# Quantum Hǒrava–Lifshitz Cosmology in the de Broglie-Bohm interpretation

G. S. Vicente<EMAIL_ADDRESS>Faculdade de Tecnologia, Universidade do
Estado do Rio de Janeiro, 27537-000 Resende, RJ, Brazil

###### Abstract

Classical and quantum Friedmann–Lemaître–Robertson–Walker universes filled
with non-interacting radiation and dust fluids are considered in the framework
of Hořava–Lifshitz gravity theory. The Hořava–Lifshitz theory is set in its
projectable version and without detailed balance condition. Canonical
quantization is performed in the Wheeler–DeWitt approach of quantum cosmology
for a minisuperspace model in the light of de Broglie-Bohm interpretation of
quantum mechanics. The main results are analytical solutions for non-singular
quantum bounce and cyclic universes for open and closed spatial sections in
terms of the parameters of Hořava–Lifshitz theory.

##  I Introduction

The standard model of particle physics successfully describes electro-weak and
strong interactions, which operate at the quantum level. General Relativity
(GR) is the most successful theory of gravitation, which is a classical theory
of space-time and matter. A first principles description of nature seems to be
quantum mechanical, then it is a natural way to assume that gravity must be
described by a quantum theory as well.

Applying the canonical quantization scheme to GR, we achieve the so called
canonical quantum gravity theory DeWitt , which is a constrained system
governed by the Wheeler-DeWitt equation QG . The solution of the Wheeler-
DeWitt equation yields the quantum state of the 3-geometry and matter fields
by a wave functional (on superspace), which describes the whole Universe.
However, this formulation faces some drawbacks. Firstly, we expect that the
wave function constrains the dynamics of the Universe, but the absence of a
momentum which is canonically conjugate to the time variable implies that the
wave function is static. This is called the problem of time time . Secondly,
the wave function describes no particular metric, but all spacelike metrics
(superspace). Both these issues compromise the definition and identification
of space-time singularities. Finally, measurement demands a collapse of the
wave function by an external observer, which is a feature of the Copenhagen
interpretation of quantum theory. This is called the measurement problem.

In order to circumvent these issues, we may consider the de Broglie-Bohm (dBB)
interpretation of quantum theory Bohm ; hollandbook  instead of the orthodox
one. The time degree of freedom can now be defined from matter degrees of
freedom if it is described by a hydrodynamical perfect fluid. This can be done
using the Schutz formalism Schutz , where the time variable is associated to
the fluid’s potentials. We can also define a metric, which evolves in time
according to guidance equations. The measurement problem is either eliminated,
because in the ontological interpretation the evolution of the Universe is
deterministic and do not demand a collapse of the wave function or the action
of an external observer. Therefore, the dBB interpretation of canonical
quantum gravity Vink:1990fm ; Shtanov:1995ie ; Goldstein:1999my ;
PintoNeto:2004uf ; Pinto-Neto:2013toa  is a suitable way to establish a
quantum cosmology theory. In this context, one can define
contraction/expansion as well as singularities in space-time.

In order to solve the Wheeler-DeWitt equation in a simple form, we restrict
the superspace to the minisuperspace ryan ; Kuchar:1989tj ; Halliwell , where
the degrees of freedom are reduced to a finite number while still preserving
the main qualitative aspects of the full picture. This is a reasonable
framework to develop quantum cosmology.

Quantum cosmology models in minisuperspace in the dBB interpretation are vast
in the literature Vink:1990fm ; KowalskiGlikman:1990df ; Squires:1992ex ;
Colistete:1997sf ; Kim:1997te ; AcaciodeBarros:1997gy ; AcaciodeBarros:1998nb
; Colistete:2000ix ; Pedram:2007mj ; Monerat:2010vf ; Vakili:2012nh ;
Das:2013oda ; Ali:2014qla ; Oliveira-Neto:2014jaa ; Peter:2006hx .
Particularly, bouncing models are a very relevant feature of quantum
cosmology, where quantum effects are responsible for avoiding the big bang
(and eventually big crunch) singularity. In this context, the authors of Refs.
Peter:2006hx ; AcaciodeBarros:1997gy ; Alvarenga:2001nm ; PintoNeto:2005gx
present bouncing models for perfect fluids using the Schutz formalism, whereas
in Ref. Falciano:2007yf  a scalar field is considered as the matter content.
In the case of perfect fluids, a very interesting result is given in Ref.
PintoNeto:2005gx , which is the quantum version of the Friedmann-Lemaître-
Robertson-Walker (FLRW) universe filled with radiation and dust, where
radiation dominates around the bounce and dust dominates in the contraction
and expansion phases far from the bounce. Cyclic universe solutions are also
present for closed spatial section, where multiple bounces are present. These
results will be very important for this work.

It is important to mention that a cosmological bounce can solve the standard
cosmological puzzles Peter:2008qz , like the horizon and flatness problems,
and are also robust to generate primordial cosmological perturbations with
almost scale invariant spectrum Peter:2008qz ; Peter:2006hx . However,
although an inflationary phase can be avoided for these models, in principle
it can still happen. Finally, dark matter has been also considered in this
context through the dynamics of a scalar field dBBscalarfield .

Recently, a new theory of gravity was proposed by Hǒrava Horava:2009uw ;
Horava:2009if , which is based on the introduction of an anisotropic scaling
between space and time. The theory is inspired in the Lifshitz work about
critical exponents in phase transitions in the context of condensed matter
physics, then referred to as Hǒrava-Lifshitz (HL) theory. The anisotropy is
introduced in order to generalize GR in the high energy ultraviolet (UV)
scale, while recovering GR in the low energy infrared (IR) scale. This
generalization is due to Lorentz symmetry breaking in the UV scale through a
Lifshitz-like process, while this symmetry is preserved in the IR scale. Due
to the anisotropy in space and time, the HL theory is usually represented in
the Arnowitt-Deser-Misner (ADM) formalism Arnowitt:1959ah , which splits the
4-metric $g_{\mu\nu}$ in a 3-metric $h_{ij}$, a shift 3-vector $N_{i}$ and a
lapse function $N$, where the latter parametrizes time.

There are two important issues to consider in HL theory. Firstly, the
variables in the ADM formalism may depend on both space and time. However,
Hǒrava pointed out Horava:2009uw  that the lapse function $N$ should only
depend on time, which is reasonable in the cosmological setting. This is
called the projectable condition. On the other hand, some authors have
considered the general case in which $N$ depends on both space and time, i.e.,
a non-projectable condition. Both projectable and non-projectable theories
suffer from problems such as ghost scalar modes and instabilities
Sotiriou:2010wn ; Visser:2011mf ; Wang:2017brl . However, a consistent
projectable HL gravity theory with a local $U(1)$ symmetry can be shown to
eliminate the ghost scalar modes Horava:2010zj ; daSilva:2010bm , so that
instability and strong coupling does not happen in the gravitational sector.
On the other hand, non-projectable HL gravity theories can also be consistent
whether the local $U(1)$ symmetry is present Zhu:2011yu ; Zhu:2011xe ;
Lin:2013tua  or not Blas , where the latter theory is the so called healthy
extension. A review of the aforementioned issues is given in Ref. Wang:2017brl
. From another point of view, in the cosmological setting, it seems risky to
impose that the lapse function only depends on time. However, in this case the
Hamiltonian constraint is no longer local but integrated over all spatial
volume. This result corresponds to Friedmann equations with an additional cold
dark matter-like component Mukohyama:2009mz . However, in homogeneous models
these spatial integrals are simply the spatial volume, then the additional
matter content vanishes Sotiriou:2009gy ; Sotiriou:2009bx . These
considerations can be extended to the quantum realm in minisuperspace
cosmology models Bertolami:2011ka ; Pitelli:2012sj , described by the
projectable version of the HL theory. Then, it is reasonable to consider the
projectable HL in this context. It is important to mention that the
aforementioned theory with local $U(1)$ symmetry can also be applied to the
cosmological setting Huang:2012ep .

