11institutetext: Helmholtz-Institut fÃr Strahlen- und Kernphysik, UniversitÃt
Bonn, 53115 Bonn, Germany

# Exotic Meson candidates in COMPASS data

David SpÃlbeck
for the COMPASS Collaboration 11<EMAIL_ADDRESS>
###### Abstract

One of the prime goals of the COMPASS experiment at CERN is the study of the
light meson spectrum, with a particular emphasis on the search for exotic
states. The focus of this paper is on signals of the lightest hybrid candidate
$\pi_{1}(1600)$ with spin-exotic quantum numbers $J^{PC}=1^{-+}$ in several
decay channels such as $\pi^{-}\pi^{+}\pi^{-}$, $\eta^{(\prime)}\pi^{-}$,
$\omega\pi^{-}\pi^{0}$, $\pi^{-}\pi^{+}\pi^{-}\eta$, and $K_{S}K_{S}\pi$. In
addition, we highlight new results for the $K^{-}\pi^{+}\pi^{-}$ final state,
which indicate a supernumerary state with respect to the constituent quark
model with $J^{P}=0^{-}$.

##  1 Introduction

The constituent quark model (QM) describes a meson as a
$q\bar{q}^{\prime}$-pair, where only the two quarks contribute to the total
quantum numbers $J^{PC}$. However, QCD predicts the existence of exotic, i.e.
non-$q\bar{q}^{\prime}$, mesons in the form of multiquark systems, glueballs
or hybrids. The latter contain an excited gluonic field, which contributes to
the total quantum numbers of the system. There are two sufficient signatures
for the observation of exotic mesons: (i) its quantum numbers are forbidden
within the QM, e.g. $0^{--}$, $(odd)^{-+}$ and $(even)^{+-}$ (spin-exotic), or
(ii) more states than predicted by the QM are observed (supernumerary). The
broad and overlapping states in the light-quark sector require large data sets
and state-of-the-art partial-wave analyses using large wave sets in order to
identify the quantum numbers of the contributing states.

##  2 Meson spectroscopy at COMPASS

The COMPASS experiment is dedicated to investigate the structure and dynamics
of light hadrons COMPASS-2008 . In the scope of its program for light meson
spectroscopy, the world’s largest data sample of diffractive dissociation
reactions has been recorded using a negative hadron beam at
$190\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{/}\textit{c}$, consisting
of mainly $\pi^{-}$ (96.8%) and $K^{-}$ (2.4%), impinging on a liquid hydrogen
target COMPASS-hadron . This allows for precision measurements of established
resonances as well as the search for new states. Isovector resonances of the
$a_{J}$ and $\pi_{J}$ families in the unflavoured sector and $K_{J}^{(*)}$
states in the strange sector can be accessed.

To disentangle the underlying resonances for a selected final state, a
partial-wave analysis (PWA) is performed, which can be separated into two
analysis stages PWAatCOMPASS . In the first stage, the partial-wave
decomposition (PWD), the data is grouped into bins of $m_{X}$, the invariant
mass, and $t^{\prime}$, the reduced four-momentum transfer to the target. For
each bin the strength and relative phase of each partial wave is determined
using an extended log-likelihood fit. In order to do so, the full amplitude
per partial wave is separated into the decay amplitude and a transition
amplitude. The decay amplitude is calculated using the isobar model, in which
the decay of a resonance $X$ is described via a sequence of two-body decays,
and the transition amplitudes are the fit parameters. In the second analysis
stage, the resonance-model fit (RMF), the transition amplitudes are
parameterized by the sum of a resonant and a non-resonant component. The
former is usually approximated by either the sum of relativistic Breit-Wigner
amplitudes or the K-matrix approach and depends only on the invariant mass
whereas the latter takes the background-dynamics in $t^{\prime}$ into account.
A phenomenological parametrization is used for the non-resonant component.

###  2.1 Exotics in the light unflavoured sector

Several models predict the lightest hybrid mesons to have quantum numbers
$(0,\textbf{1},2)^{-+}$ or $1^{--}$ HybridPred , which includes one spin-
exotic configuration known as the $\pi_{1}$. Since signals in this $1^{-+}$
sector have been observed in several channels and experiments, recent lattice
QCD simulations focused on the decay-channels of the $\pi_{1}$. The result was
a dominant branching to $b_{1}(1235)\pi$ and comparatively suppressed
branchings to $f_{1}(1285)\pi$, $\rho\pi$, $\eta^{(\prime)}\pi$,
$f_{1}(1420)\pi$ and $K^{*}\bar{K}$. With COMPASS data the $\pi_{1}$ and its
decay via these channels is currently being studied.

$\boldsymbol{\rho(770)\pi}$ \- The flagship is the $\pi^{-}\pi^{+}\pi^{-}$
final state. Based on $46\text{\,}\mathrm{M}$ selected events the full PWA has
been performed in 11 $t^{\prime}$-bins taking 88 waves into account
CompassRhoPiResModFit . The analysis of the $\pi^{-}\pi^{+}\pi^{-}$ COMPASS
data allowed for three important findings regarding the $\pi_{1}(1600)$-signal
in the $\rho(770)\pi$-channel.

Figure 1: PWA result of the $1^{-+}1^{+}\rho(770)\pi^{-}P$-wave: Main analysis
and RMF in the lowest (left) and highest (middle) $t^{\prime}$-bins
CompassRhoPiResModFit , Argand diagram (right) showing an exemplary result of
the freed-isobar ($\rho$) PWA CompassRhoPiFreed .

