# Interaction of inhomogeneous axions with magnetic fields in the early
universe

Maxim Dvornikov
Pushkov Institute of Terrestrial Magnetism, Ionosphere
and Radiowave Propagation (IZMIRAN),
108840 Moscow, Troitsk, Russia<EMAIL_ADDRESS>
###### Abstract

We study the system of interacting axions and magnetic fields in the early
universe after the quantum chromodynamics phase transition, when axions
acquire masses. Both axions and magnetic fields are supposed to be spatially
inhomogeneous. We derive the equations for the spatial spectra of these
fields, which depend on conformal time. In case of the magnetic field, we deal
with the spectra of the energy density and the magnetic helicity density. The
evolution equations are obtained in the closed form within the mean field
approximation. We choose the parameters of the system and the initial
condition which correspond to realistic primordial magnetic fields and axions.
The system of equations for the spectra is solved numerically. We compare the
cases of inhomogeneous and homogeneous axions. The evolution of the magnetic
field in these cases is different only within small time intervals. Generally,
magnetic fields are driven mainly by the magnetic diffusion. We find that the
magnetic field instability takes place for the amplified initial wavefunction
of the homogeneous axion. This instability is suppressed if we account for the
inhomogeneity of the axion.

##  1 Introduction

The existence of axions, which are pseudoscalar particles, was theoretically
proposed in Ref. [1] to provide the conservation of the CP symmetry in the
quantum chromodynamics (QCD). The conserved CP symmetry is required, in
particular, because of the experimental non-observation of an electric dipole
moment of a neutron (see, e.g., Ref. [2]). There are numerous studies of the
models for QCD axions which are summarized in Ref. [3].

Currently, there are active searches for axions in laboratory experiments.
These particles can be revealed in their interactions with electromagnetic
fields, fermion spins, and nuclear electric dipole moments. The difficulty of
a direct axion observation is because of the smallness of corresponding
coupling constants. Major present experimental studies of axions are reviewed
in Ref. [4]. There are recent claims (see, e.g., Ref. [5]) that some
experimental data can be interpreted as the axions observation. However, a
more careful analysis to confirm such findings is required.

Besides laboratory studies, there are attempts to detect axions from
astrophysical sources and cosmological media. One can try to capture axions
produced in the Sun using existing and future helioscopes (see, e.g., Ref.
[6]). Additionally, axions created in neutron stars can result in their X-ray
emission [7], which is observed by satellite based telescopes. Axions, which
the cold dark matter halo in our Galaxy consists of, can be detected using a
haloscope device. There are many haloscopes techniques, which are described in
Ref. [8].

The interest to the experimental study of astrophysical and cosmological
axions is motivated by the fact that these particles can both solve the CP
symmetry problem in QCD and be one of the most plausible candidate for the
dark matter [9]. For example, an axion can be the candidate for the dark
matter using the misalignment idea [10]. There are numerous sources of axions
in the early universe. These particles can be produced in decays of more
massive particles or topological defects, be thermally emitted by other
constituents of the standard model etc. Some of these mechanisms are reviewed
in Ref. [11].

Cosmological axions are believed to be spatially homogeneous. However, if the
Peccei-Quinn phase transition, which takes place at $T_{\mathrm{PQ}}\approx
f_{a}=(10^{10}-10^{12})\,\text{GeV}$, where $f_{a}$ is the Peccei-Quinn
constant, happens after the reheating during the inflation, then nonzero
spatial modes of axions can appear. Such a scenario is also not excluded (see,
e.g., Refs. [9, 11]). Inhomogeneous axions were proposed in Ref. [12] to
explain the formation of Bose stars. The contribution of inhomogeneous axions
to the cold dark matter was discussed in Ref. [13]. The evolution of axion
miniclusters was recently studied in Ref. [14].

In the present work, we study the evolution of the system of cosmological
axions and magnetic fields. For the first time, analogous problem was
discussed in Ref. [15]. However, nontrivial spectra of neither axions nor
magnetic fields were considered in Ref. [15]. Moreover, the universe expansion
was not accounted for properly there. In Ref. [16], we refined the analysis of
Ref. [15] by considering the spectral distribution of a primordial magnetic
field and taking into account the time dependent scale factor $a\neq 1$. Now,
we generalize the results of Ref. [16] by accounting for an arbitrary spatial
spectrum of axions.

As mentioned in Refs. [15, 16], a magnetic field in the presence of an axion
can be unstable. Indeed, in this case, the induction equation gets the
contribution, $\propto(\nabla\times\mathbf{B})$, analogous to that for the
chiral magnetic effect (the CME) [17]. The axion wavefunction contributes to
the $\alpha$-dynamo parameter responsible for the magnetic field instability.
Thus, the consideration of inhomogeneous axions is equivalent to the study of
chiral media with a coordinate dependent chiral imbalance. This problem was
studied in Ref. [18]. However, unlike the CME case with a coordinate dependent
chiral imbalance, here, we have a well established wave equation for the axion
wavefunction interacting with an electromagnetic field.

Our work is organized in the following way. In Sec. 2, we derive the general
equations for the spectra of the magnetic energy and helicity densities in the
presence of an inhomogeneous axion. The equation for the Fourier transform of
the axion wavefunction was considered in Sec. 3. Applying the mean field
approximation, we combine the obtained equations to the closed system in Sec.
4. In Sec. 5, we choose the parameters of the system, corresponding to the
realistic cosmological medium, and define the initial condition. We present
the results of the numerical simulations in Sec. 6. In Sec. 6.1, we study the
possibility to trigger the magnetic field instability in the system. Finally,
we conclude in Sec. 7.

##  2 Magnetic field in the presence of inhomogeneous axions

The equations for the electromagnetic field $F_{\mu\nu}$, interacting with a
scalar axion field $\varphi$, in a curved spacetime with the coordinates
$x^{\mu}=(t,\mathbf{x})$ read

$\displaystyle\frac{1}{\sqrt{-g}}\partial_{\nu}(\sqrt{-g}F^{\mu\nu})+g_{a\gamma}\partial_{\nu}\varphi\tilde{F}^{\mu\nu}+J^{\mu}$
$\displaystyle=0,$
$\displaystyle\frac{1}{\sqrt{-g}}\partial_{\nu}(\sqrt{-g}\tilde{F}^{\mu\nu})$
$\displaystyle=0,$ (2.1)

where $\tilde{F}^{\mu\nu}=\tfrac{1}{2}E^{\mu\nu\alpha\beta}F_{\alpha\beta}$ is
the dual counterpart of $F_{\mu\nu}$,
$E^{\mu\nu\alpha\beta}=\tfrac{1}{\sqrt{-g}}\varepsilon^{\mu\nu\alpha\beta}$ is
the covariant antisymmetric tensor, $\varepsilon^{0123}=+1$,
$g=\det(g_{\mu\nu})$, $g_{\mu\nu}$ is the metric tensor, $g_{a\gamma}$ is the
coupling constant, and $J^{\mu}=(\rho,\mathbf{J}/a)$ is the external current.
We consider the Friedmann-Robertson-Walker (FRW) metric
$g_{\mu\nu}=\text{diag}(1,-a^{2},-a^{2},-a^{2}),$where $a(t)$ is the scale
factor. Using the conformal variables [19] $\mathbf{E}_{c}=a^{2}\mathbf{E}$,
$\mathbf{B}_{c}=a^{2}\mathbf{B}$, $\rho_{c}=a^{3}\rho$, and
$\mathbf{J}_{c}=a^{3}\mathbf{J}$, we rewrite Eq. (2) in the form,

$\displaystyle(\nabla\times\mathbf{B}_{c})$
$\displaystyle=\mathbf{E}_{c}^{\prime}+\mathbf{J}_{c}+g_{a\gamma}\varphi^{\prime}\mathbf{B}_{c}+g_{a\gamma}(\nabla\varphi\times\mathbf{E}_{c}),$
(2.2) $\displaystyle(\nabla\times\mathbf{E}_{c})$
$\displaystyle=-\mathbf{B}_{c}^{\prime},$ (2.3)
$\displaystyle(\nabla\mathbf{E}_{c})$
$\displaystyle=-g_{a\gamma}(\mathbf{B}_{c}\nabla)\varphi+\rho_{c},$ (2.4)
$\displaystyle(\nabla\mathbf{B}_{c})$ $\displaystyle=0,$ (2.5)

where the prime means the derivative with respect to the conformal time $\eta$
defined by $\mathrm{d}t=a\mathrm{d}\eta$.

