# Thin accretion disks around traversable wormholes

Farook Rahaman<EMAIL_ADDRESS>Department of Mathematics, Jadavpur
University, Kolkata 700032, West Bengal, India  Tuhina Manna
<EMAIL_ADDRESS>Department of Mathematics, St.Xavier’s
College(Autonomous), 30 Mother Teresa Sarani, Kolkata-700016, India  Rajibul
Shaikh<EMAIL_ADDRESS>Department of Physics, IIT, Kanpur, UP  Somi Aktar
<EMAIL_ADDRESS>Department of Mathematics, Jadavpur University, Kolkata
700032, West Bengal, India  Monimala Mondal<EMAIL_ADDRESS>Department of Mathematics, Jadavpur University, Kolkata 700032, West Bengal,
India  Bidisha Samanta<EMAIL_ADDRESS>Department of Mathematics,
Jadavpur University, Kolkata 700032, West Bengal, India

###### Abstract

In this paper, we aim to investigate various physical properties and
characteristics of radiation emerging from the surface of accretion disks, in
a rotating traversable axially symmetric wormhole spacetime of the Teo class.
We have studied the marginally stable orbits and accretion efficiency
graphically, corresponding to different values of dimensionless spin parameter
$J/M^{2}$ ranging from 0.2 to 1.5 and some values of the throat radius
$r_{0}$, in comparison to the Kerr black hole with the same parameter values,
and also tabulated the results. The energy flux radiated by the accretion disk
$F(r)$, the temperature distribution $T(r)$ and the emission spectra $\nu
L(\nu)$ is plotted, corresponding to varying values of the dimensionless spin
parameter $J/M^{2}$ and throat radius $r_{0}$. Also, the critical frequency at
which the luminosity attains its maximum value, for various values of the
angular momentum of the wormhole $J/M^{2}$ and $r_{0}$ is tabulated. Lastly,
we have employed ray-tracing technique, to produce the intensity map of the
image of an accretion disk, as observed by an asymptotic observer, under two
conditions: firstly when the disk is on the same side as the observer and we
have also compared those with the images of an accretion disk in case of Kerr
black hole with same parameters. Secondly, the images have been provided when
the disk and observer are on opposite sides of the throat. This study may help
to detect and distinguish wormhole geometries from other compact objects.

###### pacs:

04.40.Nr, 04.20.Jb, 04.20.Dw

Keywords : Wormhole; Thin accretion disk; Marginally Stable Circular Orbits

##  I Introduction:

It is argued that the mass of almost all astrophysical bodies develop through
accretion. The current observation of the center of the galaxy M87 through the
Event Horizon Telescope suggests that in most of the active galactic nuclei
(AGN’s), there exist gas clouds around the galactic center, together with an
accompanying accretion disc. These gas clouds (scales from a tenth of a parsec
to a few hundred parsecs ) are expected to develop a geometrically and
optically distorted disc that engrosses maximum of the ultraviolet radiation
and the soft X-rays. Significant astrophysical evidence can be obtained from
the observation of the motion of the gas flows in the gravitational field of
super massive objects. The temperature in the innermost region of the disk is
about $10^{7}K$ and from this hot region, the X-ray emission arises. To study
the strong field gravity, one can use these highly compact objects, as the
detected emission arises from regions close to these compact objects.
According to the standard theory, the accretion disk is presumed to be an
optically thick Newtonian type. This model suggests that the local emergent
flux ( just like a black body ) is associated with the energy decadence at a
specific radial point in the disk. The sum of black body components coming
from various radial positions in the disk yields the observed spectrum.
Usually, general relativistic issues change this Newtonian spectrum by
incorporating Doppler broadening, gravitational redshifts, and light bending.

Since wormholes kg1 , kg14  are a specific type of astrophysical objects, so
it has been an active area of research to study accretion phenomenon around
wormholes. The phenomenon of accretion around compact objects is an
outstanding example for studying the signatures of wormholes versus black
holes. Initial research on accretion disks using a Newtonian approach was
conducted by Shakura and R.A. Sunyaev.SS  Later Thorne et. al. first
developed a general relativistic model of the thin accretion disk, where the
accreting matter had a Keplerian motion. They assumed the disk is in a steady-
state, i.e., the mass accretion rate $\dot{M}$ does not change with time, as
well as it is independent of the radius of the disk. From a theoretical point
of view, the efficient cooling due to radiation transport, maintains this disk
in hydrodynamic and thermodynamic equilibrium, thus ensuring a black body
electromagnetic spectrum. The spectra from the sources may provide some
information on the geometry and dynamics of innermost regions of accretion
disks. Following Page-Thorne Page  model, one can study emissivity properties
such as the luminosity spectra, flux of radiation, temperature profile using
accretion around wormholes Luminet . Further radiation properties of thin
accretion disks were analysed in Thorne ; Gair , along with the effects of the
photon capture by the black hole on the spin evolution. The emissivity
properties of the accretion disks, which helps us to study the conversion of
rest mass into emitting radiation in the accretion disk were studied in Bhat ,
Torres , Yuan , Guzman . The work in these papers have helped to distinguish
between the accretion properties around different classes of exotic central
objects, such as non-rotating or rotating quark, boson or fermion star, by
studying the radiation power per unit area, the temperature of the disk and
the emission spectrum of the emitted radiation and comparing them with the
Schwarzschild and Kerr black hole. Through the investigation of differences in
the thermodynamic and electromagnetic properties of the accretion disks,
Kovacs et. al. K , identified the observation signatures that may help to
distinguish between black holes and naked singularities. Image of the Janis-
Newman-Winicour naked singularity with a thin accretion disk was studied in
Gyul . They concluded that for some specific values of the spin parameter and
of the scalar charge, the radiation energy flux and conversion efficiency from
the disk around a rotating naked singularity is higher than that around a Kerr
black hole. Accretion around rotating wormholes is an active area of current
research. Recently, gravitational wave echoes of the Damour-Solodukhin
wormhole (RDSWH) have been studied by Bueno et al. Bueno . Nandi et.al.
further studied a rotating generalization of the RDSWHNandi  and concluded
that RDSWH are experimentally indistinguishable from Kerr black hole by
accretion characteristics. Similar works on comparative study of thin
accretion disks around rotating gravastars and Kerr-type black holes, in order
to distinguish between them, was undertaken byHarko2 . It was concluded that
gravastars provide a less efficient mechanism for converting mass to radiation
than black holes. Many major works in modified theories of gravity are also
available in literature. The physical properties of a thin accretion disk in
the static and spherically symmetric spacetime metric of vacuum $f(R)$ gravity
was analysed in Pun . Thus astrophysical observations of the emission spectra
from accretion disks may lead us to the possibility of directly testing
modified gravity models. Several classes of static and rotating brane world
black holes have been investigated in Pun2 , and compared with the general
relativistic case. The thin accretion disk features in dynamical Chern-Simons
modified gravity have been studied extensively in Harko3  and in Horava-
Lifshitz gravity in Harko4 . An interesting work on thin accretion disk around
a rotating Kerr-like black hole in Einstein-bumblebee gravity model has been
done by Bumblebee . Research on the properties of the radiation emerging from
the surface of the accretion disk around static and rotating axially symmetric
wormhole geometries were conducted in Harko5 , Harko6 . In Paul  numerically
constructed images of thin accretion disks in rotating wormhole backgrounds,
for the Kerr-like and the Teo class of wormholes was studied, and
interestingly it showed that there are dramatic differences between accretion
disk images in wormhole backgrounds, compared to black hole ones, owing to the
fact that a wormhole can theoretically have accretion disks on both sides of
its throat. Although, Paul  gives a full explanation of using ray-tracing of a
general Teo wormhole, they have considered specific forms of the metric
functions in order to generate results and images. In our present work, we
have not merely changed parameters, but have taken completely different forms
of the metric functions, thereby giving a wormhole with completely different
spacetime geometry or gravitational field within the Teo class. Therefore,
within the Teo class, depending on the forms of the metric functions, we could
have various wormholes with various spacetime geometries or gravitational
field, provided the forms of the metric functions satisfy the conditions
obtained by Teo. The observational signatures in these various cases in
principle will be different. The specific example which we have considered is
completely different from the one considered in Paul . Moreover, the focus of
Paul  was on constructing the images, whereas our focus was not only on the
images but also on the other properties such as luminosity, critical frequency
etc. We believe that the results presented in the paper will indeed be
relevant in future observations for the reasons explained there.

The motivation of the study is as follows. It is well-known that, according to
the Kerr Black Hole Hypothesis, the spacetime geometry around astrophysical
compact objects is well described by the Kerr solution of General Relativity
(GR). However, in modified gravity theories or in presence of quantum effect
or exotic matter, one can expect deviation from the Kerr spacetime. Such
deviations should be tested using observations. Therefore, it is
astrophysically relevant to study various observational aspects of other
possible geometries beside those of the Kerr solution. The motivation behind
such studies is to see whether to what extent we can distinguish between a
Kerr black hole and a deviated non-Kerr spacetime. In cases where differences
in the observational signatures arise, we can distinguish a Kerr black hole
from a non-Kerr geometry. However, in cases where differences do not arise, a
non-Ker geometry mimics a Kerr black hole. In our work, we have studied a
similar aspect by considering a wormhole and a Kerr black hole. Although in
the literature, there are several wormhole spacetimes which are obtained as
exact solutions of GR as well as other modified gravity theories, and matter
for such wormholes are known, most of these wormholes are non-rotating
solutions. This is due to difficulties associated in obtaining rotating
solutions as exact solutions. Given this difficulty, researchers often
consider rotating wormhole models which are not exact solutions of any gravity
theory, but can be used in modelling horizonless compact objects. The
motivation behind this latter approach is comparative studies between a black
hole and a wormhole through their observational aspects. In other words, the
objective in this case is not the matter on the RHS of the field equations,
but to see whether and to what extent we could distinguish between a wormhole
and a black hole or to test the Kerr Black Hole Hypothesis through
observational aspects. The literature is replete with several such kind of
approaches. In our work also, the motivation was to explore comparative
studies between a wormhole and a black hole through observational signatures
of accretion disks around them.

