∎

e1e-mail<EMAIL_ADDRESS>e2e-mail<EMAIL_ADDRESS>
11institutetext: Dipartimento di Fisica ’G. Occhialini’, Università degli
Studi di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy
22institutetext: Institute of Physics, University of Graz, NAWI Graz,
Universitätsplatz 5, 8010 Graz, Austria33institutetext: Silicon Austria Labs
GmbH, Inffeldgasse 25F, 8010 Graz, Austria

# Neutron Stars in Palatini $R+\alpha R^{2}$ and $R+\alpha R^{2}+\beta Q$
Theories

Georg Herzoge1,addr1  Hèlios Sanchis-Alepuze2,addr2,addr3

(Received: date / Accepted: date)

###### Abstract

We study solutions of the stellar structure equations for spherically
symmetric objects in modified theories of gravity, where the Einstein-Hilbert
Lagrangian is replaced by $f(R)=R+\alpha R^{2}$ and $f(R,Q)=R+\alpha
R^{2}+\beta Q$, with $R$ being the Ricci scalar curvature,
$Q=R_{\mu\nu}R^{\mu\nu}$ and $R_{\mu\nu}$ the Ricci tensor. We work in the
Palatini formalism, where the metric and the connection are assumed to be
independent dynamical variables. We focus on stellar solutions in the mass-
radius region associated to neutron stars. We illustrate the potential impact
of the $R^{2}$ and $Q$ terms by studying a range of viable values of $\alpha$
and $\beta$. Similarly, we use different equations of state (SLy, FPS, HS(DD2)
and HS(TMA)) as a simple way to account for the equation of state uncertainty.
Our results show that for certain combinations of the $\alpha$ and $\beta$
parameters and equation of state, the effect of modifications of general
relativity on the properties of stars is sizeable. Therefore, with increasing
accuracy in the determination of the equation of state for neutron stars,
astrophysical observations may serve as discriminators of modifications of
General Relativity.

††journal: Eur. Phys. J. C

##  1 Introduction

General Relativity (GR) as the simplest realisation of a geometric description
of gravity has so far shown, perhaps unexpectedly, total agreement with
astrophysical and cosmological observations Will:2014kxa ; Berti:2015itd , as
long as the general picture is accepted that the dark sectors, as needed for
the $\Lambda$CDM model, are related to the particle and energy content of the
Universe. Despite this tremendous success, there are reasons to investigate
modifications of GR. First, GR breaks down in the high curvature regime.
Second, it is only a classical field theory without any quantum effects and
which is not (perturbatively) renormalisable as a quantum field theory unless
quadratic curvature corrections are added to the Lagrangian Stelle:1976gc .
Moreover, it is conceivable that all or part of the effects currently
attributed to dark sectors are in reality a manifestation of different
gravitational dynamics Clifton:2011jh ; deMartino:2015zsa ; Ferreira:2019xrr ;
Frusciante:2019xia .

Some of the (theoretical) problems of GR can be ameliorated in modified
gravitational theories, like $f(R)$ and $f(R,Q)$ theories in which the
Einstein-Hilbert Lagrangian is modified by adding polynomial terms in the
Ricci scalar $R$ or in the contraction of the Ricci curvature
$Q=R_{\mu\nu}R^{\mu\nu}$ (see e.g. Sotiriou:2008rp ; DeFelice:2010aj ;
Capozziello:2010zz ; Olmo:2011uz ; Capozziello:2015lza ; Nojiri:2017ncd  and
references therein). From a different point of view, if the different terms in
a gravity action are considered as quantum operators in the spirit of
effective quantum field theories, studying their renormalisation group
evolution indicates that $f(R)$ and $f(R,Q)$ terms render the theory
asymptotically safe (and hence viable as a quantum theory) in the Planckian
and super-Planckian regime Codello:2007bd ; Hindmarsh:2012rc ; Falls:2013bv ;
Falls:2017lst ; Alkofer:2018fxj . Finally, classical theories of modified
gravity can also serve as effective descriptions of their more fundamental
quantum counterparts (see e.g. Olmo:2008nf ; Bonanno:2012jy ).

Studying the compatibility of modified theories of gravity with phenomenology
is a highly non-trivial issue since an interpretation of observations involves
a combination of many different and uncertain physical mechanisms. As
mentioned above, cosmological observations compatible with GR and dark sectors
can be also made compatible with certain modifications of gravity and
different contributions from dark sectors Pinto:2018rfg . Inflationary
scenarios can also be expressed in terms of modified theories of gravity (see
e.g. Antoniadis:2018ywb ; Antoniadis:2018yfq ; Tenkanen:2020dge ;
Gialamas:2019nly ; Gialamas:2020snr ). It has been suggested that
modifications of GR can be identified via the detection of new modes in
gravitational waves Capozziello:2008rq ; Alves:2009eg ; Corda:2010zza ;
DeLaurentis:2011tp ; Rizwana:2016qdq ; Liang:2017ahj ; Capozziello:2019klx ;
Katsuragawa:2019uto . Another possibility to investigate deviations from GR is
with astrophysical observations. Different gravitational dynamics reflects
into different stellar equations of structure Kainulainen:2006wz ; Olmo:2012er
; Berti:2015itd ; Paschalidis:2016vmz ; Olmo:2019flu ; Astashenok:2020isy  and
hence, in particular, in mass to radius ratios of neutron stars different to
those expected from GR (see e.g. Arapoglu:2010rz ; Deliduman:2011nw ;
Astashenok:2013vza ; Astashenok:2014nua ; Astashenok:2014dja ;
Capozziello:2015yza ; Arapoglu:2016ozr ; Astashenok:2017dpo ; Pannia:2016qbj ;
Astashenok:2020qds ; Astashenok:2021peo ; Odintsov:2021qbq ; Odintsov:2021nqa
). This is particularly interesting with the growing volume of observational
data on neutron stars and the discovery of compact objects that are difficult
to accomodate in a standard GR picture (see e.g. TheLIGOScientific:2017qsa ;
Alsing:2017bbc ; Abbott:2018wiz ; Tsokaros:2020hli ). These measurements are,
unfortunately, riddled with enormous uncertainties coming from the equation of
state (EoS) describing the interior of neutron stars and can as well be
interpreted as providing a handle on EoS assuming the theory of gravity is GR
Abbott:2018exr . However, as we will show in this paper, the changes in the
mass-radius relation for neutron stars coming from the uncertainty in the EoS
are sometimes comparable to the differences generated by the modifications of
the gravity theory, which implies that if, in the future, reliable ab-initio
calculations of the EoS were available, astrophysical observations could be
used as discriminators for gravity theories.

In this paper we shall investigate the influence of two particular models of
$f(R)$ and $f(R,Q)$ theories, namely $f(R)=R+\alpha R^{2}$ and
$f(R,Q)=R+\alpha R^{2}+\beta Q$ on the mass-radius relation for neutron stars.
We study those theories in the Palatini formalism in which the dynamical
degrees of freedom are the metric and the affine connection Olmo:2011uz . We
will perform our calculations using a small number of representative EoS with
the goal of estimating the uncertainty stemming from the EoS and from the
theory of gravity (in our case simply encoded in the values of $\alpha$ and
$\beta$).

The paper is organised as follows. We sketch the derivation of the equations
of stellar structure for Palatini theories in Sec. 2 and discuss several
aspects of the EoS chosen in Sec. 3. Our results are shown in Sec. 4 and we
conclude a discussion of the implications of our results on the validity of
theories studied herein and of possible future work.

##  2 Stellar Structure Equations for $f(R,Q)$ Palatini Theories

We discuss here the main aspects of the equations of stellar structure solved
in this work. We consider only the simplified case of non-rotating and static
stars, whose only observables are their mass and radius. The equations for
$f(R,Q)$ theories were derived in Olmo:2012er  and we refer to that paper for
further details (see also Kainulainen:2006wz ; Pannia:2016qbj ). In GR, the
equivalent equations are the well-known Tolman-Oppenheimer-Volkoff (TOV)
equations.