Secondly, Hǒrava also considered a simplification in order to reduce the
number of terms of his theory, which is called the detailed balance condition.
However, although the detailed balance is a simplifying assumption, it is not
really necessary Sotiriou:2009gy ; Sotiriou:2009bx . Therefore, in this paper
I consider the projectable version of HL theory without detailed balance
condition.

Non-singular classical HL cosmological models have been considered in the
literature for the matter bounce scenario Brandenberger:2009yt , where the
non-singular behavior is due to the presence of spatial curvature. However,
non-singular classical cosmologies can also be achieved by other mechanisms,
such as from matter contents BounceMatter  and other modified gravity theories
BounceOthers . On the other hand, additionally to the aforementioned quantum
non-singular cosmologies in minisuperspace in the dBB interpretation, the
literature is also vast for non-singular quantum cosmological models
nonsingularquantum  and also heuristically motivated ones
Brandenberger:2017pjz . Particularly, there is also a great literature for
quantum non-singular cosmologies in the framework of Loop Quantum Cosmology
Ashtekar:2021kfp ; ElizagaNavascues:2020uyf ; Li:2021mop ; Li:2018vzr ;
Zhu:2017jew .

Hǒrava-Lifshitz quantum cosmology in minisuperspace has already been
considered in the literature Sotiriou:2009bx ; Bertolami:2011ka ;
Pitelli:2012sj ; Vakili:2013wc ; Oliveira-Neto:2017yui ; Oliveira-Neto:2019tvo
; Tavakoli:2021kyc . Particularly, exact solutions have been obtained in Refs.
Vakili:2013wc ; Oliveira-Neto:2017yui  for perfect fluids for some values of
the equation of state parameter and for some of the parameters of HL theory.
In this work, I present exact solutions for non-singular universes in the
cases of closed and open FLRW quantum cosmologies in HL theory, where the
matter content is composed of non-interacting radiation and dust fluids.

This paper is organized as follows. In Sec. II, I introduce the FLRW cosmology
in HL gravity in the ADM formalism and the Schutz formalism for the matter
content. The gravitational and matter Hamiltonians are presented and the full
Hamiltonian for a universe filled with non-interacting radiation and dust
fluids is obtained. In Sec. III, classical FLRW cosmology in HL theory is
considered and analytical solutions for the scale factor are presented for
open and closed universes. Some particular cases are also shown. These
solutions contains all parameters of HL theory except the cosmological
constant-like term, which is neglected in this work. In Sec. IV, I have
obtained non-singular solutions for quantum FLRW cosmology in HL theory, which
are the results of this paper. From the canonical quantization of the
Hamiltonian, a Wheeler-DeWitt equation was obtained and analytically solved.
From the solutions for the wave function, using the dBB interpretation of
quantum mechanics I have obtained analytical solutions for the scale factor
for non-interacting radiation and dust fluids for open and closed universes.
Single (contracting/expanding) bounce universe solutions were obtained as well
as (multiple bounce) cyclic ones. The analytical quantum potential was also
derived for these solutions, which is the responsible for the occurrence of
non-singular behavior in the scale factor evolution. In the Appendix A, the
motivation for the initial condition used in Sec. IV is discussed. Finally,
the concluding remarks and future perspectives are presented in Sec. V.
Throughout this work I am using the natural units system, in which
$c=\hbar=1$.

##  II FLRW Cosmology in Hǒrava–Lifshitz Gravity

The FLRW metric for a homogeneous and isotropic space-time is written as:

$\displaystyle
ds^{2}=-N(t)dt^{2}+a(t)^{2}\left(\frac{dr^{2}}{1-kr^{2}}+r^{2}d\Omega^{2}\right),$
(1)

where $N(t)$ is the lapse function, $a(t)$ is the scale factor, $d\Omega^{2}$
is the line element of a 2-sphere with unitary radius and $k$ is the curvature
constant of spatial sections, which is $k=1$, $0$ and $-1$ for closed, flat
and open universes, respectively 111I will not discuss about the possibility
of the Universe being open or closed. However, Refs. Renevey:2020zdj ;
Handley:2019tkm  provide more details about the possibility of a closed
Universe.. The 4-metric is defined as $g_{\mu\nu}$, where Greek indexes run
from $0$ to $3$, and spatial 3-metric is defined as $h_{ij}$, where the Latin
indexes run from $1$ to $3$.

In the following subsections, I introduce the Hamiltonian densities for the
gravitational and matter sections in HL theory.

###  II.1 Gravitational Hamiltonian

In the HL theory, an anisotropic scaling between space and time is introduced
as:

$\displaystyle t\rightarrow b^{z}t\quad,\quad\vec{x}\rightarrow b\vec{x},$ (2)

where $b$ is a scale parameter and $z$ is a dynamical critical exponent. While
the UV sector requires $z=3$, which breaks Lorentz invariance, in the IR
sector it is recovered for $z=1$ (see Refs. Horava:2009uw ; Horava:2009if  for
details).

The gravitational action of HL theory is composed of kinetic and potential
parts. The former generalizes the Einstein-Hilbert action and is given in
terms of the extrinsic curvature (and derivatives) and a free parameter
$\lambda$, which reduces to GR kinetic term in the limit $\lambda\rightarrow
1$. On the other hand, the potential part depends only on the 3-metric (and
spatial derivatives). The detailed balance condition refers to the choice of
potential part, which simplifies the theory by reducing the number of terms
and facilitating its renormalization. However, the authors of Refs.
Sotiriou:2009bx ; Sotiriou:2009gy  show that one should avoid this condition.

Following the lines of Refs. Vakili:2013wc ; Oliveira-Neto:2017yui ; Oliveira-
Neto:2019tvo , in this paper I consider the projectable HL gravity without the
detailed balance condition, for $z=3$ in $(3+1)$ dimensions Bertolami:2011ka ,
whose action reads:

$\displaystyle S_{\rm HL}$ $\displaystyle=$ $\displaystyle\frac{M_{\rm
Pl}^{2}}{2}\int\limits_{\mathcal{M}}d^{3}x\,dtN\sqrt{h}\left[K_{ij}K^{ij}-\lambda
K^{2}-g_{0}M_{\rm Pl}^{2}-g_{1}R-M_{\rm
Pl}^{-2}\left(g_{2}R^{2}+g_{3}R_{ij}R^{ij}\right)+\right.$ (3)
$\displaystyle-$ $\displaystyle\left.M_{\rm
Pl}^{-4}\left(g_{4}R^{3}+g_{5}RR^{i}_{j}R^{j}_{i}+g_{6}R^{i}_{j}R^{j}_{k}R^{k}_{i}+g_{7}R\nabla^{2}R+g_{8}\nabla_{i}R_{jk}\nabla^{i}R^{jk}\right)\right]+M_{\rm
Pl}^{2}\int\limits_{\partial\mathcal{M}}d^{3}x\sqrt{h}K,$

where $M_{\rm Pl}=1/\sqrt{8\pi G}$ is the Planck mass, $h_{ij}$ is defined on
the boundary $\partial\mathcal{M}$ of the 4-dimensional manifold
$\mathcal{M}$, $h$ is the determinant of $h_{ij}$, $K_{ij}$ is extrinsic
curvature tensor, $K$ is the trace of $K_{ij}$, $R_{ij}$ is the Ricci tensor,
$R$ is the Ricci scalar and $\lambda$ and $g_{i}$ ($i=0,..,8$) are HL
parameters involved in corrections to GR.

