11institutetext: School of Physics and Astronomy, University of Nottingham,
University Park, Nottingham NG7 2RD, United Kingdom22institutetext: Nottingham
Centre of Gravity, University of Nottingham, Nottingham NG7 2RD,
UK33institutetext: Dipartimento di Fisica e Astronomia, Università di Bologna,
via Irnerio 46, 40126 Bologna, Italy44institutetext: INFN, Sezione di Bologna,
viale Berti Pichat 6/2, 40127 Bologna, Italy

# The cosmological constant is probably still zero

Yang Liu 1,2 Antonio Padilla 3,4 and Francisco G. Pedro
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We consider a wide class of four-dimensional effective field theories in which
gravity is coupled to multiple four-forms and their dual scalar fields, with
membrane sources charged under the corresponding three-form potentials. Four-
form flux, quantised in units of the membrane charges, generically generates a
landscape of vacua with a range of values for the cosmological constant that
is scanned through membrane nucleation. We list various ways in which the
landscape can be made sufficiently dense to be compatible with observations of
the current vacuum without running into the empty universe problem. Further,
we establish the general criteria required to ensure the absolute stability of
the Minkowski vacuum under membrane nucleation and the longevity of those
vacua that are parametrically close by. This selects the current vacuum on
probabilistic grounds and can even be applied in the classic model of Bousso
and Polchinski, albeit with some mild violation of the membrane weak gravity
conjecture. We present other models where the membrane weak gravity conjecture
is not violated but where the same probabilistic methods can be used to tackle
the cosmological constant problem.

##  1 Introduction

The cosmological constant problem has haunted theoretical physics ever since
Pauli calculated the effect of the zero-point energy of the electron on the
curvature of spacetime, declaring that the universe would “not even reach to
the moon” Enz . The situation hasn’t really improved in the century that
followed. When we apply standard quantum field theory methods, radiative
corrections to the vacuum energy are extremely sensitive to ultra-violet
physics, scaling like the fourth power of the cut-off. Vacuum energy
gravitates just like a cosmological constant presenting a major issue for
cosmology. The observed value of the cosmological constant lies sixty orders
of magnitude below the expected value predicted from vacuum energy
calculations with a TeV cut-off, set by the the scale of modern collider
experiments. If we push the cut-off all the way up to the Planck scale, the
discrepancy extends to 120 orders of magnitude. For reviews of the
cosmological constant problem, see Weinberg:1988cp ; Polchinski:2006gy ;
Burgess:2013ara ; Padilla:2015aaa .

Recently, Kaloper Kaloper:2022oqv  and later, Kaloper and Westphal
Kaloper:2022jpv  have constructed a simple dynamical model in which the
cosmological constant is relaxed to near zero through the nucleation of
membranes, charged under a pair of three-forms. Membrane nucleation triggers a
jump in the effective cosmological constant that is controlled by the
corresponding charges. Crucially, these charges are not assumed to be
exponentially small. Instead, they are assumed to have an irrational ratio,
which guarantees a dense landscape of vacua, including those whose
cosmological constant is close to zero. Given a more modest upper bound on the
charge (which may be in some tension with the weak gravity conjecture Arkani-
Hamed:2006emk ; Ibanez:2015fcv ) transitions between vacua are rapid, at least
until the system finds a vacuum with a small positive cosmological constant,
at which point, the vacuum is very long lived. This latter result has its
roots in Hawking’s famous solution to the cosmological constant problem in
which he computes the Euclidean action for a landscape of vacua and argues
that the most probable configuration is the one with the smallest absolute
curvature Hawking:1984hk .

Dynamical neutralization of the cosmological constant via membrane nucleation
was first explored by Brown and Teitelboim Brown:1987dd ; Brown:1988kg ,
although there the membrane charge is assumed to be extremely small. As a
result, the transition from a very high energy de Sitter vacuum to near
Minkowski occurs slowly via a series of small steps, resulting in the so-
called empty universe problem Abbott:1984qf . Thanks to the irrational charge
ratio in Kaloper:2022oqv ; Kaloper:2022jpv  the transition from a very high
energy de Sitter vacuum to near Minkowski can occur quickly via a single
membrane nucleation and the empty universe problem is said to be avoided.

The simple set-up presented in Kaloper:2022jpv  is a clever generalisation of
Henneaux and Teitelboim’s covariant formulation of unimodular gravity
Henneaux:1989zc , with a pair of four-form field strengths that enter only
through a bilinear mixing with a dual scalar field, and charged membranes (for
a review of unimodular gravity, see Padilla:2014yea ). Although the model is
phenomenologically interesting, its embedding within fundamental theory is
likely to prove challenging. However, in this paper we will show that key
aspects of the relaxation mechanism apply far more generally, increasing the
likelihood that it could apply in some of the four-dimensional effective
theories obtained from string theory compactifications. Remarkably, this is
true of the famous Bousso-Polchinski set-up Bousso:2000xa , offering a
probabilistic explanation for the small observed value of the cosmological
constant instead of an anthropic one.

We consider a very general set-up involving Einstein-Hilbert gravity coupled
to three-forms fields and their dual scalars, with membrane sources charged
under the three-forms. This includes the Kaloper-Westphal model
Kaloper:2022jpv  as a special case but also other effective theories including
those expected to appear in compactifications of string theory, such as the
Bousso-Polchinski set-up Bousso:2000xa . A key feature of the entirely family
of models is that in vacuum, the scalars and the three-forms always gravitate
like a cosmological constant, even in the presence of a non-trivial flux. The
flux of the four-field strength is quantised, giving rise to a landscape of
vacua. To have any hope of solving the cosmological constant problem, this
landscape must be sufficiently dense in and around Minkowski space in order to
admit vacua that match observations.

Tunnelling between vacua is achieved through membrane nucleation. This is
described by instantons, solutions to the Euclidean equations of motion with
vacua of different vacuum energies, separated by charged membranes. The
geometry of the instantons depends on the curvature of the corresponding vacua
and the membrane tensions. Many configurations are physically irrelevant
because the membranes would have to have negative tension or because the
corresponding tunnelling rate is infinitely suppressed. As in Kaloper:2022oqv
; Kaloper:2022jpv , those instanton solutions that are physically relevant can
be dissected even further. In our general setting, this dissection depends on
the value of a parameter $X=4M_{pl}^{4}\Delta k^{2}/T^{2}$, where $\Delta
k^{2}$ is the jump in vacuum curvature and $T$ is the membrane tension. We
consider several example theories where the bounds on this parameter can be
related to bounds on other microscopic parameters such as the membrane
charges. This is important since the value of $X$ not only controls which
instanton geometries are allowed but also the corresponding tunnelling rates.
In particular, when $|X|<1$, any near Minkowski vacua become very long lived
and robust against the nucleation of anti-de Sitter bubbles. Thus, if you find
yourself in a near Minkowski vacuum, chances are you will stay there.

Of course, by itself, this is not enough to solve the cosmological constant
problem. One must show that we avoid the so-called empty universe problem.
This occurs when the universe descends to the current vacuum incrementally,
via a slow cascade of de Sitter vacua, the curvature jumping by a small amount
each time, forever diluting away any matter excitations. The density of the
landscape and the abundance of dangerous de Sitter vacua render this a real
possibility in generic theories of this type. However, through a detailed
study of the kinematics and an analysis of tunnelling rates we show that the
empty universe problem can often be avoided: one is able to transition quickly
from a high scale de Sitter vacuum to near Minkowski via a single transition.
All of this suggests a wide-ranging mechanism for solving the cosmological
constant problem, inspired by, but extending far beyond, the original proposal
of Kaloper:2022oqv ; Kaloper:2022jpv .

The rest of this paper is organised as follows: in section 2, we present the
generalised set-up along with the corresponding field equations and junction
conditions. Since they will play an important role in the evaluation of
tunnelling rates later in the paper, we dwell a little on the choice of
boundary conditions and corresponding boundary terms. In section 3, we present
the landscape of Lorentzian vacuum solutions. In section 4, we Wick rotate to
Euclidean signature and solve the Euclidean field equations to find the
corresponding instanton solutions and compute the corresponding transition
rates. In section 5, we demonstrate the role of the parameter $X$ in
controlling the stability of near Minkowski vacua, protecting them from decay
into anti de Sitter. We also discuss conditions to avoid the empty universe
problem. In section 6, we run through four different models, including Brown-
Teitelboim, Bousso-Polchinski, Kaloper-Westphal, and a fourth model including
a single three-form (with standard kinetic term) and its dual scalar. In the
latter three cases, we show how the parameters of the theory can be chosen to
successfully implement the mechanism for solving the cosmological constant
problem originally proposed in Kaloper:2022oqv . In section 7, we conclude.

