# Cosmological particle production in quantum gravity

Yaser Tavakoli<EMAIL_ADDRESS>Department of Physics, University
of Guilan, 41335-1914 Rasht, Iran  School of Astronomy, Institute for Research
in Fundamental Sciences (IPM), 19395-5531, Tehran, Iran

(August 28, 2024)

###### Abstract

Quantum theory of a test field on a quantum cosmological spacetime may be
viewed as a theory of the test field on an emergent classical background. In
such a case, the resulting dressed metric for the field propagation is a
function of the quantum fluctuations of the original geometry. When the
backreaction is negligible, massive modes can experience an anisotropic
Bianchi type I background. The field modes propagating on such a quantum-
gravity-induced spacetime can then unveil interesting phenomenological
consequences of the super-Planckian scales, such as gravitational particle
production. The aim of this paper is to address the issue of gravitational
particle production associated to the massive modes in such an anisotropic
dressed spacetime. By imposing a suitable adiabatic condition on the vacuum
state and computing the energy density of the created particles, the
significance of the particle production on the dynamics of the universe in
Planck era is discussed.

###### pacs:

04.60.-m, 04.60.Pp, 98.80.Qc

##  I Introduction

Observation of cosmic microwave background (CMB) implies that the classical
universe is homogeneous and isotropic on scales larger than 250 million light
years. Based on the standard $\Lambda$CDM model of cosmology, the Friedmann-
Lemaître-Robertson-Walker (FLRW) solution of Einstein’s field equations,
provides a suitable explanation of such near perfect isotropy of the CMB and
other astrophysical observables. Nevertheless, there is not yet a decisive
answer to the question whether or not the quantum structure of the universe in
the super-Planckian regime has the same symmetry as observed in the CMB, and
how such structure can be traced in the observational data.

According to the standard model of cosmology, the structure formation at large
scales are described via (inhomogeneous) perturbations at smaller scales, in
the early universe. Quantum field theory in classical, curved spacetime
provides a good approximate description of such phenomena in a regime where
the quantum effects of gravity are negligible Birrell and Davies (1984);
Parker and Toms (2009). Within such framework, the issue of the gravitational
particle production induced by the time-dependent background in an expanding
universe and its backreaction effect is of significant importance Hu and
Parker (1978); Zeldovich (1970). Nevertheless, when tracing further back in
time, where the curvature of the universe reaches the Planck scales, the
quantum effects of gravity becomes important. This might lead to additional
phenomena which are expected to be important when exploring the dynamics of
the early universe.

Loop quantum cosmology (LQC) is a promising candidate to investigate the
quantum gravity effects in Planckian regime Ashtekar et al. (2003). It follows
the quantization scheme of loop quantum gravity (LQG), which is a background
independent, non-perturbative approach to quantization of general relativity
Ashtekar and Lewandowski (2004); Rovelli (2004); Thiemann (2001). This
approach has provided a number of concrete results: the classical big bang
singularity is resolved and is replaced by a quantum bounce Ashtekar (2009);
Ashtekar et al. (2006a); Ashtekar and Wilson-Ewing (2009); Ashtekar and Singh
(2011); the standard theory of cosmological perturbations has been extended to
a self-consistent theory from the bounce in the super-Planckian regime to the
onset of slow-roll inflation Ashtekar et al. (2009); Agullo et al. (2013a, b,
2012); Ashtekar and Barrau (2015). Further phenomenological consequences and
observational predictions of LQC can be found e.g., in Refs. Dapor et al.
(2012); Dapor and Lewandowski (2013); Lewandowski et al. (2017); Tavakoli and
Fabris (2015); Graef et al. (2020); Garcia-Chung et al. (2021); Rastgoo et al.
(2020); Lankinen and Vilja (2017); Scardua et al. (2018); Assanioussi et al.
(2015); Celani et al. (2017) and Ashtekar and Barrau (2015); Bolliet et al.
(2016), respectively.

The cosmological, quantum backgrounds in LQC establish a fruitful ground to
explore the quantum theory of inhomogeneous fields propagating on it. In a
dressed metric approach, when the full quantum Hamiltonian constraint of the
gravity-matter system is solved, an effective dressed geometry emerges for the
field modes. Depending on whether the field is massless or massive, various
spacetime metrics can raise for the dressed background geometry. The emergent
metric components, if the backreaction between the fields and the geometry is
discarded, depend, generically, on the fluctuations of the original quantum
metric operator only. For massive modes, quantum gravity effects may induce a
small deviation from the initial isotropy Dapor and Lewandowski (2013);
Rastgoo et al. (2020). If the backreaction is considered, other properties,
such as emerging a mode-dependent background, can raise, which lead to the
violation of the local Lorentz symmetry Dapor et al. (2012); Lewandowski et
al. (2017). Even in the absence of the backreaction between the states of the
gravity and matter fields, a backreaction effect may arise due to the
gravitational particle production which can challenge the validity of the test
field approximations employed in the dressed metric scenario Graef et al.
(2020).

Motivated from the above paragraphs, our purpose in this article is to address
some phenomenological issues associated to the massive test field propagating
on a quantum cosmological spacetime. In particular, we will consider a
scenario in which an anisotropic dressed geometry emerges for the massive
modes due to quantum gravity effects, and then revisit the occurrence of the
gravitational particle production on such an anisotropic dressed background.
Such effects will have important consequences in the super-Planckian regime
and at the onset of the classical inflationary epoch. We thus organize this
paper as follows. In Sec. II, we will consider the propagation of a massive
test field on a quantized FLRW spacetime. We will derive an effective
evolution equation for the field and obtain a suitable anisotropic dressed
background for the field propagation. In Sec. III, we will study the theory of
quantum field, by considering the infinitely many modes, on the emergent
anisotropic spacetime. We will then explore the problem of the gravitational
particle production by choosing a convenient, adiabatic vacuum state in the
super-Planckian regime. Finally, in Sec. V we will present the conclusion and
discussion of our work.

##  II Quantum Fields on a quantum FLRW spacetime

In this section, we study the quantum theory of a massive scalar field on a
quantum FLRW background. Firstly, we will analyze the quantum theory of a
scalar field in the classical Friedmann universe. Next, we will quantize the
background and obtain a quantum evolution equation for the composite state of
the gravity-perturbation system. Finally, in the last step, we will extract an
effective evolution equation for the field mode degrees of freedom due to
which we can extract an effective anisotropic background for propagation of
the massive modes.

###  II.1 Quantum field on a cosmological classical spacetime

We start with considering a massive scalar field propagating on a more general
classical, _anisotropic_ background spacetime. In particular, we take a
Bianchi type I model whose line element is represented by

$g_{ab}dx^{a}dx^{b}=-N_{x_{0}}^{2}\left(x_{0}\right)\left(dx_{0}\right)^{2}+\sum_{i}^{3}a_{i}^{2}\left(x_{0}\right)\left(dx^{i}\right)^{2},$
(1)

where, $N_{x_{0}}$ is the lapse function and $a_{i}$ the scale factor in the
$x^{i}$-direction. The coordinates $(x_{0},\mathbf{x})$ are chosen such that,
$x_{0}\in\mathbb{R}$ is a generic time coordinate, and
$\mathbf{x}\in\mathbb{T}^{3}$ (i.e., a 3-torus with the coordinates
$x^{j}\in(0,\ell_{j})$).

We consider a real, minimally coupled scalar field,
$\varphi(x_{0},\mathbf{x})$, with a mass $m$, propagating on the background
(1). Having the Lagrangian density for $\varphi(x_{0},\mathbf{x})$, with a
quadratic potential,

$\mathcal{L}_{\varphi}=-\frac{1}{2}\left[g^{ab}\nabla_{a}\varphi\nabla_{b}\varphi+m^{2}\varphi^{2}\right],$
(2)

the equation of motion is obtained as

$(g^{ab}\nabla_{a}\nabla_{b}-m^{2})\varphi=0.$ (3)

On the $x_{0}=const$ slice, by performing the Legendre transformation, we take
a canonically conjugate momentum $\pi_{\varphi}$ to the field $\varphi$. Then,
the classical solutions of Eq. (3) for the pair $(\varphi,\pi_{\varphi})$ can
be Fourier expanded as111We consider an elementary cell $\mathcal{V}$ by
fixing its edge to lie along the coordinates $(\ell_{1},\ell_{2},\ell_{3})$.
We then denote the volume of $\mathcal{V}$ by
$\mathring{V}=\ell_{1}\ell_{2}\ell_{3}\equiv\ell^{3}$, and restrict all
integrations in the Fourier integral to this volume.

$\varphi(x_{0},\mathbf{x})=\frac{1}{\ell^{3/2}}\sum_{\mathbf{k}\in\mathscr{L}}\varphi_{\mathbf{k}}(x_{0})\,e^{i\mathbf{k}\cdot\mathbf{x}},$
(4a)
$\pi_{\varphi}(x_{0},\mathbf{x})=\frac{1}{\ell^{3/2}}\sum_{\mathbf{k}\in\mathscr{L}}\pi_{\mathbf{k}}(x_{0})\,e^{i\mathbf{k}\cdot\mathbf{x}},$
(4b)

with $\mathscr{L}=\mathscr{L}_{+}\cup\mathscr{L}_{-}$ being a 3-dimensional
lattice:

$\displaystyle\mathscr{L}_{+}$ $\displaystyle=$
$\displaystyle\\{\mathbf{k}:\leavevmode\nobreak\
k_{3}>0\\}\cup\\{\mathbf{k}:\leavevmode\nobreak\
k_{3}=0,k_{2}>0\\}\cup\\{\mathbf{k}:\leavevmode\nobreak\
k_{3}=k_{2}=0,k_{1}>0\\},$ (5a) and $\displaystyle\mathscr{L}_{-}$
$\displaystyle=$ $\displaystyle\\{\mathbf{k}:\leavevmode\nobreak\
k_{3}<0\\}\cup\\{\mathbf{k}:\leavevmode\nobreak\
k_{3}=0,k_{2}<0\\}\cup\\{\mathbf{k}:\leavevmode\nobreak\
k_{3}=k_{2}=0,k_{1}<0\\},$ (5b)

spanned by $\mathbf{k}=(k_{1},k_{2},k_{3})\in(2\pi\mathbb{Z}/\ell)^{3}$,
where, $\mathbb{Z}$ is the set of integers Ashtekar et al. (2009); Dapor et
al. (2012).

Let us decompose the field modes into the real and imaginary parts as

$\varphi_{\mathbf{k}}(x_{0})=\frac{1}{\sqrt{2}}\left[\varphi_{\mathbf{k}}^{(1)}(x_{0})+i\varphi_{\mathbf{k}}^{(2)}(x_{0})\right].$
(6)

Then, reality condition implies that
$\varphi_{\mathbf{k}}^{(1)}=\varphi_{-\mathbf{k}}^{(1)}$ and
$\varphi_{-\mathbf{k}}^{(2)}=-\varphi_{\mathbf{k}}^{(2)}$. By introducing a
new variable $Q_{\mathbf{k}}$,

$Q_{\mathbf{k}}(x_{0})=\begin{cases}\varphi_{\mathbf{k}}^{(1)}(x_{0}),&\text{if
$\mathbf{k}\in\mathscr{L}_{+}$},\vspace{2mm}\\\
\varphi_{-\mathbf{k}}^{(2)}(x_{0}),&\text{if
$\mathbf{k}\in\mathscr{L}_{-}$},\end{cases}$ (7)

associated to the real variables $\varphi_{\mathbf{k}}^{(1)}$ and
$\varphi_{\mathbf{k}}^{(2)}$, we can rewrite the field modes
$\varphi_{\mathbf{k}}$ as

$\varphi_{\mathbf{k}}(x_{0})=\frac{1}{\sqrt{2}}\left[Q_{\mathbf{k}}(x_{0})+iQ_{-\mathbf{k}}(x_{0})\right].$
(8)

Likewise, decomposition of the momentum $\pi_{\mathbf{k}}$ as

$\pi_{\mathbf{k}}(x_{0})=\frac{1}{\sqrt{2}}\left[\pi_{\mathbf{k}}^{(1)}(x_{0})+i\pi_{\mathbf{k}}^{(2)}(x_{0})\right],$
(9)

in terms of the real variables $\pi_{\mathbf{k}}^{(1)}$ and
$\pi_{\mathbf{k}}^{(2)}$, and introducing the new variable $P_{\mathbf{k}}$
(conjugate to $Q_{\mathbf{k}}$ above), as

$P_{\mathbf{k}}(x_{0})=\begin{cases}\pi_{\mathbf{k}}^{(1)}(x_{0}),&\text{if
$\mathbf{k}\in\mathscr{L}_{+}$},\vspace{2mm}\\\
\pi_{-\mathbf{k}}^{(2)}(x_{0}),&\text{if
$\mathbf{k}\in\mathscr{L}_{-}$}\,,\end{cases}$ (10)

yields,

$\pi_{\mathbf{k}}(x_{0})=\frac{1}{\sqrt{2}}\left[P_{\mathbf{k}}(x_{0})+iP_{-\mathbf{k}}(x_{0})\right].$
(11)

The conjugate variables $(Q_{\mathbf{k}},P_{\mathbf{k}})$ satisfy the Poisson
bracket,
$\\{Q_{\mathbf{k}},P_{\mathbf{k}^{\prime}}\\}=\delta_{\mathbf{k},\mathbf{k}^{\prime}}$.