Firstly, by covering a wide $t^{\prime}$-range and using a much larger wave
set, COMPASS could reconcile apparent contradictions of previous experiments
as analysis artefacts CompassRhoPiFreed . Secondly, by splitting the data into
11 $t^{\prime}$-bins, a separation of resonant and non-resonant components in
this particular wave was possible, which is necessary for a reliable
extraction of the resonance parameters. From a Breit-Wigner model, a mass of
$m_{\pi_{1}}=1600^{+110}_{-60}$
$\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\textit{c}^{2}$ (sys) and a
width of $\Gamma_{\pi_{1}}=590^{+100}_{-230}$
$\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\textit{c}^{2}$ (sys) was
extracted. The result of the RMF for the lowest and highest $t^{\prime}$-bin
is shown in Fig. 1 (left and middle, respectively) for spin projection quantum
numbers $M^{\epsilon}=1^{+}$ CompassRhoPiResModFit . Thirdly, a freed-isobar
PWA was performed for this spin-exotic wave by removing the isobar ($\rho$)
line-shape from the decay amplitude and binning the isobar mass as well. For
each $m_{2\pi}$-bin a complex fit parameter was introduced accounting for the
dynamics of the isobar. The result for one ($m_{3\pi}$,$t^{\prime}$)-bin is
shown in form of the Argand diagram in Fig. 1 (right) where three values of
$m_{2\pi}$ are given in
$\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{/}\textit{c}^{2}$. It shows
the clear signature for the decay via the $\rho(770)\pi$ channel and by that
the validity of the isobar model is demonstrated.

$\boldsymbol{\eta^{(\prime)}\pi}$ \- These two channels have been analysed by
other experiments before with a surprising result: the presence of two spin-
exotic states with rather close masses: the $\pi_{1}(1400)$, observed only in
the $\eta\pi^{-}$-channel, and the $\pi_{1}(1600)$, observed only in the
$\eta^{\prime}\pi^{-}$-channel. At COMPASS these two channels are measured via
the decays $\eta^{(\prime)}\shortrightarrow\pi^{-}\pi^{+}\pi^{0}(\eta)$ and
$\pi^{0}(\eta)\shortrightarrow\gamma\gamma$. The PWD was performed on
$115\text{\,}\mathrm{k}$ ($\eta\pi^{-}$) and $40\text{\,}\mathrm{k}$
($\eta^{\prime}\pi^{-}$) events without any $t^{\prime}$-binning
COMPASSEtaPrimePi . The RMF of the COMPASS data, performed in JPAC  as a
coupled channel fit using the K-Matrix approach, requires only one pole to
describe the data. An analysis of Crystal Barrel data confirmed this finding
Kopf . The resulting resonance parameters ($m_{\pi_{1}}=1564\pm 24\pm 86$
$\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\textit{c}^{2}$ and
$\Gamma_{\pi_{1}}=492\pm 54\pm 102$
$\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\textit{c}^{2}$ JPAC ) are
compatible with the ones of the $\pi_{1}(1600)$ extracted by COMPASS for the
$\pi^{-}\pi^{+}\pi^{-}$ final state. Currently the analysis is redone
including the full data set, which doubles the number of exclusive events and
allows us to take into account the $t^{\prime}$-dynamics. In addition, we will
extend the mass range to higher masses, allowing us to apply constraints from
Regge theory JPAC-regge  in the analysis.

$\boldsymbol{b_{1}(1235)\pi}$ \- Based on much smaller data sets, previous
experiments observed a clear signal in the spin-exotic sector for this
channel. In BNLb1  even a second state at higher masses
($\sim$$2.0\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{/}\textit{c}^{2}$)
was suggested which still needs confirmation. At COMPASS this channel is
studied by performing the PWA on the $\omega\pi^{-}\pi^{0}$ system.
Experimentally the $\omega$ is detected via the decay into
$\pi^{+}\pi^{-}\pi^{0}$. Based on $720\text{\,}\mathrm{k}$ exclusive events,
which are grouped into four $t^{\prime}$-bins, the PWD has been performed
resulting in clear signals in the spin-exotic sector and in the expected mass
region of the $\pi_{1}(1600)$. For the highest $t^{\prime}$-bin the $b_{1}\pi$
S-wave intensity and relative phase with respect to the $\rho\omega$ D-wave
are shown in Fig. 2 (left) and (middle) for the isobar decay
$b_{1}\shortrightarrow\omega\pi$ via S-wave. For the same isobar but decay via
D-wave, the $b_{1}\pi$ S-wave intensity is shown in Fig. 2 (right). A RMF is
currently being performed in order to extract resonance parameters.

Figure 2: Result of the $\omega\pi^{-}\pi^{0}$ PWD in the highest
$t^{\prime}$-bin for the $1^{-+}b_{1}(1235)\pi S$ wave. The intensities
(phases) are sums (averages) over both charge combinations, $b_{1}^{-}\pi^{0}$
and $b_{1}^{0}\pi^{-}$.