##  2 The generalised set-up

We begin with a general four-dimensional effective theory on a manifold,
$\mathcal{M}$, with a dynamical metric $g_{\mu\nu}$ and a family of three-form
fields, $A^{i}$, and dual scalars $\phi_{i}$,

$S=\int_{\mathcal{M}}d^{4}x\sqrt{|g|}\left[\frac{M_{pl}^{2}}{2}R-\frac{1}{2}\omega^{ij}(\phi)\nabla_{\mu}\phi_{i}\nabla^{\mu}\phi_{j}-V(\phi)\right]\\\
+\int_{\mathcal{M}}\left[-\frac{1}{2}Z_{ij}(\phi)F^{i}\wedge\star
F^{j}+\sigma_{i}(\phi)F^{i}\right]+S_{\text{boundary}}+S_{\text{membranes}}$
(1)

where $R$ is the Ricci scalar. The four-form field strengths are given in
terms of the three-form fields $F^{i}=dA^{i}$ and $\star$ denotes the Hodge
star operator on the manifold $\mathcal{M}$. In components, the three-form is
written as $A^{i}=\frac{1}{3!}A^{i}_{\mu\nu\alpha}dx^{\mu}\wedge
dx^{\nu}\wedge dx^{\alpha}$ and the corresponding field strength as
$F^{i}=\frac{1}{4!}F^{i}_{\mu\nu\alpha\beta}dx^{\mu}\wedge dx^{\nu}\wedge
dx^{\alpha}\wedge dx^{\beta}$ where
$F^{i}_{\mu\nu\alpha\beta}=4\partial_{[\mu}A^{i}_{\nu\alpha\beta]}$. When
acting on the four-form, the Hodge star operator yields $\star
F^{i}=\epsilon_{\mu\nu\alpha\beta}F^{i\mu\nu\alpha\beta}$, where
$\epsilon_{\mu\nu\alpha\beta}$ is the Levi-Civita tensor on the manifold.

The action is also equipped with boundary terms which depend on the choice of
boundary conditions. These are integrals over the boundary,
$\partial\mathcal{M}$, which we take to be a co-dimension one surface
described by the embedding $x^{\mu}=X^{\mu}(\xi^{a})$. The induced metric and
the pullback of the three-forms on the boundary are given respectively by
$\gamma_{ab}=g_{\mu\nu}X^{\mu}_{,a}X^{\nu}_{,b}$ and
$\alpha^{i}=\frac{1}{3!}A^{i}_{\mu\nu\alpha}X^{\mu}_{,a}X^{\nu}_{,b}X^{\alpha}_{,c}d\xi^{a}\wedge
d\xi^{b}\wedge d\xi^{c}$, where $X^{\mu}_{,a}=\partial
X^{\mu}/\partial\xi^{a}$ are the boundary tangent vectors. We also define the
extrinsic curvature of the boundary,
$K_{ab}=\frac{1}{2}\mathcal{L}_{n}\gamma_{ab}$, as the Lie derivative of the
induced metric with respect to the outward pointing unit normal, $n^{\mu}$. If
$K=\gamma^{ab}K_{ab}$ is the trace of the extrinsic curvature on the boundary,
the boundary action is given by

$S_{\text{boundary}}=M_{pl}^{2}\int_{\partial\mathcal{M}}d^{3}\xi\sqrt{|\gamma|}K-\int_{\partial\mathcal{M}}\mu
p^{i}\phi_{i}+\lambda\chi_{i}\alpha^{i}$ (2)

where we define “conjugate momenta”,

$p^{i}=-d^{3}\xi\sqrt{|\gamma|}\omega^{ij}n^{\mu}\nabla_{\mu}\phi_{j},\quad\chi_{i}=\sigma_{i}-Z_{ij}(\star
F^{j}).$ (3)

The extrinsic curvature piece ensures that the action can be extremised under
metric variations with Dirichlet boundary conditions, $\delta\gamma_{ab}=0$
Gibbons:1976ue . Meanwhile, the parameter $\lambda$ allows us to interpolate
between Dirichlet ($\lambda=0$) and Neumann ($\lambda=1$) boundary conditions
on the three-forms, while the parameter $\mu$ allows us to interpolate between
Dirichlet ($\mu=0$) and Neumann ($\mu=1$) boundary conditions on the scalars.
To see this, note that variation of the action with respect to all fields
yields a boundary contribution Gibbons:1976ue ; Duncan:1989ug ,

$-\frac{M_{pl}^{2}}{2}\int_{\partial\mathcal{M}}d^{3}\xi\sqrt{|\gamma|}(K^{ab}-K\gamma^{ab})\delta\gamma_{ab}+\int_{\partial\mathcal{M}}(1-\lambda)\chi_{i}\delta\alpha^{i}-\lambda\delta\chi_{i}\alpha^{i}+(1-\mu)p^{i}\delta\phi_{i}-\mu\delta
p^{i}\phi_{i}$ (4)

that is required to vanish under the appropriate choice of boundary
conditions.

Finally we consider the membrane contributions. We can include contributions
from membranes and anti-membranes, $\Sigma_{I}$, charged under any of the
three-forms, such that

$S_{\text{membranes}}=-\sum_{I}\left\\{\eta^{i}_{I}q_{i}\
\int_{\Sigma_{I}}\alpha_{I}^{i}+|\eta^{i}_{I}|\tau_{i}\int_{\Sigma_{I}}d^{3}\xi\sqrt{|\gamma_{I}}|\right\\}.$
(5)

Membranes charged under $A^{i}$ carry a fundamental charge $\pm q_{i}$
depending on whether they are branes or antibranes and tension $\tau_{i}$. In
the action (5), $\eta^{i}_{I}=0,\pm 1$ depending on whether the membrane
$\Sigma_{I}$ carries positive ($\eta^{i}_{I}=1$), negative ($\eta^{i}_{I}=-1$)
or vanishing charge ($\eta^{i}_{I}=0$) under $A^{i}$. The induced metric and
the pullback of the three-forms on $\Sigma_{I}$ are given in a similar way to
the boundary, by
$\gamma_{I}{}_{ab}=g_{\mu\nu}X_{I}{}^{\mu}_{,a}X_{I}{}^{\nu}_{,b}$ and
$\alpha_{I}^{i}=\frac{1}{3!}A^{i}_{\mu\nu\alpha}X_{I}{}^{\mu}_{,a}X_{I}{}^{\nu}_{,b}X_{I}{}^{\alpha}_{,c}d\xi^{a}\wedge
d\xi^{b}\wedge d\xi^{c}$, where $X_{I}{}^{\mu}_{,a}=\partial
X_{I}{}^{\mu}/\partial\xi^{a}$ are the tangent vectors on $\Sigma_{I}$.
Throughout this paper we will restrict attention to timelike membranes so that
their unit normal, $n^{\mu}_{I}$ is spacelike.

Away from the membranes, the field equations resulting from variation of the
action (1) with respect to the metric, three-forms and scalars are given by

$\displaystyle M_{pl}^{2}G^{\mu\nu}$ $\displaystyle=$ $\displaystyle
T^{\mu\nu}_{F}+T^{\mu\nu}_{\phi}$ (6) $\displaystyle\nabla_{\mu}\chi_{i}$
$\displaystyle=$ $\displaystyle 0$ (7)
$\displaystyle\nabla_{\mu}\left(\omega^{kj}\nabla^{\mu}\phi_{j}\right)$
$\displaystyle=$ $\displaystyle
V^{,k}+\frac{1}{2}\omega^{ij}{}^{,k}\nabla_{\mu}\phi_{i}\nabla^{\mu}\phi_{j}-\left[\frac{1}{2}Z_{ij}{}^{,k}(\star
F^{i})(\star F^{j})-\sigma_{i}{}^{,k}(\star F^{i})\right]\qquad$ (8)

where $V^{,k}=\partial V/\partial\phi_{k}$ etc. The energy momentum tensors
for the three-forms, the scalar and the membranes are

$\displaystyle
T^{\mu\nu}_{F}=\frac{Z_{ij}}{3!}\left(F^{i}{}^{\mu\alpha\beta\gamma}F^{j}{}^{\nu}{}_{\alpha\beta\gamma}-\frac{1}{8}g^{\mu\nu}F^{i}_{\alpha\beta\gamma\delta}F^{j}{}^{\alpha\beta\gamma\delta}\right)\qquad$
(9) $\displaystyle
T^{\mu\nu}_{\phi}=\omega^{ij}\left(\nabla^{\mu}\phi_{i}\nabla^{\nu}\phi_{j}-\frac{1}{2}g^{\mu\nu}\nabla_{\alpha}\phi_{i}\nabla^{\alpha}\phi_{j}\right)-Vg^{\mu\nu}.\qquad$
(10)

At the membranes, we can derive junction conditions which determine how the
conjugate momenta of the system jump across the membranes. The momentum
conjugate to $\gamma_{Iab}$ is proportional to the following combination of
the extrinsic curvature
$\pi_{I}^{ab}=M_{pl}^{2}(K_{I}^{ab}-K^{I}\gamma_{I}^{ab})$. As we pass over
$\Sigma_{I}$ this jumps in accordance with the Israel junction conditions
Israel

$M_{pl}^{2}\Delta\pi_{I}^{ab}=-|\eta^{i}_{I}|\tau_{i}\gamma_{I}^{ab}.$ (11)

The momentum conjugate to the three-forms also jumps

$\Delta\chi_{i}=-\eta^{i}_{I}q_{i}\qquad(\text{no sum over $i$})$ (12)

while the scalar momenta remains continuous $\Delta p^{i}=0$. The metric and
the three-form potentials are assumed to be continuous at $\Sigma_{I}$ in
order for the membrane action to be well defined. Note that the junction
condition (12) is consistent with the momentum $\chi_{i}$ quantised in units
of the fundamental membrane charge, $\chi_{i}=-N_{i}q_{i}$ (no sum) where
$N_{i}\in\mathbb{Z}$.

