (e.g., see the asym- metric terms in the study of oscillons in Ref. [46]). ω2 β ≡ However, we can instead pass to the low momentum ðrÞ 2 : ð24Þ m B¯ ðrÞ effective theory; this is valid since we will only be studying ϕ boson star solutions whose size is much larger than the By scanning different values of β0 ≡ βð0Þ, or equivalently inverse mass of the scalar mϕ. Here one integrates out Φ ϕ ∼λ ϕ3 by scanning different values for 0, and numerically intermediate exchange processes from 3mϕ . This solving the equations, one finds a range of boson star ∼λϕ4 replaces interactions by only contact interactions; i.e., solutions. An example is given in Fig. 1 (orange curve) for at large distances the interaction between scalars acts a kind of delta-function interaction. In the low momentum effec- 0.030 tive theory this leads simply to a shift in the effective quartic coupling, which one can readily show is (e.g., see 0.025 Ref. [47]) Exact Solution for g=100 5 0.020 λ λ − λ2 Large g Approximation 4eff ¼ 4 3 3: ð18Þ 0.015 λ 0 In the effective theory one needs 4eff > in order to have the repulsive interaction needed to support the stars of 0.010 interest in this work (stars supported by quantum pressure are discussed briefly in Sec. VIII). The corresponding time 0.005 averages are then (we truncate the potential to quartic order here for simplicity) 0.000 1 λ 0 5 10 15 20 25 30 2 Φ2 4eff Φ4 hVi¼2 mϕ þ 16 ; ð19Þ pffiffiffi λ 0 2 4eff 3 FIG. 1. Exact versus large g approximation profile. Boson star 2hcosðωtÞV i¼mϕΦ þ Φ : ð20Þ 4 profile ΦðrÞ for the core value Φ0 ≈ 0.028MPl and parameter g ¼ 100. The orange is the exact numerical solution, while the Bound state solutions are found by numerically search- purple is from scaling the large g approximation (this is similar in ing for configurations that obey the boundary conditions form to a plot in Ref. [34]).
023536-5 HERTZBERG, ROMPINEVE, and YANG PHYS. REV. D 103, 023536 (2021) the choice Φ0 ≈ 0.028M , where we have defined the where pffiffiffiffiPl ≡ 1 Planck mass MPl = G. Z rˆ 1 1 ˆ ˆ0 ˆ02 β 1 Φˆ 2 Φˆ 4 IðrÞ¼ dr r 2 ð þ Þ þ 4 : ð31Þ D. Large coupling regime 0 As shown in Ref. [34], the structure of the solutions is In these dimensionless variables, one can anticipate controlled by the following dimensionless parameter: that the maximum value of the integral here, when integrating over the whole star, is I O 1 (in addition 2 max ¼ ð Þ λ4 M ≡ eff Pl to a family of lighter stars with I ≪ 1). In fact, a precise g 2 : ð25Þ ≈ 0 2 16πmϕ calculation reveals that Imax . . Thus when multiplying by the prefactor inffiffiffiffiffiffiffiffi Eq. (30) gives the maximum star Λ Λ ≈ 0 03pλ 3 2 (Note: our g is what Ref. [34] calls , but we have used as mass Mmax . 4effMPl=mϕ (see ahead to Fig. 7), λ 4 the strong coupling scale, and our 4eff= is effectively consistent with our earlier discussion in the Introduction. λ g ∼ playing the role of in Ref. [34].) If is small, then the self- The corresponding radius of the star is R few × GMmax; interactions are negligible and the solution is just provided i.e., it is comparable to, though a little larger than, the by gravity balanced by quantum pressure of the bosons Schwarzschild radius. (which is classical pressure within the field formalism); we The Klein-Gordonpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi equation (29) can now be trivially shall study this in Sec. VIII.However,ifg is large, then the solved, Φˆ ðrÞ¼ βðrÞ − 1. This evidently only makes behavior changes: there exist massive solutions in which sense in the regime β > 1.Forβ < 1 the above approx- gravity is balanced by the repulsive self-interaction, which imations break down. This is because even though g is will be the main focus of this work (in addition, there always large, the validity of this scaling rests upon the assumption exist very light solutions in which one can once again ignore that Φˆ remains appreciable, say Oð1Þ. However, at a large self-interactions, but they are less interesting to us here). For radius the exact solution has an exponentially small tail, as ⋘ λ glueballs, one anticipates mϕ MPl and 4eff a parameter seen in Fig. 1 (orange curve), and so this assumption no λ ∼ 4π 2 that is not especially small, with 4eff ð =NÞ . If we take longer holds. Instead, this approximation is accurate in the 40 N ¼ Oð2Þ and mϕ ∼ 0.1 GeV, then g ∼ 10 ; i.e., it is bulk of the star. One can easily solve these approximate extremely large. equations to compare. Figure 1 (purple curve) shows a plot In this large g regime, one can further simplify the for g ¼ 