Supposing that $\mathbf{J}_{c}=\sigma_{c}\mathbf{E}_{c}$, where
$\sigma_{c}=10^{2}$ is the conformal conductivity of ultrarelativistic plasma,
and neglecting the displacement current $\mathbf{E}_{c}^{\prime}$ in Eq.
(2.2), we get that the electric field has the form,

$\mathbf{E}_{c}=\frac{1}{\sigma_{c}}\left[(\nabla\times\mathbf{B}_{c})-g_{a\gamma}\varphi^{\prime}\mathbf{B}_{c}\right]-\frac{1}{\sigma_{c}^{2}}\left[g_{a\gamma}\left(\nabla\varphi\times(\nabla\times\mathbf{B}_{c})\right)+g_{a\gamma}^{2}\varphi^{\prime}(\mathbf{B}_{c}\times\nabla\varphi)\right].$
(2.6)

Here we keep the terms linear in $\nabla\varphi$. Setting $\rho_{c}=0$ in Eq.
(2.4), we derive the equation for $\mathbf{B}_{c}$,

$\displaystyle\mathbf{B}_{c}^{\prime}=$
$\displaystyle\frac{1}{\sigma_{c}}\left[\Delta\mathbf{B}_{c}+g_{a\gamma}\varphi^{\prime}(\nabla\times\mathbf{B}_{c})+g_{a\gamma}(\nabla\varphi^{\prime}\times\mathbf{B}_{c})\right]$
$\displaystyle+\frac{g_{a\gamma}}{\sigma_{c}^{2}}\left\\{\left((\nabla\times\mathbf{B}_{c})\cdot\nabla\right)\nabla\varphi-\left(\nabla\varphi\cdot\nabla\right)(\nabla\times\mathbf{B}_{c})-(\nabla\times\mathbf{B}_{c})\Delta\varphi\right\\}$
$\displaystyle+\frac{g_{a\gamma}^{2}}{\sigma_{c}^{2}}\big{\\{}\mathbf{B}_{c}\left[\varphi^{\prime}\Delta\varphi+(\nabla\varphi^{\prime}\cdot\nabla\varphi)\right]$
$\displaystyle+\varphi^{\prime}\left[(\nabla\varphi\cdot\nabla)\mathbf{B}_{c}-(\mathbf{B}_{c}\cdot\nabla)\nabla\varphi\right]-\nabla\varphi(\mathbf{B}_{c}\cdot\nabla)\varphi^{\prime}\big{\\}},$
(2.7)

where we use Eqs. (2.3) and (2.6).

Equation (2) can be simplified if we assume that axions are inhomogeneous but
isotropic. It means that, after the spatial averaging, one has that
$\left\langle\nabla\varphi\right\rangle=0$, whereas
$\left\langle(\nabla\varphi)^{2}\right\rangle\neq 0$ and
$\left\langle\Delta\varphi\right\rangle\neq 0$. Keeping the terms linear in
$g_{a\gamma}$ in Eq. (2) and spatially averaging it, we get the modified
induction equation,

$\displaystyle\mathbf{B}_{c}^{\prime}=$
$\displaystyle\frac{1}{\sigma_{c}}\Delta\mathbf{B}_{c}+\alpha(\mathbf{x})(\nabla\times\mathbf{B}_{c}),\quad\alpha(\mathbf{x})=\frac{g_{a\gamma}}{\sigma_{c}}\left(\varphi^{\prime}-\frac{2}{3\sigma_{c}}\Delta\varphi\right),$
(2.8)

where we omit the averaging brackets for the sake of brevity. In Eq. (2.8),
the $\alpha$-dynamo parameter $\alpha(\mathbf{x})$, which is the coefficient
in the magnetic instability term $\propto(\nabla\times\mathbf{B}_{c})$, is
different from that in Refs [15, 16]. It gets the contribution from the
inhomogeneous axion field $\propto\Delta\varphi$.

It is convenient to deal with a Fourier transform, which has the form,
$f_{\mathbf{k}}=\smallint\mathrm{d}^{3}xe^{\mathrm{i}\mathbf{kx}}f(\mathbf{x}),$
and $f_{\mathbf{k}}^{*}=f_{-\mathbf{k}}$ since $f(\mathbf{x})$ is real. Then
we define the spectra of the magnetic energy density and the magnetic
helicity,

$\displaystyle\rho_{\mathbf{p}}(\eta)$
$\displaystyle=\frac{1}{2}\int\mathrm{d}^{3}xe^{\mathrm{i}\mathbf{px}}\mathbf{B}_{c}^{2}(\mathbf{x})=\frac{1}{2}\int\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}(\mathbf{B}_{\mathbf{k}}^{(c)*}\mathbf{B}_{\mathbf{k}+\mathbf{p}}^{(c)}),$
$\displaystyle h_{\mathbf{p}}(\eta)$
$\displaystyle=\int\mathrm{d}^{3}xe^{\mathrm{i}\mathbf{px}}(\mathbf{A}_{c}(\mathbf{x})\mathbf{B}_{c}(\mathbf{x}))=\int\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}(\mathbf{A}_{\mathbf{k}}^{(c)*}\mathbf{B}_{\mathbf{k}+\mathbf{p}}^{(c)}).$
(2.9)

Note that the total magnetic helicity,

$H(\eta)=\int\mathrm{d}^{3}x(\mathbf{A}_{c}\mathbf{B}_{c})=\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}(\mathbf{A}_{\mathbf{p}}^{(c)}\mathbf{B}_{\mathbf{p}}^{(c)*}),$
(2.10)

has the standard form.

Taking the Fourier transform of Eq. (2.8) and forming the binary combinations
of the fields, one gets the following differential equations:

$\displaystyle\rho^{\prime}_{\mathbf{q}}=$
$\displaystyle-\frac{1}{2\sigma_{c}}\int\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}k^{2}\left[(\mathbf{B}_{\mathbf{k}}^{(c)*}\mathbf{B}_{\mathbf{k}+\mathbf{q}}^{(c)})+(\mathbf{B}_{\mathbf{k}}^{(c)}\mathbf{B}_{\mathbf{k}-\mathbf{q}}^{(c)*})\right]$
$\displaystyle+\frac{1}{2}\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}p^{2}\left[\alpha_{\mathbf{p}-\mathbf{k}}^{*}\mathbf{A}_{\mathbf{p}}^{(c)}\mathbf{B}_{\mathbf{k}-\mathbf{q}}^{(c)*}+\alpha_{\mathbf{p}-\mathbf{k}}\mathbf{A}_{\mathbf{p}}^{(c)*}\mathbf{B}_{\mathbf{k}+\mathbf{q}}^{(c)}\right],$
$\displaystyle h^{\prime}_{\mathbf{q}}=$
$\displaystyle-\frac{1}{\sigma_{c}}\int\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}k^{2}\left[(\mathbf{A}_{\mathbf{k}}^{(c)*}\mathbf{B}_{\mathbf{k}+\mathbf{q}}^{(c)})+(\mathbf{A}_{\mathbf{k}-\mathbf{q}}^{(c)*}\mathbf{B}_{\mathbf{k}}^{(c)})\right]$
$\displaystyle+\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}\bigg{\\{}\alpha_{\mathbf{p}-\mathbf{k}}^{*}\mathrm{i}\left(\mathbf{p}\times\mathbf{A}_{\mathbf{k}-\mathbf{q}}^{(c)*}\right)\mathbf{B}_{\mathbf{p}}^{(c)}$
$\displaystyle+\alpha_{\mathbf{p}-\mathbf{k}}\frac{1}{k^{2}}\left[(\mathbf{kp})\left(\mathbf{B}_{\mathbf{p}}^{(c)*}\mathbf{B}_{\mathbf{k}+\mathbf{q}}^{(c)}\right)-\left(\mathbf{p}\mathbf{B}_{\mathbf{k}+\mathbf{q}}^{(c)}\right)\left(\mathbf{k}\mathbf{B}_{\mathbf{p}}^{(c)*}\right)\right]\bigg{\\}},$
(2.11)

for the spectra $\rho_{\mathbf{p}}(\eta)$ and $h_{\mathbf{p}}(\eta)$ in Eq.
(2). In Eq. (2),
$\alpha_{\mathbf{k}}=\smallint\mathrm{d}^{3}xe^{\mathrm{i}\mathbf{kx}}\alpha(\mathbf{x})$
is the Fourier transform of the $\alpha$-dynamo parameter in Eq. (2.8). We
mention that the equations for the spectra in Eq. (2) cannot be written in a
closed form, analogous to that, e.g., in Refs. [20, 21], for a general
coordinate dependent $\alpha$-dynamo parameter.

##  3 Axions in the presence of an electromagnetic field

The evolution of an axion field under the influence of an electromagnetic
field obeys the following equation in a curved spacetime:

$\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}\partial^{\mu}\varphi\right)+m^{2}\varphi+\frac{g_{a\gamma}}{4}F_{\mu\nu}\tilde{F}^{\mu\nu}=0,$
(3.1)

where $m$ is the axion mass. We can rewrite Eq. (3.1) in the form,

$\varphi^{\prime\prime}+\frac{2a^{\prime}}{a}\varphi^{\prime}-\nabla^{2}\varphi+a^{2}m^{2}\varphi=\frac{g_{a\gamma}}{a^{2}}(\mathbf{E}_{c}\mathbf{B}_{c}),$
(3.2)

where we take the FRW metric and use the conformal variables.