In Palatini $f(R,Q)$ theories the action is given by

$S[g,\Gamma,\psi_{m}]=\frac{1}{2\kappa}\int
d^{4}x\sqrt{-g}f(R,Q)+S_{m}[g,\psi_{m}],$ (1)

where $R$ is the Ricci scalar, $Q=R_{\mu\nu}R^{\mu\nu}$ the aforementioned
contraction of two Ricci tensors, with $R_{\mu\nu}=R_{\mu\rho\nu}^{\rho}$ and
the Riemann tensor given by
$R^{d}_{abc}\equiv\partial_{b}\Gamma^{d}_{ac}-\partial_{c}\Gamma^{d}_{ab}+\Gamma^{e}_{ac}\Gamma^{d}_{eb}-\Gamma^{e}_{ab}\Gamma^{d}_{ec}$
with $\Gamma$ the connection coefficients, $g_{\mu\nu}$ is the metric,
$\kappa\equiv 8\pi G$ and $S_{m}[g,\psi_{m}]$ is the matter action. In the
Palatini formalism the equations of motion are obtained after varying the
action with respect to both, the metric and the connection

$\displaystyle
f_{R}R_{\mu\nu}-\frac{f}{2}g_{\mu\nu}+2f_{Q}R_{\mu\alpha}{R^{\alpha}}_{\nu}$
$\displaystyle=$ $\displaystyle\kappa T_{\mu\nu}$ (2)
$\displaystyle\nabla_{\beta}\left[\sqrt{-g}\left(f_{R}g^{\mu\nu}+2f_{Q}R^{\mu\nu}\right)\right]$
$\displaystyle=$ $\displaystyle 0\ ,$ (3)

with $f_{R}\equiv\frac{\partial f}{\partial R}$ and $f_{Q}\equiv\frac{\partial
f}{\partial Q}$. One can introduce an auxiliary metric $h_{\mu\nu}$ via the
(matter-content dependent) mapping
$\sqrt{-g}\left(f_{R}g^{\mu\nu}+2f_{Q}R^{\mu\nu}\right)\equiv\sqrt{-h}h^{\mu\nu}$
such that the equation for the connection becomes
$\nabla_{\beta}\left[\sqrt{-h}h^{\mu\nu}\right]=0$ and the connection $\Gamma$
can thus be written as the Levi-Civita connection for $h_{\mu\nu}$. Therefore,
the Ricci tensor in Eqs. 2 and 3 can be written as the standard metric Ricci
tensor of the metric $h$ which we denote as $R_{\mu\nu}(h)$. Introducing
${\Sigma_{\alpha}}^{\nu}:=\left(f_{R}\delta_{\alpha}^{\nu}+2f_{Q}{B_{\alpha}}^{\nu}\right)$,
with ${B_{\alpha}}^{\nu}$ given as
${B_{\alpha}}^{\nu}=R_{\alpha\beta}(h)g^{\beta\nu}$, the relation between
$g_{\mu\nu}$ and $h_{\mu\nu}$ is Olmo:2012er

$h_{\mu\nu}=\sqrt{\det\Sigma}{[\Sigma^{-1}]_{\mu}}^{\alpha}g_{\alpha\nu}\
\text{ },\text{ }\
h^{\mu\nu}=\frac{g^{\mu\alpha}{{\Sigma_{\alpha}}^{\nu}}}{\sqrt{\det\Sigma}}\
.$ (4)

The Ricci tensor is given by

$R_{\mu}^{\nu}(h)=R_{\mu\alpha}(h)h^{\alpha\nu}=\frac{1}{\sqrt{\mathrm{det}\Sigma}}\left(\frac{f}{2}\delta_{\mu}^{\nu}+\kappa
T_{\mu}^{\nu}\right).$ (5)

Note here that, when calculating $R$ and $Q$ from $R(h)$ in $f(R,Q)$, the
Ricci tensor $R_{\mu\nu}(h)$ must be contracted with the physical metric
$g^{\mu\nu}$, that is, via the matrix $B_{\alpha}^{\nu}$ above Olmo:2012er

$R=B_{\alpha}^{\alpha}\quad\mathrm{and}\quad
Q=B_{\alpha}^{\nu}B_{\nu}^{\alpha}~{}.$ (6)

In this paper we will only consider energy-momentum tensors of the form

$T_{\mu\nu}=\left(\rho+P\right)u_{\mu}u_{\nu}+Pg_{\mu\nu}~{},$ (7)

and Lagrangians of the form $f(R)=R+\alpha R^{2}$ and $f(R,Q)=R+\alpha
R^{2}+\beta Q$. In these cases we can write the scalars $R$ and $Q$ in terms
of $\rho$ and $P$ as $R=-\kappa T$, as in GR, and Olmo:2012er

$\beta
Q=-\left(\tilde{f}+\frac{\tilde{f}_{R}^{2}}{4f_{Q}}+2\kappa^{2}P\right)+\frac{{f_{Q}}}{16}\left[3\left(R+\frac{\tilde{f}_{R}}{f_{Q}}\right)\pm\sqrt{\left(R+\frac{\tilde{f}_{R}}{f_{Q}}\right)^{2}-\frac{4\kappa^{2}(\rho+P)}{f_{Q}}}\right]^{2}~{},$
(8)

where $\tilde{f}=f(R)=R+\alpha R^{2}$, $\tilde{f_{R}}=1+2\alpha R$ and
$f_{Q}=\beta$. The sign in front of the square root must be chosen so as to
obtain the right limit at low curvatures.

The stellar structure equations are the dynamical equations for the star’s
interior metric, parametrised as

$g_{\mu\nu}=\mathrm{diag}\left(-A(r)e^{2\psi(r)},\frac{1}{A(r)},r^{2},r^{2}\mathrm{sin}^{2}\theta\right)~{},$
(9)

where $A(r)$ will be written as $A(r)=1-\frac{2M(r)}{r}$. After a tedious but
straightforward derivation (see Olmo:2012er  for details) one arrives at the
stellar structure equations for Palatini in the case of $f(R,Q)=R+\alpha
R^{2}+\beta Q$, which read

$\left(\frac{\Omega_{r}}{\Omega}+\frac{2}{r}\right)\psi_{r}=\frac{1}{A}\left(\tau_{r}^{r}-\frac{\Omega}{S}\tau_{t}^{t}\right)-\frac{1}{2}\frac{\Omega_{r}}{\Omega}\left(2\frac{\Omega_{r}}{\Omega}+\frac{S_{r}}{S}\right)-\frac{1}{r}\left(\frac{S_{r}}{S}-\frac{\Omega_{r}}{\Omega}\right)+\frac{\Omega_{rr}}{\Omega}~{},$
(10)
$\left(\frac{\Omega_{r}}{\Omega}+\frac{2}{r}\right)\frac{M_{r}}{r}=\frac{3\tau_{r}^{r}-\frac{\Omega}{S}\tau_{t}^{t}}{2}+A\left(\frac{\Omega_{rr}}{\Omega}+\frac{\Omega_{r}}{\Omega}\left(\frac{2r-3M}{r(r-2M)}-\frac{3}{4}\frac{\Omega_{r}}{\Omega}\right)\right)~{},$
(11)
$P_{r}=-\frac{P^{(0)}_{r}}{\left[1-\alpha(r)\right]}\frac{2}{\left(1\pm\sqrt{1-\beta(r)P^{(0)}_{r}}\right)}~{},$
(12)

where, as before, a subscript denotes partial derivation, i.e.
$M_{r}=\frac{\partial M}{\partial r}$ and
$\Omega_{rr}=\frac{\partial^{2}\Omega}{\partial r^{2}}$, and we introduced for
compactness

$\alpha(r)=\frac{(\rho+P)}{2}\left(\frac{\Omega_{P}}{\Omega}+\frac{S_{P}}{S}\right)~{},$
(13)
$\beta(r)=(2r)\frac{\Omega_{P}}{\Omega}\left[1-\frac{(\rho+P)}{2}\left(\frac{3}{2}\frac{\Omega_{P}}{\Omega}-\left\\{\frac{\Omega_{P}}{\Omega}-\frac{S_{P}}{S}\right\\}\right)\right]~{},$
(14)
$P^{(0)}_{r}=\frac{(\rho+P)}{r(r-2M)}\left[M-\left(\tau_{r}^{r}+\frac{\Omega}{S}\tau_{t}^{t}\right)\frac{r^{3}}{4}\right]~{}.$
(15)

Here, $\Omega$ and $S$ are given by the relation between the auxiliary metric
$h_{\mu\nu}$ and the physical metric $g_{\mu\nu}$

$h_{tt}=\frac{\sigma_{2}^{2}}{\sqrt{\sigma_{1}\sigma_{2}}}g_{tt}\equiv
Sg_{tt}\quad\mathrm{and}\quad
h_{ij}=\sqrt{\sigma_{1}\sigma_{2}}g_{ij}\equiv\Omega g_{ij}~{},$ (16)

where $\sigma_{1}$ and $\sigma_{2}$ appear upon rewriting the matrix $\Sigma$
as
$\Sigma_{\alpha}^{\nu}=\text{diag(}\sigma_{1},\sigma_{2},\sigma_{2},\sigma_{2}\text{)}$
from Eq. 5 and are given by Olmo:2012er

$\sigma_{1}=\frac{f_{R}}{2}\pm\sqrt{2f_{Q}}\sqrt{\lambda^{2}-\kappa(\rho+P)},$
(17) $\sigma_{2}=\frac{f_{R}}{2}+\sqrt{2f_{Q}}\lambda~{},$ (18)

where

$\lambda=\frac{\sqrt{2f_{Q}}}{8}\left(3\left(R+\frac{f_{R}}{f_{Q}}\right)\pm\sqrt{\left(R+\frac{f_{R}}{f_{Q}}\right)^{2}-\frac{4\kappa(\rho+P)}{f_{Q}}}\right)~{}.\\\
$

In order to recover the GR limit one has to use the positive sign in front of
the square root in Eq. 12 and for $\sigma_{1}$ but the negative sign in
$\lambda$ Olmo:2012er . Finally, $\tau$ refers to the right-hand side of Eq.
5, namely $R_{\mu}^{\nu}\equiv\tau_{\mu}^{\nu}$.

