Constraints on ultra-light axions from compact binary systems

Ultra light particles $(m_a \sim 10^{-21}eV-10^{-22}eV)$ with axion-like couplings to other particles can be candidates for fuzzy dark matter (FDM) if the axion decay constant $f_a\sim 10^{17}GeV$. If a compact star is immersed in such a low mass axionic potential it develops a long range field outside the star. This axionic field is radiated away when the star is in a binary orbit. The orbital period of a compact binary decays mainly due to the gravitational wave radiation, which was confirmed first in the Hulse-Taylor binary pulsar. The orbital period can also decay by radiation of other light particles like axions and axion like particles(ALPs). For axionic radiation to take place, the orbital frequency of the periodic motion of the binary system should be greater than the mass of the scalar particle which can be radiated. This implies that, for most of the observed binaries, particles with mass $m_a<10^{-19}eV$ can be radiated, which includes FDM particles. In this paper, we consider four compact binary systems: PSR J0348+0432, PSR J0737-3039, PSR J1738+0333, and PSR B1913+16 (Hulse Taylor Binary) and show that the observations of the decay in orbital period put the bound on axion decay constant, $f_a\lesssim \mathcal{O}(10^{11}GeV)$. This implies that Fuzzy Dark Matter cannot couple to gluons.


I. INTRODUCTION
Axion was first introduced to solve the strong CP problem [1][2][3][4]. The most stringent probe of the strong CP violation is the electric dipole moment of neutron. Quantum chromodynamics (QCD) is one of the possible theories which can explain the strong interaction.
However, the theory has a problem known as the strong CP problem. We can write the QCD Lagrangian where the dual of the gluon field strength tensor is, The last term in the QCD Lagrangian violates the discrete symmetries P, T, CP . Since all the quark masses are non-zero, the θ term in the Lagrangian must be present. The QCD depends on θ through some combination of parameters,θ = θ + arg(det(M)), where M is the quark mass matrix [5,6]. The neutron electric dipole moment (EDM) depends onθ and from chiral perturbation theory we can obtain the neutron EDM as d n few × 10 −16θ e.cm.
However the current experimental constraint on the neutron EDM is d n < few × 10 −26 e.cm, which impliesθ 10 −10 [7]. The smallness ofθ is called the strong CP problem. To solve this, Peccei and Quinn, in 1977 [1], came up with an idea thatθ is not just a parameter but it is a dynamical field driven to zero by its own classical potential. They postulated a global U P Q (1) quasi symmetry which is a symmetry at the classical level but explicitly broken by the non perturbative QCD effects which produces the θ term, and spontaneously broken at a scale f a . Thus the pseudo-Nambu-Goldstone bosons appear and these are known as the axions. The QCD axion mass (m a ) is related to the axion decay constant (f a ) by m a = 5.7 × 10 −12 eV 10 18 GeV fa . So if we need axion decay constant less than the Planck scale (M pl ) then the mass of the axion is m a 10 −12 eV [8]. Also there are other pseudo scalar particles which are not the actual QCD axions, but these particles have many similar properties like the QCD axions. These are called the axion like particles (ALPs). For ALPs, the mass and decay constant are independent of each other. These ALPs are motivated from the string theory [9]. The interaction of ALPs with the standard model particles is governed by the Lagrangian [10] where g's are the coupling constants which depend on the model. The first term is the dynamical term of ALPs. The second, third, and last terms denote the coupling of ALPs with the gluons, photons, and fermion fields respectively. ALPs couple with the SM particles very weakly because the couplings are suppressed by 1 fa , where f a is called the axion decay constant and for ALPs, it generally takes larger value.
There is no direct evidence of axions in the universe. However, there are lots of experimental and astrophysical bounds on axion parameters. There are some ongoing searches for solar axions which correspond to f a ∼ 10 7 GeV having sub-eV masses [11,12]. If solar axions were there, then it would violate the supernova 1987A result which requires f a 10 9 GeV.
Axions with f a 10 8 GeV provide the component of hot dark matter [13][14][15]. Large value of f a is allowed in the anthropic axion window and can be studied by isocurvature fluctuations [16]. The laboratory bounds for the axions are discussed in [17][18][19][20][21][22][23][24]. The cosmological bounds for the cold axions produced by the vacuum realignment mechanism are discussed in [25,26]. The bounds on axion mass and decay constant are discussed in [27][28][29], if cold axions are produced by the decay of axion strings.