In the Fourier representation, Eq. (3.2) takes the form,

$\varphi_{\mathbf{k}}^{\prime\prime}+\frac{2a^{\prime}}{a}\varphi_{\mathbf{k}}^{\prime}+k^{2}\varphi_{\mathbf{k}}+a^{2}m^{2}\varphi_{\mathbf{k}}=\frac{g_{a\gamma}}{a^{2}}\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}(\mathbf{E}_{\mathbf{p}+\mathbf{k}}^{(c)}\mathbf{B}_{\mathbf{p}}^{(c)*}).$
(3.3)

The integral in the right hand side of Eq. (3.3) differs from the derivative
of the magnetic helicity,

$\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}(\mathbf{E}_{\mathbf{p}+\mathbf{k}}^{(c)}\mathbf{B}_{\mathbf{p}}^{(c)*})\neq-\frac{1}{2}H^{\prime}(\eta)=\int\mathrm{d}^{3}x(\mathbf{E}{}_{c}\mathbf{B}_{c})=\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}(\mathbf{E}_{\mathbf{p}}^{(c)}\mathbf{B}_{\mathbf{p}}^{(c)*}),$
(3.4)

which is given in Eq. (2.10).

Nevertheless, we assume that the electromagnetic field is inhomogeneous but
isotropic. Thus all odd correlators, like $\left\langle k_{i}\right\rangle$,
$\left\langle k_{i}k_{j}k_{n}\right\rangle$ etc., vanish. In this case, we
rewrite Eq. (3.4) as

$\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}(\mathbf{E}_{\mathbf{p}+\mathbf{k}}^{(c)}\mathbf{B}_{\mathbf{p}}^{(c)*})=\int\mathrm{d}^{3}x\exp\left(-\frac{k^{2}x^{2}}{6}\right)(\mathbf{E}_{c}(\mathbf{x})\mathbf{B}_{c}(\mathbf{x}))=-\frac{1}{2}\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}h^{\prime}_{\mathbf{p}}(\eta)f_{\mathbf{p}}^{(k)*},$
(3.5)

where $h_{\mathbf{p}}(\eta)$ is the spectrum of the magnetic helicity density
in Eq. (2) and

$f_{\mathbf{p}}^{(k)}=\int\mathrm{d}^{3}xe^{\mathrm{i}\mathbf{px}}\exp\left(-\frac{k^{2}x^{2}}{6}\right)=\frac{6\sqrt{6}\pi^{3/2}}{k^{3}}\exp\left(-\frac{3p^{2}}{2k^{2}}\right),$
(3.6)

is the scalar function.

##  4 Mean field approximation

Equations (2) and (3.3) are difficult to analyze in general case. Thus, we
apply the mean field approximation. It implies the consideration of nontrivial
spectra which are attributed, however, to the zero mode only.

First, we assume in Eq. (2) that
$\alpha_{\mathbf{k}}=\alpha_{0}\delta^{3}(\mathbf{k})$, where

$\alpha_{0}(\eta)=\int\mathrm{d}^{3}k\alpha_{\mathbf{k}}(\eta)=\frac{4\pi
g_{a\gamma}}{\sigma_{c}}\int
k{}^{2}\mathrm{d}k\left[\varphi^{\prime}_{k}(\eta)+\frac{2k^{2}}{3\sigma_{c}}\varphi_{k}(\eta)\right].$
(4.1)

In this approximation, we set $\mathbf{q}=0$ in Eq. (2) and consider the mean
spectra of the magnetic energy and the magnetic helicity,
$\rho(k,\eta)=\tfrac{k^{2}}{4\pi^{2}}(\mathbf{B}_{\mathbf{k}}^{(c)*}\mathbf{B}_{\mathbf{k}}^{(c)})$
and
$h(k,\eta)=\tfrac{k^{2}}{2\pi^{2}}(\mathbf{A}_{\mathbf{k}}^{(c)*}\mathbf{B}_{\mathbf{k}}^{(c)})$,
which were introduced, e.g., in Refs. [20, 21]. They are related to the total
spectra by the expressions,

$\rho_{\mathbf{q}=0}=\int\mathrm{d}k\rho(k,\eta),\quad
h_{\mathbf{q}=0}=\int\mathrm{d}kh(k,\eta),$ (4.2)

where $\rho_{\mathbf{p}}$ and $h_{\mathbf{p}}$ are given in Eq. (2). Using Eq.
(2) in the considered approximation, we get that the mean spectra obey the
equations,

$\displaystyle\frac{\partial\rho(k,\eta)}{\partial\eta}$
$\displaystyle=-\frac{2k^{2}}{\sigma_{c}}\rho(k,\eta)+\alpha_{0}k^{2}h(k,\eta),$
$\displaystyle\frac{\partial h(k,\eta)}{\partial\eta}$
$\displaystyle=-\frac{2k^{2}}{\sigma_{c}}h(k,\eta)+4\alpha_{0}\rho(k,\eta),$
(4.3)

which formally coincide with those derived in Refs. [20, 21]. However, Eq. (4)
involves a nontrivial spectrum of the axion field in Eq. (4.1).

Then, we consider the limit $k\to 0$ in the integrand in Eq. (3.3). In this
case, $f_{\mathbf{p}}^{(k\to 0)}=(2\pi)^{3}\delta^{3}(\mathbf{p})$ in Eq.
(3.6). Using Eq. (3.5) in this approximation and Eq. (4.2), we rewrite Eq.
(3.3) in the form,

$\varphi_{k}^{\prime\prime}+\frac{2a^{\prime}}{a}\varphi_{k}^{\prime}+k^{2}\varphi_{k}+a^{2}m^{2}\varphi_{k}=-\frac{g_{a\gamma}}{2a^{2}}\int\mathrm{d}k^{\prime}\frac{\partial
h(k^{\prime},\eta)}{\partial\eta},$ (4.4)

which is similar to that used in Ref. [16]. However, we keep the term $\propto
k^{2}\varphi_{k}$ in the left hand side of Eq. (4.4).

Equations (4.4) and (4), together with Eq. (4.1), form the closed set of the
evolution equations, which define the evolution of the magnetic and axion
fields. We shall solve them numerically in Secs. 5 and 6.

##  5 Initial condition and the parameters of the system

We consider the evolution of the system after the QCD phase transition which
happens at $T_{\mathrm{QCD}}=160\,\text{MeV}$. In this situation, the axion
mass is taken to be constant, $m=10^{-6}\,\text{eV}$. We use the
parameterization in which the scale factor $a=1/T$ is dimensional, where $T$
is the primordial plasma temperature. The conformal time is defined up to a
constant. We take that
$\eta=\tilde{M}_{\mathrm{Pl}}(T^{-1}-T_{\mathrm{QCD}}^{-1})$, where
$\tilde{M}_{\mathrm{Pl}}=M_{\mathrm{Pl}}/1.66\sqrt{g_{*}}$,
$M_{\mathrm{Pl}}=1.2\times 10^{19}\,\text{GeV}$ is the Planck mass, and
$g_{*}=17.25$ is the number of the relativistic degrees of freedom at
$T_{\mathrm{QCD}}$ [24]. In this case $\eta(T_{\mathrm{QCD}})=0$.

We assume that the seed spectrum of the magnetic energy is Kolmogorov,
$\rho_{0}(k)\propto k^{n}$ with $n=-5/3$. The seed spectrum of the magnetic
helicity is $h_{0}(k)=2q\rho_{0}(k)/k$, where $0\leq q\leq 1$ is the parameter
fixing the initial helicity. Note that $k$ is the dimensionless conformal
momentum. It spans the range $k_{\mathrm{min}}<k<k_{\mathrm{max}}$, where
$k_{\mathrm{min}}=T_{\mathrm{QCD}}/\tilde{M}_{\mathrm{Pl}}=9.1\times 10^{-20}$
is the reciprocal horizon size at $T_{\mathrm{QCD}}$. The maximal conformal
momentum $k_{\mathrm{max}}$ is a free parameter in our model. We take that
$k_{\mathrm{max}}<r_{\mathrm{D}}^{-1}=0.1$, where $r_{\mathrm{D}}$ is the
conformal Debye length, to guarantee the plasma electroneutrality.

There is not so much knowledge about the seed axion spectrum. We assume that
$\varphi_{k}^{(0)}=8\pi^{3/2}f_{a}\exp(-k^{2}/\delta^{2})/\delta^{3}$, where
$\delta$ is the phenomenological parameter. There is the approximate relation
between $m$ and $f_{a}$ [9]: $m=(5.7\pm
0.007)\,\mu\text{eV}\left(\tfrac{10^{12}\,\text{GeV}}{f_{a}}\right)$. Thus, we
take $f_{a}=10^{12}\,\text{GeV}$ since we have chosen $m=10^{-6}\,\text{eV}$.
The suggested values of $m$ and $f_{a}$ are in the sensitivity range of the
operating and future lumped element detectors [8] (see also Refs. [22, 23]).
We take that $0\leq k<k_{\mathrm{max}}$ in the axion seed spectrum to be able
to probe small momenta. Note that
$\varphi_{k}^{(0)}\to(2\pi)^{3}f_{a}\delta^{3}(\mathbf{k})$ at $\delta\to 0$.
In this case, we can reproduce the zero mode approximation studied in Ref.
[16]. We shall consider $\delta\sim k_{\mathrm{max}}$ as a rule. Following
Ref. [16], we estimate the initial first derivative
$\partial_{\eta}\varphi_{k}^{(0)}$ basing on the virial theorem. Computing the
zero component of the energy-momentum tensor of an axion, we get that
$\partial_{\eta}\varphi_{k}^{(0)}=\sqrt{k^{2}+m^{2}/T_{\mathrm{QCD}}^{2}}\varphi_{k}^{(0)}$.

