# New models and Big Bang Nucleosynthesis constraints in $f(Q)$ gravity

Fotios K. Anagnostopoulos<EMAIL_ADDRESS>Department of
Informatics and Telecommunications, University of Peloponnese, Karaiskaki 70,
22100, Tripoli, Greece  Viktor Gakis<EMAIL_ADDRESS>Department of
Physics, National Technical University of Athens, Zografou Campus GR 157 73,
Athens, Greece  Emmanuel N. Saridakis<EMAIL_ADDRESS>National
Observatory of Athens, Lofos Nymfon, 11852 Athens, Greece  Department of
Astronomy, School of Physical Sciences,
University of Science and Technology of China, Hefei, Anhui 230026, China  CAS
Key Laboratory for Research in Galaxies and Cosmology,
University of Science and Technology of China, Hefei, Anhui 230026, China
Spyros Basilakos<EMAIL_ADDRESS>National Observatory of Athens,
Lofos Nymfon, 11852 Athens, Greece  Academy of Athens, Research Center for
Astronomy and Applied Mathematics, Soranou Efesiou 4, 11527, Athens, Greece
School of Sciences, European University Cyprus, Diogenes Street, Engomi, 1516
Nicosia, Cyprus

###### Abstract

The $f(Q)$ theories of modified gravity arise from the consideration of non-
metricity as the basic geometric quantity, and have been proven to be very
efficient in describing the late-time Universe. We use the Big Bang
Nucleosynthesis (BBN) formalism and observations in order to extract
constraints on various classes of $f(Q)$ models. In particular, we calculate
the deviations that $f(Q)$ terms bring on the freeze-out temperature $T_{f}$
in comparison to that of the standard $\Lambda$CDM evolution, and then we
impose the observational bound on $\left|\frac{\delta{T}_{f}}{{T}_{f}}\right|$
to extract constraints on the involved parameters of the considered models.
Concerning the polynomial model, we show that the exponent parameter should be
negative, while for the power-exponential model and the new hyperbolic tangent
- power model we find that they pass the BBN constraints trivially. Finally,
we examine two DGP-like $f(Q)$ models, and we extract the bounds on their
model parameters. Since many gravitational modifications, although able to
describe the late-time evolution of the Universe, produce too-much
modification at early times and thus fall to pass the BBN confrontation, the
fact that $f(Q)$ gravity can safely pass the BBN constraints is an important
advantage of this modified gravity class.

##  I Introduction

The accelerated expansion of the late Universe is commonly attributed to the
existence of an extra cosmic constituent known collectively as “Dark Energy”.
In the context of the concordance $\Lambda$CDM scenario, dark energy is just
the standard cosmological constant and this scenario is very successful in
interpreting observations such as Cosmic Microwave Background (CMB) [1],
Supernovae Ia (SNIa) [2], Large Scale Structure (LSS) [3], etc.

However, recently various observational issues have gain attention, for
example the Hubble constant tension, that is the discrepancy between the value
of the Hubble constant from the CMB and the corresponding one from Cepheids up
to 5$\sigma$ (see [4] for a thorough and recent review). Another example is
the difference of the root-mean-square amplitude of matter over-density
perturbation $\sigma_{8}$, between the value extracted from the CMB (imposing
$\Lambda$CDM) and the corresponding value extracted by fitting on the Large
Scale power spectrum [5]. Furthermore, the value of the cosmological constant
in the late Universe seems to be inconsistent with the corresponding value at
the early Universe [6].

On the other hand, the current gravitational theory, namely General Relativity
(GR), cannot be re-normalized and thus it is difficult to be consistent with a
quantum description [7]. Hence, a large portion of the literature follows the
direction of modified gravity, which seems promising to solve the latter
problem [8], and in addition it is very efficient in describing the two phases
of accelerated expansion of the universe [9, 10]. Finally, note that modified
gravity can be efficient in alleviating the two aforementioned tensions of
$\Lambda$CDM cosmology, namely the $H_{0}$ and the $\sigma_{8}$ ones [11].

Along these lines and due to the fact that the Einstein-Hilbert Lagrangian is
just one of many possible Lagrangians that could lead to the same field
equations, the community has developed many frameworks of modified gravity. A
class of gravitational modification arises from the extension of the Einstein-
Hilbert Lagrangian with extra terms, however maintaining its geometrical
structure, i.e. its Riemannian formulation, resulting for instance in $f(R)$
gravity [12], in Gauss-Bonnet and $f(G)$ gravity [13, 14], in Weyl gravity
[15], etc. A more radical approach to follow is to modify the building blocks
of General Relativity, that is the underlying geometrical structure. An
instance of this kind of modification arises from the torsional formulation of
gravity, namely the modified teleparallel theories, such as the $f(T)$ gravity
[16, 17], the $f(T,T_{G})$ gravity [18], the scalar-torsion theories [19, 20]
etc. In contrast to the standard GR and the corresponding $f(R)$ extensions,
which are based on curvature, in modified teleparallel formulation gravity is
manifested via the torsion tensor, with curvature being zero.

There is yet another possibility, which is formulated using the non-metricity
$Q$ of the connection. In this setup, only the non-metricity is non-zero while
curvature and torsion both vanish. Hence, non-metricity can be used in order
to describe gravity through geometry, leading to the so-called Symmetric
Teleparallel Equivalent to General Relativity (STEGR) [21]. It is interesting
to note that although Teleparallel Equivalent of General Relativity (TEGR) as
well as STEGR, are completely equivalent to General Relativity at the level of
equations, when we modify them the resulting theories $f(R)$, $f(T)$ and
$f(Q)$ are different theories. These types of modifications have in general
more degrees of freedom than the two of standard GR, unless specific
constraints are imposed in the functional forms. Thus, during the last years
$f(Q)$ gravity, its application to cosmology and generalizations has attracted
the interest of the literature [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32,
28, 29, 30, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42].

A very strong constraint on modified gravity models in the early Universe
comes from the era of Big Bang Nucleosynthesis (BBN) [43, 44, 45, 46, 47, 48].
In the present work we are interested in imposing BBN constraints on $f(Q)$
gravity, within an analytical approach, following [49, 50, 51, 52, 53].
Specifically, we use observational data on primordial abundances of
$\ce{{}^{4}He}$ to constrain all known $f(Q)$ models along some that are
proposed for the first time inspired by the successful $f(T)$ models. The plan
of the manuscript is the following. In Section II we provide a formal
description of Symmetric Teleparallel gravity and its $f(Q)$ extension, while
in Section III we present the cosmological solutions of the theory alongside
the specific $f(Q)$ models that are going to be studied. In Section IV we
briefly review the formalism of BBN constraints, and then in Section V we
apply it in the framework of $f(Q)$ gravity. Finally, in Section VI we derive
our conclusions and we point out further directions.