Using the variables (7) and (10), the Hamiltonian of the scalar test field,
$\varphi$, can be written as the sum of the Hamiltonians,
$H_{\mathbf{k}}\left(x_{0}\right)$, of the decoupled harmonic oscillators,
each given in terms of $(Q_{\mathbf{k}},P_{\mathbf{k}})$:

$H_{\varphi}(x_{0})\coloneqq\sum_{\mathbf{k}\in\mathscr{L}}H_{\mathbf{k}}(x_{0})=\frac{N_{x_{0}}(x_{0})}{2|a_{1}a_{2}a_{3}|}\sum_{\mathbf{k}\in\mathscr{L}}\Big{(}P_{\mathbf{k}}^{2}+\omega_{k}^{2}(x_{0})Q_{\mathbf{k}}^{2}\Big{)},$
(12)

where, $\omega_{k}\left(x_{0}\right)$ is a time-dependent frequency, defined
by

$\omega_{k}^{2}\left(x_{0}\right)\coloneqq|a_{1}a_{2}a_{3}|^{2}\left[\sum_{i=1}^{3}\left(\frac{k_{i}}{a_{i}}\right)^{2}+m^{2}\right].$
(13)

Note that, $Q_{\mathbf{k}}$ is the field amplitude for the mode characterized
by $\mathbf{k}$.

To quantize the field, we follow the Schrödinger representation. While the
background spacetime is left as classical, quantization of $Q_{\mathbf{k}}$,
for a fixed mode $\mathbf{k}$, resembles that of a quantum harmonic oscillator
with the Hilbert space ${\cal
H}_{Q}^{(\mathbf{k})}=L^{2}(\mathbb{R},dQ_{\mathbf{k}})$, where
$(Q_{\mathbf{k}},P_{\mathbf{k}})$ are promoted to operators on ${\cal
H}_{Q}^{(\mathbf{k})}$ as

$\displaystyle\hat{Q}_{\mathbf{k}}\psi(Q_{\mathbf{k}})=Q_{\mathbf{k}}\psi(Q_{\mathbf{k}})\quad\text{and}\quad\hat{P}_{\mathbf{k}}\psi(Q_{\mathbf{k}})=-i\hbar\big{(}\partial/\partial
Q_{\mathbf{k}}\big{)}\psi(Q_{\mathbf{k}}).$ (14)

Then, the Hamiltonian operator $\hat{H}_{\mathbf{k}}$ generates the time
evolution of the state $\psi(Q_{\mathbf{k}})$ via the Schrödinger equation

$i\hbar\partial_{x_{0}}\psi(x_{0},Q_{\mathbf{k}})=\frac{\ell^{3}N_{x_{0}}}{2V}\Big{(}\hat{P}_{\mathbf{k}}^{2}+\omega_{k}^{2}\hat{Q}_{\mathbf{k}}^{2}\Big{)}\psi(x_{0},Q_{\mathbf{k}}),$
(15)

where, $V=\ell^{3}\left|a_{1}a_{2}a_{3}\right|$ denotes the physical volume of
the universe.

By setting $x_{0}=\phi$ in Eq. (15), as an internal time parameter, the
evolution of the state $\psi(Q_{\mathbf{k}})$, with respect to $\phi$, on a
Bianchi-I background with components
$\big{(}\tilde{N}_{\phi},\tilde{a}_{i}(\phi)\big{)}$, reads

$i\hbar\partial_{\phi}\psi\left(\phi,Q_{\mathbf{k}}\right)=\frac{\tilde{N}_{\phi}}{2\left|\tilde{a}_{1}\tilde{a}_{2}\tilde{a}_{3}\right|}\Big{[}\hat{P}_{\mathbf{k}}^{2}+\tilde{\omega}_{k}^{2}(\phi)\hat{Q}_{\mathbf{k}}^{2}\Big{]}\psi\left(\phi,Q_{\mathbf{k}}\right),$
(16)

where,

$\tilde{\omega}_{k}^{2}(\phi)\,=\,\left(\sum_{i}^{3}\frac{\tilde{k}_{i}^{2}}{\tilde{a}_{i}^{2}}+\tilde{m}^{2}\right)\left(\tilde{a}_{1}\tilde{a}_{2}\tilde{a}_{3}\right)^{2}.$
(17)

Clearly, one gets an isotropic background for the field, by
$\tilde{a}_{1}(\phi)=\tilde{a}_{2}(\phi)=\tilde{a}_{3}(\phi)\equiv\tilde{a}(\phi)$.

###  II.2 Quantization of the background

In our model herein this paper, we will assume that the field, $\varphi$,
propagates on an isotropic FLRW spacetime in a super-Planckian regime, so that
this isotropic background has to be quantized. However, the reason of
constructing a general formalism of the field on an anisotropic background in
the previous subsection, is for the purpose of comparison, when an effective
dressed spacetime emerges from the isotropic quantum background. We will show
that, the emergent effective spacetime can have the same structure of an
anisotropic Bianchi-I geometry for the field propagation.

Let us assume a harmonic time gauge, $x_{0}=\tau$, and set the isotropic
components, $a_{1}=a_{2}=a_{3}=a(\tau)$, in Eq. (1). Then,
$N_{\tau}=a^{3}(\tau)$, and the Hamiltonian (12) becomes

$H_{\varphi}^{{\rm(iso)}}=\sum_{\mathbf{k}}H_{\tau,\mathbf{k}}\coloneqq\frac{1}{2}\sum_{\mathbf{k}}\Big{(}P_{\mathbf{k}}^{2}+\omega_{\tau,k}^{2}\,Q_{\mathbf{k}}^{2}\Big{)},$
(18)

where,

$\omega_{\tau,k}^{2}(\tau)=k^{2}a^{4}(\tau)+m^{2}a^{6}(\tau).$ (19)

We note that, the massless scalar field, $\phi(\tau)$, still serves as an
internal time parameter.

In quantum theory, we will quantize not only the test field, but also the
background geometry. We will assume that the backreaction of the quantum field
on the background quantized spacetime is negligible. This yields an evolution
for the wave function, $\psi(Q_{\mathbf{k}})$, of the test field, with respect
to the internal time $\phi$. Let ${\cal H}_{{\rm kin}}^{o}={\cal H}_{{\rm
grav}}\otimes{\cal H}_{\phi}$ denotes the background Hilbert space, which
consists of the Hilbert space of the gravity sector and that of the scalar
clock variable $\phi$; the matter sector is quantized according to the
Schrödinger representation, ${\cal H}_{\phi}=L^{2}(\mathbb{R},d\phi)$.
Likewise, the Hilbert space of the field modes reads ${\cal
H}_{\varphi}^{(\mathbf{k})}=L^{2}(\mathbb{R},dQ_{\mathbf{k}})$, as before.
Subsequently, the full kinematical Hilbert space of the system, for a single
mode $\mathbf{k}$, is given by ${\cal H}_{{\rm kin}}^{(\mathbf{k})}={\cal
H}_{{\rm kin}}^{o}\otimes{\cal H}_{\varphi}^{(\mathbf{k})}$.

In LQC, the background Hamiltonian constraint operator,

$\displaystyle\hat{{\cal C}}_{o}=\hat{{\cal C}}_{{\rm grav}}+\hat{{\cal
C}}_{\phi},$ (20)

is well-defined on ${\cal H}_{{\rm kin}}^{o}$; the physical states
$\Psi_{o}(\phi,\nu)\in{\cal H}_{{\rm kin}}^{o}$ are those lying on the kernel
of $\hat{{\cal C}}_{o}$, and are solutions to the self-adjoint constraint
equation Ashtekar et al. (2006b)

$N_{\tau}\hat{{\cal
C}}_{o}\Psi_{o}(\nu,\phi)=-\frac{\hbar^{2}}{2\ell^{3}}(\partial_{\phi}^{2}+\Theta)\Psi_{o}(\nu,\phi)=0.$
(21)

The quantum number $\nu$ is the eigenvalue of the background volume operator,
$\hat{V}_{o}=\widehat{\ell^{3}a^{3}}$, which acts on the states
$\Psi_{o}(\phi,\nu)\in{\cal H}_{{\rm kin}}^{o}$ as

$\hat{V}_{o}\,\Psi_{o}(\nu,\phi)=2\pi\gamma\ell_{{\rm
Pl}}|\nu|\,\Psi_{o}(\nu,\phi).$ (22)

Moreover, $\Theta$ is a difference operator that acts on $\Psi_{o}(\nu)$,
involving only the volume sector $\nu$. Taking only the positive frequency
solutions to Eq. (21), we get a Schrödinger equation for the background as

$-i\hbar\partial_{\phi}\Psi_{o}(\nu,\phi)=\hbar\sqrt{\Theta}\Psi_{o}(\nu,\phi)\eqqcolon\hat{H}_{o}\Psi_{o}(\nu,\phi).$
(23)

The solutions yield a physical Hilbert space, ${\cal H}_{{\rm phys}}^{o}$,
equipped by the inner product

$\langle\Psi_{o}|\Psi_{o}^{\prime}\rangle=\sum_{\nu}\Psi_{o}^{\ast}(\nu,\phi_{0})\Psi_{o}^{\prime}(\nu,\phi_{0}),$
(24)

for an “instant” $\phi_{0}$, of the internal time.