$\boldsymbol{f_{1}(1285)\pi^{-}}\&\boldsymbol{K^{*}\bar{K}}$ \- The
$f_{1}(1285)\pi^{-}$ channel is analysed at COMPASS via a four-body PWA of the
$\pi^{-}\pi^{+}\pi^{-}\eta$ system based on $620\text{\,}\mathrm{k}$ events.
For the events with at least one $\pi^{-}\pi^{+}\eta$ combination inside the
$f_{1}(1285)$-range, the invariant mass is plotted in Fig. 3 showing a broad
and peaking structure in the resonant region. The $K^{*}\bar{K}$ channel is
analysed by performing a PWA of the $K_{S}K_{S}\pi^{-}$ system where the two
$K_{S}$ are experimentally detected via the decay into $\pi^{+}\pi^{-}$. We
have selected $240\text{\,}\mathrm{k}$ exclusive events. The invariant mass
distribution is shown in Fig. 4. Conclusions on the presence of the
$\pi_{1}(1600)$ shall be drawn based on the results of the PWA, which is
currently work in progress for both systems.

Figure 3: Invariant mass distribution of $f_{1}(1285)\pi^{-}$ events. The
insert shows the selection of $f_{1}(1285)$ from the $\pi^{-}\pi^{+}\eta$
subsystem.

Figure 4: Invariant mass distribution of $K_{S}K_{S}\pi^{-}$ events.

Figure 5: Intensity (data points) and RMF (lines): resonances (brown and blue
(2x)), background (orange and green), full model (red).

###  2.2 Exotics in the strange sector

Using the $K^{-}$ component of the beam, the light strange sector can be
accessed as well. Our new analysis of the $K^{-}\pi^{+}\pi^{-}$ final state
identifies eight resonance-like signals. The most interesting one is
compatible with the not PDG-established $K(1630)$ and has $J^{P}=0^{-}$. In
this sector only two excited states are predicted by the QM below
$2\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{/}\textit{c}^{2}$, but three
states are observed experimentally. The lightest $K(1460)$ and the heaviest
$K(1830)$ can be assigned to ordinary $q\bar{q}^{\prime}$ states, which makes
the $K(1630)$ a supernumerary candidate. In order to achieve a stable fit for
the $0^{-}\rho(770)KP$-wave, which is the only robust one in this sector, the
parameters of the well-known $K(1460)$ were fixed to the PDG values. In a
systematic study the third resonance was removed from the RMF, resulting in a
worse $\chi^{2}_{\text{red.}}$. Translating this into a significance yields a
value of $8.3\sigma$ for the presence of three states. The intensity
distribution as well as the result of the RMF is shown in Fig. 5 for the
$0^{-}\rho(770)KP$-wave in the lowest $t^{\prime}$-bin.

##  3 Conclusions

Recording the world’s largest data set of diffractive dissociation reactions
with a $\pi^{-}$ beam, COMPASS entered the era of precision spectroscopy of
light mesons. Including all data recorded allows us to include large wave sets
and to perform a binning in both the invariant mass and squared momentum
transfer. We report on new investigations on the spin-exotic $\pi_{1}(1600)$
in different decay channels. The decay branching will be crucial for the model
interpretation of this state. Using data taken with a $K^{-}$ beam, we perform
a first PWA of the strange meson spectrum in COMPASS and find evidence for a
supernumerary state with $0^{-}$ quantum numbers.

## References

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  * (4) B. Ketzer, Proceedings of Science, Volume 157, 025 (2012)
  * (5) Antoni J. Woss et al., Phys. Rev. D 103, 054502 (2021)
  * (6) M. Aghasyan et al. (COMPASS Collaboration), Phys. Rev. D 98, 092003 (2018)
  * (7) G. D. Alexeev et al. (COMPASS Collaboration), Phys. Rev. D 105, 012005 (2022)
  * (8) C. Adolph et al. (COMPASS Collaboration), Phys. Lett. B, Volume 740, 303-311 (2015)
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# Non-singular cosmology from non-supersymmetric AdS instability conjecture

Cao H. Nam<EMAIL_ADDRESS>Phenikaa Institute for Advanced
Study and Faculty of Fundamental Sciences, Phenikaa University, Yen Nghia, Ha
Dong, Hanoi 12116, Vietnam

###### Abstract

We show that the non-supersymmetric AdS instability conjecture can point to
how quantum gravity removes the initial Big Bang singularity, leading to a
potential resolution for the past-incomplete inflationary universe. From the
constraints on the dynamics of the universe realized as the nucleation of a
thin-wall bubble mediating the decay of the non-supersymmetric AdS vacuum, we
find the critical temperature $T_{c}$ and the critical scale factor $a_{c}$
for which the universe exists. These critical quantities are all finite and
determined in terms of the parameters specifying the stringy 10D AdS vacuum
solutions. Additionally, we derive the prediction of quantum gravity for
$T_{c}$ and $a_{c}$ relying on the inflationary observations.

_Introduction._ —Cosmological inflation is a rapidly accelerating expansion
epoch of space in the very early universe, which can explain why the universe
is big, homogeneous, isotropic, and flat as well as the absence of the
magnetic monopoles Guth1981 ; Albrecht1982 ; Linde1983 . However, an
inflationary universe is past-incomplete because of the presence of an initial
Big Bang singularity Borde2003 ; Yoshida2018  where the size of the universe
is zero, hence the spacetime curvature and the energy density become infinite.
The Big Bang singularity is unavoidable in General Relativity (GR), which
demonstrates the breakdown of GR in the extreme regime of the very early
universe. On the other hand, the Big Bang singularity should be absent in a
full theory of quantum gravity whose final form has not been available so far.