In the case of FLRW space-time, the metric and the action are given by Eqs.
(1) and (3), respectively. From the results of Ref. Vakili:2013wc , the
Lagrangian density in HL theory reads:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\mathcal{L}_{\rm
HL}=N\left(-\frac{a\dot{a}^{2}}{N^{2}}+g_{c}ka-
g_{\Lambda}a^{3}-\frac{g_{r}k^{2}}{a}-\frac{g_{s}k}{a^{3}}\right),$ (4)

where the HL parameters are defined as Maeda:2010ke :

$\displaystyle g_{c}=\frac{2}{3\lambda-1}\quad,\quad
g_{\Lambda}=\frac{2\Lambda}{3(3\lambda-1)}\quad,\quad
g_{r}=\frac{4(3g_{2}+g_{3})}{(3\lambda-1)M_{\rm Pl}^{2}}\quad,\quad
g_{s}=\frac{8(9g_{4}+3g_{5}+g_{6})}{(3\lambda-1)M_{\rm Pl}^{4}},$ (5)

and the parameter $g_{c}$ is positive definite. The subscripts refer to the
fluid-like behavior of each term: $(g_{c},\,g_{\Lambda},\,g_{r},\,g_{s})=$
(curvature, cosmological constant, radiation, stiff matter)-like term.

We note that the scale factor is the only variable in Eq. (4). In order to
obtain the Hamiltonian formulation, we must compute $P_{a}$, the momentum
canonically conjugated to $a$, which reads:

$\displaystyle
P_{a}=\frac{\partial\mathcal{L}_{HL}}{\partial\dot{a}}=-\frac{2a\dot{a}}{N}.$
(6)

Using the definition of $P_{a}$, the Hamiltonian density reads:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!H_{\rm
HL}=N\left(-\frac{P_{a}^{2}}{4a}-g_{c}ka+g_{\Lambda}a^{3}+\frac{g_{r}k^{2}}{a}+\frac{g_{s}k}{a^{3}}\right).$
(7)

The lapse function $N$ appears in $H_{\rm HL}$ as a lagrangian multiplier
(there is no momentum canonically conjugated to $N$). Rewriting $H_{\rm HL}$
as $H_{\rm HL}=N\mathcal{H}_{\rm HL}$, when we vary $H_{\rm HL}$ with respect
to $N$, we obtain Pinto-Neto:2013toa  that $\mathcal{H}_{\rm HL}\approx 0$
($\approx$ means weakly zero). This is called the super-Hamiltonian
constraint.

###  II.2 Matter Hamiltonian

A perfect fluid can be described by a Hamiltonian using the Schutz formalism
Schutz . The action for a perfect fluid in this formalism reads:

$\displaystyle S_{\rm m}=\int d^{4}x\sqrt{-g}\,(16\pi p),$ (8)

where $g$ is the determinant of $g_{\mu\nu}$ and $p$ is the pressure of the
fluid. The equation of state for a perfect fluid is written as:

$\displaystyle p=\omega\rho,$ (9)

where $\rho$ is the energy density and $\omega$ is the equation of state
parameter, which is subject to $-1\leq\omega\leq 1$ .

The Schutz formalism consists of writing the fluid 4-velocity $U_{\mu}$ in
terms of six velocity-potentials, which in FRLW cosmology these potentials are
reduced to three. Substituting the FLRW metric (1) into the $S_{m}$ action
(8), identifying the canonical variables and performing some canonical
transformations Lapchinsky:1977vb , we obtain a simple Hamiltonian for a
single perfect fluid, which reads:

$\displaystyle H_{\rm m}=N\frac{P_{T}}{a^{3\omega}},$ (10)

where $P_{T}$ is the momentum canonically conjugated to the fluid degree of
freedom $T$, which can be interpreted as a time variable. The Hamiltonian
$H_{\rm m}$ can also be written as $H_{\rm m}=N\mathcal{H}_{\rm m}$, where
$\mathcal{H}_{\rm m}$ is a lagrangian multiplier, similarly to
$\mathcal{H}_{\rm HL}$ of the gravitational part.

###  II.3 Full Hamiltonian

From the gravitational and matter Hamiltonians, Eqs. (7) and (10),
respectively, the minisuperspace Hamiltonian for a single perfect fluid in HL
theory reads:

$\displaystyle H$ $\displaystyle=$ $\displaystyle
N\left(-\frac{P_{a}^{2}}{4a}-g_{c}ka+g_{\Lambda}a^{3}+\frac{g_{r}k^{2}}{a}+\frac{g_{s}k}{a^{3}}+\frac{P_{T}}{a^{3\omega}}\right),$
(11)

from which we can also define the super-Hamiltonian constraint
$\mathcal{H}=\mathcal{H}_{\rm HL}+\mathcal{H}_{m}$, which satisfies:

$\displaystyle\mathcal{H}\approx 0.$ (12)

In the following sections, I will show that this constraint leads to the
Friedmann equation in the classical level, whereas at the quantum level it
gives the Wheeler-DeWitt equation. In the latter, the super-Hamiltonian
constraint is essential in the procedure of canonical quantization.

In this work, I will consider a HL quantum FLRW universe filled with non-
interacting radiation ($\omega=1/3$) and dust ($\omega=0$) fluids. However,
Eq. (11) is valid only for a single fluid. I will set $\omega=1/3$ in order to
describe a radiation fluid and, following Ref. PintoNeto:2005gx , I will also
include dust as second decoupled fluid by introducing another term of the type
(10) for $\omega=0$. From these considerations, the Hamiltonian for radiation
and dust fluids in HL theory reads:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!H_{\rm rd}$ $\displaystyle=$
$\displaystyle N\mathcal{H}_{\rm
rd}=N\left(-\frac{P_{a}^{2}}{4a}-g_{c}ka+g_{\Lambda}a^{3}+\frac{g_{r}k^{2}}{a}+\frac{g_{s}k}{a^{3}}+\frac{P_{T}}{a}+P_{\varphi}\right),$
(13)

where $P_{\varphi}$ is a constant stemming from Eq. (10) for $\omega=0$ and
$\mathcal{H}_{\rm rd}$ is the super-Hamiltonian related to $H_{\rm rd}$.

Additionally, for the potential part of HL theory, I will consider from now on
the particular case of no cosmological constant-like term ($g_{\Lambda}=0$).

##  III Classical dynamics Hǒrava–Lifshitz Cosmology

In this section, I consider analytical solutions of the HL classical cosmology
in minisuperspace for non-interacting radiation and dust fluids. The equations
of motion for each system variable and its canonically conjugate momentum are
then obtained from the evaluation of the Poisson brackets of each of them with
the Hamiltonian.