##  3 Vacua

We take vacua to be real Lorentzian solutions with constant scalars, four-
forms of constant flux $\star F^{i}=c^{i}$, and a maximally symmetric metric
with constant curvature $k^{2}$, corresponding to de Sitter ($k^{2}>0$),
Minkowski ($k^{2}=0$) or anti de Sitter ($k^{2}<0$) spacetime. Away from any
membranes, the field equations imply that

$\displaystyle 3M_{pl}^{2}k^{2}=V+\frac{1}{2}Z_{ij}c^{i}c^{j}$ (13)
$\displaystyle
V^{,k}=\frac{1}{2}Z_{ij}{}^{,k}c^{i}c^{j}-\sigma_{i}{}^{,k}c^{i}$ (14)

while the conjugate momentum,

$\chi_{i}=\sigma_{i}-Z_{ij}c^{j}$ (15)

is locally constant. At membranes, we get a jump in $\chi_{i}$ according to
(12), generically triggering a jump in the spacetime curvature. This means we
have a landscape of possible vacua with different cosmological constants,
scanned through membrane nucleation.

In order to be compatible with observations, this landscape must include vacua
whose curvature is of order the Hubble scale today, $H_{0}^{2}$. This suggests
a dense landscape with vacua separated by no more than this very low scale. In
the original Brown-Teitelboim scenario Brown:1987dd ; Brown:1988kg , this
required the membrane charge to be exponentially small in Planck units. That
condition was lifted in Bousso-Polchinski Bousso:2000xa  where a sufficiently
dense landscape follows from a family of $\mathcal{O}(100)$ three-form fields
coupled to membranes with incommensurate charges. In Kaloper-Westphal
Kaloper:2022oqv ; Kaloper:2022jpv  the dense landscape follows from an
irrational ratio for the fundamental membranes charges.

Of course, the density of the landscape raises the possibility that
transitions between vacua of similar curvature are possible. If this is the
case, we may encounter an empty universe problem where the universe arrives at
a low scale vacuum via a slow cascade through higher-curvature vacua, cooling
the universe down through many many efolds of exponential expansion. Although
the empty universe problem is known to be an issue for the original Brown-
Teitelboim scenario, we will show how it is avoided in the other cases we
consider.

Membrane nucleation, necessary for scanning the landscape of vacua, is a
quantum process. When we compute transition rates we do so between eigenstates
of constant $\chi_{i}$. This suggests a path integral formalism equipped with
Neumann boundary conditions, fixing $\chi_{i}$ in both the in-state and in the
out-state Duncan:1989ug . We will compute these transition rates in the next
section. Corresponding formulae for Dirichlet boundary conditions, fixing
$A^{i}$ in in and out states, are presented in the appendix.

##  4 Nucleation rates

To compute the rate at which membranes are nucleated, mediating transitions
between vacua, we analytically continue to Euclidean signature, setting

$t\to-it_{E},\qquad S\to iS_{E}$ (16)

where $S_{E}$ is the Euclidean action. Following Brown:1987dd ; Brown:1988kg ,
we assume that scalars are unchanged under this Wick rotation, including the
dual of the four-forms $\star F\to\star F$. It follows that the three-forms
should be analytically continued as $A\to iA$ 111If $\eta$ and
$\eta_{E}=i\eta$ are the volume forms for Lorentzian and Euclidean signature
respectively, then $F=-(\star F)\eta$ and $F_{E}=(\star F)_{E}\eta_{E}$.
Setting $\star F=(\star F)_{E}$, as desired, we infer that $F=iF_{E}$. This is
consistent with our choice of $A=iA_{E}$.. The resulting Euclidean action now
takes the form

$S_{E}=-\int_{\mathcal{M}}d^{4}x_{E}\sqrt{|g|}\left[\frac{M_{pl}^{2}}{2}R-\frac{1}{2}\omega^{ij}(\phi)\nabla_{\mu}\phi_{i}\nabla^{\mu}\phi_{j}-V(\phi)\right]\\\
+\int_{\mathcal{M}}\left[-\frac{1}{2}Z_{ij}(\phi)F^{i}\wedge\star
F^{j}+\sigma_{i}(\phi)F^{i}\right]+S^{E}_{\text{boundary}}+S^{E}_{\text{membranes}}$
(17)

where the boundary terms are chosen to be consistent with Neumann boundary
conditons on the three-form fields ($\lambda=1$). Since we will only be
considering bounce configurations that transition between two vacua, we assume
that there is a single membrane, $\Sigma$, which is charged under $A^{i}$ for
some particular choice of $i=i_{*}$. The membrane charges under $A^{i}$ are
given by $Q_{i}=\delta_{ii_{*}}Q_{i_{*}}$ where $Q_{i_{*}}=\pm q_{i_{*}}$ and
the membrane tension is $T=\tau_{i_{*}}$. As a result, the (Euclidean)
membrane action is given by

$S^{E}_{\text{membranes}}=-Q_{i_{*}}\
\int_{\Sigma}\alpha_{\Sigma}^{i_{*}}+T\int_{\Sigma}d^{3}\xi_{E}\sqrt{|\gamma_{\Sigma}|}\,.$
(18)

We consider $O(4)$ symmetric Euclidean field configurations, with metric

$ds^{2}=dr^{2}+\rho(r)^{2}d\Omega_{3}$ (19)

where $d\Omega_{3}=h_{ij}d\xi^{i}d\xi^{j}$ is the metric on a unit 3-sphere,
Euclidean three-form potentials

$A^{i}=A^{i}(r)\sqrt{|h|}d^{3}\xi$ (20)

and scalars $\phi_{i}=\phi_{i}(r)$. The radial coordinate is assumed to run
from $r_{\text{min}}$ to $r_{\text{max}}$ (to be determined) and the membrane
lies at $r=0$. We label the interior geometry, where $r_{\text{min}}\leq r<0$
, by $\mathcal{M}_{-}$ and the exterior, where $0<r<r_{\text{max}}$, by
$\mathcal{M}_{+}$.

With this ansatz, the field equations away from the membrane simplify to the
following

$\displaystyle
3M_{pl}^{2}\left[\frac{1}{\rho^{2}}-\left(\frac{\rho^{\prime}}{\rho}\right)^{2}\right]$
$\displaystyle=$ $\displaystyle
V-\frac{1}{2}\omega^{ij}\phi^{\prime}_{i}\phi^{\prime}_{j}+\frac{1}{2}Z_{ij}\frac{A^{i}{}^{\prime}A^{j}{}^{\prime}}{\rho^{6}}\qquad$
(21) $\displaystyle
M_{pl}^{2}\left[\frac{1}{\rho^{2}}-\left(\frac{\rho^{\prime}}{\rho}\right)^{2}-2\frac{\rho^{\prime\prime}}{\rho}\right]$
$\displaystyle=$ $\displaystyle
V+\frac{1}{2}\omega^{ij}\phi^{\prime}_{i}\phi^{\prime}_{j}+\frac{1}{2}Z_{ij}\frac{A^{i}{}^{\prime}A^{j}{}^{\prime}}{\rho^{6}}$
(22) $\displaystyle\chi_{i}^{\prime}$ $\displaystyle=$ $\displaystyle 0$ (23)
$\displaystyle\frac{1}{\rho^{3}}\left(\rho^{3}\omega^{kj}\phi_{j}^{\prime}\right)^{\prime}$
$\displaystyle=$ $\displaystyle
V^{,k}-\frac{1}{2}Z_{ij}{}^{,k}\frac{A^{i}{}^{\prime}A^{j}{}^{\prime}}{\rho^{6}}+\frac{A^{i}{}^{\prime}}{\rho^{3}}\sigma_{i}{}^{,k}\qquad$
(24)

where $\chi_{i}=\sigma_{i}-Z_{ij}\frac{A^{j}{}^{\prime}}{\rho^{3}}$.

Assuming all scalars are locally constant and satisfy the constraints listed
from (13) to (15), this system is solved by

$\rho=\frac{\sin(k(\epsilon r+r_{0})}{k},\qquad\epsilon=\pm 1$ (25)

where the curvature $k^{2}$ matches the curvature of the Lorentzian solution
(13), and three-form potentials

$A^{i}(r)=A^{i}(0)+c^{i}\int_{0}^{r}dr\rho(r)^{3}.$ (26)

Note that the solution for $\rho$ extends to $k^{2}\leq 0$ by analytic
continuation. The parameter $r_{0}$ is an integration constant, setting the
radius of the 3-sphere at the membrane.

For $k^{2}>0$, the geometry is that of a section of a 4-sphere and it turns
out that $\rho$ is invariant under $\epsilon\to-\epsilon$,
$r_{0}\to\pi/k-r_{0}$. This allows us to set $\epsilon=+1$, WLOG, while also
assuming $r_{0}\in[0,\pi/k]$. The poles of the 4-sphere are located at
$r_{\text{min}}=-r_{0}$ and $r_{\text{max}}=\pi/k-r_{0}$.

For $k^{2}=0$ and $k^{2}<0$, the geometry is that of a section of a four-
dimensional Euclidean plane or hyperboloid respectively. Since $\rho$ is
assumed to be non-negative at the stack, we take $r_{0}\geq 0$. Note that we
can no longer fix $\epsilon$ WLOG, and must consider each sign separately. For
$\epsilon=+1$, $r_{\text{min}}=-r_{0}$ corresponding to the point where the
3-spheres shrink to zero size, while $r_{\text{max}}=\infty$ corresponding to
where they diverge. For $\epsilon=-1$, the reverse is true:
$r_{\text{min}}=-\infty$ corresponding to the point where the 3-spheres
diverge, while $r_{\text{max}}=r_{0}$ corresponding to where they shrink to
zero size.