100, where the approximation is seen to be some- equations of motion (as pointed out in Ref. [34]). To make what accurate in the bulk. However, as mentioned above, it manifest which terms are important and which terms can we are mainly interested in extremely large values of g,in be ignored at large g, it is useful to pass to the dimension- which the approximation becomes extremely accurate less variables (away from the exponentially suppressed tail). ’ ffiffiffi pffiffiffiffiffiffiffiffi Furthermore, we can compare the star s total mass M in ˆ ≡ p Φˆ ≡ Φ 4π r mϕr= g; g=MPl: ð26Þ the exact versus approximate methods as a function of the dimensionless parameter g. We have fixed β0 ¼ 1.585 and By rewriting the above Einstein-Klein-Gordon equations in numerically obtained the exact equations as well as the terms of these variables and rescaled metric component β approximate result. The ratio of the masses is plotted as a → and then neglecting all terms that are subdominant in the function of g in Fig. 2. Note that the ratio Mapprox=Mexact g → ∞ limit, one obtains the following simplified form of 1 as g increases, as expected. For very large g it becomes the equations: prohibitively difficult to solve the exact equations. This is because at fixed Φ0, one has to search for the corresponding ¯ ˆ A0 1 1 1 value of β0 to obtain a zero node solution that asymptotes to þ 1 − ¼ðβ þ 1ÞΦˆ 2 þ Φˆ 4; ð27Þ rˆA¯ 2 rˆ2 A¯ 2 a flat space at large radii. This requires continually increasing precision. So instead bypassing to the large g β0 1 1 1 equations we can readily make progress in this regime. − − 1 − ¼ðβ − 1ÞΦˆ 2 − Φˆ 4; ð28Þ rˆ A¯ β rˆ2 A¯ 2 III. QUANTUM ANNIHILATION IN CORES βΦˆ ¼ Φˆ þ Φˆ 3: ð29Þ The above analysis is all rigorous in the case of a complex scalar field ψ with a global Uð1Þ symmetry. In that One can readily integrate up the equation for A¯, as above, to case the above solution is closely related to an exactly obtain the enclosed mass in this limit as harmonic solution by writing ψðr; tÞ¼ΦðrÞe−iωt. This ffiffiffiffiffiffiffiffi p 3 corresponds to a boson star made out of a collection of λ4 M M rˆ effffiffiffi Pl I rˆ ; particles (or antiparticles). The particles are then stable encð Þ¼ 4pπ 2 ð Þ ð30Þ mϕ against annihilation into one another due to the symmetry.
023536-6 DECAY OF BOSON STARS WITH APPLICATION TO … PHYS. REV. D 103, 023536 (2021)
linearize them. At small couplings, one can then further 1 obtain solutions for δϕˆ working perturbatively. One assumes that the operator begins in the Minkowski vacuum exact and then evolves the system from there. Since the equations M / ϕ 0.5 are linear, this is doable in principle. If the background were homogeneous, the perturbations would easily be approx
M diagonalized in k space, leading to standard Floquet theory. ϕ However, since our background of interest (along with the metric components A and B) depend on space, all the k 0.2 modes are coupled to one another. This makes the analysis nontrivial, even though the system is linear, but can be Mass Ratio formulated as a generalized type of Floquet theory. 0.1 This was all studied by some of us systematically in Refs. [47,50]; other work appears in Refs. [51–57]. The 0.1 1 10 100 important result was that when one is in a small coupling Parameter g regime, the final resonance matches the result from standard perturbation theory of particles annihilating in FIG. 2. Exact versus large g approximation mass. Ratio of the vacuum; i.e., the Bose enhancement is shut off for mass of a boson star using the large g approximation scheme to sufficiently small coupling. As shown in that work, the the exact mass using the full equations (albeit always in the single requirement for this is that the maximum parametric harmonic approximation) with β0 ¼ 1.585. Note that the ratio μ approaches 1 at large g. resonance Floquet rate from a homogeneous oscillating 1 condensate is smaller than the inverse size of the star =R . The reason for this is that in this regime the resonantly One may not expect global symmetries to be exact, but we produced scalar particles escape the condensate before put aside the discussion of this issue here. For the present Bose-Einstein statistics can be effective. The regime of ϕ purposes of a glueball or single elementary scalar , the parametric resonance will be studied next in Sec. V. ϕ boson star is made of particles which are their own In any case, the perturbative decay rates normally act as a antiparticles. Their stability against annihilations is there- lower bound on the true decay