Due to the inflation scale close to the Planck scale where the effects of
quantum gravity become important, the inflation is highly sensitive to the
physics in the ultraviolet (UV) regime. The effects of UV physics can break
some essential futures assumed in the inflationary models. Hence, it is
necessary to embed the inflation into a UV complete theory from which the
inflation is obtained as an effective field theory (EFT) in the low-energy
limit of the fundamental theory. In addition, this embedding opens a window to
probe quantum gravity, beyond the higher-dimensional operators, which is based
on the inflationary observations measured from the spectrum of cosmic
microwave background (CMB) and the spectrum of primordial gravitational waves.

The recent progress exploring quantum gravity aims to determine its universal
and fundamental principles usually stated as the conjectures whose establish
is mainly based on the physics of black holes, the compactification of string
theory, and the general arguments of quantum gravity Palti2019 . These
conjectures impose the constraints on the low-energy EFTs that are UV
completed into quantum gravity. The Big Bang singularity is inconsistent with
quantum gravity, thus it is reasonable to think that universal and fundamental
principles of quantum gravity can tell us a mechanism in UV complete theory of
gravity to create a non-singular universe whose critical scale factor and
critical temperature representing the smallest size and the highest
temperature for its existence are all finite.

The non-SUSY AdS instability conjecture Ooguri2017  indicates that an anti-de
Sitter (AdS) vacuum in quantum gravity without being protected by
supersymmetry (SUSY) or (it is supported by flux) must decay as a result of
applying the weak gravity conjecture Harlow2023  for $p$-forms. This
conjecture has many implications for particle physics and cosmology like the
connection between the neutrino masses and the observed cosmological constant
Martin-Lozano2017 ; Hamada2017 ; Nam2023b ; Nam2023c . In particular, the non-
SUSY AdS instability conjecture leads to a way to embed the dS universe in
string theory considered as UV complete theory of quantum gravity where the dS
universe emerges from the decay of the false non-SUSY AdS vacuum to the true
SUSY AdS vacuum Banerjee2018 . There are obstructions to constructing dS
vacuum solutions in string theory, motivating the dS conjecture Obied2018
stating that dS vacua do not exist in quantum gravity and the studies of the
dS instability Krishnan2019 . The proposal in Banerjee2018  can provide a
resolution for matching string theory with the observations about the late-
time accelerating expansion of the universe.111An alternative to make string
theory consistent with observed dS universe, based on the AdS vacuum, is to
compactify the AdS5 factor on a circle and take into account the nontrivial
dynamics of the 4D tensor component Nam2023a  This cosmological scenario has
been extended in the presence of a cloud of strings Koga2019  and quintessence
Koga2021 , with Kerr-AdS5 spacetime Koga2023 , and for the AdS
compactification of M-theory over a squashed four-sphere Dibitetto2023 .

In this work, we show how quantum gravity removes the initial Big Bang
singularity via the non-SUSY AdS instability conjecture. Our universe is
realized as the nucleation of a thin-wall bubble mediating the decay of the
false non-SUSY AdS vacuum to the true SUSY AdS vacuum. In particular, from the
constraints on the dynamics of the bubble and Heisenberg’s uncertainty
principle, we determine the critical temperature and the critical scale factor
for the existence of the universe. These critical quantities are finite, hence
the beginning of the universe avoids the Big Bang singularity. In order to
determine the critical temperature and the critical scale factor in terms of
the parameters of quantum gravity, we study the supersymmetry and gauge
symmetry of the critical points of 5D supergravity and their uplift to the 10D
AdS vacuum solutions of string theory compactified on a 5D internal manifold.
Then, we express these critical quantities in terms of the units of the self-
dual five-form flux stabilizing the size of the 5D internal manifold.
Interestingly, as soon as the universe was created by the vacuum decay, it
would enter the inflation era. Using the inflationary observations, we derive
the prediction of quantum gravity for the critical temperature and the
critical scale factor.

_Creation of the non-singular universe._ —Let us start with two
$\text{AdS}_{5}$ vacua denoted by “+” and “-” corresponding to the following
element

$ds^{2}_{5,\pm}=-(1+k^{2}_{\pm}r^{2})d\tau^{2}_{\pm}+\frac{dr^{2}}{1+k^{2}_{\pm}r^{2}}+r^{2}d\Omega^{2}_{3},$
(1)

where $k_{-}>k_{+}$ and $d\Omega^{2}_{3}$ is the metric of three-sphere
$S^{3}$. The $\text{AdS}_{5}$ vacuum with the higher energy corresponding to
the cosmological constant $\Lambda_{+}=-6k^{2}_{+}$ is unstable and thus it
decays to the $\text{AdS}_{5}$ vacuum with the lower energy corresponding to
the cosmological constant $\Lambda_{-}=-6k^{2}_{-}$. Through the Coleman-de
Luccia transition Coleman1980 , the vacuum decay is mediated by the nucleation
of a thin-wall bubble or spherical brane which is realized as our universe.

