Stimulated Radiation from Axion Cluster Evolution in Static Spacetimes

If axions exist (for reviews of axion physics see Kim:1986ax ; Cheng:1987gp ; Raffelt:1990yz ), they are a dark matter (DM) candidate produced in a Bose-Einstein condensate (BEC) in the early universe by vacuum misalignment Abbott:1982af ; Preskill:1982cy ; Dine:1982ah and/or by cosmic strings after inflation Sikivie:2010bq where the produced axions relax into a BEC via $a+a\rightarrow a+a$ scattering and by the gravitationally scattering of axions. When the temperature is near the QCD scale ($T\sim 1$GeV), regions where the axion field $a$ is far from the minimum of its potential $V(a(T))$ have axion over densities that can decouple from the Hubble flow to form axion mini clusters. The typical mass of these objects is roughly $\sim 10^{-14}M_{\odot}$ Enander ; Fairbairn:2017sil and a fraction $f_{mc}\sim 10\%$ of all axions are expected to be in mini clusters Fairbairn:2017dmf . The evolution of axion mini clustered have been studied in Kolb:1993zz ; Kolb:1993hw and more recently in Braaten:2019knj .

Since the central density, radius and total masses of axion miniclusters are known along their evolutionary stability curves, we can check to see if these parameters are in the range where lasing can commence. If it does lase, then the cluster experiences an immediate (approximate $\delta$-function) change in those parameters due to the fast mass loss after which the cluster must rearrange itself to regain stability or quasi-stability.

Gravitationally bound and self bound axion clusters have been studied by several groups. Relativistic axions are best expressed in terms of a real scalar field and low energy axions $E<<m_{a}$ by an effective complex field, see Guth:2014hsa ; Braaten:2019knj ; Braaten:2018lmj . For a detailed review of for various bound state including quasi-bound states and including breather like states, see Braaten:2019knj . Particles in free space can interact gravitationally or through other mutual interactions to form bound or quasi-bound objects. If the particles are identical bosons in the early universe, then the objects are Bose stars. If the particles are cold or can radiate energy, then they can form a Bose-Einstein condensate (BEC) with properties that differ from low occupation number configurations. For a recent review of axion cosmology see Marsh:2015xka . Depending on invisible axion model parameters, they can form gravitationally bound clumps called axion stars or self interaction systems called axitons. For an extensive review of bound axion systems with a review of axion field theory, and definitions see again Braaten:2019knj . (We assume that the axions considered here to be is states of high occupation number, but not in BECs. They could in principle also be in a large set of overlapping BECs, but not in a single BEC, and in this case they would still behave sufficiently classically that our methods will apply to a good approximation.)

Let us call all the possible types of bound or quasibound axion configurations axion clusters. Here we will be interested in the evolution of these axion clusters, however we will not need to distinguish between stable or quasi-stable clusters, since we will be studying axion cluster decay to photons via lasing who’s time scale is much shorter than other time scales associated with cluster evolution phenomenon, like relaxation, radiative cooling, etc. Hence the clusters we consider need not even be bound, but could just be dense transients. Hence our focus will be on snap shots of otherwise slowly evolving clusters where we ask if the parameters are right for lasing to commence when we take both general and special relativistic effects into account, i.e., we allow for axion velocities near the speed of light $c$ in curved backgrounds.

The axion field $a$ couples to standard model particles, plus it self interacts via an effective potential $V(a)$, but the coupling of most interest here is to photons via the global chiral anomaly term in the Lagrangian

$$\frac{\alpha\,c_{\gamma 0}}{8\pi f_{a}}aF\wedge F$$

where $\alpha$ is the fine structure constant, $f_{a}$ is the axion decay constant, $F$ is the electromagnetic field strength and $c_{\gamma 0}$ is an $O(1)$ model dependent constant. From this term we can calculate the decay rate for $a\rightarrow\gamma+\gamma$.

Experimental and cosmological limits have now constrained the mass of the QCD axion to lie in a narrow range $10^{-3}eV>m_{a}>10^{-5}eV$. Because axions are copiously produced at rest during the QCD phase transition, they would over close the universe if their mass was lighter than the lower bound and they would have been experimentally detected if they were heavier than the upper bound. Since they are born non-relativistic they are a cold dark matter (CDM) candidate. Studies of density perturbations have shown that CDM can form structures on all scales. Consequently, assuming axions are the CDM, we expect to find them seeding galaxies or clusters of galaxies, but also perhaps being the seeds of early stars. There are many potential forms of these axion clumps. If they are diffuse the axions will remain non-relativistic. If they become dense then they can become relativistic. The clumps may or may not be spherically symmetric depending on the environment in which they were formed. If they are sufficiently dense the a curved space metric will be necessary to describe the cluster. If their number density is high enough, then they can lase and hence the resulting photons can be detectable at large distances.