We introduce the new variables in Eqs. (4) and (4.4),

$\displaystyle\rho(k,\eta)$ $\displaystyle=\frac{R(\kappa,\tau)}{4\pi
g_{a\gamma}^{2}T_{QCD}^{2}k_{\mathrm{max}}^{2}},\quad
h(k,\eta)=\frac{H(\kappa,\tau)}{2\pi
g_{a\gamma}^{2}T_{QCD}^{2}k_{\mathrm{max}}^{3}},$
$\displaystyle\varphi_{k}(\eta)$ $\displaystyle=\frac{\sigma_{c}}{8\pi
g_{a\gamma}k_{\mathrm{max}}^{4}}\Phi(\kappa,\tau),\quad\tau=\frac{2k_{\mathrm{max}}^{2}}{\sigma_{c}}\eta,\quad
k=k_{\mathrm{max}}\kappa,$ (5.1)

where
$0\leq\tau\leq\tau_{\mathrm{max}}=2k_{\mathrm{max}}^{2}\tilde{M}_{\mathrm{Pl}}/T_{\mathrm{now}}$,
$\kappa_{m}<\kappa<1$ in the magnetic spectra, whereas $0<\kappa<1$ in the
axion spectrum, $T_{\mathrm{now}}=2.7\,\text{K}$ is the current universe
temperature, and $\kappa_{m}=k_{\mathrm{min}}/k_{\mathrm{max}}$. Using Eq.
(5), Eqs. (4) and (4.4) are rewritten in the form,

$\displaystyle\frac{\partial R}{\partial\tau}=$
$\displaystyle-\kappa^{2}R+\kappa^{2}H\int_{0}^{1}\kappa^{\prime
2}\mathrm{d}\kappa^{\prime}\left[\frac{\partial\Phi(\kappa^{\prime},\tau)}{\partial\tau}+\frac{\kappa^{\prime
2}}{3}\Phi(\kappa^{\prime},\tau)\right],$ $\displaystyle\frac{\partial
H}{\partial\tau}=$ $\displaystyle-\kappa^{2}H+R\int_{0}^{1}\kappa^{\prime
2}\mathrm{d}\kappa^{\prime}\left[\frac{\partial\Phi(\kappa^{\prime},\tau)}{\partial\tau}+\frac{\kappa^{\prime
2}}{3}\Phi(\kappa^{\prime},\tau)\right],$
$\displaystyle\frac{\partial^{2}\Phi}{\partial\tau^{2}}=$
$\displaystyle-\frac{2}{\tilde{a}}\frac{\mathrm{d}\tilde{a}}{\mathrm{d}\tau}\frac{\partial\Phi}{\partial\tau}-\left(\kappa^{2}\mathcal{K}^{2}+\tilde{a}^{2}\mu^{2}\right)\Phi$
$\displaystyle+\frac{1}{\tilde{a}^{2}}\bigg{\\{}\int_{\kappa_{m}}^{1}\kappa^{\prime
2}H(\kappa^{\prime},\tau)\mathrm{d}\kappa^{\prime}$
$\displaystyle-\int_{\kappa_{m}}^{1}R(\kappa^{\prime},\tau)\mathrm{d}\kappa^{\prime}\int_{0}^{1}\kappa^{\prime
2}\mathrm{d}\kappa^{\prime}\left[\frac{\partial\Phi(\kappa^{\prime},\tau)}{\partial\tau}+\frac{\kappa^{\prime
2}}{3}\Phi(\kappa^{\prime},\tau)\right]\bigg{\\}},$ (5.2)

where $\mu=\tfrac{\sigma_{c}m}{2k_{\mathrm{max}}^{2}T_{\mathrm{QCD}}}$ is the
effective axion mass, $\mathcal{K}=\sigma_{c}/2k_{\mathrm{max}}$,
$\tilde{a}=1+\lambda\tau$, and
$\lambda=\tfrac{\sigma_{c}T_{\mathrm{QCD}}}{2k_{\mathrm{max}}^{2}\tilde{M}_{\mathrm{Pl}}}$.

The initial condition for the new variables in Eq. (5) reads

$R(\kappa,0)=C\kappa^{n},\quad H(\kappa,0)=q\frac{R(\kappa,0)}{\kappa},\quad
C=\frac{2\pi
g_{a\gamma}^{2}T_{\mathrm{QCD}}^{2}k_{\mathrm{max}}(n+1)B_{c}^{(0)2}}{(1-\kappa_{m}^{n+1})},$
(5.3)

where $B_{c}^{(0)}=B_{\mathrm{now}}/T_{\mathrm{now}}^{2}=3.6\times 10^{-4}$ is
the seed conformal magnetic field which corresponds to the present day
magnetic field $B_{\mathrm{now}}=10^{-9}\,\text{G}$. Such a strength is within
the current constrains on the extragalactic magnetic field obtained in Ref.
[25] basing on the cosmic rays observations. However, lower bounds on cosmic
magnetic fields are considered in Ref. [26]. The initial condition for the
dimensionless axion wavefunction is

$\displaystyle\Phi(\kappa,0)$
$\displaystyle=\frac{32\pi^{3/2}\alpha_{\mathrm{em}}k_{\mathrm{max}}^{4}}{\sigma_{c}\delta^{3}}\exp\left(-\kappa^{2}\frac{k_{\mathrm{max}}^{2}}{\delta^{2}}\right),$
$\displaystyle\frac{\partial\Phi(\kappa,0)}{\partial\tau}$
$\displaystyle=\sqrt{\mathcal{K}^{2}\kappa^{2}+\mu^{2}}\Phi(\kappa,0),$ (5.4)

where we take into account that [9]
$g_{a\gamma}\approx\tfrac{\alpha_{\mathrm{em}}}{2\pi f_{a}}$ and
$\alpha_{\mathrm{em}}=7.3\times 10^{-3}$ is the fine structure constant.

##  6 Results

In this section, we show the results of the numerical solution of the system
in Eq. (5) with the initial condition in Eqs. (5.3) and (5).

The system analogous to that in Eq. (5) was studied in Ref. [16] for a
spatially homogeneous axion. Here, we are mainly focused on the influence of a
nontrivial seed spectrum of an axion on the evolution of the system. That is
why we compare the inhomogeneous axion MHD with the homogeneous one. As we
mentioned in Sec. 5, the parameter $\delta\sim k_{\mathrm{max}}$. In Fig. 1,
we compare the cases of $\delta=10^{-10}$ (the inhomogeneous axion) and
$\delta=0$ (the homogeneous axion). The rest of the parameters and the initial
condition are the same for both the inhomogeneous and homogeneous cases.

Figure 1: The observable parameters of the system while it evolves in the
universe cooling down from $T_{\mathrm{QCD}}$. We take that
$k_{\mathrm{max}}=10^{-10}$, $B_{c}^{(0)}=3.6\times 10^{-4}$
($B_{\mathrm{now}}=10^{-9}\,\text{G}$), $m=10^{-6}\,\text{eV}$,
$f_{a}=10^{12}\,\text{GeV}$, and $q=1$. We choose the seed Kolmogorov
spectrum. Panels (a) and (c): the case of the inhomogeneous axion with the
initial condition in Eq. (5) and $\delta=10^{-10}$. Panels (b) and (d): the
situation of the homogeneous axion corresponding to $\delta=0$ in Eq. (5) and
$\varphi_{0}=f_{a}$. The insets in panels (b) and (d) show the corresponding
quantities at small evolution times. Panels (a) and (b): the evolution of the
magnetic field. Panel (c): the behavior of the total axion energy in Eq. (6.2)
of the inhomogeneous axion. Panel (d): the evolution of the energy density
$\rho_{a}$ of the homogeneous axion.

Unfortunately, we can proceed in numerical simulation only for $T\lesssim
T_{\mathrm{QCD}}$ for inhomogeneous axions because of the very rapid
oscillations of the axion wavefunction (see Fig. 2 below). We checked the
dependence of the evolution of the system on the seed magnetic helicity $q$
defined in Sec. 5. It turns out to be negligible. Thus, we set $q=1$ which
corresponds to the maximally helical fields.