##  II Symmetric Teleparallel Gravity

General Relativity is built upon Lorentzian Geometry which is specified by
choosing a connection that is symmetric and metric compatible. The latter is
the Levi-Civita connection and due to its properties it only produces non-zero
curvature, whilst torsion and non-metricity are zero [54]. Nevertheless, a
different kind of connection can be utilized when using geometrodynamics as
the mathematical framework for gravity. As a matter of fact the most general
connection is called metric-affine it is described by the formula

$\displaystyle\Gamma^{\alpha}{}_{\mu\nu}$
$\displaystyle=\mathring{\Gamma}^{\alpha}{}_{\mu\nu}+K^{\alpha}{}_{\mu\nu}+L^{\alpha}{}_{\mu\nu}\,,$
(1)

where $\mathring{\Gamma}^{\alpha}{}_{\mu\nu}$ represents the Christoffel
symbols of the Levi-Civita connection, $K^{\alpha}{}_{\mu\nu}$ is the
contorsion tensor and $L^{\alpha}{}_{\mu\nu}$ is the disformation tensor
related to the non-metricity [55]

$\displaystyle Q_{\alpha\mu\nu}$ $\displaystyle:=\nabla_{\alpha}g_{\mu\nu}\,.$
(2)

The disformation tensor can then be expanded as [21]

$\displaystyle L^{\alpha}{}_{\mu\nu}$
$\displaystyle:=\frac{1}{2}g^{\alpha\beta}\left(-Q_{\mu\beta\nu}-Q_{\nu\beta\mu}+Q_{\beta\mu\nu}\right)\,,$
(3)

and encapsulates all the information coming from the non-metricity tensor of
the general affine connection. In summary, these three types of geometric
deformation $\mathring{\Gamma}^{\alpha}{}_{\mu\nu}$, $K^{\alpha}{}_{\mu\nu}$
and $L^{\alpha}{}_{\mu\nu}$ form a kind of “trinity of gravity”, which
encompasses all components of the general connection
$\Gamma^{\alpha}{}_{\mu\nu}$ [56]. This practically implies that it is
possible to express the geometry of a gravitational theory in either
curvature, non-metricity or torsion. A connection which admits only non-
metricity, whilst curvature and torsion are zero, is called Symmetric
Teleparallel Gravity (STG) connection, while connections which have only zero
curvature are called teleparallel connections [57, 58].

Similarly to TEGR, which is based on a connection which has only torsion [59],
the STG connection has only non-metricity and hence STEGR is equivalent to GR
at the level of field equations, since its action differs by a boundary term
from the Einstein-Hilbert. In particular, defining the non-metricity scalar as
[21]

$Q:=-\frac{1}{4}Q_{\alpha\mu\nu}Q^{\alpha\mu\nu}+\frac{1}{2}Q_{\alpha\mu\nu}Q^{\mu\alpha\nu}+\frac{1}{4}Q_{\alpha}Q^{\alpha}-\frac{1}{2}\bar{Q}_{\alpha}Q^{\alpha}\,,$
(4)

with

$\displaystyle Q_{\mu}$
$\displaystyle:=Q_{\mu}{}^{\alpha}{}_{\alpha}\,,\quad\quad\bar{Q}_{\mu}:=Q{}_{\alpha}{}^{\alpha}{}_{\mu}\,,$
(5)

one has the relation

$\displaystyle\mathring{R}$ $\displaystyle=Q+B\,,$ (6)

where $\mathring{R}$ is the Ricci scalar of the Levi-Civita connection, and
the boundary term is

$B:=\mathring{\nabla}_{\alpha}\left(\bar{Q}^{\alpha}-Q^{\alpha}\right).$ (7)

Therefore, General Relativity, which uses $\mathring{R}$ as the Lagrangian
density, and STEGR, which uses $Q$, lead to exactly the same equations.

One can extend STEGR by extending the Lagrangian to an arbitrary function,
resulting to $f(Q)$ gravity, with action [21, 60]

$\displaystyle S_{G}$ $\displaystyle={\displaystyle\int
d^{4}x\,\sqrt{-g}\left[-\frac{1}{16\pi G}f(Q)+\mathcal{L}_{m}\right]\,,}$ (8)

where $G$ is the gravitational constant and $\mathcal{L}_{m}$ the matter
Lagrangian. The corresponding field equations are

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sqrt{-g}\left(\frac{1}{2}fg_{\mu\nu}-\frac{\partial
f}{\partial
g^{\mu\nu}}\right)-2\nabla_{\alpha}\left(\sqrt{-g}P^{\alpha}{}_{\mu\nu}\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =8\pi
G\sqrt{-g}\,T_{\mu\nu}\,,$ (9)

with

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\frac{\partial f}{\partial
g^{\mu\nu}}$
$\displaystyle=-\frac{f_{\text{Q}}}{\sqrt{-g}}\left(\frac{\partial\left(\sqrt{-g}Q\right)}{\partial
g^{\mu\nu}}-\frac{1}{2}\sqrt{-g}Qg_{\mu\nu}\right)$
$\displaystyle=-f_{\text{Q}}\biggl{[}c_{1}\left(2Q_{\alpha\beta\mu}Q^{\alpha\beta}{}_{\nu}-Q_{\mu\alpha\beta}Q^{\nu}{}_{\alpha\beta}\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \
\,+c_{2}Q_{\alpha\beta\mu}Q^{\beta\alpha}{}_{\nu}+c_{3}\left(2Q_{\alpha}Q^{\alpha}{}_{\mu\nu}\\!-\\!Q_{\mu}Q_{\nu}\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \
\,+c_{4}\widetilde{Q}_{\mu}\widetilde{Q}_{\nu}+c_{5}\widetilde{Q}_{\mu}Q^{\alpha}{}_{\mu\nu}\biggl{]},$
(10)

and where the conjugate to $f(Q)$ is defined as [21]

$\displaystyle P^{\alpha}{}_{\mu\nu}$
$\displaystyle:=\frac{1}{2\sqrt{-g}}\frac{\partial(\sqrt{-g}f(Q))}{\partial
Q{}_{\alpha}{}^{\mu\nu}}\,$
$\displaystyle\phantom{:}=-f_{\text{Q}}\biggl{\\{}c_{1}Q^{\alpha}{}_{\mu\nu}+c_{2}Q_{(\mu}{}^{\alpha}{}_{\nu)}+c_{3}g_{\mu\nu}Q^{\alpha}$
$\displaystyle\quad\,+c_{4}\delta{}^{\alpha}{}_{(\mu}\widetilde{Q}_{\nu)}+\frac{c_{5}}{2}\bigl{[}\widetilde{Q}^{\alpha}g_{\mu\nu}+\delta{}^{\alpha}{}_{(\mu}Q_{\nu)}\bigr{]}\biggl{\\}}\,,$
(11)

while as usual the matter energy momentum tensor is defined as

$\displaystyle T_{\mu\nu}$
$\displaystyle:=-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_{m})}{\delta
g^{\mu\nu}}.$ (12)

The field equations (9) are quite general with respect to both their
normalizations described by $G$ and also the defined constants $c_{i}$ of $Q$.
For instance, by choosing $Q$ as in Eq. (4) then we retrieve the standard
modified class of $f(Q)$ gravity [21]. In the rest of this work we deal with
this standard modified class of $f(Q)$ theories.

##  III Cosmological application of $f(Q)$ gravity

In this section we proceed to the cosmological application of $f(Q)$ gravity.
We impose a spatially flat Friedmann-Robertson-Walker (FLRW) metric of the
form

$\displaystyle ds^{2}$
$\displaystyle=-dt^{2}+a^{2}(t)\delta_{ij}dx^{i}dx^{j}\,,$ (13)

with $a(t)$ the scale factor. In this case Eq. (9) gives rise to the two
Friedmann equations [22]

$\displaystyle 6f_{\text{Q}}H^{2}-\frac{1}{2}f$ $\displaystyle=8\pi
G(\rho_{m}+\rho_{r})\,,$ (14)
$\displaystyle\big{(}12H^{2}f_{\text{QQ}}+f_{\text{Q}}\big{)}\dot{H}$
$\displaystyle=-4\pi G(\rho_{m}+p_{m}+\rho_{r}+p_{r})\,,$ (15)

where $H=\dot{a}/a$ is the the Hubble function and with $\rho_{m}$, $\rho_{r}$
and $p_{m}$, $p_{r}$ the energy densities and pressures of the matter and
radiation perfect fluids respectively. Additionally, note that the non-
metricity scalar $Q$ in an FRW background becomes $Q=6H^{2}$. Finally, the
equations constitute a close system by considering the matter and radiation
conservation equations, namely

$\displaystyle\dot{\rho}_{m}+3H(\rho_{m}+p_{m})$ $\displaystyle=0\,,$ (16)
$\displaystyle\dot{\rho}_{r}+3H(\rho_{r}+p_{r})$ $\displaystyle=0\,.$ (17)