Note that, in Eq. 11, we have
$\Omega_{rr}=\Omega_{PP}P_{r}^{2}+\Omega_{P}P_{rr}$ which involves the second
derivative $P_{rr}$, in contrast to the standard TOV equations. As it turns
out, it is possible to write $P_{rr}$ in terms of the first order derivative
$P_{r}$. The result is Olmo:2012er

$\frac{P_{rr}}{P_{r}}=\frac{P^{(0)}_{rr}}{P^{(0)}_{r}}\left[1+\frac{s}{2}\frac{\beta
P^{(0)}_{r}}{\sqrt{1-\beta P^{(0)}_{r}}(1\pm\sqrt{1-\beta
P^{(0)}_{r}})}\right]+\left[\frac{\alpha_{r}}{1-\alpha}+\frac{\pm
1}{2}\frac{\beta_{r}P^{(0)}_{r}}{\sqrt{1-\beta P^{(0)}_{r}}(1\pm\sqrt{1-\beta
P^{(0)}_{r}})}\right]~{},$ (19)

with $\alpha_{r}=\alpha_{P}P_{r}$ and $\beta_{r}=\beta_{P}P_{r}$ and

$\frac{P^{(0)}_{rr}}{P^{(0)}_{r}}=\left[\left(\frac{1+\rho_{P}}{\rho+P}\right)-\frac{\Phi_{P}\frac{r^{3}}{4}}{\left(M-\Phi\frac{r^{3}}{4}\right)}\right]P_{r}-\left(\frac{2(r-M)}{r(r-2M)}+\frac{\frac{3\Phi
r^{2}}{4}}{M-\Phi\frac{r^{3}}{4}}\right)+M_{r}\left(\frac{2}{r-2M}+\frac{1}{M-\Phi\frac{r^{3}}{4}}\right).$
(20)

where we introduced $\Phi\equiv(\tau_{r}^{r}+\frac{\Omega}{S}\tau_{t}^{t})$.
In this way, Eqs. 10–12 are a closed (after fixing an equation of state)
system of equations expressed in terms of $r$, $\rho(P)$, $P$, $M$ and their
first radial derivatives only.

##  3 Equations of State

The essential ingredient to solve the equations of stellar structure is an
equation of state that relates the pressure to the energy density inside the
star.

In oder to estimate the different effects on the mass and radius of neutron
stars coming from the modifications of the gravity lagrangian and from the
uncertainty in the EoS, we use a number of different EoS in analytic as well
as tabulated form. Specifically, we use two different tabulated and two
different analytic EoS. The analytic equations of state we used are the so-
called SLy and FPS Haensel:2004nu  The SLy and FPS EoS are analytic
parametrisations of the results of many-body calculations with unified
effective nuclear Hamiltonians Friedman:1981qw ; Douchin:2001sv , describing
all regions of the neutron star interior from crust to core including its
transitions. The analytic paramerisations take care that all thermodynamic
conditions involving derivatives of the EoS are fulfilled, an aspect that will
become problematic when using tabular data directly, as we will see. The SLy
and FPS EoS are both parametrised by the following function

$\displaystyle\zeta$ $\displaystyle=$
$\displaystyle\frac{a_{1}+a_{2}\xi+a_{3}\xi^{3}}{1+a_{4}\,\xi}\,f_{0}(a_{5}(\xi-
a_{6}))$ (21) $\displaystyle+(a_{7}+a_{8}\xi)\,f_{0}(a_{9}(a_{10}-\xi))$
$\displaystyle+(a_{11}+a_{12}\xi)\,f_{0}(a_{13}(a_{14}-\xi))$
$\displaystyle+(a_{15}+a_{16}\xi)\,f_{0}(a_{17}(a_{18}-\xi))~{},$

where $\xi=\text{log}(\rho/\text{g cm}^{-3})$ and
$\zeta=\text{log}(P/\text{dyn cm}^{-2})$ and $f_{0}(x)$ is given by
Haensel:2004nu

$f_{0}(x)=\frac{1}{e^{x}+1}.$ (22)

The fitted parameters $a_{i}$ in Eq. 21 are shown in Table 1.

Table 1: SLy and FPS parameters for Eq. (21), given in Haensel:2004nu . i |  $a_{i}$(FPS) |  $a_{i}$(SLy) | i |  $a_{i}$(FPS) |  $a_{i}$(SLy)
---|---|---|---|---|---
$1$ | 6.22 | 6.22 | ${10}$ | 11.8421 | 11.4950
$2$ | 6.121 | 6.121 | ${11}$ | $-22.003$ | $-22.775$
$3$ | 0.006004 | 0.005925 | ${12}$ | 1.5552 | 1.5707
$4$ | 0.16345 | 0.16326 | ${13}$ | 9.3 | 4.3
$5$ | 6.50 | 6.48 | ${14}$ | 14.19 | 14.08
$6$ | 11.8440 | 11.4971 | ${15}$ | 23.73 | 27.80
$7$ | 17.24 | 19.105 | ${16}$ | $-1.508$ | $-1.653$
$8$ | 1.065 | 0.8938 | ${17}$ | 1.79 | 1.50
$9$ | 6.54 | 6.54 | ${18}$ | 15.13 | 14.67
Figure 1: The Sly and FPS equation of state and their first and second
derivatives.

Additionally to the two analytic EoS above, we also studied neutron star
solutions based on two other EoS in tabulated form. We choose EoS available in
the CompOSE online service composeweb , namely the so called HS(DD2)
Hempel:2009mc ; Typel:2009sy  and HS(TMA) Toki:1995ya ; Hempel:2009mc  EoS.
These two EoS are both based on the the statistical model presented in
Hempel:2009mc , which includes the contribution of nuclei, nucleons,
electrons, positrons and photons (excluding neutrino contributions) and which
requires as input the masses and binding energies of nuclei and an effective
model for the nucleon-nucleon interaction, which is then treated in the
relativistic mean-field (RMF) approximation. The two EoS thus differ in the
different model used for that dynamical input. The HS(DD2) equation of state
uses the density-dependent nuclear model called DD2 Typel:2009sy , with the
nuclei properties given by the FRDM model Moller:1996uf . The HS(TMA) equation
of state uses instead the TMA model Toki:1995ya , with the masses of nuclei
taken from Geng:2005yu .

The drawback of using tabulated data to solve the equations of stellar
structure is that interpolation functions and numerical derivatives thereof
must be used. This procedure is ambiguous (as one can use many different forms
for the interpolating functions) and, as mentioned above, certain
thermodynamic relations must be preserved by the derivatives of the EoS (see
e.g. Haensel:2004nu ). This is a particularly difficult problem to tackle in
the case of modified Palatini theories, since higher derivatives of the EoS
are needed as compared to GR. Indeed, in the stellar structure equations for
Palatini $f(R)$ and $f(R,Q)$ the first and second derivative of the density
$\rho$ with respect to the pressure $P$ appear. For the analytically given EoS
above this is relatively straightforward. Remembering that
$P(\rho)=10^{\zeta(\xi(\rho))}=10^{\zeta(\text{log}_{10}(\rho))}$ we have

$\frac{dP(\rho)}{d\rho}=\frac{dP}{d\zeta}\frac{d\zeta}{d\xi}\frac{d\xi}{d\rho}=\frac{P}{\rho}\frac{d\zeta}{d\xi}~{},$
(23)

and

$\frac{d^{2}P}{d\rho^{2}}=\frac{d}{d\rho}\left(\frac{dP}{d\rho}\right)=\frac{dP}{d\rho}\frac{1}{\rho}\frac{d\zeta}{d\xi}-\frac{P}{\rho^{2}}\frac{d\zeta}{d\xi}+\frac{P}{\rho^{2}}\frac{d^{2}\zeta}{d\xi^{2}}~{},$
(24)

in order to arrive at the first and second derivative of the density with
respect to the pressure we now have to invert Eqs. 23 and 24. One gets

$\frac{d\rho}{dP}=\frac{1}{\frac{dP}{d\rho}}~{},$ (25)
$\frac{d^{2}\rho}{dP^{2}}=-\frac{\frac{d^{2}P}{d\rho^{2}}}{\left(\frac{dP}{d\rho}\right)^{3}}~{}.$
(26)

The result for the parametrization of the SLy and FPS EoS together with the
first and second derivative of the parametrisations are shown in Fig. 1. For
tabulated EoS data one can either apply a similar procedure as above after
defining appropriate interpolating functions or simply calculate the
derivatives numerically, as we did. When calculating the derivatives
numerically, the different ways in which the data can be interpolated and
hence derivatives can be taken has an impact on the results, as we discuss
below.

##  4 Results

The mass and radius of a star are obtained by integrating from inside out the
stellar structure equations after fixing a central density and until the
pressure reaches a pre-determined threshold. In our calculations, the
threshold was set to $P=P_{c}\cdot 10^{-12}$ where $P_{c}$ is the initial
central pressure (determined by the EoS from the central density) in case of
the analytic EoS. For tabulated EoS we stopped the integration when reaching
the last entry of the EoS tables. We considered central densities in the range
$\xi=[14.4,16]$ (at $r=0.01$mm for numerical stability). The integration was
performed using a fourth-order Runge-Kutta method with a fixed stepsize of
$1$m, since after exploring several adaptive-step methods we concluded that a
fixed step was accurate enough and sped up the calculations.