Spherically symmetric non-relativistic lasing axion clusters were first studied in the 1980s Kephart:1986vc ; Tkachev:1987cd ; Kephart:1994uy . More recently non-spherically symmetric non-relativistic cluster with arbitrary momentum and spacial distributions have also been studied Chen:2020ufn ; Chen:2020yvx . However, these studies focuses on self-bounded axion cluster. Our purpose here is to study axion cloud bounded by a host object in static space-times. An example is axions moving in a static geometry like Schwarzschild space-time.

QCD axion clumps with conventional couplings can undergo resonant decay for sufficiently large angular momentum Hertzberg:2018zte . Radio emission from QCD axions occurs in the context of bose stars if axion-photon coupling is large Levkov:2020txo . These are among recent studies on this topic in flat spacetime. There are many standard general relativistic treatment of axion cluster in the literature. Braaten:2019knj gives a comprehensive review of axion clusters and includes applications that considered gravitational effect on axion cluster. Ikeda:2018nhb shows that laserlike emission from axion clouds exists by numerically solving the problem at classical level in fixed Kerr background. Detailed studies of axionic instability induced by electromagnetic field deformations of the Kerr-Newman geometry, and periodic bursts of light from superradiant growth are carried out in Boskovic:2018lkj . These studies did the analysis from the perspective of field theory. What sets Kephart:1994uy apart from these studies is that it tackled the problem from statistical model using Boltzmann equation. Therefore, results from Kephart:1994uy are potentially applicable to other types of pesudoscalar particle decaying into two photons. Although Kephart:1994uy is a statistical model built in flat spacetime, other studies such as Ikeda:2018nhb and Boskovic:2018lkj suggests curved spacetime may provide more information on this issue. Therefore, we decided to rebuild this model in curved static spacetime. Hence, the methodology implemented here is tailored particularly towards Kephart:1994uy . For other approachs of superradiance-blast simulation, please see the studies mentioned above.

This is a brief summary of the formulation of axion cluster lasing using Boltzmann equation in Kephart:1994uy . Two photons emitted by decay of a spin zero particle have the same helicity, as required by angular momentum conservation. The change in the number density of photons of a given helicity $\lambda=\pm 1$ within the axion cluster, due to the process $a\leftrightarrow\gamma+\gamma$ in Minkowski spacetime, is

	$\displaystyle{dn_{\lambda}\over dt}=$	$\displaystyle\int dX^{(3)}_{\text{LIMS}}[f_{a}(1+f_{1\lambda})(1+f_{2\lambda})-f_{1\lambda}f_{2\lambda}(1+f_{a})]$		(1)
		$\displaystyle\times|M(a\rightarrow\gamma(\lambda)\gamma(\lambda))|^{2}~{},$

where $f_{a}$, $f_{1\lambda}$ and $f_{2\lambda}$ are the occupation numbers of the axion and the two photons and $M=M(a\rightarrow\gamma(+)\gamma(+))=M(a\rightarrow\gamma(-)\gamma(-))$ is the decay amplitude determined by the Abelian chiral anomalyAdler:1969gk ; Bell:1969ts and is related to the spontaneous axion decay constant by

$\displaystyle\tau_{a}^{-1}=\Gamma_{a}={1\over{8\pi}}({1\over{2m_{a}}}){1\over 2}\sum_{\lambda=\pm}|M(a\rightarrow\gamma(\lambda)\gamma(\lambda))|^{2}~{}.$

(2)

The three body Lorentz invariant momentum measure is

	$\displaystyle\int dX^{(3)}_{\text{LIMS}}=$	$\displaystyle\int{d^{3}p\over(2\pi)^{3}2p^{0}}\int{d^{3}k_{1}\over(2\pi)^{3}2k_{1}^{0}}\int{d^{3}k_{2}\over(2\pi)^{3}2k_{2}^{0}}$
		$\displaystyle\times(2\pi)^{4}\delta^{(4)}(p-k_{1}-k_{2})~{}.$

See the Appendix for details of deriving $dX_{\text{LIMS}}$. From here one obtains eq. (10) of Kephart:1994uy

	$\displaystyle 2k\frac{df_{\lambda}(\vec{k})}{dt}=\frac{4m_{a}\Gamma_{a}}{\pi}\int\frac{d^{3}k_{1}}{2k_{1}^{0}}\frac{d^{3}p}{2p^{0}}\delta^{4}(p-k-k_{1})$		(3)
	$\displaystyle\times\{f_{a}(\vec{p})[1+f_{\lambda}(\vec{k})+f_{\lambda}(\vec{k_{1}})]-f_{\lambda}(\vec{k})f_{\lambda}(\vec{k_{1}})\}~{}.$

which became the starting point for studying the lasing of axion cluster in flat spacetime.