The metric that is induced on the bubble or brane universe reads

$ds^{2}_{\text{bub.}}=-dt^{2}+\frac{a(t)^{2}}{\mathcal{K}}d\Omega^{2}_{3},$
(2)

where $a(t)$ is the scale factor related to the radial coordinate $r$ of the
AdS bulk as $r=a(t)/\sqrt{\mathcal{K}}$ and $\mathcal{K}>0$ is the curvature
of the spatial three-sphere at the present-day time $t_{0}$ with $a(t_{0})=1$.
The dynamics of the universe is governed by the second Israel junction
condition Delsate2014  as follows

$\displaystyle\left[K^{(+)}_{\mu\nu}-K^{(-)}_{\mu\nu}\right]-h_{\mu\nu}\left[K^{(+)}-K^{(-)}\right]=-\frac{S_{\mu\nu}}{M^{3}_{5}},$
(3)

where $K^{(\pm)}_{\mu\nu}=e^{M}_{\mu}e^{N}_{\nu}\nabla_{M}n^{(\pm)}_{N}$ is
the extrinsic curvature of the universe (with $e^{M}_{\mu}$ and
$n^{(\pm)}_{N}=(-\dot{a}(t)/\sqrt{\mathcal{K}},\dot{\tau}(t)_{\pm},0,0,0)$ are
the tangent vector and the unit normal vector to the universe, respectively),
$K^{(\pm)}=h^{\mu\nu}K^{(\pm)}_{\mu\nu}$, and $S_{\mu\nu}$ is the energy-
momentum tensor of the universe. From this equation, we find

$\displaystyle\left(\frac{\dot{a}}{a}\right)^{2}+\frac{\mathcal{K}}{a^{2}}$
$\displaystyle=$
$\displaystyle\frac{\left(k^{2}_{1}-\hat{\rho}^{2}\right)\left(k^{2}_{2}-\hat{\rho}^{2}\right)}{4\hat{\rho}^{2}},$
(4)

where the dot denotes the derivatives with respect to $t$, $k_{1,2}\equiv
k_{-}\pm k_{+}$, and $\hat{\rho}\equiv(\sigma+\rho_{r})/(3M^{3}_{5})$ where
$\sigma$ and $\rho_{r}$ refer to the brane tension and the energy density of
the radiation, respectively, which besides the curvature are two dominant
components over all other species at the very early universe. The usual
Friedmann equation can be restored in the regime of
$\mathcal{\sqrt{K}}/a,\dot{a}/a\ll k_{\pm}$ where we can identify the observed
Planck scale as

$\displaystyle M^{2}_{\text{Pl}}=\frac{k_{-}-k_{+}}{2k_{-}k_{+}}M^{3}_{5}.$
(5)

We observe from Eq. (4) that quantum gravity determines an initial condition
for the beginning of the universe in which the universe has a positive spatial
curvature corresponding to a closed universe. This is consistent with the
Planck 2018 data that without the lensing likelihood prefers to the universe
of the positive spatial curvature at a ratio of 50 : 1 against a spatially
flat universe Planck2018 . The presence of the positive spatial curvature can
resolve the tensions between the CMB data with other measurements like the CMB
lensing, the baryon acoustic oscillations, and Supernovae Ia Valentino2019 ;
Handley2021 . With the best-fit value of the present-day curvature density
$\Omega_{\mathcal{K},0}\equiv-\mathcal{K}/(a_{0}H_{0})^{2}=-0.0438$ and the
present-day Hubble parameter $H_{0}=67.4$ km s-1 Mpc-1 Planck2018 , we can
find $\mathcal{K}\approx 2.2\times 10^{-9}$ Mpc-2 and the present-day
curvature radius about $2.1\times 10^{4}$ Mpc.

For a non-singular universe, there is a critical temperature $T_{c}$ and a
critical scale factor $a_{c}$, which are all finite, representing the highest
temperature and the smallest (rescaled) size of the universe, respectively. We
shall show that the decay of the unstable AdS vacuum to the stable AdS vacuum
would create such a universe, allowing the universe to avoid the initial Big
Bang singularity. Because the energy density of the radiation on the brane
universe behaves in the scale factor as $\rho_{r}\propto a^{-4}$ and the
temperature as $\rho_{r}\propto T^{4}$, $\rho_{r}$ and $T$ would grow when
decreasing the size of the universe. However, in the present cosmological
scenario, the size of the universe cannot be arbitrarily small or the universe
temperature would never diverge because there is an upper bound for the energy
density of the radiation on the brane.

For the decay of the unstable AdS vacuum through the bubble nucleation
happening, the energy density on the brane should not exceed the critical
value $3M^{3}_{5}(k_{-}-k_{+})$, as depicted in Fig. 1.

---
Figure 1: The behavior of $V(\hat{\rho})$ that denotes the right-hand side of
Eq. (4) in terms of the reduced energy density $\hat{\rho}$ on the brane.

In addition, in the situation that the energy density of the radiation on the
brane grows due to decreasing the size of the universe, the brane universe
cannot transfer the energy-momentum from the brane to the bulk to guarantee
the aforementioned upper bound. This can be realized from the conservation law
along the brane given by $\nabla^{\mu}S_{\mu\nu}=R_{MN}n^{M}e^{N}_{\nu}$ where
$R_{MN}$ is the Ricci tensor of the AdS bulk Clifton2011 . The term
$R_{MN}n^{M}e^{N}_{\nu}$ which describes the interchange of the energy-
momentum between the brane and the AdS bulk vanishes identically with the bulk
geometry given by Eq. (1). Hence, there is no exchange of the energy-momentum
between the brane and the AdS bulk. As a result, the energy density of the
radiation on the brane needs to satisfy the following condition
$\rho_{r}\lesssim 6k_{-}k_{+}M^{2}_{\text{Pl}}-\sigma$ where we have used Eq.
(5). Then, we find the critical temperature for which the brane universe
exists as