We can re-write Eqs. (14)-(15) as

$\displaystyle H^{2}$ $\displaystyle=\frac{8\pi
G}{3}\left(\rho_{m}+\rho_{r}+\rho_{DE}\right)\,,$ (18) $\displaystyle\dot{H}$
$\displaystyle=-4\pi
G\left(\rho_{m}+p_{m}+\rho_{r}+p_{r}+\rho_{DE}+p_{DE}\right)\,,$ (19)

where we have defined the energy density and pressure of the effective dark
energy sector as

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\rho_{DE}=\frac{1}{16\pi
G}\left[6H^{2}\left(1-2f_{\text{Q}}\right)+f\right]\,,$ (20)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!p_{DE}=\frac{1}{16\pi
G}\left[4\left(f_{\text{Q}}-1\right)\dot{H}-f\right.$ $\displaystyle\left.\ \
\ \ \ \ \ \ \ \
+6H^{2}\left(8f_{\text{QQ}}\dot{H}+2f_{\text{Q}}-1\right)\right]\,.$ (21)

Finally, it proves convenient to introduce the density parameters for the
various sectors us

$\displaystyle\Omega_{i}=\frac{8\pi G\rho_{i}}{3H^{2}},$ (22)

where “i” stands for matter, radiation and dark energy.

Let us now focus on particular $f(Q)$ forms that are of interest for the late
Universe description. In general, any model that exhibits identical Hubble
rate with the concordance one (see [28]) can pass the BBN constraints
trivially, hence we do not examine models of this kind. Note also that we
consider versions of our models in their “bare” form, i.e. without explicitly
containing the cosmological constant, in order to avoid re-introducing the
cosmological constant problem and also to keep the smallest possible set of
free parameters. One should in principle combine any selection of the
following models, however the final model could have a large number of free
parameters.

  1. 1.

Polynomial model

The following model was introduced in [22]

$\displaystyle f(Q)$ $\displaystyle=Q-6\lambda
M^{2}\left(\frac{Q}{6M^{2}}\right)^{\alpha}\,,$ (23)

where $\lambda$ and $\alpha$ are dimensionless parameters. The parameter $M$
corresponds to a mass scale that should be of the order $\sqrt{\Lambda}$, with
$\Lambda$ the standard cosmological constant. The case of $\alpha=0$ gives
STEGR plus a cosmological constant equal to $6\lambda M^{2}$, while the case
$\alpha=1$ corresponds to STEGR with $G\rightarrow G/(1-\lambda)$. More
generally, $\alpha>1$ is mostly relevant to early Universe, while $\alpha<1$
is able to describe dark energy and thus it is relevant to late Universe.
Moreover, in the latter case the existence of an asymptotic GR limit at early
times is apparent. Finally, note that the case $\alpha=-1$ has been confronted
with late universe observations in [34].

In this model for the effective dark-energy density (20) we obtain

$\rho_{\text{DE}}=\frac{1}{16\pi
G}\left[6^{-\alpha}\Omega_{\text{F0}}H_{0}^{-2\alpha}Q^{\alpha+1}\right]\,,$
(24)

where $\Omega_{F0}=1-\Omega_{m0}-\Omega_{r0}$, and with the subscript denoting
the value of a quantity at present time. Calculating the first Friedmann (14)
at present time we find that the free parameter $\lambda$ can be eliminated in
terms of $\alpha$ and $M$ as

$\lambda=\frac{1}{12\alpha+6}\left(\Omega_{\text{F0}}H_{0}{}^{-2\alpha}M^{2\alpha-2}\right)\,.$
(25)

  2. 2.

Power - Exponential model

This model has proposed in [41] and has been proven to provide slightly better
fits to observational data than the concordance model [41]. It is
characterized by

$\displaystyle f(Q)$ $\displaystyle=Qe^{\lambda\frac{Q_{0}}{Q}}\,.$ (26)

As it is apparent, in case $\lambda=0$ GR without a cosmological constant is
recovered. In the past, where the term $Q_{0}/Q$ decreases, since the Hubble
function increases, the model at hand effectively reduces to the power-law
model, thus alleviating the cosmological constant problem. In a sense, this
model behaves as infinite number of different power-law models that coexist,
and at each moment of the cosmic history one of them becomes dominant. Another
advantage of the model is that it has the same number of free parameters with
the concordance one, while additionally containing rich phenomenology.

In this case (20) gives

$\rho_{\text{DE}}=\frac{1}{16\pi G}\left[Q-e^{\frac{\lambda
Q_{0}}{Q}}\left(Q-2\lambda Q_{0}\right)\right]\,.$ (27)

The parameter $\lambda$ is expressed from the first Friedmann Eq. (14) at
present as

$\lambda=\frac{1}{2}+W\left(\frac{\Omega_{\text{F0}}-1}{2\sqrt{e}}\right)\,,$
(28)

where $W$ is the Lambert function.

  3. 3.

Log-square-root model

Let us now propose a new $f(Q)$ model. In particular, following the approach
of [61], we introduce a logarithmic-square-root $f(Q)$ model as:

$\displaystyle f(Q)$ $\displaystyle=Q+nQ_{0}\sqrt{\frac{Q}{\lambda
Q_{0}}}\ln\left(\lambda\frac{Q_{0}}{Q}\right)\,,$ (29)

with $n$ and $\lambda>0$ the model parameters. The corresponding effective
energy density (20) gives

$\displaystyle\rho_{\text{DE}}$ $\displaystyle=\frac{1}{16\pi
G}\Omega_{\text{F0}}\sqrt{Q_{0}Q}\,,$ (30)

and the first Friedmann equation at present leads to

$\displaystyle n$
$\displaystyle=\frac{\Omega_{\text{F0}}}{2}\sqrt{\lambda}\,.$ (31)

  4. 4.

Hyperbolic tangent - power model

We consider the hyperbolic tangent power model, in similar lines with [62],
namely we choose

$\displaystyle f(Q)$ $\displaystyle=Q+\lambda
Q_{0}\left(\frac{Q}{Q_{0}}\right){}^{n}\tanh\left(\frac{Q_{0}}{Q}\right)\,.$
(32)

In this case we have

$\displaystyle\ \ \rho_{\text{DE}}=\frac{1}{16\pi G}Q_{0}^{1-n}Q^{n-1}$
$\displaystyle\ \ \ \ \ \ \
\cdot\left[(1\\!-\\!2n)Q\tanh\left(\\!\frac{Q_{0}}{Q}\\!\right)\\!+\\!2Q_{0}\text{sech}^{2}\left(\\!\frac{Q_{0}}{Q}\\!\right)\right],$
(33)

while

$\displaystyle n$
$\displaystyle=\frac{1}{2}\left[-\coth(1)\Omega_{\text{F0}}+1+4\text{csch}(2)\right]\,.$
(34)

  5. 5.

DGP-like $f(Q)$ model - I

In the recent work [63] two new $f(Q)$ models were introduced, by requiring
the extra terms in the modified Friedman equation to be of the form $\sim H$.
It was argued that a term proportional to $Q$ is responsible for the $H^{2}$
term, thus adding a term proportional to $\sqrt{Q}$ leads to a term $\sim H$
in the Friedman equation. These two models resemble the Dvali-Gabadadze-
Porrati cosmology [64] at the background level, and this is where they acquire
their names from. We mention here that this resemblance is only at the
background level, since the perturbations in the usual DGP model and $f(Q)$
gravity are fundamentally different, and hence the present scenario does not
share the known problem of usual DGP model in fitting the perturbation-related
data (such as LSS and CMB temperature/polarization) [65, 66].