At the membrane, the induced metric $ds_{\Sigma}^{2}=\rho^{2}(0)d\Omega_{3}$
and the pullback of the three-forms
$\alpha_{\Sigma}^{i}=A^{i}(0)\sqrt{|h|}d^{3}\xi$ are well defined. Note that
for the induced metric this gives us a continuity constraint

$\Delta\rho(0)=\Delta\left[\frac{\sin kr_{0}}{k}\right]=0\,,$ (27)

where we introduce the notation $\Delta x=x^{+}-x^{-}$ and $\langle
x\rangle=\frac{1}{2}(x^{+}+x^{-})$ corresponding respectively to the
difference and average of some quantity $x$ defined on either side of the
membrane. We also have the following junction conditions on the conjugate
momenta,

$\displaystyle 6M_{pl}^{2}\Delta\left[\frac{\rho^{\prime}(0)}{\rho(0)}\right]$
$\displaystyle=$ $\displaystyle-3T$ (28) $\displaystyle\Delta\chi_{i}$
$\displaystyle=$ $\displaystyle-\delta_{ii_{*}}Q_{i_{*}}.$ (29)

Physically realistic membranes always carry non-negative tension, resulting in
the following constraint on the allowed configurations

$\Delta\left[\epsilon\cos kr_{0}\right]\leq 0\,.$ (30)

The instanton solutions can also be classified according to the value of

$X=\frac{4M_{pl}^{4}\Delta k^{2}}{T^{2}}$ (31)

with $X>0$ in downward transitions and $X<0$ in upward transitions. More
importantly, however, it turns out that some configurations are kinematically
allowed only when $|X|\leq 1$ with the remainder allowed when $|X|\geq 1$. An
analogous result given explicitly in terms of membrane charges was shown to be
true in Kaloper:2022oqv ; Kaloper:2022jpv . Here we show how it is generalised
to a much wider class of models which could, in principle, include the Bousso-
Polchinski set-up Bousso:2000xa . Indeed, we can always relate $X$ to the
charge of the membrane and the curvature of the parent vacuum using equations
(13) to (15), along with the junction condition (29), although the
relationship is not always a simple one222In principle, this can be done as
follows for a generic model. First, we use (14) and (15) to express the fluxes
$c^{i}$ and the scalars $\phi_{i}$. in terms of $\chi_{i}$. Plugging this into
(13) we obtain an expression for the bulk cosmological constant of the form
$k^{2}=\mathcal{K}(\chi_{i})$. For example, for tunnelling from near Minkowski
into AdS, we then have $\chi_{i}^{+}$ fixed such that
$k^{2}_{+}=\mathcal{K}(\chi_{i}^{+})\approx 0$, with
$\chi_{i}^{-}=\chi_{i}^{+}+Q_{i}$ fixed by the junction conditions (29).
Finally we have that $X^{2}=4M_{pl}^{4}\Delta
k^{2}/T^{2}=4M_{pl}^{4}[\mathcal{K}(\chi_{i}^{+})-\mathcal{K}(\chi_{i}^{+}+Q_{i})]/T^{2}$.

To see how $X$ relates to the geometry of the instanton we use the junction
conditions,

$\Delta\rho(0)=0,\qquad\Delta[\rho^{\prime}(0)]=-\frac{T}{2M_{pl}^{2}}\rho(0)$
(32)

along with the expression $\rho^{\prime 2}=1-k^{2}\rho^{2}$ to prove the
following useful relations

$\langle\rho^{\prime}(0)\rangle=\frac{M_{pl}^{2}}{T}\rho(0)\Delta k^{2}$ (33)

and

$\rho(0)=\frac{\frac{T}{M_{pl}^{2}}}{\sqrt{(\Delta
k^{2})^{2}+\frac{T^{2}}{M_{pl}^{4}}\langle
k^{2}\rangle+\left(\frac{T^{2}}{4M_{pl}^{4}}\right)^{2}}}.$ (34)

We now take the ratio of (32) and (33) to see that

$X=-\frac{2\langle\rho^{\prime}(0)\rangle}{\Delta[\rho^{\prime}(0)]}=\frac{1+z}{1-z},\qquad
z=\frac{(\epsilon\cos kr_{0})_{+}}{(\epsilon\cos kr_{0})_{-}}.$ (35)

The value of $X$ now corresponds to a kinematic constraint on the geometry. As
an illustrative example, consider an instanton describing a
$\text{dS}_{+}\to\text{dS}_{-}$ transition with
$(kr_{0})_{+}\geq\frac{\pi}{2}\geq(kr_{0})_{-}$ and, of course,
$\epsilon_{\pm}=1$. Since $(\cos kr_{0})_{+}\geq 0\geq(\cos kr_{0})_{-}$ it
immediately follows that this solution has $T\geq 0$ and $z\leq 0$, or in
other words, $-1\leq X\leq 0$.

A summary of these classifications is presented for all potential instanton
geometries in table 1. The table also lists those geometries for which the
corresponding tunnelling rates are infinitely suppressed and so can be
ignored. As we will see presently, these correspond to configurations with
$k_{+}^{2}\leq 0$ and $\epsilon_{+}=-1$. Table 1 mirrors the ‘Baedeker’ of
Kaloper:2022oqv , although we emphasize again how it applies to a much broader
set of models.

| $\text{dS}_{+}$ | $\text{Minkowski/AdS}_{+}$ | $\text{Minkowski/AdS}_{+}$
---|---|---|---
| $\epsilon_{+}=+1$ | $\epsilon_{+}=+1$ | $\epsilon_{+}=-1$
| $\text{dS}_{-}$
---
$\epsilon_{-}=+1$
| $(kr_{0})_{+}\geq\frac{\pi}{2}\geq(kr_{0})_{-}$ | $(kr_{0})_{+}\geq(kr_{0})_{-}\geq\frac{\pi}{2}$
---|---
allowed for $|X|\leq 1$ | allowed for $X\leq-1$
$\frac{\pi}{2}\geq(kr_{0})_{+}\geq(kr_{0})_{-}$ | $(kr_{0})_{+}<(kr_{0})_{-}$
allowed for $X\geq 1$ | negative tension
negative tension |  | $(kr_{0})_{-}\geq\frac{\pi}{2}$ | $\frac{\pi}{2}\geq(kr_{0})_{-}$
---|---
kinematically | kinematically
allowed for $X\leq-1$, | allowed for $-1\leq X\leq 0$,
infinitely suppressed | infinitely suppressed
| $\text{Minkowski/AdS}_{-}$
---
$\epsilon_{-}=+1$
| $(kr_{0})_{+}\geq\frac{\pi}{2}$ | $\frac{\pi}{2}\geq(kr_{0})_{+}$
---|---
allowed for $0\leq X\leq 1$ |  allowed for $X\geq 1$
| $|k_{-}|\geq|k_{+}|$ | $|k_{-}|<|k_{+}|$
---|---
allowed for $X\geq 1$ | negative tension
| kinematically allowed for $|X|\leq 1$,
---
infinitely suppressed
| $\text{Minkowski/AdS}_{-}$
---
$\epsilon_{-}=-1$
negative tension | negative tension |  | $|k_{-}|>|k_{+}|$ | $|k_{-}|\leq|k_{+}|$
---|---
negative tension | kinematically
|  allowed for $X\leq-1$,
|  infinitely suppressed

Table 1: Summary of transitions $\mathcal{M}_{+}\to\mathcal{M}_{-}$ where
$\mathcal{M}$ is de Sitter, Minkowski or anti de Sitter, as indicated. If a
transition is forbidden because it violates the condition of non-negative
tension, we mark it accordingly. For the remaining transitions, we mark
whether they are kinematically allowed for $|X|\leq 1$ or $|X|\geq 1$ where
$X=4M_{pl}^{4}\Delta k^{2}/T^{2}$. In some of these cases, the transition may
be kinematically allowed but is ruled out because the transition rate is
infinitely suppressed. Recall that we can take $\epsilon=+1$ for all dS
configurations WLOG, so any examples contrary to this are not applicable. For
a given model, equations (29) and (32) can be used to determine, $X$ and
identify precisely where we are on the Baedeker.

In semi-classical theory of vacuum decay, transition rates between vacua
$\mathcal{M}_{+}\to\mathcal{M}_{-}$ are given by Coleman:1977py ;
Callan:1977pt ; Coleman:1980aw

$\frac{\Gamma}{\text{Vol}}\sim e^{-B/\hbar}$ (36)

where

$B=S_{E}(\text{instanton})-S_{E}(\text{parent}).$ (37)

Here $S_{E}(\text{instanton})$ is the Euclidean action evaluated on the bubble
configurations described above, interpolating between the vacua
$\mathcal{M}_{+}$ and $\mathcal{M}_{-}$. In contrast, $S_{E}(\text{parent})$
is the Euclidean action evaluated on the complete parent vacuum,
$\mathcal{M}_{+}$, with no bubbles.

The tunnelling exponent can be computed in all cases after a relatively
lengthy calculation and plenty of heart warming cancellations, giving

$B=-2M_{pl}^{2}\Omega_{3}\Delta\left\\{\frac{1}{k^{2}}\left[\rho^{\prime}(r)^{3}\right]^{0}_{r_{\text{min}}}\right\\}+\Omega_{3}T\rho(0)^{3}$
(38)

where $r_{\text{min}}$ denotes the minimal value of the radial coordinate and
$\Omega_{3}$ is the volume of the unit 3-sphere. Note that this is
quantitatively identical to the tunnelling exponent computed in General
Relativity for the same geometry, reflecting the fact that the scalars are
constants in vacuo, while the three-forms gravitate like a cosmological
constant. Whenever we have a parent AdS or Minkowski vacuum with
$\epsilon_{+}=-1$, there are configurations with non-negative membrane tension
whose tunnelling exponents contain divergent contributions at
$r_{\text{min}}$. These correspond to exotic configurations where an infinite
asymptotic space is removed by membrane nucleation. The corresponding
tunnelling rates are either infinitely suppressed, ruling out the transition,
or infinitely enhanced, signalling a catastrophic instability. As we are
imposed Neumann boundary conditions on the three-forms, these particular
configurations are always infinitely suppressed333For
$\text{AdS}_{+}\to\text{AdS}_{-}$ configurations with $\epsilon_{\pm}=-1$,
there are divergent contributions to the tunnelling exponent for both vacua.
If we assume that the divergence in the proper distance occurs at the same
rate in each case, $|r_{\text{min}}^{+}|\sim|r_{\text{min}}^{-}|\to\infty$,
then the positivity of the membrane tension ensures that $|k|_{+}>|k|_{-}$,
and the dominance of the divergent contribution of the + vacuum..