We generalize equation (1) to find the change in the number density of photons of a given helicity $=\pm 1$ within the axion cluster, due to the process $a\leftrightarrow\gamma+\gamma$ in static spacetime

$\displaystyle ds^{2}=g_{00}dt^{2}+g_{ij}dx^{i}dx^{j}~{},$

which is

	$\displaystyle{dn_{\lambda}\over d\tau}=$	$\displaystyle\int dX^{(3)}_{\text{SCMS}}[f_{a}(1+f_{1\lambda})(1+f_{2\lambda})-f_{1\lambda}f_{2\lambda}(1+f_{a})]$
		$\displaystyle\times|M(a\rightarrow\gamma(\lambda)\gamma(\lambda))|^{2}~{},$		(4)

where $dX_{\text{SCMS}}$ is the static spacetime covariant measure, the generalization of $dX_{\text{LIMS}}$ in flat spacetime. We give a pedagogical discussion of how we derive and interpret $dX_{\text{SCMS}}$ in the Appendix.

The one body static covariant measure is

$\displaystyle\int dX_{\text{SCMS}}=$

$\displaystyle\int{\sqrt{||g_{ij}||}dp^{1}dp^{2}dp^{3}\over(2\pi)^{3}2\sqrt{-g_{00}}p^{0}}~{},$

and the three body static covariant measure is

		$\displaystyle\int dX^{(3)}_{\text{SCMS}}$
	$\displaystyle=$	$\displaystyle\int{\sqrt{||g_{ij}||}dp^{1}dp^{2}dp^{3}\over(2\pi)^{3}2\sqrt{-g_{00}}p^{0}}{\sqrt{||g_{ij}||}dk_{1}^{1}dk_{1}^{2}dk_{1}^{3}\over(2\pi)^{3}2\sqrt{-g_{00}}k_{1}^{0}}$
		$\displaystyle\times{\sqrt{||g_{ij}||}dk_{2}^{1}dk_{2}^{2}dk_{2}^{3}\over(2\pi)^{3}2\sqrt{-g_{00}}k_{2}^{0}}(2\pi)^{4}\delta^{(4)}(p-k_{1}-k_{2})~{}.$

Equation (2) describes the axion decay constant $\tau_{a}$ in a local inertial frame that is comoving with the axion. The relation between the time in the comoving frame $\tau$ with the axion and the coordinate time $t$ in static spacetime is

$\displaystyle{dt\over d\tau}=$

$\displaystyle{1\over\sqrt{-g_{00}}}\sqrt{g_{ij}{p^{i}p^{j}\over m^{2}}+1}~{}.$

Specifically, in Schwarzschild spacetime

$\displaystyle ds^{2}=-(1-{2M\over r})dt^{2}+(1-{2M\over r})^{-1}dr^{2}+r^{2}d\Omega^{2}~{},$

(5)

this becomes

$\displaystyle({dt\over d\tau})=$

$\displaystyle(1-{2M\over r})^{-1/2}\sqrt{g_{ij}{p^{i}p^{j}\over m^{2}}+1}$

where

$\displaystyle g_{ij}p^{i}p^{j}=$

$\displaystyle(1-{2M\over r})^{-1}(p^{r})^{2}+r^{2}(p^{\theta})^{2}+r^{2}\sin^{2}\theta(p^{\phi})^{2}$

is the square of the magnitude of the 3-momentum. The decay constant $\tau_{a}$ in the comoving frame would change to the decay constant $\tau_{al}$ in lab frame

$\displaystyle\tau_{al}=\tau_{a}{1\over\sqrt{-g_{00}}}\sqrt{g_{ij}{p^{i}p^{j}\over m^{2}}+1}~{},$

and the decay rate in the lab frame becomes

$\displaystyle\Gamma_{al}=\Gamma_{a}\sqrt{-g_{00}}(g_{ij}{p^{i}p^{j}\over m^{2}}+1)^{-1/2}~{}.$