For a composite state $\Psi(\nu,Q_{\mathbf{k}},\phi)\in{\cal H}_{{\rm
kin}}^{(\mathbf{k})}$ of the geometry-test field system, the action of the
total quantum Hamiltonian constraint, $\hat{{\cal C}}_{\tau,\mathbf{k}}$, is
written as Ashtekar et al. (2009)

$\hat{{\cal
C}}_{\tau,\mathbf{k}}\Psi(\nu,Q_{\mathbf{k}},\phi)=\big{(}N_{\tau}\hat{{\cal
C}}_{o}+\hat{H}_{\tau,\mathbf{k}}\big{)}\Psi(\nu,Q_{\mathbf{k}},\phi)=0,$ (25)

where, $\hat{H}_{\tau,\mathbf{k}}$ is the Hamiltonian operator of the
$\mathbf{k}$th field mode,

$\hat{H}_{\tau,\mathbf{k}}=\frac{1}{2}\left[\hat{P}_{\mathbf{k}}^{2}+\left(k^{2}\hat{a}^{4}+m^{2}\hat{a}^{6}\right)\hat{Q}_{\mathbf{k}}^{2}\right].$
(26)

By replacing the expression (21) into the constraint equation (25), we obtain

$\displaystyle-i\hbar\partial_{\phi}\Psi(\nu,Q_{\mathbf{k}},\phi)=\left[\hat{H}_{o}^{2}-2\ell^{3}\hat{H}_{\tau,\mathbf{k}}\right]^{\frac{1}{2}}\Psi(\nu,Q_{\mathbf{k}},\phi),$
(27)

which represents a quantum evolution of $\Psi(\nu,Q_{\mathbf{k}},\phi)$ with
respect to the internal time $\phi$. In a test field approximation, when the
backreaction effect is omitted, the expression under the square root can be
expanded, up to the first order terms, as Ashtekar et al. (2009)

$-i\hbar\partial_{\phi}\Psi(\nu,Q_{\mathbf{k}},\phi)\,\approx\,\big{(}\hat{H}_{o}-\hat{H}_{\phi,\mathbf{k}}\big{)}\Psi(\nu,Q_{\mathbf{k}},\phi).$
(28)

In expanding the right-hand-side of Eq. (27) to derive the equation above, we
regarded $\hat{H}_{o}$ as the Hamiltonian of the heavy degree of freedom,
whereas $\hat{H}_{\phi,\mathbf{k}}$, defined by

$\hat{H}_{\phi,\mathbf{k}}\,\coloneqq\,\ell^{3}\hat{H}_{o}^{-\frac{1}{2}}\hat{H}_{\tau,\mathbf{k}}\hat{H}_{o}^{-\frac{1}{2}},$
(29)

was considered as the Hamiltonian of the light degree of freedom (i.e, a
perturbation term). In this approximation, it is suitable to separate the
total state of the system as

$\Psi(\nu,Q_{\mathbf{k}},\phi)=\Psi_{o}(\nu,\phi)\otimes\psi(Q_{\mathbf{k}},\phi).$
(30)

To explore the quantum evolution of a pure test field state,
$\psi(Q_{\mathbf{k}},\phi)$, on a time-dependent background, it is more
convenient to employ an interaction picture. Thus, we introduce

$\Psi_{{\rm
int}}(\nu,Q_{\mathbf{k}},\phi)=e^{-(i\hat{H}_{o}/\hbar)(\phi-\phi_{0})}\Psi(\nu,Q_{\mathbf{k}},\phi),$
(31)

In this picture, the geometry evolves by $\hat{H}_{o}$, through Eq. (23), for
any $\Psi_{o}\in{\cal H}_{{\rm kin}}^{o}$ in the Heisenberg picture,

$\Psi_{o}(\nu,\phi)=e^{(i\hat{H}_{o}/\hbar)(\phi-\phi_{0})}\Psi_{o}(\nu,\phi_{0}).$
(32)

Plugging this into Eq. (31), we get

$\Psi_{{\rm
int}}(\nu,Q_{\mathbf{k}},\phi)=\Psi_{o}(\nu,\phi_{0})\otimes\psi(Q_{\mathbf{k}},\phi).$
(33)

Thereby, the geometrical sector in the composite state $\Psi_{{\rm
int}}(\nu,Q_{\mathbf{k}},\phi)$ becomes frozen in the instant of time
$\phi_{0}$, so that $\Psi_{{\rm int}}$ represents a time-dependent test field
state, $\psi$, solely.

By replacing (33) into the evolution equation (28), and tracing out the
geometrical state, $\Psi_{o}(\nu,\phi_{0})$, a quantum evolution for
$\psi(Q_{\mathbf{k}},\phi)$ is obtained as

$\displaystyle i\hbar\partial_{\phi}\psi=$
$\displaystyle\frac{1}{2}\left[\langle\hat{H}_{o}^{-1}\rangle\hat{P}_{\mathbf{k}}^{2}+\left(k^{2}\big{\langle}\hat{H}_{o}^{-\frac{1}{2}}\hat{a}^{4}(\phi)\hat{H}_{o}^{-\frac{1}{2}}\big{\rangle}+m^{2}\big{\langle}\hat{H}_{o}^{-\frac{1}{2}}\hat{a}^{6}(\phi)\hat{H}_{o}^{-\frac{1}{2}}\big{\rangle}\right)\hat{Q}_{\mathbf{k}}^{2}\right]\psi,$
(34)

where, $\left\langle\cdot\right\rangle$ denotes the expectation value with
respect to $\Psi_{o}(\nu,\phi_{0})$. It is clear that, the use of the
interaction picture in Eq. (34) provided a the Heisenberg description for the
quantum geometrical elements; that is, the geometry state,
$\Psi_{o}(\nu,\phi_{0})$, is fixed at time $\phi=\phi_{0}$, while the
geometrical operator, $\hat{a}(\phi)=\hat{V}_{o}^{1/3}(\phi)/\ell$, evolves in
time as

$\hat{a}(\phi)=e^{-(i\hat{H}_{o}/\hbar)(\phi-\phi_{0})}\,\hat{a}\,e^{(i\hat{H}_{o}/\hbar)(\phi-\phi_{0})}.$
(35)

Therefore, Eq. (34) represents a $\phi$-evolution of the field,
$\psi(Q_{\mathbf{k}},\phi)$, on a time-dependent (classical) background,
similar to the one we had in classical spacetime (cf. Eq. (16)).

###  II.3 Emergence of anisotropic dressed spacetimes

Eq. (34) can be interpreted as an evolution equation for the field modes on a
(effective) classical spacetime, whose components are generated by the
expectation values of the original isotropic quantum geometry operators with
respect to the unperturbed state $\Psi_{o}$. To explore the properties of such
effective spacetime, we can compare Eq. (34), for evolution of the state
$\psi$, with the corresponding equation (16), for the same state $\psi$, on an
anisotropic classical background. This comparison yields a set of relations
between parameters of the Bianchi I geometry,
$(\tilde{N}_{\phi},\tilde{a}_{i},\tilde{k}_{i},\tilde{m})$, and those of the
isotropic quantum geometry, $(\hat{a},\hat{H}_{o},k,m)$, as

$\displaystyle\tilde{N}_{\phi}$ $\displaystyle=$
$\displaystyle\ell^{3}\left|\tilde{a}_{1}\tilde{a}_{2}\tilde{a}_{3}\right|\big{\langle}\hat{H}_{o}^{-1}\big{\rangle},$
(36a)
$\displaystyle\sum_{i=1}^{3}\frac{\tilde{k}_{i}^{2}}{\tilde{a}_{i}^{2}}\,\tilde{N}_{\phi}\,\left|\tilde{a}_{1}\tilde{a}_{2}\tilde{a}_{3}\right|$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{3}k_{i}^{2}\ell^{3}\big{\langle}\hat{H}_{o}^{-1/2}\hat{a}^{4}\hat{H}_{o}^{-1/2}\big{\rangle},$
(36b)
$\displaystyle\tilde{N}_{\phi}\,\tilde{m}^{2}\,\left|\tilde{a}_{1}\tilde{a}_{2}\tilde{a}_{3}\right|$
$\displaystyle=$
$\displaystyle\ell^{3}m^{2}\big{\langle}\hat{H}_{o}^{-1/2}\hat{a}^{6}\hat{H}_{o}^{-1/2}\big{\rangle}.$
(36c)

Note that, $\hat{a}\equiv\hat{a}(\phi)$ and
$\langle\hat{H}_{o}^{-1}\rangle=(\tilde{p}_{\phi})^{-1}$. Eqs. (36) provide an
underdetermined system of five equations with eight unknowns
$(\tilde{N}_{\phi},\tilde{a}_{i},\tilde{k}_{i},\tilde{m})$. Thus, to be able
to solve this system, we need to impose some arbitrary conditions on these
parameters to reduce the number of unknowns to five. Different classes of
solutions, by imposing various conditions on the variables, and their physical
consequences were discussed in Rastgoo et al. (2020). As an example, we will
present two classes of such solutions; one is produced by a massless test
field, $m=\tilde{m}=0$, and the other is provided by the dressed massive
modes, $\tilde{m},m\neq 0$.

For a massless test field, $m=0$, we will immediately obtain $\tilde{m}=0$. In
this case, we will have four equations for the seven unknown parameters
$(\tilde{N}_{\phi},\tilde{a}_{i},\tilde{k}_{i})$. Therefore, we will still
need three more conditions to be able to solve the system (36a) and (36b). The
simplest choice is $k_{i}^{2}=\tilde{k}_{i}^{2}$ (for each $i$), so that
$\tilde{a}_{1}^{2}=\tilde{a}_{2}^{2}=\tilde{a}_{3}^{2}=\tilde{a}^{2}$. Then,
we obtain

$\displaystyle\tilde{a}^{4}$ $\displaystyle=$
$\displaystyle\frac{\big{\langle}\hat{H}_{o}^{-1/2}\hat{a}^{4}(\phi)\hat{H}_{o}^{-1/2}\big{\rangle}}{\big{\langle}\hat{H}_{o}^{-1}\big{\rangle}},\vspace{1mm}$
(37a) $\displaystyle\tilde{N}_{\phi}$ $\displaystyle=$
$\displaystyle\ell^{3}\big{\langle}\hat{H}_{o}^{-1}\big{\rangle}^{\frac{1}{4}}\big{\langle}\hat{H}_{o}^{-1/2}\hat{a}^{4}(\phi)\hat{H}_{o}^{-1/2}\big{\rangle}^{\frac{3}{4}}=\ell^{3}\tilde{p}_{\phi}^{-1}\tilde{a}^{3}\,\equiv\,\bar{N}_{\phi}.$
(37b)

If $\,m,\tilde{m}\neq 0$ and $\tilde{m}\neq m$, we will have different ranges
of solutions [cf. Rastgoo et al. (2020)]. However, for our purpose in this
paper, we will consider only a specific solution by imposing the condition
$\tilde{k}_{i}=\alpha_{ii}k_{i}$ (with $i=1,2,3$). This yields

$\displaystyle\tilde{N}_{\phi}$ $\displaystyle=$
$\displaystyle\frac{\bar{N}_{\phi}}{\lambda}\,,$ (38a)
$\displaystyle\tilde{a}_{i}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{ii}}{\lambda}\,\tilde{a},$ (38b)
$\displaystyle\tilde{m}^{4}$ $\displaystyle=$ $\displaystyle
m^{4}\lambda^{4}\,\frac{\big{\langle}\hat{H}_{o}^{-1/2}\,\hat{a}^{6}(\phi)\,\hat{H}_{o}^{-1/2}\big{\rangle}^{2}\big{\langle}\hat{H}_{o}^{-1}\big{\rangle}}{\big{\langle}\hat{H}_{o}^{-1/2}\hat{a}^{4}(\phi)\hat{H}_{o}^{-1/2}\big{\rangle}^{3}},$
(38c)

where, $\lambda\equiv(\alpha_{11}\alpha_{22}\alpha_{33})^{1/2}$ is a constant.
Note that, as a special subcase, when $\alpha_{ii}=1$ (so $\lambda=1$), the
wave vector becomes undressed, $\tilde{k}_{i}=k_{i}$, and an isotropic dressed
scale factor, $\tilde{a}_{1}=\tilde{a}_{2}=\tilde{a}_{3}=\tilde{a}$, identical
to Eq. (37a), is obtained. But here, differently from the isotropic case above
for $m=0$, a nonzero dressed mass is obtained as

$\displaystyle\bar{m}^{2}$ $\displaystyle=$ $\displaystyle
m^{2}\,\frac{\big{\langle}\hat{H}_{o}^{-1/2}\hat{a}^{6}(\phi)\hat{H}_{o}^{-1/2}\big{\rangle}\big{\langle}\hat{H}_{o}^{-1}\big{\rangle}^{\frac{1}{2}}}{\big{\langle}\hat{H}_{o}^{-1/2}\hat{a}^{4}(\phi)\hat{H}_{o}^{-1/2}\big{\rangle}^{\frac{3}{2}}}=\frac{\tilde{m}^{2}}{\lambda^{2}}.$
(39)

This solution represents an isotropic dressed spacetime, with the scale factor
$\bar{a}(\phi)$, over which a massive mode with the mass $\bar{m}$, and an
undressed wave-vector $\mathbf{k}=(k_{1},k_{2},k_{3})$ propagates.

##  III QFT on the dressed spacetime

So far we have seen that the massive field modes propagating on an isotropic
quantum geometry can explore a classical, anisotropic dressed background,
$\tilde{g}_{ab}$ (see e.g., solutions (38)), which is the solution of Eqs.
(36). In this section, we will study the QFT and the issue of the
gravitational particle productions on such an emergent anisotropic spacetime.