A problem arising whenever a modified theory of gravity is used, is to
determine the relative strength of the additional terms of the Lagrangian.
Since in the present paper we use theories of the type $f(R)=R+\alpha R^{2}$
and $f(R,Q)=R+\alpha R^{2}+\beta Q$, the problem reduces to finding reasonable
values for $\alpha$ and $\beta$, consistent with phenomenology. However until
now no solid experimental bounds on the values of $\alpha$ and $\beta$ exist
for theories in the Palatini formalism (see Sotiriou:2005cd  for a study in
Palatini). There exist, however, more restrictive experimental bounds on
$\alpha$ for theories in the metric formalism (where only the metric is
considered a dynamical degree of freedom). The Gravity Probe B experiment
constrains $\alpha$ to $\alpha\lesssim 5\cdot 10^{15}$ cm2, while the
constraint coming the Pulsar B in PSR J0737-3039 is four orders of magnitude
higher Naf:2010zy . In Naf:2010zy  they derived an even more stringent
constraint on $\alpha$ from the Eöt-Wash experiment, constraining $\alpha$ to
$\alpha\lesssim 10^{-6}$ cm2. However, this bound was derived in the low
curvature regime of a laboratory on earth. Since possible modifications to GR
will likely only be relevant in the high curvature regime, we do not take the
bound from this experiment into consideration. Recently another bound for
$\alpha$ has been derived by calculating the stability of stars in metric
$f(R)=R+\alpha R^{2}$ gravity Pretel:2020rqx . Using polytropic EoS and
looking for the maximum value of $\alpha$ which still fulfills the stability
criteria they found $\alpha\lesssim 2.4\cdot 10^{8}$ cm2. To the best of our
knowledge, no bounds on $\beta$ exist at the moment.

Since the dynamics of metric and Palatini theories can be very different, the
bounds discussed above may not apply in our case. Lacking a thorough analysis
along the lines of Naf:2010zy ; Pretel:2020rqx  for Palatini theories, we used
values for $\alpha$ and $\beta$ for which an appreciable change in the mass-
radius curve was obtained and that, for the case of $\alpha$, are still within
the bounds obtained in the metric formalism and in Sotiriou:2005cd .

###  4.1 $R+\alpha R^{2}$ case

We begin with the study of $f(R)=R+\alpha R^{2}$ theories and show the mass-
radius relation for a range of positive and negative values of $\alpha$. Note
that an analogous study was performed in Pannia:2016qbj  for the SLy and FPS
EoS. However, our resuls are in contradiction with those presented in
Pannia:2016qbj . While in Pannia:2016qbj  the authors found only minor
variations with respect to GR of the mass and radius of neutron stars, our
calculations show instead a significant deviation from GR in some cases. We
show in Fig. 2 our results for the mass-radius relation using the SLy and FPS
EoS, compared to the corresponding GR result. As one can see, significant
deviations from GR appear in the low-radius region even though smaller
deviations are also observed for less compact stars. Interestingly, for
negative values of $\alpha$, $f(R)=R+\alpha R^{2}$ theories generate heavier
stars than GR and thus push the mass limit for the SLy and FPS EoS.

Figure 2: Mass-radius relation for GR and $f(R)=R+\alpha R^{2}$, the SLy and
FPS EoS and different values of the parameter $\alpha$.

Figure 3: Mass-radius relation for GR and $f(R)=R+\alpha R^{2}$, the HS(DD2)
and HS(TMA) EoS and different values of the parameter $\alpha$.

The situation is less pronounced in the case of the tabulated HS(DD2) EoS,
Fig. 3. In this case, the differences in the mass-radius relation with respect
to GR are still visible but are significantly smaller than in the previous
case. What remains true is that negative values of $\alpha$ allow for a larger
maximum mass. In the case of the HS(TMA) EoS, Fig. 3, the deviations with
respect to GR are dramatic for the largest values of $\alpha$ we used. This is
certainly an unexpected behaviour that we were able to trace back to a small
difference in the HS(TMA) EoS with respect to the other EoS we used111To
confirm this hypothesis, we performed calculations using the HS(TM1) and SFHO
EoS as well, which all show the same behaviour in their derivatives as HS(DD2)
and do not show as large deviations from GR as HS(TMA)., which induces a
larger difference in the first and second derivative of the EoS. As can be
seen in the first panel of Fig. 9 at around $P=10^{32}\ \frac{g}{cm\ s}$ the
HS(TMA) EoS differs slightly from all other EoS but the difference is
amplified for the first and second derivatives as can be seen in the second
and third panel of Fig. 9. Note finally, that the range of radii obtained in
the case of tabulated EoS is limited by the maximum and minimum values of the
data provided, unlike in the case where we used analytic expressions.

In addition to the mass-radius plots, it is instructive to analyse the results
as in Figs. 4–7, were we show the difference in percentage between the $f(R)$
and the GR solutions, for the mass and the radius of a star separately. In
this way we observe that the effect of the $R^{2}$ term on the radius of stars
is generally smaller than on the mass, except for the lowest central-density
regions. The shift in the mass is thus the main responsible of the change of
the mass-radius ratios discussed above. The changes, moreover, are strongly
dependent on the EoS; SLy, FPS and HS(TMA) exhibit large deviations with
respect to GR, but the latter shows a qualitatively different behaviour as a
function of the central density. For HS(DD2), the deviations with respect to
GR are significantly smaller than for the other EoS, for both mass and radius.

Figure 4: Difference between the mass (upper panel) and the radius (lower
panel) obtained, given a central density value $\xi$, with $f(R,Q)=R+\alpha
R^{2}$ and GR, for the SLY EoS.

Figure 5: Difference between the mass (upper panel) and the radius (lower
panel) obtained, given a central density value $\xi$, with $f(R,Q)=R+\alpha
R^{2}$ and GR, for the FPS EoS.

Figure 6: Difference between the mass (upper panel) and the radius (lower
panel) obtained, given a central density value $\xi$, with $f(R,Q)=R+\alpha
R^{2}$ and GR, for the HS(DD2) EoS.

Figure 7: Difference between the mass (upper panel) and the radius (lower
panel) obtained, given a central density value $\xi$, with $f(R,Q)=R+\alpha
R^{2}$ and GR, for the HS(TDA) EoS.

It is clear from our discussion so far, that as long as the ambiguities in the
EoS of neutron stars are not resolved, it is hardly possible to draw
conclusions from mass and radius observations on the theory of gravity, as for
a fixed $f(R)$ (here, for a given $\alpha$ value) the mass-radius relation
changes dramatically for different EoS. However, it is still interesting to
speculate how relevant would astrophysical observations be, if we knew the
correct EoS of a neutron star. For example, the observation of the
2.2$M_{\odot}$ neutron star MSP J0740+6620 Cromartie:2019kug  has been used as
an argument to rule out certain EoS, including some used in this paper. This
is can be seen in Fig. 8, where we show the constraints put by MSP J0740+6620
on EoS, assuming GR is the theory of gravity and also using $f(R)=R+\alpha
R^{2}$ with $\alpha=-4\cdot 10^{9}$cm2 (for which the increase of mass is the
largest in our calculations). Clearly, some EoS can be rescued adding an
$R^{2}$ term to the theory of gravity. Reversing the argument, with increasing
certainty in the EoS of neutron stars, observations could be used to
constraint deviations of the theory of gravity from GR.

Figure 8: Maximum mass allowed by an $f(R)=R+\alpha R^{2}$ theory with
$\alpha=-4\cdot 10^{9}$cm2 for the different EoS used in this paper, compared
with the observed mass of the MSP J0740+6620 neutron star. Figure 9: The EoS
used in this paper (first panel) together with their first and second
derivatives (second and third panel respectively).  Figure 10: Difference
between the exact and numeric derivative for the first (upper panel) and
second derivative (lower panel) of the EoS.

At this point, we would like to discuss in some detail the differences in the
results induced by the different methods used to calculate derivatives of the
EoS in the case of tabulated data. To this end, we repeated the calculations
with the analytic EoS SLy, this time generating tabulated data from the
analytic expressions as follows. In one case, the derivatives were calculated
analytically as outlined in Sec. 3 and separate data tables were generated
from them; these values then had to be interpolated (in the same way as the
EoS data is interpolated). As a second method we calculated the derivatives
numerically with a finite difference approach, using second order central
differences. In Fig. 10 the difference between the derivatives using those
methods can be seen. While the difference for the first derivative is rather
small and never exceeds $4\%$, the second derivative shows very large
differences, of about $200\%$ which rises up to $400\%$, at some points. To
exemplify the impact of such differences in the mass-radius relations, we
repeated the calculation for the SLy EoS with the different methods of
calculating derivatives. The results are shown in Fig. 11. In the upper panel
we show the calculation where the tabulated data for exact derivatives is
interpolated, which clearly leads to the same results as when using the fully
analytic version. However, in the lower panel, where the derivatives have been
calculated with a finite difference method, shows only a minor difference
between $f(R)$ and GR, in fact in agreement with the results published in
Pannia:2016qbj . This discussion points to a potential problem appearing when
calculating with tabulated EoS and theories with high-order derivatives of
matter; all calculations that rely on numerical differentiation must be taken
with due caution.