The first model of this kind reads as

$f(Q)=\alpha\sqrt{Q}\log Q+2\beta Q,$ (35)

where $\alpha,\ \beta$ are free parameters. We mention that in the original
parametrization the free parameter $\alpha$ is not dimensionless, and we use
the form (35) in order to maintain compatibility with the results of [63]
(although it would be more convenient to impose a re-scaling
$\alpha\rightarrow\sqrt{Q_{0}}\alpha$ to ensure that $\alpha$ is
dimensionless). Nevertheless, we stress that inside the logarithm one should
explicitly include a constant value of dimension $H_{0}^{-2}$, and for
consistency in the present work we do apply this modification.

In this case (20) gives

$\displaystyle\rho_{\text{DE}}=\frac{1}{16\pi
G}\big{[}\sqrt{6}H_{0}\sqrt{Q}\left(2\beta+\Omega_{\text{F0}}-1\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \,-2\beta Q+Q\big{]},$ (36)

while the first Friedmann equation gives

$\displaystyle\alpha=-\sqrt{\frac{3}{2}}H_{0}\left(2\beta+\Omega_{\text{F0}}-1\right).$
(37)

Finally, note that near the current time, the term $\sim H$ ceases to be
negligible in comparison to $H^{2}$, thus giving rise to new phenomenology.
However, in the past this term is negligible and the model is very close to
the concordance one.

  6. 6.

DGP-like $f(Q)$ model - II

The second model introduced in [63] reads as

$\ \ \ \ \,\
f(Q)=\frac{Q\sqrt{u(Q)}\left[\\!\sqrt{u(Q)}\\!-\\!\sqrt{\gamma}\text{arctanh}\left(\\!\frac{\sqrt{u(Q)}}{\sqrt{\gamma}}\right)\\!\right]}{8\pi
G\sqrt{Qu(Q)}}\,,$ (38)

where $u(Q)=\gamma+\beta^{2}Q$, and $\beta$,$\gamma$ are free parameters,
different than zero. The particular choice of $u$ is motivated from the
previous case and includes also a mixing between $H^{2}$ and $H$ terms [63].
In this case we acquire

$\displaystyle\rho_{\text{DE}}$ $\displaystyle=\frac{1}{16\pi
G}\left[Q-\sqrt{Q}\sqrt{\gamma+\beta^{2}Q}\right]\,,$ (39)

while

$\displaystyle\gamma$
$\displaystyle=-6H_{0}{}^{2}\left(\beta-\Omega_{\text{F0}}+1\right)\left(\beta+\Omega_{\text{
F0}}-1\right)\,.$ (40)

Similarly to the previous case, in order to maintain dimensional consistency
within the square root, we include an $H_{0}^{-2}$ normalization (i.e. we
apply the re-scaling $\gamma\rightarrow Q_{0}\gamma$).

##  IV Big Bang Nucleosynthesis constraints

In this Section we review the Big Bang Nucleosynthesis (BBN) formalism
following [43, 44, 45, 46]. The BBN takes place during the radiation era and
thus the energy density of relativistic particles needs to be taken into
account, namely

$\displaystyle{\displaystyle\rho_{r}}$
$\displaystyle=\frac{\pi^{2}}{30}g_{*}T^{4}\,,$ (41)

where

$g_{*}\sim 10\,$ (42)

is the effective number of degrees of freedom and $T$ is the temperature (for
more details of the BBN framework used in this work see the appendix of [47]).
The calculation of the neutron abundance is realized by taking into account
the protons-neutron conversion rate

$\lambda_{pn}(T)=\lambda_{(n+\nu_{e}\to p+e^{-})}+\lambda_{(n+e^{+}\to
p+\bar{\nu}_{e})}+\lambda_{(n\to p+e^{-}+\bar{\nu}_{e})}\,,$ (43)

and its inverse $\lambda_{np}(T)$, thus the total rate is

$\lambda_{tot}(T)=4A\,T^{3}(4!T^{2}+2\times 3!{\cal Q}T+2!{\cal Q}^{2})\,,$
(44)

where ${\cal Q}=m_{n}-m_{p}=1.29\times 10^{-3}$GeV is the neutro-proton mass
difference and $A=1.02\times 10^{-11}$GeV-4.

Regarding the primordial mass fraction of ${}^{4}He$, it can be estimated [44]
as

$Y_{p}:=\lambda\,\frac{2x(t_{f})}{1+x(t_{f})}\,,$ (45)

with $\lambda=e^{-(t_{n}-t_{f})/\tau}$, $t_{f}$ the freeze-out time of the
weak interactions, $t_{n}$ the corresponding freeze-out time of
nucleosynthesis, $\tau$ the neutron mean lifetime and $x(t_{f})=e^{-{\cal
Q}/T(t_{f})}$ the neutron-to-proton equilibrium ratio. The role of the
function $\lambda(t_{f})$ is to account for the fraction of neutrons that
decay into protons during the time interval $t\in[t_{f},t_{n}]$.

In case of modified gravity models, in general the Friedmann equations will
contain extra terms than the standard GR ones. The BBN is realized in the
radiation epoch and according to observations these extra contributions have
to be small compared to the radiation sector in the Standard Model of
particles physics in the framework of General Relativity, whilst we can safely
neglect the matter sector as we are deep in the radiation era. Thus, the first
Friedmann equation can be approximated as

$\displaystyle H^{2}$ $\displaystyle\approx\frac{8\pi
G}{3}\rho_{r}=:H_{GR}^{2}\,,$ (46)

where the scale factor evolves as $a\sim t^{1/2}$, with $t$ the cosmic time.
Consequently, temperature and time are related by
${\displaystyle\frac{1}{t}\simeq\left(\frac{32\pi^{3}g_{*}}{90}\right)^{1/2}\frac{T^{2}}{M_{P}}}$
(or $T(t)\simeq(t/\text{sec})^{-1/2}$MeV), which can further lead to

$\displaystyle H$
$\displaystyle\approx\left(\frac{4\pi^{3}g_{*}}{45}\right)^{1/2}\frac{T^{2}}{M_{P}}\,,$
(47)

where

$M_{P}=(8\pi G)^{-1}=1.22\times 10^{19}\text{GeV}\,,$ (48)

is the Planck mass.

Assuming that the expansion time is much smaller than the interaction time,
then the interaction rate $\lambda_{tot}(T)$ given in (44) satisfies
$\frac{1}{H}\ll\lambda_{tot}(T)$, which means that all processes can be
approximated as being in thermal equilibrium [44, 67]. In contrast, if the
particles do not have the necessary time intervals to interact then
$\frac{1}{H}\gg\lambda_{tot}(T)$ and thus they decouple. The temperature at
which the particles decouple is called freeze-out temperature and is denoted
as $T_{f}$, and it is defined through $H=\lambda_{tot}(T)|_{T=T_{f}}$. Using
$H\approx\left(\frac{4\pi^{3}g_{*}}{45}\right)^{1/2}\frac{T^{2}}{M_{P}}$,
while $\lambda_{tot}(T)\approx qT^{5}$, with

$q=4A4!\simeq 9.8\times 10^{-10}\text{GeV}{}^{-4}\,,$ (49)

the freeze-out temperature is provided as

$\displaystyle T_{f}$
$\displaystyle=\left(\frac{4\pi^{3}g_{*}}{45M_{P}^{2}q^{2}}\right)^{1/6}\sim
0.0006\,\text{GeV}\,.$ (50)

In the realm of modified gravity, the Hubble function $H$ will, in general,
deviate from $H_{GR}$. This in turn means that the corresponding freeze-out
temperatures will also deviate by an amount of $\delta T_{f}$. This deviation
induces a difference $\delta Y_{p}$ of the fractional mass $Y_{p}$

$\displaystyle\delta Y_{p}$
$\displaystyle=Y_{p}\left[\left(1-\frac{Y_{p}}{2\lambda}\right)\ln\left(\frac{2\lambda}{Y_{p}}-1\right)-\frac{2t_{f}}{\tau}\right]\frac{\delta
T_{f}}{T_{f}}\,,$ (51)

where $\delta T(t_{n})=0$ was imposed due to the fact that $T_{n}$ is fixed by
the binding energy of deuterium [49, 50, 51, 52, 53]. Thus, the observational
imprints of the mass fraction $Y_{p}$ of baryons that convert to ${}^{4}He$
during the BBN epoch, are [68, 69, 70, 71, 72, 73, 74]

$\displaystyle Y_{p}$ $\displaystyle=0.2476\,,\qquad|\delta Y_{p}|<10^{-4}\,.$
(52)

By replacing these into (51), the upper bound of $\frac{\delta T_{f}}{T_{f}}$
is obtained as

$\displaystyle\left|\frac{\delta T_{f}}{T_{f}}\right|$
$\displaystyle<4.7\times 10^{-4}\,,$ (53)

which quantifies the allowed deviation from the cosmology of GR.

