Axion stars in the infrared limit

Journal of High Energy Physics, Mar 2015

Following Ruffini and Bonazzola, we use a quantized boson field to describe condensates of axions forming compact objects. Without substantial modifications, the method can only be applied to axions with decay constant, f a , satisfying δ = (f a /M P )2 ≪ 1, where M P is the Planck mass. Similarly, the applicability of the Ruffini-Bonazzola method to axion stars also requires that the relative binding energy of axions satisfies \( \varDelta =\sqrt{1-{\left({E}_a/{m}_a\right)}^2}\ll 1 \), where E a and m a are the energy and mass of the axion. The simultaneous expansion of the equations of motion in δ and Δ leads to a simplified set of equations, depending only on the parameter, \( \lambda =\sqrt{\delta }/\varDelta \) in leading order of the expansions. Keeping leading order in Δ is equivalent to the infrared limit, in which only relevant and marginal terms contribute to the equations of motion. The number of axions in the star is uniquely determined by λ. Numerical solutions are found in a wide range of λ. At small λ the mass and radius of the axion star rise linearly with λ. While at larger λ the radius of the star continues to rise, the mass of the star, M , attains a maximum at λmax ≃ 0.58. All stars are unstable for λ > λmax. We discuss the relationship of our results to current observational constraints on dark matter and the phenomenology of Fast Radio Bursts.

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Axion stars in the infrared limit

Received: December Axion stars in the infrared limit Joshua Eby 0 1 Peter Suranyi 0 1 Cenalo Vaz 0 1 L.C.R. Wijewardhana 0 1 Open Access 0 1 c The Authors. 0 1 0 2600 Clifton Ave , Cincinnati, OH 45221 , U.S.A 1 Dept. of Physics, University of Cincinnati Following Ruffini and Bonazzola, we use a quantized boson field to describe condensates of axions forming compact objects. Without substantial modifications, the method can only be applied to axions with decay constant, fa, satisfying = (fa / MP )2 1, where MP is the Planck mass. Similarly, the applicability of the Ruffini-Bonazzola method to axion stars also requires that the relative binding energy of axions satisfies of the star continues to rise, the mass of the star, M , attains a maximum at max ' 0.58. All stars are unstable for > max. We discuss the relationship of our results to current observational constraints on dark matter and the phenomenology of Fast Radio Bursts. Cosmology of Theories beyond the SM; Classical Theories of Gravity - 1, where Ea and ma are the energy and mass of the axion. The equations, depending only on the parameter, = marginal terms contribute to the equations of motion. The number of axions in the star is 1 Introduction 3 4 2.1 Axion field dynamics in the condensed state Expectation values Equations of motion Expansion in the binding energy of axions Analytic approximations to the physical parameters of axion stars A Calculation of the quantum potential in tree approximation Scalar fields, which give rise to spin-zero quanta satisfying Bose statistics, are natural in quantum field theories of phenomenological importance. The recently discovered Higgs field, which generates masses for all the other particles, is the prime example. The axion, a yet to be discovered pseudoscalar, postulated to solve the strong-CP problem endemic to QCD, is another well motivated spin-zero particle. It is also generic in string theory for 4-dimensional axion-like degrees of freedom to arise as Kaluza-Klein zero modes from the compactification of anti-symmetric tensor fields defined in 10 space-time dimensions. Quintessence, the almost massless scalar field invoked to drive the late-time acceleration of the universe, is another hypothetical scalar degree of freedom. It would be quite interesting to determine if such spin-zero degrees of freedom could form stable compact structures due to their self-gravitation. Starting with the seminal works of Kaup [1] and Ruffini and Bonazzola [2], the study of gravitationally bound bosonic degrees of freedom has received a great deal of attention. It has been seen that complex scalar field configurations in the presence of gravity can form stable compact objects, termed boson stars [310]. Kaup in his original work solved the Klein-Gordon free field equation in an asymptotically flat spherically symmetric spacetime background and found localized spherically symmetric field configurations which are energy eigenstates. Adopting a slightly different approach, Ruffini and Bonazzola used the expectation value of the energy-momentum tensor operator of a second quantized free hermitian scalar field system, evaluated in an N -particle state where all the particles occupy the lowest energy state, to be the source of gravitational interactions. Here the basis states for second quantization were the wave functions of the Klein-Gordon equation. This approach yielded an equation of motion for the wave function of the N -particle state, which represented a condensate of N bosons in a single state. They found stable localized wave functions for the condensate, indicating the interesting possibility that lumps of selfgravitating scalar field configurations could exist in nature. In