Explaining the nature of dark matter and the dark energy is a major unsolved problem in modern cosmology. An interesting dark matter model is fuzzy dark matter (FDM) [30,31]. The FDM are axion like particles (ALPs) with mass(10 −21 eV − 10 −22 eV ) such that the associated de Broglie wavelength is comparable to the size of the dwarf galaxy (∼ 2kpc). Axions and ALPs can be possible dark matter candidates [32] or can be dynamical dark energy [33]. Axions can also form clouds around black hole or neutron star from superradiance instabilities and change the mass and spin of the star [34,35]. Cold FDM can be produced by an initial vacuum misalignment and, to have the correct relic dark matter density, the axion decay constant should be f a ∼ 10 17 GeV [31]. This ultra light FDM was introduced to solve the cuspy halo problem.
ALPs are pseudo-Nambu Golstone bosons which have a spin-dependent coupling with nucleons so that, in an unpolarized macroscopic body, there is no net long range field for ALPs outside the body. However, if the ALPs also have a CP violating coupling, then they can mediate long range forces even in unpolarized bodies [36,37].
It has been pointed out recently [38] that if a compact star is immersed in an axionic potential (which will take place if the ALPs are FDM candidates), a long range field is developed outside the star.
The ALPs can be sourced by compact binary systems such as neutron star-neutron star (NS-NS), neutron star-white dwarf (NS-WD), and can have very small mass (< 10 −19 eV ).
They can be possible candidates of FDM. The FDM density arises from a coherent oscillation of an axionic field in free space. If such axionic FDM particles have a coupling with nucleons, then the compact objects (NS, WD) immersed in the dark matter potential develop long range axionic hair. When such compact stars are in a binary orbit, they can lose orbital period by radiating the axion hair in addition to the gravitational wave [38,39].
In this paper, we study a model of ALPs sourced by the compact stars and put bounds on f a from the observations of the orbital period decay of compact binaries.
The paper is organised as follows. In section II, we compute in detail the axionic charge (including GR corrections) of compact stars immersed in a (ultra) low-mass axionic background potential. In section III, we show how the axionic scalar Larmor radiation can change the orbital period of compact binary systems. There may also be an axion mediated long ranged fifth force between the stars in a binary system. In section IV, we put constraints on f a for four compact binaries: PSR J0348+0432 [40], PSR J0737-3039 [41], PSR J1738+0333 [42], and PSR B1913+16 (Hulse Taylor Binary) [39,43], available in the literature. In section V, we discuss the implication of the ALPs sourced by the compact binaries as the FDM. Finally, we summarize our results.
We use the units = c = 1 throughout the paper. shift symmetry a → a + δ is used to remove the QCD theta angle. Suppose the fermions are quarks and we give a chiral rotation to the quark field, so that only the non derivative coupling appears through the quark mass term. Such a field redefinition allow us to move the non derivative couplings into the two lightest quarks and all other quarks are integrated out. So we can work in the effective 2 flavour theory. Thus, in the chiral expansion, all the non derivative dependence of axion is contained in the pion mass term of the Lagrangian where U = e iΠ fπ and Π =  , B 0 is related with the chiral condensate and it is determined by the pion mass term. f π is called the pion decay constant. We can obtain the effective axion potential from the neutral pion sector. On the vacuum, the neutral pion attains a vacuum expectation value and trivially be integrated out leaving the effective where m u and m d are the up and down quark masses respectively, and m π is the mass of pion.