As expressed in table 1, there are now only three configurations that avoid
any problems with negative brane tension or infinitely suppressed tunnelling
rates. These are the configurations of physical interest corresponding to

  * •

$\text{dS}_{+}\to\text{dS}_{-}$

  * •

$\text{dS}_{+}\to\text{Minkowski/AdS}_{-}$

  * •

$\text{Minkowski/AdS}_{+}\to\text{Minkowski/AdS}_{-}$ $(|k_{-}|\geq|k_{+}|)$

each with $\epsilon_{\pm}=1$ and so $\rho^{\prime}(r_{\text{min}})=1$. To
simplify the corresponding formulae for the tunnelling rates, we make use of
equations (32) to (34). The tunnelling exponents for the three configurations
of interest can now be written as

$B=\frac{2M_{pl}^{2}\Omega_{3}}{k_{+}^{2}k_{-}^{2}}\left\\{-\Delta
k^{2}+\frac{M_{pl}^{2}}{T}\rho(0)\left[(\Delta
k^{2})^{2}+\frac{T^{2}}{2M_{pl}^{4}}\left\langle
k^{2}\right\rangle\right]\right\\}$ (39)

If we recall that $X=4M_{pl}^{4}\Delta k^{2}/T^{2}$ and introduce

$Y(X)=\sqrt{(X-1)^{2}+16k_{+}^{2}M_{pl}^{4}/T^{2}}$ (40)

we can rewrite the tunnelling exponent in the following simple form

$B=\frac{4M_{pl}^{2}\Omega_{3}}{k_{+}^{2}}\left[\frac{1+Y-X}{Y(1+Y+X)}\right]\,.$
(41)

##  5 Cascades, bungees and the stability of the Minkowski vacuum

Armed with a general expression for the tunnelling exponent and a detailed
understanding of the instanton geometries as shown in table 1, we are ready to
take a more in-depth look at the tunnelling dynamics. In particular, we are
interested in what happens if the universe begins in a high scale de Sitter
vacuum with curvatures far above the current Hubble scale. Can we descend into
the current low scale vacuum in a single jump, avoiding the empty universe
problem, or do we need to cascade into it via a series of incremental changes
in curvature? Is there a danger we fall deep into an anti-de Sitter vacuum, so
that the universe is driven to a apocalyptic crunch? How long is the current
vacuum expected to survive before it gives way to these apocalyptic AdS vacua?

Consider tunnelling from a parent vacuum with a non-negative curvature, so
that $k_{+}^{2}\geq 0$. It follows that $Y\geq|X-1|\geq 0$, and so rewriting
the tunnelling exponent as

$B=\frac{4M_{pl}^{2}\Omega_{3}}{k_{+}^{2}}\left[\frac{Y-(X-1)}{Y(2+Y+X-1)}\right],$
(42)

we see that it is everywhere positive444This contradicts Kaloper:2022oqv
where it is claimed that the bounce action can be negative when the membrane
radius is close to the de Sitter radius of the two vacua. This involved
neglecting terms that we said to be small compared to a “leading” term, equal
to the difference in curvature. However, this “leading” term is just as small
as the neglected terms in the relevant limit.. Further, for $k_{+}$ fixed, it
is monotonically decreasing from $4M_{pl}^{2}\Omega_{3}/k_{+}^{2}$ as
$X\to-\infty$ to zero as $X\to+\infty$555The quickest way to see this is to
define $\zeta\in(-\infty,\infty)$ such that $X\equiv\frac{4M_{pl}^{4}\Delta
k^{2}}{T^{2}}=1+\frac{4M_{pl}^{2}k_{+}}{T}\sinh\zeta$. The tunnelling exponent
then reduces to the following simple form
$B=\frac{4M_{pl}^{2}\Omega_{3}}{k_{+}^{2}(1+e^{2\zeta})\left(1+\frac{2M_{pl}^{2}k_{+}}{T}e^{\zeta}\right)}~{}.$
This is clearly monotonically decreasing between the two limits stated in the
text. . For a given parent de Sitter vacuum, this means that transitions
towards smaller values of the curvature are likely to be the most rapid,
provided they are kinematically allowed. This raises the possibility of decay
deep into anti de Sitter space, which would be incompatible with the current
low scale vacuum, with curvature no larger than the current Hubble scale.

Generically we find that descent into AdS is kinematically forbidden for
parent vacua with very large de Sitter curvatures. This is because the
membrane doesn’t carry enough charge to trigger a sufficiently large jump in
the cosmological constant. For parent vacua with curvature less than some
critical value, $k_{+}^{2}\leq k_{c}^{2}$, descent into AdS is kinematically
allowed. This is potentially dangerous if the tunnelling rates are too high.
One way to avoid this is as follows: first we choose the parameters of the
model such that $0<X<1$ for all downward transitions from parent vacua below
the critical curvature, $k_{+}^{2}\leq k_{c}^{2}$. It follows that

$B>B(X=1)=\frac{2M_{pl}^{2}\Omega_{3}}{k_{+}^{2}\left(1+\frac{2M_{pl}^{2}k_{+}}{T}\right)}>\frac{M_{pl}^{2}\Omega_{3}}{k_{c}^{2}}=B_{c}$
(43)

where we have also used the fact that
$4M_{pl}^{4}k_{+}^{2}/T^{2}<1+4M_{pl}^{4}k_{-}^{2}/T^{2}\leq 1$ for
$k_{+}^{2}\leq k_{c}^{2}$ and $k_{-}^{2}\leq 0$. Since we typically expect
$k_{c}\lesssim M_{pl}$, we can choose parameters such that $B_{c}$ is large,
and all vacua in danger of decay into anti de Sitter are long lived.
Furthermore, the lifetime of these vacua increases as we reduce the curvature
of the parent vacuum, with $B$ containing a pole as $k_{+}\to 0$.

We can see the importance of the proposed bound on $X$ by taking a closer look
at transitions from parent vacua near Minkowski space. For these transitions,
we have

$X_{M_{+}\to AdS_{-}}\approx-\frac{4M_{pl}^{4}k_{-}^{2}}{T^{2}}.$ (44)

Note that $X_{M_{+}\to AdS_{-}}=|X_{M_{+}\to AdS_{-}}|$ since the daughter
vacuum is AdS and so $k_{-}^{2}<0$. The tunnelling exponent for the transition
goes as

$B_{M_{+}\to
AdS_{-}}\sim\frac{2M_{pl}^{2}\Omega_{3}}{k_{+}^{2}}(1-S(X))+\frac{8M_{pl}^{6}\Omega_{3}}{T^{2}X(X-1)^{2}}\left[(X-1)^{2}(1-S(X))+2S(X)\right]$
(45)

where $S(X)=\text{sgn}(X-1)$ and we understand that $X=X_{M_{+}\to AdS_{-}}$
here and in the remainder of this section. The lifetime of the parent near
Minkowski vacuum is clearly determined by the sign of $X-1$. In particular,
when $X>1$, we have

$B_{M_{+}\to AdS_{-}}\sim\frac{16M_{pl}^{6}\Omega_{3}}{T^{2}X(X-1)^{2}},\qquad
X>1.$ (46)

For tunnelling into deep AdS so that $X$ is large, this exponent is suppressed
and the transition is quick, rendering the near Minkowski vacuum short lived.
In contrast, if $X<1$, the tunnelling exponent goes as

$B_{M_{+}\to AdS_{-}}\sim\frac{4M_{pl}^{2}\Omega_{3}}{k_{+}^{2}},\qquad X<1.$
(47)

As the parent vacuum approaches Minkowski, this exponent diverges, suppressing
any transition into AdS and ensuring a long lived Minkowski vacuum. This is
the key ingredient for addressing the cosmological constant advocated in
Kaloper:2022oqv ; Kaloper:2022jpv , although we now see how it can apply more
generally, as long as the microscopic details of transition from Minkowski
into AdS enforce $X_{M_{+}\to AdS_{-}}=-\frac{4M_{pl}^{2}k_{-}^{2}}{T^{2}}<1$.

Of course, it is not enough for the near Minkowski vacuum to be long lived. We
need to show that we are likely to reach it from a generic initial state
corresponding to a high scale de Sitter vacuum. There are two ways in which
this might happen. The first involves a bungee jump, a single transition from
a high scale de Sitter vacuum towards a near Minkowski vacuum, $k_{+}^{2}\gg
H_{0}^{2}\gtrsim|k_{-}|^{2}$. If a bungee jump is kinematically possible, the
tunnelling exponent goes as

$B_{dS_{+}\to
M_{-}}\sim\frac{4M_{pl}^{2}\Omega_{3}}{k_{+}^{2}\left(1+\frac{4M_{pl}^{4}k_{+}^{2}}{T^{2}}\right)^{2}}.$
(48)

For de Sitter vacua, the larger the value of $k_{+}$, the more this exponent
is suppressed and the faster the transition. Bungee jumps avoid the empty
universe problem Abbott:1984qf . As explained in Bousso:2000xa , the inflaton
is significantly displaced from its minimum by quantum diffusion before the
transition. After the transition, the inflaton is in slow roll and follows the
classical evolution towards the minimum, at which point it begins to oscillate
and reheating occurs. However, because the vacuum energy is now no larger than
the current Hubble scale, there is no rapid cooling.