The HL super-Hamiltonian constraint containing radiation and dust fluids, Eq.
(13), together with the equation for $P_{a}$, Eq. (6), result in the classical
Friedmann equation:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\left(\frac{\dot{a}}{a}\right)^{2}=N^{2}\left(\frac{g_{s}k}{a^{6}}+\frac{g_{r}k^{2}}{a^{4}}-\frac{g_{c}k}{a^{2}}+\frac{P_{T}}{a^{4}}+\frac{P_{\varphi}}{a^{3}}\right).$
(14)

From the Hamiltonian, Eq. (13), the equations of motion for $T$ and $\varphi$
read:

$\displaystyle\dot{T}$ $\displaystyle=$ $\displaystyle\\{T,H_{\rm
rd}\\}=\frac{N}{a}\quad\rightarrow\quad a\,dT=N\,dt,$
$\displaystyle\dot{\varphi}$ $\displaystyle=$ $\displaystyle\\{\varphi,H_{\rm
rd}\\}=N\quad\,\rightarrow\quad\ \ d\varphi=N\,dt.$

One needs to choose a gauge, which corresponds to choosing the lapse function
$N$ in order to define the time variable. In principle, both $T$ and $\varphi$
can be the time variable. If we choose $N=1$, $\varphi$ is the cosmic time,
whereas for $N=a$ we obtain that $T$ is the conformal time. Analytical
solutions of the Friedmann equation, Eq. (14), can only be obtained for the
latter, so I will consider the gauge $N=a$ throughout this paper, which means
that $T=\eta$, where $\eta$ is the conformal time. Therefore, Eq. (14) now
reads:

$\displaystyle\\!\\!\\!\\!\\!\\!\left(\frac{a^{\prime}}{a^{2}}\right)^{2}=\frac{g_{s}k}{a^{6}}+\frac{g_{r}k^{2}}{a^{4}}-\frac{g_{c}k}{a^{2}}+\frac{P_{\eta}}{a^{4}}+\frac{P_{\varphi}}{a^{3}},$
(15)

where the prime denotes derivative with respect to conformal time.

The analytical solutions for Eq. (15) with the initial condition $a(0)=0$
read:

$\displaystyle
a(\eta)=\left\\{\begin{array}[]{lllll}\sqrt{\sqrt{\frac{g_{s}}{g_{c}}}\sin\left(2\sqrt{g_{c}}\eta\right)+\frac{(P_{\eta}+g_{r})}{g_{c}}\sin^{2}\left(\sqrt{g_{c}}\eta\right)}&+&\frac{P_{\varphi}}{2g_{c}}\left[1-\cos\left(\sqrt{g_{c}}\eta\right)\right]&,&k=1,\\\
\sqrt{\sqrt{\frac{|g_{s}|}{g_{c}}}\sinh\left(2\sqrt{g_{c}}\eta\right)+\frac{(P_{\eta}+g_{r})}{g_{c}}\sinh^{2}\left(\sqrt{g_{c}}\eta\right)}&+&\frac{P_{\varphi}}{2g_{c}}\left[\cosh\left(\sqrt{g_{c}}\eta\right)-1\right]&,&k=-1,\end{array}\right.$
(18)

where $g_{c}$ and the product $g_{s}k$ are both positive definite. The latter
is positive when $g_{s}>0$ ($g_{s}<0$) for $k=1$ ($k=-1$). From now on the
notation $g_{s}k=|g_{s}|$ represents the case $g_{s}<0,\,k=-1$ or both
$g_{s}>0,\,k=1$ and $g_{s}<0,\,k=-1$ cases when the results both for $k=\pm 1$
can be written in a single expression. From Eq. (15), the stiff matter-like
term involving $g_{s}$ initially dominates near the singularity at $\eta=0$,
followed by radiation dominance, which consists of a radiation fluid plus a
“HL radiation” term involving $g_{r}$. Far from the singularity, radiation
domination is followed by dust domination, and ending up with domination of
the curvature term involving $g_{c}$. If the cosmological constant term were
not neglected, as in the Hamiltonian given by Eq. (13), far from the
singularity it would dominate after the curvature term.

Now I consider some particular cases. In the limit where the curvature-like
term is negligible ($g_{c}\to 0$), Eqs. (18) read:

$\displaystyle
a(\eta)=\sqrt{2\sqrt{|g_{s}|}\,\eta+\left(P_{\eta}+g_{r}\right)\eta^{2}}+\frac{P_{\varphi}\eta^{2}}{4},$
(19)

which is valid for both $k=\pm 1$. Also, when we additionally consider that
the stiff matter-like time is negligible ($g_{s}\to 0$), the latter result
reduces to:

$\displaystyle
a(\eta)=\sqrt{P_{\eta}+g_{r}k^{2}}\,\eta+\frac{P_{\varphi}}{4}\eta^{2}.$ (20)

The remaining term from HL theory is the radiation-like constant, $g_{r}$,
which adds up to usual radiation. When $k=0$ or in the limit $g_{r}\to 0$, HL
reduces to GR and one obtains a flat universe filled with radiation and dust
fluids, where radiation dominates near the singularity and dust dominates far
from it.

On the other hand, when stiff matter like time is negligible ($g_{s}\to 0$) in
Eqs. (18), one obtains:

$\displaystyle
a(\eta)=\left\\{\begin{array}[]{lll}\sqrt{\frac{(P_{\eta}+g_{r})}{g_{c}}}\sin\left(\sqrt{g_{c}}\eta\right)+\frac{P_{\varphi}}{2g_{c}}\left[1-\cos\left(\sqrt{g_{c}}\eta\right)\right]&,&k=1,\\\
\sqrt{\frac{(P_{\eta}+g_{r})}{g_{c}}}\sinh\left(\sqrt{g_{c}}\eta\right)+\frac{P_{\varphi}}{2g_{c}}\left[\cosh\left(\sqrt{g_{c}}\eta\right)-1\right]&,&k=-1.\end{array}\right.$
(23)

In the following I will consider the HL quantum solutions.

##  IV Quantum dynamics Hǒrava–Lifshitz Cosmology

In this section, I consider analytical solutions of the HL quantum cosmology
in minisuperspace for non-interacting radiation and dust fluids. The cyclic
and bounce universe solutions presented in this section are obtained for the
first time for HL theory for nonzero $g_{c},\,g_{r},\,g_{s}$ and for radiation
and dust fluids.

The HL super-Hamiltonian constraint containing radiation and dust fluids, from
Eq. (13), reads:

$\displaystyle\mathcal{H}_{rd}=-\frac{P_{a}^{2}}{4a}-g_{c}ka+\frac{g_{r}k^{2}}{a}+\frac{g_{s}k}{a^{3}}+\frac{P_{\eta}}{a}+P_{\varphi}\approx
0,$ (24)

where I have considered the lapse function $N=a$ and $g_{\Lambda}=0$ as in the
Sec. III. Using the Dirac formalism for constrained systems, the super-
Hamiltonian constraint is promoted to an operator, which annihilates the
quantum wave function of the universe, $\Psi$, in the form:

$\displaystyle\hat{\mathcal{H}}_{\rm rd}\Psi=0.$ (25)