The paper is organized as follows: in section II we have described some
physical properties of thin accretion disks, followed by the investigation of
marginally stable circular orbits in section III. Next in section IV, we have
studied the accretion disk properties of a rotating traversable wormhole of
the Teo class in details. Following this, in section V, the images of an
accretion disk around a wormhole is analysed as observed by an asymptotic
observer whose location is taken first on the same side of the throat as the
disk; and again on the side of the throat opposite to that of the disk.
Lastly, in section VI we have mentioned some concluding remarks.

##  II  Some physical properties of thin accretion disks

Usually, it is claimed that compact X-ray sources are binary systems. One is a
normal star and the other is a highly compact object and former landfills
material onto its latter companion. The majority of the standard models for
this mass transfer suggest that the relocated material forms a thin disk
around the highly compact objects. The viscous stresses handover angular
momentum outside over the disk, causing the material to spiral steadily
inward. Also, these viscous stresses, functioning against the disk’s
differential rotation, warm the disk thus producing a large flux of X-rays.
According to steady-state models, the disk accretion ( in which accreting
material is interstellar gas and magnetic fields ) onto a highly compact
object may happen in the nuclei of galaxies. The gravitational energy of the
accreting material converts to heat and some parts of the heat transform into
radiation. This radiation partly seeps out and chills the accretion disc.
After receiving this radiation and analyzing its electromagnetic spectrum
through radio, optical and X-ray telescopes , we get information about the
disk as well as compact object surrounded by the disk. In the analysis of mass
accretion, we adopt a geometrically thin accretion disk. In the thin accretion
disk, the vertical size of the disk is insignificant compared to its
horizontal extension i.e. most of the matter lies close to the radial plane.
This means that the disk height $H$ ( described by the maximum half-thickness
of the disk ), is considerably lesser than any characteristic radii $R$ of the
disk, $\frac{H}{R}<<1$. Here, the vertical size in cylindrical coordinates
$(r,\phi,z)$ defined along the z-axis is negligible, Also for this
consideration, stress and dynamic friction in the disk generates heat that can
be spread through the radiation above its surface. This cooling process
confirms that the disk could be in hydrodynamical equilibrium and as a result,
the mass accretion rate $\dot{M}$ in the disk does not change with time i.e.
constant. Since the disk is steadied at hydrodynamic equilibrium, the pressure
and vertical entropy gradient should be negligible. This thin disk has an
inner edge defined by the marginally stable circular radius $r_{ms}$ and the
accreting matter at higher radii has a Keplerian motion. This thin accretion
disk supposition implies that spacetime metric components as well as the
accretion particles move in Keplerian orbits around the compact objects with a
rotational velocity, $\Omega=\frac{d\phi}{dt}$ and its specific energy E $\&$
specific angular momentum L depends only on the radii of the orbits i.e. on
the radial coordinate. The accretion particles with the four-velocity
$u_{\mu}$ form a disk and the disk has an averaged surface density $\Sigma$,

$\Sigma=\int_{-H}^{H}<\rho_{0}>dz,$ (1)

where $\rho_{0}$ is the density of rest mass and H is the disk height during
the time $\Delta t$. Here, $<\rho_{0}>$ is defined as the averaged rest mass
density in time $\Delta t$ with angle $2\pi$. The orbiting particles in the
disk is modeled by an anisotropic fluid source which is given by,

$T^{\mu\nu}=\rho_{0}u^{\mu}u^{\nu}+2u^{(\mu}q^{\nu)}+t^{\mu\nu},$ (2)

where the energy flow vector $q_{\mu}$ and the stress tensor $t^{\mu\nu}$ are
measured in the averaged rest-frame with
$u_{\mu}q^{\mu}=0,~{}u_{\mu}t^{\mu\nu}=0.$ Here we neglect specific heat.
Another characteristic of the disk structure is the torque which is given by,

$W_{\phi}^{r}=\int_{-H}^{H}<t_{\phi}^{r}>dz,$ (3)

with the averaged component $<t_{\phi}^{r}>$ in time $\Delta t$ with angle
$2\pi$. The most significant physical measure recounting the disk is the
radiation flux $F(r)$ over the disk surface that can be obtained from the time
and orbital average of the energy flux vector as,

$F(r)=<q^{z}>.$ (4)

Using stress energy tensor of the disk $T^{\mu\nu}$, one can define the four-
vectors of the energy and of the angular momentum flux in a local basis as,

$-E^{\mu}=T^{\mu}_{\nu}\left(\frac{\partial}{\partial
t}\right)^{\nu}~{}and~{}J^{\mu}=T^{\mu}_{\nu}\left(\frac{\partial}{\partial\phi}\right)^{\nu},$
(5)

respectively.

Now, the structure equations of the thin disk could be obtained from the
conservation laws of the rest mass, of the energy, and of the angular
momentum. Rest mass conservation,

$\nabla_{\mu}(\rho_{0}u^{\mu})=0$ (6)

yields the time averaged rate of the rest mass accretion, which is independent
of the disk radius,

$\dot{M}\equiv-2\pi\sqrt{-g}\Sigma u^{r}=~{}\text{constant},$ (7)

where dot implies the differentiation with respect to the time coordinate t.
The energy conservation law, $\nabla_{\mu}E^{\mu}=0$ yields,

$\dot{M}E-2\pi\sqrt{-g}\Omega W_{\phi}^{r}]_{,r}=4\pi\sqrt{-g}FE,$ (8)

where a comma represents the derivative with respect to the radial coordinate
r. This is an equilibrium equation, which suggests that the energy conveyed by
the rest mass flow, $\dot{M}E$, and the energy conveyed by the torque in the
disk, $2\pi\sqrt{-g}\Omega W_{\phi}^{r}$, is well-adjusted by the energy
emitted from the surface of the disk, $4\pi\sqrt{-g}FE$. Angular momentum
conservation law, $\nabla_{\mu}J^{\mu}=0$, yields,

$[\dot{M}L-2\pi\sqrt{-g}W_{\phi}^{r}]_{,r}=4\pi\sqrt{-g}FL.$ (9)

As before, the first term characterizes the angular momentum supported by the
rest mass of the disk; the second term is the angular momentum conveyed by the
torque in the disk; the right hand side term is the angular momentum transport
from the disk’s surface by radiation. Using above two equations, one can find
an expression for the energy flux F by eliminating $W_{\phi}^{r}$. Now,
utilizing the relation between energy and angular momentum $dE=QdJ$ for which
the circular geodesic orbits assumes the form $E_{,r}=\Omega L_{,r}$, one can
obtain the flux $F$ of the radiant energy over the disk in terms of the
specific energy, angular momentum and of the angular velocity of the disk
matter as,

$F(r)=-\frac{\dot{M}}{4\pi\sqrt{-g}}\frac{\Omega_{,r}}{(E-\Omega
L)^{2}}\int_{r_{ms}}^{r}(E-\Omega L)L_{,r}dr.$ (10)

Here, the “no torque” inner boundary condition was assumed which means torque
disappears at the inner edge of the disk as the accreting matter at the
marginally stable orbit $r_{ms}$ drops spontaneously into the centre of the
compact objects, and cannot wield significant torque on the disk. Note that
since the radiant energy flux of the disk is proportional to the accretion
rate $\dot{M}$, hence the radiation flux from the disk increases linearly with
the accretion rate $\dot{M}$.

One more significant feature of the mass accretion activity is the efficiency
$\epsilon$ ( computed by the specific energy of a particle at the marginally
stable orbit $r_{ms}$ when all the radiated photons can seepage to infinity )
with which the central object changes rest mass into outward radiation. The
accretion efficiency $\epsilon$ (should be non-negative) is given by,

$\epsilon=1-E(r_{ms}).$ (11)

In the steady-state thin disk accretion model, it is argued that the accreting
matter is in thermodynamical equilibrium. As a result, the radiation produced
by the entire disk surface can be recounted as a perfect black body radiation,
where the energy flux is given,

$F(r)=\sigma T^{4}(r),$ (12)

with $T$ is the effective temperature of the geometrically thin black-body
disk, and $\sigma=5.670374419\times 10^{-8}~{}J~{}s^{-1}m^{-2}K^{-4}$ is the
Stefan-Boltzmann constant. Allowing for that the disk radiates as a black
body, then the observed luminosity $L(\nu)$ of the disk can be exemplified by
a redshifted black body spectrum as,

$L(\nu)=4\pi d^{2}I(\nu)=\frac{8\pi h\cos
i}{c^{2}}\int_{r_{in}}^{r_{out}}\int_{0}^{2\pi}\frac{\nu_{e}^{3}rdrd\phi}{\exp[\frac{h\nu_{e}}{\kappa_{B}T}]-1}.$
(13)

Here, d is the distance between the observer and the center of the disk ,
$r_{in}$ and $r_{out}$ stand for the inner and outer radii of the disc
respectively ( usually, one takes $r_{in}=r_{ms}$ and $r_{out}=\infty$ ) , $i$
is the disk inclination angle to the vertical, $h$ is the Planck constant,
$k_{B}$ is the Boltzmann constant, $\nu_{e}$ is the emission frequency and
$I(\nu)$ indicates the thermal energy flux radiated by the disk. The observed
photons are redshifted and the detected frequency $\nu$ is associated with the
discharged radiation frequency from the disk as,

$\nu_{e}=(1+z)\nu.$ (14)

For a common axisymmetric metric that represents the space-time geometry
around a rotating compact object, the redshift factor $(1+z)$ which comprises
the effects of both gravitational redshift and Doppler shift, takes the
following form,

$1+z=\frac{1+\Omega r\sin i\sin\phi}{\sqrt{-g_{tt}-2\Omega
g_{t\phi}-\Omega^{2}g_{\phi\phi}}}~{}~{}.$ (15)

Here, the light bending effect has been neglected.