The second way in which we might finally arrive at the current vacuum is via a
cascade, several transitions in which the cosmological constant descends from
a high scale in small increments, each with $\Delta k^{2}\lesssim H_{0}^{2}$.
For small incremental transitions where the vacuum energy changes by a small
amount, $k_{+}^{2}\to k_{-}^{2}=k_{+}^{2}(1-\epsilon)$, for $\epsilon\ll 1$,
the tunnelling exponent goes as

$B_{dS_{+}\to dS_{-}\approx
dS_{+}}\sim\frac{4M_{pl}^{2}\Omega_{3}}{k_{+}^{2}\sqrt{1+\frac{16M_{pl}^{4}k_{+}^{2}}{T^{2}}}}.\qquad$
(49)

Unfortunately, these cascades do run into the empty universe problem
Abbott:1984qf . As the cosmological constant slowly descends, quantum
diffusion effects are scaled down, allowing the inflaton to settle into its
minimum and reheat the universe. However, this happens too soon, before the
vacuum energy has settled into its current value. The universe then spends far
too long in de Sitter vacua at scales above the current Hubble scale, and any
matter produced after reheating is diluted away by exponential expansion.

To avoid the empty universe problem, it is clear that bungee jumps with
$k_{+}^{2}\gg H_{0}^{2}\gtrsim|k_{-}|^{2}$ must be kinematically allowed.
Furthermore, a cascade of incremental transitions with $\Delta k^{2}\lesssim
H_{0}^{2}$, should either be kinematically forbidden, or suppressed relative
to a bungee jump. If this is the case, the universe initially in a high scale
de Sitter vacuum is expected to find the current vacuum before inflation has
ended and reheating has begun. Furthermore, if we can consistently impose a
condition $|X|<1$ for a parent vacuum close to Minkowski, then the current
vacuum is long lived, and the catastrophe of descent into AdS is delayed until
the far future, or avoided altogether.

##  6 Examples

As we have seen, the key features for addressing the cosmological constant
problem proposed in Kaloper:2022oqv ; Kaloper:2022jpv  could apply quite
generally to a large class of models under consideration. This is because the
scalars and the four-form field strengths all gravitate as cosmological
constants in the vacuum. Consequently, the tunnelling exponent, $B$, takes a
universal form which we can express in terms of the tension of the membrane
and the curvature of the two vacua involved in the transition. Things only
become model dependent when we ask how the quantities relate to other model
parameters, such as the membrane charge. In particular, the condition $|X|<1$
will mean different things for different models. Kinematic considerations can
also vary. We shall now explore a few cases of particular interest.

###  6.1 The Brown-Teitelboim set-up

We begin with the pioneering model of this class, originally proposed by Brown
and Teitelboim Brown:1987dd ; Brown:1988kg . As is well known, this model runs
into problems with the empty universe and cannot be taken seriously as a
solution to the cosmological constant problem. However, we include a
discussion of its main features as a warm up to the more interesting models.
In the Brown-Teitelboim set-up, there are no scalars and a single four-form,
corresponding to the case where $i,j=1$ only and

$\omega^{ij}=\sigma_{i}=0,\quad Z_{ij}=1,\quad V=V_{QFT}$ (50)

where $V_{QFT}$ is the renormalised vacuum energy. In the absence of scalars,
the bulk constraints (13) to (15) yield

$\displaystyle 3M_{pl}^{2}k^{2}=V_{QFT}+\frac{1}{2}c^{2}$ (51)
$\displaystyle\chi=-c$ (52)

from which it follows that

$k^{2}=\frac{V_{QFT}+\frac{1}{2}\chi^{2}}{3M_{pl}^{2}}\ .$ (53)

Membrane nucleation triggers a jump $\Delta\chi=-Q=-\eta q$, where $q$ is the
fundamental membrane charge and $\eta=+1$ for branes and $\eta=-1$ for anti-
branes. This is consistent with the quantisation condition $\chi=-Nq$ for
$N\in\mathbb{Z}$, so that membrane nucleation gives $\Delta N=\eta$. The
spectra of vacua are now characterised by the following curvature

$k^{2}(N)=\frac{V_{QFT}+\frac{1}{2}N^{2}q^{2}}{3M_{pl}^{2}}\ .$ (54)

For this to include near Minkowski vacua, we assume $V_{QFT}<0$, which is easy
to engineer in string compactifications. The smallest possible separation in
curvature, $\delta k^{2}$, occurs between neighbouring vacua and takes the
value

$\delta k^{2}=k^{2}(N\pm 1)-k^{2}(N)=\frac{(1\pm 2N)q^{2}}{6M_{pl}^{2}}\ .$
(55)

This density varies with where we are in the landscape, through the dependence
on $N$. For a near Minkowski vacuum,
$N=N^{\text{Mink}}\approx\pm\sqrt{2|V_{QFT}|}/q$ and the separation takes two
possible values

$\delta k^{2}|_{k^{2}=0}\approx\frac{q^{2}\pm
q\sqrt{8|V_{QFT}|}}{6M_{pl}^{2}}\,.$ (56)

To be compatible with observation, we require the absolute value of both to be
no larger than the current Hubble scale, $H_{0}^{2}$. Assuming $|V_{QFT}|\gg
M_{pl}^{2}H_{0}^{2}$, this imposes an upper bound on the membrane charge

$q\lesssim\frac{6M_{pl}^{2}H_{0}^{2}}{\sqrt{8|V_{QFT}|}}\ .$ (57)

Nucleation of a membrane triggers a jump $N^{+}\to N^{-}=N^{+}\pm 1$, where we
take the $+$ for a brane and the $-$ for an anti-brane. In the Brown-
Teitelboim setup, this corresponds to a minimal change in curvature

$\Delta k^{2}=\frac{(1\mp 2N^{-})q^{2}}{6M_{pl}^{2}}=\frac{(-1\mp
2N^{+})q^{2}}{6M_{pl}^{2}}\ .$ (58)

In a general set-up, adjacent vacua in flux space are separated by a
curvature, $\Delta k^{2}$, which is generically not the same as the smallest
jump in curvature, $\delta k^{2}$. Crucially, however, in the Brown-Teitelboim
set-up, the simplicity of the model renders the two to be equivalent: $\Delta
k^{2}=\delta k^{2}$. By the density assumption, nearest neighbour jumps are
now no larger than the current Hubble scale whenever the transitions include
the current vacuum. More explicitly, consider the transition from a parent de
Sitter vacuum to a near Minkowski vacuum. For this to the arise from the
nucleation of a single membrane, the parent vacuum will have curvature

$k_{+}^{2}=\frac{(1+2|N^{\text{Mink}}|)q^{2}}{6M_{pl}^{2}}\approx\frac{q}{6M_{pl}^{2}}\sqrt{8|V_{QFT}|}\lesssim
H_{0}^{2}$ (59)

where we have used the fact that
$|N^{\text{Mink}}|\gtrsim|V_{QFT}|/M_{pl}^{2}H_{0}^{2}\gg 1$ along with the
density constraint (57) on the fundamental charge. This demonstrates the fact
that bungee jumps to the current vacuum cannot be mediated by single membrane.
As is well known, this suggests that the Brown-Teitelboim set-up suffers from
an empty universe problem Bousso:2000xa . Tunnelling rates in this set-up are
easily computed and can be shown to agree with Brown:1987dd ; Brown:1988kg .
This can also be obtained from the corresponding analysis in the next section
by taking the limit of a single three-form potential.

###  6.2 The Bousso-Polchinski set-up

We now switch to a set-up with no scalars and a large number of four-forms as
originally proposed by Bousso and Polchinski Bousso:2000xa  as a way to
overcome the limitations of Brown:1987dd ; Brown:1988kg . This corresponds to
the case where

$\omega^{ij}=\sigma_{i}=0,\quad Z_{ij}=\delta_{ij},\quad V=V_{QFT}$ (60)

with $i,j$ running from $1$ to $\mathcal{N}$. In the absence of scalars, the
bulk constraints (13) to (15) yield

$\displaystyle 3M_{pl}^{2}k^{2}=V_{QFT}+\frac{1}{2}\delta_{ij}c^{i}c^{j}$ (61)
$\displaystyle\chi_{i}=-\delta_{ij}c^{j}$ (62)

from which it follows that

$k^{2}=\frac{V_{QFT}+\frac{1}{2}\delta^{ij}\chi_{i}\chi_{j}}{3M_{pl}^{2}}\ .$
(63)

Recall the quantisation condition $\chi_{i}=-N_{i}q_{i}$ (no sum), for some
$N_{i}\in\mathbb{Z}$. Assuming nucleation of a brane or anti-brane of type
$i_{*}$, this is consistent with the membrane junction condition (29),
$\Delta\chi_{i}=-\delta_{ii_{*}}Q_{i_{*}}$, with $Q_{i_{*}}=\pm q_{i_{*}}$ (no
sum), or equivalently, $\Delta N_{i}=\pm\delta_{ii_{*}}$. The spectra of vacua
is therefore characterised by curvature

$k^{2}=\frac{V_{QFT}+\frac{1}{2}\sum_{i=1}^{\mathcal{N}}N_{i}^{2}q_{i}^{2}}{3M_{pl}^{2}}\
.$ (64)