The measurable quantities in the axion MHD are the conformal magnetic field,

$B_{c}=2\left[\int_{k_{\mathrm{min}}}^{k_{\mathrm{max}}}\mathrm{d}k\rho(k,\eta)\right]^{1/2},$
(6.1)

and the total axion energy in the cooling universe,
$E_{a}\propto\smallint\mathrm{d}^{3}xT_{00}$,

$E_{a}=\int_{0}^{1}\kappa^{2}\mathrm{d}\kappa\left[\left(\frac{\partial\Phi}{\partial\tau}\right)^{2}+\left(\kappa^{2}\mathcal{K}^{2}+\tilde{a}^{2}\mu^{2}\right)\Phi^{2}\right],$
(6.2)

where $T_{00}$ is the zeroth component of the axion energy-momentum tensor. We
normalize $E_{a}$ on its initial value $E_{a}^{(0)}$. In case of a homogeneous
axion, we deal with the axion energy density
$\rho_{a}=(\partial_{\tau}\Phi)^{2}+\tilde{a}^{2}\mu^{2}\Phi^{2}$, which is
also normalized on
$\rho_{a}^{(0)}=(\partial_{\tau}\Phi_{0})^{2}+\mu^{2}\Phi_{0}^{2}$.

We show $B_{c}$ versus $T$ in the universe cooling down from
$T_{\mathrm{QCD}}$ in Figs. 1 and 1 in inhomogeneous and homogeneous cases
respectively. The magnetic field decays in both cases, mainly because of the
diffusion terms, $\propto-\kappa^{2}R$ and $\propto-\kappa^{2}H$, in Eq. (5).

The evolution time in Fig. 1 approximately corresponds to that in the inset in
Fig. 1. We can see the ripple in the inset in Fig. 1 (homogeneous axion),
whereas the line in Fig. 1 (inhomogeneous axion) is smooth. This ripple is not
noticeable on great evolution times in Fig. 1. Comparing Fig. 1 and the inset
in Fig. 1, we can see that the inhomogeneous case approximately corresponds to
the mean value of the homogeneous axion wavefunction.

We depict $E_{a}$ in Fig. 1 and $\rho_{a}$ in Fig. 1 versus $T$. By the end of
the evolution, both quantities are about $0.1$ from their initial values.
However, the decay in the inhomogeneous case in Fig. 1 happens much faster
than in the homogeneous situation in Fig. 1. Such a behavior is owing to the
additional term $\propto\kappa^{\prime 2}\Phi$ in the integrand in the last
line of Eq. (5). Therefore the damping of the inhomogeneous axion is greater
effectively.

In Fig. 2, we show the evolution of the axion wavefunction integrated over the
spectrum,

$\bar{\varphi}(\eta)=\int_{0}^{k_{\mathrm{max}}}\mathrm{d}k\,k^{2}\varphi_{k}(\eta),$
(6.3)

which is again normalized on its initial value at $T_{\mathrm{QCD}}$. One can
see that it oscillates vary rapidly. As mentioned above, this fact makes the
solution of the system in Eq. (5) more complicated than for the homogeneous
axion studied in Ref. [16]. Moreover, as one can see in the inset in Fig. 2,
there are both rapid oscillations and a sort of beatings in the evolution of
$\bar{\varphi}$. These beatings appear since we integrate different
wavenumbers over the spectrum in Eq. (6.3).

Figure 2: The evolution of the wavefunction of the inhomogeneous axion
integrated over the spectrum in Eq. (6.3). The parameters of the system are
the same as in Figs. 1 and 1.

We take that $k_{\mathrm{max}}=10^{-10}$ in our simulations. It means that the
present length scale obeys
$l_{\mathrm{now}}>(k_{\mathrm{max}}T_{\mathrm{now}})^{-1}\approx
10^{-2}R_{\odot}$, where $R_{\odot}$ is the solar radius. Thus the
perturbations of the axion field considered in our work, can result in the
formation of hypothetical astronomical objects like Bose stars; cf. Ref. [12].

###  6.1 Magnetic field instability

We showed above that there are certain differences in the behavior of the
measurable quantities in MHD in the presence of inhomogeneous and homogeneous
axions. Nevertheless, these differences are smeared at low temperatures $\sim
T_{\mathrm{now}}$. The axion wavefunction contributes to the $\alpha$-dynamo
parameter in Eq. (4.1), which is known to result in the instability of the
magnetic field. However, we did not observe this instability in the cases
studied above. The behavior of $B_{c}$ was mainly affected by the magnetic
diffusion.

It happens since we take that the initial axion wavefunction is
$\varphi_{0}\sim f_{a}$ at $T_{\mathrm{QCD}}$. This initial condition is
realistic (see, e.g., Ref. [15]). However, the impact of the axion on the
magnetic field is small in this case.

Now we consider the situation when the initial axion wavefunction is
increased. Namely, we take that $\varphi_{k}^{(0)}=8\pi^{3/2}\Xi
f_{a}\exp(-k^{2}/\delta^{2})/\delta^{3}$, where the factor $\Xi=5\times
10^{7}$. We also consider the homogeneous axion with $\varphi_{0}=\Xi f_{a}$,
where $\Xi$ has the same value as in the inhomogeneous case. The initial axion
wavefunctions, which are greater than $f_{a}$, were mentioned in Ref. [15] to
potentially influence magnetic fields. We shall see soon that it is the case.
The rest of the parameters and the initial condition are the same as in Fig.
1.

In Fig. 3, we show the evolution of the magnetic field defined in Eq. (6.1).
We study the inhomogeneous and homogeneous cases in the same time interval.
One can see in Fig. 3 that the magnetic field becomes unstable. The evolution
of the magnetic field in the inhomogeneous case shown in Fig. 3 does not
reveal the instability.

Figure 3: The evolution of the magnetic field for (a) the inhomogeneous axion;
and (b) the homogeneous one. The parameters are the same as in Fig. 1 except
for the initial axion wavefunction which is amplified by the factor
$\Xi=5\times 10^{7}$ in both panels now.

To illustrate the development of the instability of the magnetic field in the
homogeneous case, we plot the spectra of the densities of the magnetic energy
and the magnetic helicity in Fig. 4. Unlike Eq. (5), here we use different new
variables,
$R^{\prime}(\kappa,\tau)=g_{a\gamma}^{2}T_{\mathrm{QCD}}^{2}\rho(k,\eta)/k_{\mathrm{max}}$
and $H^{\prime}(\kappa,\tau)=g_{a\gamma}^{2}T_{\mathrm{QCD}}^{2}h(k,\eta)/2$.
We studied the evolution of magnetic fields with $\Xi<5\times 10^{7}$ and did
not reveal the magnetic field instability, at least in the considered time
interval.

In Fig. 4, by dashed lines, we show the seed Kolmogorov spectra which
correspond to $T_{\mathrm{QCD}}$. We also depict the final spectra which are
at the end of the evolution of the system. One can see that parts of the
spectra at great momenta, i.e. at small scales, increase. One can say that it
is equivalent to the direct energy cascade triggered by the homogeneous axion.

Figure 4: The spectra of the densities of (a) the magnetic energy; and (b) the
magnetic helicity in case of the homogeneous axion. The parameters of the
system and the initial condition are the same as in Fig. 3. Solid lines are
final spectra at the end of the system evolution. Dashed lines are the seed
Kolmogorov spectra.

The $\alpha$-dynamo parameter in Eq. (4.1) has two axion contributions in the
integrand: (i) $\propto\varphi^{\prime}_{k}$; and (ii) $\propto
k^{2}\varphi_{k}$. Despite we take a greater axion wavefunction, the
inhomogeneous axion oscillates more rapidly; cf. Fig. 2. Thus, the term (i)
prevails over the term (ii). Since the term (i) is sign changing, the
consideration of the inhomogeneous axion prevents the development of the
magnetic field instability.

##  7 Conclusion

In the present work, we have studied the evolution of the system which
consists of axions and magnetic fields. Both axions and magnetic fields are
spatially inhomogeneous. The evolution of such a system was considered in the
early universe cooling down from the QCD phase transition, with the universe
expansion rate being accounted for exactly.

First, in Sec. 2, we have started with the general equations for an
electromagnetic field, which interacts with axions, in an arbitrary curved
spacetime. Then, taking the FRW metric, we have derived the equations for the
conformal electric and magnetic fields. Basing on this result, we have
obtained the general induction Eq. (2) for the magnetic field, which accounts
for the spatial distribution of axions. This equation has been further
modified to Eq. (2.8) in the isotropic medium. Then, we have obtained the
general Eq. (2) for the spectra of the magnetic energy and the magnetic
helicity.

To describe the evolution of axions in the presence of an electromagnetic
field, in Sec. 3, we have considered the general Eq. (3.1) for $\varphi$ in a
curved spacetime. This equation has been rewritten in the Fourier space in Eq.
(3.3). An arbitrary magnetic field contributes to the evolution of axions via
the conformal time derivative of the magnetic helicity spectrum in Eq. (3.5).

In order to get the closed system of the evolution equations for a magnetic
field and an axion, in Sec. 4, we have applied the mean field approximation
which consists in the consideration of zero mode perturbations. In this
situation, we could define the mean spectra, $\rho(k,\eta)$ and $h(k,\eta)$,
in Eq. (4.2), as well as derive Eqs. (4) and (4.4). Formally, Eqs. (4) and
(4.4) coincide with those obtained in Ref. [16]. However, they account for the
nontrivial spectrum of the $\alpha$-dynamo parameter.