In a static spacetime, the number density given by

$\displaystyle n(x^{\alpha})=\int f(p^{i},x^{\alpha}){d^{3}p\over(2\pi)^{3}}=\int f(p^{i},x^{\alpha}){\sqrt{||g_{ij}||}\over(2\pi)^{3}}dp^{1}dp^{2}dp^{3}~{},$

where $|g_{ij}|$ is the determinant of the 3-surface metric

$\displaystyle ds^{2}=g_{ij}dx^{i}dx^{j}$

at constant coordinate time $t$. The total number of the particle is

$\displaystyle N(t)=\int n(x^{\alpha})d^{3}x=\int n(x^{\alpha})\sqrt{||g_{ij}||}dx^{1}dx^{2}dx^{3}~{},$

In static spacetime, the rate of change in number density is related to the rate of change in occupation number by

$\displaystyle{dn_{\lambda}\over dt}=$

$\displaystyle\int 2\sqrt{-g_{00}}k^{0}{df_{\lambda}\over dt}{\sqrt{||g_{ij}||}dk^{1}dk^{2}dk^{3}\over(2\pi)^{3}2\sqrt{-g_{00}}k^{0}}~{}.$

In Minkowski spacetime $k^{0}=k$, but the magnitude of energy and momentum are no longer equal in general in a static spacetime. Instead, we have a static energy-momentum relation for photon:

$\displaystyle g_{ij}k^{i}k^{j}=$

$\displaystyle-g_{00}({k^{0}})^{2}~{},$

(6)

and for massive particle, the corresponding relation is

$\displaystyle m^{2}+g_{ij}p^{i}p^{j}=$

$\displaystyle-g_{00}(p^{0})^{2}~{}.$

(7)

Multiplying equation (4) by ${d\tau\over dt}$, we have an equation for the rate of change in the number density of photons measured by coordinate time $t$,

	$\displaystyle{dn_{\lambda}\over dt}=$	$\displaystyle\int dX^{(3)}_{\text{SCMS}}\{[f_{a}[1+f_{\lambda}(k^{i})+f_{\lambda}(k_{1}^{i})]$
		$\displaystyle-f_{\lambda}(k^{i})f_{\lambda}(k_{1}^{i})\}\times 16\pi m_{a}\Gamma_{a}{d\tau\over dt}~{}.$

Writing $dn_{\lambda}\over dt$ in terms of $f_{a}$ we have one body measure $dX_{\text{SCMS}}$ on the LHS of this equation that we use to cancel one of the three measures in $dX^{(3)}_{\text{SCMS}}$ on the RHS. There remains two measures, one for the axion and one for a photon, plus the $\delta$ function,

	$\displaystyle 2\sqrt{-g_{00}}k^{0}{df_{\lambda}\over dt}=$	$\displaystyle\int{\sqrt{||g_{ij}||}dp^{1}dp^{2}dp^{3}\over(2\pi)^{3}2\sqrt{-g_{00}}p^{0}}{\sqrt{||g_{ij}||}dk_{1}^{1}dk_{1}^{2}dk_{1}^{3}\over(2\pi)^{3}2\sqrt{-g_{00}}k_{1}^{0}}$
		$\displaystyle\times(2\pi)^{4}\delta^{4}(p^{\alpha}-k^{\alpha}-k_{1}^{\alpha})$
		$\displaystyle\times\{[f_{a}[1+f_{\lambda}(k^{i})+f_{\lambda}(k_{1}^{i})]-f_{\lambda}(k^{i})f_{\lambda}(k_{1}^{i})\}$
		$\displaystyle\times 16\pi m_{a}\Gamma_{a}\sqrt{-g_{00}}(g_{ij}{p^{i}p^{j}\over m_{a}^{2}}+1)^{-1/2}~{}.$

After simplification, we obtained the evolution equation

	$\displaystyle 2k^{0}\frac{df_{\lambda}}{dt}=$	$\displaystyle\frac{4m_{a}\Gamma_{a}}{\pi}\int{\sqrt{||g_{ij}||}dp^{1}dp^{2}dp^{3}\over 2\sqrt{-g_{00}}p^{0}}{\sqrt{||g_{ij}||}dk_{1}^{1}dk_{1}^{2}dk_{1}^{3}\over 2\sqrt{-g_{00}}k_{1}^{0}}$
		$\displaystyle\times\{[f_{a}[1+f_{\lambda}(k^{i})+f_{\lambda}(k_{1}^{i})]-f_{\lambda}(k^{i})f_{\lambda}(k_{1}^{i})\}$
		$\displaystyle\times\delta^{4}(p^{\alpha}-k^{\alpha}-k_{1}^{\alpha})(g_{ij}{p^{i}p^{j}\over m_{a}^{2}}+1)^{-1/2}~{}.$		(8)

which resembles equation (3), and reduces to it in the flat spacetime limit.