rate. The only reasonable fore not protected by any symmetry. Since the above very ways they could be shut off is (i) if the decay products were λ ϕn dense stars exist due to the self-interactions n , one fermions, then there is the issue of Pauli blocking [58] should be concerned that these same interactions may lead (however, this is irrelevant here since our decay products ’ to rapid annihilation in the star s core. Previous work on are the bosons ϕ themselves); or (ii) if the star carried this topic has often ignored these details. A notable sufficient binding energy to make the process kinematically exception is the work of Ref. [43], which studied 3ϕ → 1ϕ impossible. This is not an issue for dilute stars which carry annihilations (with a focus on axion stars); we very small binding energy, while highly compact stars will shall return to a discussion of that work in Sec. VII. be addressed in Sec. VI. These perturbative rates can be readily calculated. The A. Perturbative rates boson star (at least away from very high compactness) can To discuss this issue, suppose we had the full classical be viewed as a condensate of nonrelativistic particles. Their field boson star solution, including its full time dependence. large de Broglie wavelengths mean that they overlap with As we will return to in Sec. VII, this can never be exact due one another and can therefore annihilate through the to outgoing classical radiation, but that turns out to be contact interactions of the potential. This leads to the star typically very small. On the other hand, at any order in the emitting energy in the form of pairs of scalar particles. As harmonic expansion, one can find a periodic solution that shown in Ref. [47] the normalized perturbative rates for ϕ Γ we can refer to as the classical star solution ðr; tÞ; that energy output d ¼jdE =dtj=E (where E ¼ M is the will suffice in this section. The approximate classicality can energy of the star) for 3ϕ → 2ϕ and 4ϕ → 2ϕ annihilation be justified by various considerations, including averaging processes are given by and decoherence at high occupancy [48,49]. To discuss the R inevitable quantum fluctuations on top of this, we can work 3 5=2 3 3 λ π k32 d xn ðxÞ in the Heisenberg picture and write the field operator as Γ 3ϕ → 2ϕ 32 R ; 33 dð Þ¼ 3! 224 Γ 3 2π 3 6 3 x ð Þ ð Þ ð2Þð Þ mϕ d xn ð Þ ϕˆ ðx;tÞ¼ϕ ðr; tÞþδϕˆ ðx;tÞ: ð32Þ R 4 5=2 3 4 λ π k42 d xn ðxÞ By treating the quantum fluctuations as small δϕˆ ≪ ϕ ,we Γ 4ϕ → 2ϕ 42 R ; 34 dð Þ¼ 4! 225 Γ 3 2π 3 9 3 x ð Þ can write down the Heisenberg equations of motion and ð Þ ð2Þð Þ mϕ d xn ð Þ
023536-7 HERTZBERG, ROMPINEVE, and YANG PHYS. REV. D 103, 023536 (2021) pffiffiffi pffiffiffi where k32 ≈ 5mϕ=2 and k42 ≈ 3mϕ are the momenta of The radius-mass relationship for the condensate arises the two outgoing particles in each process, respectively. Here from minimizing the energy from kinetic energy (pressure), x repulsive self-interaction, and gravitation. It can be shown n ð Þ is the number density of particles within the boson star, x ρ x this leads to [59–61] which can be approximated as n ð Þ¼ ð Þ=mϕ, where ρ x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ is the local energy/mass density. Here the coefficients 3 2 2 3 a a2 3bcM˜ 2 are given by λ32 ≡ ð5λ3ðλ3 þ 3λ4Þ=12 þ λ5Þ ∼ λ and ˜ þ þ 4 2 2 4 R ¼ ˜ ; ð36Þ λ42 ≡ ðλ4 − λ6Þ ∼ λ , where for completeness we have bM 6 2 included the sextic term λ6ϕ =ð6!mϕÞ in Eq. (34) (although where a, b, c are numerical constants that depend on the we did not include it earlier when discussing the profile of the choice of ansatz. For example, for the above sech ansatz star). We note that for special choices of couplings, such as 12 π2 6π2 6 12ζ 3 − π2 π4 λ2 λ they are a ¼ð þ Þ=ð Þ, b ¼ ð ð Þ Þ= , 4 ¼ 6, these rates can be shut off; but this is not the generic π2 − 6 8π5 λ2 λ and c ¼ð Þ=ð Þffiffiffiffiffiffiffiffi. Here we are using dimensionlessffiffiffiffiffiffiffiffi situation. Interestingly, this condition 4 ¼ 6 occurs when ˜ 2 p ˜ p variables R ≡ m R=ð λ4 M Þ and M ≡ λ4 M =M . one Taylor expands a cosine potential, which may describe ϕ eff Pl eff Pl ˜ ≫ 1 some kinds of axions; however, since it has attractive Notep thatffiffiffiffiffiffiffiffiffiffi at large M the radius approaches a constant ˜ interactions, it is not of direct relevance here. R → 3c=b, giving a star size of pffiffiffiffiffiffiffiffiffiffiffiffiffi d 3cλ M pffiffiffiffiffiffiffiffi B. Application to boson stars → ffiffiffi4eff Pl ≫ λ R p 2 M MPl= 4eff : ð37Þ ’ x bmϕ To evaluate these rates we need the star s density n