gr short=GR , long=general relativity uv short=UV , long=ultraviolet ir
short=IR , long=infrared wdw short=WDW , long=Wheeler-DeWitt hl short=HL ,
long=Hořava-Lifshitz rg short=RG , long=renormalization group adm short=ADM ,
long=Arnowitt, Deser and Misner

# DeWitt wave function in Hořava-Lifshitz cosmology
with tensor perturbation

Paul Martens<EMAIL_ADDRESS>Center for Gravitational
Physics and Quantum Information, Yukawa Institute for Theoretical Physics,
Kyoto University, 606-8502, Kyoto, Japan  Hiroki Matsui
<EMAIL_ADDRESS>Center for Gravitational Physics and
Quantum Information, Yukawa Institute for Theoretical Physics, Kyoto
University, 606-8502, Kyoto, Japan  Shinji Mukohyama
<EMAIL_ADDRESS>Center for Gravitational Physics and
Quantum Information, Yukawa Institute for Theoretical Physics, Kyoto
University, 606-8502, Kyoto, Japan  Kavli Institute for the Physics and
Mathematics of the Universe (WPI), The University of Tokyo Institutes for
Advanced Study, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan

(August 27, 2024)

###### Abstract

We present a well-tempered DeWitt wave function, which vanishes at the
classical big-bang singularity, in Hořava-Lifshitz (HL) cosmology with tensor
perturbation, both analytically and numerically. In general relativity, the
DeWitt wave function is ill-behaved once the tensor perturbation is taken into
account. This is essentially because the amplitude of the perturbation
diverges at the singularity and the perturbative expansion completely breaks
down. On the other hand, in HL gravity it is known that the higher dimensional
operators required by the perturbative renormalizability render the tensor
perturbation scale-invariant and regular all the way up to the singularity. In
this paper we analytically show that in $d+1$ dimensional HL gravity the
DeWitt wave function for tensor perturbation is indeed well-defined around the
classical big-bang singularity. Also, we numerically demonstrate the well-
behaved DeWitt wave function for tensor perturbation from the big-bang to a
finite size of the Universe.

††preprint: YITP-22-52, IPMU22-0028

###### Contents

  1. I Introduction
  2. II Framework
    1. II.1 Hořava-Lifshitz gravity
    2. II.2 Cosmological setup
    3. II.3 Canonical quantization
  3. III DeWitt wave function with tensor perturbations
    1. III.1 Analytical estimation
    2. III.2 Numerical solution
  4. IV Conclusion
  5. A Hořava-Lifshitz DeWitt wave function without tensor perturbations

##  I Introduction

Attempts to build a quantum field theory of gravity have encountered many
serious issues. As is well known, gr is not renormalizable, essentially
because Newton’s constant is dimensionfull (with the mass dimension
$\left[G_{N}\right]=-2$). The non-renormalizability leads to uncontrollable uv
divergences. Adding higher curvature terms to the Einstein-Hilbert action can
make the theory renormalizable Stelle (1977), or even super-renormalizable
Asorey et al. (1997), but leads to massive ghosts and makes the corresponding
quantum theory non-unitary in the UV.

That is where hl gravity Horava (2009) enters. This quantum gravity theory is
power-counting renormalizable, thanks to the presence of higher-order spatial
curvature terms. The action and equations of motion in hl gravity contain only
terms up to second order in time derivatives and thus the theory is free from
Ostrogradsky ghosts associated with higher time derivatives. The hl theory was
recently proved to be perturbatively renormalizable Barvinsky et al. (2016,
2018), and, therefore, has come to be regarded as a valid uv completion path
of quantum gravity.

One of the fundamental principles of the hl gravity is the so-called
anisotropic scaling, or Lifshitz scaling, i.e.

$t\to b^{z}t\,,\quad\vec{x}\to b\vec{x}\,,$ (1)

where $t$ is the time coordinate, $\vec{x}$ represents the spatial coordinates
vector and $z$ is a number called dynamical critical exponent. In $3+1$
dimensions, an anisotropic scaling of $z=3$ in the uv regime breaks Lorentz
symmetry, but ensures renormalizability, while the usual $z=1$ scaling is
recovered in the ir regime. An anisotropic scaling of $z=3$ also handily
provides a mechanism for generating scale-invariant cosmological
perturbations, simultaneously solving the horizon problem Mukohyama (2009a)
without inflation and providing the so-called anisotropic instanton, which
could be an answer to the flatness problem Bramberger et al. (2018).

Although there have been numerous works on the early cosmology based on the HL
gravity, the early Universe cannot be treated classically when quantum gravity
is dominant, and we have to adopt a quantum cosmology approach. A common
approach, in quantum cosmology, is to separate space and time using the adm
formalism and work with the Hamiltonian formulation. In this framework, the
Hamiltonian constraint is interpreted as an operator equation, the wdw
equation DeWitt (1967), $\hat{H}[g]\Psi[g]=0$ where $\hat{H}[g]$ is the
Hamiltonian operator and $\Psi[g]$ is called the wave function of the Universe
(see Halliwell (1989); Wiltshire (1995) for a review).

Whether this wave function of the Universe gives finite correlation functions
of physical perturbations has a long history of debate among quantum
cosmologists. In particular, in recent years, in the Lorentzian path integral
formulation of the no-boundary Hartle and Hawking (1983) and tunneling
proposals Vilenkin (1984) —well-known boundary conditions of quantum
cosmology—, it has been argued that the wave function of small perturbations
around the background may take an inverse Gaussian form and become
uncontrollable Feldbrugge et al. (2017, 2018). In this work, we adopt instead
the so-called the DeWitt boundary condition DeWitt (1967), which states that
the wave function of the Universe should vanish at the classical big-bang
singularity. In a homogeneous and isotropic Universe, the DeWitt boundary
condition can be concisely expressed as $\Psi(a=0)=0$ and is known to
successfully regularize the behavior of the wave function near the classical
singularity. In a mini-superspace, where the dynamics of the Universe is
parameterized only by the scale factor $a(t)$, it is easy to find an analytic
expression for the DeWitt wave function, i.e. a solution to the wdw equation
with the DeWitt boundary condition. However, how to generalize this result
beyond this mini-superspace is not obvious.