##  III Marginally Stable Circular Orbits :

The thin accretion disk is made by the timelike particles moving in stable
circular timelike geodesics. These timelike geodesics around compact
astrophysical objects are very significant for theoretical as well as
experimental point of view as they convey the information about the compact
object where very strong gravity takes place. For an arbitrary stationary and
axially symmetric geometry, let us consider the metric in a general form as

$ds^{2}=-g_{tt}dt^{2}+2g_{t\phi}dtd\phi+g_{rr}dr^{2}+g_{\theta\theta}d\theta^{2}+g_{\phi\phi}d\phi^{2}.$
(16)

In the equatorial approximation $(|\theta-\frac{\pi}{2}|<<1)$, the metric
functions
$g_{tt},~{}g_{rr},g_{\phi\phi},g_{t\phi}~{}and~{}g_{\theta\theta}~{}$ will be
the function of the radial coordinate r.

Now we consider the geodesic of radial timelike particles in the planes
$\theta=constant$, for which the Lagrangian assumes the following form ( with
$\tau$ is an affine parameter of the world line of the particle ),

$L^{*}=-g_{tt}\dot{t}^{2}+2g_{t\phi}\dot{t}\dot{\phi}+g_{rr}\dot{r}^{2}+g_{\phi\phi}\dot{\phi}^{2}.$
(17)

Here the dot means derivative with respect to $\tau$.

Euler -Lagrangian equation $\left(\frac{d}{d\tau}\left(\frac{\partial
L^{*}}{\partial\dot{x}}\right)-\frac{\partial L^{*}}{\partial
x}=0,~{}~{}x=t,\phi\right)$ yields,

$\displaystyle-g_{tt}\dot{t}+g_{t\phi}\dot{\phi}=$ $\displaystyle-E,$
$\displaystyle g_{t\phi}\dot{t}+g_{\phi\phi}\dot{\phi}=$ $\displaystyle L.$
(18)

Here, $E$ and $L$ be two constants of motion for particles which are known as
the specific energy and the specific angular momentum , respectively of the
particles moving on circular orbits around the compact objects. Solving the
above two equations , we obtain the orbiting particle’s four-velocity
components as,

$\displaystyle u^{t}\equiv\dot{t}$
$\displaystyle=\frac{Lg_{t\phi}+g_{\phi\phi}E}{g_{tt}g_{\phi\phi}+g_{t\phi}^{2}},$
$\displaystyle u^{\phi}\equiv\dot{\phi}$
$\displaystyle=\frac{-g_{t\phi}E+Lg_{tt}}{g_{tt}g_{\phi\phi}+g_{t\phi}^{2}}.$
(19)

For time-like particles the first integral of geodesics equation
($g_{\mu\nu}\dot{x^{\mu}}\dot{x^{\nu}}=-1$) yields,

$\displaystyle
g_{rr}\dot{r}^{2}=-1+g_{tt}\dot{t}^{2}-2g_{t\phi}\dot{t}\dot{\phi}-g_{\phi\phi}\dot{\phi}^{2}.$

Putting the values of $\dot{t}$, $\dot{\phi}$, one obtains,

$\displaystyle
g_{rr}\bigg{(}\frac{dr}{d\tau}\bigg{)}^{2}=-1+\frac{E^{2}g_{\phi\phi}+2ELg_{t\phi}-L^{2}g_{tt}}{g_{t\phi}^{2}+g_{tt}g_{\phi\phi}}.$

From the above equation, one can define an effective potential term as,

$V_{eff}(r)=-1+\frac{E^{2}g_{\phi\phi}+2ELg_{t\phi}-L^{2}g_{tt}}{g_{t\phi}^{2}+g_{tt}g_{\phi\phi}}.$
(20)

For stable circular orbits in the equatorial plane, the time-like geodesic
satisfies $\dot{r}=0,\ddot{r}=0~{}and~{}\dddot{r}<0$. In terms of the
effective potential, these conditions take the forms as
$V_{eff}=0,~{}~{}\frac{dV_{eff}}{dr}=0~{}~{}and~{}~{}\frac{d^{2}V_{eff}}{dr^{2}}<0.$
The first two conditions provide the kinematic parameters such as the specific
energy, the specific angular momentum and the angular velocity of the
particles moving in the stable circular orbits .

Here,

$\displaystyle\frac{d\phi}{dt}=\frac{\dot{\phi}}{\dot{t}}=\Omega=\frac{Lg_{tt}-g_{t\phi}E}{Lg_{t\phi}+g_{\phi\phi}E}.$
$\Rightarrow L=\frac{(g_{t\phi}+\Omega g_{\phi\phi})E}{g_{tt}-\Omega
g_{t\phi}}.$ (21)

Now, $V_{eff}(r)=0$ implies

$\displaystyle
E^{2}g_{\phi\phi}+2ELg_{t\phi}-L^{2}g_{tt}=(g_{t\phi})^{2}+g_{tt}g_{\phi\phi}.$

Putting the value of $L$ in the above equation, from eq. (21) one can get

$E=\frac{g_{tt}-\Omega g_{t\phi}}{\sqrt{g_{tt}-2\Omega
g_{t\phi}-g_{\phi\phi}\Omega^{2}}}.$ (22)

So, using the above expression in eq. (21) we obtain,

$L=\frac{g_{t\phi}+\Omega g_{\phi\phi}}{\sqrt{g_{tt}-2\Omega
g_{t\phi}-g_{\phi\phi}\Omega^{2}}}.$ (23)

The condition, $\frac{dV_{eff}(r)}{dr}=0$, yields,

$\Omega=\frac{-g_{t\phi,r}\pm\sqrt{(g_{t\phi,r})^{2}+g_{tt,r}g_{\phi\phi,r}}}{g_{\phi\phi,r}}.$
(24)

The marginally stable circular orbits around the compact object could be found
from the following condition:

$\frac{d^{2}V_{eff}}{dr^{2}}=0.$ (25)

##  IV Accretion disk properties of a rotating traversable wormhole of Teo
class

We first study a rotating generalization of the static Morris-Thorne wormhole
kg14 . For that consider the following stationary, axisymmetric spacetime
metric describing a rotating traversable wormhole of Teo class Teo , given as,

$ds^{2}=-N^{2}dt^{2}+\frac{dr^{2}}{1-\frac{b}{r}}+r^{2}K^{2}\left[d\theta^{2}+\sin^{2}\theta\left(d\varphi-\omega
dt^{2}\right)\right],$ (26)

where $-\infty<t<\infty$, and $r_{0}\leq r<\infty$ ,$0\leq\theta\leq\pi$ and
$0\leq\varphi\leq 2\pi$ are in spherical coordinates. $N,b,K$, and $\omega$
are functions of $r$ and $\theta$ only, such that they are regular on the axis
of symmetry $\theta=0,\pi$. This spacetime describes two identical,
asymptotically flat regions, with the throat, $r=r_{0}=b>0$ as the boundary.
Since the spacetime geometry of the two regions of the wormhole connected
through the throat are exact copies, the wormhole is symmetric about the
throat. The function $N$ is interpreted as the redshift function while $b$ as
the shape function of the wormhole. In order to represent a wormhole the
metric eq.(26) must be devoid of any event horizons or curvature
singularities. For that the redshift function N must be non-zero and finite
everywhere in $r_{0}\leq r<\infty$. Also we must have,
$\partial_{\theta}b\mid_{r=r_{0}}=0$ to avoid any event horizon or curvature
singularity at the throat, and the shape function $b$ must satisfy the flare-
out condition at the throat, $\partial_{r}b\mid_{r=r_{0}}<1$ and $b\leq r$.
The function $K$ determines the area radius given by $R=rK$ while the function
$\omega$ measures the angular velocity of the wormhole Teo . As an specific
example, we consider a wormhole considered in Konoplya_2016  with the metric
functions given by:

$\displaystyle N$ $\displaystyle=\sqrt{1-\frac{2M}{r}},$ $\displaystyle b$
$\displaystyle=\frac{r_{0}^{2}+2M\left(r-r_{0}\right)}{r},$ $\displaystyle K$
$\displaystyle=1,$ $\displaystyle\omega$
$\displaystyle=\frac{2Ma}{r^{3}}=\frac{2J}{r^{3}}.$ (27)

where J is the angular momentum of the wormhole and $M$ is the mass. The
throat is at $r=r_{0}>2M$.

From eq.(20) we can write the expression of effective potential as :

$V_{eff}=\frac{-E^{2}r^{6}+4ELJr^{3}-2L^{2}Mr^{3}+L^{2}r^{4}-2r^{5}M+r^{6}-4J^{2}L^{2}}{r^{5}\left(-r+2M\right)},$
(28)

which is obtained from the geodesic motion of the test particles in the
equatorial plane of the wormhole. From the conditions, $V_{eff}(r)=0$ and
$\frac{dV_{eff}(r)}{dr}=0$, the expressions of the various kinematic
properties can be obtained as below. From eq. (22) we get the specific energy
as,

$E=\frac{r^{4}-2Mr^{3}-6J^{2}+2J\sqrt{Mr^{3}+9J^{2}}}{r^{2}\sqrt{r^{4}-3Mr^{3}-18J^{2}+6J\sqrt{Mr^{3}+9J^{2}}}},$
(29)

and from eq. (23) we obtain the specific angular momentum as,

$L=\frac{r\left(\sqrt{Mr^{3}+9J^{2}}-3J\right)}{\sqrt{r^{4}-3Mr^{3}-18J^{2}+6J\sqrt{Mr^{3}+9J^{2}}}}.$
(30)

The angular velocity of the particles in the accretion disk, moving in the
stable circular orbit are given by eq. (24) as,

$\Omega=\frac{\sqrt{Mr^{3}+9J^{2}}-J}{r^{3}}.$ (31)

Figure 1: The Marginally stable circular orbits ($r_{ms}$) and the efficiency
of an accretion disk ($\epsilon$), for different values of $J/M^{2}$ and
$r_{0}$.