Figure 11: Mass-radius relation for fake tabulated SLy EoS using the exact
values for the derivative (left panel) and the values derived by the finite
difference method (right panel).

###  4.2 $R+\alpha R^{2}+\beta Q$ case

We proceed now to discuss the results for the $f(R,Q)=R+\alpha R^{2}+\beta Q$.
As already indicated above, we consider values of $\beta$ for which an
appreciable change in the mass-radius curve is obtained. This means that, even
the small changes in the stars masses that we observe, as discussed next, may
be artificially large.

Figure 12: Mass-radius relation for $f(R,Q)=R+\alpha R^{2}+\beta Q$ for SLY
(upper panel) and FPS (lower panel) EoS.

Figure 13: Mass-radius relation for $f(R,Q)=R+\alpha R^{2}+\beta Q$ for
HS(DD2) (upper panel) and HS(TDA) (lower panel) EoS.

We show in Figs. 12 and 13 the mass-radius relation for two values of $\beta$,
with and without the $R^{2}$ term (here the $R^{2}$ term is excluded by
setting $\alpha$ to a very small value, for computational convenience). The
effect of the $Q$ term appears to be generally small, the deviation from GR
mostly coming from the $R^{2}$ term as we have seen in the previous section.
The effect of the $Q$ term seems to be more pronounced for analytic EoS than
for the tabulated ones, which may point to the problem with numerical
derivatives that we discussed above.

As in the $f(R)$ case, it is interesting to plot separately the differences
between the $f(R,Q)$ and the GR solutions, for the mass and the radius of a
star. We show them in Figs. 14–17. In contrast to the $R^{2}$ term, for which
the dependence of the mass and radius as a function of the central density
showed a very different qualitative behaviour for different EoS, in the case
of the $Q$ term these show a similar dependence on $\xi_{c}$ for all studied
EoS. In all four cases the effect is somewhere below $5\%$ on the mass, with
the exception of HS(DD2) which is up to $8\%$ for low central densities. As
for the $R^{2}$ case, it is interesting to note that the $Q$ terms influences
the mass of the star much more than its radius, again with the exception of
lowest values of the central density. In that region, the radius and mass
effects compensate in a way such that they are not visible in the mass-radius
plots.

To conclude, we wish to mention a peculiar feature of the solutions we
obtained for $f(R)$ gravities. In Fig. 18, we show a star profile (that is,
the metric function $M(r)$ as a function of the radial coordinate $r$) for the
SLy EoS and $\xi=15.1$. The metric function $M(r)$ shows an unexpected cusp
near the surface of the star. This feature, absent in GR, was already seen in
Pannia:2016qbj  and predicted in Barausse:2007pn  (see also Pani:2012qd ),
where it was argued that it invalidates Palatini $f(R)$ theories as physically
viable. In Barausse:2007pn  it was also speculated that a $Q$ term may
ameliorate this odd behaviour of the metric inside the star. As we see in Fig.
18, it is true that an $f(Q)$ term (without $R^{2}$) does not show a cusp222We
have studied changes of $\beta$ by many orders of magnitude, and the cusp was
absent in all cases.. However, in an $f(R,Q)$ model (i.e. with both $R^{2}$
and $Q$ terms) the cusp remains; that is, the $Q$ term is not able to
compensate for the unexpected behaviour generated by the $R^{2}$ term. Note
that in Barausse:2007pn  the authors argue that for nonlinear $f(R)$ one
always finds divergences at the surface of the star, which may in turn
generate curvature divergences. For our numerical results, we have carefully
investigated the appeareance of divergences in all terms of the equations of
structure (e.g. divergences in $P_{r}$ or $P_{rr}$ which may result, upon
integration, in finite values of $M(r)$) but found none. From our
calculations, thus, we can only conclude that the cusp in Fig. 18 is simply a
dynamical effect with no clear relation to the arguments in Barausse:2007pn .
Note finally that, even though in Fig. 18 we showed one particular solution
only , in all cases we have studied in this paper the qualitative behaviour of
the metric function $M(r)$ was analogous.

Figure 14: Difference between the mass (upper panel) and the radius (lower
panel) obtained, given a central density value $\xi$, with $f(R,Q)=R+\alpha
R^{2}+\beta Q$ and GR, for the SLy EoS.

Figure 15: Difference between the mass (upper panel) and the radius (lower
panel) obtained, given a central density value $\xi$, with $f(R,Q)=R+\alpha
R^{2}+\beta Q$ and GR, for the FPS EoS.

Figure 16: Difference between the mass (upper panel) and the radius (lower
panel) obtained, given a central density value $\xi$, with $f(R,Q)=R+\alpha
R^{2}+\beta Q$ and GR, for the HS(DD2) EoS.

Figure 17: Difference between the mass (upper panel) and the radius (lower
panel) obtained, given a central density value $\xi$, with $f(R,Q)=R+\alpha
R^{2}+\beta Q$ and GR, for the HS(TDA) EoS. Figure 18: Metric function $M(r)$
as a function of the radial coordinate $r$, for a neutron star with a central
density $\xi_{c}=15.1$ and the SLy EoS.

##  5 Summary

In this paper we studied neutron stars solutions in Palatini $f(R)=R+\alpha
R^{2}$ and $f(R,Q)=R+\alpha R^{2}+\beta Q$ gravities. We performed
calculations using four different EoS and different values for $\alpha$ and
$\beta$.

Our study shows that the differences in the masses and radius of stars induced
by modifications of the gravity Lagrangian can in fact be as large as the
uncertainty induced by the use of different EoS. Notably, it is the mass of
the star and not its radius that is mostly affected by modifications of GR. We
have also discussed separately the effect of the $R^{2}$ and the $Q$ terms.
The deviations from GR induced by the $R^{2}$ term are qualitatively different
for different EoS. The $Q$ term instead shows a similar behaviour for all EoS,
and in general smaller than the effect from the $R^{2}$ term.

Even though it was not a goal of our investigations, we found that some of the
EoS which have previously been ruled out by neutron star observations, might
become viable again if the $\alpha R^{2}$ correction is indeed part of the
right theory of gravity. For negative $\alpha$ the maximum mass allowed by a
given EoS tends to be increased. For some EoS, such as SLy or HS(TMA), it
turns out that an $\alpha R^{2}$ term can produce solutions as heavy as the
MSP J0740+6620, the heaviest neutron star found so far Cromartie:2019kug .

To further highlight the relevance of studies like the present one it is
interesting to mention the 2.6$M_{\odot}$ secondary component of GW190814
LIGOScientific:2020zkf . Calculations using modified gravity theories indicate
that it could be described as a neutron star (replacing MSP J0740+6620 as the
heaviest known neutron star) Indeed, as shown in Astashenok:2020qds ;
Astashenok:2021peo  for metric $f(R)=R+\alpha R^{2}$ gravities, it is possible
for certain EoSs to support neutron stars in the 2.6$M_{\odot}$ mass range.
Interestingly, the authors showed that studying rotating stars also increases
the maximum mass of neutron stars for those modified gravities. In this paper
we observe a similar effect for Palatini $f(R)=R+\alpha R^{2}$ and
$f(R,Q)=R+\alpha R^{2}+\beta Q$, obtaining neutron star solutions in the
2.6$M_{\odot}$ mass range for the HS(TMA) EoS (see Fig. 3). Alternatively,
allowing a sufficiently negative value for $\alpha$ also leads to neutron
stars in this mass range when using other EoS, like the analytic SLy EoS.

In this work we focused on the relatively simple problem of calculating the
mass, radius and profiles of spherically symmetric stars. In the future, it is
necessary to perform more detailed studies. For example, it is crucial to
study the stability of a solution in order to understand whether it is viable
as a physical solution. For example, it could be that the solutions beyond the
GR mass limit are unstable and hence not realised in Nature. This is an even
more pressing issue, considering the odd behaviour of the metric function
$M(r)$ for $f(R)$ theories, as discussed in last section. Also, realistic
neutron stars rotate and the system of equations solved herein provides only
an estimate of their properties. Finally, on the technical side, it is
necessary to establish reliable and unambiguous methods to calculate high-
order derivatives of EoS, as we have seen.

Clearly, no conclusive results can be obtained yet, at least until the ongoing
efforts to nail down the EoS for a neutron star from first principles are
successful. However, our study indicates that, in the future, astrophysical
observations may serve as discriminators of modified gravity.

###### Acknowledgements.

We thank G. Olmo for a critical reading of the manuscript.

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# Eliciting Knowledge from Language Models for Event Extraction

Jiaju Lin,1 Jin Jian, 2 Qin Chen 3

###### Abstract

Eliciting knowledge contained in language models via prompt-based learning has
shown great potential in many natural language processing tasks, such as text
classification and generation. Whereas, the applications for more complex
tasks such as event extraction are less studied, since the design of prompt is
not straightforward due to the complicated types and arguments. In this paper,
we explore to elicit the knowledge from pre-trained language models for event
trigger detection and argument extraction. Specifically, we present various
joint trigger/argument prompt methods, which can elicit more complementary
knowledge by modeling the interactions between different triggers or
arguments. The experimental results on the benchmark dataset, namely ACE2005,
show the great advantages of our proposed approach. In particular, our
approach is superior to the recent advanced methods in the few-shot scenario
where only a few samples are used for training.