As we mentioned above, the effective dark energy $\rho_{DE}$ of modified
gravity is present during the BBN times too, and compared to $\rho_{r}$,
$\rho_{DE}$ should be smaller and thus it can be considered as a first-order
deviation. Hence, the Hubble function is

$\displaystyle H$
$\displaystyle=H_{GR}\sqrt{1+\frac{\rho_{DE}}{\rho_{r}}}=H_{GR}+\delta H,$
(54)

where

$\displaystyle\delta H$
$\displaystyle=\left(\sqrt{1+\frac{\rho_{DE}}{\rho_{r}}}-1\right)H_{GR}\,.$
(55)

The deviation $\delta H$ from standard $H_{GR}$ will induce the deviation
$\delta T_{f}$ from $T_{f}$, and according to the relation
$H_{GR}=\lambda_{tot}\approx qT^{5}$ it turns out that

$\left(\sqrt{1+\frac{\rho_{DE}}{\rho_{r}}}-1\right)H_{GR}=5qT_{f}^{4}\delta
T_{f}\,.$ (56)

Since $\rho_{DE}\ll\rho_{r}$ we finally obtain

$\frac{\delta
T_{f}}{T_{f}}\simeq\frac{\rho_{DE}}{\rho_{r}}\frac{H_{GR}}{10qT_{f}^{5}}\,.$
(57)

We have now all the information needed to proceed to the investigation of the
BBN bounds on the parameters of the $f(Q)$ models introduced above.

##  V BBN constraints on $f(Q)$ gravity

In this section, the models introduced in Sec. III will be tested against the
constraint (53). This will be realized by calculating (57) for each of the
models. In general, we are interested in the regions of the parameter space
where the constraint (57) is satisfied. Obviously, all models are expected to
satisfy the constraint trivially for the case $\Omega_{\text{F0}}=0$, i.e. in
the case where there is no effective dark energy. In the following, in all
figures the drawn parts correspond to satisfaction of the aforementioned BBN
constraint. For the standard free parameters we impose the ranges
$h\in[0.6,0.9]$ and $\Omega_{\text{F0}}\in[0,1]$. For the case of free model
parameters we use the mathematically allowed ranges.

### 1\. Polynomial model

Substituting Eq. (25) along with Eq. (24) in Eq. (57) we obtain

$\displaystyle\frac{\delta T_{f}}{T_{f}}$
$\displaystyle\simeq\frac{1}{q}2^{4\alpha+3}3^{\frac{1}{2}-\alpha}5^{-\alpha-\frac{3}{2}}\pi^{3\alpha+2}H_{0}\Omega_{\text{F0}}\text{}$
$\displaystyle\phantom{\simeq}\cdot
g_{*}^{\alpha+\frac{1}{2}}T_{f}^{4\alpha-3}\left(H_{0}M_{p}\right){}^{-2\alpha-1}\,.$
(58)

Imposing the constraint (53) on the parameter space of the polynomial model
using (58), we obtain Fig. 1. Note that we imposed the relation
$H_{0}=2.12\cdot 10^{-42}h$ GeV and we have also replaced the numerical values
(49), (42), (50) and (48).

From Fig. 1 we deduce that the model parameter $\alpha$ should be in the range
$\alpha\lesssim 0.88$ in order for the model to satisfy the BBN constraints.
However, if we additionally desire the model to be able to describe dark
energy in the late-time Universe, then we obtain the combined constraint
$\alpha<0$.

### 2\. Power-Exponential model

Substituting Eq. (28) alongside with (27) into (57) we acquire

$\frac{\delta
T_{f}}{T_{f}}\simeq\frac{8\sqrt{\frac{3}{5}}\pi^{2}\sqrt{g}T_{0}^{4}\left\\{e^{\lambda\left(\frac{T_{0}}{T_{f}}\right)^{4}}\left[2\lambda-\left(\frac{T_{f}}{T_{0}}\right)^{4}\right]+\left(\frac{T_{f}}{T_{0}}\right)^{4}\right\\}}{5qT_{f}^{7}M_{p}}\,.$
(59)

Inserting this expression into (53) we find that for the range of values
$0.5\leq h\leq 0.9$ and $0\leq\Omega_{\text{F0}}\leq 1$, the constraint (53)
is satisfied trivially for all parameter values. This result is intuitively
expected, since as we mentioned above the past asymptotic behavior of the
model recovers pure GR at early times, and thus during the BBN epoch [41].
This is a great advantage of this particular model.

Figure 1: The parameter space of the polynomial model (23) that is consistent
with the BBN constraints. If we also desire the model to be able to describe
dark energy in the late-time Universe we obtain the combined constraint
$\alpha<0$.

### 3\. Log-square-root

In this case, inserting (31) and (30) in (57) we find

$\frac{\delta T_{f}}{T_{f}}\simeq
8\pi^{2}\sqrt{\frac{3g}{5}}\frac{\Omega_{\text{F0}}T_{0}^{2}}{5qT_{f}^{5}M_{p}}\,.$
(60)

Therefore, substituting (60) in (53) we observe that for the range of values
$0.5\leq h\leq 0.9$ and $0\leq\Omega_{\text{F0}}\leq 1$, the constraint (53)
is always satisfied. This is an advantage of the model, and supports its
viability. Definitely, one should confront the model with late-time
observational data, too.

### 4\. Hyperbolic tangent - power model

In the case of the hyperbolic tangent - power model (32), substituting (34)
and (33) into (57) we obtain

$\displaystyle\\!\\!\\!\\!\\!\\!\frac{\delta
T_{f}}{T_{f}}\simeq\sqrt{\frac{3}{5}}\frac{8\pi^{2}\sqrt{g}}{5qM_{p}T_{f}^{7}}\left(\frac{T_{f}}{T_{0}}\right)^{4n-4}$
$\displaystyle\\!\\!\\!\\!\\!\\!\cdot\left\\{\\!(1\\!-\\!2n)T_{f}^{4}\tanh\\!\left[\\!\left(\frac{T_{0}}{T_{f}}\right)^{4}\right]\\!+\\!2T_{0}^{4}\text{sech}^{2}\\!\left[\\!\left(\frac{T_{0}}{T_{f}}\right)^{4}\right]\\!\right\\}.$
(61)

In Fig. 2 we depict the constraint (53) by using (61). The procedure is the
same as in case of Fig. 1 above. As we observe, the BBN constraints are
satisfied for $n\lesssim 1.88$.

Figure 2:  The parameter space of the hyperbolic tangent - power model (32)
that is consistent with the BBN constraints. If we also desire the model to be
able to describe dark energy in the late-time Universe we obtain the combined
constraint $n\lesssim 1.88$.