###  III.1 Field equation on the dressed anisotropic background

We suppose that the scalar field, $\varphi$, propagates on the dressed Bianchi
I background

$\tilde{g}_{ab}dx^{a}dx^{b}=-\tilde{N}_{\phi}^{2}\left(\phi\right)d\phi^{2}+\sum_{i=1}^{3}\tilde{a}_{i}^{2}\left(\phi\right)\left(dx^{i}\right)^{2},$
(40)

whose components are given as the solutions to Eqs. (36). For conveniences, we
introduce the new variables

$\tilde{c}(\phi)\coloneqq(\tilde{a}_{1}\tilde{a}_{2}\tilde{a}_{3})^{\frac{2}{3}}=(\tilde{c}_{1}\tilde{c}_{2}\tilde{c}_{3})^{\frac{1}{3}}\quad{\rm
and}\quad\tilde{c}_{i}(\phi)\coloneqq\tilde{a}_{i}^{2}(\phi),$ (41)

and consider a conformal time parameter, $\tilde{\eta}$, defined by

$\tilde{N}_{\phi}d\phi=\tilde{N}_{\tilde{\eta}}d\tilde{\eta}=\tilde{c}^{1/2}d\tilde{\eta}\quad\Rightarrow\quad
d\tilde{\eta}=\ell^{3}\langle\hat{H}_{o}^{-1}\rangle\tilde{c}(\phi)d\phi.$
(42)

Let’s now take an auxiliary field,
$\chi_{\mathbf{k}}\equiv\sqrt{\tilde{c}(\tilde{\eta})}\,\varphi_{\mathbf{k}}$.
Then, in terms of the above variables, the equation of motion reads

$\chi_{\mathbf{k}}^{\prime\prime}+\left[\tilde{\omega}_{\tilde{\eta},k}^{2}(\tilde{\eta})-\frac{\tilde{c}^{\prime\prime}}{2\tilde{c}}+\frac{\tilde{c}^{\prime
2}}{4\tilde{c}^{2}}\right]\chi_{\mathbf{k}}=0,$ (43)

where, a prime stands for a differentiation with respect to $\tilde{\eta}$,
and $\tilde{\omega}_{\tilde{\eta},k}$ is given by

$\tilde{\omega}_{\tilde{\eta},k}^{2}(\tilde{\eta})=\tilde{c}^{-2}\left[k^{2}\frac{\big{\langle}\hat{H}_{o}^{-\frac{1}{2}}\hat{a}^{4}\hat{H}_{o}^{-\frac{1}{2}}\big{\rangle}}{\langle\hat{H}_{o}^{-1}\rangle}+m^{2}\frac{\big{\langle}\hat{H}_{o}^{-\frac{1}{2}}\hat{a}^{6}\hat{H}_{o}^{-\frac{1}{2}}\big{\rangle}}{\langle\hat{H}_{o}^{-1}\rangle}\right].$
(44)

Notice that, in simplifying equations above, we have used the relations
between components according to Eqs. (36).

The frequency (44) is specified only when the solutions for
$\tilde{c}(\tilde{\eta})$ are determined. It turns out that, only two classes
of solutions for $\tilde{c}$ exist, which depend on the conditions on the
field mass:

  1. i)

For massive field with undressed mass, $\tilde{m}=m\neq 0$, the relation for
$\tilde{c}(\tilde{\eta})$ has the form

$\tilde{c}(\tilde{\eta})=\frac{\big{\langle}\hat{H}_{o}^{-\frac{1}{2}}\hat{a}^{6}\hat{H}_{o}^{-\frac{1}{2}}\big{\rangle}^{\frac{1}{3}}}{\langle\hat{H}_{o}^{-1}\rangle^{\frac{1}{3}}}.$
(45)

  2. ii)

For a massive field with the dressed mass, $\tilde{m}\neq m$, or a massless
field, $\tilde{c}(\tilde{\eta})$ has the solutions of the form

$\tilde{c}(\tilde{\eta})=\xi^{2}\frac{\big{\langle}\hat{H}_{o}^{-\frac{1}{2}}\hat{a}^{4}\hat{H}_{o}^{-\frac{1}{2}}\big{\rangle}^{\frac{1}{2}}}{\langle\hat{H}_{o}^{-1}\rangle^{\frac{1}{2}}}=\xi^{2}\tilde{a}^{2}(\tilde{\eta}),$
(46)

where, $\xi$ is a parameter which distinguishes the anisotropic solutions from
the isotropic one, and depends on the conditions imposed to the additional
unknown variables $\tilde{a}_{i}$’s and $\tilde{k}_{i}$’s of the system (36).
The case $\xi^{2}=1$ associates with the isotropic solution, whereas the case
$\xi\neq 1$ denotes the anisotropic solutions [cf. Eqs. (37) and (39) for
different classes of the solutions].

###  III.2 The adiabatic condition and the vacuum state

Any solution $\chi_{\mathbf{k}}(\tilde{\eta})$ to the Klein-Gordon (43) can be
expanded as

$\chi_{\mathbf{k}}(\tilde{\eta})=\frac{1}{\sqrt{2}}\left[a_{\mathbf{k}}u_{k}^{\ast}(\tilde{\eta})+a_{-\mathbf{k}}^{\ast}u_{k}(\tilde{\eta})\right],$
(47)

where, $a_{\mathbf{k}}$ and $a_{\mathbf{k}}^{\ast}$ are constants of
integration, and $u_{k}(\tilde{\eta})$ are the general mode solutions of Eq.
(43),

$u_{k}^{\prime\prime}+\Big{(}\tilde{\omega}_{\tilde{\eta},k}^{2}(\tilde{\eta})-{\cal
Q}(\tilde{\eta})\Big{)}u_{k}=0,$ (48)

with, ${\cal Q}\equiv\tilde{c}^{\prime\prime}/2\tilde{c}-\tilde{c}^{\prime
2}/4\tilde{c}^{2}$. These modes, together with their complex conjugates,
$u_{k}^{\ast}(\tilde{\eta})$, form a complete set of orthonormal basis, under
the scalar product

$W(u_{k}^{\ast},u_{k})\coloneqq
u_{k}u_{k}^{\ast\prime}-u_{k}^{\ast}u_{k}^{\prime}=2i.$ (49)

In quantum theory, $a_{\mathbf{k}}$ and $a_{\mathbf{k}}^{\ast}$ become
annihilation and creation operators, $\hat{a}_{\mathbf{k}}$ and
$\hat{a}_{\mathbf{k}}^{\dagger}$, satisfying the commutation relation

$\big{[}\hat{a}_{\mathbf{k}},\hat{a}_{\mathbf{k}^{\prime}}^{\dagger}\big{]}=\hbar\ell^{3}\delta_{\mathbf{k},\mathbf{k}^{\prime}},$
(50)

where, all other commutation relations vanish. For the _positive frequency_
solutions, a choice of the basis, $u_{k}(\tilde{\eta})$, determines a vacuum
state, $|0\rangle$, which is defined as the eigenstate of the annihilation
operators obeying the equation $\hat{a}_{\mathbf{k}}|0\rangle=0$. Different
families of the mode solutions distinguish different vacuum states.
Accordingly, a Fock space is generated by repeatedly acting the operator
$\hat{a}_{\mathbf{k}}^{\dagger}$ on a chosen vacuum state. Thereby, the
evolution of the quantum fields on the quantum-gravity-induced dressed
spacetime is determined. Now, the general (auxiliary) field solutions,
$\chi(\tilde{\eta},\mathbf{x})$, can be expanded as

$\chi(\tilde{\eta},\mathbf{x})=\frac{1}{\ell^{3}}\sum_{\mathbf{k}}\chi_{\mathbf{k}}(\tilde{\eta},\mathbf{x}),$
(51)

where,

$\chi_{\mathbf{k}}=\frac{1}{\sqrt{2}}\left[a_{\mathbf{k}}u_{k}^{*}(\tilde{\eta})e^{i\mathbf{k}\cdot\mathbf{x}}+a_{\mathbf{k}}^{\ast}u_{k}(\tilde{\eta})e^{-i\mathbf{k}\cdot\mathbf{x}}\right].$
(52)

To have a well-defined vacuum state for the mode solutions, the (effective)
frequency

$\Omega_{k}^{2}(\tilde{\eta})=\tilde{\omega}_{\tilde{\eta},k}^{2}(\tilde{\eta})-{\cal
Q}(\tilde{\eta}),$ (53)

should be positive. Thus, the wave-number must satisfy $k^{2}\geq
k_{\ast}^{2}$, where

$k_{\ast}^{2}(\tilde{\eta})\coloneqq\frac{{\cal
Q}(\tilde{\eta})\tilde{c}^{2}(\tilde{\eta})\langle\hat{H}_{o}^{-1}\rangle}{\big{\langle}\hat{H}_{o}^{-\frac{1}{2}}\hat{a}^{4}\hat{H}_{o}^{-\frac{1}{2}}\big{\rangle}}-m^{2}\,\frac{\big{\langle}\hat{H}_{o}^{-\frac{1}{2}}\hat{a}^{6}\hat{H}_{o}^{-\frac{1}{2}}\big{\rangle}}{\big{\langle}\hat{H}_{o}^{-\frac{1}{2}}\hat{a}^{4}\hat{H}_{o}^{-\frac{1}{2}}\big{\rangle}},$
(54)

introduces a physical wavelength, $\lambda_{\ast}(\tilde{\eta})\equiv
2\pi\sqrt{\tilde{c}(\tilde{\eta})}/k_{\ast}$, which provides an upper bound
for the wavelengths of the physical mode functions, i.e.,
$\lambda\leq\lambda_{\ast}$. Therefore, modes with short wavelengths (i.e.,
large momenta), $\lambda\ll\lambda_{\ast}$, characterize the vacuum states in
short distances, i.e., the _ultra-violet_ (UV) regimes. This regime contains a
limit of arbitrary slow time, $\tilde{\eta}$, variation of the metric
functions, called the _adiabatic_ regime. We will, henceforth, explore the
mode functions describing the physical vacuum and particles, associated with
this adiabatic regime.

A positive-frequency, adiabatic vacuum mode can be defined by a generalized
WKB approximate solution to the Klein-Gordon equation, as Parker and Fulling
(1974)

$\underline{u}_{k}(\tilde{\eta})=\frac{1}{\sqrt{W_{k}(\tilde{\eta})}}\exp\Big{(}-i\int^{\tilde{\eta}}W_{k}(\eta)d\eta\Big{)}.$
(55)

Substituting this relation in Eq. (48) we obtain a relation for $W_{k}(\eta)$
as

$\frac{W_{k}^{\prime\prime}}{W_{k}}-\frac{W^{\prime}_{k}}{W_{k}^{2}}-\frac{1}{2}\frac{W^{\prime
2}_{k}}{W_{k}^{2}}+2\left(W_{k}^{2}-\Omega_{k}^{2}\right)=0.$ (56)

An appropriate function can be chosen for $W_{k}(\tilde{\eta})$, such as
Chakraborty (1973)

$W_{k}(\tilde{\eta})=\big{[}Y(1+\underline{\epsilon_{2}})(1+\underline{\epsilon_{4}})\big{]}^{\frac{1}{2}},$
(57)

where, $Y\equiv\Omega_{k}^{2}=\tilde{\omega}_{\tilde{\eta},k}^{2}-{\cal Q}$,
and

$\displaystyle\underline{\epsilon_{2}}$
$\displaystyle\coloneqq-Y^{-\frac{3}{4}}\partial_{\tilde{\eta}}\Big{(}Y^{-\frac{1}{2}}\partial_{\tilde{\eta}}Y^{\frac{1}{4}}\Big{)},$
(58) $\displaystyle\underline{\epsilon_{4}}$
$\displaystyle\coloneqq-Y^{-\frac{1}{2}}(1+\underline{\epsilon_{2}})^{-\frac{3}{4}}\partial_{\tilde{\eta}}\Big{\\{}[Y(1+\underline{\epsilon_{2}})]^{-\frac{1}{2}}\partial_{\tilde{\eta}}(1+\underline{\epsilon_{2}})^{\frac{1}{4}}\Big{\\}}.$
(59)

To get the solutions for $\underline{u}_{k}(\tilde{\eta})$, one needs to solve
the equation (56) for $W(\tilde{\eta})$. However, instead of solving (56)
exactly, one way is to generate asymptotic series in orders of time
$\tilde{\eta}$ derivatives of the background dressed metric. Terminating this
series at a given order will specify an adiabatic mode
$\underline{u}_{k}(\tilde{\eta})$ to that order. Up to an order $n$ (being the
power of $\tilde{\eta}$-derivatives of $\tilde{a}_{i}$), denoted ${\cal
O}(\sqrt{\tilde{c}}/k\lambda_{n+\varepsilon})^{n+\varepsilon}$ (with positive
real number $\varepsilon$), the asymptotic adiabatic expansion of $W_{k}$ is
defined to match

$W_{k}(\tilde{\eta})=W_{k}^{(0)}+W_{k}^{(2)}+\cdot\cdot\cdot+W_{k}^{(n)},$
(60)

and, is obtained for higher order estimates by iteration.