The momentum space integrations needed for the evaluation of (8) are rather tedious and can be found in the Appendix. Specifically, doing the $k_{1}$ integration first leads to

		$\displaystyle 2k^{0}\frac{df_{\lambda}(k^{i})}{dt}$
	$\displaystyle=$	$\displaystyle\frac{4m_{a}\Gamma_{a}}{\pi(-g_{00})}\int{m_{a}\over p^{0}}{\sqrt{||g_{ij}||}dp^{1}dp^{2}dp^{3}\over 2\sqrt{-g_{00}}p^{0}}$
		$\displaystyle\times{1\over 2\sqrt{g_{ij}(p^{i}-k^{i})(p^{j}-k^{j})}}$
		$\displaystyle\times\delta[p^{0}-k^{0}-\sqrt{g_{ij}(p^{i}-k^{i})(p^{j}-k^{j})\over-g_{00}}]$
		$\displaystyle\times\{f_{a}(p^{i})[1+f_{\lambda}(k^{i})+f_{\lambda}(p^{i}-k^{i})]-f_{\lambda}(k^{i})f_{\lambda}(p^{i}-k^{i})\}~{}.$		(9)

and then we assume that the occupation numbers of both axion and photon are isotropic,

$\displaystyle f_{a}(p^{i})=f_{a}(\sqrt{g_{ij}p^{i}p^{j}})~{},\qquad f_{\lambda}(k^{i})=f_{\lambda}(\sqrt{g_{ij}k^{i}k^{j}})~{},$

and the $p$ integration given in the Appendix results in

		$\displaystyle\frac{df_{\lambda}(\sqrt{g_{ij}k^{i}k^{j}})}{dt}$		(10)
	$\displaystyle=$	$\displaystyle{m_{a}\Gamma_{a}\sqrt{-g_{00}}\over g_{ij}k^{i}k^{j}}\int{m_{a}\over p^{0}}dp^{0}\times\{f_{a}(\sqrt{g_{ij}p^{i}p^{j}})$
		$\displaystyle[1+f_{\lambda}(\sqrt{g_{ij}k^{i}k^{j}})+f_{\lambda}\left(\sqrt{-g_{00}}(p^{0}-k^{0})\right)]$
		$\displaystyle-f_{\lambda}(\sqrt{g_{ij}k^{i}k^{j}})f_{\lambda}\left(\sqrt{-g_{00}}(p^{0}-k^{0})\right)\}~{}.$

where we have a factor of $\sqrt{-g_{00}}$ from gravitational redshift which corrects the time difference between the clock in the lab frame and the clock at the location the axion. The factor ${m_{a}\over p^{0}}$ is due to the special relativistic correction.

In the Appendix, we find that at event ${\mathcal{P}_{0}}$, there are upper and lower limits on the $0^{th}$ component of photon momentum. This is true for every event at any point in spacetime. The bounds are

	$\displaystyle p^{0}\geq$	$\displaystyle k^{0}+{m_{a}^{2}\over 4k^{0}(-g_{00})}$
	$\displaystyle k^{0}_{\text{max/min}}=$	$\displaystyle{1\over 2}[\ p^{0}\pm\sqrt{(p^{0})^{2}-{m_{a}^{2}\over-g_{00}}}\ ]$
	$\displaystyle=$	$\displaystyle{1\over 2}(p^{0}\pm\sqrt{g_{ij}p^{i}p^{j}\over-g_{00}})$
	$\displaystyle\sqrt{g_{ij}k^{i}k^{j}}_{\text{max/min}}=$	$\displaystyle{1\over 2}(\sqrt{-g_{00}}p^{0}\pm\sqrt{g_{ij}p^{i}p^{j}})~{}.$

Switching from variable $\sqrt{g_{ij}p^{i}p^{j}}$ to $p^{0}$, the evolution equation becomes

		$\displaystyle\frac{df_{\lambda}(\sqrt{g_{ij}k^{i}k^{j}})}{dt}$
	$\displaystyle=$	$\displaystyle{m_{a}\Gamma_{a}\sqrt{-g_{00}}\over g_{ij}k^{i}k^{j}}\int_{k^{0}-{m_{a}^{2}\over 4k^{0}g_{00}}}{m_{a}\over p^{0}}dp^{0}\{f_{a}\left(\sqrt{(-g_{00})(p^{0})^{2}-m_{a}^{2}}\right)$
		$\displaystyle\times[1+f_{\lambda}(\sqrt{g_{ij}k^{i}k^{j}})+f_{\lambda}\left(\sqrt{-g_{00}}(p^{0}-k^{0})\right)]$
		$\displaystyle-f_{\lambda}(\sqrt{g_{ij}k^{i}k^{j}})f_{\lambda}\left(\sqrt{-g_{00}}(p^{0}-k^{0})\right)\}~{}.$