a recent set of publications, [1114] analyzed the self-gravitating field theoretical system consisting of a real scalar field with an interaction potential of the form a potential represents the interactions of the axion field at low energies, as derived using the dilute instanton gas approximation, where fa represents the axion decay constant, and ma the axion mass [1517]. In this paper we analyze the same system but present an analytic expansion, which simplifies the equations of motion. We also consider a different range of input parameters. This leads to interesting physical consequences. Without substantial modification, our expansion method can only be applied to axions with fa theoretic method of Ruffini and Bonazzola [2] can only be applied in this regime when fa . We find analytic than the central density of the star, has the advantage of being able to compare total energies of solutions with equal numbers of axions as shown in figure 2. As expected, the critical temperature of the condensed axion star is very large, Tc > 1010 GeV. ma2 Ea2 / ma, is small. Axion field dynamics in the condensed state The dynamics of a self-gravitating, free hermitian scalar field was first analyzed in [2]. More recently, the authors of [11, 12] used a similar procedure to describe the condensation of interacting hermitian Bose fields. We revisit this analysis by evaluating the expectation value of the axion potential in the N particle condensed state and simplifying the equations of motion using the expansion method described in the previous section. As stated earlier the range of parameters we explore differ from that of [1114]. Expectation values with potential energy We will see that the leading-order deviation of the space-time metric from flat space is We start with the standard definition of the axion potential (2.1), but replace it by the expectation value of a quantum potential in which the quantum field, expanded in modes having definite radial, polar and azimuthal quantum numbers, is where an,l,m is the creation operator for an axion with the appropriate quantum numbers. To a very good approximation every axion in an axion star is in the ground state, so the star is an almost perfect condensate. As we will see later, for QCD axions the critical temperature of the condensate is 1011 GeV. Consequently, in what follows, we will use the expression and, in appendix A, we calculate the expectation value of the axion potential exactly in the tree approximation. Note that, to justify using the tree approximations one needs to consider the fact that negative energy states and with them loop corrections can be ma. We obtain in the large N limit an effective potential that is different from the classical potential. We thus obtain the equations of motion by taking the expectation value of the Einstein equation and the scalar field equation. For this, we will also need the expectation values Equations of motion We write the spherically symmetric metric as N R(r) ei E t. ds2 = B(r) dt2 A(r) dr2 r2 d. 1. When = 0 then the solution of order contributions can be safely neglected. The expectation value of the Einstein equation and of the nonlinear equation of motion A 1 A fa2 + m2 [1 J0(X)] , 2 B 2 A function X(z) = 2 N R(r) / fa, and defining z a0 = z b0 = X02 + 1 J0(X) , X02 1 + J0(X) , X00 = + (a0 b0) X0 2(1 + a b) X + 2 (1 + a) J1(X). Expansion in the binding energy of axions of the Compton wave length of the axion. We will show now that for axion stars with equations depending on 1 2 ' p2(m E)/m and through the combination = 1, (2.8) can be reduced to a system of radial wave function, as Substituting (2.9) into (2.8) the axion equation of motion takes the form V (X) = 16 192 from each term are the engineering dimensions of the corresponding operator. All terms of (2.10) are of dimension three, except for the irrelevant, non-renormalizable terms, like X5 and even higher powers of X, and of the relevant last term on the right hand side a0(x) = b0(x) = Y (x)2 Y 00(x) = Y (x) x Y 0(x) 8 relevant and marginal terms only is tantamount to taking the infrared limit of the theory. The leading order equations for the dimensionless field Y (x), depending on the dimensionless coordinate x are number of axions in the system. N is the natural physical input parameter. of (2.11) as fa / MP 1. These constraints can be combined together to give the range of validity the system (2.11) provides a correct solution for a wide range of sizes of axion stars. We used the shooting method to integrate (2.11) and calculate the function, Y (x). which can be chosen as the value of Y (x) and b(x) at the center of the star. The boundary conditions are that Y (x), a(x), and b(x) tend to zero at x . a(x) and b(x) have implement in the numerical calculations. However, notice that (2.11) implies a(x) + b(x) = 2 x 1 Z has similar behavior. Such a boundary condition can be imposed easily in a numerical example, in figure 1 we plot our solution for the wave function, Y (x), as a function of x shape. The initial values required for attaining the necessary asymptotic behavior are Analytic approximations to the physical parameters of axion stars The most important parameters describing axion stars are the total mass, M , the radius