### 5\. DGP-like model - I

For the DGP-like model - I (35), if we insert (37) and (36) into (57), we
acquire

$\frac{\delta
T_{f}}{T_{f}}\simeq\frac{8\sqrt{\frac{3}{5}}\pi^{2}\sqrt{g}T_{0}{}^{2}\left[2\beta\\!+\\!\Omega_{\text{F0}}\\!-\\!2\beta\left(\frac{T_{f}}{T_{0}}\right)^{2}\\!+\\!\left(\frac{T_{f}}{T_{0}}\right)^{2}\\!-\\!1\right]}{5qM_{p}T_{f}{}^{5}}.$
(62)

Furthermore, numerical exploration of this equation for the standard parameter
ranges $0.5\leq h\leq 0.9$ and $0\leq\Omega_{\text{F0}}\leq 1$, shows that the
constraint (53) is satisfied only in the tuned interval
$0.499984<\beta<0.500016$.

### 6\. DGP-like model - II

Finally, let us come to the DGP-like model - II of (38). The constraint (57),
after inserting (40) and (39), reduces to

$\displaystyle\\!\\!\\!\\!\\!\frac{\delta
T_{f}}{T_{f}}\simeq\sqrt{\frac{3}{5}}\frac{8\pi^{2}\sqrt{g}T_{0}{}^{2}}{5qM_{p}T_{f}^{5}}$
$\displaystyle\\!\\!\\!\\!\\!\cdot\left\\{\\!\\!\left(\frac{T_{f}}{T_{0}}\right)^{2}\\!\\!-\\!\sqrt{\\!\left(\Omega_{\text{
F0
}}\\!-\\!2\right)\\!\Omega_{\text{F0}}\\!+\\!\beta^{2}\\!\left[\\!\left(\frac{T_{f}}{T_{0}}\right)^{4}\\!-\\!1\\!\right]\\!+\\!1}\right\\}.$
(63)

Examining this expression for the range of values $0.5\leq h\leq 0.9$ and
$0\leq\Omega_{\text{F0}}\leq 1$, we deduce that the constraint (53) is
satisfied in the tuned window $0.999969<|\beta|<1.00003$.

##  VI Conclusions

We use the Big Bang Nucleosynthesis formalism and observations in order to
extract constraints on $f(Q)$ gravity. The latter is a modified gravitational
theory arising from the consideration of non-metricity as the basic geometric
quantity. We investigated various classes of $f(Q)$ models, and we
additionally introduced a number of new ones, inspired by other modified
gravities, that are capable of describing the late-time Universe evolution. We
applied the semi-analytic approach of [50, 51, 52] for the physics of the BBN
epoch, and we calculated the deviations that $f(Q)$ terms bring on the freeze-
out temperature $T_{f}$ in comparison to that of the standard $\Lambda$CDM
evolution. We then imposed the observational bound on
$\left|\frac{\delta{T}_{f}}{{T}_{f}}\right|$ to extract constraints on the
involved parameters of the considered models.

Concerning the polynomial model, we showed that the exponent parameter
$\alpha$ should be in the interval $\alpha<0$ if we desire the model to
simultaneously pass the BBN constraints and be able to describe the late-time
Universe acceleration. On the other hand, the power-exponential model is found
to pass the BBN constraints trivially, which was expected since this model
recovers general relativity at early times. This is a significant advantage,
since this particular model is known to be compatible with late-time
observational datasets slightly more efficiently than $\Lambda$CDM scenario
[41], and thus it can be considered as a good candidate for the description of
Nature.

In the case of the new proposed Log-square-root models we showed that BBN
constraints are also trivially satisfied. However, in the new hyperbolic
tangent - power model we showed that the combined BBN and late-time
constraints require the model parameter to lie in the range $n\lesssim 1.88$.
Finally, we examined the two DGP-like $f(Q)$ models that have recently
appeared in the literature. In both cases we found that the BBN constraints
are satisfied for a very narrow window for the model parameter $\beta$.

In summary, we showed that $f(Q)$ gravity can safely pass the BBN constraints,
and in some cases this is obtained trivially. This is an important advantage,
since many gravitational modifications, although able to describe the late-
time evolution of the Universe, fall to pass the BBN confrontation since they
produce too-much modification at early times. Hence, $f(Q)$ gravity proves to
be a class of modified gravity that deserves further investigation. One could
try to proceed beyond the semi-analytical approach of this work, using
numerical codes like Parthenope [75] and AlterBBN [76]. Such a detailed
investigation lies beyond the scope of the present work and it is left for a
future project.

###### Acknowledgements.

The authors acknowledge participation in the COST Association Action CA18108
“Quantum Gravity Phenomenology in the Multimessenger Approach (QG-MM)”.

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# OpenTapioca: Lightweight Entity Linking for Wikidata

Antonin Delpeuch
Department of Computer Science
University of Oxford, UK
<EMAIL_ADDRESS>
###### Abstract

We propose a simple Named Entity Linking system that can be trained from
Wikidata only. This demonstrates the strengths and weaknesses of this data
source for this task and provides an easily reproducible baseline to compare
other systems against. Our model is lightweight to train, to run and to keep
synchronous with Wikidata in real time.

##  1 Introduction

Named Entity Linking is the task of detecting mentions of entities from a
knowledge base in free text, as illustrated in Figure 1.

Most of the entity linking literature focuses on target knowledge bases which
are derived from Wikipedia, such as DBpedia (Auer et al., 2007) or YAGO
Suchanek et al. (2007). These bases are curated automatically by harvesting
information from the info-boxes and categories on each Wikipedia page and are
therefore not editable directly.

Wikidata Vrandečić and Krötzsch (2014) is an editable, multilingual knowledge
base which has recently gained popularity as a target database for entity
linking (Klang and Nugues, 2014; Weichselbraun et al., 2018; Sorokin and
Gurevych, 2018; Raiman and Raiman, 2018). As these new approaches to entity
linking also introduce novel learning methods, it is hard to tell apart the
benefits that come from the new models and those which come from the choice of
knowledge graph and the quality of its data.

We review the main differences between Wikidata and static knowledge bases
extracted from Wikipedia, and analyze their implactions for entity linking. We
illustrate these differences by building a simple entity linker,
OpenTapioca111The implementation and datasets are available at
https://github.com/wetneb/opentapioca and the demo can be found at
https://opentapioca.org/., which only uses data from Wikidata, and show that
it is competitive with other systems with access to larger data sources for
some tasks. OpenTapioca can be trained easily from a Wikidata dump only, and
can be efficiently kept up to date in real time as Wikidata evolves. We also
propose tools to adapt existing entity linking datasets to Wikidata, and offer
a new entity linking dataset, consisting of affiliation strings extracted from
research articles.

Associated Press (Q40469)Julie Pace (Q34666768)Washington D.C. (Q61)employer
(P108) Associated Press writer Julie Pace
---
contributed from Washington.
Figure 1: Example of an annotated sentence

##  2 Particularities of Wikidata

Wikidata is a wiki itself, meaning that it can be edited by anyone, but
differs from usual wikis by its data model: information about an entity can
only be input as structured data, in a format that is similar to RDF.

Wikidata stores information about the world in a collection of _items_ , which
are structured wiki pages. Items are identified by ther Q-id, such as Q40469,
and they are made of several data fields. The _label_ stores the preferred
name for the entity. It is supported by a _description_ , a short phrase
describing the item to disambiguate it from namesakes, and _aliases_ are
alternate names for the entity. These three fields are stored separately for
each language supported by Wikidata. Items also hold a collection of
_statements_ : these are RDF-style claims which have the item as subject. They
can be backed by _references_ and be made more precise with _qualifiers_ ,
which all rely on a controlled vocabulary of _properties_ (similar to RDF
predicates). Finally, items can have _site links_ , connecting them to the
corresponding page for the entity in other Wikimedia projects (such as
Wikipedia). Note that Wikidata items to not need to be associated with any
Wikipedia page: in fact, Wikidata’s policy on the notability of the subjects
it covers is much more permissive than in Wikipedia. For a more detailed
introduction to Wikidata’s data model we refer the reader to Vrandečić and
Krötzsch (2014); Geißet al. (2017).