This is the so called Wheeler-DeWitt equation. From Eqs. (24) and (25), one
obtains:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!i\partial_{\eta}\Psi=\left(\frac{1}{4}\partial_{a}^{2}-g_{c}ka^{2}+g_{r}k^{2}+\frac{g_{s}k}{a^{2}}-i\,a\,\partial_{\varphi}\right)\Psi,$
(26)

where $\hat{P}_{a}=-i\partial_{a}$, $\hat{P}_{\eta}=-i\partial_{\eta}$ and
$\hat{P}_{\varphi}=-i\partial_{\varphi}$ (see Refs. PintoNeto:2005gx ;
Lapchinsky:1977vb ) and $\Psi=\Psi(a,\varphi,\eta)$. However, two important
considerations must be made. Firstly, there is an operator-ordering ambiguity
in the Wheeler-DeWitt equation, which is related to the choice of measure in
the path integral in the canonical quantization procedure when one replaces
the momentum $P_{a}$ by its corresponding operator Halliwell:1988wc . In order
to account for this ambiguity, the kinetic term must be rewritten properly.
Secondly, in order to get rid of the derivative $\partial_{\varphi}$,
following Ref. PintoNeto:2005gx  I will consider that the wave function $\Psi$
is an eigenstate of the dust matter operator, such that
$\hat{P}_{\varphi}|\Psi\rangle=P_{\varphi}|\Psi\rangle$, which implies that
dust matter is conserved 222As a first attempt, I consider this simple case,
where the evolution is non-unitary. However, one can manage to obtain an
unitary solution in which dust matter creation is possible PintoNeto:2005gx ..
Therefore, the wave function $\Psi$ in an eigenstate of $\hat{P}_{\varphi}$
with eigenvalue $P_{\varphi}$ and the wave function
$\Psi=\Psi(a,\varphi,\eta)$ can be written as:

$\displaystyle\Psi(a,\varphi,\eta)=\Psi(a,\eta)e^{iP_{\varphi}\varphi}.$ (27)

From these considerations, Eq. (26) now reads:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!i\partial_{\eta}\Psi(a,\eta)$
$\displaystyle=$
$\displaystyle\left(-\frac{1}{4}\partial_{a}^{2}+\frac{\alpha}{4a}\partial_{a}+g_{c}ka^{2}-g_{r}k^{2}-\frac{g_{s}k}{a^{2}}-P_{\varphi}\,a\right)\Psi(a,\eta),$
(28)

where the parameter $\alpha$ represents the ambiguity in the ordering of $a$
and $P_{a}$ in the kinetic term of Eq. (24) and the transformation
$\eta\to-\eta$ was also considered. The appropriate choice of $\alpha$ will be
useful in the following calculations, although the results must not depend on
it. On the other hand, the sign change in the time variable was done in order
to write the Wheeler-DeWitt equation as a Schrödinger-type equation (except
for the term involving $\alpha$, at this point).

###  IV.1 De Broglie-Bohm Interpretation

The wave function $\Psi$ can be written in the polar form as
$\Psi=R\,e^{i\,S}$, such that imaginary and real parts give evolution
equations for $R$ and $S$, respectively, which are real functions. In order to
introduce the dBB interpretation, I write the Lagrangian for Eq. (28), which
reads:

$\displaystyle\mathcal{L}_{\rm rd}$ $\displaystyle=$ $\displaystyle
a^{-\alpha}\left[i\Psi^{*}\partial_{\eta}\Psi-\frac{1}{4}\partial_{a}\Psi^{*}\partial_{a}\Psi-V\Psi^{*}\Psi\right],$
(29)

where

$\displaystyle V=g_{c}ka^{2}-\frac{g_{s}k}{a^{2}}-g_{r}k^{2}-P_{\varphi}a,$
(30)

is an external classical potential. From Noether’s theorem PESKIN , invariance
of $\Psi$ under internal symmetry ($\Psi\to e^{i\theta}\Psi$) results in a
conserved charge $\rho$ and a conserved current $J$, which are related by the
following continuity equation:

$\displaystyle\partial_{\eta}\left(a^{-\alpha}R^{2}\right)+\partial_{a}\left(a^{-\alpha}R^{2}\frac{\partial_{a}S}{2}\right)=0,$
(31)

where $\rho=a^{-\alpha}R^{2}$ and $J=a^{-\alpha}R^{2}(\partial_{a}S)/2$. This
equation is exactly the imaginary part of Eq. (28) mentioned before and is
interpreted as the equation of conservation of probability.

In the dBB interpretation, particles have a deterministic trajectory, which is
given by $a(\eta)$ in the present case. Therefore, an equation of motion must
be postulated. This equation can be built out of $J=\rho\,v$, where
$v=a^{\prime}$ is the velocity of the particle. Therefore,

$\displaystyle a^{\prime}=\frac{\partial_{a}S}{2},$ (32)

which is known as the guidance equation 333For arbitrary lapse function $N$
and equation of state parameter $\omega$, and undoing the transformation
$\eta\to-\eta$, $a^{\prime}=-\frac{N}{2a}(\partial_{a}S)$.. From the knowledge
of $S$ and an initial condition for $a$, one can integrate it to obtain
$a(\eta)$. Also, one can notice the trajectory of the particle is independent
of the choice of $\alpha$.

On the other hand, the real part of Eq. (28) in terms of $R$ and $S$ reads:

$\displaystyle\partial_{\eta}S+\frac{(\partial_{a}S)^{2}}{4}+V+Q=0.$ (33)

This equation is a Hamilton–Jacobi-type equation, where the last term,

$\displaystyle
Q=-\frac{1}{4}\frac{\partial_{a}^{2}(a^{-\alpha}R)}{(a^{-\alpha}R)},$ (34)

is the so called quantum potential. Therefore, from Eq. (33), one concludes
that deterministic trajectories are subject to classical and quantum
potentials.

Therefore, in order to obtain a solution for $a(\eta)$ from Eq. (32), one
needs to solve both Eqs. (31) and (33), which are coupled equations for $R$
and $S$.

###  IV.2 Analytical Results

The main goal of this paper is to present analytical solutions for Eq. (28).
There is a solution for a similar problem in the literature, which can be
adapted for the present case. However, some changes of variables must be done
in order to use this solution. These details will be presented in the
following. From the analytical solutions of the Wheeler-DeWitt equation,
analytical solutions can be obtained for the scale factor, solving Eq. (32).

The aforementioned solution is given in Ref. PintoNeto:2005gx . This solution
was given for the particular case of Eq. (28) when $g_{c}=1$, $g_{r}=0$ and
$g_{s}=0$. Additionally, the authors set $\alpha=0$. The Wheeler-DeWitt
equation for this case (Eq. (25) from Ref. PintoNeto:2005gx ) reads:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!i\partial_{\eta}\psi(a,\eta)$
$\displaystyle=$
$\displaystyle\left(-\frac{1}{2m}\partial_{a}^{2}+\frac{m\omega_{0}^{2}}{2}a^{2}-P_{\varphi}\,a\right)\psi(a,\eta).$
(35)

This is a one-dimensional Schrödinger equation for a particle with mass $m$
and a potential which contains an harmonic oscillator term with frequency
$\omega_{0}$ and a constant force $P_{\varphi}$. In order to use this solution
for the present case, one needs to reduce Eq. (28) to Eq. (35) performing some
changes of variables. One also needs to set $\alpha\neq 0$, which is in fact
the key point here and, as shown in Sec. IV.1, it does not affect the result
for the scale factor.