$\displaystyle
T_{c}\simeq\left[\frac{30\left(6k_{-}k_{+}M^{2}_{\text{Pl}}-\sigma\right)}{\pi^{2}g_{*}}\right]^{1/4},$
(6)

where $g_{*}$ is the relativistic number of degrees of freedom near the
critical temperature. The critical scale factor $a_{c}$ can be estimated from
Heisenberg’s uncertainty principle where the momentum of a relativistic
particle is given by $T_{c}$ and the size of the volume containing the
particle is about $2\pi a_{c}/\sqrt{\mathcal{K}}$. This leads to

$\displaystyle
a_{c}\simeq\frac{1}{2\pi}\left[\frac{\pi^{2}g_{*}\mathcal{K}^{2}}{30\left(6k_{-}k_{+}M^{2}_{\text{Pl}}-\sigma\right)}\right]^{1/4}.$
(7)

_$T_{c}$ and $a_{c}$ in UV complete theory of gravity._—Because the non-
singular universe is created in the context of quantum gravity, an ultraviolet
(UV) complete theory of gravity shall tell us in a more precise way about the
critical temperature and the critical scale factor for the existence of the
universe.

The non-SUSY/SUSY AdS5 vacua correspond to critical points of the supergravity
potential of 5D supergravity $V(\alpha,\chi)$ given by Pilch2000

$\displaystyle
V(\alpha,\chi)=\frac{g^{2}}{3}\left(\frac{3}{8}\left|\partial_{\chi}W\right|^{2}+\frac{1}{16}\left|\partial_{\alpha}W\right|^{2}-\left|W\right|^{2}\right),$
(8)

where $\alpha$ and $\chi$ are the supergravity fields, $g$ is the gauge
coupling, and $W$ is the superpotential given as follows

$\displaystyle
W(\alpha,\chi)=\frac{1}{4e^{2\alpha}}\left[\cosh(2\chi)(e^{6\alpha}-2)-(3e^{6\alpha}+2)\right].$
(9)

All critical points of the supergravity potential and their supersymmetry and
gauge symmetry are analyzed in Gunaydin1985 ; Gunaydin1986 ; Khavaev2000 . The
potential at these critical points is negative, they thus admit the AdS
metrics. Because the 5D supergravity is a consistent truncation of the 10D IIB
supergravity, we can uplift the 5D AdS vacua to the 10D AdS vacuum solutions
of string theory compactified on a proper 5D internal manifold. The gauge
symmetry of the critical point would be the isometry of the 5D internal
manifold.

The stable SUSY AdS vacuum corresponds to a critical point at $\alpha=0$ and
$\chi=0$ which preserves all supersymmetries (i.e. $\partial_{i}V=0$ and
$\partial_{i}W=0$ with $i=\alpha,\chi$). This critical point leads to the SUSY
AdS vacuum with the cosmological constant $\Lambda_{-}=-3g^{2}/2$. Also, this
critical point has the SO(6) gauge symmetry which is the isometry of the round
five-sphere $S^{5}$, hence this AdS vacuum can be uplifted to the compactified
solution of the IIB string theory on the round five-sphere, i.e.
$\text{AdS}_{5}\times S^{5}$. As a result, the parameter $k_{-}$ or the gauge
coupling $g$ is determined by $N_{5}$ units of self-dual five-form flux which
stabilizes $S^{5}$ as follows

$\displaystyle k_{-}=\frac{g}{2}=2M_{5}\left(\frac{\pi}{N_{5}}\right)^{2/3},$
(10)

where the 5D Planck scale $M_{5}$ is related to the string coupling $g_{s}$,
the string length $l_{s}$, and the volume of the internal manifold
$\text{Vol}_{5}$ as
$M^{3}_{5}=(2\pi)^{-7}\text{Vol}_{5}/(g_{s}l^{2}_{s})^{4}$.

The unstable non-SUSY AdS vacuum corresponds to another critical point at
$\alpha=0$ and $\chi=\text{arccosh}(2)/2$ which breaks completely
supersymmetry (i.e. $\partial_{i}V=0$ and $\partial_{i}W\neq 0$ with
$i=\alpha,\chi$). This critical point has the SU(3)$\times$U(1) gauge symmetry
which is the isometry of the squashed five-sphere which is treated as a U(1)
bundle over the base $\mathbb{CP}^{2}$ Imamura2013 . Thus, this critical point
can be uplifted to the compactified solution of the IIB string theory on the
squashed five-sphere with the squashed parameter $\nu=\sqrt{2/3}$ Romans1985 .
All supersymmetries are broken by the presence of the three-form flux sourced
by the brane mediating the vacuum decay. In addition, the three-form flux
squashes $S^{5}$, leading to break the SO(6) isometry to the
SU(3)$\times$U(1). In general, the deformation of the internal manifold does
not preserve the number of five-form flux. On the other hand, the number of
five-form flux in the non-SUSY AdS vacuum may be smaller than that in the SUSY
AdS vacuum if the nucleated brane does not source the five-form flux. Then, we
determine $k_{+}$ in terms of $N^{\prime}_{5}$ units of five-form flux in the
non-SUSY AdS vacuum as follows

$\displaystyle k_{+}$ $\displaystyle=$
$\displaystyle\frac{3M_{5}}{\sqrt{2}}\left(\frac{\pi}{2N^{\prime}_{5}}\right)^{2/3}.$
(11)