In Sec. 5, we have chosen the initial conditions and have defined all the
parameters in Eqs. (4) and (4.4). We have considered the evolution of the
system below the QCD phase transition when the axion mass takes its vacuum
value. It should be noted that there is not so much information about the seed
spectrum of axions. We have adopted the initial Gaussian distribution for
$\varphi_{k}$ which is characterized by the phenomenological parameter
$\delta$. One can reproduce, e.g., the zero mode case, discussed in Refs. [15,
16], by setting $\delta\to 0$. For the magnetic field, we have chosen the seed
Kolmogorov spectrum with realistic parameters. Finally, we have rewritten the
evolution equations in the form convenient for numerical simulations; cf. Eq.
(5).

The numerical solution of Eq. (5) has been present in Sec. 6. We have been
mainly interested in the dependence of the results on the seed axion spectrum.
For this purpose we have compared the cases of the inhomogeneous, with
$\delta=10^{-10}$, and the homogeneous, with $\delta=0$, axions. In both
cases, the magnetic field evolution is different only in small time intervals.
It is smooth for inhomogeneous axions and reveal a ripple for homogeneous
axions; cf. Figs. 1 and 1. Globally, the evolution of magnetic fields is
driven mainly by the diffusion terms in Eq. (5). The energy of the
inhomogeneous axion decays faster than for the homogeneous one; cf. Figs. 1
and 1.

The possibility to get the magnetic field instability has been considered in
Sec. 6.1. For this purpose we have taken the initial axion wavefunction
amplified by the factor $\Xi=5\times 10^{7}$. The impact of axions on the
magnetic field was mentioned in Ref. [15] to be possible if the initial axion
wavefunction is greater than $f_{a}$. We have obtained that the magnetic field
can be unstable in the case of a homogeneous axion; cf. Fig. 3. This
instability is washed out for the inhomogeneous axion (see Fig. 3) owing to
the very rapid sign alteration of the $\alpha$-dynamo parameter.

## Acknowledgments

I am thankful to P.M. Akhmetev and S.V. Troitsky for useful discussions.

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11institutetext: Indian Institute of Technolgy Guwahti, Kamrup, Assam - 781039

# Budget Sensitive Reannotation of Noisy Relation Classification Data Using
Label Hierarchy

Akshay Parekh  Ashish Anand  Amit Awekar

###### Abstract

Large crowd-sourced datasets are often noisy and relation classification (RC)
datasets are no exception. Reannotating the entire dataset is one probable
solution however it is not always viable due to time and budget constraints.
This paper addresses the problem of efficient reannotation of a large noisy
dataset for the RC. Our goal is to catch more annotation errors in the dataset
while reannotating fewer instances. Existing work on RC dataset reannotation
lacks the flexibility about how much data to reannotate. We introduce the
concept of a reannotation budget to overcome this limitation. The immediate
follow-up problem is: Given a specific reannotation budget, which subset of
the data should we reannotate? To address this problem, we present two
strategies to selectively reannotate RC datasets. Our strategies utilize the
taxonomic hierarchy of relation labels. The intuition of our work is to rely
on the graph distance between actual and predicted relation labels in the
label hierarchy graph. We evaluate our reannotation strategies on the well-
known TACRED dataset. We design our experiments to answer three specific
research questions. First, does our strategy select novel candidates for
reannotation? Second, for a given reannotation budget is our reannotation
strategy more efficient at catching annotation errors? Third, what is the
impact of data reannotation on RC model performance measurement? Experimental
results show that our both reannotation strategies are novel and efficient.
Our analysis indicates that the current reported performance of RC models on
noisy TACRED data is inflated.

###### Keywords:

Noisy Dataset Dataset Reannotation Relation Classification.

##  1 Introduction

Relation Classification (RC) is the task of predicting a relation label
between a pair of real-world entities in a given natural language sentence.
For example in Figure 1, the sentence has subject entity Satya Nadella and
object entity Microsoft. Their types are PERSON and ORGANIZATION respectively.
The assigned relation label for this sentence is $org:top\\_member/employee$.
In general, the RC task considers a five-tuple $I=<S,e_{1},e_{2},T_{1},T_{2}>$
as input where $S$ is a sentence in a natural language, and $e_{1}$, $e_{2}$
are a pair of entities present in $S$. Entity $e_{1}$ is the subject and
entity $e_{2}$ is the object. The type labels for these entities are $T_{1}$
and $T_{2}$ respectively. The task of Relation Classification (RC) is to
assign a relation label $r$ to $I$ where $r\in R$ (the set of relation
labels). Relation classification is an important NLP task with potential
application in multiple domains such as knowledge graph completion [16],
question answering [6], and reasoning [15].

Figure 1: A sampled example from TACRED (top) and a part of a taxonomic-
hierarchy for TACRED relation labels.

The performance of state-of-the-art (SOTA) RC models is throttled in the range
of 70-80% accuracy only [37, 36, 13, 33, 40]. Various paradigms such as
feature based methods, kernel based methods, and Deep Learning have been
explored for designing supervised models for the RC task. All the SOTA RC
models are based on Deep Learning. Performance of such models depends
significantly on the quality of the dataset used for training and testing.
Recent studies have shown that quality of dataset is a major bottleneck in
improving the performance of supervised machine learning models for RC [1,
28].

One possible solution for improving the data quality is to reannotate the
data. Existing methods for RC dataset reannotation follow two extremes. They
either select only a tiny fraction for reannotation [1] or go for reannotation
of the complete data [28]. The existing methods also ignore the taxonomic
hierarchy that exists between the relation labels.

Our work overcomes both these limitations with the following specific research
contributions

  * •

We introduce the concept of reannotation budget to provide flexibility about
what fraction of dataset to reannotate.

  * •

This is the first work that uses relation label hierarchy while selecting
datapoints for reannotation for the RC task.

  * •

We perform extensive experiments using the popular RC dataset TACRED. We show
that our reannotation strategies are novel and more efficient when compared
with the existing approaches.

  * •

Our experiments suggest that the reported performance of existing RC models on
the noisy dataset is inflated. The F1 score of these models drops from the
range of 60%-70% to as low as below 50% when tested on clean test data
generated using our reannotation strategy.

For the complete reproducibility, all data, models, relation hierarchy, and
code from our work will be made available publicly on the Web111Link
eliminated for double-blind review. Our current experimental results are
specific to the RC task. However, our reannotation strategy can be applied to
any task and dataset where the label hierarchy is available. Rest of the paper
is organized as follows. We review the related work in Section 2. Our
reannotation approach along with experimental results is presented in Section
3. Limitations of our work are discussed in Section 4. Finally, we conclude
and point the future work in Section 5.

##  2 Related Work

Some of the earliest methods for the RC task were based on pattern extraction.
They [12, 20] use syntactic patterns to construct rule for extracting
relational facts from texts. To reduce the human effort in identifying
relation facts, statistical relational relation extraction has been
extensively explored in two directions, namely, feature-based methods [14, 38,
21] and kernel-based methods [7, 4, 5, 31]. Following the success of embedding
models in other NLP challenges, [32, 9] have made efforts in utilising low-
dimensional textual embedding for relational learning.

Neural Relation Extraction models introduce neural networks for capturing
relational facts within text, such as recursive neural networks [27, 19],
convolutional neural networks (CNN) [17, 34, 22], recurrent neural networks
(RNN) [37, 35], and attention-based neural networks [39, 30]. Recently,
Transformer [29] and pre-trained language models [8] have also been explored
for relation extraction [2, 3, 13, 25, 33, 40] and have achieved new state-of-
the-arts performance.

Out of all the RC datasets [18, 26, 11, 37, 10], TACRED [37] is the largest
and most widely used dataset. It contains more than 100 thousand sentences for
42 relation classes (Train: 68124 sentences, Dev: 22631 sentences, and Test:
15509 sentences). Sentences are collected from TAC knowledge base population
2009-2014 tasks [37]. Each sentence contains a pair of real-world entities
representing subject and object entities and is annotated with either one of
the 41 relations or no_relation. Sentences are manually annotated with
relation labels using the Amazon Mechanical Turk crowdsourcing platform, where
each annotator was provided with sentence, subject and object entities, and a
set of relations to choose from.

A recent study by Alt et. al. (referred to as TACRev) trained 49 different RC
models and selected 5000 most miss-classified instances from TEST and DEV
partitions for reannotation [1]. Their expert annotators were trained
linguists. Out of 5000 instances, they ended up modifying 960 sentences from
TEST and 1610 sentences from DEV. The revised dataset resulted in an average
8% improvement in f1-score, suggesting the noisy nature of TACRED dataset.
However, their work has two main bottlenecks. First, their method has a fixed
set of sentences to reannotate. Second, they ignore the label hierarchy and
use only model confidence to select sentences for reannotation. We compare our
strategies against TACRev by extending their method for the reannotation
budget. Given a specific reannotation budget, we select top sentences from the
dataset where RC models have the highest confidence.