ð Þ. It is difficult to provide exact analytical results as one needs This is as expected from the scalings of the full Einstein- to solve nonlinear differential equations to obtain the star’s Klein-Gordon equations in the previous section. profile ΦðrÞ; important work includes Refs. [59,60].To Let us focus on this asymptotic regime of the more proceed it is useful to use an approximate form for the massive stars, since they are of most interest astrophysi- shape of a star, which captures the following two basic cally. We can readily carry out the above integrals to obtain ideas: (i) it is flat near its core, i.e., Φ0ð0Þ¼0, and (ii) it the decay rates as falls off exponentially at large distances. For example, as used in Ref. [61], a useful representation that has these λ3 5 2 32mϕM properties and is somewhat accurate (at least for stars that Γ 3ϕ → 2ϕ α3 ; 38 dð Þ¼ λ3 6 ð Þ are not too compact) is a sech ansatz 4effMPl
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ4 m7 M3 Γ 4ϕ → 2ϕ α 42 ϕ 3M dð Þ¼ 4 ; ð39Þ Φ ≈ λ9=2 9 ðrÞ 3 2 3sechðr=RÞ; ð35Þ 4 MPl π mϕR
where α3 and α4 are Oð1Þ prefactors. The sech ansatz where R is a length scale that should be adjusted to estimates their values to be α3 ≈ 0.04 and α4 ≈ 0.05. Note Γ 4ϕ → 2ϕ minimize the energy to obtain the most accurate solution. that in the denominator of the dð Þ result we have 2 2 λ → λ The corresponding energy/mass density is ρ ðxÞ¼m jΦj , replaced 4eff 4 since it is relevant only if the cubic term ϕ ∼λ ϕ3 and the normalization ensures that the mass of the star is 3mϕ is negligible. λ ∼ 4π n−2 M . This analysis is quite accurate in the nonrelativistic Recall that for glueballs we anticipate n ð =NÞ . λ3 λ3 O 1 Γ 3ϕ → 2ϕ regime of low compactness stars, as shown in Ref. [61].For So 32= 4eff ¼ ð Þ in the expression for dð Þ, compact stars, they involve orbital boson speeds that are making it essentially independentffiffiffi of the size of the gauge 4 9=2 p Oð1=2Þ that of light. In that regime these estimates are group. While λ42=λ4 ∼ 1= λ ∼ N=ð4πÞ in the expression Γ 4ϕ → 2ϕ anticipated to still be correct to within a factor of a few, for dð Þ, making it increase linearly with the size which suffices for our main results. of the gauge group (if other parameters are kept fixed). Star solutions involve a relationship between the total mass/energy of the star M and its physical radius R .A IV. ASTROPHYSICAL BOUNDS precise definition of the physical radius of a star is the region that contains a fixed percentage of the mass of the If the above annihilation rates (38) and (39) are much star. For definiteness we will take this to be the radius that bigger than the current Hubble rate H0 of the universe, encloses 90% of the mass. This turns out to be related to the then the stars are unlikely to be cosmologically relevant. scale that appears in the argument of the above sech profile As an example, consider the case with λ ¼ Oð1Þ and ≈ ≈ 2 8 ∼ 0 1 by R dR, with d . for the sech profile. However, an mϕ . GeV, giving rise to Mmax of the order of 10s Oð1Þ change in this definition will not be important for our of solar masses. Then, in order for the decay rates main results. Eqs. (38) and (39) to be smaller than today’s Hubble rate
023536-8 DECAY OF BOSON STARS WITH APPLICATION TO … PHYS. REV. D 103, 023536 (2021)
−33 H0 ≈ 10 eV, we obtain the following bound on the ≲ 10−18 3ϕ → 2ϕ mass of the star: M Msun if processes are ≲ 10−11 4ϕ → 2ϕ active and M Msun if processes are active. Such stars have very low compactness and are no- where close to being relevant to the LIGO/LISA bands. This already undermines the claims of Refs. [27,38–40],as well as other work on real scalars, including Ref. [62], since the heavy stars of these analyses would be too short-lived. A much more general exploration of the constrained parameter space is provided in Figs. 3–6, where we have included detailed information in their captions. The decay Γ rates d are plotted versus star mass M in Figs. 3 and 4, with the coupling λ ¼ð2πÞ2 fixed in Fig. 3, and the scattering cross section fixed in Fig. 4 (see the next subsection for an explanation). Then in Figs. 5 and 6 we ’ Γ fix the decay rate to be today s Hubble rate d ¼ H0, with contours of fixed mass M indicated; these provided upper bounds on the allowed mass for the star to live to present time. We also indicated the compactness (see the upcoming subsection for an explanation). In the remainder of this section and the next section we would like to provide more details on the ingredients that have gone into these figures.