I consider the following change of variables for the wave function of Eq.
(28):

$\displaystyle\Psi(a,\eta)=e^{ig_{r}k^{2}\eta}a^{\alpha/2}\mu(a,\eta),$ (36)

where $\alpha=-1\pm\sqrt{1+16g_{s}k}$. Both values of $\alpha$ are suitable to
absorb the term $g_{s}/a^{2}$. I choose:

$\displaystyle\alpha=-1+\sqrt{1+16g_{s}k},$ (37)

where one obtains $\alpha\to 0$ when $g_{s}\to 0$. Also, the term
$-g_{r}k^{2}$ is also absorbed in the complex exponential. From these change
of variables, Eq. (28) is now an equation for $\mu(a,\eta)$, which reads:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!i\partial_{\eta}\mu(a,\eta)$
$\displaystyle=$
$\displaystyle\left(-\frac{1}{4}\partial_{a}^{2}+g_{c}ka^{2}-P_{\varphi}\,a\right)\mu(a,\eta).$
(38)

This equation is exactly Eq. (35) for the particular case when:

$\displaystyle\quad m$ $\displaystyle=$ $\displaystyle 2,$ (39)
$\displaystyle\omega_{0}$ $\displaystyle=$ $\displaystyle\sqrt{g_{c}\,k}.$
(40)

For these values of $\omega_{0}$ and $m$, the analytical result for the wave
function, given by Eq. (41) of Ref. PintoNeto:2005gx , reads:

$\displaystyle\mu(a,\eta)$
$\displaystyle=\left(\frac{8\bar{\sigma}}{\pi}\right)^{1/4}\sqrt{\frac{1}{\cos\left(\sqrt{g_{c}k}\eta\right)\left[1+i\bar{\sigma}\frac{\tan\left(\sqrt{g_{c}k}\eta\right)}{\sqrt{g_{c}k}}\right]}}\exp{\left\\{\frac{i\sqrt{g_{c}k}}{\tan\left(\sqrt{g_{c}k}\eta\right)}\right.}$
$\displaystyle\times\left.\left[a^{2}-\frac{\left(a-\frac{P_{\varphi}}{2}\frac{\left[1-\cos\left(\sqrt{g_{c}k}\eta\right)\right]}{g_{c}k}\right)^{2}}{\cos\left(\sqrt{g_{c}k}\eta\right)^{2}\left[1+i\bar{\sigma}\frac{\tan\left(\sqrt{g_{c}k}\eta\right)}{\sqrt{g_{c}k}}\right]}+aP_{\varphi}\frac{1-\cos\left(\sqrt{g_{c}k}\eta\right)}{g_{c}k\cos\left(\sqrt{g_{c}k}\eta\right)}\right.\right.$
$\displaystyle\\!\\!\\!\left.\left.+\frac{P_{\varphi}^{2}}{2g_{c}k}\left(\eta\frac{\tan\left(\sqrt{g_{c}k}\eta\right)}{\sqrt{g_{c}k}}-\frac{1-\cos\left(\sqrt{g_{c}k}\eta\right)}{g_{c}k\cos\left(\sqrt{g_{c}k}\eta\right)}\right)\right]\right\\}.$
(41)

The solution for $\mu(a,\eta)$ was obtained using the propagator of the forced
quantum harmonic oscillator using the following initial state:

$\displaystyle\mu_{0}(a,0)=\left(\frac{8\,\bar{\sigma}}{\pi}\right)^{1/4}e^{-\bar{\sigma}a^{2}},$
(42)

where $\bar{\sigma}=\sqrt{\sigma^{2}-g_{s}k}+i\sqrt{g_{s}k}$. One must notice
that both the solution and its initial condition contain the parameter $g_{s}$
in the constant $\bar{\sigma}$, although there is no term involving $g_{s}$ in
Eq. (38). In fact, only the parameters $\sigma$ and $P_{\varphi}$, which are
related to radiation and dust fluids, respectively, were expected.
Additionally, the dynamics for this matter content is radiation domination
near $\eta=0$ and dust domination far from it. Therefore, an initial condition
at $\eta=0$ must involve only the parameter $\sigma$. However, the full
problem considered in this section before the changes of variables involves a
$g_{s}$ term, which, from the classical dynamics of Sec. III, behaves as a
stiff matter-like fluid and dominates over radiation near $\eta=0$. Then, one
should expect that an initial condition for this problem involves uniquely
$g_{s}$. Nonetheless, due to the fact that the $g_{s}$ term does not represent
a fluid, radiation fluid degrees of freedom must also be present in the
initial condition, i.e., it must be a function of both $g_{s}$ and $\sigma$.
In other words, although changes of variables are performed and $g_{s}$ is no
longer explicit in Eq. (38), the $g_{s}$ information is encoded in
$\bar{\sigma}$ due to its relevance near $\eta=0$. In the Appendix A, I derive
an ansatz solution for the particular case of a radiation dominated quantum
FLRW universe in the framework of HL theory where only the $g_{s}$ parameter
is nonzero. The obtained ansatz justifies this choice of initial condition.
Analogously, the $g_{r}$ radiation fluid-like term is encoded in radiation
fluid degree of freedom, $\sigma$.

From this analytical result, the expression for $\Psi(a,\varphi,\eta)$, from
Eqs. (27), (36) and (37), reads:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\Psi(a,\varphi,\eta)=e^{ig_{r}k^{2}\eta}e^{iP_{\varphi}\varphi}a^{\left(-1+\sqrt{1+16g_{s}k}\right)/2}\mu(a,\eta),$
(43)

where $\mu(a,\eta)$ is given by the solution of Eq. (41),

From Eq. (43), one can obtain $S(a,\eta)$ and substitute it in Eq. (32).
Solving this equation for the initial condition $a(0)=a_{B}$, one obtains:

$\displaystyle
a(\eta)=\left\\{\begin{array}[]{lll}a_{B}\sqrt{\cos^{2}\left(\sqrt{g_{c}}\eta\right)+\sqrt{\frac{g_{s}}{g_{c}}}\sin\left(2\sqrt{g_{c}}|\eta|\right)+\frac{\sigma^{2}}{g_{c}}\sin^{2}\left(\sqrt{g_{c}}\eta\right)}+\frac{P_{\varphi}}{2g_{c}}\left[1-\cos\left(\sqrt{g_{c}}\eta\right)\right]&,&k=1,\\\
a_{B}\sqrt{\cosh^{2}\left(\sqrt{g_{c}}\eta\right)+\sqrt{\frac{|g_{s}|}{g_{c}}}\sinh\left(2\sqrt{g_{c}}|\eta|\right)+\frac{\sigma^{2}}{g_{c}}\sinh^{2}\left(\sqrt{g_{c}}\eta\right)}+\frac{P_{\varphi}}{2g_{c}}\left[\cosh\left(\sqrt{g_{c}}\eta\right)-1\right]&,&k=-1.\end{array}\right.$
(46)

These are quantum bounce solutions, which, compared to the classical ones, the
squared trigonometric and hyperbolic cosine novel terms play the role of
avoiding singularities. These results enable the study of the background
quantum cosmology for this model from an analytical perspective.

The positive spatial section solution represents a cyclic universe (no big
bang and big crunch singularities), whereas the negative spatial section
solution has a unique bounce, which connects a contraction and an expansion
phase. However, one must consider some limits for the constants $g_{c}$ and
$g_{s}$ in order to appreciate the overlap of these effects. Some plots will
be presented later in this section in order to illustrate the behavior of
these solutions for different values of its constants.