Our goal is to evaluate the usefulness of this crowdsourced structured data
for entity linking. We will therefore refrain from augmenting it with any
external data (such as phrases and topical information extracted from
Wikipedia pages), as is generally done when working with DBpedia or YAGO. By
avoiding a complex mash-up of data coming from disparate sources, our entity
linking system is also simpler and easier to reproduce. Finally, it is
possible keep OpenTapioca in real-time synchronization with the live version
of Wikidata, with a lag of a few seconds only. This means that users are able
to fix or improve the knowledge graph, for instance by adding a missing alias
on an item, and immediately see the benefits on their entity linking task.
This constrasts with all other systems we are aware of, where the user either
cannot directly intervene on the underlying data, or there is a significant
delay in propagating these updates to the entity linking system.

##  3 Related work

We review the dominant architecture of entity linking heuristics following
Shen et al. (2015), and assess its applicability to Wikidata.

Entities in the knowledge base are associated with a set (or probability
distribution) of possible surface forms. Given a text to annotate, candidate
entities are generated by looking for occurrences of their surface forms in
the text. Because of homonymy, many of these candidate occurrences turn out to
be false matches, so a classifier is used to predict their correctness. We can
group the features they tend to use in the following categories:

  * •

local compatibility: these features assess the adequacy between an entity and
the phrase that refers to it. This relies on the dictionary of surface forms
mentioned above, and does not take into account the broader context of the
phrase to link.

  * •

topic similarity: this measures the compatibility between the topics in the
text to annotate and the topics associated with the candidate entity. Topics
can be represented in various ways, for instance with a bag of words model.

  * •

mapping coherence: entities mentioned in the same text are often related, so
linking decisions are inter-dependent. This relies on a notion of proximity
between entities, which can be defined with random walks in the knowledge
graph for instance.

###  3.1 Local compatibility

These features compare the phrase to annotate with the known surface forms for
the entity. Collecting such forms is often done by extracting mentions from
Wikipedia (Cucerzan, 2007). Link labels, redirects, disambiguation pages and
bold text in abstracts can all be useful to discover alternate names for an
entity. It is also possible to crawl the web for Wikipedia links to improve
the coverage, often at the expense of data quality (Spitkovsky and Chang,
2012).

Beyond collecting a set of possible surface forms, these approaches count the
number of times an entity $e$ was mentioned by a phrase $w$. This makes it
possible to use a Bayesian methodology: the compatibility of a candidate
entity $e$ with a given mention $w$ is $P(e|w)=\frac{P(e,w)}{P(w)}$, which can
be estimated from the statistics collected.

In Wikidata, items have labels and aliases in multiple languages. As this
information is directly curated by editors, these phrases tend to be of high
quality. However, they do not come with occurence counts. As items link to
each other using their Wikidata identifiers only, it is not possible to
compare the number of times USA was used to refer United States of America
(Q30) or to United States Army (Q9212) inside Wikidata.

Unlike Wikipedia’s page titles which must be unique in a given language, two
Wikidata items can have the same label in the same language. For instance
Curry is the English label of both the item about the Curry programming
language (Q2368856) and the item about the village in Alaska (Q5195194), and
the description field is used to disambiguate them.

Manual curation of surface forms implies a fairly narrow coverage, which can
be an issue for general purpose entity linking. For instance, people are
commonly refered to with their given or family name only, and these names are
not systematically added as aliases: at the time of writing, Trump is an alias
for Donald Trump (Q22686), but Cameron is not an alias for David Cameron
(Q192). As a Wikidata editor, the main incentive to add aliases to an item is
to make it easier to find the item with Wikidata’s auto-suggest field, so that
it can be edited or linked to more easily. Aliases are not designed to offer a
complete set of possible surface forms found in text: for instance, adding
common mispellings of a name is discouraged.222The guidelines are available at
https://www.wikidata.org/wiki/Help:Aliases

###  3.2 Topic similarity

The compatibility of the topic of a candidate entity with the rest of the
document is traditionally estimated by similarity measures from information
retrieval such as TFIDF (Štajner and Mladenić, 2009; Ratinov et al., 2011) or
keyword extraction (Strube and Ponzetto, 2006; Mihalcea and Csomai, 2007;
Cucerzan, 2007).

Wikidata items only consist of structured data, except in their descriptions.
This makes it difficult to compute topical information using the methods
above. Vector-based representations of entities can be extracted from the
knowledge graph alone (Bordes et al., 2013; Xiao et al., 2016), but it is not
clear how to compare them to topic representations for plain text, which would
be computed differently. In more recent work, neural word embeddings were used
to represent topical information for both text and entities (Ganea and
Hofmann, 2017; Raiman and Raiman, 2018; Kolitsas et al., 2018). This requires
access to large amounts of text both to train the word vectors and to derive
the entity vectors from them. These vectors have been shown to encode
significant semantic information by themselves (Mikolov et al., 2013), so we
refrain from using them in this study.

###  3.3 Mapping coherence

Entities mentioned in the same context are often topically related, therefore
it is useful not to treat linking decisions in isolation but rather to try to
maximize topical coherence in the chosen items. This is the issue on which
entity linking systems differ the most as it is harder to model.

First, we need to estimate the topical coherence of a sequence of linking
decisions. This is often done by first defining a pairwise relatedness score
between the target entities. For instance, a popular metric introduced by
Witten and Milne (2008) considers the set of wiki links $|a|,|b|$ made from or
to two entities $a$, $b$ and computes their relatedness:

$\text{rel}(a,b)=1-\frac{\log(\max(|a|,|b|))-\log(|a|\cap|b|)}{\log(|K|)-\log(\min(|a|,|b|))}$

where $|K|$ is the number of entities in the knowledge base.

When linking to Wikidata instead of Wikipedia, it is tempting to reuse these
heuristics, replacing wikilinks by statements. However, Wikidata’s linking
structure is quite different from Wikipedia: statements are generally a lot
sparser than links and they have a precise semantic meaning, as editors are
restricted by the available properties when creating new statements. We
propose in the next section a similarity measure that we find to perform well
experimentally.

Once a notion of semantic similarity is chosen, we need to integrate it in the
inference process. Most approaches build a graph of candidate entities, where
edges indicate semantic relatedness: the difference between the heuristics lie
in the way this graph is used for the matching decisions. Moro et al. (2014)
use an approximate algorithm to find the _densest subgraph_ of the semantic
graph. This determines choices of entities for each mention. In other
approaches, the initial evidence given by the local compatibility score is
propagated along the edges of the semantic graph (Mihalcea and Csomai, 2007;
Han et al., 2011) or aggregated at a global level with a Conditional Random
Field (Ganea and Hofmann, 2017).

##  4 OpenTapioca: an entity linking model for Wikidata

We propose a model that adapts previous approaches to Wikidata. Let $d$ be a
document (a piece of text). A _spot_ $s\in d$ is a pair of start and end
positions in $d$. It defines a phrase $d[s]$, and a set of candidate entities
$E[s]$: those are all Wikidata items for which $d[s]$ is a label or alias.
Given two spots $s,s^{\prime}$ we denote by $|s-s^{\prime}|$ the number of
characters between them. We build a binary classifier which predicts for each
$s\in d$ and $e\in E[s]$ if $s$ should be linked to $e$.

###  4.1 Local compatibility

Although Wikidata makes it impossible to count how often a particular label or
alias is used to refer to an entity, these surface forms are carefully curated
by the community. They are therefore fairly reliable.

Given an entity $e$ and a phrase $d[s]$, we need to compute $p(e|d[s])$.
Having no access to such a probability distribution, we choose to approximate
this quantity by $\frac{p(e)}{p(d[s])}$, where $p(e)$ is the probability that
$e$ is linked to, and $p(d[s])$ is the probability that $d[s]$ occurs in a
text. In other words, we estimate the popularity of the entity and the
commonness of the phrase separately.