In Fig. 1, we have shown the dependence of the marginally stable circular
orbits $r_{ms}$ and the efficiency $\epsilon$ of the accretion disk on of the
angular momentum of the wormhole $J$ for different wormhole throat size
$r_{0}$, along with the corresponding results for a Kerr black hole. From a
physical point of view, the marginally stable circular orbits form the
innermost stable circular orbits for the wormhole, beyond which the particle
will fall into the throat and will be lost to the other universe. Note that,
for a given angular momentum and throat size, $r_{ms}$ cannot be less than
$r_{0}$. If all the roots of $\frac{d^{2}V_{eff}}{dr^{2}}=0$ are found to be
smaller than $r_{0}$, then we take $r_{ms}=r_{0}$ as the throat acts as the
position of the marginal stable orbits in that case. See (Paul, ) for more
details. In Table 1, the numerical values of the position of marginally stable
circular orbits $r_{ms}$ and the efficiency $\epsilon$ of the accretion disk,
corresponding to various values of the angular momentum of the wormhole $J$
and throat size $r_{0}$, are tabulated, along with the corresponding results
for a Kerr black hole.

Table 1: The Marginally stable circular orbits ($r_{ms}$) and the efficiency of the accretion disk ($\epsilon$), for different values of $J/M^{2}$ and $r_{0}$. |  $r_{ms}$ (in M) | Efficiency $\epsilon$
---|---|---
| Wormhole | Kerr Black | Wormhole | Kerr Black
$J/M^{2}$ | $r_{0}=2.5M$ | $r_{0}=4M$ | hole | $r_{0}=2.5M$ | $r_{0}=4M$ | hole
0 | 6.0 | 6.0 | 6.0 | 0.05719 | 0.05719 | 0.05719
0.2 | 5.32761 | 5.32761 | 5.32944 | 0.06474 | 0.06474 | 0.064634
0.5 | 4.28536 | 4.28536 | 4.23300 | 0.08254 | 0.08254 | 0.08212
0.7 | 3.67576 | 4.0 | 3.39313 | 0.10112 | 0.09940 | 0.10361
0.9 | 3.25027 | 4.0 | 2.32088 | 0.12446 | 0.11299 | 0.15575
1.5 | 2.75805 | 4.0 | — | 0.18496 | 0.13821 | —

To obtain the flux radiated, temperature and emission spectra of the disk, we
consider the mass of the wormhole to be $M=15M_{\odot}$ (Imp, ) and the mass
accretion rate to be $\dot{M}\approx 3\times 10^{-12}M$/year. Also, to obtain
the results in the physical units, we restore $G$ and $c$ by replacing $M$ by
$GM/c^{2}$ and $\dot{M}$ by $\dot{M}c^{2}$, where $G$ and $c$ are,
respectively, Newton’s gravitational constant and speed of light in vacuum.
The flux $F(r)$ and the temperature $T(r)$ of the disk for different values of
the dimensionless spin $J/M^{2}$ and throat size $r_{0}$ are, respectively,
plotted in Figs. 2 and 3, along with the corresponding results for a Kerr
black hole. From the figures, we can note that the flux and the temperature
sharply increase just near the throat $r_{0}$ and then decrease gradually.
Similarly, the emission spectra of the disk for various values of the
dimensionless spin $J/M^{2}$ is plotted in Fig. 4, along with the emission
spectra of a Kerr black hole. Note that, when both the spin and throat size
are small, the flux, the temperature and the emission spectra of the disk
mimic those of a Kerr black hole. However, with increasing either the spin
value or the throat size, the results for the wormhole start differing from
those of a Kerr black hole. In Table 2, following (Imp, ), we have shown the
critical frequency $\nu_{crit}$ at which the spectra of the disk have a
maximum values $\nu L(\nu)$.

Figure 2: The energy flux radiated by the accretion disk around the wormhole
(solid) for dimensionless spin values $J/M^{2}=0.2$ (black), $0.5$ (blue),
$0.7$ (magenta), $0.9$ (red), $1.2$ (orange) and $1.5$ (green). The first and
second row, respectively, corresponds to the throat size $r_{0}=2.5M$ and
$4M$. For comparison, we also have shown the corresponding plots for a Kerr
black hole (dashed) of same mass and spin. Some solid and dashed plots
coincide with one another.

Figure 3: The temperature distribution of the accretion disk around the
wormhole (solid) for dimensionless spin values $J/M^{2}=0.2$ (black), $0.5$
(blue), $0.7$ (magenta), $0.9$ (red), $1.2$ (orange) and $1.5$ (green). The
first and second row, respectively, corresponds to the throat size
$r_{0}=2.5M$ and $4M$. For comparison, we also have shown the corresponding
plots for a Kerr black hole (dashed) of the same mass and spin. Some solid and
dashed plots coincide with one another.

Figure 4: The emission spectra of the accretion disk around the wormhole (solid) for dimensionless spin values $J/M^{2}=0.2$ (black), $0.5$ (blue), $0.7$ (magenta), $0.9$ (red), $1.2$ (orange) and $1.5$ (green). The first and second row, respectively, corresponds to the throat size $r_{0}=2.5M$ and $4M$. For comparison, we also have shown the corresponding plots for a Kerr black hole (dashed) of the same mass and spin. Some solid and dashed plots coincide with one another. Table 2: The critical frequency $\nu_{crit}$ at which the spectra has a maximum value $\nu_{crit}L_{max}(\nu_{crit})$ for different values of $J/M^{2}$ and $r_{0}$. |  $\nu_{crit}$($\times 10^{16}$ Hz) |  $\nu_{crit}L_{max}(\nu_{crit})$ ($\times 10^{25}$erg $\text{s}^{-1}$)
---|---|---
| Wormhole | Kerr Black | Wormhole | Kerr Black
$J/M^{2}$ | $r_{0}=2.5M$ | $r_{0}=4M$ | hole | $r_{0}=2.5M$ | $r_{0}=4M$ | hole
0.2 | 2.663 | 2.631 | 2.665 | 3.608 | 3.559 | 3.612
0.5 | 3.110 | 3.048 | 3.110 | 4.544 | 4.433 | 4.539
0.7 | 3.520 | 3.387 | 3.588 | 5.499 | 5.216 | 5.648
0.9 | 3.975 | 3.644 | 4.503 | 6.683 | 5.869 | 8.195
1.2 | 4.555 | 3.904 | — | 8.432 | 6.618 | —
1.5 | 4.942 | 4.072 | — | 9.769 | 7.168 | —

##  V Images of an accretion disk around the wormhole

(a) $J/M^{2}=0.2$ (Kerr black hole)

(b) $J/M^{2}=0.5$ (Kerr black hole)

(c) $J/M^{2}=0.9$ (Kerr black hole)

(d) $J/M^{2}=0.2$, $r_{0}=2.5$ (Wormhole)

(e) $J/M^{2}=0.5$, $r_{0}=2.5$ (Wormhole)

(f) $J/M^{2}=0.9$, $r_{0}=2.5$ (Wormhole)

(g) $J/M^{2}=0.5$, $r_{0}=3.0$ (Wormhole)

(h) $J/M^{2}=0.5$, $r_{0}=4.0$ (Wormhole)

(i) $J/M^{2}=0.5$, $r_{0}=5.0$ (Wormhole)

Figure 5: The images of an accretion disk in a Kerr black hole and rotating
wormhole [(a)-(i)]. For wormhole, the disk is on the same side of the
observer. The outer edge of the disk is at $r=20M$ , and the observer’s
inclination angle is $\theta_{o}=80^{\circ}$. The observer is placed at the
radial coordinate $r=10^{4}M$, which corresponds effectively to the asymptotic
infinity. In order to get rid of the parameters $M$ and $\dot{M}$, we have
normalized the fluxes by the maximum flux observed for the Schwarzschild black
hole. Also, we have plotted square-root of the normalized flux for a better
visualization. The color bars show the values the flux. All spatial
coordinates are in units of $M$.

(a) $J/M^{2}=0.2$, $r_{0}=2.5$ (Wormhole) (b) $J/M^{2}=0.5$, $r_{0}=2.5$
(Wormhole) (c) $J/M^{2}=0.9$, $r_{0}=2.5$ (Wormhole)

Figure 6: The images of an accretion disk in the rotating wormhole [(a)-(c)]
when the disk is on the other side of the throat (i.e. the disk and the
observer are on two opposite sides of the throat). The outer edge of the disk
is at $r=20M$ , and the observer’s inclination angle is
$\theta_{o}=80^{\circ}$. The observer is placed at the radial coordinate
$r=10^{4}M$, which corresponds effectively to the asymptotic infinity. In
order to get rid of the parameters $M$ and $\dot{M}$, we have normalized the
fluxes by the maximum flux observed for the Schwarzschild black hole. Also, we
have plotted square-root of the normalized flux for better looking. The color
bars show the values of the flux. All spatial coordinates are in units of $M$.