inside which 99% of the matter contained in the star is concentrated, R99, and the number of axions in the star, N . The mass of the star is given in leading order of the infrared limit M = 3 V (). V (y) = V () R99 = x99 = R99 = V () Furthermore, using the standard definition for boson stars, we define x99 as Then restoring the appropriate scale, the radius of the axion star becomes Combining (3.1) and (3.4) we obtain the relationship between its radius and its mass. 2 103 where RS is the Schwarzschild radius. This relationship, which implies gravitational sta 1 / 500. As shown by (3.6), our formalism could even be applied to axions with fa 1015 1016 GeV, because for axions with decay constant 1 a linear fit gives an excellent representation of R99, are represented by the solid (dashed) line. The dotted line connects sample states with identical numbers of axions. The mass difference of these stars is a small fraction of the mass. 106 105. M = V () ' 15 We plotted the dependence of the mass, scaled by the factor fa ma / MP , as a function axion wave function have already been found in previous work [3, 612], but the maximum in figure 2 has a physical significance due to the following considerations.2 Note that the number of axions in the condensate is approximately equal to 2Seidel and Suen [5] have discussed the question of instability in a complex scalar field model of boson N = maximal number of axions in an axion star is relationship between the masses is illustrated by the dotted line in figure 2. As a result, dashed line, is unstable. In fact, using (3.12) it is easy to estimate the mass difference between the two states of the axion star containing N axions. We obtain macroscopic energy. present paper we do not attempt to calculate the decay rate. That will be the subject of In table 1 we provide the mass, mass difference between stars containing the same 101 where our approximations are valid.) For the input values of fa and ma we use QCD of the choice of fa. A final comment concerns the critical temperature of the axion condensate. To see that the axion star is a pure condensate, with very little contribution from excited states one 3 M / (4 R939 ma), and the expression for the radius in (3.6) we obtain the following esti mate for the critical temperature (neglecting interactions) Tc = As it has been pointed out in the previous section, to describe compact objects formed from axions one needs to consider that, except possibly in the early universe, the temperature of the object is many orders magnitudes smaller than the critical temperature of boson condensation. In other words, in a good approximation one may assume that all the axions are in the ground state. Therefore, along with Ruffini and Bonazzola [2] we substitute the wave function of the axions by a quantized field. As the resulting complicated nonlinear field theory cannot be solved analytically, one is restricted to use perturbation theory. A similar approach was used by Barranco and Bernal [11, 12] in their application to axion The most important observations of this paper are that the application of the method of [2] to axion stars must be restricted in two different ways: (i) the decay constant of the axions, fa must be much smaller than the Planck mass, or using the notations of this 1; (ii) only weakly bound axions should be considered, i.e. the axion binding energy should satisfy ma E ma. The reason for requiring small binding energy is a requirement of using the Ruffini-Bonazzola method [2] for an interacting field theory: if the binding energy is of O(ma) then the perturbation expansion of the axion field theory breaks down. Pairs of positive and negative energy states contribute to the ground state and to the expectation values of physical quantities, as well. To extend calculations theory, which is beyond the scope of this paper. 1 requires using non-perturbative field 1 has allowed us to expand the equations of motion in the scale transcendental axion potential do not contribute. We have calculated the masses, radii and densities of axion stars as a function of the 1. We found that the radius of the star increases approximately linearly independent of fa, while the mass is an increasing function of fa. The maximal number of In the future we will also investigate rotating axion stars and analyze the maximal mass and the question of stability as a function of angular momentum. We will investigate collisions of axion stars in view of the existence of the maximal mass. Furthermore if an stable state. We will compute rate of transition, and possible signatures of their decay. The masses of compact axion star solutions found in our work are consistent with the mass bounds derived by Tkachev for condensate formation through gravity [20]. Axion stars, along with free axions may form all or part of dark matter. Various production mechanisms of axions in the early universe are discussed in [2123]. Bounds on the axion decay constant resulting from astrophysical and cosmological observations are discussed in [21] and [2426]. We will investigate how our conclusions change if axion stars are in equilibrium within a cloud of free axions after reaching the maximal mass. We will also investigate the consequence if axions