Comparing Eqs. (46) with the classical solutions, Eqs. (18), one obtains that
in the classical limit $\sigma^{2}=P_{\eta}+g_{r}k^{2}$, which means that
$\sigma$ is a degree of freedom related to a radiation fluid. Therefore,
$g_{r}$ only contributes to an effective radiation fluid density and does not
appear explicitly in $a(\eta)$. Also, the classical behavior is obtained when

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left.\begin{array}[]{rl}&|\tan\left(\sqrt{g_{c}}\eta\right)|\\\
&|\tanh\left(\sqrt{g_{c}}\eta\right)|\end{array}\right\\}\gg\frac{\sqrt{g_{c}}(\sqrt{|g_{s}|+\sigma^{2}}-\sqrt{|g_{s}|})}{\sigma^{2}},$
(49)

for $k=1$ and $k=-1$, respectively. In the limit where $g_{c}\to 1$ and
$g_{s}\to 0$, these results reduce to $1/\sigma$ for both values of $k$, where
$\sigma$ contains the $g_{r}k^{2}$ term. Also, both results reduce to
$|\eta|\gg 1/\sigma$ when one additionally considers the limit $k\to 0$ (see
Ref. PintoNeto:2005gx ).

One must draw attention to the module $|\eta|$ in the terms involving $g_{s}$
in Eqs. (46). This is because the wave function given in Eq. (41) was obtained
by the propagation of the initial condition only for negative values of
$\eta$. Therefore, one needs to solve also for positive values of $\eta$
(setting $\eta\to-\eta$ in both Eqs. (32) and (41)) and incorporate it in the
solutions. Due to the presence of odd trigonometric and hyperbolic sines, the
module ensures that the solution is symmetric, which can be confirmed from the
classical solutions of Sec. III. This observation was not necessary in the
calculations of Ref. PintoNeto:2005gx , because only even functions appear in
the results for $a(\eta)$.

As in Sec. III, I will consider some particular cases of Eqs. (46). In the
limit where $g_{c}\to 0$, for both values of $k$ it reduces to:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!a(\eta)=a_{B}\sqrt{1+2\sqrt{|g_{s}|}\,|\eta|+\sigma^{2}\eta^{2}}+\frac{P_{\varphi}\eta^{2}}{4}.$
(50)

Also considering in the latter equation the limit where $g_{s}\to 0$, it
reduces to:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!a(\eta)=a_{B}\sqrt{1+\sigma^{2}\eta^{2}}+\frac{P_{\varphi}\eta^{2}}{4},$
(51)

where now $\bar{\sigma}=\sigma$. This result resembles the case of a FLRW flat
universe in GR with radiation and dust PintoNeto:2005gx , except for the HL
radiation term correction for $k\neq 0$. When $k=0$, HL theory reduces to GR
and the results are exactly the same.

On the other hand, when $g_{s}\to 0$, Eqs. (46) reduce to:

$\displaystyle
a(\eta)=\left\\{\begin{array}[]{lll}a_{B}\sqrt{\cos^{2}\left(\sqrt{g_{c}}\eta\right)+\frac{\sigma^{2}}{g_{c}}\sin^{2}\left(\sqrt{g_{c}}\eta\right)}+\frac{P_{\varphi}}{2g_{c}}\left[1-\cos\left(\sqrt{g_{c}}\eta\right)\right]&,&k=1,\\\
a_{B}\sqrt{\cosh^{2}\left(\sqrt{g_{c}}\eta\right)+\frac{\sigma^{2}}{g_{c}}\sinh^{2}\left(\sqrt{g_{c}}\eta\right)}+\frac{P_{\varphi}}{2g_{c}}\left[\cosh\left(\sqrt{g_{c}}\eta\right)-1\right]&,&k=-1.\end{array}\right.$
(54)

These results are qualitatively similar to the ones of Ref. PintoNeto:2005gx
when $g_{c}=1$, which are given by:

$\displaystyle
a(\eta)=\left\\{\begin{array}[]{lll}a_{B}\sqrt{\cos^{2}\left(\eta\right)+\sigma^{2}\sin^{2}\left(\eta\right)}+\frac{P_{\varphi}}{2}\left[1-\cos\left(\eta\right)\right]&,&k=1,\\\
a_{B}\sqrt{\cosh^{2}\left(\eta\right)+\sigma^{2}\sinh^{2}\left(\eta\right)}+\frac{P_{\varphi}}{2}\left[\cosh\left(\eta\right)-1\right]&,&k=-1.\end{array}\right.$
(57)

In order to illustrate the results, I present some plots of the scale factor
for representative values, which capture the effect of each constant in its
evolution. I will consider $P_{\varphi}=0$ for all plots and focus on the
effects of $g_{c}$ and $g_{s}$, which are introduced by HL theory.
Additionally, in all plots there is a gray curve which corresponds to the
solution with respect to GR, where the HL parameters are null, given by Eq.
(51).

In Figure 1, the scale factor is considered in the limit where $g_{s}\to 0$,
given by Eqs. (54). In panel (a), one notices that for $k=1$ nonzero values of
$g_{c}$ produce oscillating universes, which are cyclic and with multiple
bounces. Starting from $g_{c}=0$, for increasing values of $g_{c}$ the
oscillation frequency grows and the oscillation amplitude becomes smaller. In
panel (b), for $k=-1$, increasing values of $g_{c}$ make the bounce more
curved.

In Figure 2, the scale factor is considered in the limit where $g_{c}\to 0$,
given by Eq. (50). In this case, for both values of the spatial section, the
bouncing universes become more curved for increasing values of $g_{s}$. This
is qualitative similar to the case $g_{s}\to 0$ for $k=-1$ of panel (b) of
Fig. 1, where it is the increase of $g_{c}$ that progressively curves the
universes. However, their difference is that in the present case
$a(\eta)/a_{B}\approx 1+\sqrt{g_{s}}|\eta|+\mathcal{O}\left(\eta^{2}\right)$
near the bounce, whereas $a(\eta)/a_{B}\approx
1+\left(g_{c}+\sigma^{2}\right)\eta^{2}/2+\mathcal{O}\left(\eta^{3}\right)$ in
panel (b) of Fig. 1. Therefore, the bounce solution is more abrupt in the
present case.

In Figure 3, the scale factor is considered for nonzero values of both $g_{c}$
and $g_{s}$, given by Eqs. (46). The effects are overlapped, which is the more
general case. In panel (a), where $k=1$, one notices that the oscillatory
behavior due to nonzero values of $g_{c}$ is affected by nonzero values of
$g_{s}$ in two aspects: (i) in the oscillatory regime, the frequency and the
oscillation amplitude increase (far from the bounce) and (ii) the bounce
around $\eta=0$ is linear, whereas the other bounces are qualitatively similar
and deeper. In panel (b), where $k=-1$, the effects of both increasing values
of $g_{c}$ and $g_{s}$, as I have shown in Fig. 1 panel (b) and Fig. 2, make
the bounce more curved. Therefore, the effects add up. However, near the
bounce the scale factor reads $a(\eta)/a_{B}\approx
1+\sqrt{g_{s}}|\eta|+\left(g_{c}-|g_{s}|+\sigma^{2}\right)\eta^{2}/2+\mathcal{O}\left(\eta^{3}\right)$,
which means that $g_{s}$ dominates near the bounce, whereas as we move away
from the bounce both effects become relevant and $g_{c}$ eventually dominates
over $g_{s}$.

Figure 1:  Analytical scale factor in quantum HL theory as a function of the
conformal time in the limit $g_{s}\to 0$. The plots are given for $a_{B}=1$,
$P_{\varphi}=0$, $\sigma=0.5$ and some representative values of $g_{c}$. In
panels (a) and (b) the spatial sections $k=1$ and $k=-1$ are considered,
respectively. Figure 2: Analytical scale factor in quantum HL theory as a
function of the conformal time in the limit $g_{c}\to 0$. The plots are given
for $a_{B}=1$, $P_{\varphi}=0$, $\sigma=0.5$ and some representative values of
$g_{s}$. The plots are valid for both spatial sections, $k=\pm 1$, where
$g_{s}>0\,(g_{s}<0)$ for $k=1\,(k=-1)$.