We now consider how the image of an accretion disk around the wormhole looks
like. In order to produce the intensity map of the image, we implement ray-
tracing techniques discussed in (Paul, ; Shaikh_disk, ). We do not discuss the
detailed procedure here as it can be found in (Paul, ; Shaikh_disk, ). Figure
5 shows how the image of an accretion disk looks like to an asymptotic
observer when the disk and the observer are on the same side of the wormhole
throat. For comparison, we also have shown the images for a Kerr black hole.
Note that, when both the spin and the wormhole throat size are small, the
images almost mimic those of the black hole (compare among Figs. 5(a),5(b),
5(d) and 5(e)). However, with increasing spin value or the throat size, the
image starts differing both qualitatively and quantitatively from that of a
Kerr black hole (compare the flux values and the inner thin rings in Figs.
5(c) and 5(f) and in Figs. 5(b), 5(h) and 5(i)). This indicates that, when the
disk and the observer are on the same side, wormholes with larger spin or
throat size can be distinguished from a Kerr black hole with the same spin
through the observation of their disk images. Figure 6 shows the image of an
accretion disk when the disk and the observer are on the two opposite sides of
the wormhole throat. Note that, irrespective of the spin or the throat size,
the images in this case are strikingly different from those of a Kerr black
hole. In this case, a single intensity map of a disk image contains multiple
images. This unique characteristic feature of the images can in principle be
used to distinguish a wormhole from a black hole.

##  VI Concluding Remarks

In this paper, we have investigated thin accretion disk around a rotating
traversable wormhole of Teo class Teo , of mass $M=15M_{\odot}$. We have
studied the variation of the physical properties of our wormhole, with the
spin parameter $J/M^{2}$ ranging from 0.2 to 1.5 and for two specific values
of throat radius: $r_{0}=2.5M$ and $r_{0}=4M$; and hence compared those
results with that of Kerr black hole. In classical theory of general
relativity, traversable wormholes are supported by exotic matter in their
throat, which is characterized by a stress energy tensor that violates the
null energy condition. This unique feature may help to distinguish it and
compare it with the Kerr black hole geometry. The expressions of the basic
kinematic properties of the accretion disk, like angular velocity $\Omega$,
specific energy $E(r)$, specific angular momentum $L(r)$ and effective
potential term $V_{eff}$, are investigated analytically. The effect of
changing the spin parameter $J/M^{2}$ on the marginally stable circular orbits
$r_{ms}$ are studied graphically in figure 1.(a) corresponding to three values
of throat radius: $r_{0}=2.5M$ (black),$3.0M$ (red) and $4.0M$ (blue);
compared to the dashed curve representing the Kerr black hole. Note that the
curves are almost overlapping for small values of $J/M^{2}$, but as $J/M^{2}$
increases beyond $0.5$ they separate out, though the value of $r_{ms}$ can
never be less than the throat radius $r_{0}$, which explains the fact that
each of the curve corresponding to different throat radius ends in a straight
line with the constant value of $r_{ms}=r_{0}$ . The values of $r_{ms}$ are
found to be decreasing with increase in $J/M^{2}$, but are more than the
values obtained in case of Kerr black hole. The efficiency of accretion
$\epsilon$ is plotted in figure 1(b). In general, for all the different values
of throat radius $r_{0}$, efficiency increases with increase in $J/M^{2}$, but
is found to be lesser than the Kerr black hole. For small value of $J/M^{2}$
the curves overlap but for $J/M^{2}>0.5$, we find that the efficiency for the
wormhole decreases for increasing values of throat radius $r_{0}$. These
features of marginally stable circular orbits $r_{ms}$ and efficiency
$\epsilon$ can also be studied in details from table I, corresponding to the
different $J/M^{2}$ values and for the two specific throat radius $r_{0}=2.5M$
and $4.0M$ . Note that the efficiency $\epsilon$ of accretion ranges between
5.7% to 18.5 % approximately.

Now to analyze the properties of the radiation emerging from the surface of
the disk, we have plotted the energy flux of perfect black body radiation
$F(r)$ [in $erg$ $sec^{-1}cm^{-2}$] and temperature distribution $T(r)$ [in
$K$], in figures 2 and 3, respectively, for six different values of the
dimensionless spin parameter $J/M^{2}$ ranging from 0.2 to 1.5 and for throat
$r_{0}=2.5M$ (along first row) and $r_{0}=4M$ (along second row) and compared
those with that of Kerr black hole (in dashed curves). The radiation flux is
of the order of $10^{18}$ $ergsec^{-1}cm^{-2}$ while temperature is of order
$10^{5}K$. In figure 4, the emission spectra $\nu L(\nu)$, [in $ergs^{-1}$] of
the accretion disk around the wormhole is plotted corresponding to the
different values of $J/M^{2}$ and $r_{0}$ as mentioned before. In table II, we
have tabulated the value of frequency, called the critical frequency
$\nu_{crit}$ where the emission spectra has a maximum value of luminosity of
the disk $\nu_{crit}L_{max}(\nu_{crit})$, corresponding to the different
values of the dimensionless spin parameter $J/M^{2}$ and the throat radius
$r_{0}$; in order to understand in details the effect of these two parameters
on the luminosity of the disk. From the table it is evident that, along each
column corresponding to the two throat radius $r_{0}$ values, the critical
frequency $\nu_{crit}$ increases with increase in spin parameter $J/M^{2}$;
but along any row, corresponding to any particular value of $J/M^{2}$, the
$\nu_{crit}$ value is slightly higher in case of lower value of throat radius
$r_{0}$. Same trend is noted in case of the maximum luminosity values
$\nu_{crit}L_{max}(\nu_{crit})$ column. It increases with the increase in
$J/M^{2}$ along each column, when the $r_{0}$ values are kept constant; but
along any particular row, corresponding to a constant value of $J/M^{2}$, the
values are higher in case of the lower value of $r_{0}$. Also we can find that
for lower values of both spin parameter $J/M^{2}$ and throat $r_{0}$, the
critical frequency and maximum luminosity $\nu_{crit}L_{max}(\nu_{crit})$ are
closer to the Kerr black hole case, but their difference gradually increases
when the two parametric values increases.

In section V, we have tried to visualize the image of an accretion disk around
the wormhole, by the intensity map of the image using ray-tracing techniques
discussed in (Paul, ; Shaikh_disk, ). Figures 5.(d)-(i) shows how the image of
an accretion disk looks like to an asymptotic observer when the disk and the
observer are on the same side of the wormhole throat. Note that for smaller
values of both the spin and the wormhole throat size , the images almost mimic
those of the Kerr black hole shown in figures 5. (a)-(c), but when the spin
parameter values or the throat radius is increased, the image starts differing
qualitatively and quantitatively from that Kerr black hole. In figures
6.(a)-(c), the images of an accretion disk in the rotating wormhole are shown,
as observed by an asymptotic observer , when the disk is on the other side of
the throat (i.e.the disk and the observer are on two opposite sides of the
throat). Note that, irrespective of the spin or the throat size, the images of
the accretion disk when the observer is located at the opposite side of the
throat, are strikingly different from those of a Kerr black hole; and a single
intensity map of a disk image contains multiple images. This unique
characteristic feature of the images can in principle be used to distinguish a
wormhole from a black hole. Thus the detailed analysis of the physical and
geometrical properties of these images, studied in this paper, could yield
invaluable information on the nature of the accretion flows, in case of a
rotating wormhole. Also the striking results from image analysis of its
emission spectra could help to detect and distinguish wormhole geometries from
other compact objects. We hope our work could be an interesting study and
could motivate similar research in this field. For the Kerr black hole
geometry, the horizons exist for $a<M$ case. For $a>M$, the horizons do not
exist, and we have a Kerr naked singularity. While we hope that future
observations might tell whether or not a Kerr naked singularity exists, on the
theoretical front, there has been a lot of works on what could be
observational aspects of a Kerr naked singularity if it exists in nature
(new1, ; new2, ; new3, ; new4, ; new5, ). Moreover, in presence of other
parameters (or hairs), a spacetime can represent a black hole even when $a>M$.
For example, a tidally charged spacetime with $a>M$ can represent a black hole
[ see (new6, ) for a work on its shadow in light of M87∗ ]. Therefore, some
black holes with parameters other than mass and spin or some horizonless
object might have $a>M$, although their existence and otherwise should be
confirmed through future observations.

## Acknowledgments

FR would like to thank the authorities of the Inter-University Centre for
Astronomy and Astrophysics, Pune, India for providing research facilities.
This work is a part of the project submitted by FR in SERB, DST. We are also
thankful to the referees for their valuable comments and constructive
suggestions.

## Appendix

One can note that our Lorentzian wormhole has a throat at $r=r_{0}$. When
$M=0$, we have the massless wormhole found by Ellis and Bronnikov (new7, ;
new8, ). Also one can check that , $\partial_{\theta}b|_{r=r_{0}}=0$. Thus no
event horizon or curvature singularity at the throat. For this choice of b the
flare out condition $\partial_{r}b|_{r=r_{0}}<1$ and $b<r$ have been
satisfied. The wormhole considered in eq. 27 was considered in reference [27]
of our paper to study quasinormal ringing of a wormhole. It has been shown
that, depending on the parameters, the wormhole may ring similarly or
completely differently from that of a Kerr black hole. We took this metric to
explore it in the context of accretion disk properties.

To check the traversability of this wormhole, let us find out the tidal
acceleration at the throat. It shall be less than one earth gravity, i.e $\leq
9.8m/s^{2}$ for traversable. But, for simplicity, we consider the spin zero
case below.