In a previous work Matsui et al. (2021), we have analytically shown that, in
GR and many other theories of gravity, the DeWitt wave function for a
homogeneous and isotropic background with tensor perturbations is not well-
behaved. The supposedly small perturbations are not suppressed near the
classical big-bang singularity. This is a serious problem if the DeWitt wave
function is adopted as a description of the very early Universe in theories of
gravity, including gr. Therefore, the DeWitt boundary condition, proposed to
overcome the classical singularity in quantum cosmology, requires a
description of gravity that goes beyond gr. Fortunately, in the hl gravity,
the wave function of the tensor perturbation is found to behave well, thanks
to the presence of higher dimensional operators that are required for
perturbative renormalizability.

We here extend this result from $3+1$ dimensions to $d+1$ dimensions, and show
that the DeWitt wave function for tensor perturbation exhibits scale
invariance near the classical big-bang singularity and thus is regular. We
also demonstrate the well-behaved DeWitt wave function for tensor perturbation
numerically. This numerical demonstration for the first time illustrates the
regular behavior of the DeWitt wave function for tensor perturbation in hl
gravity all the way from the classical big-bang singularity to finite values
of the scale factor.

The rest of the present paper is organized as follows: in section II we give a
brief review of the quantum cosmology framework in the $d+1$ dimensional HL
gravity. In section III, we derive the DeWitt wave function for a $d+1$
dimensional Universe described by a homogeneous and isotropic background and
tensor perturbation around it, and show that the wave function for the tensor
perturbation is normalizable and scale-invariant on constant-$a$ hypersurfaces
near $a=0$, where $a$ is the scale factor. This result is due to the
anisotropic scaling with $z=d$. Numerical results are demonstrated for $d=3$
for concreteness. Finally, we conclude our work in section IV.

##  II Framework

###  II.1 Hořava-Lifshitz gravity

In this section we will briefly review the construction of hl gravity in $d+1$
dimensions and introduce the wdw equation. Similar to the adm decomposition of
the metric in gr Arnowitt et al. (2008), we can write the $d+1$ dimensional
metric as

$\displaystyle\mathrm{d}s^{2}=-N^{2}\mathrm{d}t^{2}+g_{ij}(\mathrm{d}x^{i}+N^{i}\mathrm{d}t)(\mathrm{d}x^{j}+N^{j}\mathrm{d}t)\,,$
(2)

where $i=1,\dots,d$, $N$ is the lapse function, $N^{i}$ the shift vector, and
$g_{ij}$ is the spatial metric with positive definite signature
$(+,+,\cdots)$. The $d+1$ dimensional action $S$ describing hl gravity is the
sum of a kinetic part $\mathcal{L}_{K}$ and a potential part $\mathcal{L}_{V}$
Mukohyama (2010); that is

$S=\frac{{\cal M}_{\rm
HL}^{d-1}}{2}\int\mathrm{d}t\mathrm{d}^{d}xN\sqrt{g}\left({\cal L}_{K}+{\cal
L}_{V}\right)\,,$ (3)

where ${\cal M}_{\rm HL}$ sets the overall mass scale. The kinetic part
$\mathcal{L}_{K}$ can be expressed in terms of the extrinsic curvature
$K_{ij}$ as

${\cal L}_{K}=(K^{ij}K_{ij}-\lambda K^{2})\,,\quad\text{where}\quad
K_{ij}=\frac{1}{2N}(\partial_{t}g_{ij}-g_{jk}\nabla_{i}N^{k}-g_{ik}\nabla_{j}N^{k})\,,$
(4)

with $\lambda$ being a coupling constant, and $\nabla_{i}$ the spatial
covariant derivative compatible with $g_{ij}$. The indices $i,j$ run from $1$
to $d$ and the inverse of $g_{ij}$ is written as $g^{ij}$. Accordingly, we
wrote $K^{ij}=g^{ik}g^{jl}K_{kl}$ and $K=g^{ij}K_{ij}$. For simplicity, we
will consider the projectable hl gravity where the lapse function depends only
on time, i.e. $N=N(t)$. The results of the present paper can be directly
applied to the non-projectable theory as well, by simply setting
$C_{\alpha}=0$ (and thus $\mathcal{C}=0$).

As for the potential part, it shall be built out of any possible
renormalizable operators without parity violating terms, and must be expressed
in terms of invariants assembled from the products of the Riemann tensor
(schematically denoted as Rm below) and its derivatives Sotiriou et al.
(2009a, b). For the highest-order potential part, denoted as
$\mathcal{L}_{z}$, the list of possible terms is

$\\{(\textrm{Rm})^{d},\,(\nabla\textrm{Rm})^{2}(\textrm{Rm})^{d-3},\,(\nabla\textrm{Rm})^{4}(\textrm{Rm})^{d-6},\,(\nabla\textrm{Rm})^{6}(\textrm{Rm})^{d-9},\cdots\\}\,.$
(5)

Now, including lower-dimensions terms as well, we come to write

$\displaystyle\begin{split}{\cal L}_{V}&={\cal L}_{0}+{\cal L}_{1}+{\cal
L}_{2}+{\cal L}_{3}+\cdots+{\cal L}_{z-1}+{\cal L}_{z}\,,\end{split}$ (6)

where ${\cal L}_{i}$ denotes a collection of invariant terms with $2i$ spatial
derivatives acted on the spatial metric. Here, the first two terms ${\cal
L}_{0}+{\cal L}_{1}$ are necessary to recover the $d+1$-dimensional gr in the
ir limit, the third term ${\cal L}_{2}$ includes all possible quadratic and
spatial curvature terms, and so on. Explicitly, the four five terms
$\mathcal{L}_{i}$ read

$\displaystyle{\cal L}_{0}$ $\displaystyle=-2\Lambda\,,$ (7a)
$\displaystyle{\cal L}_{1}$ $\displaystyle=c^{2}_{g}R\,,$ (7b)
$\displaystyle{\cal L}_{2}$
$\displaystyle=c_{2,1}R^{2}+c_{2,2}R_{i}^{j}R_{j}^{i}+c_{2,3}R^{ijkl}R_{ijkl}\,,$
(7c) $\displaystyle{\cal L}_{3}$
$\displaystyle=c_{3,1}R^{3}+c_{3,2}RR_{i}^{j}R_{j}^{i}+c_{3,3}R_{i}^{j}R_{j}^{k}R_{k}^{i}+c_{3,4}\nabla_{i}R\nabla^{i}R+c_{3,5}\nabla_{i}R_{jk}\nabla^{i}R^{jk}+\cdots\,,$
(7d) $\displaystyle\begin{split}{\cal
L}_{4}&=c_{4,1}R^{4}+c_{4,2}R^{2}R_{i}^{j}R_{j}^{i}+c_{4,3}RR_{i}^{j}R_{j}^{k}R_{k}^{i}+c_{4,4}R_{i}^{j}R_{j}^{k}R_{k}^{l}R_{l}^{i}\\\
&\phantom{=c_{4,1}R^{4}+c_{4,2}R^{2}R_{i}^{j}R_{j}^{i}}+c_{4,5}R\,\nabla_{i}R\nabla^{i}R+c_{4,6}R\,\nabla_{i}R_{jk}\nabla^{i}R^{jk}+\cdots\,,\end{split}$
(7e)

where $R$, $R_{ij}$ and $R_{ijkl}$ respectively denote the Ricci scalar, the
Ricci tensor and the Riemann tensor, and dots represent terms depending on the
Riemann tensor and its derivatives. The constants $\Lambda$ and $c_{g}$ are
the cosmological constant and the propagation speed of tensor gravitational
waves, and $c_{m,n}$ are constants of appropriate dimensions. All those
constants are subject to running under the rg flow.