Note that $p^{0}=k^{0}+k_{1}^{0}$ and $\sqrt{-g_{00}}k_{1}^{0}=\sqrt{g_{ij}k_{1}^{i}k_{1}^{j}}$. The $p^{0}$ integral can be rewritten as an integral over $\sqrt{g_{ij}k_{1}^{i}k_{1}^{j}}$,

		$\displaystyle\frac{df_{\lambda}(\sqrt{g_{ij}k^{i}k^{j}})}{dt}$
	$\displaystyle=$	$\displaystyle{m_{a}\Gamma_{a}\sqrt{-g_{00}}\over g_{ij}k^{i}k^{j}}\int_{m_{a}^{2}\over 4\sqrt{g_{ij}k^{i}k^{j}}}{m_{a}d(\sqrt{g_{ij}k_{1}^{i}k_{1}^{j}})\over\sqrt{g_{ij}k^{i}k^{j}}+\sqrt{g_{ij}k_{1}^{i}k_{1}^{j}}}$
		$\displaystyle\times\{f_{a}\left(\sqrt{(\sqrt{g_{ij}k^{i}k^{j}}+\sqrt{g_{ij}k_{1}^{i}k_{1}^{j}})^{2}-m_{a}^{2}}\right)$
		$\displaystyle\times[1+f_{\lambda}(\sqrt{g_{ij}k^{i}k^{j}})+f_{\lambda}(\sqrt{g_{ij}k_{1}^{i}k_{1}^{j}})]$
		$\displaystyle-f_{\lambda}(\sqrt{g_{ij}k^{i}k^{j}})f_{\lambda}(\sqrt{g_{ij}k_{1}^{i}k_{1}^{j}})\}~{}.$

We convert the LHS of this equation into the changing rate of the number density by integrating over the momentum space,

$\displaystyle\frac{dn_{\lambda}}{dt}=\int\frac{df_{\lambda}(\sqrt{g_{ij}k^{i}k^{j}})}{dt}{g_{ij}k^{i}k^{j}\over 2\pi^{2}}d(\sqrt{g_{ij}k^{i}k^{j}})~{}.$

The rate of change in photon number density is then

	$\displaystyle\frac{dn_{\lambda}}{dt}={m_{a}\Gamma_{a}\sqrt{-g_{00}}\over 2\pi^{2}}\int\int_{k^{0}-{m_{a}^{2}\over 4k^{0}g_{00}}}{m_{a}\over p^{0}}\{f_{a}\left(\sqrt{(-g_{00})(p^{0})^{2}-m_{a}^{2}}\right)$
	$\displaystyle\times[1+f_{\lambda}(\sqrt{g_{ij}k^{i}k^{j}})+f_{\lambda}(\sqrt{-g_{00}}p^{0}-\sqrt{g_{ij}k^{i}k^{j}})]$
	$\displaystyle-f_{\lambda}(\sqrt{g_{ij}k^{i}k^{j}})f_{\lambda}(\sqrt{-g_{00}}p^{0}-\sqrt{g_{ij}k^{i}k^{j}})\}$
	$\displaystyle\times dp^{0}d(\sqrt{g_{ij}k^{i}k^{j}})~{},$

or alternatively

		$\displaystyle\frac{dn_{\lambda}}{dt}={m_{a}\Gamma_{a}\sqrt{-g_{00}}\over 2\pi^{2}}\int\int_{m_{a}^{2}\over 4\sqrt{g_{ij}k^{i}k^{j}}}{m_{a}d(\sqrt{g_{ij}k_{1}^{i}k_{1}^{j}})d(\sqrt{g_{ij}k^{i}k^{j}})\over\sqrt{g_{ij}k^{i}k^{j}}+\sqrt{g_{ij}k_{1}^{i}k_{1}^{j}}}$		(11)
		$\displaystyle\times\{f_{a}\left(\sqrt{(\sqrt{g_{ij}k^{i}k^{j}}+\sqrt{g_{ij}k_{1}^{i}k_{1}^{j}})^{2}-m_{a}^{2}}\right)[1+f_{\lambda}(\sqrt{g_{ij}k^{i}k^{j}})$
		$\displaystyle+f_{\lambda}(\sqrt{g_{ij}k_{1}^{i}k_{1}^{j}})]-f_{\lambda}(\sqrt{g_{ij}k^{i}k^{j}})f_{\lambda}(\sqrt{g_{ij}k_{1}^{i}k_{1}^{j}})\}~{}.$