$\displaystyle a^{\hat{j^{\prime}}}$ $\displaystyle=$
$\displaystyle-c^{2}R^{\hat{j}^{\prime}}_{\hat{0}^{\prime}\hat{k}^{\prime}\hat{0^{\prime}}}\xi^{\hat{k}}\textrm{[
see eq. 46 Morris and Thorne paper in reference [2] ]}$
${}^{\prime}a^{\prime}$ $\displaystyle=$
$\displaystyle-c^{2}R^{\hat{1}^{\prime}}_{\hat{0}^{\prime}\hat{1}^{\prime}\hat{0}^{\prime}}\xi^{\hat{1}}\textrm{(
radial acceleration)}.$

Now,

$\displaystyle
R^{\hat{1}^{\prime}}_{\hat{0}^{\prime}\hat{1}^{\prime}\hat{0}^{\prime}}=-\left(1-\frac{b}{r}\right)\bigg{[}-\phi^{\prime\prime}+\frac{b^{\prime}r-b}{2r(r-b)}\phi^{\prime}-{\phi}^{\prime
2}\bigg{]}.$

Now,

$\displaystyle e^{\phi}$ $\displaystyle=$
$\displaystyle\sqrt{1-\frac{2M}{r}}\Rightarrow\phi=\frac{1}{2}\log\bigg{(}1-\frac{2M}{r}\bigg{)},$
$\displaystyle\Rightarrow\phi^{\prime}$ $\displaystyle=$
$\displaystyle\frac{M/r^{2}}{1-\frac{2M}{r}}\Rightarrow\phi^{\prime\prime}=\frac{(1-\frac{2M}{r})(-\frac{2M}{r^{3}})-\frac{M}{r^{2}}\times\frac{2M}{r^{2}}}{(1-\frac{2M}{r})^{2}}$
$\displaystyle=$
$\displaystyle\frac{-\frac{2M}{r^{3}}(1-\frac{2M}{r})}{(1-\frac{2M}{r})^{2}}.$
$\displaystyle b$ $\displaystyle=$
$\displaystyle\frac{r_{0}^{2}+2M(r-r_{0})}{r}=2M+\frac{r_{0}^{2}-2Mr_{0}}{r},$
$\displaystyle b^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{r_{0}^{2}-2Mr_{0}}{r^{2}}.$ $\displaystyle R^{1}_{010}$
$\displaystyle=$
$\displaystyle-\left(1-\frac{b}{r}\right)\bigg{[}\frac{-\frac{2M}{r^{3}}(1-\frac{2M}{r})}{(1-\frac{2M}{r})^{2}}+\frac{\frac{M^{2}}{r^{4}}}{(1-\frac{2M}{r})^{2}}\bigg{]}-\frac{b^{\prime}r-b}{2r^{2}}\phi^{\prime}.$

At throat $r=r_{0}$, first term is zero. Now

$\displaystyle
R^{\hat{1}^{\prime}}_{\hat{0}^{\prime}\hat{1}^{\prime}\hat{0}^{\prime}}|_{r=r_{0}}$
$\displaystyle=$
$\displaystyle-\frac{b^{\prime}r-b}{2r^{2}}\phi^{\prime}|_{r=r_{0}}$
$\displaystyle=$
$\displaystyle\frac{1}{2r^{2}_{0}}\bigg{[}\frac{r_{0}^{2}-2Mr_{0}}{r_{0}}+\frac{r^{2}_{0}}{r_{0}}\bigg{]}\times\frac{\frac{M^{2}}{r^{2}_{0}}}{1-\frac{2M}{r_{0}}}$
$\displaystyle=$
$\displaystyle\frac{1}{2r^{2}_{0}}[2r_{0}-2M]\times\frac{\frac{M^{2}}{r^{2}_{0}}}{1-\frac{2M}{r_{0}}}$
$\displaystyle=$
$\displaystyle\frac{1}{r^{2}_{0}}\frac{\frac{M}{r_{0}}(1-\frac{M}{r_{0}})}{1-\frac{2M}{r_{0}}}.$

Radial tidal acceleration of throat

$\displaystyle\mid
a^{\hat{i}^{\prime}}\mid|_{r=r_{0}}=c^{2}\xi^{\prime}\times\frac{1}{r^{2}_{0}}\frac{\frac{M}{r_{0}}(1-\frac{M}{r_{0}})}{1-\frac{2M}{r_{0}}}.$

For a traveler of size $\xi^{\prime}=2$ meter and $r_{0}=2.5M=\frac{5}{2}M$,

$\displaystyle\mid
a^{\hat{i}^{\prime}}\mid|_{r=r_{0}}=2c^{2}\times\frac{4}{25M^{2}}\frac{\frac{2}{5}\times\frac{3}{5}}{\frac{1}{5}}=\frac{48c^{2}}{125M^{2}}.$

Now $r_{0}=2.5M$ is in $G=c=1$ unit. In dimension full unit
$M=\frac{GM}{c^{2}}$,

$\displaystyle\mid
a^{\hat{i}^{\prime}}\mid|_{r=r_{0}}=\frac{48}{125}\frac{c^{2}}{\frac{G^{2}M^{2}}{c^{4}}}=\frac{48}{125}\times\frac{c^{6}}{G^{2}M^{2}}.$

If we take $M$ to be of that of $M87^{\star}$ as

$M=6.5\times 10^{9}M_{0}=6.5\times 10^{9}\times 2\times 10^{30}kg=13\times
10^{39}kg,$

then

$\mid a^{\hat{i}^{\prime}}\mid|_{r=r_{0}}=\frac{48}{125}\times\frac{(3\times
10^{8})^{6}}{(6.67\times 10^{-11})^{2}(13\times 10^{39})^{2}}$
$\simeq~{}3.7\times 10^{-10}m/s^{2}$ $\Rightarrow\textrm{negligible
because}~{}~{}a^{\prime}\sim\frac{1}{M^{2}}~{}~{}\textrm{and}~{}M~{}\textrm{is
large.}$

Similarly, $a^{2}$ and $a^{3}$ will be small $\Rightarrow$ Traversable.

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# Axion-like particle oscillations

Francesca Chadha-Day

###### Abstract

String theory compactifications may generate many light axion-like particles
(ALPs) with weak couplings to electromagnetism. In general, a large number of
ALPs may exist, with a linear combination having a potentially observable
coupling to electromagnetism. The basis in which only one ALP couples to
electromagnetism is in general misaligned with the mass basis. This leads to
mixing between the ‘electromagnetic’ ALP and a number of ‘hidden’ ALPs that do
not interact directly with the photon. The process is analagous to neutrino
oscillations. I will discuss the phenomenological consequences of this mixing
for astrophysical ALP signals, in particular showing that it may significantly
reduce the predicted signal in experiments such as the CERN Axion Solar
Telescope.

##  1 Introduction

The existence of the QCD axion is well motivated by the strong CP problem [1,
2]. The QCD axion also arises in string theory models, along with a
potentially large number of axion-like particles (ALPs) that do not couple
directly to gluons [3, 4, 5, 6]. Significant experimental and theoretical
effort has been dedicated to detecting these ALPs via their coupling to
photons, either through astrophysical observations or through ground based
experiments [7]. In particular, in the presence of a background magnetic
field, ALPs and photons may interconvert, leading to striking and potentially
detectable signatures [8].

Recent work has also explored the distinctive phenomenology of the string
axiverse [9] in astrophysical and cosmological contexts - many ALPs with
different masses. For example, string axiverse phenomenology has been studied
in the context of black hole superradiance [10, 11] and the cosmology of ALP
dark matter [12, 13]. The phenomenology of multiple axions or ALPs has also
been studied in the context of inflation [14, 15], kinetic mixing between the
QCD axion and ALPs [16], isocurvature perturbations [17], cosmic birefringence
[18, 11] and decay of axion dark matter [19].

In this work, I will consider astrophysical string axiverse phenemenology in
the context of electromagnetic detection of ALPs. As shown in [20], we can
move to a basis in which a linear combination of ALPs couple to
electromagnetism. The remaining ALPs are ‘hidden’ with no direct coupling to
photons. I will show that this basis transformation leads to a mass mixing
between the electromagnetic ALP and the hidden ALPs. This mixing may have a
significant effect on a subset of ALP detection experiments.

This paper is structured as follows. In section 2, I will introduce the
Lagrangian for ALPs in the string axiverse. In section 3, I will consider
moving to the electromagnetic basis, in which only one ALP couples directly to
photons, and discuss ALP oscillations in this basis. In section 4, I will
discuss the observational consequences of these oscillations. In section 5, I
will conclude and discuss the implications of this work.

##  2 The String Axiverse

We will consider a string axiverse scenario with $N$ axion fields $\phi^{i}$.
Each $\phi^{i}$ would typically interact rather weakly with electromagnetism,
but the combined effect of many such fields leads to a potentially observable
coupling between the axion sector and the photon [20]. The $\phi^{i}$ also in
general couple to gluons as $\mathcal{L}\supset
g_{i}^{g}\phi^{i}G_{\mu\nu}\tilde{G}^{\mu\nu}$. We therefore have:

$\begin{split}\mathcal{L}\supset\sum_{i}\Bigl{(}&-\frac{1}{2}\partial^{\mu}\phi^{i}\partial_{\mu}\phi^{i}-\frac{1}{2}m_{i}^{2}(\phi^{i})^{2}\\\
&-g^{\gamma}_{i}\phi^{i}\tilde{F}^{\mu\nu}F_{\mu\nu}-g^{g}_{i}\phi^{i}\tilde{G}^{\mu\nu}G_{\mu\nu}\Bigl{)}\end{split}$
(2.1)

We will move to a basis in which only one field, $\phi_{\rm QCD}$, couples to
gluons. This field is then the QCD axion [5]. The mass and coupling to photons
of $\phi_{\rm QCD}$ are set by its interaction with the QCD sector, in
particular by mixing with the neutral pion [21]. For the remaining fields,
which do not couple to directly to gluons, their mass and coupling to photons
are in principle independent parameters. We will therefore refer to these as
axion-like particles or ALPs. Note that we are no longer in the mass basis,
and so we expect some mixing between $\phi_{\rm QCD}$ and the other
$\phi^{i}$. However, as the mass of the QCD axion is generated by QCD effects,
we will assume that the QCD basis is approximately aligned with the mass
basis, and therefore mass mixing between the QCD axion and the ALPs is
negligible. We are free to choose a basis where the mass matrix for the ALPs
is diagonal. We therefore have the ALP Lagrangian

$\mathcal{L}\supset\sum_{i}\left(-\frac{1}{2}\partial^{\mu}\phi^{i}\partial_{\mu}\phi^{i}-\frac{1}{2}m_{i}^{2}(\phi^{i})^{2}-g^{\gamma}_{i}\phi^{i}\tilde{F}^{\mu\nu}F_{\mu\nu}\right),$
(2.2)

where the $\phi_{i}$ have been redefined such that they are orthogonal to
$\phi_{\rm QCD}$. In this paper, we will discuss the electromagnetic
phenomenology of the Lagrangian in equation 2.2. The parameters $m_{i}$ and
$g^{\gamma}_{i}$, as well as the total number of ALPs depend on the details of
the string compactification. The purpose of this paper is therefore to draw
attention to the high level of model dependence in interpreting and comparing
different bounds on ALPs.