In the uv regime, terms with two time derivatives and those with $2z$ spatial
derivatives are dominant. If $z=d$, the theory is renormalizable and if $z>d$,
the theory is super-renormalizable. In the ir regime, on the other hand,
higher derivative terms are subdominant and the theory naturally converges to
$z=1$. Moreover, if $\lambda$ (of eq. 4) goes to $1$ in the ir limit, and if
it does sufficiently quickly, then gr is recovered thanks to an analogue of
the Vainshtein mechanism Mukohyama (2010); Izumi and Mukohyama (2011);
Gumrukcuoglu et al. (2012). For simplicity, in the rest of the present paper,
we restrict our considerations to the renormalizable theory, i.e. $z=d$.

###  II.2 Cosmological setup

We shall assume the $d$-dimensional space of the model to be the union of
connected pieces $\Sigma_{\alpha}$ ($\alpha=1,\dots$), each of which we shall
call a local $d$-dimensional Universe. The union of all $\Sigma_{\alpha}$
would thus represent the Universe in its entirety. In such a configuration,
while the lapse function $N=N(t)$ is common for all pieces, we have a set of a
shift vector $N^{i}=N_{\alpha}^{i}(t,x)$ and a spatial metric
$g_{ij}=g^{\alpha}_{ij}(t,x)$, which is different for different local Universe
$\Sigma_{\alpha}$. In the ADM formalism, the lapse and the shift vectors
basically act as Lagrange multipliers. The variation of the projectable hl
action (eq. 3) with respect to $N$ yields the following Hamiltonian constraint

$\sum_{\alpha}\int_{\Sigma_{\alpha}}\mathrm{d}^{d}x\,\mathcal{H}_{g\perp}=0\,,\quad\text{where}\quad\mathcal{H}_{g\perp}=\frac{{\cal
M}_{\rm HL}^{d-1}}{2}\sqrt{g}({\cal L}_{K}-{\cal L}_{V})\,.$ (8)

While this constrain sets the sum of contributions from all $\Sigma_{\alpha}$
to vanish, each contribution does not have to vanish, i.e.

$\int_{\\{\Sigma_{\alpha}\\}}d^{d}x\,\mathcal{H}_{g\perp}\neq 0\,.$ (9)

Therefore, if we are interested in one particular local Universe, i.e. an
element of $\\{\Sigma_{\alpha}\\}$, the Hamiltonian constraint does not need
to be enforced.

Hereafter, we further assume that each connected space $\Sigma_{\alpha}$ is a
closed universe, and that each closed universe is described by a closed
Friedmann-Lemaître-Robertson-Walker (FLRW) metric and perturbations around it.

$N_{\alpha}^{i}=0\,,\quad
g^{\alpha}_{ij}=a^{2}_{\alpha}(t)\left[\Omega_{ij}({\bf
x})+h_{ij}^{\alpha}(t\,,{\bf x})\right]\,,$ (10)

where $\Omega_{ij}$ is the metric of the unit $d$-sphere with the curvature
constant set to $1$, i.e. the Riemann curvature of $\Omega_{i}j$ is simply
$\delta^{i}_{k}\delta^{j}_{l}-\delta^{i}_{l}\delta^{j}_{k}$. Given this
definition, the spatial indices $i,j,\dots$ are thus raised and lowered by
$\Omega^{ij}$ and $\Omega_{ij}$ respectively. The tensor perturbation
$h^{\alpha}_{ij}$ must satisfy the transverse and traceless condition, i.e.
$\Omega^{ij}h^{\alpha}_{ij}=\Omega^{ki}D_{k}h^{\alpha}_{ij}=0$, where $D_{i}$
is the spatial covariant derivative compatible with $\Omega_{ij}$.
Furthermore, the perturbation $h_{ij}^{\alpha}$ can be expanded in terms of
the hyper-spherical harmonics Gerlach and Sengupta (1978) as

$h_{ij}^{\alpha}(t,x^{i})=\sum_{snlm}h^{\alpha s}_{nlm}(t)Q^{snlm}_{ij}\,,$
(11)

where $s=\pm$ is the polarization label and the triplet $(n,l,m)$ is comprised
within the ranges $n\geq 3$, $l\in[0,n-1]$ and $m\in[-l,l]$. The tensor
eigenfunctions $Q^{snlm}_{ij}$ of the Laplacian operator $D^{2}[\Omega]$ on
the unit $d$-sphere Higuchi (1987) satisfy

$D^{2}[\Omega]\,Q^{snlm}_{ij}=-\left[n^{2}+(d-3)n-d\right]Q^{snlm}_{ij}\,,$
(12)

and are normalized via the following relation

$\int
d^{d}\sqrt{\Omega}\Omega^{ik}\Omega^{jl}Q^{snlm}_{ij}Q^{s^{\prime}n^{\prime}l^{\prime}m^{\prime}}_{kl}=V_{d}\,\delta^{ss^{\prime}}\delta^{nn^{\prime}}\delta^{ll^{\prime}}\delta^{mm^{\prime}}\,.$
(13)

With these assumptions and the harmonic expansion (11), the $d+1$-dimensional
hl action is expanded up to the second order in perturbations as
$S=S^{(0)}+S^{(2)}+\mathcal{O}(h^{3})$, where