## Introduction

Event extraction aims to transform text into structural records. For example,
in Figure 1, the raw sentence would be transformed into the structural
informative records shown in the table on the right side. The table-like
records include event types, trigger words and corresponding arguments.

Mainstream methods can be roughly categorized into two groups: (1) The
sequence tagging based approaches that treat the event extraction task as
trigger/argument identification and classification.(Lin et al. 2020)(Wadden et
al. 2019) (2) The generation based approaches that generate trigger and
argument tokens given the event descriptions or event schema(Lu et al.
2021)(Li, Ji, and Han 2021). Recent studies focus on enhancing the performance
with the pretrained language models (PLMs)(Du and Cardie
2020)(EventExtractionasMulti-turnQuestionAnswering)(liu-etal-2020-event). In
particular, QA based MRC models are proposed for event extraction, which
design the related questions manually and provide the answer as the
trigger/argument.

Despite these inspiring progress, current methods still struggle due to the
following critical issues. (1) The knowledge contained in the pretrained
language models is not fully exploited. (Lu et al. 2021) modifies event
extraction as a generation task, and introduces a sequence-to-structure model
to extract trigger and arguments jointly. But the event schema they used lacks
informative notion, making the potential of PLMs can not be completely
unleashed. (2) Mechanisms to capture the interactions among arguments are not
elegant and optimal. (Xiangyu et al. 2021) attempted to model the relations
among both inter and intra event arguments via an RNN model. Whereas, the
performance was heavily dependent on named entity recognition, which was prone
to error propagation. (Wei et al. 2021) introduced a teacher-student mechanism
to distill the knowledge from various teacher models. Specifically, the
teacher models were trained to predict argument given relevant roles of a
specific event, while the student model was trained for learn the knowledge
from teachers for argument prediction. To impart all-around knowledge for
argument extraction, many well-trained teacher models are needed, which poses
big challenge for fine-tuning.

Figure 1: An example of event extraction from ACE205 corpus. There are three
events happening in this sentence. While only one trigger word, ‘killed’,
links to two arguments directly, other triggers’ arguments need to be inferred
via the information provided by ‘killed’ event.

To address the above issues, we propose a PrOmpt-based Knowledge Eliciting
approach (PoKE), which elicits specific knowledge from PLMs for event
extraction. Specifically, we first design trigger/argument prompts based on
the annotation guidelines, and then ask the model to fill the blanks with
trigger words or argument texts. In this way, the event extraction is
transformed into a conditional generation task, where the knowledge obtained
during pretraining are elicited to generate the answer. In order to better
capture the relations between different events and arguments, we present
various joint prompt strategies to gauge the complementary knowledge about
triggers and arguments that occur in the same context.

Our approach is superior to the existing methods in three folds. (1) We mingle
description understanding and interaction modeling task together. Not only can
we elicit knowledge embedded in PLM, but also making full use of the PLM’s
understanding ability. For instance, it is challenging for most existing
models to extract all information correctly from the sentence in Figure1,
where the arguments of three events overlap each other and some trigger-
argument linkages are not direct. However, our model is able to infers the
‘place’ and ‘target’ argument of ‘suicide bombing’ with the help of trigger
‘killed’, capturing the interactions between triggers efficiently. (2)
Different from current generation models that generate arguments according to
a pre-defined schema, we generate arguments separately. Thus our results would
not suffer from the sub-optimal order of generation. Moreover, our model can
take related arguments into consideration with the joint prompt. (3) Instead
of introducing extra modules, we simply adjust the format of input data and
realize the purpose of capturing interactions among triggers and arguments.
Such brief mechanism decreases the amount of model parameters and alleviates
the difficulty of fine tuning.

We evaluate our approach on the benchmark event extraction dataset, namely
ACE2005. Experimental results show that our approach outperforms the state-of-
the-art methods for argument extraction. Besides we accomplish competitive
performance with our event extraction pipeline. It is also notable that our
approach is capable of better learning in the low resource set, and
outperforms the recent advanced methods under the few-shot scenario.

Our contributions can be summarized as follows:

  * •

As far as we know, it is the first attempt to explore the effectiveness of
eliciting knowledge from PLMs for event extraction, where the prompt design is
not straightforward due to the complicated types and arguments.

  * •

We develop various joint prompt strategies, which can capture the interactions
among different triggers, intra and inter-event arguments, without any
redundant modules.

  * •

Extensive experiments on the most widely used ACE-2005 event extraction
dataset demonstrate the superiority of our approach under both fully
supervised and few-shot learning scenarios.

Figure 2: Architecture of PoKE for coarse-grained event detection. It is the
first part of event detection, in this step, we extract triggers for eight
main types. For this purpose, we introduce external joint trigger prompt and
internal joint trigger prompt. We develop the former by setting masks in the
prompts, to teach model how to infer trigger via the combination of passage
and prompts. To generate the latter, we mask the triggers in the passage,
asking the model to fill the mask via the information in the prompts. During
this process, our model would model the interactions among triggers.

## The Proposed Approach

In this section, we first present an overview of our framework. Then, we
introduce various prompt methods to elicit knowledge from PLMs for trigger
identification and argument extraction. After that we describe the details of
prompt-based event extraction.

### Framework Overview

Our PoKE framework depends on the natural language understanding ability of
T5. To be specific, the different prompts we design would make our model
understand the purpose of our task, i.e. comprehend linkage between answer and
given context, rather than remember the distribution of answers. For event
detection task, we split it into two parts. First, we generate the trigger
words corresponding to 8 main types in each sentence,i.e., Movement,
Personnel, Conflict, Contact, Life, Transaction, Justice, Business. With that
result, we further classify triggers into 33 subtypes, also model it as a
generation task. For event argument extraction task, our model would fill the
bank in each prompt with argument span from the sentence, with trigger words
and event type provided.

#### Event Detection

Owing to the diversity of event subtypes, we first extract triggers according
to 8 main types, and then classify them into subtypes. For extracting triggers
for main type, we develop two joint trigger prompts to elicit more related
knowledge by modeling the event interactions.

External Joint Trigger Prompt. Given a sentence, we add 8 prompts of all main
types after it, asking the model to generate the trigger words jointly. In
this manner, model can detect the discrepancy between various event types.
Intuitively, we use the event type as an inevitable part of the prompt.
Besides, considering the fact that contexts may render cues for the event, we
expand the boundary of sentence. With a window of size=100 words, our sentence
is set at the middle position and includes contexts in the window into the
final passage. To illuminate which sentence triggers should be extracted from,
two special tokens ‘$\langle e\rangle$’ and ‘$\langle/e\rangle$’ are applied
to mark the original boundary of the raw sentence. Furthermore, to make the
paragraph more natural sounding, we insert a bridge section, ‘In the passage
above’, between the passage and prompts. A typical trigger external joint
trigger prompt is shown in Figure 2. If an event does not occur in the
sentence, we use ‘None’ to represent the NA answer.

Internal Joint Trigger Prompt. Given the fact that relevant events occur close
to each other in a paragraph of text, it is sensible to teach our model how to
capture the interaction between triggers. Due to that reason, we present
internal joint trigger prompt for trigger extraction. Events within one
sentence has certain relation with each other. Besides, we assume that the
events appearing within a span of three sentences have potential
interdependence. Thus we filter ineligible samples and concatenate adjoin
eligible sentences as raw sentences. Similar to trigger prompt, we employ a
window with size = 120 words to incorporate the contextual information. Unlike
external joint trigger prompt that the masks are outside of the raw sentence,
the internal joint trigger prompt masks the trigger words inside the raw
sentence as shown in Figure 2

With the identified triggers for the main event type, we need to further
classify them into 33 subtypes. We inherit the trigger prompts and make
classifications among the subtypes. As shown in Figure 2, with the identified
triggers ‘convicted’ and ‘sentenced’ for the main event type ‘Justice’, we
perform fine-grained classifications to classify them into the subtypes of
events, namely ‘Convivt’ and ‘Sentence’.

#### Argument Extraction

After event detection, we need to extract the event related arguments for
better understanding. For example, given the description of ‘convicted’ event,
it is necessary to identify different argument roles as ‘defendant’ and
‘adjudicator’ to make further decisions. In this section, we present a single
and joint argument based prompts for argument extraction.

Single Argument Prompt. For each role in an event, to incorporate more
semantic information into our templates, we first utilize the descriptions in
the ACE annotation guidelines for events. Particularly, we make a masked
position in the argument description text and urge the model to generate the
corresponding words. In the case where multiple entities in a passage
represent the same argument, we insert ‘$|$’ in the gold answer to split them.
In addition, it is necessary for the model to have the knowledge of the
trigger and its type. To provide the trigger and event information, we stick
the event type and the corresponding trigger at end of the sentence. Both the
trigger word in the sentence and template is surrounded by a pair of special
tokens as $\rm\langle t\rangle$ $\rm\langle/t\rangle$. An example of the
single argument prompt is presented in Figure 3.