It has been pointed out in [38] that, if we consider ALPs which couple to nucleons, then compact stars such as neutron stars and white dwarfs can be the source of long range axionic force. The reason for this long range force is as follows. In the vacuum, the potential for the ALPs is For simplicity we choose m u = m d and, therefore, the mass of the ALPs in vacuum becomes Inside a compact star, the quark masses are corrected by the nucleon density and the potential inside the star changes to and where n N is the nucleon number density, m q is the quark mass, m π is the pion mass, and f π is the pion decay constant. σ N ∼ 59M eV from lattice simulation [45] and we consider the parameter space where ≤ 0.1 [38]. The tachyonic mass of the ALPs is the square root of the second derivative of the potential Eq. (8) at a = 0. Inside of the neutron star, σ N n N /m 2 π f 2 π is not equal to zero and m T m a . Thus the magnitude of the tachyonic mass of the ALPs inside the compact star becomes where r N S is the radius of the compact star. The compact star can be the source of ALPs if its size is larger than the critical size given by [38] r c 1 m T .
For a typical neutron star and white dwarf, the condition Eq. (11) is satisfied. By matching the axionic field solution inside and outside of the compact star, we get the long range behaviour of the axionic field. The axionic potential has degenerate vacua and this degeneracy can be weakly broken by higher dimensional operators suppressed by the Planck scale [46].
The degeneracy can also be broken by a finite density effect like the presence of a NS and WD. At the very high nuclear density, the axionic potential changes its sign which allows the ALPs to be sourced by the compact stars. Due to the very small size of the nuclei, it cannot be the source of the ALPs and long range axion fields arise only in large sized objects like NS and WD.
Using Eq. (7) in Eq. (10) we can write the tachyonic mass as Putting values of all the parameters and m a ∼ 10 −19 eV, we get the upper bound of the axion decay constant (using Eq. (11) Compact stars with large nucleon number density can significantly affect the axion potential. The second derivative of the potential Eq. (8) with respect to the field value is Outside of the compact star, σ N = 0 which implies that Therefore, outside of the compact star (r > r N S ), the potential attains minima ( ∂ 2 V ∂a 2 > 0) corresponding to the field values a = 0, ±4πf a , ... and maxima ( ∂ 2 V ∂a 2 < 0) corresponding to the field values a = ±2πf a , ±6πf a ... etc.
Inside of the compact star (r < r N S ), σ N = 0 and σ N n N m 2 π f 2 π > . Therefore, inside of the compact star, the potential has maxima at a = 0, ±4πf a , ... and minima at the field values a = ±2πf a , ±6πf a ... etc.
The axionic field becomes tachyonic inside of a compact star and reside on one of the local maxima of the axionic potential and, outside of the star, the axionic field rolls down to the nearest local minimum and stabilizes about it. The axionic field asymptotically reaches zero value a = 0 at infinity. Therefore, throughout interior of the compact star the axionic field assumes a constant value a = 4πf a , the nearest local maximum.
For an isolated compact star of constant density the equation of motion for the axionic where θ = a/f a . The sgn function is required to take care of the absolute value | cos(θ/2)| in the potential. Note that the equation of motion for the axionic field inside the compact star is satisfied by the field value a = 4πf a .
Assuming the exterior spacetime geometry due to the compact star to be the Schwarzschild, the axionic field equation Eq. (15) becomes where M is the mass of the compact star, G is the Newton's gravitational constant and we have used the approximation sin(θ/2) ≈ θ/2 for small θ.
At a large distance (r >> 2GM ) from the compact star, the axionic field Eq. (16) becomes Assuming a = ξ(r)/r, the above equation reduces to ξ − m 2 a ξ = 0 (where prime denotes derivative with respect to r). This has the solution ξ = C 1 e mar +C 2 e −mar . Since a → 0 in the limit r → ∞, C 1 = 0. Thus, a behaves as a ∼ q ef f e −mar /r where we rename the integration constant C 2 as q ef f . Further, for sufficiently light mass (m a << 1/D << 1/r N S where D is the distance between the stars in a binary system), the scalar field has a long range behaviour with an effective charge q ef f . For scalar Larmor radiation, the orbital frequency (ω) of the binary pulsar should be greater than the mass of the particle that is radiated (i.e. ω > m a ).