A. Scattering in galaxies Apart from the boson stars, one expects there to be a large collection of diffuse ϕ particles acting as a form of dark matter. Beccause of the above self-interactions they will undergo scattering in the galaxy. The 2ϕ → 2ϕ scattering cross section for nonrelativistic particles is readily obtained as
λ2 σ 4eff 2→2 ¼ 2 : ð40Þ 128πmϕ FIG. 3. Fixing coupling λ to its maximum value. Decay Γ −1 rates d (in units Gyr ) of boson stars as a function of the ’ If the scalar particles ϕ make up a significant fraction, or star smassM (units Msun) for different choices of particle all, of the dark matter, then there are bounds on this mass mϕ (solid red lines) and compactness C (dotted gray λ 2π 2 scattering. Some of the best bounds come from observa- lines). Here we have imposed the coupling ¼ð Þ ,which tions of collisions of galaxies, such as the bullet cluster, is on the order of the maximum allowed by unitarity (appropriate for small gauge groups). The black dashed line which are essentially consistent with noninteracting dark is the present Hubble rate H0. In the upper right orange region matter. This imposes an observational upper bound on the the stars would collapse to black holes. The lower right scattering cross section of [28,29] green shaded region is excluded by bullet cluster bounds. The σ yellow band is LISA, and the blue band is LIGO, which is 2→2 2 only appreciable at the top right where the stars are compact. ≲ ζu cm =g; ð41Þ mϕ The light blue shaded region indicates where parametric resonance may occur according to a simple analysis. Upper 3ϕ → 2ϕ ϕ where ζu is argued to be Oð1Þ, depending on the analysis. panel: processes when there are odd powers of in For concreteness, we will take the bound with ζ ¼ 1 in this V. Lower panel: 4ϕ → 2ϕ processes when there are only even u ϕ work. In Figs. 3, 5, and 6, we have imposed this bound, powers of in V. which rules out the green region. We add that if the scalar particles ϕ only make up a tiny fraction of the dark matter (yet there is still enough to provide some boson stars), then On the other hand, it has been argued that the cores of these bounds are significantly weakened; in this case the galaxies are explained precisely by the presence of such green region can largely be ignored. scattering [63]. To do so requires a lower bound on the
023536-9 HERTZBERG, ROMPINEVE, and YANG PHYS. REV. D 103, 023536 (2021)
FIG. 4. Fixing scattering to core-cusp preferred value. Γ −1 Decay rates d (in units Gyr )ofbosonstarsasafunction ’ of the star smassM (units Msun) for different choices of coupling λ ¼ð4π=NÞ2 (solid blue lines). Here we have 2 imposed the scattering cross section σ2→2=mϕ ¼ 1 cm =gto possibly explain the cores of galaxies. The black dashed line is the present Hubble rate H0. In the upper right orange region FIG. 5. Fixing perturbative decay rate to Hubble and the stars would collapse to black holes. In the upper left green exploring fmϕ; λg parameter space. Parameter space of region there are no stars, since once would need couplings solutions after imposing the decay rate is equal to the current λ ≫ 1 violating unitarity. The yellow band is LISA, and the Hubble rate Γ ¼ H0. The solid brown lines are contours blue band is LIGO, which is only appreciable at the top right d of fixed star mass M and the dotted black lines are contours where the stars are compact. The light blue shaded region of fixed compactness C. The lower right green shaded indicates where parametric resonance may occur according to 3ϕ → 2ϕ region is excluded by bullet cluster bounds. The lower left asimpleanalysis.Upperpanel: processes when shaded orange region would have compactness so high the there are odd powers of ϕ in V. Lower panel: 4ϕ → 2ϕ ϕ stars would collapse to black holes. The yellow band is processeswhenthereareonlyevenpowersof in V. LISA, and the blue band is LIGO, which is only appreciable at the lower left where the stars are compact. The light blue σ ≳ ζ 2 ζ shaded region indicates where parametric resonance may cross section of 2→2=mϕ l cm =g, where l has occur according to a simple analysis. Upper panel: 3ϕ → also argued to be Oð1Þ. Altogether this provides some 2ϕ ϕ 2 processes when there are odd powers of in V.Lower motivation to consider the case of σ2→2=mϕ ∼ cm =g, panel: 4ϕ → 2ϕ processes when there are only even powers although it is still controversial. (See Ref. [64] for a critical of ϕ in V.
023536-10 DECAY OF BOSON STARS WITH APPLICATION TO … PHYS. REV. D 103, 023536 (2021)
examination of using ultralight scalars to solve this prob- 2 lem.) We have fixed the scattering to be σ2→2=mϕ ∼ cm =g in Fig. 4. In that plot we are showing decay rate versus star λ mass M for different choices of the coupling . In the upper left region one would have λ ≫ 1, which is forbidden by unitarity. For concreteness we take λ ¼ð2πÞ2 as the upper value (although one could argue for even smaller values to be safer from unitarity considerations).