When computing the expectation value of the energy-momentum operator of the
scalar field, $\langle\tilde{0}|\hat{T}_{ab}|\tilde{0}\rangle$, with respect
to the adiabatic vacuum, $|\tilde{0}\rangle$, associated to the mode function
(55), one encounters the UV divergences issues. All of these UV divergences
are contained within the terms of adiabatic order equal to and smaller than
four [cf. Fulling et al. (1974) and the appendix of Ref. Rastgoo et al.
(2020)]. Thus, the (adiabatic) regularization of the energy density and
pressure of the scalar field are obtained from subtractions of the divergences
contained within the adiabatic terms up to fourth order; e.g., the
renormalized energy density of the field is given by

$\langle\tilde{0}|\hat{\rho}_{\varphi}|\tilde{0}\rangle_{\rm
ren}=\frac{\hbar}{\ell^{3}\tilde{c}^{2}}\sum_{k}\left(\rho_{k}[u_{k}(\tilde{\eta})]-\underline{\rho_{k}}[\underline{u_{k}}(\tilde{\eta})]\right),$
(61)

where, $\rho_{k}$ is the energy density of each mode, $u_{k}(\tilde{\eta})$,
and $\underline{\rho_{k}}$ is the energy density associated with the mode,
$\underline{u_{k}}$, up to 4th adiabatic order. We will, thus, restrict
ourselves to the fourth order adiabatic states only222Note that, our aim at
the moment is just to avoid the UV divergence terms when regularizing the
energy-momentum of the field. However, to have a more complete mode solution,
in particular when exploring the issues of the particle productions or
backreaction effects, as we will see later, higher order terms in
$W_{k}(\tilde{\eta})$ should be taken into account.. Then, by setting $n=4$,
we define $W_{k}(\tilde{\eta})$ to match the terms in (60) that fall slowly in
$\tilde{\omega}_{\tilde{\eta},k}$, rather than considering the exact mode
solutions of (48). Accordingly, we consider an appropriate asymptotic
condition (up to the order four) at which the field’s exact mode functions,
$u_{k}(\tilde{\eta})$, match the adiabatic functions,
$\underline{u}_{k}(\tilde{\eta})$, as

$\displaystyle|u_{k}(\tilde{\eta}_{b})|=$
$\displaystyle\big{|}\underline{u}_{k}(\tilde{\eta}_{b})\big{|}\Big{(}1+{\cal
O}\big{(}\sqrt{\tilde{c}}/k\lambda_{4+\varepsilon}\big{)}^{4+\varepsilon}\Big{)},$
(62a) $\displaystyle|u_{k}^{\prime}(\tilde{\eta}_{b})|=$
$\displaystyle\big{|}\underline{u}_{k}^{\prime}(\tilde{\eta}_{b})\big{|}\Big{(}1+{\cal
O}\big{(}\sqrt{\tilde{c}}/k\lambda_{4+\varepsilon}\big{)}^{4+\varepsilon}\Big{)},$
(62b)

where, $\tilde{\eta}=\tilde{\eta}_{b}$ is a natural choice for the preferred
instant of time, given at the quantum bounce, in LQC. When a mode function,
$u_{k}(\tilde{\eta})$, satisfies the conditions (62), at some initial time
$\tilde{\eta}_{b}$, it will satisfy it for all times $\tilde{\eta}$.
Therefore, an observable vacuum, $|0\rangle$, associated to
$u_{k}(\tilde{\eta})$, will be of the 4th order, for all times.

Following the above discussion, we thus discard the adiabatic order terms
higher than four in Eq. (57), and write

$W_{k}(\tilde{\eta})=\tilde{\omega}_{\tilde{\eta},k}(1+\epsilon_{2}+\epsilon_{4})^{\frac{1}{2}},$
(63)

where, $\epsilon_{2}$, $\epsilon_{4}$ are defined by

$\displaystyle\epsilon_{2}$
$\displaystyle=\underline{\epsilon_{2}}-\tilde{\omega}_{\tilde{\eta},k}^{-2}\,{\cal
Q}$
$\displaystyle=-\frac{1}{4}Y^{-2}Y^{\prime\prime}+\frac{5}{16}Y^{-3}(Y^{\prime})^{2}-\tilde{\omega}_{\tilde{\eta},k}^{-2}\,{\cal
Q},$ (64) $\displaystyle\epsilon_{4}$
$\displaystyle=\underline{\epsilon_{4}}-\underline{\epsilon_{2}}\,\tilde{\omega}_{\tilde{\eta},k}^{-2}\,{\cal
Q}$
$\displaystyle=-\frac{1}{4}Y^{-1}(1+\underline{\epsilon_{2}})^{-2}\Big{[}\underline{\epsilon_{2}}^{\prime\prime}-\frac{1}{2}Y^{-1}Y^{\prime}\underline{\epsilon_{2}}^{\prime}-\frac{5}{4}(1+\underline{\epsilon_{2}})^{-1}(\underline{\epsilon_{2}}^{\prime})^{2}\Big{]}-\underline{\epsilon_{2}}\,\tilde{\omega}_{\tilde{\eta},k}^{-2}\,{\cal
Q},$ (65)

and, $\underline{\epsilon_{2}}$ is given by

$\underline{\epsilon_{2}}=-\frac{1}{4}Y^{-2}Y^{\prime\prime}+\frac{5}{16}Y^{-3}(Y^{\prime})^{2}.$
(66)

The leading order term of $\underline{\epsilon_{2}}$ is of second order,
whereas the leading order in $\underline{\epsilon_{4}}$ is four. Thus,
$\epsilon_{2}$ contains terms of orders two, four and higher, i.e,

$\epsilon_{2}=\epsilon_{2}^{(2)}+\epsilon_{2}^{(4)}+{\rm
Higher\leavevmode\nobreak\ order\leavevmode\nobreak\ terms},$ (67)

and, $\epsilon_{4}$ contains leading order term four, i.e.,

$\displaystyle\epsilon_{4}$
$\displaystyle=\epsilon_{4}^{(4)}\leavevmode\nobreak\ +\leavevmode\nobreak\
{\rm Higher\leavevmode\nobreak\ order\leavevmode\nobreak\ terms}.$ (68)

It should be noticed that, we have considered only terms until the fourth
order in their expressions [for more details, see Rastgoo et al. (2020)].

##  IV Gravitational particle production

Our aim in this section is to study the gravitational particle production in
the herein quantum gravity regime due to quantum field on the anisotropic,
dressed background.

Once a vacuum state, $|\tilde{0}\rangle$, is specified due to the positive
frequency solutions, $\underline{v}_{k}(\tilde{\eta})$, of Eq. (48), a Fock
space, ${\cal H}_{{\rm F}}$, for the quantum field $\varphi$ is generated. Let
$\\{v_{k}\\}$ and $\\{\underline{v}_{k}\\}$ be two sets of WKB solutions,
given by Eq. (55), in the herein adiabatic regime. These mode functions form
two normalized bases333They satisfy the normalization condition (49) for the
mode functions. for ${\cal H}_{{\rm F}}$, so they can be related to each other
through the time-independent Bogolyubov coefficients, $\alpha_{k}$ and
$\beta_{k}$, as

$\underline{v}_{k}(\tilde{\eta})=\alpha_{k}v_{k}(\tilde{\eta})+\beta_{k}v_{k}^{\ast}(\tilde{\eta}).$
(69)

The Bogolyubov coefficients satisfy the relation
$|\alpha_{k}|^{2}-|\beta_{k}|^{2}=1$ via condition (49). Comparing Eqs. (69)
and (52), it follows that the creation and annihilation operators associated
with two families of mode functions (i.e, those with and without ‘underline’),
are related as

$\hat{a}_{\mathbf{k}}=\alpha_{k}\underline{\hat{a}}_{\mathbf{k}}+\beta_{k}^{\ast}\underline{\hat{a}}_{\mathbf{k}}^{\dagger}.$
(70)

Working in Heisenberg picture, an initial vacuum state of the system, say
$|\tilde{0}\rangle$ connected to the ‘underlined’ modes,
$\underline{v}_{k}(\tilde{\eta})$, is the vacuum state of the system for all
times. Then, the number operator,
$\hat{N}_{\mathbf{k}}=(\hbar\ell^{3})^{-1}\hat{a}_{\mathbf{k}}^{\dagger}\hat{a}_{\mathbf{k}}$,
associated to the particles in $v_{k}$ mode, gives the average number of
particles in the $|\tilde{0}\rangle$ vacuum. Thus, the $v_{k}$-mode related
vacuum state contains

${\cal
N}_{k}\coloneqq\langle\tilde{0}|\hat{N}_{\mathbf{k}}|\tilde{0}\rangle=|\beta_{k}|^{2},$
(71)

particles in the $\underline{v}_{k}$-mode vacuum.

Let us rewrite the mode solution, $\underline{v}_{k}$, in the WKB
approximation (55), as

$\underline{v}_{k}(\tilde{\eta})=\frac{1}{\sqrt{W_{k}(\tilde{\eta})}}\Big{[}\alpha_{k}e_{k}(\tilde{\eta})+\beta_{k}e_{k}^{\ast}(\tilde{\eta})\Big{]},$
(72)

where,

$e_{k}(\tilde{\eta})\coloneqq\exp\Big{(}-i\int^{\tilde{\eta}}d\eta\,W_{k}(\eta)\Big{)}.$
(73)

Taking the time ($\tilde{\eta}$) derivative of
$\underline{v}_{k}(\tilde{\eta})$, we obtain

$\underline{v}_{k}^{\prime}(\tilde{\eta})=-i\sqrt{W_{k}}\Big{[}\alpha_{k}e_{k}(\tilde{\eta})-\beta_{k}e_{k}^{\ast}(\tilde{\eta})\Big{]}.$
(74)

Inverting the Eqs. (72) and (74) yields a relation for $\beta_{k}$ as

$\beta_{k}=\frac{\sqrt{W_{k}}}{2}\left(\underline{v}_{k}-\frac{i}{W_{k}}\underline{v}_{k}^{\prime}\right)e_{k}.$
(75)

Since the initial condition in LQC is fixed at the quantum bounce,
$\tilde{\eta}=\tilde{\eta}_{b}$, by setting $\alpha_{k}=1$ and $\beta_{k}=0$
at $\tilde{\eta}=\tilde{\eta}_{b}$, we assume that no particle is created at
bounce. This yields the following conditions on $\underline{v}_{k}$:

$\underline{v}_{k}(\tilde{\eta}_{b})=1/W_{k}(\tilde{\eta}_{b})\quad\text{and}\quad\underline{v}_{k}^{\prime}(\tilde{\eta}_{b})=-iW_{k}(\tilde{\eta}_{b})\,\underline{v}_{k}(\tilde{\eta}_{b}),$
(76)

where $e_{k}(\tilde{\eta}_{b})=1$. Then, for any time
$\tilde{\eta}>\tilde{\eta}_{b}$, the number of particle production becomes

${\cal
N}_{k}(\tilde{\eta})=\frac{1}{4}\Big{(}W_{k}\,|v_{k}|^{2}+W_{k}^{-1}|v_{k}^{\prime}|^{2}-2\Big{)}.$
(77)

Here, for conveniences, we have dropped the ‘underline’ for the mode functions
after the bounce ($\tilde{\eta}>\tilde{\eta}_{b}$), i.e., we assume
$\underline{v}_{k}\equiv\underline{v}_{k}(\tilde{\eta}_{b})$ and
$v_{k}(\tilde{\eta})\equiv\underline{v}_{k}(\tilde{\eta})$ for
$\tilde{\eta}>\tilde{\eta}_{b}$.