##  3 ALP oscillations in the electromagnetic basis

As described in [20], we will now move to the electromagnetic basis in which
only one ALP couples directly to $\tilde{F}^{\mu\nu}F_{\mu\nu}$. Let this
electromagnetic ALP be $\phi^{1}$. We then have

$\phi^{1}=\frac{\sum_{i}g^{\gamma}_{i}\phi^{i}}{\sqrt{\sum_{i}{g^{\gamma}_{i}}^{2}}}.$
(3.1)

We redefine the other ALP fields such that they are orthogonal to $\phi^{1}$.
In this electromagnetic basis, the mass matrix is not diagonal. The Lagrangian
is

$\mathcal{L}\supset-\sum_{i}\frac{1}{2}\partial^{\mu}\phi^{i}\partial_{\mu}\phi^{i}-\sum_{i,j}\frac{1}{2}M_{ij}\phi^{i}\phi^{j}-g^{\gamma}\phi^{1}\tilde{F}^{\mu\nu}F_{\mu\nu},$
(3.2)

where $g^{\gamma}=\sqrt{\sum_{i}(g^{\gamma}_{i})^{2}}$. As shown in [20],
$g^{\gamma}$ may be large enough to be observable. Note that $\phi_{1}$ now
has a mass mixing with each of the other hidden ALPs. We may, however, choose
a basis such that there is no mass mixing within the hidden ALP sector.

Moving to the electromagnetic basis leads to oscillations between the
electromagnetic ALP and the hidden ALPs. The effect is analagous to neutrino
oscillations. This effect occurs when an ALP is produced via an
electromagnetic process such as the Primakoff effect. In the case where ALP
masses are negligible, as considered in [20], the produced ALP will be
$\phi^{1}$ \- the linear combination of ALP fields that couple to
electromagnetism. This is analagous to the fact that weak processes produce
neutrinos in flavour eigenstates. However, $\phi^{1}$ is not a mass
eigenstate, and therefore will oscillate into the hidden ALPs as it
propagates.

The transformation between the mass and electromagnetic bases is given by

$\ket{\phi^{i}_{\rm mass}}=U_{ji}\ket{\phi^{j}_{\rm EM}},$ (3.3)

where $U$ is a unitary matrix that includes the transformation in equation
(3.1). In the ultra-relativistic limit, the mass eigenstates $\phi^{i}_{\rm
mass}$ propagate as $\ket{\phi^{i}_{\rm mass}(L)}={\rm
e}^{-i\frac{m_{i}^{2}L}{2E}}\ket{\phi^{i}_{\rm mass}(0)}$. Following the
standard derivation for neutrino oscillations, if the probability of
interconverting with the photon is negligible, the survival probability of the
electromagnetic ALP $\ket{\phi^{1}_{\rm EM}}$ after propagating a distance $L$
in vacuum is then

$P_{1\rightarrow 1}=|\braket{\phi^{1}_{\rm EM}(L)}{\phi^{1}_{\rm
EM}(0)}|^{2}=\left|\sum_{i}U^{\star}_{1i}U_{1i}{\rm
e}^{-i\frac{m_{i}^{2}L}{2E}}\right|^{2}.$ (3.4)

This gives

$P_{1\rightarrow 1}=1-4\sum_{i>j}|U_{1i}|^{2}|U_{1j}|^{2}{\rm
sin}^{2}\left(\frac{\Delta m_{ij}^{2}L}{4E}\right),$ (3.5)

where $\Delta m_{ij}^{2}\equiv m_{i}^{2}-m_{j}^{2}$.

##  4 The phenomenology of ALP oscillations

Calculations of the phenomenology of the ALP-photon interaction have in
general considered only one ALP coupled to photons, and have not considered
the mass mixing between this electromagnetic ALP and any hidden ALPs. However,
in some experiments this mass mixing may significantly affect the predicted
signal. In particular, this effect is relevant in situations where an ALP is
produced via an electromagnetic interaction and then propagates a large
distance before being detected electromagnetically. In this case, we must take
into account the probability that the electromagnetic ALP will oscillate into
an undetectable hidden ALP.

We will calculate the effect of mixing with hidden ALPs on the predicted ALP
signal from the Cern Axion Solar Telescope (CAST) [22] and from the supernova
SN1987A [23]. In both cases, we will assume that the QCD axion is too heavy to
lead to a significant signal, such that only the ALPs are relevant. In models
with a light QCD axion and an ALP coupled to electromagnetism, this would lead
to an additional contribution to the signal. CAST seeks to detect axions and
ALPs produced in the Sun via their interconversion with photons in a
laboratory magnetic field. In the case of ALP production in the Sun via the
ALP coupling to photons, the electromagnetic ALP produced may oscillate into a
hidden ALP as it propagates towards Earth. As these hidden ALPs would produce
no signal in CAST, we expect this effect to lead to a reduction in the
predicted CAST signal for a given $g^{\gamma}$ when compared to the case with
only a single ALP field with coupling $g^{\gamma}$ to photons and no hidden
ALPs.

The supernova SN1987A was observed in the nearby Large Magellanic Cloud in
1987. Low mass ALPs would have been copiously produced in the supernova,
primarily via their electromagnetic interaction. The produced ALPs would
therefore again be the linear combination of light ALP fields that couple to
photons. These ALPs would interconvert with photons in the Milky Way magnetic
field, leading to the production of observable gamma rays. The non-observation
of these gammas rays is used in [23] to place bounds on $g^{\gamma}$. However,
if the electromagnetic ALP oscillates into a hidden ALP between the Large
Magellanic cloud and the Milky Way, this hidden ALP would not mix with the
photon in the Milky Way magnetic field, and hence the expected gamma ray
signal for a given $g^{\gamma}$ would be reduced.

Other observations would be much less sensitive to ALP oscillations into
hidden ALPs. For example, ALP induced stellar cooling [24] is sensitive
primarily to the production rate of ALPs, and will not be significantly
affected if these ALPs subsequently oscillate into hidden ALPs. Similarly, ALP
induced modulations in point source spectra [25, 26, 27] in the low conversion
probability regime result primarily from photon to ALP conversion, and hence
the effect of subsequent oscillations into hidden ALPs will be small.

I will now turn to consider the expected ALP signal from CAST and from SN1987A
in the case of a single hidden ALP and in the case of a large number of hidden
ALPs.

###  4.1 The two ALP case

We will first consider the case where there are only two ALPs with non-
negligible coupling to photons. The Lagrangian in the mass basis is

$\begin{split}\mathcal{L}\supset&-\frac{1}{2}\partial^{\mu}\phi_{1}\partial_{\mu}\phi_{1}-\frac{1}{2}\partial^{\mu}\phi_{2}\partial_{\mu}\phi_{2}\\\
&-\frac{1}{2}m_{1}^{2}\phi_{1}^{2}-\frac{1}{2}m_{2}^{2}\phi_{2}^{2}-g^{\gamma}_{1}\phi_{1}\tilde{F}^{\mu\nu}F_{\mu\nu}-g^{\gamma}_{2}\phi_{2}\tilde{F}^{\mu\nu}F_{\mu\nu}.\end{split}$
(4.1)

We will now move to the basis where only one ALP couples to photons. The
electromagnetic ALP is

$\phi_{\gamma}=\frac{g^{\gamma}_{1}\phi_{1}+g^{\gamma}_{2}\phi_{2}}{\sqrt{{g^{\gamma}_{1}}^{2}+{g^{\gamma}_{2}}^{2}}}.$
(4.2)

The hidden ALP is

$\phi_{h}=\frac{g^{\gamma}_{2}\phi_{1}-g^{\gamma}_{1}\phi_{2}}{\sqrt{{g^{\gamma}_{1}}^{2}+{g^{\gamma}_{2}}^{2}}}.$
(4.3)

The mass basis and the electromagnetic basis are related by

$\begin{pmatrix}\phi_{1}\\\ \phi_{2}\\\ \end{pmatrix}=\begin{pmatrix}{\rm
cos}\theta&-{\rm sin}\theta\\\ {\rm sin}\theta&{\rm
cos}\theta\end{pmatrix}\begin{pmatrix}\phi_{\gamma}\\\ \phi_{h}\end{pmatrix},$
(4.4)

with

$\theta={\rm
cos}^{-1}\left(\frac{g_{1}^{\gamma}}{\sqrt{{g_{1}^{\gamma}}^{2}+{g_{2}^{\gamma}}^{2}}}\right).$
(4.5)

In the absence of a background magnetic field (and therefore mixing with the
photon), the probability of an electromagnetic ALP oscillating into a hidden
ALP is

$P_{\phi_{\gamma}\rightarrow\phi_{h}}={\rm sin}^{2}{2\theta}{\rm
sin}^{2}\left(\frac{\Delta m^{2}L}{4E}\right),$ (4.6)

where $\Delta m^{2}=m_{1}^{2}-m_{2}^{2}$, $L$ is the propagation length and
$E$ is the ALP’s energy. This oscillation probability is exactly analogous to
that for neutrino oscillations with two flavours.