B. Compactness Of particular interest for the production of gravitational waves is the compactness C of a star. We shall define this as the ratio of the corresponding Schwarzschild radius 2 RS ¼ GM and the physical radius of the star R , R C ≡ S : ð42Þ R If the compactness is Oð1Þ, then it is undergoing strong → gravity in its vicinity. When M Mmax, as defined in Sec. II D, then we are indeed in this regime. The compact- ness there is roughly C ∼ 1=2, and mergers can produce gravitational waves with significant amplitudes. Of course, the compactness parameter cannot be larger than 1 or the system would have collapsed to a black hole. In Figs. 3–5 we have indicated this in the orange region (in Fig. 6 this occurs at much higher values of C than those displayed). On the other hand, the compactness can be small C ≪ 1, describing the dilute boson star. In this regime the gravitational wave signal from mergers is expected to be suppressed. We can consider a family of solutions that exist at some compactness C. For example, one may imagine that the stars will continue to accrete until they have achieved their maximum compactness C ∼ 1=2. For any particle mass mϕ we can consider this possibility. So we use Eqs. (37) and (42), eliminate mϕ in favor of C, and insert into the decay rate formulas to obtain
λ3 3=2 5=2 32MPl C Γ ð3ϕ → 2ϕÞ¼β3 pffiffiffiffiffi ; ð43Þ d λ7=4 4eff M
λ4 3=2 7=2 42MPl C FIG. 6. Fixing perturbative decay rate to Hubble and Γ ð4ϕ → 2ϕÞ¼β4 pffiffiffiffiffi ; ð44Þ λ d λ11=4 exploring fC; g parameter space. Parameter space of sol- 4 M utions after imposing the decay rate is equal to the current Γ Hubble rate d ¼ H0. The solid brown lines are contours of where β3, β4 are Oð1Þ prefactors. The sech ansatz estimates fixed star mass M . The upper right green shaded region is their values to be β3 ≈ 0.005, β4 ≈ 0.0001. Contours of excluded by bullet cluster bounds. In this figure we have not fixed compactness are provided in Fig. 3 for fixed coupling indicated the LISA or LIGO bands or the black hole regime and in Fig. 5 for fixed decay rate. since these are only important at high compactness, which is far off the top of the plot. The light blue shaded region C. Implications for gravitational waves indicates where parametric resonance may occur according to asimpleanalysis.Upperpanel:3ϕ → 2ϕ processes when Importantly, by fixing the decay rate to be Hubble, we ϕ 4ϕ → 2ϕ there are odd powers of in V.Lowerpanel: plot contours of maximum allowed star mass M in the processes when there are only even powers of ϕ in V. fC; λg plane in Fig. 6. This shows that for reasonable
023536-11 HERTZBERG, ROMPINEVE, and YANG PHYS. REV. D 103, 023536 (2021) couplings, and to obey the bullet cluster bound, the mass interested in is to imagine a star has formed, and we are and compactness need to be small for longevity. This has only interested in these annihilation processes inside its significant implications for gravitational waves from pos- core. These are particle number changing processes, are sible mergers of these stars. It implies that one is well mediated by self-interactions of the potential, and are not outside of both the LIGO and LISA bands. Gravitational mediated by gravity (one could consider resonance into wave detection requires Oð10–106Þ solar mass objects with gravitons but this is normally highly suppressed). Hence we high compactness. However, these results indicate that the can focus on flat space fluctuations for the purpose of this decay rates are too fast to achieve this. For completeness, discussion. The perturbed equation of motion is we have indicated both the LIGO and the LISA bands in δϕ̈ − ∇2δϕ 2 δϕ 00 ϕ δϕ 0 Figs. 3. In Figs. 3 and 4 the signals are only appreciable in þ mϕ þ VI ð 0ðtÞÞ ¼ ; ð47Þ the upper right-hand region, which involves rapid decay. While in Fig. 5, where the decay is fixed to Hubble, the where VI is the interaction potential. The advantage of signal is only appreciable in the lower left-hand region, studying this homogeneous condensate is that it can be involving extremely tiny self-couplings, and may be readily diagonalized by passing to Fourier space δϕ → δϕk 2 2 unlikely to persist due to the phenomenon of parametric and −∇ δϕ → k δϕk. This makes Eq. (47) a form of Hill’s resonance, which we turn to now. equation since it is a linear differential equation with a 00 ϕ periodically changing prefactor VI ð 0ðtÞÞ. V. PARAMETRIC RESONANCE A. Floquet exponents In addition to the quantum mechanical perturbative decays discussed above, one can also enter a regime in Let us begin with the case when both odd and even which the coherently oscillating boson star condensate powers in the potential are included. The most obvious λ ϕ3 3! λ ϕ4 4! drives the parametric resonance of its own field fluctua- version is the cubic term 3mϕ = and quartic 4 = . However to obtain the relevant resonance is slightly tions. Since the stars of interest are wideffiffiffi compared to the ∼ p ≫ 1 complicated (as can be appreciated by drawing all the inverse mass of the particle (R g=mϕ =mϕ), they are susceptible to parametric resonance. This would re- corresponding Feynman diagrams). So for the sake of present a type of instability against linear perturbations; simplicity, let us focus on an interaction provided by λ ϕ5 5! 