The decay of the non-SUSY AdS vacuum to the SUSY AdS vacuum is mediated by the
nucleation of a spherical $(p,q)$-five brane Banerjee2018  which is a bound
state of $p$ D5-branes and $q$ NS5-branes Aharony1998  and sources the three-
form flux breaking all supersymmetries and squashing $S^{5}$ . A BPS bound for
the tension of the domain wall interpolating two vacua is related to the
change of the superpotential across the domain wall corresponding to the
change in the fluxes jumping from one side to another side Gukov2000 . This
implies the BPS bound $\sigma_{\text{BPS}}$ for the tension of the
$(p,q)$-five brane mediating the vacuum decay as Banerjee2018

$\displaystyle\sigma_{\text{BPS}}$ $\displaystyle=$ $\displaystyle
M^{3}_{5}g\left|W(0,0)-2^{-2/3}W\left(0,\text{arccosh}(2)/2\right)\right|.$
(12) $\displaystyle=$ $\displaystyle
M^{4}_{5}\left(\frac{\pi}{N_{5}}\right)^{2/3}\left(6-\frac{7}{2^{2/3}}\right).$

The $(p,q)$-five brane with the tension saturating the BPS bound describes the
vacuum decay which preserves the number of five-form flux, i.e.
$N^{\prime}_{5}=N_{5}$. On the contrary, when the deformation of the internal
manifold leads to $N^{\prime}_{5}<N_{5}$, it would push the brane away from
the BPS bound. In this situation, the vacuum decay is mediated by the
nucleation of the $(p,q)$-five brane with the tension
$\sigma<\sigma_{\text{BPS}}$.

In general, the critical temperature $T_{c}$ and the critical scale factor
$a_{c}$ depend on $N_{5}$ units of five-from flux in the true SUSY AdS vacuum,
the $(p,q)$-five brane tension, the distortion of the internal manifold that
can change the number of five-form flux in the false non-SUSY AdS vacuum. In
Fig. 2, we show the values of $T_{c}$ (as the solid curves) and those of
$a_{c}$ (as the dashed curves) in terms of $N_{5}$ and $N^{\prime}_{5}$ with
$\sigma/\sigma_{\text{BPS}}=0.8$ and $g_{*}\simeq 106.75$ (the relativistic
number of degrees of freedom for the Standard Model).

---
Figure 2: The contours of $T_{c}\times 10^{-16}/\text{GeV}$ (as the solid
curves) and $a_{c}\times 10^{59}$ in the $(N_{5},N^{\prime}_{5})$ plane, with
$\sigma/\sigma_{\text{BPS}}=0.8$ and $g_{*}\simeq 106.75$. The magenta band
refers to the inflation constraint.

The light gray region is excluded because the $(p,q)$-five brane does not
source the five-form flux. Whereas, the light orange region does not lead to
the decay of the non-SUSY AdS vacuum to the SUSY AdS vacuum due to
$k_{-}<k_{+}$ yielding $N^{\prime}_{5}/N_{5}<3\sqrt{3}/2^{13/4}$. The allowed
values of $T_{c}$ and $a_{c}$ belong to the white region. When incorporating
inflation, there is additionally a constraint shown by the magenta band, which
we shall study below. In the case that the number of five-form flux is
untouched in the vacuum decay, the critical temperature $T_{c}$ and the
critical scale factor $a_{c}$ are completely determined by a unique quantity
in the UV complete theory as follows

$\displaystyle T_{c}\sim\frac{M_{\text{Pl}}}{N^{1/2}_{5}},\ \ a_{c}\sim
N^{1/2}_{5}\left(\frac{\mathcal{K}}{M^{2}_{\text{Pl}}}\right)^{1/2}.$ (13)

_Constraint from inflation._ —The energy density of the radiation decreases
with the expansion of the universe leading to the right-hand side of Eq. (4)
being nearly constant given by
$(k^{2}_{1}-\hat{\sigma}^{2})(k^{2}_{2}-\hat{\sigma}^{2})/(4\hat{\sigma}^{2})$
with $\hat{\sigma}\equiv\sigma/(3M^{3}_{5})$. This results in a phase of
inflation shortly after the universe was created by the decay of the false AdS
vacuum to the true AdS vacuum.

Inflation ends and is followed by the reheating as a vacuum energy part of the
false non-SUSY AdS vacuum is converted to the matter in the bulk. Inflation
stops when the vacuum energy of the false non-SUSY AdS vacuum decreases from
the initial value ($\propto-k^{2}_{+}$) to a lower one such that the right-
hand side of Eq. (4) is near zero. This requires the backreaction of
additional fields in the exterior region of the AdS bulk. In analogy to the
modulus stabilization by the Goldberger-Wise mechanism Goldberger1999 , the
presence of such bulk fields would modify the scalar potential by which it
leads to the change in the vacuum energy of the false non-SUSY AdS vacuum.
Whereas, the matter in the exterior region of the AdS bulk contributes an ADM
mass term in the line element of the false AdS vacuum which plays the role of
the radiation with respect to the observer on the bubble universe Kraus1999 ;
Danielsson2019 . This can allow us to realize the reheating after inflation.