$\displaystyle S^{(0)}$ $\displaystyle={\cal
V}\sum_{\alpha}\int\mathrm{d}t\,\left({Na_{\alpha}^{d}}\right)\left[\frac{d(1-d\lambda)}{2}\left(\frac{\dot{a}_{\alpha}}{Na_{\alpha}}\right)^{2}+\frac{\alpha_{d}}{a^{2d}_{\alpha}}\cdots+\frac{\alpha_{3}}{a^{6}_{\alpha}}+\frac{\alpha_{2}}{a^{4}_{\alpha}}+c_{\rm
g}^{2}\frac{d(d-1)}{2a^{2}_{\alpha}}-\Lambda\right]\,,$ (14a) $\displaystyle
S^{(2)}$ $\displaystyle={\cal
V}\sum_{\alpha}\int\mathrm{d}t\left({Na_{\alpha}^{d-2}}\right)\sum_{snlm}\left[\frac{1}{8}\left(\frac{a_{\alpha}}{N}\dot{h}^{\alpha
s}_{nlm}\right)^{2}+\left(\beta_{1}+\frac{\beta_{2}}{a^{2}_{\alpha}}+\frac{\beta_{3}}{a^{4}_{\alpha}}\cdots+\frac{\beta_{d}}{a^{2d-2}_{\alpha}}\right)\left(h^{\alpha
s}_{nlm}\right)^{2}\right]\,.$ (14b)

Here, $\alpha_{i}$ and $\beta_{i}$ are constants that are linear combinations
of coupling constants in the action, ${\cal V}={\cal M}_{\rm HL}^{d-1}V_{d}$
and $V_{d}=\int\mathrm{d}^{d}x\sqrt{\Omega}$ is the volume of the unit
$d$-sphere. For the tensor perturbation of each connected space
$\Sigma_{\alpha}$, in the following for simplicity we restrict our attention
to the dynamics of only one mode $h^{\alpha s}_{nlm}$ with
$(s,n,l,m)=(s_{\alpha},n_{\alpha},l_{\alpha},m_{\alpha})$, which we denote by
$h_{\alpha}$. With this reduction, the system is thus described by
$\\{a_{\alpha},h_{\alpha}\\}$.

###  II.3 Canonical quantization

Now, we turn attention to quantum cosmology, equipped with hl gravity and
assuming that the dynamics of the homogeneous and isotropic Universe
(including any tensor perturbations) is governed by the wave function of the
wdw equation. From the hl action (eq. 14), we can compute the corresponding
Hamiltonian and obtain

$\displaystyle H$
$\displaystyle=\sum_{\alpha=1}\left(\Pi_{a_{\alpha}}\dot{a}_{\alpha}+\Pi_{h_{\alpha}}\dot{h}_{\alpha}\right)-\mathcal{L}$
(15a) $\displaystyle\begin{split}&=\sum_{\alpha=1}{\cal
V}\left(\frac{N}{a_{\alpha}^{d-2}}\right)\biggl{[}-\frac{1}{2\gamma}\Pi_{a_{\alpha}}^{2}-\frac{\alpha_{d}}{a^{2}_{\alpha}}\cdots-\frac{\alpha_{3}}{a^{8-2d}_{\alpha}}-\frac{\alpha_{2}}{a^{6-2d}_{\alpha}}-c_{\rm
g}^{2}\frac{d(d-1)}{2a^{4-2d}_{\alpha}}+\frac{\Lambda}{a^{2-2d}_{\alpha}}\\\
&\quad\quad+\frac{2}{{\cal
V}^{2}a^{2}_{\alpha}}\Pi_{h_{\alpha}}^{2}-\left(\frac{\beta_{1}}{a^{4-2d}_{\alpha}}+\frac{\beta_{2}}{a^{6-2d}_{\alpha}}+\frac{\beta_{3}}{a^{8-2d}_{\alpha}}\cdots+\frac{\beta_{d}}{a^{2}_{\alpha}}\right)h_{\alpha}^{2}\biggr{]}\,,\end{split}$
(15b)

where $\mathcal{L}$ is defined by
$S^{(0)}+S^{(2)}=\int\mathrm{d}t\mathcal{L}$,
$\gamma=d\left(d\lambda-1\right){\cal V}^{2}$ and the canonical momenta
$\Pi_{a_{\alpha}}$ and $\Pi_{h_{\alpha}}$ conjugate respectively to
$a_{\alpha}$ and $h_{\alpha}$ are given by

$\Pi_{a_{\alpha}}=\frac{\partial\mathcal{L}}{\partial\dot{a}_{\alpha}}=-d\left(d\lambda-1\right){\cal
V}\frac{a_{\alpha}^{d-2}}{N}\dot{a}_{\alpha}\,,\quad\Pi_{h_{\alpha}}=\frac{\partial\mathcal{L}}{\partial\dot{h}_{\alpha}}=\frac{{\cal
V}}{4}\frac{a_{\alpha}^{d}}{N}\dot{h}_{\alpha}\,.$ (16)

We can now carry out the canonical quantization of hl gravity. This is done by
transforming the canonical variables of Hamiltonian mechanics into Hermitian
operators that satisfy the canonical commutation relations. The algebra
generated by commutative quantities ($c$-numbers) becomes an algebra generated
by non-commutative quantities ($q$-numbers) in quantum mechanics. Therefore,
an ambiguity in the operator ordering here arises. In performing the canonical
quantization, the canonical conjugate momenta are transformed into Hermitian
operators, i.e.

$\begin{array}[]{r}\Pi_{a_{\alpha}}\mapsto-i\dfrac{\partial}{\partial
a_{\alpha}}\\\\[10.0pt] \Pi_{h_{\alpha}}\mapsto-i\dfrac{\partial}{\partial
h_{\alpha}}\end{array}\quad\Longrightarrow\quad\begin{array}[]{l}\Pi_{a_{\alpha}}^{2}=-\dfrac{1}{a_{\alpha}^{p}}\dfrac{\partial}{\partial
a_{\alpha}}\left(a_{\alpha}^{p}\dfrac{\partial}{\partial
a_{\alpha}}\right)\\\\[10.0pt]
\Pi_{h_{\alpha}}^{2}=-\dfrac{\partial^{2}}{\partial
h_{\alpha}^{2}}\end{array}\,,$ (17)

where we should take into account the operator ordering ambiguity of
$a_{\alpha}$ Hartle and Hawking (1983). In quantum cosmology, there exists two
well-known orderings: the Laplace-Beltrami operator ordering ($p=1$) and the
Vilenkin ordering ($p=-1$) Steigl and Hinterleitner (2006).