3ϕ → 2ϕ differing views on this have been expressed in the literature VI ¼ 5 =ð mÞ, which provides with no inter- [65,66]. As mentioned earlier, the criteria for this to occur is mediate propagators. Although one should also anticipate the presence of cubic terms, the final result will have a μ 1 λ ∼ λðn−2Þ=2 R > ; ð45Þ similar scaling (assuming n ). To first approximation, the oscillating condensate is where μ is the maximum exponential growth rate (“Floquet mainly driven by the mass term. So the oscillations of exponent”) within the homogeneous background approxi- the background are mation. The intuition behind this is that when this inequal- ϕ ≈ ϕ ω ity is not satisfied, the produced particles escape the 0ðtÞ a cosð 0tÞ; ð48Þ condensate before Bose-Einstein statistics are effective. ω ≈ ϕ In any case, it was earlier established in Ref. [47] (also see where 0 mϕ and a is the amplitude of oscillations. By earlier work in the context of axion photons in inserting this back into Eq. (47) one obtains the interaction Refs. [51,52,57]), so we shall not repeat the derivation term as a pair of harmonics due to the factor here. Hence we do not need to consider the full compli- ϕ3 cations of expanding around the inhomogeneous star 00 3 a V ðϕ0ðtÞÞ ∝ ϕ ðtÞ¼ ð3 cosðω0tÞþcosð3ω0tÞÞ: ð49Þ background; we can focus on the corresponding homo- I 0 4 geneous configuration with amplitude matching the star’s δϕ core amplitude, obtain μ, and check on this inequality. One can then expand in harmonics, too. One can have Let us perturb around the background as resonance from long wavelength perturbations, which can be driven by the leading harmonic term. However, as mentioned above this is not important for us. We know that ϕðx;tÞ¼ϕ0ðtÞþδϕðx;tÞ: ð46Þ this only leads to the destabilization of the homogeneous Here one should, in principle, also allow for fluctuations in condensate toward a boson star, etc. Instead, we are the metric. However, the metric fluctuations are primarily interested in the possible resonance at the higher harmonic 3ω ≈ 3 δϕ ∝ i3ω0=2 only important to describe long wavelength perturbations 0 mϕ. This is potentially resonant for k e , around the homogeneous background. This leads very since when we insert this into the equation of motion, importantly to the collapse of homogeneous structure the driving term will have a frequency that matches the and is the source of structure formation, and ultimately input frequency.ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi In turn this matches the natural frequency p 2 2 to the formation of boson stars, etc. However, what we are for ωk ≡ k þ m ¼ 3ω0=2 ≈ 3mϕ=2, which means
023536-12 DECAY OF BOSON STARS WITH APPLICATION TO … PHYS. REV. D 103, 023536 (2021) pffiffiffi k ≈ k32 ≈ 5mϕ=2 is the outgoing wave number men- B. Application to boson stars tioned earlier in our perturbative analysis in Eq. (33). We now would like to apply these results to the boson Hence in the vicinity of this resonance of interest, we can star. First, we need to relate ϕa to the properties of the star. μ 1 write the equation of motion as The idea of the criteria R > for resonance is that one replaces the amplitude of the homogeneous configuration λ ϕ3 ϕ δϕ̈ ω2δϕ 5 a 3ω δϕ a by the corresponding amplitude at the core of the k þ k k þ cosð 0tÞ k ¼ n:r; ð50Þ 4 · 3!mϕ star, i.e., sffiffiffiffiffiffiffiffiffiffiffiffi “ ” ffiffiffi 2 where n.r refers to the nonresonant term from the p M ϕ → 2Φ0 ¼ f ; ð55Þ cos ω0t in Eq. (49). Ignoring the nonresonant piece, this a 2 3 ð Þ mϕR is a form of the Mathieu equation O 1 2 where f is yet anotherpffiffiffiffiffiffiffiffiffiffiffiffiffiffið Þ prefactor; in the sech ansatz it d δϕ 2 2τ 0 is given by f ¼ 3d3=π3 ≈ 1.46. Hence the product of τ2 k þðAk þ B cosð ÞÞ ¼ ; ð51Þ d interest is found to be which is known to possess exponential growth in λ3=2 2 3=2 some band of wave numbers. Here we can identify 32 mϕM μ3→2R ¼ γ3 ; ð56Þ ω2 3ω 2 2 λ ϕ3 8 3! 3ω 2 2 λ7=4 7=2 Ak ¼ k=ð 0= Þ , B ¼ 5 a=ð · mϕÞ=ð 0= Þ .For 4effMPl small amplitudes, the Floquet exponent is known to be [67] λ2 m3 M2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μ γ 42 ϕ 3ω 4→2R ¼ 4 5 2 ; ð57Þ μ 0 2 − − 1 2 λ = M5 k ¼ 4 B ðAk Þ : ð52Þ 4 Pl pffiffiffi 3 7=4 3=4 7=4 7=2 4 5=2 where γ3 f b = 108 23 c d and γ4 f b = As we scan over different k values this is clearly maximal pffiffiffi¼ ð Þ ¼ 3456 3 5 5=2 when A ¼ 1, i.e., for ω ¼ 3ω0=2 as expected. This gives ð d c Þ. In the sech ansatz they are given by k k γ ≈ 2 1 γ ≈ 1 8 the maximum Floquet exponent of (using ω0 ≈ mϕ) 3 . and 4 . . The region in which the inequality for parametric μ 1 λ ϕ3 λ3=2ϕ3 resonance is obeyed R > is indicated by the light j 5j a 32 a – μ3→2 → ; blue region in Figs. 3 6. We see that it can constrain the ¼ 144 2 144 2 ð53Þ mϕ mϕ allowed parameter space considerably. However, its scaling is different from the perturbative decays, so it is often where we have indicated in the final step that when