####  4.1.1 CAST: The two ALP case

The Cern Axion Solar Telescope (CAST) [22] seeks to observe axions and ALPs
produced in the Sun by inducing their conversion to photons in a magnetic
field on Earth. We will assume that $m_{1}$ and $m_{2}$ are both negligible
compared to the temperature and effective photon mass in the Sun. In this
case, Primakoff production in the Sun produces the state $\phi_{\gamma}$.
There will be no significant Primakoff production of the hidden ALP as it has
no direct coupling to photons. As described in [20], the production of the
electromagnetic ALP in the Sun in this case will not be significantly altered
from the case of a single mass eigenstate ALP with negligible mass and with
coupling to photons
$g^{\gamma}=\sqrt{{g_{1}^{\gamma}}^{2}+{g_{2}^{\gamma}}^{2}}$. As described
above, ALPs produced electromagnetically in the Sun may oscillate into hidden
ALPs as they travel to Earth, and therefore be unobservable to CAST. For a
single hidden ALP, the electromagnetic ALP survival probability is given by

$P_{\phi_{\gamma}\rightarrow\phi_{\gamma}}=1-{\rm
sin}^{2}2\theta\frac{\int_{2\,{\rm keV}}^{7\,{\rm keV}}{\rm
sin}^{2}\left(\frac{\Delta
m^{2}L}{4E}\right)\frac{d\Phi_{a}}{dE}dE}{\int_{2\,{\rm keV}}^{7\,{\rm
keV}}\frac{d\Phi_{a}}{dE}dE},$ (4.7)

where $L$ is the Earth-Sun distance and $\frac{d\Phi_{a}}{dE}$ is the
differential solar ALP flux as given in [22]. We integrate this flux over the
CAST energy range of interest. The expected CAST signal will be reduced in
proportion to $P_{\phi_{\gamma}\rightarrow\phi_{\gamma}}$. Note we have
neglected the extremely small probability of ALP to photon conversion during
propagation from the Sun to the Earth [28].

The resulting electromagnetic ALP survival probability is shown in Figure 1
for a mixing angle $\theta=\frac{\pi}{3}$ and a range of mass differences. We
see that the expected CAST signal is significantly reduced for $\Delta
m^{2}\gtrsim 10^{-14}\,{\rm eV}^{2}$. For $\Delta m^{2}\gtrsim 10^{-12}\,{\rm
eV}^{2}$, oscillations in the survival probability are washed out by averaging
over the ALP flux, and we have
$P_{\phi_{\gamma}\rightarrow\phi_{\gamma}}\simeq 1-\frac{1}{2}{\rm
sin}^{2}2\theta$.

Figure 1: The survival probability for electromagnetic ALPs observed by CAST
for a mixing angle $\theta=\frac{\pi}{3}$.

####  4.1.2 SN1987A: The two ALP case

A similar effect may be seen in observations of ALPs from SN1987A, the 1987
supernova in the nearby Large Magellanic Cloud. ALPs would have been produced
during the supernova via their interaction with the photon, and subsequently
converted to gamma-rays in the Milky Way magnetic field. The non-observation
of these gamma-rays has been used to place bounds on the ALP-photon coupling
[23].

If the electromagnetic ALPs produced in the supernova oscillated into hidden
ALPs between the Large Magellanic Cloud and the Milky Way, the gamma-ray
signal would be correspondingly reduced. The electromagnetic ALP survival
probability in the case of a single hidden ALP is shown in Figure 2. We again
average over the differential axion flux as given in [23], which has typical
energies $\mathcal{O}(100\,{\rm MeV})$. In this case, we have
$P_{\phi_{\gamma}\rightarrow\phi_{\gamma}}\simeq 1-\frac{1}{2}{\rm
sin}^{2}2\theta$ for $\Delta m^{2}\gtrsim 10^{-18}\,{\rm eV}^{2}$.

Figure 2: The survival probability for electromagnetic ALPs travelling from
the Large Magellanic Cloud to the Milky Way for a mixing angle
$\theta=\frac{\pi}{3}$.

###  4.2 The many ALP case

Let us now consider a situation where many ALP mass eigenstates have non-
negligible coupling to electromagnetism, leading to many hidden ALPs. We will
remain agnostic as to the specifics of the ALP mass spectrum. The
electromagnetic ALP survival probability is then given by equation (3.5). In
the long $L$ limit for any given $\Delta m_{ij}^{2}$,
$L\gg\frac{4E_{0}}{\Delta m_{ij}^{2}}$, we obtain

$\frac{\int{\rm sin}^{2}\left(\frac{\Delta
m_{ij}^{2}L}{4E}\right)\frac{d\Phi_{a}}{dE}dE}{\int\frac{d\Phi_{a}}{dE}dE}\sim\frac{1}{2},$
(4.8)

where $\frac{d\Phi_{a}}{dE}$ is the differential ALP flux of the source and
$E_{0}$ is the average ALP energy. In the long $L$ limit for all $\Delta
m_{ij}^{2}$, we therefore have an electromagnetic ALP survival probability

$P_{1\rightarrow 1}\sim 1-2\sum_{i>j}|U_{1i}U_{1j}|^{2}.$ (4.9)

Now, from the definition of the electromagnetic ALP we have

$U_{1i}=\frac{g^{\gamma}_{i}}{\sqrt{\sum_{i}{g^{\gamma}_{i}}^{2}}},$ (4.10)

giving

$\begin{split}P_{1\rightarrow 1}&\sim
1-2\sum_{i>j}\left|\frac{g^{\gamma}_{i}g^{\gamma}_{j}}{\sum_{k}{g^{\gamma}_{k}}^{2}}\right|^{2}\\\
&=1-2\frac{\sum_{i>j}\left(g^{\gamma}_{i}g^{\gamma}_{j}\right)^{2}}{\sum_{i>j}\left(g^{\gamma}_{i}g^{\gamma}_{j}\right)^{2}+\sum_{j>i}\left(g^{\gamma}_{i}g^{\gamma}_{j}\right)^{2}+\sum_{i=j}\left(g^{\gamma}_{i}g^{\gamma}_{j}\right)^{2}},\end{split}$
(4.11)

assuming the $g_{i}^{\gamma}$ are real. For large numbers of hidden ALPs, the
third term in the denominator is negligible, and so we have $P_{1\rightarrow
1}\rightarrow 0$ for $L\gg\frac{4E_{0}}{\Delta m_{ij}^{2}}$. For CAST
observations, this corresponds to $\Delta m_{ij}^{2}\gg 10^{-14}\,{\rm
eV}^{2}$ for a large number of ALP mass eigenstates that contribute to the
electromagnetic ALP. For observations of SN1987A, this corresponds to $\Delta
m_{ij}^{2}\gg 5\times 10^{-20}\,{\rm eV}^{2}$ for a large number of ALP mass
eigenstates that contribute to the electromagnetic ALP.

##  5 Discussion

In string axiverse scenarios, the axion-like particle that mixes with the
photon may not be a mass eigenstate but a linear combination of many mass
eigenstate ALPs [20]. We have shown that this would lead to mass mixing
between the electromagnetically coupled ALP and other ‘hidden’ ALPs that do
not couple directly to electromagnetism. Depending on the parameters of the
ALP fields, we have shown that the observational effects are potentially
dramatic. Note that we have not included any additional interaction between
the ALP fields - the effect is purely due to basis misalignment.

We expect the most significant effects in settings where an ALP is produced
electromagnetically and then travels a large distance before being (hopefully)
detected electromagnetically. In these cases, mixing of the electromagnetic
ALP with undetectable hidden ALPs during propagation may significantly reduce
the expected signal. We have shown how this effect is applied to both
helioscope and supernova bounds on the ALP-photon coupling. Where there is
significant mixing between the electromagnetic and hidden ALPs, the bounds on
$g^{\gamma}$ from these experiments could be substantially reduced.

Further study of the effects of mixing with hidden ALPs on other search
strategies such as stellar cooling and spectral modulations is required.
However, there is no obvious mechanism for a dramatic reduction in signal
strength in these cases. This means that oscillations with hidden ALPs could
reconcile the possible observation of ALP induced modulations in galactic
pulsar spectra with the bound on $g^{\gamma}$ from CAST [29, 30]. We also
emphasise that this work does not apply to the QCD axion, whose mass and
coupling to photons each derive in part from the coupling to QCD.

An analogous effect could occur with other ALP couplings, for example the ALP-
fermion coupling. In fact, the mass, electromagnetic coupling and fermion
coupling bases may all be mutually misaligned. This could have interesting
phenomenological consequences, particularly for effects where both the
electromagnetic and fermionic couplings are relevant, such as considered in
[31, 32].

We have shown that the phenomenology of a string axiverse may be quite
different to that of a single ALP. We have shown here that mixing with hidden
ALPs may lead to significant signal reductions in some experiments. More
broadly, mass mixing in the string axiverse scenario means that bounds on
$g^{\gamma}$ from different observations cannot be trivially compared. The
interpretation of these bounds is highly model dependent. String axiverse
phenomenology clearly warrants further study.

## Acknowledgements

I am supported by Stephen Hawking Fellowship EP/T01668X/1 and STFC grant
ST/P001246/1. I am very grateful to Sven Krippendorf for helpful comments on a
draft of this paper. I further thank Rick Gupta, David J. E. Marsh, Jack
Shergold and Jessica Turner for helpful discussions.

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