Once the canonical quantization of the Hamiltonian of eq. 15 is realized, we
obtain the wdw equation

$\displaystyle\sum_{\alpha}\frac{1}{a_{\alpha}^{d-2}}\,\Biggl{\\{}$
$\displaystyle\frac{1}{2\gamma}\left(\frac{\partial^{2}}{\partial
a^{2}_{\alpha}}+\frac{p}{a_{\alpha}}\frac{\partial}{\partial
a_{\alpha}}\right)-\frac{\alpha_{d}}{a^{2}_{\alpha}}-\cdots-\frac{\alpha_{2}}{a^{6-2d}_{\alpha}}-c_{\rm
g}^{2}\frac{d(d-1)}{2a^{4-2d}_{\alpha}}+\frac{\Lambda}{a^{2-2d}_{\alpha}}$
(18) $\displaystyle-\frac{2}{{\cal
V}^{2}a^{2}_{\alpha}}\frac{\partial^{2}}{\partial
h_{\alpha}^{2}}-\left(\frac{\beta_{1}}{a^{4-2d}_{\alpha}}+\frac{\beta_{2}}{a^{6-2d}_{\alpha}}+\cdots+\frac{\beta_{d}}{a^{2}_{\alpha}}\right)h_{\alpha}^{2}\Biggr{\\}}\Psi\left(\\{a_{\alpha},h_{\alpha}\\}\right)=0\,,$

where $\Psi\left(\\{a_{\alpha},h_{\alpha}\\}\right)$ is the wave function of
the entire Universe. By separating the variables, one can easily obtain
special solutions of the form
$\Psi=\prod_{\alpha}\Psi_{\alpha}(a_{\alpha},h_{\alpha};C_{\alpha})$, where
$\\{C_{\alpha}\\}$ are separation constants satisfying
$\sum_{\alpha}C_{\alpha}=0$. A general solution can be then written as a
linear combination of these special solutions, explicitly

$\Psi\left(\\{a_{\alpha},h_{\alpha}\\}\right)=\int\Biggl{(}\prod_{\beta}\mathrm{d}C_{\beta}\Biggr{)}A_{\\{C_{\beta}\\}}\prod_{\alpha}\Psi_{\alpha}\left(a_{\alpha},h_{\alpha};C_{\alpha}\right)\,.$
(19)

where each $\Psi_{\alpha}\left(a_{\alpha},h_{\alpha};C_{\alpha}\right)$
represents the wave function of one local Universe $\Sigma_{\alpha}$.

We now redefine the variables and constants in eq. 18 as well as the
separation constant $C_{\alpha}$.

$\begin{gathered}\mathfrak{h}=\frac{h_{\alpha}}{2\sqrt{\gamma}}\,;\qquad\mathcal{C}=\gamma
C_{\alpha}\,;\\\ g_{d}=\gamma\alpha_{d}\,,\quad\cdots\,,\quad
g_{2}=\gamma\alpha_{2}\,,\quad g_{1}=\frac{\gamma c_{\rm
g}^{2}d(d-1)}{2}\,,\quad g_{0}=\gamma\Lambda\,;\\\
f_{d}=-8\gamma^{2}\beta_{d}\,,\quad\cdots\,,\quad
f_{2}=-8\gamma^{2}\beta_{2}\,,\quad
f_{1}=-8\gamma^{2}\beta_{1}\,;\end{gathered}$

The WDW equation for each local universe can then be re-written as

$\displaystyle\left\\{\frac{1}{2}\left(\frac{\partial^{2}}{\partial
a^{2}}+\frac{p}{a}\frac{\partial}{\partial
a}\right)+\left(\mathcal{C}a^{d-2}-\frac{g_{d}}{a^{2}}-\cdots-\frac{g_{2}}{a^{6-2d}}-\frac{g_{1}}{a^{4-2d}}+\frac{g_{0}}{a^{2-2d}}\right)\right.$
$\displaystyle\quad\quad\quad\left.-\frac{1}{2{\cal
V}^{2}a^{2}}\frac{\partial^{2}}{\partial\mathfrak{h}^{2}}+\frac{\mathfrak{h}^{2}}{2}\left(\frac{f_{1}}{a^{4-2d}}+\frac{f_{2}}{a^{6-2d}}+\cdots+\frac{f_{d}}{a^{2}}\right)\right\\}\Psi(a,\mathfrak{h})=0\,,$
(20)

where we have dropped the index $\alpha$ and abbreviated
$\Psi_{\alpha}\left(a_{\alpha},h_{\alpha};C_{\alpha}\right)$ as
$\Psi(a,\mathfrak{h})$.

##  III DeWitt wave function with tensor perturbations

In this section, we compute the $d+1$-dimensional DeWitt wave function for a
universe represented by a homogeneous and isotropic background and tensor
perturbation around it. In particular we show that the DeWitt wave function
near the classical big-bang singularity features scale-invariant tensor
perturbations due to the anisotropic scaling with $z=d$.

###  III.1 Analytical estimation

We will start by extending the results of our previous work Matsui et al.
(2021) to $d+1$ dimensions. In order to obtain a solution of the WDW equation
(section II.3) that satisfies the DeWitt boundary condition, i.e.
$\Psi(0,\mathfrak{h})=0$ for all $\mathfrak{h}$, we adopt the following
expansion around $a=0$,

$\Psi(a,\mathfrak{h})=a^{c}\sum_{i=0}^{\infty}F_{i}(\mathfrak{h})\,a^{i}\,,$
(21)

where $F_{0}(\mathfrak{h})$ is not identically zero and $c$ is a positive
constant to be determined. For $\Psi(a,\mathfrak{h})$ to be normalizable on
constant-$a$ hypersurfaces, we also impose the following condition on
$\Psi(a,\mathfrak{h})$

$\lim_{\mathfrak{h}\to\pm\infty}F_{i}(\mathfrak{h})=0\,.$ (22)

If this condition (eq. 22) is not satisfied, the correlation functions such as
the power spectrum of $\mathfrak{h}$ on constant-$a$ hypersurfaces would
diverge. In order to determine the positive constant $c$, we use the above
mentioned requirement that $F_{0}(\mathfrak{h})$ be a non-trivial smooth
function satisfying the condition of eq. 22 (with $i=0$).

In gr, we previously demonstrated that there are no parameter set for which
$F_{0}(\mathfrak{h})$ can satisfy the condition of eq. 22 (with $i=0$) and the
tensor perturbations are not smoothly suppressed Matsui et al. (2021). In
other words, no DeWitt wave function is normalizable on constant-$a$
hypersurfaces around the classical big-bang singularity, unless higher spatial
derivative terms are present, as we will see in the following.