Having the number of particle production per mode, ${\cal
N}_{k}(\tilde{\eta})$, at a given time $\tilde{\eta}>\tilde{\eta}_{b}$, we can
compute the total number density of production, ${\cal N}(\tilde{\eta})$, as
the limit of $\sum_{k}{\cal N}_{k}(\tilde{\eta})$ in a box of volume
$\ell^{3}\to\infty$, divided by the volume of the universe,
$V=\ell^{3}|\tilde{a}_{1}\tilde{a}_{2}\tilde{a}_{3}|=\ell^{3}\tilde{c}^{3/2}$,
as

${\cal N}(\tilde{\eta})=\frac{\ell^{3}}{V}\sum_{k}{\cal
N}_{k}(\tilde{\eta})=\frac{1}{4\tilde{c}^{3/2}}\sum_{k}\Big{(}W_{k}\,|v_{k}|^{2}+W_{k}^{-1}|v_{k}^{\prime}|^{2}-2\Big{)}.$
(78)

The energy density of the created particles, for each mode, reads
$\varrho_{k}(\tilde{\eta})\equiv W_{k}(\tilde{\eta})\,{\cal
N}_{k}(\tilde{\eta})$. Thus, the total energy density is obtained by summing
over all modes, as

$\rho_{{\rm par}}=\frac{1}{\tilde{c}^{2}}\sum_{k}\,\varrho_{k}(\tilde{\eta}),$
(79)

where,

$\displaystyle\varrho_{k}(\tilde{\eta})=\frac{1}{4}\Big{(}|v_{k}^{\prime}|^{2}+W_{k}^{2}\,|v_{k}|^{2}-2W_{k}\Big{)}.$
(80)

At the bounce, $\tilde{\eta}=\tilde{\eta}_{b}$, the energy density of
production is zero, $\varrho_{k}(\tilde{\eta}_{b})=0$, as expected. However,
for $\tilde{\eta}>\tilde{\eta}_{b}$, particles will be produced as the
universe expands. In the following, we will analyze the energy density of
production in the assumed adiabatic regime.

Following the (adiabatic) regularization scheme, the energy density of created
particles is obtained as

$\rho_{\rm
par}^{\rm(ren)}=\frac{1}{\tilde{c}^{2}}\sum_{k}\Big{(}\varrho_{k}[v_{k}(\tilde{\eta})]-\underline{\varrho_{k}}(\tilde{\eta})\Big{)},$
(81)

where,

$\displaystyle\underline{\varrho_{k}}(\tilde{\eta})$
$\displaystyle=\frac{1}{16}\left[\frac{(\tilde{\omega}_{k,\tilde{\eta}}^{\prime})^{2}}{\tilde{\omega}_{k,\tilde{\eta}}^{3}}+\tilde{\omega}_{k,\tilde{\eta}}\big{(}\epsilon_{2}^{(2)}\big{)}^{2}-\frac{(\tilde{\omega}_{k,\tilde{\eta}}^{\prime})^{2}}{2\tilde{\omega}_{k,\tilde{\eta}}^{3}}\epsilon_{2}^{(2)}+\frac{\tilde{\omega}_{k,\tilde{\eta}}^{\prime}}{\tilde{\omega}_{k,\tilde{\eta}}^{2}}\epsilon_{2}^{\prime(3)}\right]$
$\displaystyle=\varrho_{k}^{(0)}+\varrho_{k}^{(2)}+\varrho_{k}^{(4)}.$ (82)

It turns out that, $\underline{\varrho_{k}}$ does not fall off faster than
$k^{-4}$ when $k\rightarrow\infty$. The zeroth adiabatic order term in (82) is
zero, $\varrho_{k}^{(0)}=0$, and the divergences are included in the second
and fourth order terms (for massive modes). For the massless modes, the
divergences are included only in the fourth order term. Therefore, the
renormalized energy density of created particles, Eq. (81), can be obtained by
subtraction of the adiabatic vacuum energy of the particle productions up to
fourth order. When considering the higher order terms (more than 4th order) in
the adiabatic mode functions, $v_{k}(\tilde{\eta})$, associated to the vacuum
state $|0\rangle$, particles are produced and the total energy density of the
created particles is proportional to $1/\tilde{c}^{2}(\tilde{\eta})$. For an
isotropic case (either massive or massless), where $\tilde{c}=\tilde{a}^{2}$,
we get $\rho_{\rm par}^{\rm(ren)}\propto 1/\tilde{a}^{4}(\tilde{\eta})$ which
scales as radiation. This is a similar situation derived in Ref. Graef et al.
(2020). Therefore, unlike the classical FLRW cosmology, even massless modes
contain nonzero particle production due to quantum gravity effects.

In the pre-inflationary scenario considered in Graef et al. (2020), by
assuming a massless field in an isotropic quantum cosmological background,
$\tilde{a}(\tilde{\eta})=\tilde{c}^{1/2}(\tilde{\eta})$, an integration range
were chosen in Eq. (79) by the window of observable modes of the CMB. Therein,
it has been argued that, when taking a UV cutoff at the characteristic
momentum
$k_{b}=k_{\ast}(\tilde{\eta}_{b})=\sqrt{\tilde{a}^{\prime\prime}(\tilde{\eta})/\tilde{a}(\tilde{\eta})}|_{\tilde{\eta}_{b}}\approx
3.21\,m_{\rm Pl}$, and an infra-red (IR) cutoff at
$k_{\ast}(\tilde{\eta})=\sqrt{\tilde{a}^{\prime\prime}(\tilde{\eta})/\tilde{a}(\tilde{\eta})}=\lambda^{-1}_{\ast}$
for all $\tilde{\eta}>\tilde{\eta}_{b}$, being the physical energy of
particles after they reenter the effective horizon, the energy density of
created particles becomes

$\rho_{\rm par}=0.012\,m_{\rm Pl}^{4}/\tilde{a}^{4}(\tilde{\eta}).$ (83)

This gives the energy density of particles produced in the region
$k_{\ast}\leq k\leq k_{b}$. Thus, the main contribution to the particle
productions are the modes whose wavelengths, $\lambda$, hold the range
$\lambda_{b}\leq\lambda\leq\lambda_{\ast}$, i.e., the modes that reenter
$\lambda_{\ast}$ after the bounce, during the pre-inflationary phase, and will
only reexit $\lambda_{\ast}$ again in the slow-roll inflationary phase.

This implies that the energy density of particles produced in the quantum
gravity regime, which scales as relativistic matter, is significant comparing
to the background energy density. The background energy density during the
slow-roll inflation is dominant over the backreaction of particle productions.
However, it should be guaranteed that the production density is still dominant
in the pre-inflationary phase before beginning of the slow-roll phase. By
assuming that, the elapsed e-foldings between the bounce and the onset of
inflation is about $4$-$5$ e-folds, the energy density of the backreaction is
smaller than the background energy density as $\rho_{\rm
par}\lesssim\langle\hat{\rho}_{\varphi}\rangle\approx 2\times 10^{-5}\,m_{\rm
Pl}^{4}$. However, the analysis in Graef et al. (2020) indicates that, the
energy density (83) is 2 orders of magnitude larger than the required upper
bound, $2\times 10^{-5}\leavevmode\nobreak\ m_{\rm Pl}^{4}$, estimated for the
density of particle creation during the pre-inflationary phase. It turns out
that, the backreaction of produced particles cannot be neglected, so that a
more careful analysis of backreacted wave function for the background quantum
geometry is needed.

The above argument indicates that, when starting from Eq. (27), the
backreaction of the field modes on the quantum background is not negligible,
so that the total wave function, $\Psi(\nu,Q_{\mathbf{k}},\phi)$, cannot be
decomposed as $\Psi=\psi_{o}\otimes\psi$. It may even make a further
constraint on expanding the right-hand-side of Eq. (27) to derive the
evolution equation (28). Nevertheless, if we suppose that the approximation
(28) is valid, the presence of the backreaction would lead to a modification
of the total state as $\Psi=\psi_{o}\otimes\psi+\delta\Psi$ [cf. Dapor et al.
(2012); Lewandowski et al. (2017)]. Taking into account such modification, the
dispersion relation of the field, propagating on the emergent effective
geometry, will be modified such that the local Lorentz symmetry becomes
violated. In such a scenario, each mode feels a distinctive background
geometry which depends on that mode; a rainbow dressed background emerges. On
such backgrounds, the standard approach for studying the infinite number of
field modes fails and an alternative procedure should be employed [e.g., see
Martin and Brandenberger (2001)]. For other approaches in canonical quantum
gravity, where the quantum theory of cosmological perturbations and their
backreactions are implemented, see e.g. Bojowald and Ding (2021); Fernández-
Méndez et al. (2013); Gomar et al. (2014, 2015); Bojowald et al. (2008);
Cailleteau et al. (2012a, b).

##  V Conclusion and Discussion

In this paper, the quantum theory of a (inhomogeneous) massive test field,
$\varphi$, propagating on a quantized FLRW geometry is addressed. The
background geometry constitutes of a homogeneous massless scalar field,
$\phi$, as matter source, which plays the role of internal time in quantum
theory. From an effective point of view, due to quantum gravity effects on the
background geometry, quantum modes of the field can experience a dressed
spacetime whose geometry differs from the original FLRW metric. If the
backreaction of the field modes, $Q_{\mathbf{k}}$, on the background is
discarded, within a test field approximation, the full quantum state of the
system can be decomposed as
$\Psi(\nu,Q_{\mathbf{k}},\phi)=\Psi_{o}(\nu,\phi)\otimes\psi(Q_{\mathbf{k}},\phi)$;
$\Psi_{o}$ and $\psi$ are the quantum states of the (unperturbed) background
and the field modes, respectively.

In the interaction picture, an evolution equation emerges for
$\psi(Q_{\mathbf{k}},\phi)$ which resembles the Schrödinger equation for the
same field modes propagating on a time-dependent dressed spacetime, whose
metric components are functions of the quantum fluctuations of the FLRW
geometry. For massive and massless modes, there exists a wide class of
solutions for the effective dressed background metric. The _massless_ modes
can only experience an isotropic and homogeneous dressed background with a
dressed scale factor, $\tilde{a}(\phi)$ [cf. see Ashtekar et al. (2009)]. The
_massive_ modes, however, yield a general class of solutions for the emergent
dressed geometries which resembles the anisotropic Biachi I spacetimes.
Likewise, the scale factors, $(\tilde{a}_{1},\tilde{a}_{2},\tilde{a}_{3})$, of
the consequent dressed Bianchi-I metric are functions of fluctuations of the
isotropic quantum geometry.

Given a dressed anisotropic spacetime, as a solution discussed above, we
reviewed the standard quantum field theory on such background. More precisely,
we investigated the issue of gravitational particle production associated to
the field modes on the dressed Bianchi I geometry in a suitably chosen
adiabatic regime. This led to some backreaction issues in the super-Planckian
regime which may affect the dynamics of the early universe. To have a
regularized energy-momentum operator of the test field, the adiabatic vacuum
state was chosen up to 4th order terms. We computed the energy density of the
particle production within this adiabatic limit. The divergences in the energy
density of the produced particles were regularized within the fourth order
adiabatic terms and the remaining terms change as $\rho_{\rm
par}^{\rm(ren)}\propto 1/(\tilde{a}_{1}\tilde{a}_{2}\tilde{a}_{3})^{4/3}$.
Some phenomenological issues related to such particle production were
discussed. It was demonstrated that the backreaction due to the particle
production in the super-Planckina regime may have significant effects on the
evolution of the universe and may subsequently modify the existing pre-
inflationary scenario of LQC Agullo et al. (2012) [cf. Graef et al. (2020)]

###### Acknowledgements.

The results of this review article are mainly based on the past collaborations
with my colleagues, J. C. Fabris and S. Rastgoo. This paper is based upon work
from European Cooperation in Science and Technology (COST) action CA18108 –
Quantum gravity phenomenology in the multi-messenger approach supported by
COST.