come in a multitude of flavors, as expected in theories derived from compact extra dimensions. The collisions of axion condensates with neutron stars have been studied in the past [13, 14, 28]. Recently axion stars have been proposed as progenitors for Fast Radio Bursts (FRBs) [2932] via collisions with neutron star atmospheres [3335]. Whatever produces FRBs should be able to generate large amounts of energy, on the order of approximately 1042 ergs/s, in a fairly tight frequency range around 1.4 GHz over time scales on the order of milliseconds. Assuming that all of the mass of an axion star is converted into radiation during a putative collision with a neutron star, one finds that an axion star mass on the order of 1012 M would be required. This mass is indeed compatible with our results listed in table 1. However, as the radius of our boson star is at least an order of magnitude larger that that of a neutron star we feel that the possibility of converting all the axions to radiation in such a collision is highly unlikely. Therefore we plan to perform an accurate estimation of this conversion rate in the near future. The radius of the axion stars may be estimated from the duration of the bursts and was found in [2932] to be on the order of 100 km, which is also compatible with our results in table 1. Moreover, [2932] also showed that the frequency of the radiation can be in greater detail and will report on the results of our investigation elsewhere. 3Note that Rmax is the radius of the heaviest axion star but not the axion star of largest radius. The radii of unstable axion stars are always larger than Rmax. As mentioned, one result of our calculations is that for a fixed number of axions consequence is the possibility of tunneling from the state with a higher mass to the one with a lower mass. We will investigate whether collisions with neutron stars could induce such a tunneling process resulting in a fainter companion burst of approximately 1029 ergs, detected, such a companion burst could serve to distinguish between axion star progenitors of FRBs and other proposals [3638]. However, considering the faintness of the companion burst, it is unlikely to be observed from events that occur outside our galaxy. The observed this phenomenon even more difficult to observe. Finally, we would like to emphasize again that there is no fundamental reason to limit calculations to theories with decay constants satisfying fa MP and stars with sufficiently small binding energy. We just state that the method of taking the expectation value of the equations of motion in free particle states, which are defined in flat space is not permissible if the axion decay constant is comparable with the Planck mass and/or the relative binding energy is not much smaller than 1. Using the current formalism to extend to false results. Such an extension would require using exact solutions of an interacting field theory, possibly in curved space, a calculation, which is beyond the scope of our current analysis. Restricting ourselves to leading order contributions has the added benefit of simplifying results to the extent of being able to obtain a clearer physical interpretation of the properties of axion stars. We thank P. Argyres, P.Esposito, A. Kagan, C. Kouvaris, and J. Zupan for discussions. Calculation of the quantum potential in tree approximation X Rn,l(r) [Ylm(, ) an,l,m + Ylm ?(, ) an,l,m], where an,l,m is the annihilation operator of a state with radial quantum number n and with the rest of the commutators vanishing. particle condensate can be calculated exactly, without resorting to the Taylor expansion. (k!)2 (N k)! S = at fixed R / fa, the sum is dominated by k multiplier on the right hand side of (A.5) tends to N k giving our final result N . Then the last The Baker-Campbell-Hausdorff lemma implies that, if [X +[ + cos(X + + X ) = e 21 [X + ,X ] ei X + ei X + c.c. If we introduce the rescaled field, X(z) = 2 N R / fa the exponent S acquires a factor N in the denominator, while the N -dependence cancels in all other terms of the equations of motion. Thus, the in the limit of N we can set S 0 leaving the tree level different for the two potentials. For small X (i.e. for large distances), where quadratic contributions dominate, Vc(X) / Vq(X) ' 2. approximation one should evaluate When one calculates the expectation value of the scalar equation in the semiclassical Taking the expectation value of the operator of (A.3) between N particle condensates using (A.1) one obtains4 hN | cos[(+ + ) / fa]|N i = e 2 f1a2 [+ ,]hN |ei + / fa ei / fa |N i + c.c Omitting loop corrections and using free particle states, the right hand side of (A.4) can be readily calculated after expanding the two exponentials into power series of the creation and annihilation operators. Owing to the fact that in the condensate only ground state particles can be annihilated in the expectation values all contributions containing operators of the excited states vanish on the right hand side of (A.4). It follows that with V 0(X) = J1(X). Open Access. 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Joshua Eby, Peter Suranyi, Cenalo Vaz. Axion stars in the infrared limit, Journal of High Energy Physics, 2015, 80, DOI: 10.1007/JHEP03(2015)080