This translates the mass spectrum of radiated ALPs for a typical neutron star-neutron star (NS-NS) or a neutron star-white dwarf (NS-WD) binary system into m a 10 −19 eV. Also, the axion Compton wavelength should be much larger than the binary distance in order to use the massless limit in the computation of scalar radiation and effective charge, i.e. m −1 a >> D. The critical value of axion mass required for the scalar radiation and the binary distance for four compact binary systems are given in Table I  To identify the effective charge q ef f , we exploit the continuity of the axion field across the surface of the compact star. Therefore, we solve Eq. (16) in the massless limit (m a → 0), Integrating Eq. (18) we get a = −C 3 /r 2 (1 − 2GM/r) and further integration yields a = − C 3 2GM ln (1 − 2GM/r) + C 4 , where C 3 and C 4 are integration constants. For r >> 2GM limit, a → q ef f /r and, therefore, C 3 = q ef f and C 4 = 0. Therefore, we get the axionic field profile outside the compact star The behaviour of the axionic potential as a function of the axionic field and distance are At the surface of the compact star, a(r N S ) = 4πf a . Thus we identify If GM r N S << 1, q ef f ∼ 4πf a r N S [38]. However, for a typical neutron star (M = 1.4M and r N S = 10 km) the above correction is not negligible. For white dwarf the effect is negligible.
The charges can be both positive as well as negative depending on the sign of the axionic field values at the surface of the compact star. If q 1 and q 2 are the charges of two compact stars, then if q 1 q 2 > 0 the two stars attract each other and, if q 1 q 2 < 0, then they repel each other [38]. For neutron star, the new effective axion charge Eq. (20) is smaller than 4πf a r N S by 21.46%. The effect of new axion charge is illustrated in Fig. 3 where the plot of axion profile inside and outside of a neutron star is shown.

PACT BINARIES
Such a long range axionic field mediates a "fifth" force (in addition to the Newtonian gravitational force) between the stars of a binary system (NS-NS or NS-WD), where q 1,2 are effective charges of the stars in the binary system. Due to the presence of this scalar mediated fifth force the Kepler's law is modified by [47] where α = q 1 q 2 4πGm 1 m 2 is the ratio of the scalar mediated fifth force to the gravitational force, ω is the angular frequency of orbital motion of the stars, m 1 and m 2 are the masses of the stars and µ = m 1 m 2 /(m 1 + m 2 ) is the reduced mass of the binary system. There are constraints on the fifth force from either scalar-tensor theories of gravity [47][48][49] or the dark matter components [49][50][51]. In this paper we show that the constraint on α from time period loss  by scalar radiation is more stringent than the measured change in orbital period Eq. (22) due to fifth force.
The orbital period of the binary star system decays with time because of the energy loss primarily due to the gravitational quadrupole radiation and about one percent due to ultra light scalar or pseudoscalar Larmor radiation. The total power radiated for such quasi-periodic motion of a binary system is (1 + e 2 /2) where e is the eccentricity of the elliptic orbit and E is the total energy of the binary system.
The first term on the r.h.s. is the gravitational quadrupole radiation formula [39,50] and the second term is the massless scalar dipole radiation formula [38,39,52]. There is the radiation of the ALPs if the orbital frequency is greater than the mass of the ALPs. The dipole moment in the centre of mass frame of the binary system can be written as or, where r 1,2 are the radial distances of the stars in the binary system from the centre of mass along the semi-major axis. For nonzero scalar radiation the charge-to-mass ratio (q/m) should be different for two stars. Thus for the companion star in a binary system with the equal effective charge, there should be some mass difference of the two stars. The decay of the orbital time period is given by [39,53] where P b = 2π/ω. NS-NS binaries (with different mass components) as well as NS-WD binaries are the sources for the scalar Larmor radiation and also for the axion mediated fifth force. On the other hand, NS-BH systems can be the source of scalar radiation but there is no long range fifth force in between, as the scalar charges for the black holes (BH) are zero [54].
In the next section, we consider four compact binaries and put constraints on f a .

IV. CONSTRAINTS ON AXION PARAMETERS OF DIFFERENT COMPACT BINARIES
A. PSR J0348+0432 This binary system is consist of a neutron star and a low mass white dwarf companion.
The orbital period of the quasi-periodic binary motion is P b = 2.46h. The mass of the neutron star in this binary system is M p = 2.01M and the mass of the white dwarf is 172M . The radius of the white dwarf is r W D = 0.065R and we assume the radius of the neutron star r N S = 10km. We compute the semi-major axis of the orbit using Kepler's law Eq. (22). The observed decay of the orbital period isṖ b = 0.273×10 −12 ss −1 [40]. This is primarily due to gravitational quadrupole radiation from the binary NS-WD system.