Next, let us turn our attention to hl gravity. By substituting the small-$a$
expansion (eq. 21) into the wdw equation (section II.3), and solving it order
by order, we can explicitly determine the functions $F_{i}$. For $i=0,1,2$, we
find the following three equations

$\displaystyle\partial_{\mathfrak{h}}^{2}F_{0}-{\cal
V}^{2}\left[f_{d}\mathfrak{h}^{2}+(c+p-1)c-2g_{d}\right]F_{0}$
$\displaystyle=0\,,$ (23a)
$\displaystyle\partial_{\mathfrak{h}}^{2}F_{1}-{\cal
V}^{2}\left[f_{d}\mathfrak{h}^{2}+(c+p)(c+1)-2g_{d}\right]F_{1}$
$\displaystyle=0\,,$ (23b)
$\displaystyle\partial_{\mathfrak{h}}^{2}F_{2}-{\cal
V}^{2}\left[f_{d}\mathfrak{h}^{2}+(c+p+1)(c+2)-2g_{d}\right]F_{2}$
$\displaystyle={\cal V}^{2}(f_{d}\mathfrak{h}^{2}-3g_{d-1})F_{0}\,.$ (23c)

Let us first consider the first relation, i.e. eq. 23a. By requiring (22)
(with $i=0$) and the continuity of both $F_{0}(\mathfrak{h})$ and
$\partial_{\mathfrak{h}}F_{0}(\mathfrak{h})$ at $\mathfrak{h}=0$, we obtain

$F_{0}(\mathfrak{h})=\left\\{\begin{array}[]{ll}\frac{A}{\sqrt{\mathfrak{h}}}W_{N+1/4,1/4}(w)\,,&\
(\mathfrak{h}>0)\\\ -\frac{A}{\sqrt{-\mathfrak{h}}}W_{N+1/4,1/4}(w)\,,&\
(\mathfrak{h}<0)\end{array}\right.\,,$ (24)

where $W_{\mu,\nu}(w)$ is the Whittaker function, $A$ is some constant, $N$
some non-negative integer, and $w$ and $c$ are defined by

$w={\cal V}\sqrt{f_{d}}\mathfrak{h}^{2}\,,\quad
c^{2}+(p-1)c+\left[\frac{\sqrt{f_{d}}}{{\cal V}}(4N+1)-2g_{d}\right]=0\,,$
(25)

which, in turn, constrains $f_{d}$ to be strictly positive.

For eq. 23b, by again requiring (22) (with $i=1$) and the continuity of both
$F_{1}(\mathfrak{h})$ and $\partial_{\mathfrak{h}}F_{1}(\mathfrak{h})$ at
$\mathfrak{h}=0$, and refraining from fine-tuning, we find
$F_{1}(\mathfrak{h})\equiv 0$ as the next-to-leading order solution in $a$.
However, at the next-to-next-to-leading order, $F_{2}$ cannot be identically
zero. Once $F_{0}(\mathfrak{h})$ is given, $F_{2}$ is determined by eqs. 23c
and 22 (with $i=2$). While the computation is straightforward, the explicit
expression for $F_{2}$ is fairly long. Hence, we show only its structure, for
$N=0,1,2$, without explicit expressions. Using the leading-order solution for
different values of $N$, i.e.

$F_{0}(\mathfrak{h})=\begin{cases}A_{0}\exp\left(-\frac{1}{2}w\right)\,,&(N=0)\\\
(1-2w)A_{1}\exp\left(-\frac{1}{2}w\right)\,,&(N=1)\\\
\left(1-4w+\frac{4}{3}w^{2}\right)A_{2}\exp\left(-\frac{1}{2}w\right)\,,&(N=2)\end{cases}\,,$
(26)

where $A_{N}$ ($N=0,1,2$) is some integration constant, the solution for
$F_{2}$ takes the form

$F_{2}(\mathfrak{h})=\sum_{\tilde{N}=0}^{N+1}a_{N,\tilde{N}}w^{\tilde{N}}A_{N}\exp\left(-\frac{1}{2}w\right)\,.$
(27)

Here, $a_{N,\tilde{N}}$ ($\tilde{N}=0,\cdots,N+1$) are constant coefficients
determined by the parameters of the wdw equation (section II.3).

If $N=0$, we obtain the ground state of the DeWitt wave function, which, in
the $a\to+0$ limit, can be expressed as

$\Psi(a,\mathfrak{h})=A({\cal V}\sqrt{f_{d}})^{1/4}a^{c}\left[e^{-\frac{{\cal
V}\sqrt{f_{d}}\mathfrak{h}^{2}}{2}}+\mathcal{O}(a^{2})\right]\,.$ (28)

Since $f_{d}\propto n^{2d}$ for large $n$, the corresponding correlation
function for the tensor perturbations then reads

$\lim_{a\to+0}\langle\mathfrak{h}^{2}\rangle=\lim_{a\to+0}\frac{\int\mathrm{d}\mathfrak{h}\,\mathfrak{h}^{2}\left|\Psi(a,\mathfrak{h})\right|^{2}}{\int\mathrm{d}\mathfrak{h}\,\left|\Psi(a,\mathfrak{h})\right|^{2}}=\frac{1}{2{\cal
V}\sqrt{f_{d}}}\propto n^{-d}\,,\quad\mbox{for large }\,n\,\mbox{ and
}\,N=0\,.$ (29)

This clearly shows the scale-invariance and the finiteness of the power
spectrum of $\mathfrak{h}$ for $f_{d}>0$ (see Matsui et al. (2021) for the
$d=3$ case). Also in the general case of eq. 24 with general $N$ ($\geq 0$),
the power spectrum and other correlation functions of $\mathfrak{h}$ are
scale-invariant and finite in the UV limit ($a\to+0$) as far as $f_{d}>0$. In
particular,

$\mathcal{M}_{\rm
HL}^{3d-3}\lim_{a\to+0}n^{d}\langle\mathfrak{h}^{2}\rangle=\mathcal{O}(1)\times{\cal
M}^{d-1}\,,\quad\mbox{for large }\,n\,\mbox{ and any }\,N\geq 0\,,$ (30)

where $\mathcal{M}$ is the mass scale such that $\beta_{d}\sim
n^{2d}/\mathcal{M}^{2d-2}$ (and thus ${\cal V}\sqrt{f_{d}}\sim
n^{d}\mathcal{M}_{\rm HL}^{3d-3}/\mathcal{M}^{d-1}$) for large $n$. This is
completely consistent with the result of Mukohyama (2009a).