Another recent study by Stoica et. al. (referred to as ReTACRED) reannotated
the complete TACRED dataset using crowd-sourcing [28]. For more effective
crowd-sourcing, they have made slight modifications to the original TACRED
relation list. They corrected 3936 and 5326 annotation errors in the TEST and
DEV partitions respectively. They have also eliminated 2091, 3047, and 9659
sentences from TEST, DEV, and TRAIN set respectively for various reasons. We
use the data from their reannotation experiment to simulate our reannotation
strategies. Our work cannot be directly compared against ReTACRED as they
simply use the brute-force method of complete reannotation. However, for a
meaningful comparison with ReTACRED, we extend their approach by selecting a
random set of sentences for a given reannotation budget.

Recently Parekh et. al. [23], have collected more than 600 relations between
Person, Organization, and Location entity types from multiple knowledge bases
and arranged them in a taxonomical hierarchy following is-a relationship. They
manually created the relation hierarchy but did not show any applications of
the generated hierarchy. In this work, we create a similar relation hierarchy
for TACRED relation labels, and further use that hierarchy for calculating
distance between ground-truth and prediction.

The main takeaway points from the existing work can be summarized as follows:

  * •

Deep Learning is the paradigm of SOTA RC models. Hence, we will focus only on
Deep Learning based models for our work.

  * •

TACRED is the RC dataset used by SOTA models. Also, reannotation data for the
complete TACRED dataset is availble. Hence, for our experiments we will used
the TACRED dataset.

  * •

Both the existing works on RC dataset reannotation are rigid. They fail to
provide flexibility about how much data to reannotate. Hence, we will
introduce the concept of reannotation budget to overcome this limitation.

  * •

The concept of taxonomic hierarchy of relation labels is not yet used for the
RC dataset reannotation. We will build such a hierarchy of relation labels in
the TACRED dataset and use it for selecting datapoints for reannotation.

##  3 Our Work

Considering the insights from related work, our goal is to build a budget
sensitive reannotation method and show its effect on TACRED dataset and recent
deep learning based models. Consider an RC dataset $D$. Let $N$ be the sent of
noisy or mislabelled sentences in $D$. As $N$ is a subset of $D$,
$|N|\leq|D|$. Let $R$ represent the set of sentences that we are going to
reannotate from $D$. Now, our reannotation budget is $|R|$. In other words, we
can afford to reannotate only $|R|$ sentences out of $|D|$. The goal of any
reannotation method should be to maximize the $|N\cap R|$. In the ideal case,
$N$ and $R$ should be the identical sets.

Given a reannotation budget $|R|$, the immediate follow-up problem is: what
subset of the data should we reannotate? If our reannotation budget permits us
to reannotate a significant part of the data ($|R|\approx|D|$) then the
strategy for selecting data points for reannotation does not matter. However,
in the real world, the reannotation budget is far small as compared to the
dataset size ($|R|\ll|D|$) for two reasons. First, reannotation is a costly
and time-consuming task. Second, redundant efforts in reannotating a correctly
labeled data point should be avoided. In the context of the RC task, this
paper proposes two strategies for selecting data points for reannotation. Our
approach capitalizes on the taxonomic hierarchy of relation labels which is
largely an ignored aspect of the RC datasets. For each data point, we compute
the graph distance between the actual label provided in the dataset and the
predicted label using an ensemble of RC models. Data points with a higher
value for this distance are given higher priority for the reannotation task.
The overview of our work is presented in Figure 2. Our work is divided into
three steps: Model Training, Reannotation, and Evaluation. These steps are
described in the following subsections.

Figure 2: Overview of our work

###  3.1 Model Training

The first step trains multiple RC models using the original TACRED dataset.
However, we transform the relation labels to match the ReTACRED labels. This
transformation is necessary as we have reannotation data available from
ReTACRED only. For our experiments we have selected following five Deep
Learning based RC models: PA-RNN, LSTM, and Bi-LSTM following [37], CGCN, and
GCN [36]. We have trained these models using the same set of hyper-parameters
as mentioned in their original papers. This step can be further improved by
training more RC models. However, we were limited by available computing
infrastructure.

All the input word vectors are initialised using pre-trained 300-dimensional
Glove vectors [24] for CGCN and GCN and 200-dimensional Glove vectors for
LSTM, BiLSTM, and PA-LSTM. For training GCN and CGCN, we used the hyper-
parameters following [36]. We used LSTM hidden size and feedforward hidden
size as 200. We used two GCN layers and two feedforward layers, SGD as
optimizer, initial learning rate of 1.0 which is reduced by a factor of 0.9
after epoch five. We trained the model for 100 epochs. We used word dropout of
0.04 and dropout of 0.5 to LSTM layers. All other embedding (such as NER, POS)
size is fixed as 30. For training PALSTM, LSTM, and BiLSTM, we followed [37]
for hyper-parameters. We have used 2 layer stacked LSTM layers for all the
models with hidden size of 200. We used AdaGrad with learning rate of 1.0
which is reduced by a factor 0.9 after 20th epoch. We have trained the model
for 30 epochs. We used word dropout of 0.04 and dropout of 0.5 to LSTM layers.
All other embedding (such as NER, POS) size is fixed as 30.

###  3.2 Reannotation

The second step simulates the reannotation process by creating a specific
permutation of the data points in the reannotation pool. For our experiments,
we have considered TEST and DEV partitions of the TACRED dataset as the
reannotation pool. The reannotation data is not available for the TRAIN
partition. In our simulation, the crowd-sourced workers will reannotate only a
subset of data points from the reannotation pool depending on the reannotation
budget. We have experimented with four reannotation strategies: two from the
literature (TACRev[1] and ReTACRED[28]) and our proposed two strategies. Given
a particular budget for reannotation, we have to select a set of sentences
from the reannotation pool to perform the reannotation task. A reannotation
strategy creates a ranked list of the reannotation pool. The goal of a
reannotation strategy is to ensure that sentences having annotation errors
should appear at the top of the ranked list. Such ranking is expected to use a
given reannotation budget effectively. Reannotating all the sentences in the
reannotation pool will require a large amount of resources. In the real world,
we can typically afford to reannotate only a fraction of the reannotation
pool. We expect that a good reannotation strategy should be able to catch most
of the annotation errors with a relatively small reannotation budget.

A trained RC model when presented with a data point for relation
classification, assigns a relation label along with the confidence score. This
score indicates how confident the model is while assigning the relation label.
The TACRev strategy uses multiple RC models for ranking the data points based
on the confidence of the RC models. In our experiments, we have used five RC
models mentioned in subsection 3.1. The ReTACRED method does not have the
concept of ranking data points. It is a brute-force strategy that simply
reannotates the whole reannotation pool. To simulate the ReTACRED strategy
with the reannotation budget, we simply pick up random sentences from the
reannotation pool to perform the reannotation task.

Our reannotation strategy is based on the taxonomic-hierarchy prepared by
Parekh et al. [23]. Following their work, we arranged all relations from the
TACRED dataset in a taxonomic hierarchy. The relation labels are arranged as a
tree. Each relation label has a parent and multiple or zero children. Please
refer to Figure 1 for an excerpt of the relation hierarchy. With such a
hierarchy, it becomes feasible to measure the error in the model prediction.
The complete taxonomic hierarchy is available for download along with the
code. Both our reannotation strategies utilize this hierarchy for selecting
candidates for reannotation.

Consider a sentence $S$ with its relation label annotated in the dataset as
$AR(S)$. Here $AR$ stands for Actual Relation. We have a list of $K$ different
RC models: $M_{1}$ to $M_{K}$. Each model $M_{i}$ assigns relation label
$PR_{i}(S)$ to the sentence $S$. Here $PR$ stands for Predicted Relation. Both
$AR(S)$ and $PR_{i}(S)$ can be located as nodes in the relation hierarchy
tree. Our first reannotation strategy measures the length of the path between
these two nodes in the tree. We refer to this strategy as the Graph Distance
(GD) strategy. For each sentence $S$, we compute the following score:

$GD(S)=(\sum_{i=1}^{K}f_{i}(S))/K$,where

$f_{i}(S)=Distance(AR(S),PR_{i}(S))$

The $Distance$ function computes the length of shortest path between given two
tree nodes. This is a basic problem in graph algorithms and can be solved in
time complexity $O(h)$, where $h$ is the height of the tree. Please refer to
Figure 1. Let us assume that for a given sentence $S$, $AR(S)$ is $per:parent$
and $PR(S)$ is $per:age$. In this case, the shortest path between these two
labels is ($per:parent$, $per:family$, $per-per$, $per$, $per-misc$,
$per:age$) with the $Distance$ as five. The function $f_{i}(S)$ measures the
disagreement between the prediction of model $M_{i}$ and the label given in
the dataset. High value for $f_{i}(s)$ indicates high disagreement between
model $M_{i}$ and the currently assigned label in the dataset. The $GD(S)$
score computes the average across $K$ models. Our Graph Distance reannotation
strategy considers sentences with a high $GD$ score as preferred candidates
for reannotation. Intuitively, we are selecting sentences where multiple RC
models have strong disagreement with the currently assigned label.