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# Document-level Clinical Entity and Relation Extraction via Knowledge Base-
Guided Generation

Kriti Bhattarai1,2, Inez Y. Oh1, Zachary B. Abrams1, Albert M. Lai1,2
Institute for Informatics, Data Science & Biostatistics, Washington University
School of Medicine
Department of Computer Science, Washington University in St. Louis

###### Abstract

Generative pre-trained transformer (GPT) models have shown promise in clinical
entity and relation extraction tasks because of their precise extraction and
contextual understanding capability. In this work, we further leverage the
Unified Medical Language System (UMLS) knowledge base to accurately identify
medical concepts and improve clinical entity and relation extraction at the
document level. Our framework selects UMLS concepts relevant to the text and
combines them with prompts to guide language models in extracting entities.
Our experiments demonstrate that this initial concept mapping and the
inclusion of these mapped concepts in the prompts improves extraction results
compared to few-shot extraction tasks on generic language models that do not
leverage UMLS. Further, our results show that this approach is more effective
than the standard Retrieval Augmented Generation (RAG) technique, where
retrieved data is compared with prompt embeddings to generate results.
Overall, we find that integrating UMLS concepts with GPT models significantly
improves entity and relation identification, outperforming the baseline and
RAG models. By combining the precise concept mapping capability of knowledge-
based approaches like UMLS with the contextual understanding capability of
GPT, our method highlights the potential of these approaches in specialized
domains like healthcare.

Document-level Clinical Entity and Relation Extraction via Knowledge Base-
Guided Generation

Kriti Bhattarai1,2, Inez Y. Oh1, Zachary B. Abrams1, Albert M. Lai1,2
Institute for Informatics, Data Science & Biostatistics, Washington University
School of Medicine Department of Computer Science, Washington University in
St. Louis

##  1 Introduction

Generative pre-trained transformer (GPT) models have shown significant
potential across various clinical tasks, including information extraction,
summarization, and question-answering (Agrawal et al., 2022; Tang et al.,
2023a; Yang et al., 2022, Singhal et al., 2023). Generative models are able to
generate contextually relevant text given a prompt. However, for real-world
clinical use, in tasks that require high precision, it is equally important to
understand the context and minimize the errors that come from GPT models.
However, accuracy of these models is limited to their training data. While GPT
models are great at capturing nuanced contextual information, they often fall
short in accurately identifying all medical concepts, possibly due to limited
or outdated domain-specific data (Tang et al., 2023b, Singhal et al., 2023).
Knowledge bases store domain-specific data. Medical knowledge bases, such as,
Unified Medical Language System (UMLS) knowledge base (Bodenreider, 2004),
include comprehensive information about medical concepts. Integrating
knowledge bases with language models is an open research area with multiple
works exploring different ways of integrating them with language models, such
as BERT (Devlin et al., 2019). There are limited studies on the integration of
medical knowledge bases, particularly UMLS, with most recent large language
models (LLMs), such as GPT.
To address this limitation, we introduce an approach for clinical entity
extraction that leverages UMLS for knowledge augmentation. While GPT can
identify nuanced contextual information, UMLS includes a comprehensive
repository of domain-specific clinical concepts that GPT may not recognize,
such as brand names for drugs, abbreviations, acronyms, and aliases (Agrawal
et al., 2022).
Our contributions in this paper are summarized as follows:
(1) we introduce a framework to integrate UMLS concepts into the default
generative models to facilitate few-shot information extraction of biomedical
entities and relations.
(2) we explore current state-of-the-art knowledge augmentation techniques,
such as Retrieval Augmented Generation (RAG) aimed at improving extraction,
and
(3) we conduct evaluation of our framework, comparing the performance of
models augmented with UMLS knowledge with and without RAG, and against those
without augmentation.

##  2 Related Work

###  2.1 Few-shot in-context learning

With the introduction of GPT models, there have been several works around few-
shot in-context learning for clinical entity extraction where prompts guide
information extraction in a contextually relevant manner (Agrawal et al.,
2022; Hu et al., 2024; Shyr et al., 2024, Brown et al., 2020). Generative
models can provide nuanced contextual understanding to extract clinical
concepts, but cannot identify all domain-specific terminologies, especially in
the clinical domain (Tang et al., 2023b). While recent language models have
demonstrated improvement over prior language models (Guevara et al., 2024),
there remains room for performance improvement.

###  2.2 Knowledge base-guided models

Previous research has explored the integration of knowledge bases to enhance
information extraction tasks. (Sastre et al., 2020) proposed a Bi-LSTM model
to identify drug-related information and integrate it into knowledge graph
embeddings to evaluate drug identification accuracy. (Gilbert et al., 2024)
addressed how knowledge bases complement language models for medical
information identification tasks. Recently, a RAG model, Almanac, demonstrated
significant performance improvements compared to the standard LLMs across
various metrics (Zakka et al., 2024), further showing the benefits of access
to domain-specific corpora for information extraction.

##  3 Methods

Figure 1: (A) Step-by-step approach to integrating UMLS and extracting
relation pairs. (B) Example of UMLS concepts mapped from the text. Some of the
concepts, such as Prednisone, are recognized by GPT, as they are concepts GPT
model is inherently trained on. However, concepts such as ASA, Cipro, Plavix
are not recognized by GPT; UMLS facilitates their recognition.

###  3.1 Overview of the Framework

Our approach leverages the context-capturing capability of GPT and knowledge-
capturing capability of UMLS. UMLS contains a comprehensive list of more than
1 million biomedical concepts from over 100 source vocabularies. By using the
concepts in the prompts in a few-shot learning setting, we attempt to improve
GPT’s ability to identify entities with the specified context that it may
otherwise fail to extract independently. We map UMLS concepts to each text
instance to create dynamic prompts unique to the specific context of the
clinical text.The overview of the proposed framework is displayed in Figure 1.

###  3.2 UMLS Integration in Large Language Model

UMLS Concept Mapping
We first map UMLS concepts from clinical text using MetaMap (Aronson, 2001).
Given clinical text $X=\\{x_{1},x_{2},\ldots,x_{n}\\}$ where $x_{i}$
represents the $i$th clinical text, we map
$C_{i}=\\{c_{i1},c_{i2},\ldots,c_{in}\\}$, where $C_{i}$ denote the set of
concepts identified by MetaMap from $x_{i}$. $n$ denotes the number of
concepts identified from $x_{i}$. These concepts are extracted leveraging
MetaMap’s lexical parsing, syntactic analysis, semantic mapping, and concept
mapping techniques.
Next, we filter the mapped concepts to include only those categorized as
’organic chemical’, ’antibiotic’, or ’pharmacologic substance’ within the UMLS
concept hierarchy as these groups contains the medications. For this work, we
only target and filter medication-related concepts for augmentation and for
further analysis. Let’s denote the filtered set of concepts for the $i$th
input clinical text $x_{i}$ as $C_{{filtered}}$, such that
$C_{filtered,i}=\\{c_{ij}\in C_{i}|c_{ij}\in\text{filtered groups}\\}$.

Figure 2: An example of a prompt used to extract dosage information from the
text using the UMLS concepts. The ‘note_text’ represents each text instance
from ADE or n2c2 corpus. The ‘medication_list’ represents the UMLS concepts
extracted from MetaMap.

Prompt Strategy and Large Language Model Implementation
Next, we prompt the GPT model to extract entity-relation pairs from the text,
leveraging the mapped UMLS concepts from MetaMap, and employing a few-shot
prompt strategy. Let $P_{i}$ represent the prompt generated for each input
text $x_{i}$, incorporating the relevant UMLS concepts $C_{filtered,i}$. The
final prompt $P_{i}$ is constructed as the concatenation of the initial prompt
and the set of UMLS concepts, i.e., $P_{i}=\text{Concat}(P,C_{filtered,i})$.
We use OpenAI’s GPT-4-32k (Version 0613) and GPT-3.5-turbo (Version 0301) via
HIPPA-compliant Microsoft Azure’s OpenAI REST
API111https://learn.microsoft.com/en-us/azure/ai-
services/openai/quickstart?tabs=command-line%2Cpython-new&pivots=programming-
language-pythonendpoint. A sample prompt and hyperparameters used by the
models for this task are available in Figure 2 and A.2 respectively. As our
goal for the project was not to explore different prompting strategies, we
tested a few prompts and selected the prompt that generated more specific
result. We used the same format for all relation pairs replacing only the
entity type for every run.

Retrieval Augmented Generation
We also explored another approach-RAG to leverage UMLS in a language model,
which is a more conventional method involving the use of external data. RAG
was chosen for its potential to enhance the generation process by
incorporating domain-specific knowledge from sources like the UMLS knowledge
base. Appendix A.4 includes details on our RAG implementation.

###  3.3 Datasets Description

We used the n2c2 and ADE datasets for our experiments.

n2c2 Dataset
We used a curated National NLP Clinical Challenges (n2c2) dataset (Henry et
al., 2019) consisting of 303 deidentified discharge summaries obtained from
the MIMIC-III (Medical Information Mart for Intensive Care-III) critical care
database (Table 1A) (Johnson et al., 2016). The data also contained
annotations of medication-related entities and their relationship to other
entities. Annotations conducted by 3 subject matter experts served as a gold
standard to evaluate model performance.

ADE Dataset
The Adverse Drug Events (ADE) dataset annotated by 5 individuals consists of
MEDLINE222https://www.nlm.nih.gov/medline/medline_home.html case reports with
information on medications, dosages and adverse effects associated with the
medications (Gurulingappa et al., 2012) (Table 1B). It also contains
relationships between medications, dosages, and adverse effects. For our
experiments, we used the second version of the dataset downloaded from
Huggingface333https://huggingface.co/datasets/ade_corpus_v2.

Figure 3: Sample text of discharge summaries in the (A) n2c2 dataset and (B)
ADE Corpus. The text highlighted in red are the targeted entities for
extraction Table 1: Statistics on the relation pairs in the (A) n2c2 dataset
and the (B) ADE dataset

A. n2c2 Dataset
---
Entity-Entity Relation | Total instances
Strength-Drug | 13338
Duration-Drug | 643
Route-Drug | 11038
Form-Drug | 6636
ADE-Drug | 2214
Dosage-Drug | 4207
Reason-Drug | 5160
Frequency-Drug | 6288

B. ADE Dataset
---
Entity-Entity Relation | Total instances
Drug-ADE | 6821
Drug-Dosage | 279

Table 2: Comparison of models on n2c2 Dataset and ADE Corpus. The reported results are micro-averaged precision, recall, and F-1 scores across all entity-entity relation pairs within the datasets. Model | n2c2 Dataset |  | ADE Corpus
---|---|---|---
Precision | Recall | F1 |  | Precision | Recall | F1
GPT-3.5-turbo | 0.73 | 0.74 | 0.73 |  | 0.625 | 0.57 | 0.596
GPT-3.5-turbo + UMLS | 0.77 | 0.77 | 0.77 |  | 0.83 | 0.70 | 0.75
RAG w/ GPT-3.5-turbo | 0.73 | 0.74 | 0.74 |  | 0.65 | 0.63 | 0.64
GPT-4-32k | 0.75 | 0.76 | 0.76 |  | 1.0 | 0.70 | 0.82
GPT-4-32k + UMLS | 0.79 | 0.80 | 0.80 |  | 1.0 | 0.89 | 0.94
RAG w/ GPT-4-32k | 0.77 | 0.76 | 0.76 |  | 1.0 | 0.74 | 0.85

##  4 Results

###  4.1 Experimental Setup

We evaluated two generative models, GPT-4-32k and GPT-3.5-turbo, with and
without UMLS integration, and the RAG model to access the quality of generated
outputs. All models were evaluated against the gold-standard annotations using
precision, recall, and micro-F1 score.