We have seen that eq. 25 and eqs. (23c) and (22) (with $i=0$) requires
$f_{d}>0$. Therefore, if $f_{d}\leq 0$ then there is no non-trivial smooth
solution satisfying eq. 22 (with $i=0$). In particular, this is the case in
the absence of $z=d$ terms (for which $f_{d}=0$). This no-go result applies to
many gravitational theories (including gr) for which the action does not
contain terms with $2d$ spatial derivatives.

###  III.2 Numerical solution

Figure 1:  Numerical integration result for the parameter set $f_{2}=0$,
$\mathcal{C}=0$, $g_{0}=0$. For any $N$, while the amplitude of $\Psi$ here
exponentially increases (central column) with $a$, the initial shape is
conserved unchanged (right column).

Figure 2:  Numerical integration result for the parameter set $f_{2}=1$,
$\mathcal{C}=0$, $g_{0}=0$. While overall similar, unlike the case
$f_{2}=\mathcal{C}=g_{0}=0$ of fig. 1, the shape of the wave function evolves
significantly through the “evolution” towards larger $a$, as the rightmost
columns exhibits. Essentially, the amplitude’s growth is slowed down the
further it is from $\mathfrak{h}=$0$$ (central and right columns) as $f_{2}$
actually enables a term in $\mathfrak{h}^{2}$ in eq. 32.

Figure 3:  Numerical integration result for the parameter set $f_{2}=1$,
$\mathcal{C}=10$, $g_{0}=0$. Now that $\mathcal{C}$ is set to $10$, a term in
$a$ now enters eq. 32 and will quickly dominate over the previous $-a^{-2}$
term. In so doing, it brings the system in a regime of damped oscillation
(central column). The effect of $f_{2}$ (see fig. 2) remains, though now
slightly muted (right column).

Figure 4:  Numerical integration result for the parameter set $f_{2}=1$,
$\mathcal{C}=10$, $g_{0}=10$. The $g_{0}$ coefficient now brings a $a^{4}$
term, that will eventually overcome the previous $a$ and $-a^{-2}$
contributions, pushing the system into an even more strongly damped
oscillation regime (central column). The influence of $f_{2}$ remains visible,
though now barely visible (right column).

Numerically integrating the wdw equation (section II.3) is fairly
straightforward. The equation is hyperbolic, and one can use the expansion
around $a=0$ obtained in the previous section III.1 as the initial condition
on a constant-$a$ hypersurface, provided that the initial value of $a$ is
sufficiently small. The initial condition is uniquely specified by the non-
negative integer $N$ up to an overall amplitude. For concreteness, we fix the
overall amplitude by

$\lim_{a\to+0}a^{-c}\Psi(a,0)=1\,.$ (31)

This translates into $A_{N}=1$ for all $N=0,1,2$.

In the numerical study, we shall only consider the $3+1$ dimensional case,
employ the second-order finite difference scheme for derivatives with respect
to $\mathfrak{h}$, and adopt two different integration methods to evolve the
system with respect to $a$. Both the iterated Crank-Nicolson method with two
iterations and the fourth-order Runge-Kutta method have shown identical
satisfactory results 111The two independent codes were previously used in
Berti et al. (2022) and one of them was used also in Bhattacharjee et al.
(2018); Mukohyama and Namba (2021).. To this end, the “space” $\mathfrak{h}$
is sampled on a grid of $40\,000$ points spaced by
$\Delta\mathfrak{h}=$1\text{⋅}{10}^{-4}$$. As for the “time”, the range in $a$
is discretized into $10^{6}$ steps, each of size $\Delta
a=10^{-1}\cdot\Delta\mathfrak{h}=$1\text{⋅}{10}^{-5}$$. We start from the
small initial “time” $a=a_{0}$ by using the solution obtained in III.1 —the
expansion of eq. 21 with eqs. 26 and 27—, which is valid for sufficiently
small $a_{0}$. Here, we choose $a_{0}=10^{-1}\cdot\Delta
a=$1\text{⋅}{10}^{-6}$$.

For simplicity we set $p=1$, $g_{3}=2$, $f_{3}=1$, and $g_{2}$, $g_{1}$,
$f_{1}$ and $f_{0}$ to $0$ (with $d=3$) in all simulations shown hereafter.
Then, the Wheeler-DeWitt equation (II.3) is specified by the remaining
parameters $f_{2}$, $\mathcal{C}$ and $g_{0}$. It can now be compactly written
as

$\left\\{\frac{1}{2}\left(\frac{\partial^{2}}{\partial
a^{2}}+\frac{1}{a}\frac{\partial}{\partial
a}\right)+\left[\mathcal{C}a-\frac{2}{a^{2}}+g_{0}a^{4}+\left(f_{2}+\frac{1}{a^{2}}\right)\frac{\mathfrak{h}^{2}}{2}\right]-\frac{1}{2{\cal
V}^{2}a^{2}}\frac{\partial^{2}}{\partial\mathfrak{h}^{2}}\right\\}\Psi(a,\mathfrak{h})=0\,.$
(32)

From thereon, we show the numerical results in figs. 1, 2, 3 and 4, and make
some comments.

In the four parameter sets exhibited here, we turn on one after the other each
of the three parameters we left free, i.e. $f_{2}$, $\mathcal{C}$ and $g_{0}$.
In fig. 1, we set $f_{0}=\mathcal{C}=g_{0}=0$ and there are only two non-
derivative terms, both of which are proportional to $a^{-2}$. The system in
this case exhibits the exact $z=3$ anisotropic scaling. The initial input here
simply exponentially grows, without its form being altered. This is easily
understood by observing the separability of eq. 32 with respect to
$\mathfrak{h}$ and $a$. For $f_{2}=1$, on the other hand, a term
$\propto\mathfrak{h}^{2}\cdot a^{0}$ enters and, as seen in fig. 2, visibly
modifies the shape of the wave function.

In the next and last two figures, figs. 3 and 4, as we set $\mathcal{C}$ then
$g_{0}$ to 10, we enable, in turn, terms $\propto a$ and $\propto a^{4}$,
respectively, in eq. 32. They will drive the system into a damped oscillation
regime; $g_{0}a^{4}$ more strongly so than $\mathcal{C}a$. As they come to
dominate, the effects of $f_{2}\mathfrak{h}^{2}$ and $\mathfrak{h}^{2}/a^{2}$,
while remaining, are proportionately muted.

Overall, the afore-described behaviors are as expected from eq. 32 (or section
II.3). Most importantly, the wave function is obviously well-defined and
normalizable on constant-$a$ hypersurfaces, all the way from the classical
big-bang singularity to finite values of $a$.