Joint Argument Prompt. When trying to understand an event from a passage, it
is natural for a human to inference the arguments from other co-occurred
arguments. To stimulate such learning process, we present a joint argument
prompt method to elicit more related knowledge by modeling the interactions
among the intra-event arguments and inter-event arguments.

For each sentence that contains an event, we provide the corresponding
arguments information behind it, while mask the arguments words in the
sentence. The ACE annotation guidelines are applied to generate the arguments
description information. Also, the trigger word and event type are added after
the argument descriptions. Besides, to prevent our model from cheating by
heuristic relations, like filling the blanks according to the order of
arguments mentioned in prompts, we shuffle the list of prompts randomly.
Figure 3 shows an example of our joint argument prompt. With this prompt, more
complementary knowledge can be elicited from the pre-trained language models,
building solid relations between the descriptions in prompts and the concrete
argument role for more accurate argument extraction.

Figure 3: Architecture of PoKE for argument extraction. In the training stage,
we design two varieties of prompts: Joint Argument Prompt and Single Argument
Prompt. To generate the former, we mask the arguments in the sentence, asking
the model to fill the mask via the information in the prompts. During this
process, our model would model the interactions among arguments and triggers.
Meanwhile, we develop Single Argument Prompt, whose masks are in the prompts,
to teach model how to infer arguments in the inference stage.

### Prompt-based Event Extraction

Since our base model architecture is an encoder-decoder design. We employed T5
for the generation task. The generation process follows T5 pre-training task,
a slew of extra tokens, from ‘$\langle extra\\_id\\_0\rangle$’ to ‘$\langle
extra\\_id\\_99\rangle$’ are exploited as masks. We replace the specific word
spans with these special tokens. And the generation output is the answers
joined by these extra tokens, like $\langle extra\\_id\\_0\rangle$ Argument_0
$\langle extra\\_id\\_1\rangle$ Argument_1 $\langle
extra\\_id\\_2\rangle\langle/s\rangle$, here $\langle/s\rangle$ means the end
of the sentence.

The generation process models the conditional probability of selecting a new
token given the previous tokens and the input to the encoder.

$p(x|prompt)=\prod\limits_{i=1}^{|x|}p(x_{i}|x_{i},prompt)$ (1)

The model is trained by minimizing the cross entropy loss over vocabulary $V$.

$-\sum_{i=1}^{V}y_{i}\log(p_{i}(x|prompt))$ (2)

In inference stage, we use greedy decoding, i.e. choosing the highest-
probability logit at every times step. Given the fact that the vocabulary
space is huge and it is possible for the model to generate words out of the
passage. We constrain the candidate vocabulary to $V_{s}$ the set of tokens in
the input ,by setting logits of extraneous words to -inf.

$p=\left\\{\begin{array}[]{cc}p(x|prompt)&w\in V_{s}\\\ -inf&w\notin
V_{s}\end{array}\right.$ (3)

| Trigger-C | Arg-C |  |
---|---|---|---|---
Model | P | R | F1 | P | R | F1 | PLM | Annotation
OneIE | - | - | 74.7 | - | - | 56.8 | BERT-large | Token+Entity
Text2Event | - | - | 69.2 | 46.7 | 53.4 | 49.8 | T5-base | TextRecord
TANL | - | - | 68.5 | - | - | 48.5 | T5-base | Token
EEQA | 71.1 | 73.7 | 72.3 | 56.7 | 50.2 | 53.3 | 2*BERT | Token
CondiGen | - | - | 71.1 | - | - | 53.7 | BART-large | Token
PoKE | 64.1 | 76.2 | 69.6 | 47.7 | 58.1 | 52.4 | T5-base | Token
Table 1: Experiment results of event extraction on ACE2005. Trig-C indicates trigger identification and classification. Arg-C indicates argument identification and classification. The column PLM indicates the pre-trained language models used by each method.  Model | P | R | F1 | Encoder/PLM
---|---|---|---|---
HMEAE | 66.04 | 68.58 | 67.28 | CNN
CondiGen | 69.66 | 67.63 | 68.83 | BART-large
TANL | 65.19 | 64.21 | 64.69 | T5-base
EEQA | 67.88 | 63.02 | 65.36 | 2*BERT
PoKE single only | 60.79 | 64.92 | 62.79 |
PoKE w.o. inter | 61.52 | 67.13 | 64.20 | T5-base
PoKE | 66.20 | 74.48 | 70.10 |
Table 2: Results of event argument extraction. PoKE single only means we train
the model only with single argument prompts. PoKE w.o. inter indicates we
replace joint argument prompts with the single prompts but mask in the
passage, which means no interaction information is impart to the model.

## Experiments

### Experimental Setup

Data set. We evaluate our work on the most widely used ACE2005 dataset,
containing 599 documents annotated with 33 event subtypes and 35 argument
types. For fair competition, we follow the same split and preprocessing step
as the previous work(Yang et al. 2018) (Wadden et al. 2019) (Du and Cardie
2020). The data set splits are as follows, 40 newswire documents as test set,
30 randomly selected documents for development and a training set including
the rest 529 documents.
Baselines. For complete event extraction task we compare our method with five
baselines. Considering the difference in the annotation methods and label-
intensity, we categorize them into three types following (Lu et al. 2021)
1.Baseline using token annotation and entity annotation: OneIE (Lin et al.
2020), an end-to-end system, employs global features to conduct named entity
recognition ,relation extraction and event extraction at the same time.

2.Baselines using token annotation: CondiGen (Li, Ji, and Han 2021) is one of
the cutting-edge method for argument extraction task. It utilizes the
conditionally generation ability of BART. Given description of arguments and
their relations then generate the corresponding words in the sentence. The
candidate arguments are predicted jointly.However, it gives up generation
model and introduces a modified TapNet for event detection, formulating it as
classic sequence tagging task. EEQA which is one of the earliest work that
introduces a new paradigm for event extraction by formulating it as a question
answering task. It generates questions from annotation guidelines and raise
question for each argument under one event type respectively(Du and Cardie
2020). TANL, a sequence generation-based method, frame event extraction as a
translation task between augmented natural languages. Multi-task TANL extends
TANL by transferring structure knowledge from other tasks.(Paolini et al.
2021)

3\. Baseline using Text-Record Annotation: TexT2Event, an end-to-end
generation-based, modeling event-detection and argument extraction tasks in a
single model. Compared with normal generation methods, it outputs a structural
event record with event type, trigger, and arguments(Lu et al. 2021).

Implementation. We optimized our model using AdamW (Kalchbrenner and Blunsom
2013) with different hyperparameters for each task. For coarse-grained event
detection, we set learning rate=1e-4 training epoch=6. For fine-grain trigger
classification, learning rate=5e-4 batch size=8 training epoch=3. For argument
extraction task, learning rate=1e-3 and batch size=64. The experiments are
conducted on a single NVIDIA GeForce RTX 3090. The random seed set for all
tasks is 42.

### Supervised Learning Results

#### Event Extraction Results

The performance of PoKE and all baselines on ACE05 are presented in Table 1,
where ‘Trig-C’ indicates trigger identification and classification and ‘Arg-C’
reflects the result of argument identification and classification. We can
observe that by leveraging interactions among triggers and arguments and
taking advantage of prompts, our model achieve comparable or better
performance with the baseline models. Compared with the other two global-
generation model, TANL and Text2Event, our method overtake them both in event
detection and argument extraction task. Notice that the results of argument
extraction is directly affected by the output of event detection. An error
prediction of event type would render wrong candidate classes for argument
extractor. Hence the reduction in F1 after argument extraction indicates our
merits in argument extraction.

#### Argument Extraction Results

Since the current comparisons are based on the trigger predictions of previous
event detection work(Wang et al. 2019a). It maybe reasonable when no advance
can be achieved for event detection task. However, with the rapid evolution in
event detection methods(Pouran Ben Veyseh et al. 2021), we believe such
previous criterion can not show the whole picture of each architecture’s
ability. Hence, following the hyper parameter settings mentioned in their
paper, we reproduce the results of several cutting-edge models with gold
triggers as input to do a fair comparison. Meanwhile, we introduce another
baseline developed for event argument extraction specifically. HMEAE which
applies a concept hierarchy indicating argument roles. With the help of
hierarchical modular attention, correlation between argument roles sharing the
same high-level unit can be utilized.(Wang et al. 2019b). It should be noticed
that HMEAE takes use of the named entity information as candidate arguments.
Also, we do some ablation study to examine the efficiency of our model. ‘PoKE
single only’ means we train the model only with single argument prompts. ‘PoKE
w.o. inter’ indicates we replace joint argument prompts with the single
prompts but mask in the passage, which means no interaction information is
impart to the model.
The results are shown in Table 2, from which we can observe that. (1) Our
method surpasses all baseline methods, with an absolute improvement of at
least 1.3 in F1. (2) Models which consider the interactions between arguments
perform better. For instance, CondiGen is trained to generate all arguments of
a single event jointly. Although the manual templates used for its inference
are sub-optimal, it overtakes EEQA with 3.1 in F1. Taking advantages of both
joint prediction and informative description, our model achieves the sate-of-
the-art performance. (3) Named entity annotation is not an absolute necessity.
Although without the assistance of named entity recognition results, models
utilize PLMs achieve competitive performances. Particularly, input of HMEAE
are sentences with named entity annotation, however, its encoder, a vanilla
CNN, restricts its ability. (4) Joint argument prompts enhance model’s
performance dramatically, with almost 8 in F1 , from 62.79 to 70.10 .