Figure 3: Analytical scale factor in quantum HL theory as a function of the
conformal time for representative values of $g_{c}$ and $g_{s}$. The plots are
given for $a_{B}=1$, $P_{\varphi}=0$, $\sigma=0.5$. In panels (a) and (b) the
spatial sections $k=1$ and $k=-1$ are considered, respectively.

From the solution for the wave function, Eq. (43), one can also obtain the
quantum potential, Eq. (34), which is responsible for the quantum effects. The
full analytical expression is lengthy, but one can obtain a reasonable
approximation noticing from Eqs. (46) that the dust fluid contribution
dominates only far from the bounce. The quantum effects are present in the
whole evolution for $k=1$ when $g_{c}\neq 0$, but the dust fluid contribution
does not significantly affect the qualitative quantum potential behavior.
Therefore, it is a good approximation to consider Eqs. (46) when $P_{\varphi}$
is negligible. From these considerations, the quantum potential, Eq. (34),
reads:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!Q(\eta)=\frac{a_{B}^{2}\left[\sqrt{\sigma^{2}-|g_{s}|}-4a_{B}^{2}\left(\sigma^{2}-|g_{s}|\right)\right]}{a(\eta)^{2}},$
(58)

where $a(\eta)$ is given by Eqs. (46) for negligible $P_{\varphi}$. This
result is a generalization of Eq. (22) of Ref. Oliveira-Neto:2017yui . One
explicitly notices that when $a_{B}=0$, i.e., the universe is singular, the
quantum potential vanishes.

In Fig. 4, one can notice that the behavior of $Q(\eta)$ for both spatial
sections is in agreement with the panels of Fig. 3. For $k=1$, the cyclic
universe solutions with multiple bounces show that the quantum potential is
also oscillatory and non-negligible in the whole time evolution. On the other
hand, for $k=-1$ there is a unique bounce at $\eta=0$ dominated by the $g_{s}$
effect, which becomes more curved as we increase the values of $g_{c}$ and
$g_{s}$.

Figure 4: Quantum potential in HL theory as a function of the conformal time
for both $k=\pm 1$. The plots are given for the representative values
$a_{B}=1$, $P_{\varphi}=0$ and $\sigma=g_{c}=|g_{s}|=0.5$.

##  V Conclusions

In this work I have presented a quantum minisuperspace model of FLRW cosmology
in the framework HL theory of gravity. For this purpose, I have considered the
HL theory in its projectable version and without detailed balance condition.
The Universe is filled with non-interacting radiation and dust fluids, which
were introduced using the Schutz approach. Canonical quantization is then
performed and a Wheeler-DeWitt equation is obtained, Eq. (28). Performing a
separation of variables and a suitable choice of parameters, an analytical
solution of Eq. (28) for the wave function was first obtained in this context
for nonzero values of $g_{c}$, $g_{r}$ and $g_{s}$ present in the HL
Hamiltonian, Eq. (13). I have set $g_{\Lambda}=0$, which will be considered in
a future work.

From the solution of the wave function, I have considered the dBB
interpretation of quantum mechanics in order to derive analytical solutions
for the scale factor, given by Eqs. (46). These are quantum bounce solutions
in the cases of closed and open FLRW quantum cosmologies in HL theory. The
closed universe solutions are cyclic and with multiple bounces, which avoid
the big bang and big crunch singularities. On the other hand, the open
universe solutions are contracting/expanding unique bounce solutions, which
get more curved as one increases the values of the $g_{c}$ and $g_{s}$ HL
parameters.

I have also obtained the quantum potential, Eq. (58), whose qualitative
behavior confirms its role in the quantum evolution of each solution, i.e.,
that it is the responsible for the avoidance of the singularities.

Classical results were also presented in Sec. III, which were important in
order to identify $\bar{\sigma}$ and construct the quantum results with a
well-defined classical limit. In the Appendix A, an ansatz for the wave
function was derived in the case where only the radiation fluid degree of
freedom and the $g_{s}$ parameter of HL gravity are nonzero, which confirms
the value of $\bar{\sigma}$.

As future perspectives, these results will be important to study gravitational
particle production and baryogenesis in this context. On the other hand, it
would also be interesting to consider a stiff matter fluid ($\omega=1$) in
this context as well $g_{\Lambda}\neq 0$, as aforementioned.

## ACKNOWLEDGMENTS

I thank Yves E. Chifarelli for useful discussions during the elaboration of
this paper.

##  Appendix A Ansatz solution for a quantum FLRW universe filled with
radiation and nonzero $g_{s}$ HL parameter

In this section, I consider the quantum cosmology for HL theory in the
particular case where $g_{c}=g_{r}=0$ and $g_{s}\neq 0$ (apart from
$g_{\Lambda}$, which is neglected in the whole paper) filled with a radiation
fluid. This case corresponds to a particular case of Eq. (28), which reads:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!i\partial_{\eta}\Psi(a,\eta)$
$\displaystyle=$
$\displaystyle\left(-\frac{1}{4}\partial_{a}^{2}+\frac{\alpha}{4a}\partial_{a}-\frac{g_{s}k}{a^{2}}\right)\Psi(a,\eta).$
(59)

A solution for this equation can be obtained from the following ansatz:

$\displaystyle\Psi_{\rm
ansatz}(a,\eta)=a^{\alpha/2}f(\eta)\,e^{-g(\eta)a^{2}},$ (60)

where $f(\eta)$ and $g(\eta)$ are arbitrary functions of $\eta$. From Sec. IV,
the quantum analytical solution for this case, where the particular choice
$\alpha=-1+\sqrt{1+16g_{s}k}$ was considered, is given by:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!a(\eta)=a_{B}\sqrt{1+2\sqrt{g_{s}k}\,|\eta|+\sigma^{2}\eta^{2}}.$
(61)

The expressions for $f(\eta)$ and $g(\eta)$ of the wave-function ansatz which
reproduce this analytical solution read:

$\displaystyle f(\eta)=\frac{1}{\sqrt{1+i\bar{\sigma}\eta}}\qquad,\qquad
g(\eta)=\frac{\bar{\sigma}}{1+i\bar{\sigma}\eta}.$ (62)

where $\bar{\sigma}=\sqrt{\sigma^{2}-g_{s}k}+i\sqrt{g_{s}k}$. In terms of
these functions, the ansatz results:

$\displaystyle\Psi_{\rm ansatz}(a,\eta)\propto
a^{(-1+\sqrt{1+16g_{s}k})/2}\frac{e^{-\frac{\bar{\sigma}a^{2}}{1+i\bar{\sigma}\eta}}}{\sqrt{1+i\bar{\sigma}\eta}},$
(63)

From the ansatz, the initial condition at $\eta=0$ reads:

$\displaystyle\Psi_{\rm ansatz}(a,0)\propto
a^{(-1+\sqrt{1+16g_{s}k})/2}e^{-\bar{\sigma}a^{2}}.$ (64)

Comparing this result with Eqs. (36) and (41), this is exactly the initial
dependence on the scale factor $a$. One can also notice the presence of the
parameter $\bar{\sigma}$ in the exponential. Therefore, Eq. (42) is the
appropriate initial condition, noticing that the term
$a^{(-1+\sqrt{1+16g_{s}k})/2}$ was factorized by a previous change of
variables.

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