##  IV Conclusion

We have investigated the behavior of a $d+1$-dimensional Universe near the
classical big-bang singularity based on the Wheeler-DeWitt equation and the
DeWitt boundary condition, which amounts to a vanishing wave function of the
Universe at the classical singularity. For concreteness we have studied a
homogeneous and isotropic background and tensor perturbation around it
described respectively by the scale factor $a$ and the amplitude of tensor
perturbation $\mathfrak{h}$ as a simple model of the Universe. In general
relativity, the DeWitt wave function for $\mathfrak{h}$ on constant-$a$
hypersurfaces is not normalizable near $a=0$, meaning that the perturbative
expansion breaks down. On the contrary, in Hořava-Lifshitz gravity the higher
dimensional operators required by the perturbative renormalizability of the
theory render the wave function of the tensor perturbation normalizable, all
the way from the classical big-bang singularity at $a=0$ to finite values of
$a$. The DeWitt wave function for $\mathfrak{h}$ on constant-$a$ hypersurfaces
is of a Gaussian shape for the ground state and a Gaussian multiplied by an
even polynomial of $\mathfrak{h}$ for excited states. We have analytically
proved these behaviors of the wave function near $a=0$ for any $d$, and
numerically demonstrated them in a finite interval from $a=0$ to a finite
value of $a$. These results provide strong evidences for close connections
between the regularity of the DeWitt wave function of the Universe and the
renormalizability of HL gravity.

###### Acknowledgements.

Two of us (H.M. and S.M.) would like to thank Atsushi Naruko for collaborating
on the previous work Matsui et al. (2021) on the subject. P.M. acknowledges
support from the Japanese Government (MEXT) scholarship for Research Student.
The work of H.M. was supported by JSPS KAKENHI Grant No. JP22J01284. S.M.’s
work was supported in part by JSPS Grants-in-Aid for Scientific Research No.
17H02890, No. 17H06359, and by World Premier International Research Center
Initiative, MEXT, Japan.

##  Appendix A Hořava-Lifshitz DeWitt wave function without tensor
perturbations

We review quantum cosmology based on the hl gravity and consider the DeWitt
wave function in the wdw equation (section II.3) for a homogeneous and
isotropic universe without any perturbations. For only $a$ in $3+1$
dimensions, the hl DeWitt wave function has already been investigated in
previous works e.g. Refs. Bertolami and Zarro (2011); Ali et al. (2012);
Vakili and Kord (2013). However, the hl gravity introduces an extra component
that behaves like a pressure-less dust and that is called dark matter as an
integration constant Mukohyama (2009b, c). The new dark matter component was
derived in the classical cosmology. Correspondingly, when one considers the
wdw equation, the hl gravity naturally introduces a dark matter as a
separation constant (instead of an integration constant) into the local
universe Matsui et al. (2021). Therefore, the extra component potentially
alters the quantum cosmology based on the HL gravity.

Hereafter, we will consider the evolution of the early universe in the hl
quantum cosmology. In the anisotropic scaling regime, the hl term of $g_{3}$
dominates and the solution of the wdw equation (section II.3 with
$\partial/\partial h$ and $h$ removed) approximately takes the from
$\Psi(a)=\mathcal{A}_{1}a^{\tilde{c}}$, where $\mathcal{A}_{1}$ is a
normalization constant Bertolami and Zarro (2011) and $\tilde{c}$ is a
constant determined by $\tilde{c}(\tilde{c}-1)+p\tilde{c}-2g_{3}=0$, i.e.

$\tilde{c}=\frac{1}{2}\left(1-p+\sqrt{(1-p)^{2}+8g_{3}}\right)\,,$ (33)

where we must impose $g_{3}>-(1-p)^{2}/8$ to ensure the regularity of the
solution. Recalling that the DeWitt wave function is regular at the classical
big-bang singularity, the wave function describes a universe that has emerged
from the initial singularity to the anisotropic scaling regime.

In order to highlight the effects of the dark matter as an integration
constant (or as a separation constant), i.e. the ${\cal C}$ term, we set
$g_{2}=g_{1}=g_{0}=0$. In this case the wdw equation (section II.3 with
$\partial/\partial h$ and $h$ removed) reads,

$\displaystyle\left\\{\frac{1}{2}\left(\frac{\partial^{2}}{\partial
a^{2}}+\frac{p}{a}\frac{\partial}{\partial
a}\right)+\mathcal{C}a-\frac{g_{3}}{a^{2}}\right\\}\Psi\left(a\right)=0\,.$
(34)

Thus, imposing the DeWitt boundary condition on the wave function and using
the asymptotic form of $J_{\nu}(z)$, we can get the following wave function

$\displaystyle\begin{split}&\Psi(a)=\mathcal{A}_{2}\,3^{\frac{p-1}{3}}a^{\frac{1-p}{2}}(2\mathcal{C})^{\frac{1-p}{6}}\Gamma\left(1+\frac{1}{3}\sqrt{(1-p)^{2}+8g_{3}}\right)J_{\frac{1}{3}\sqrt{(1-p)^{2}+8g_{3}}}\left(\frac{2}{3}a^{3/2}\sqrt{2\mathcal{C}}\right)\,,\end{split}$
(35)

whose asymptotic forms are given by

$\displaystyle\begin{split}&\Psi(a\ll
1)\sim\mathcal{A}_{2}\,(\mathcal{C}a)^{\frac{1-p+\sqrt{(1-p)^{2}+8g_{3}}}{6}},\\\
&\Psi(a\gg
1)\sim\mathcal{A}_{2}\,a^{-\frac{1+2p}{4}}(2\mathcal{C})^{-\frac{1+2p}{12}}\cos\left(\frac{2}{3}a^{3/2}\sqrt{2\mathcal{C}}-\frac{1+\frac{2}{3}\sqrt{(1-p)^{2}+8g_{3}}}{4}\pi\right)\,.\end{split}$
(36)

Here, $\mathcal{A}_{2}$ is a normalization constant. These expressions show
that the DeWitt wave function is suppressed and oscillates, respectively, in
the anisotropic scaling regime and the dark-matter dominated regime. After the
Universe emerges through the classical big-bang singularity and the
anisotropic scaling regime, the dark matter as an integration constant (or as
a separation constant) represented by the parameter $\mathcal{C}$ dominates
the energy density of the universe. Subsequently, in such a local universe,
the dark matter component may eventually decay into ordinary matter and
radiation Bramberger et al. (2018), and in this case the cosmological
evolution would be smoothly connected to the usual big-bang cosmology.

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