In this way, the backreaction of additional fields on the false non-SUSY AdS
vacuum would modify the metric outside the bubble. As a result, in Eq. (4), we
need to modify $k_{+}$ in a general way as follows

$\displaystyle k^{2}_{+}$ $\displaystyle\longrightarrow$ $\displaystyle
k^{2}_{+}+U_{0}\xi(a),$ (14)

where $U_{0}$ satisfies $k_{-}-\sqrt{k^{2}_{+}+U_{0}}\simeq\sigma$ such that
the inflation ends and $\xi(a)$ is a function of the scale factor. When the
backreaction is small as seen by an observer on the bubble universe,
corresponding to the inflation era, the function $\xi(a)$ is close to zero.
However, when the backreaction becomes significant, the inflation ends and it
is followed subsequently by the reheating, which implies the behavior of the
$\xi(a)$ as $\xi(a)\simeq 1-\rho_{\text{re}}a^{-4}$ where $\rho_{\text{re}}$
is the energy density of the radiation at the reheating. Phenomenologically,
we simply adopt $\xi(a)$ as follows

$\displaystyle\xi(a/a_{e})=1-q\times\left(\frac{a_{e}}{a}\right)^{4[\omega(a/a_{e})+1]},$
(15)

where $q$ is constant, $a_{e}$ is the scale factor at the end of inflation,
and $\omega$ is a function of the scalar factor. During the inflation,
$q(a_{e}/a)^{4(\omega+1)}$ is close to one, and thus $\omega$ is near to $-1$.
The inflation stops when $\omega$ changes drastically from $-1$ to zero around
$a_{e}$.

With $\Omega_{\mathcal{K},0}=-0.0438$ and the pivot scale given by
$k_{*}=a_{*}H_{*}=0.05$ Mpc-1 Planck2018 , we can calculate the curvature
density parameter at the time which $k_{*}$ crosses the comoving Hubble
horizon as $\Omega_{\mathcal{K},*}=8.84\times 10^{-7}$. This implies that the
mode $k_{*}$ is unaffected by the non-zero spatial curvature and hence we can
calculate primordial cosmological parameters using the flat approximation.

The scalar spectrum index $n_{s}$ and the tensor-to-scalar ratio $r$ are
determined as $n_{s}=1-4\epsilon_{*}+2\eta_{*}$ and $r=16\epsilon_{*}$ where
$\epsilon_{*}$ and $\eta_{*}$ are the values of two Hubble slow-roll
parameters defined by $\epsilon=-\dot{H}/H^{2}$ and $\eta=\ddot{H}/(\dot{H}H)$
at the horizon crossing time (denoted by $*$). They are given by

$\displaystyle\epsilon_{*}$ $\displaystyle\simeq$
$\displaystyle\frac{4U_{0}(k_{1}k_{2}+\hat{\sigma}^{2})}{(k^{2}_{1}-\hat{\sigma}^{2})(k^{2}_{2}-\hat{\sigma}^{2})}(1+\omega_{*})qe^{4N_{*}(\omega_{*}+1)},$
$\displaystyle\eta_{*}$ $\displaystyle\simeq$
$\displaystyle-\epsilon_{*}-4(1+\omega_{*}),$ (16)

where $N_{*}$ refers to the number of e-folds after the mode $k_{*}$ crosses
the comoving Hubble horizon. Note that, $qe^{4N_{*}(\omega_{*}+1)}$ is
approximately one during inflation. We represent the inflation prediction due
to the decay of the non-SUSY AdS space with an additional source described by
the function $\xi(a)$ given by Eq. (15) in Fig. 3, expressed by the blue band.

---
Figure 3: The tensor-to-scalar ratio versus the scalar spectral index,
predicted in the present cosmological scenario represented by the blue band.

In addition, we show the $1\sigma$ and $2\sigma$ confidence contours for
$(n_{s},r)$ from the PLANCK 2018 Planck2019  and BICEP2 BICEP2018  with an
extension of the positive spatial curvature for $\Lambda$CDM model Hergt2022 .
We observe that the non-conservation of the number of five-form flux due to
the deformation of the internal manifold is less to favor the data compared to
the case of the conservation.

The scalar power spectrum amplitude $A_{s}\approx 2.1\times 10^{-9}$
Planck2019  is calculated in terms of the Hubble parameter $H_{i}$ during
inflation and the tensor-to-scalar ratio $r$ as
$A_{s}=2H_{i}^{2}/(\pi^{2}M^{2}_{\text{Pl}}r)$. Using this relation and the
inflation constraint just studied above, we can derive the prediction of
quantum gravity for the critical temperature and critical scale factor. In the
case of $N^{\prime}_{5}<N_{5}$, the prediction is represented by the magenta
band given in Fig. 2, whereas for $N^{\prime}_{5}=N_{5}$, the prediction is
given in Fig. 4.

---
Figure 4: The prediction of $T_{c}$ and $a_{c}$ for the case that the number
of five-form flux is untouched, which agrees with the $1\sigma$ and $2\sigma$
contours of $(n_{s},r)$.

Accordingly, the critical temperature and the critical scale factor of the
universe are estimated as follows

$\displaystyle T_{c}\sim 10^{16}\ \text{GeV},\ \ a_{c}\sim 10^{-60}.$ (17)

This critical value is in the order of the grand unification scale. In
addition, the critical radius of the universe is above the Planck length
scale, hence the universe can be quite well described by GR.

I thank N. A. Ky, N. T. H. Van, P. V. Ky, and T. N. Hung for comments and
discussions.

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