We assume that the various dependences of the axion occupation number are separable, and of the form

		$\displaystyle f_{a}(\sqrt{g_{ij}p^{i}p^{j}},r,t)$
	$\displaystyle=$	$\displaystyle\Theta(p_{\text{max}}-\sqrt{g_{ij}p^{i}p^{j}})[f_{ac}(t)\Theta(r_{+}-r)\Theta(r-r_{-})$
		$\displaystyle+f_{ad}(t)d(r)]~{}.$		(12)

We let $r_{+}\sim r_{-}$ both be far beyond the event horizon if the host object is a black hole, with the small distortion $d(r)$ in the region $(r_{-}\leq r\leq r_{+})$ away from a uniform distribution. In the Appendix, we find that the maximum momentum of an axion is

$$p_{\text{max}}=m_{a}\beta^{\prime}=m_{a}\sqrt{{-g_{00}(r_{+})\over-g_{00}(r_{-})}-1}~{}.$$

Since we set the maximum axion momentum to be $m_{a}\beta^{\prime}$, according to the bounds mentioned in the previous section, extremes of photon momentum become

	$\displaystyle\sqrt{g_{ij}k^{i}k^{j}}_{\pm}=$	$\displaystyle{1\over 2}(\sqrt{-g_{00}}p^{0}\pm\sqrt{g_{ij}p^{i}p^{j}})\bigg{|}_{\sqrt{g_{ij}p^{i}p^{j}}=m_{a}\beta^{\prime}}$
	$\displaystyle=$	$\displaystyle{1\over 2}[\sqrt{m_{a}^{2}+(m_{a}\beta^{\prime})^{2}}\pm\sqrt{(m_{a}\beta^{\prime})^{2}}]$
	$\displaystyle=$	$\displaystyle{m_{a}\over 2}(\sqrt{1+\beta^{\prime 2}}\pm\beta^{\prime})~{}.$

This is somewhat different from the extreme photon momentum values in Kephart:1994uy , because there the escape velocity was defined in the non-relativistic sense.
Suppose that the photon occupation number which correspondences to that of axions (12), is

		$\displaystyle f_{\lambda}(\sqrt{g_{ij}k^{i}k^{j}},r,t)$
	$\displaystyle=$	$\displaystyle[f_{\lambda c}(t)\Theta(r_{+}-r)\Theta(r-r_{-})+f_{\lambda d}(t)d(r)]$		(13)
		$\displaystyle\times\Theta(\sqrt{g_{ij}k^{i}k^{j}}_{+}-\sqrt{g_{ij}k^{i}k^{j}})\Theta(\sqrt{g_{ij}k^{i}k^{j}}-\sqrt{g_{ij}k^{i}k^{j}}_{-})~{}.$

The volume of the shell $\Theta(\sqrt{g_{ij}k^{i}k^{j}}_{+}-\sqrt{g_{ij}k^{i}k^{j}})\Theta(\sqrt{g_{ij}k^{i}k^{j}}-\sqrt{g_{ij}k^{i}k^{j}}_{-})$ is

	$\displaystyle V_{k}\approx$	$\displaystyle 4\pi[{1\over 2}(\sqrt{g_{ij}k^{i}k^{j}}_{+}+\sqrt{g_{ij}k^{i}k^{j}}_{-})]^{2}$
		$\displaystyle\times(\sqrt{g_{ij}k^{i}k^{j}}_{+}-\sqrt{g_{ij}k^{i}k^{j}}_{-})=\pi m_{a}^{3}\beta^{\prime}(1+\beta^{\prime 2})~{}.$

The $k$ and $k_{1}$ integrations of (11) are both long and tedious and so have been relegated to the Appendix. They result in the following expression

		$\displaystyle\frac{dn_{\lambda}}{dt}={m_{a}\Gamma_{a}\over 2\pi^{2}}\sqrt{-g_{00}}$
		$\displaystyle\times\{{m_{a}^{2}\beta^{\prime 3}\over 3}[f_{ac}(t)\Theta(r_{+}-r)\Theta(r-r_{-})+f_{ad}(t)d(r)$
		$\displaystyle+2f_{ac}(t)f_{\lambda c}(t)\Theta(r_{+}-r)\Theta(r-r_{-})$
		$\displaystyle+2(f_{ac}(t)f_{\lambda d}(t)+f_{ad}(t)f_{\lambda c}(t))d(r)$
		$\displaystyle-f_{\lambda c}^{2}(t)\Theta(r_{+}-r)\Theta(r-r_{-})-2f_{\lambda c}(t)f_{\lambda d}(t)d(r)]$
		$\displaystyle-{m_{a}^{2}\beta^{\prime 2}\over 2}[f_{\lambda c}^{2}(t)\Theta(r_{+}-r)\Theta(r-r_{-})+2f_{\lambda c}(t)f_{\lambda d}(t)d(r)]\}~{},$		(14)