For this to include near Minkowski vacua, we must once again assume that
$V_{QFT}<0$. The vacua span an $\mathcal{N}$ dimensional grid with spacing
$q_{i}$. Surfaces of constant vacuum curvature correspond to spheres centred
at $N_{i}=0$. Consider two such surfaces of curvature
$k^{2}=\frac{V_{QFT}+\frac{1}{2}r^{2}}{3M_{pl}^{2}}$ and $k^{\prime
2}=\frac{V_{QFT}+\frac{1}{2}r^{\prime 2}}{3M_{pl}^{2}}$. The corresponding
spheres form the boundary of a shell of volume
$V_{\text{shell}}\approx\frac{1}{2}\Omega_{\mathcal{N}-1}r^{\mathcal{N}-2}\delta
r^{2}$ where $\delta r^{2}=r^{\prime 2}-r^{2}$ and
$\Omega_{\mathcal{N}-1}=2\pi^{\mathcal{N}/2}/\Gamma(\mathcal{N}/2)$ is the
volume of a unit $(\mathcal{N}-1)$-sphere. To estimate the density of vacua,
we require that this shell should contain at least one grid point (or if there
is degeneracy $D$, it should contain $D$ grid points), setting a bound
$V_{\text{shell}}\gtrsim D\prod_{i=1}^{\mathcal{N}}q_{i}$. This now translates
into vacua with a typical curvature separation of Bousso:2000xa

$\delta
k^{2}\approx\frac{D\prod_{i=1}^{\mathcal{N}}q_{i}}{3M_{pl}^{2}\Omega_{\mathcal{N}-1}r^{\mathcal{N}-2}},\qquad
r=\sqrt{6M_{pl}^{2}k^{2}-2V_{QFT}}\ .$ (65)

For this model, we assume the $q_{i}$ are incommensurate so that $D\sim
2^{\mathcal{N}}$, arising from the vacuum degeneracy $N_{i}\to-N_{i}$. As in
the Brown-Teitelboim set-up, the density of vacuum energies depends on where
you are in the landscape. Near Minkowski, the density is

$\delta
k^{2}|_{k^{2}=0}\approx\frac{2^{\mathcal{N}}\prod_{i=1}^{\mathcal{N}}q_{i}}{3M_{pl}^{2}\Omega_{\mathcal{N}-1}(2|V_{QFT}|)^{\mathcal{N}/2-1}}\
.$ (66)

If we assume that there are no hierarchies in the membrane charges,
$q_{i}\sim\mathcal{O}(q)$ for all $i$, whilst remaining incommensurate, it
turns out that

$\delta
k^{2}|_{k^{2}=0}\sim\frac{2|V_{QFT}|}{3M_{pl}^{2}}\sqrt{\frac{\pi}{\mathcal{N}}}\left(\frac{\mathcal{N}q^{2}}{e\pi|V_{QFT}|}\right)^{\mathcal{N}/2}$
(67)

Provided $\mathcal{N}q^{2}<e\pi|V_{QFT}|$, by taking $\mathcal{N}$ sufficently
large, we can achieve a density well within the desired observational bound
$\delta k^{2}|_{k^{2}=0}\lesssim H_{0}^{2}$ where $H_{0}\sim 10^{-60}M_{pl}$
is the current Hubble scale. For example, if $|V_{QFT}|\sim M_{pl}^{4}$ and
$q\sim 0.02M_{pl}^{2}$ then we can achieve the desired density of vacuum
energies for $\mathcal{N}\approx 100$.

Nucleation of a membrane of type $i_{*}$ triggers a jump in one of the flux
integers, $N^{+}_{i_{*}}\to N_{i_{*}}^{-}=N^{+}_{i_{*}}\pm 1$, while leaving
the remaining $N_{i\neq i_{*}}$ unchanged. This translates into a jump in
vacuum curvature

$\Delta k^{2}=\frac{(1\mp
2N_{i_{*}}^{-})q_{i_{*}}^{2}}{6M_{pl}^{2}}=\frac{(-1\mp
2N_{i_{*}}^{+})q_{i_{*}}^{2}}{6M_{pl}^{2}}.$ (68)

As in the previous example, this is sensitive to where we are in the landscape
through the dependence on $N^{\pm}_{i_{*}}$, a consequence of the curvature
having quadratic four-form dependence.

Let us consider descent from a parent de Sitter vacuum with curvature
$k_{+}^{2}\geq 0$ and compute the important quantity, $X=4M_{pl}^{4}\Delta
k^{2}/T^{2}$. To do this, we first relate the curvature to the integer fluxes
using the relation

$\sum_{i=1}^{\mathcal{N}}(N_{i}^{+})^{2}q_{i}^{2}=r_{+}^{2},\quad
r_{+}=\sqrt{6M_{pl}^{2}k_{+}^{2}-2V_{QFT}}.$ (69)

Since $\Delta k^{2}\geq 0$, it now follows that

$X=\frac{2M_{pl}^{2}q_{i_{*}}^{2}}{3\tau_{i_{*}}^{2}}(2|N_{i_{*}}^{+}|-1)\sim\frac{2M_{pl}^{2}q^{2}}{3\tau^{2}}(2|N_{i_{*}}^{+}|-1)$
(70)

where we have used the fact there are no hierarchies,
$q_{i}\sim\mathcal{O}(q)$, $\tau_{i}\sim\mathcal{O}(\tau)$ for all $i$.
Further, equation (69) now implies that $|\vec{N}|^{2}\sim r^{2}_{+}/q^{2}$,
where $|\vec{N}|^{2}=\sum_{i=1}^{\mathcal{N}}N_{i}^{2}\geq|N_{i_{*}}|^{2}$ and
so

$X\lesssim\frac{4M_{pl}^{2}qr_{+}}{3\tau^{2}}\equiv X_{\text{max}}(k_{+})\,.$
(71)

Dangerous transitions into AdS are kinematically forbidden if
$X_{\text{max}}(k_{+})<X_{0}(k_{+})$, where
$X_{0}(k_{+})=4M_{pl}^{4}k_{+}^{2}/\tau^{2}$ represents the critical value of
$X$ for which the daughter vacuum is approximately Minkowski. For the largest
values of $k_{+}^{2}\gg|V_{QFT}|/3M_{pl}^{2}$, it is clear that
$r_{+}\approx\sqrt{6}M_{pl}k_{+}$ and so

$X_{\text{max}}\approx\frac{4\sqrt{6}M_{pl}^{3}qk_{+}}{3\tau^{2}}<\frac{4M_{pl}^{2}k_{+}}{3\tau^{2}}\sqrt{\frac{6e\pi|V_{QFT}|}{\mathcal{N}}}\ll
X_{0}(k_{+})$ (72)

where we have used the density condition $\mathcal{N}q^{2}<e\pi|V_{QFT}|$. It
is, therefore, kinematically impossible for parent vacua with
$k_{+}^{2}\gg|V_{QFT}|/3M_{pl}^{2}$ to decay directly into AdS.

What about parent vacua with $k_{+}^{2}\lesssim|V_{QFT}|/3M_{pl}^{2}$? These
have $r_{+}\sim\sqrt{2|V_{QFT}|}$ and so

$X_{\text{max}}\sim\frac{2M_{pl}^{2}q}{3\tau^{2}}\sqrt{8|V_{QFT}|}.$ (73)

For sufficiently small $k_{+}^{2}<k_{c}^{2}=q\sqrt{8|V_{QFT}|}/6M_{pl}^{2}$,
we can have $X_{\text{max}}>X_{0}(k_{+})$ and so decay into AdS is now
kinematically possible.

The potential for decay into AdS might be troubling at first glance. However,
as we hinted in the previous section, we can impose the condition

$\frac{2M_{pl}^{2}q}{3\tau^{2}}{\sqrt{8|V_{QFT}|}}<1\ $ (74)

ensuring that $0<X<1$ whenever the parent vacuum has $k^{2}_{+}<k_{c}^{2}$ and
is therefore vulnerable to this dangerous decay. However, the lifetime of
these vacua can be made large by ensuring that $B_{c}\gg 1$, where $B_{c}$ is
defined by equation (43). Inputting the value of $k_{c}$ we have just derived
for the Bousso-Polchinski set-up, we find that

$B_{c}\gtrsim\frac{6M_{pl}^{4}\Omega_{3}}{q\sqrt{8|V_{QFT}|}}$ (75)

For our canonical example, with $|V_{QFT}|\sim M_{pl}^{4}$, $q\sim
0.02M_{pl}^{2}$ and $\mathcal{N}\approx 100$, we have $B_{c}\gtrsim 2000$, and
so vacuum decay into AdS is heavily suppressed in all instances. Furthermore,
the condition (74) ensures that $|X_{M_{+}\to AdS_{-}}|<1$, guaranteeing the
presence of a pole in the transition rate as the parent vacuum approaches
Minkowski, rendering it arbitrarily long-lived.

All of this suggests that the current vacuum can be selected using
probabilistic arguments, similar to those presented in Kaloper:2022oqv ;
Kaloper:2022jpv . This is a significant departure from the anthropic arguments
usually presented for the Bousso-Polchinski set-up Bousso:2000xa . There is a
small price to pay for this success: the condition (74) is in violation of the
weak gravity conjecture for membranes Ibanez:2015fcv , which requires

$\tau_{i}\lesssim g_{i}M_{pl}q_{i}$ (76)

where $g_{i}$ is the strength of the three-form coupling. To see this, note
that we have two inequalities on the charge,

$\sqrt{\frac{e\pi|V_{QFT}|}{\mathcal{N}}}>q>\frac{2M_{pl}^{2}q^{2}}{3\tau^{2}}\sqrt{8|V_{QFT}|}$
(77)

and so

$\frac{M_{pl}q}{\tau}<\left(\frac{9e\pi}{32\mathcal{N}}\right)^{1/4}<1$ (78)

in violation of (76).