we complementary. μ include the cubic and quartic couplings, we can generalize If we now eliminate the particle mass mϕ to rewrite R 2 3=2 the result to jλ5j → jλ5 þ 5λ3ðλ3 þ 3λ4Þ=12j¼λ32 , since in terms of the compactness C, as we did in Sec. IV B,we we know this arises from the scattering amplitudes. obtain If there are no odd powers of ϕ in the potential, we can ffiffiffiffiffiffiffi λ3=2 p still have parametric resonance from the quartic term μ δ 32 C M 4 3→2R ¼ 3 5 4 pffiffiffiffiffiffiffiffi ; ð58Þ V ¼ λ4ϕ =4!. In this case the analysis is rather more λ = I 4 MPl complicated. This can be seen by the fact that there are now eff pffiffiffiffiffiffiffi several Feynman diagrams associated with the 4ϕ → 2ϕ λ2 3=2 42C M μ4→2R ¼ δ4 ffiffiffiffiffiffiffiffi ; ð59Þ process. The full details of this analysis was carried out by λ7=4p some of us in Ref. [67]. A simpler calculation arises from a 4 MPl 6 2 sextic term VI ¼ λ6ϕ =ð6!mϕÞ because it occurs through where δ3, δ4 are Oð1Þ prefactors. The sech ansatz estimates a process without any intermediate processes. For this their values to be δ3 ≈ 0.4, δ4 ≈ 0.1. This result suggests we can again follow the above analysis, this time expanding something important: By imposing the bound on mass 00 ϕ ∝ ϕ4 pffiffiffiffiffiffiffiffi VI ð 0ðtÞÞ 0ðtÞ in harmonics and focusing on the M ≫ M = λ4 , so that we are in the self-interaction 4ω Pl eff cosð 0tÞ term. The result is supported regime (as opposed to the quantum pressure supported regime), and using λ ∼ λðn−2Þ=2, we have λ ϕ4 λ2 ϕ4 n j 6j a 42 a μ ≫ μ ≫ 3=2 μ4→2 → ; 3→2R C and 4→2R C . This means that if one ¼ 1536 3 1536 3 ð54Þ mϕ mϕ considers stars of large compactness, say C ∼ 1=2, then the inequality Eq. (45) is satisfied and parametric resonance is where we have indicated that we can generalize this to expected to occur. This suggests very compact stars that 2 2 λ6 → jλ6 − λ4j¼λ42, since we know this is the relevant could undergo mergers and be relevant to LIGO/LISA contribution from both processes. are likely to be quite unstable to parametric resonance.
023536-13 HERTZBERG, ROMPINEVE, and YANG PHYS. REV. D 103, 023536 (2021)
However, since we have not solved for the time dependence fully, we cannot be certain of this conclusion. This leads us 0.030 blue) red) ]( to search for additional clues, as we do in the next section. ]( 2 0.025 3 m m / / 3 3 Pl VI. BINDING ENERGY Pl 0.020 M M The results of the previous sections indicate that any 4eff massive compact stars will decay rapidly. However, one 4eff 0.015 [ [ * might consider the possibility that the boson stars carry so * E 0.010 N much binding energy that these processes are shut off. The above analyses were done reliably in the weak gravitational 0.005 field regime, where the binding energy is small and is Energy Number unlikely to be able to shut off these processes. However, in 0.000 the strong gravity regime this becomes at least conceivable. 1 5 10 50 100 500 1000 To examine this fully for a real scalar, we would really Parameter 0 need to solve the full set of time dependents equations by ’ expanding in a tower of harmonics, as we described in FIG. 7. Boson star s energy/mass (red line) and number (blue line) as a function of parameter β0 within the large g approxi- Sec. II C. However, as we also discussed there, this is a mation. (The left solid curves are the solutions of mostffiffiffiffiffiffiffiffi interest in rather difficult task and will be left for future work. p 3 2 this work, with the maximum value M ≈ 0.03 λ4 M =m . For now we will simply take our clues from the much max eff Pl ϕ The right dashed curves might suffer from other sorts of simpler case of a complex scalar field theory. To define this, instabilities.) we can return to our starting action Eqs. (4), replace the real scalar ϕ by a complex scalar ψ, and endow the theory with a global Uð1Þ symmetry. To be concrete, the kinetic term is scalar. The right-hand dashed curves may have instabilities now ΔL ¼ −j∂ψj2=2, and we choose the potential now as of a variety we have not discussed in this work; we leave 2 2 4 their analysis for future work. V ¼ mϕjψj =2 þ λ4jψj =16. The Uð1Þ symmetry ensures there is a conserved current [68] Now we would like to infer any ramifications for the case of a real scalar. We can be concrete about this in the pffiffiffiffiffi μ μν following way: Suppose we have the above complex scalar J ¼ i jgjg ðψ ∂νψ − ψ∂νψ Þ=2 ð60Þ with a Uð1Þ symmetry, and now we introduce small Uð1Þ breaking terms, which can in principle mediate particle with a corresponding conserved particle number number changing processes. For gravity to prevent an Z annihilation process of the form nϕ → 2ϕ, the change in 3 0 N ¼ d xJ ðx; tÞ: ð61Þ energy with respect to the particle number would need to be less than 2mϕ=n. So to kinematically forbid 3 → 2 anni- 2 3 One can now rigorously define the boson star as a state that hilations one would need dE =dN < mϕ= , while to minimizes the energy subject to the constraint of a fixed kinematically forbid 4ϕ → 2ϕ annihilations one would 2 particle number. It is simple to show that such solutions need dE =dN