The contribution from the radiation of some scalar or pseudoscalar particles must be within It is a double neutron star binary system whose average orbital period is P b = 2.4h. Its In Table II,   The ALPs that are radiated from the compact binaries can be possible candidates of FDM whose mass is ∼ O(10 −21 eV − 10 −22 eV ). At the very early universe the axionic field evolves with a cosine potential The equation of motion for the axionic field is where R(t) is the scale factor in the FRW spacetime. Taking the Fourier transform of Eq. (28), the modes decouple and we have, For non relativistic (small k) or zero modes, the third term becomes zero and the equation of motion of the axionic field is damped harmonic oscillatory. The axionic field takes a constant value as long as H m a which fixes the initial misalignment angle and then the axionic field starts oscillating with a frequency ∼ m a . When the oscillation starts at H ∼ m a , then the energy density of axionic field is of the order of m 2 a a 2 0 , where a 0 is the initial field value during inflation. The oscillation modes are damped as R − 3 2 . The energy density of the axionic field when it is oscillating, goes as 1 R 3 . Hence, at the late time, the axionic energy density redshifts like a cold dark matter. The ratio of dark matter to radiation energy densities increases as 1 T with the expansion of the universe and the dark matter starts dominating over radiation at T ∼ 1eV . Using these facts the dark matter relic density becomes [31] Ω DM ∼ 0.1 a 0 10 17 GeV where a 0 = θ 0 f a and θ 0 is the initial misalignment angle which can take values in the range −π < θ 0 < π. Since the coupling of ALPs with matter is proportional to 1 fa , large values of f a corresponds to weaker coupling with matter. Therefore, direct detection of the ALPs in this scale is much more difficult. However, the ALPs in this large f a scale has some theoretical motivations [9]. Axion decay constant in the GUT scale implies that a single axion condensate For the NS-WD binaries PSR J0348+0432 and PSR J1738+0333, the bound on the axion decay constant (f a O(10 11 GeV )) is well below the GUT scale and this gives the stronger bound. This implies that if the ultra-light ALPs has to be FDM then they do not couple with gluons.

VI. CONCLUSIONS AND DISCUSSIONS
In this paper, we have obtained upper bounds on the decay constant of the ultralight ALPs from the study of decay in orbital period of the compact binary stars (NS-NS, NS-WD). Compact stars such as neutron stars and white dwarfs can be the source of ALPs.
We have assumed that the mass of the ALPs is sufficiently low such that the axionic field has a long range behaviour over a distance between the binary companions. Due to such axionic field, the binary system will emit scalar Larmor radiation. Although the gravitational quadrupole radiation mainly contributes to the decay of orbital period, the contribution of scalar radiation is not negligible. However, its contribution must be within the excess value of the observed decay in the orbital period. For the NS-NS and NS-WD binary systems, an additional axionic "fifth" force arises which is not relevant as much as the scalar radiation in our study.
We have obtained the axionic profile for an isolated compact star assuming it to be a spherical object of uniform mass density. We have identified the form of effective axionic charge of the compact star [38] and its GR correction. We have also considered eccentricity of the orbit of binary system-a generalization of previous result for axionic scalar radiation [38]. Using the updated formula for the total power radiated, we have studied four compact binary systems: PSR J0348+0432, PSR J0737-3039, PSR J1738+0333, and PSR B1913+16 ALPs can give rise to isocurvature fluctuations during inflation which are tightly constrained from CMB observation. The Hubble scale during inflation (for single field slow roll models) is H I = 8 × 10 13 r/0.1GeV [55] and, therefore, for our bound f a O(10 11 GeV ), it is possible to have 2πf a < H I which means that the ALPs symmetry breaking takes place after inflation and there will be no iso-curvature perturbations from ALPs. However observations of Lyman-α disfavor FDM [56].
ALPs with larger mass range (m a > 10 −19 eV) can be probed from the observation of the gravitational wave signals from binary merger events at the LIGO-Virgo detectors. For this, detailed analysis of gravitational wave-form and phase are required which will take into account the energy loss by axionic emission.