###  4.2 Dataset

We identified 8 different entity-entity relation pairs within the n2c2
dataset, and 2 entity-entity relation pairs within the ADE corpus, each with
varying instances of the relation pairs (Table 1, Figure 3). Token length
distributions of the text, and example of individual entities in n2c2 and ADE
dataset are available in A.1, A.3.

###  4.3 Performance Results

Results on n2c2 and ADE Dataset
Our results suggests that integrating prior knowledge from UMLS in the prompts
have significant performance improvement as demonstrated by the higher average
F-1 scores across both n2c2 and ADE datasets (Table 4). The reported results
are average across all entity-entity relation pairs across models and for 2
datasets. GPT-4-32k model with UMLS show 4% improvement of F-1 score on the
n2c2 dataset, and 12% improvement on the ADE dataset from the F-1 score of
GPT-4-32k model without knowledge integration. For every entity-entity
relation pairs, there was a performance improvement by a few percentages for
both models and across both datasets. Additional detailed results for each
entity-entity relation pairs can be found in Appendices A.6 through A.11.
Upon a closer look at the results, we identified that prompts with UMLS
resulted in additional concepts verifying that UMLS is able to identify
medications from the text that GPT may not identify independently(Appendix
A.5).

Comparison with Retrieval Augmented Generation
RAG model and GPT-3.5-turbo had low F-1 score and it improved with UMLS for
both models, but it did not have higher score compared to the GPT-4-32k+UMLS.
We observed performance variations across entity-entity relation pairs with
retrieval augmented generation (A.8, A.11). While some entity pairs showed
performance enhancements, others did not show significant improvements. This
discrepancy might arise from the limitations of RAG, particularly its
inability to utilize entire UMLS thesaurarus in the generation process. Since
UMLS data is partitioned into chunks for indexing and embedding and embedding
models can only take 8192 tokens per index, some concepts may not be in the
top-k extracted documents used for generation, potentially limiting the scope
of augmentation and its impact on final relation pairs. Further experiments
are required to confirm this hypothesis.

##  5 Discussion and Conclusion

Our study highlights the significance of merging the strengths of domain-
specific knowledge bases, such as UMLS, with the contextual understanding
capabilities of LLMs, such as GPT. Our hybrid approach, integrating mapped
UMLS concepts with GPT, shows improvement in the model’s ability to identify
specific entities not inherently within its training data.

Our results on entity and relation extraction task indicated that leveraging
mapped UMLS concepts as additional guidance to the GPT model, helps create
focused and unique prompts that significantly enhances GPT’s performance. This
approach proves more effective than the standard RAG technique.

In conclusion, the ability to generate tailored prompts based on UMLS concepts
offers a promising avenue for improving accuracy and relevance of extracted
entities, ultimately enhancing the utility of LLMs in biomedical text analysis
tasks.

##  6 Limitations and Future Work

While our framework has shown significant improvements, we acknowledge several
limitations in this study. Firstly, our work focused solely on medication
concepts, which may restrict the generalizability of our findings to other
concepts. However, our approach is adaptable to incorporate additional UMLS
entities through prompt adjustments. Future research will explore harnessing
UMLS’s rich semantic metadata to leverage additional concept relationships,
enabling the extraction of a broader spectrum of entity groups beyond
medications.

Secondly, our comparison was limited to two generative models, GPT-4-32k and
GPT-3.5-turbo. Though they have good performance, we have not included recent
models that could have comparable performance. Future work will explore
additional models, such as BioGPT, and LAMA for comprehensive comparison and
evaluation. This expanded comparison will provide a more nuanced understanding
of the performance and capabilities of various generative models in relation
to UMLS integration and RAG techniques.
These future tasks will advance our understanding of the role of domain-
specific knowledge in enhancing LLM capabilities and facilitating more
effective clinical information extraction.

##  7 Ethics Statement

IRB approval was not required for this task. To input our text data into the
language models, we use Microsoft’s Azure OpenAI REST API Service within the
Washington University tenant to access OpenAI’s language models . We are on a
HIPPA-compliant subscription and exempted from content filtering, data review
and human review for our use of the Azure OpenAI service.

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##  Appendix A Appendix

###  A.1 Token length of the text in (A) n2c2 and (B) ADE dataset

A  B

###  A.2 Hyperparameters for the GPT models

Hyperparameters |  Value
---|---
Tokenization and Context
Window |  200 tokens
Temperature (Randomness of the model output) |  0
Top p (Top-K Sampling
Technique) |  0.95
Presence Penalty (Penalty to
discourage model from
generating responses that contain certain specified tokens) |  -1.0

###  A.3 Example of the individual entities within the n2c2 and ADE dataset

Table 3: Example of the individual entities within the n2c2 and ADE dataset Entities |  Examples
---|---
Drug |  Morphine, ibuprofen, antibiotics (abx), chemotherapy (carboplatin)
ADE/Reason |  Nausea, rash, seizures, vitamin K deficiency
Strength |  10 mg, 60 mg
Form |  Capsule, syringe, tablet, topical (apply topical)
Dosage |  60 mg/0.6 mL
Frequency |  Daily, twice a day, Q4H (every 4 hours)
Route |  Transfusion, oral, intravenous (IV)
Duration |  For 10 days, 2 cycles, for a week

###  A.4 Retrieval Augmented Generation

Method:
1\. Split UMLS data
(MRCONSO.RRF444https://www.ncbi.nlm.nih.gov/books/NBK9685/table/ch03.T.concept_names_and_sources_file_mr/)
into manageable chunks (8192 tokens) to facilitate processing. MRCONSO.RRF
file contains the UMLS concepts.
2\. Generate embeddings for each chunk, capturing its semantic represetations
2\. Store the embeddings in a vector database for efficient retrieval
3\. Compare each prompt with the stored data in the vector database.
4\. Extract the top 30 results with the highest similarity scores between the
query and the UMLS data.
5\. Concatenate the retrieved results with the prompt to generate the final
extraction output.

###  A.5 Qualitative Results

Table 4: Some of the qualitative results for the Strength-Drug Pair. (A) Without UMLS integration. (B) With UMLS integration |  Examples
---|---
A |  [(‘aspirin’, ‘81 mg’) (‘atorvastatin’, ‘20 mg’), (‘amiodarone’, ‘200 mg’), (‘metoprolol tartrate’, ‘50 mg’), (‘spironolactone’, ‘25 mg’), (‘acetaminophen’, ‘325 mg’), (‘ranitidine HCl’, ‘150 mg’), (‘prednisone’, ‘60 mg’)]
B |  [(‘aspirin’, ‘81 mg’), (‘atorvastatin’, ‘20 mg’), (‘amiodarone’, ‘200 mg’), (‘metoprolol tartrate’, ‘50 mg’), (‘spironolactone’, ‘25 mg’), (‘acetaminophen’, ‘325 mg’), (‘ranitidine HCl’, ‘150 mg’), (‘prednisone’, ‘60 mg’), (‘Plavix’, ‘75 mg’), (‘ASA’, ‘325’), (‘Cipro’, ‘250 mg’)]

###  A.6 Comparison of GPT-3.5-turbo for all Entity-Entity Relation pairs with
and without UMLS Integration for the n2c2 dataset

Entity-Entity | GPT-3.5-turbo | GPT-3.5-turbo+UMLS
---|---|---
P | R | Micro F-1 | P | R | F-1
Dosage-Drug | 0.75 | 0.75 | 0.75 | 0.80 | 0.80 | 0.80
Duration-Drug | 0.76 | 0.76 | 0.76 | 0.81 | 0.81 | 0.81
Route-Drug | 0.74 | 0.73 | 0.73 | 0.76 | 0.75 | 0.75
Form-Drug | 0.72 | 0.73 | 0.72 | 0.75 | 0.76 | 0.75
ADE-Drug | 0.69 | 0.71 | 0.70 | 0.74 | 0.75 | 0.75
Reason-Drug | 0.73 | 0.74 | 0.74 | 0.76 | 0.77 | 0.777
Frequency-Drug | 0.73 | 0.74 | 0.73 | 0.75 | 0.76 | 0.77
Average | 0.73 | 0.74 | 0.73 | 0.77 | 0.77 | 0.77

###  A.7 Comparison of GPT-4-32k for all Entity-Entity Relation pairs without
UMLS Integration for the n2c2 dataset

Entity-Entity | GPT-4-32k | GPT-4-32k+UMLS
---|---|---
P | R | Micro F-1 | P | R | F-1
Dosage-Drug | 0.77 | 0.77 | 0.77 | 0.82 | 0.82 | 0.82
Duration-Drug | 0.78 | 0.77 | 0.78 | 0.83 | 0.82 | 0.83
Route-Drug | 0.79 | 0.77 | 0.78 | 0.81 | 0.78 | 0.79
Form-Drug | 0.74 | 0.76 | 0.74 | 0.77 | 0.79 | 0.77
ADE-Drug | 0.69 | 0.73 | 0.71 | 0.75 | 0.78 | 0.77
Reason-Drug | 0.74 | 0.75 | 0.735 | 0.77 | 0.78 | 0.76
Frequency-Drug | 0.78 | 0.77 | 0.78 | 0.80 | 0.79 | 0.79
Average | 0.75 | 0.76 | 0.76 | 0.79 | 0.79 | 0.79

###  A.8 Comparison of Models for all Entity-Entity Relation pairs with UMLS
for RAG on the n2c2 dataset

Entity-Entity | GPT-4-32k | GPT-3.5-turbo
---|---|---
P | R | F-1 | P | R | F-1
Dosage-Drug | 0.77 | 0.77 | 0.77 | 0.75 | 0.75 | 0.75
Duration-Drug | 0.79 | 0.78 | 0.78 | 0.76 | 0.77 | 0.77
Route-Drug | 0.79 | 0.77 | 0.78 | 0.74 | 0.73 | 0.73
Form-Drug | 0.74 | 0.73 | 0.74 | 0.73 | 0.74 | 0.74
ADE-Drug | 0.72 | 0.73 | 0.72 | 0.70 | 0.70 | 0.70
Reason-Drug | 0.76 | 0.76 | 0.76 | 0.75 | 0.78 | 0.76
Frequency-Drug | 0.81 | 0.80 | 0.80 | 0.70 | 0.71 | 0.71
Average | 0.77 | 0.76 | 0.76 | 0.73 | 0.74 | 0.74

###  A.9 Comparison of GPT-4-32k for all Entity-Entity Relation pairs with and
without UMLS on the ADE dataset

Entity-Entity | GPT-4-32k | GPT-4-32k+UMLS
---|---|---
P | R | F-1 | P | R | F-1
Dosage-Drug | 1.0 | 0.66 | 0.795 | 1.00 | 0.85 | 0.91
ADE-Drug | 1.0 | 0.73 | 0.84 | 1.00 | 0.93 | 0.97
Average | 1.0 | 0.70 | 0.82 | 1.0 | 0.89 | 0.94

###  A.10 Comparison of GPT-3.5-turbo for all Entity-Entity Relation pairs
with and without UMLS on the ADE dataset

Entity-Entity | GPT-3.5-turbo | GPT-3.5-turbo+UMLS
---|---|---
P | R | F-1 | P | R | F-1
ADE-Drug | 0.57 | 0.53 | 0.55 | 0.60 | 0.65 | 0.62
Dosage-Drug | 0.68 | 0.61 | 0.64 | 0.70 | 0.75 | 0.72
Average | 0.625 | 0.57 | 0.596 | 0.83 | 0.70 | 0.75

###  A.11 Comparison of the models for all Entity-Entity Relation pairs with
UMLS for RAG on the ADE dataset

Entity-Entity | GPT-4-32k | GPT-3.5-turbo
---|---|---
P | R | F-1 | P | R | F-1
ADE-Drug | 1.0 | 0.73 | 0.84 | 0.62 | 0.61 | 0.60
Dosage-Drug | 1.0 | 0.75 | 0.86 | 0.68 | 0.65 | 0.66
Average | 1.0 | 0.74 | 0.85 | 0.65 | 0.63 | 0.64