where the subscripts $c$ and $d$ refer to the flat and distorted part of the spectrum. (See the next two sections for more discussion of the distortion function $d(r)$.) This result will be used to analyze the Schwarzschild case in the next section. Later we apply the results to a specific model and extract the distortion caused by the relativistic corrections of the flat space case. Equation (14) is one of the main results of this work.

The rate of change in axion number density is the opposite of that of photon, and is

$\displaystyle\frac{dn_{a}}{dt}=-{1\over 2}\sum_{\lambda=\pm}\frac{dn_{\lambda}}{dt}~{}.$

as we find from (14).

The rate equations we have just developed applies to all static spacetime metrics, includes those from metric based theories other than general relativity. E.g., there could be static host objects in other metric based gravitational theory that allow spontaneous particle creation which serve as a source of the axion production. But for now we set the question of axion production aside and focus on the gravitational correction to stimulated radiation in axion clusters.

The major difference between this model and Kephart:1994uy is that here the axion self gravity is ignored but the gravity from the host objects(star, black hole, etc) is taken into account. If these axions were produced by perturbative black hole processes, we assume the total energy in axions be much smaller than the mass of the black hole.

We want to investigate static stationary spacetime where

$\displaystyle g_{00}=g_{00}(r)\qquad g_{ij}=g_{ij}(r,\theta)\qquad\lim_{r\rightarrow\infty}g_{\mu\nu}=\eta_{\mu\nu}~{}.$

Schwarzschild and Reissner-Nordström metric belong to this category. Because of the factor $\sqrt{-g_{00}}$, it is difficult to maintain a uniform distribution for photons along the radial direction. Thus we introduce a small distortion function $d(r)$ in the following calculations.

The axion number density at event $x^{\alpha}$ is an integration of $f_{a}(\sqrt{g_{ij}p^{i}p^{j}},r)$ over $p^{i}$,

	$\displaystyle n_{a}(t,x^{i})=$	$\displaystyle\int_{0}^{m_{a}\beta^{\prime}}\Theta(p_{\text{max}}-\sqrt{g_{ij}p^{i}p^{j}}){g_{ij}k^{i}k^{j}\over 2\pi^{2}}d(\sqrt{g_{ij}k^{i}k^{j}})$
		$\displaystyle\times[f_{ac}(t)\Theta(r_{+}-r)\Theta(r-r_{-})+f_{ad}(t)d(r)]$
	$\displaystyle=$	$\displaystyle{(m_{a}\beta^{\prime})^{3}\over 6\pi^{2}}[f_{ac}(t)\Theta(r_{+}-r)\Theta(r-r_{-})+f_{ad}(t)d(r)]$
	$\displaystyle=$	$\displaystyle[n_{ac}(t)\Theta(r_{+}-r)\Theta(r-r_{-})+n_{ad}(t)d(r)]~{}.$

The occupation numbers can be converted to number densities with

	$\displaystyle f_{ac}(t)=$	$\displaystyle{6\pi^{2}\over(m_{a}\beta^{\prime})^{3}}n_{ac}(t)$
	$\displaystyle f_{ad}(t)=$	$\displaystyle{6\pi^{2}\over(m_{a}\beta^{\prime})^{3}}n_{ad}(t)$

The number density of photon

$\displaystyle n_{\lambda}(t,x^{i})=$

$\displaystyle n_{\lambda c}(t)\Theta(r_{+}-r)\Theta(r-r_{-})+n_{\lambda d}(t)d(r)$

(15)

is an integration of photon occupation number $f_{\lambda}$ over $k^{i}$. The coefficients of occupation number and number density are related by

	$\displaystyle f_{\lambda c}(t)=$	$\displaystyle{(2\pi)^{3}n_{\lambda c}(t)\over V_{k}}={8\pi^{2}\over m_{a}^{3}\beta^{\prime}}n_{\lambda c}(t)$
	$\displaystyle f_{\lambda d}(t)=$	$\displaystyle{(2\pi)^{3}n_{\lambda d}(t)\over V_{k}}={8\pi^{2}\over m_{a}^{3}\beta^{\prime}}n_{\lambda d}(t)~{},$

which we note is different from the axion relations.