Figure 4: Few-shot experiment results. K-shot corresponds to the number of
data in training and validation set.

### Few Shot Scenario

Imitating few-shot settings in LM-BFF(Gao, Fisch, and Chen 2021), we conduct
8,16,32 shot experiments to examine the few-shot ability of our model.
Specifically, for each event type, we sample k instances from initial training
and validation set. After training, we test the model performance on standard
scale test set. The experiment results are in TableLABEL:tab:fewshot. It can
be observed that our model perform better than baselines in few-shot scenario.
Although EEQA and CondiGen both have outstanding zero-shot ability, when it
comes to few-shot task, they are exceeded by our PoKE. Even a 32-shot trained
EEQA model can not overtake PoKE trained by 8-shot data. Our mechanism
leverages naturalness into prompts which ensure its few-shot performance.

Method | P | R | F1
---|---|---|---
EDA | 54.46 | 77.43 | 63.94
BackTranslation | 54.03 | 76.95 | 63.48
PoKE single only | 60.79 | 64.92 | 62.79
PoKE | 66.20 | 74.48 | 70.10
Table 3: Compare PoKE with Data Augment Methods

### Compared with Data Augment Methods

PoKE increases the number and variety of training samples, which would enhance
model’s ability naturally, while data augment methods can improve model’s
performance in the same way. Hence we compare our method with two most common
seen data augment methods, back translation and EDA(Wei and Zou 2019),which
consists of four word-level operations, replacement, insertion, swap, and
deletion, to boost diversity of sentences. Experiments are conducted on
argument extraction task. We discard the interaction learning data for two
data augment methods and expand the argument inference data set to the same
number of training data used for PoKE. The results are demonstrated in Table
3. From the result, we can see that compared with using inference data only,
data augment methods do improve model’s performance. Nonetheless, the
improvement is limited. It further demonstrates that our mechanism unleash
PLM’s understanding ability.

Figure 5: Case study of PoKE and EEQA

## Case Study and Error Analysis

### Case Study

To demonstrate how our model utilize information from co-occuring events, we
show extraction results of samples with multi events in Figure5. Sentence S1
contains two trigger words: ‘destroyed’ and ‘killed’, indicating ‘Die’ and
’Attack’ event respectively. It is a common fact that the agent responsible
for a death is usually the attacking agent in an attack. Also, word
‘commandos’ is the subject of the sentence, strongly syntactically related to
both two triggers. Given that, our model is capable of exploiting the clues
hidden behind and makes the correct prediction of ‘Die_Agent’ and
‘Attack_Attacker’. Besides, although there is a false prediction of ‘Victim’,
i.e., our model identifies ‘soldiers’ as victims while no ‘Victim’ is
annotated. We believe it is the gold answer that contains a mistake. In many
cases, our model makes more sensible argument extraction that human annotated
data.

### Error Analysis

To facilitate further research, we investigate the error predictions of our
model. We randomly sampled 50 wrong answers and attributed them to 5 types.
Here we list 3 most common problems caused by our model design.

Shallow Heuristics obtained in training set(24%) Data in the training set
always contains some bias that induce heuristic relations between entity
mentions and corresponding argument types. Model learns these heuristic
relations to enhance its performance, while neglect the intended task. We
define the event argument extraction task as utilizing the sentence $s$,
trigger $t$, event type $e$ to infer corresponding arguments $a$,
$\\{s,t,e\\}\rightarrow\\{a\\}$. However, we observed that, in some sense,
extraction of arguments relies on the entity mention in the sentence and event
type,$\\{s,e\\}\rightarrow\\{a\\}$, omitting the relation between entity and
trigger word. For example, in the training set, when an event about ‘Justice’
and entity ‘court’ appear in the same sentence, ‘court’ always plays a role of
‘Adjudicator’. Thus, when the same situation occurs in the inference period,
our model identifies ‘court’ in the sentence as ‘Adjudicator’, no matter
whether it is linked to the trigger word.

One of the error predictiond is as follows:
The Belgrade district $\rm[court]_{Adjudicator}$ said that Markovic will be
tried along with 10 other Milosevic-era officials who face similar $\rm\langle
t0\rangle charges\langle/t0\rangle$ of inappropriate use of state property
that carry a sentence of up to five years in jail.

In this instance, ‘court’ has no direct relation to the trigger ‘charge’.Thus
the gold annotation of ‘Adjudicator’ is ‘None’. However, due to the wrong
experience accumulated in training stage, our model classifies it as a
adjudicator because the sentence contains a Justice-Sentence event.

Ambiguous expression (20%) Owing to the ambiguity of natural language, even
human reader would make some mistakes, especially under the situation where no
context are given. Ambiguous expression error refers to the case when our
model makes a prediction inconsistent with gold answer, but reasonable from
some perspectives. These mistakes usually occur when inferring ‘Place’
arguments or arguments expressed implicitly. For example, in the sentence
‘Kelly , the US assistant secretary for East Asia and Pacific Affairs ,
arrived in $\rm[Seoul]_{Place}$ from Beijing Friday to $\rm\langle t0\rangle
brief\langle/t0\rangle$ Yoon , the foreign minister.’ Seoul is predicted as
the ‘Place’ of ‘Contact.Meet’ event, but no ‘Place’ argument is annotated. It
is true that the place of meeting is not mentioned clearly, but for humans,
applying rhetoric to imply implicit arguments is not rare sense.
We also find the gold sample expressing place argument implicitly: $\rm\langle
t0\rangle Former\langle/t0\rangle\ [Illinois]_{Place}$ Senator Carol Moseley -
Braun recently decided to run for president . Here ‘Illinois’ is annotated as
the place where employment relationship ends.

Coreference & head of entity mistake(10%) Errors ascribed to this kind refers
to the situation when the model predicts a coreferent of gold annotation or
the whole entity span rather than the head of it. For example, “
$\rm[Erdogan]_{Person}$ , a leader of Turkey ’s pro - Islamic movement when he
was $\rm\langle t0\rangle jailed\langle/t0\rangle$ , said he moderated his
policies in prison .” ‘Erdogan’ is predicted as ’Person’ argument of
’Justice.Arrest-Jail’ event, while the gold annotation is ’leader’. For head
of entity mistake, the model fails to figure out the head word of entity span
correctly. Here is a concrete example: “ Anwar will be taken to the
$\rm\langle t0\rangle appeal\langle/t0\rangle$ court early Friday for a bail
application pending his appeal to the country ’s highest
$[FederalCourt]_{Adjudicator}$ against his sodomy conviction , counsel Sankara
Nair said .” The gold answer of ’Adjudicator’ is ’Court’ but ‘Federal Court’
is predicted by our model.

## Related Work

### Event extraction

Most efforts in event extraction field are concentrated on the ACE2005 corpus.
Various of neural models have been exploited for better performance. (Nguyen
and Grishman 2015) first introduced Convolutional Meural Metwork, (CNN) for
event detection for better representation of features. And (Nguyen, Cho, and
Grishman 2016) employed recurrent neural network to build robust
model.Besides, Graph Convulutional Neural Networks (Liu, Luo, and Huang 2018)
and plenty of fancy ideas have been applies to handle difficulties we meet(Lu
et al. 2021). Recently, there is a trend of modeling event extraction into QA-
based tasks (Du and Cardie 2020)(liu-etal-2020-event). These researches employ
reading comprehension abilities of natural language models. Despite these
advantages, current methods still suffers from the bottlenecks mentioned in
Introduction.

### Prompt-based learning

Prompt-based learning is based on language models that model the probability
of text directly. Prompt is a piece of string that contains descriptive
information about the answer of the task. The language model is used to
probabilistically fill the unfilled slot to obtain a final string as the
prediction of the task. Exploration in prompt methods can be categorized into
prompt engineering which refers to creating a prompting function $f_{p}(x)$
that results in efficient performance on the downstream task(Shin et al.
2020)(Jiang et al. 2020), answer engineering which targets at looking for an
optimal answer space and a map to the original output (Shin et al. 2020) and
multi-prompt learning which involves prompt ensemble(Yuan, Neubig, and Liu
2021), prompt augmentation(Gao, Fisch, and Chen 2021), prompt composition(Han
et al. 2021) and prompt decomposition(Cui et al. 2021). In this work, we
utilize manual prompts and take the idea of prompt ensemble, but only modify
the format of prompts in training stage to increase the PLM’s understanding of
the task.

## Conclusion and Future Work

We propose a prompt-based approach (PoKE), which includes various
trigger/argument prompts to elicit knowledge from language models for event
extraction. In particular, the prompts model the interactions between
different events and arguments, which gauges more complementary knowledge and
yields better performance. The experimental results demonstrate the
superiority of our approach in achieving comparable performance without
additional model parameters or modules. It is also notable that we can still
perform well under that few-shot scenario. In the future, we will introduce
automatic prompt methods for this task and explore the effectiveness of prompt
for joint event extraction.

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