We estimate the popularity of an entity $e$ by a log-linear combination of its
number of statements $n_{e}$, site links $s_{e}$ and its PageRank $r(e)$. The
PageRank is computed on the entire Wikidata using statement values and
qualifiers as edges.

The probability $p(d[s])$ is estimated by a simple unigram language model that
can be trained either on any large unannotated dataset333For the sake of
respecting our constraint to use Wikidata only, we train this language model
from Wikidata item labels..

The local compatibility is therefore represented by a vector of features
$F(e,w)$ and the local compatibility is computed as follows, where $\lambda$
is a weights vector:

$\displaystyle F(e,w)$ $\displaystyle=(-\log p(d[s]),\log p(e),n_{e},s_{e},1)$
$\displaystyle p(e|d[s])$ $\displaystyle\propto e^{F(e,w)\cdot\lambda}$

###  4.2 Semantic similarity

The issue with the features above is that they ignore the context in which a
mention in found. To make it context-sensitive, we adapt the approach of Han
et al. (2011) to our setup. The general idea is to define a graph on the
candidate entities, linking candidate entities which are semantically related,
and then find a combination of candidate entities which have both high local
compatibility and which are densely related in the graph.

For each pair of entities $e,e^{\prime}$ we define a similarity metric
$s(e,e^{\prime})$. Let $l(e)$ be the set of items that $e$ links to in its
statements. Consider a one-step random walks starting on $e$, with probability
$\beta$ to stay on $e$ and probability $\frac{1-\beta}{|l(e)|}$ to reach one
of the linked items. We define $s(e,e^{\prime})$ as the probability that two
such one-step random walks starting from $e$ and $e^{\prime}$ end up on the
same item. This can be computed explicitly as

$\displaystyle s(e,e^{\prime})$
$\displaystyle=\beta^{2}\delta_{e=e^{\prime}}+\beta(1-\beta)(\frac{\delta_{e\in
l(e^{\prime})}}{|l(e^{\prime})|}$ $\displaystyle+\frac{\delta_{e^{\prime}\in
l(e)}}{|l(e)|}+(1-\beta)^{2}\frac{|l(e)\cap
l(e^{\prime})|}{|l(e)||l(e^{\prime})|}$

We then build a weighted graph $G_{d}$ whose vertices are pairs $(s\in d,e\in
E[s])$. In other words, we add a vertex for each candidate entity at a given
spot. We fix a maximum distance $D$ for edges: vertices $(s,e)$ and
$(s^{\prime},e^{\prime})$ can only be linked if $|s-s^{\prime}|\leq D$ and
$s\neq s^{\prime}$. In this case, we define the weight of such an edge as
$(\eta+s(e,e^{\prime}))\frac{D-|s-s^{\prime}|}{D}$, where $\eta$ is a
smoothing parameter. In other words, the edge weight is proportional to the
smoothed similarity between the entities, discounted by the distance between
the mentions.

The weighted graph $G_{d}$ can be represented as an adjacency matrix. We
transform it into a column-stochastic matrix $M_{d}$ by normalizing its
columns to sum to one. This defines a Markov chain on the candidate entities,
that we will use to propagate the local evidence.

###  4.3 Classifying entities in context

Han et al. (2011) first combine the local features into a local evidence
score, and then spread this local evidence using the Markov chain:

$\displaystyle G(d)=(\alpha I+(1-\alpha)M_{d})^{k}\cdot LC(d)$ (1)

We propose a variant of this approach, where each individual local
compatibility feature is propagated independently along the Markov chain.444It
is important for this purpose that features are initially scaled to the unit
interval. Let $F$ be the matrix of all local features for each candidate
entity: $F=(F(e_{1},d[s_{1}]),\dots,F(e_{n},d[s_{n}]))$. After $k$ iterations
in the Markov chain, this defines features $M_{d}^{k}F$. Rather than relying
on these features for a fixed number of steps $k$, we record the features at
each step, which defines the vector

$(F,M_{d}\cdot F,M_{d}^{2}\cdot F,\dots,M_{d}^{k}\cdot F)$

This alleviates the need for an $\alpha$ parameter while keeping the number of
features small. We train a linear support vector classifier on these features
and this defines the final score of each candidate entity. For each spot, our
system picks the highest-scoring candidate entity that the classifier predicts
as a match, if any.

##  5 Experimental setup

Most entity linking datasets are annotated against DBpedia or YAGO. Wikidata
contains items which do not have any corresponding Wikipedia article (in any
language), so these items do not have any DBpedia or YAGO URI either.555This
is the case of Julie Pace (Q34666768) in Figure 1. Therefore, converting an
entity linking dataset from DBpedia to Wikidata requires more effort than
simply following owl:sameAs links: we also need to annotate mentions of
Wikidata items which do not have a corresponding DBpedia URI.

We used the RSS-500 dataset of news excerpts annotated against DBpedia and
encoded in NIF format (Usbeck et al., 2015). We first translated all DBpedia
URIs to Wikidata items666We built the nifconverter tool to do this conversion
for any NIF dataset.. Then, we used OpenRefine (Huynh et al., 2019) to extract
the entities marked not covered by DBpedia and matched them against Wikidata.
After human review, this added 63 new links to the 524 converted from DBpedia
(out of 476 out-of-KB entities).

We also annotated a new dataset from scratch. The ISTEX dataset consists of
one thousand author affiliation strings extracted from research articles and
exposed by the ISTEX text and data mining service777The original data is
available under an Etalab license at https://www.istex.fr/. In this dataset,
only 64 of the 2,624 Wikidata mentions do not have a corresponding DBpedia
URI.

We use the Wikidata JSON dump of 2018-02-24 for our experiments, indexed with
Solr (Lucene). We restrict the index to humans, organizations and locations,
by selecting only items whose type was a subclass of (P279) human (Q5),
organization (Q43229) or geographical object (Q618123). Labels and aliases in
all languages are added to a case-sensitive FST index.

We trained our classifier and its hyper-parameters by five-fold cross-
validation on the training sets of the ISTEX and RSS datasets. We used GERBIL
(Usbeck et al., 2015) to evaluate OpenTapioca against other approaches. We
report the InKB micro and macro F1 scores on test sets, with GERBIL’s weak
annotation match method.888The full details can be found at
http://w3id.org/gerbil/experiment?id=201904110006

| AIDA-CoNLL | Microposts 2016
---|---|---
| Micro | Macro | Micro | Macro
AIDA | 0.725 | 0.684 | 0.056 | 0.729
Babelfy | 0.481 | 0.421 | 0.031 | 0.526
DBP Spotlight | 0.528 | 0.456 | 0.053 | 0.306
FREME NER | 0.382 | 0.237 | 0.037 | 0.790
OpenTapioca | 0.482 | 0.399 | 0.087 | 0.515
| ISTEX-1000 | RSS-500
| Micro | Macro | Micro | Macro
AIDA | 0.531 | 0.494 | 0.455 | 0.447
Babelfy | 0.461 | 0.447 | 0.314 | 0.304
DBP Spotlight | 0.574 | 0.575 | 0.281 | 0.261
FREME NER | 0.422 | 0.321 | 0.307 | 0.274
OpenTapioca | 0.870 | 0.858 | 0.335 | 0.310
Figure 2: F1 scores on test datasets

##  6 Conclusion

The surface forms curated by Wikidata editors are sufficient to reach
honourable recall, without the need to expand them with mentions extracted
from Wikipedia. Our restriction to people, locations and organizations
probably helps in this regard and we anticipate worse performance for broader
domains. Our approach works best for scientific affiliations, where spelling
is more canonical than in newswire. The availability of Twitter identifiers
directly in Wikidata helps us to reach acceptable performance in this domain.
The accuracy degrades on longer texts which require relying more on the
ambiant topical context. In future work, we would like to explore the use of
entity embeddings to improve our approach in this regard.

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