Our second strategy fine-tunes the computation of disagreement between model
prediction and currently assigned label in the dataset. We locate both $AR(S)$
and $PR_{i}(S)$ in the relation hierarchy and compute their Lowest Common
Ancestor (LCA). The $LCA$ computation in the tree is also a basic graph
problem and can be solved in time complexity $O(h)$, where $h$ is the height
of the tree. Let us consider the same example with $AR(s)$ as $per:parent$ and
$PR(S)$ as $per:age$. In this case, the $LCA$ of these two labels is the node
$per$. Consider the path from the root of the relation hierarchy to the node
$AR(S)$. The LCA node is the point at which the model prediction starts
differing from the currently assigned label in the dataset. We compute the
following score for each sentence $S$:

$LD(S)=(\sum_{i=1}^{K}g_{i}(S))/K$,where

$g_{i}(S)=Distance(AR(S),LCA(AR(S),PR_{i}(S)))$

The function $g_{i}(S)$ is an improvement over the previously defined function
$f_{i}(S)$. Instead of considering the complete path between $AR(S)$ and
$PR_{i}(S)$, we are considering only the part up to the LCA of these two
nodes. We call this reannotation strategy as LCA Distance (LD) strategy. Each
reannotation strategy creates a permutation of the reannotation pool by
sorting sentences in the descending order of the corresponding scores.

###  3.3 Evaluation

After Step 2, each reannotation strategy will create its own permutation of
the reannotation pool. Based on the reannotation budget $|R|$, we will select
the top $|R|$ data points from each permutation. While comparing our
strategies with the TACRev and ReTACRED, we need to answer two important
questions. First about novelty: do our strategies provide novel candidates for
reannotation? Second about efficiency: do our strategies catch more annotation
errors for a given reannotation budget?

Figure 3: Similarity of a reannotation strategy with TACRev.

####  3.3.1 Novelty in Candidate Selection

Novelty of GD and LD against ReTACRED is trivial because ReTACRED selects
random data points for reannotation. Please refer to Figure 3 for the analysis
of novelty against TACRev. The X-axis indicates the reannotation budget.
Y-axis indicates the Jaccard similarity, that is the size of the intersection
divided by the size of the union. The similarity of TACRev with itself is
always 100%. We can observe that for a low reannotation budget, the similarity
score grows quickly and approaches 100%. This indicates that initially, all
strategies choose similar candidates for reannotation. For medium budget
value, the similarity falls significantly. This indicates that our strategies
differ significantly from TACRev for medium-budget value. For a higher
reannotation budget, the similarity score again approaches 100% as expected.
When the reannotation budget is high, we are going to reannotate almost all
the dataset. In such a scenario, there is hardly any chance for novel
candidate selection.

####  3.3.2 Efficiency

ReTACRED performed complete reannotation of TACRED data [28]. They observed
that the set of noisy datapoints ($N$) in TACRED are: 5326 sentences from DEV
partition and 3936 sentences from TEST partition. To evaluate the performance
of our reannotation strategies, the obvious way is to carry out the
reannotation task. However, we simulated the performance of our reannotation
strategies using the set $N$ for two main reasons. First, the set $N$ is
carefully prepared by designing a large and multi-stage crowdsourcing
experiment by ReTACRED authors. Second, we did not have the financial budget
available to carry out the large-scale reannotation task. We take the labels
from ReTACRED as gold standard labels.

Please refer to Figure 4. The X-axis indicates the reannotation budget. Y-axis
depicts the percentage of sentences from the set $N$ that we can find in the
given budget using a particular reannotation strategy. For example, when we
reannotate the top 1000 sentences from the TEST partition using the LD
strategy, it contains 566 sentences, around 14.4% from the set $N$. We can
observe that for a low reannotation budget, GD, LD, and TACRev have the same
efficiency. This is expected as their similarity was close to 100% for the low
reannotation budget. However, for the medium reannotation budget our
strategies outperform TACRev. For the very high value of the reannotation
budget, TACRev performs slightly better than our strategies. The efficiency of
GD and LD strategies is almost the same as their lines are overlapping in the
figure. Efficiency of ReTACRED grows linearly as it chooses random sentences
for reannotation.

Figure 4: Annotation errors corrected by various reannotation strategies.

####  3.3.3 Impact on Model Performance Measurement

Please refer to Figure 5. The X-axis indicates the reannotation budget. Y-axis
indicates the F1 score of each model. Please note that we are only
reannotating the TEST data here. The models are trained on the noisy TACRED
dataset. We can observe that for three reannotation strategies (TACRev, GD,
and LD), initially, the model performance improves with reannotation. This
indicates that RC models agree with the initial reannotations. However, the
performance of all models falls significantly with a higher reannotation
budget. The test dataset is the cleanest when the reannotation budget is
maximum. The least value of the F1 score for all models at this point
indicates that the original performance reported in the literature for these
models was inflated. The main reason for this inflated value is the
significant noise in the TRAIN partition of the TACRED dataset.

Figure 5: Performance of RC models at different Reannotation Budget.

None of the previous works could catch this trend for the following reason. We
have conducted experiments here with noisy training data and a varying degree
of noise reduction in the test data. However previous works can be classified
in the following categories

Noisy training and noisy test data: Various works that directly used the
TACRED data are training their RC models on noisy TRAIN partition of TACRED
[36]. They are also testing their RC models on noisy TEST data. As a result,
their reported F1 score is in the range of 60% to 70%. This corresponds to
reannotation budget zero in Figure 5.

Noisy training and partially clean test data: The TACRev study had reported an
average 8% improvement in the F1 score with the reannotation budget of about
2300 sentences from the TEST partition [1]. Our results for TACRev in Figure 5
correlate with their reported numbers. We can observe that while using the
TACRev reannotation strategy, performance of all models peaks with the
reannotation budget of around 2300. However, their results are a special case
of our experiments with a fixed reannotation budget of 2300. We can observe a
similar trend for our reannotation strategies GD and LD in Figure 5. Further
cleaning of TEST dataset beyond the budget of 2300, actually results in the
performance drop for all RC models and all reannotation strategies.

Partially clean training and completely clean test data: ReTACRED performed
complete reannotation of the DEV and TEST partitions of the TACRED dataset
[28]. They did not perform any reannotation of the TRAIN partition. However,
they eliminated around ten thousand potentially noisy sentences from the TRAIN
partition. As a result, they report at least 10% improvement in F1 score for
various RC models as compared to previous works. This case cannot be captured
in Figure 5 because we are not making any changes to the TRAIN partition of
the TACRED dataset.

We have extended the ReTACRED approach as a reannotation strategy while
keeping the training data unchanged and progressively cleaning the test data
depending on the reannotation budget. In Figure 5, we can observe that for
ReTACRED strategy, the model performance goes on monotonically decreasing with
the reannotation budget. This is expected as the order of the sentences for
reannotation is random. From other three reannotation strategies we can
observe that reannotation of only about top 2500 sentences helps RC models to
boost their performance. All these 2500 sentences are ordered randomly in the
ReTACRED strategy. Reannotation of the rest of the sentences from the TEST
partition does not agree with RC model prediction and brings down the model
performance.

##  4 Limitations of Our Work

The first limitation of our work is that we are trusting the annotations from
ReTACRED as the ground truth. In reality, the ReTACRED annotations are also
done using crowd-sourcing. However, as compared to the original TACRED
dataset, they have taken more precautions to avoid annotations errors. For
example, they have rearranged the relation labels to avoid confusion in
labeling. They have also created better instructions for the crowd-source
workers. One way to further reduce the possibility of errors in the data is to
get annotations from domain experts. However, considering the scale of
dataset, it is a challenging task.

The second limitation of our work is that we have not yet dealt with
reannotation of TRAIN partition. It will be interesting to see if the concept
of reannotation budget helps to significantly improve the quality of training
data while annotating only a small fraction of the training data. Currently,
we are working on an automatic method for reannotating the training data using
an ensemble of RC models.

The third limitation of our work is that it requires a taxonomic hierarchy of
relation labels. We have created the relation hierarchy manually for the
TACRED dataset. It will be helpful if such a hierarchy can be created
automatically from the RC dataset.

##  5 Conclusion and Future Work

We presented two reannotation strategies in this paper: GD and LD. We also
introduced the concept of reannotation budget to provide flexibility about how
much data to reannotate. As compared to existing strategies, our strategies
are novel and efficient. These reannotation strategies also help us understand
the true performance of RC models trained over the noisy TACRED data. We plan
to extend this work by designing strategies for selective reannotation of
training data and evaluating its impact on model training.

The important conclusions from this work are: (i) relation between labels is
an important but underutilized aspect of the data. It can helpful in numerous
ways to improve dataset. (ii) A high rate of reannotation indicates that
crowd-sourced labels are not trustworthy, especially for complex NLP tasks
such as Relation Classification. Hence, data annotation should not be
considered as a one-time task. Rather it should be treated as a budget-
sensitive and iterative process where data is labelled in multiple iterations.

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