If we assume sub-Planckian membrane tension consistent with a well-defined
effective theory and the maximum charge, $q_{\text{max}}$, tolerated by the
density constraint $\delta k^{2}\lesssim H_{0}^{2}$ (where $\delta k^{2}$ is
given by equation (67)), the stability condition (74) is also incompatible
with an arbitrarily large underlying vacuum energy $|V_{QFT}|$. To see this,
we note that

$q_{\text{max}}=\sqrt{\frac{e\pi|V_{QFT}|}{\mathcal{N}}}\left(\frac{3M_{pl}^{2}H_{0}^{2}}{2|V_{QFT}|}\sqrt{\frac{\mathcal{N}}{\pi}}\right)^{\frac{1}{\mathcal{N}}}\approx\sqrt{\frac{e\pi|V_{QFT}|}{\mathcal{N}}}\epsilon(\mathcal{N}),\quad$
(79)

where, in the last step, we have taken a limit of large $\mathcal{N}$ and
defined
$\epsilon(\mathcal{N})=\left(\frac{H_{0}^{2}}{M_{pl}^{2}}\right)^{\frac{1}{\mathcal{N}}}<1$.
We include the $\epsilon(\mathcal{N})$ correction to the large $\mathcal{N}$
limit to allow for the fact there is a large hierarchy between $H_{0}$ and
$M_{pl}$. For $q\sim q_{\text{max}}$, the stability condition (74) now implies
that

$\frac{2|V_{QFT}|}{3M_{pl}^{4}}\sqrt{\frac{8e\pi}{\mathcal{N}}}\left(\frac{3M_{pl}^{2}H_{0}^{2}}{2|V_{QFT}|}\sqrt{\frac{\mathcal{N}}{\pi}}\right)^{\frac{1}{\mathcal{N}}}\lesssim\frac{\tau^{2}}{M_{pl}^{6}}<1$
(80)

where we have also used the condition of sub-Planckian membrane tensions. This
can be used to place a limit on the scale of the QFT contribution to the
cosmological constant. For large $\mathcal{N}$, this bound on $|V_{QFT}|$
takes on a particularly simple form

$|V_{QFT}|\lesssim\frac{3}{2\epsilon(\mathcal{N})}\sqrt{\frac{\mathcal{N}}{8e\pi}}\times\frac{\tau^{2}}{M_{pl}^{2}}.$
(81)

This is quadratically sensitive to the brane tension and not especially
restrictive as can be seen from the numerical examples in table 2. Note, in
particular, the case where $\mathcal{N}=100$, corresponding to the benchmark
point in Bousso:2000xa . Here we can have Planckian $|V_{QFT}|$ with the
current vacuum being both natural and quantum mechanically stable if the brane
tensions, $\tau\sim 0.2M_{pl}^{3}$ and the charges, $q\leq 0.02M_{pl}^{2}$.

$\mathcal{N}$ | Upper bound on $|V_{QFT}|$ | Upper bound on $q_{\text{max}}$
---|---|---
10 | $10^{12}\times\frac{\tau^{2}}{M_{pl}^{2}}$ | $10^{-7}\times\frac{\tau}{M_{pl}}$
60 | $10^{2}\times\frac{\tau^{2}}{M_{pl}^{2}}$ | $10^{-2}\times\frac{\tau}{M_{pl}}$
100 | $10\times\frac{\tau^{2}}{M_{pl}^{2}}$ | $10^{-1}\times\frac{\tau}{M_{pl}}$
Table 2: Maximum scale of QFT contribution to the cosmological constnat (in
units of $\tau^{2}/M_{pl}^{2}$) compatible with the quantum mechanical
stability of low scale de Sitter space. We also show the corresponding upper
bound on the brane charges (in units of $\tau/M_{pl}$). The quoted numbers are
order of magnitude estimates.

The last thing we need to check is already well known: is it possible to
transition from a high scale de Sitter vacuum to the current vacuum in a
single jump? Near Minkowski daughter vacua satisfy

$\sum_{i=1}^{\mathcal{N}}(N_{i}^{\text{Mink}})^{2}q_{i}^{2}\approx
2|V_{QFT}|.$ (82)

If this is the daughter vacuum generated by the nucleation of a brane of type
$i_{*}$, then the parent vacuum has curvature

$k_{+}^{2}=\frac{\sum_{i=1}^{\mathcal{N}}(1+2|N_{i_{*}}^{\text{Mink}}|)q_{i_{*}}^{2}}{6M_{pl}^{2}}\sim\frac{q\sqrt{8|V_{QFT}|}}{6M_{pl}^{2}}=k_{c}^{2}$
(83)

For physically allowed configurations with positive brane tension this will be
positive, corresponding to a high scale de Sitter vacuum provided $q$ is not
tuned to exponentially small values. In particular, for our canonical
parameter values, $|V_{QFT}|\sim M_{pl}^{4}$, $q\sim 0.02M_{pl}^{2}$ and
$\mathcal{N}\approx 100$, we have a bungee jump from vacua of curvature
$k_{+}^{2}\sim 0.01M_{pl}^{2}$ to the current vacuum. This enables us to avoid
the empty universe problem in the Bousso-Polchinski set-up, as is well known.

To summarise, if we are prepared to accept some violation of the membrane weak
gravity conjecture, we can select the near Minkowski vacuum using
probabilistic methods even for the Bousso-Polchinski set-up. This offers a
tantalising alternative to the standard anthropic arguments used to recover a
vacuum consistent with observation.

###  6.3 The Kaloper-Westphal set-up

We now switch to the original motivation for our work: the Kaloper-Westphal
set-up Kaloper:2022jpv . This corresponds to the case where there are just two
three-forms (so $i,j$ runs from $1$ to $2$) and

$\omega^{ij}=Z_{ij}=0,\quad
V=V_{QFT}+M_{pl}^{2}(\phi_{1}+\phi_{2}),\quad\sigma_{i}=-2\phi_{i}.$ (84)

Crucially, the membrane charges are assumed to have an irrational ratio,
$q_{1}/q_{2}=w\notin\mathbb{Q}$. The bulk constraints on the vacua (13) to
(15) now give

$\displaystyle 3M_{pl}^{2}k^{2}=V_{QFT}+M_{pl}^{2}(\phi_{1}+\phi_{2})$ (85)
$\displaystyle M_{pl}^{2}=2c^{k}$ (86) $\displaystyle\chi_{i}=-2\phi_{i}$ (87)

from which it follows that

$k^{2}=\frac{V_{QFT}-\frac{M_{pl}^{2}}{2}(\chi_{1}+\chi_{2})}{3M_{pl}^{2}}.$
(88)

Recall that the nucleation of the membrane of type $i_{*}$ triggers a jump
$\Delta\chi_{i}=-\delta_{ii_{*}}Q_{i_{*}}$, where $Q_{i_{*}}=\pm q_{i_{*}}$,
consistent with the quantisation condition on $\chi_{i}=-N_{i}q_{i}$ (no sum)
for some $N_{i}\in\mathbb{Z}$. The spectra of vacua is characterised by the
curvature

$k^{2}=\frac{V_{QFT}+\frac{M_{pl}^{2}}{2}(N_{1}q_{1}+N_{2}q_{2})}{3M_{pl}^{2}}.$
(89)

The irrational ratio in membrane charge ensures that this landscape of vacua
is dense and passes arbitrarily close to the Minkowski vacuum. The nucleation
of our membrane triggers a consistent jump $N^{+}_{i_{*}}\to
N_{i_{*}}^{-}=N^{+}_{i_{*}}\pm 1$ leaving $N_{i\neq i_{*}}$ unchanged. This
gives rise to a jump in the vacuum curvature

$\Delta k^{2}=\pm\frac{q_{i_{*}}}{6}$ (90)

giving

$X=\pm\frac{2M_{pl}^{4}q_{i_{*}}}{3\tau^{2}_{i_{*}}}.$ (91)

If we start in a parent de Sitter vacuum with curvature $k_{+}^{2}$,
transitions to AdS are only possible for $k_{+}^{2}<k_{c}^{2}=q/6$, where
$q=\text{max}\\{q_{1},q_{2}\\}$. In Kaloper:2022jpv , the authors impose the
condition,

$\frac{2M_{pl}^{4}q_{i}}{3\tau_{i}^{2}}<1,\qquad i=1,2$ (92)

which is equivalent to $|X|<1$ in all cases. In particular, as we have seen in
the previous section, this type of condition ensures that those vacua
vulnerable to decay into anti de Sitter can be long lived. More precisely,
when $k_{+}^{2}<k_{c}^{2}=q/6$, we recall that the tunnelling exponent $B\geq
B_{c}$, where $B_{c}$ is computed using (43),

$B_{c}\sim\frac{6M_{pl}^{2}\Omega_{3}}{q}.$ (93)

As long $q$ is sub-Planckian, this remains large. Indeed, for
$q_{\text{max}}\sim 0.1M_{pl}^{2}$, the low scale de Sitter vacua can be made
very long lived, with $B_{c}\gtrsim 1000$. Since $|X|<1$, we also have
absolute stability of the Minkowski vacuum thanks to the presence of the pole
in the tunnelling exponent as $k_{+}\to 0$. Note that unlike in the Bousso-
Polchinski setup, the condition (92) is independent of $|V_{QFT}|$ or, in
other words, the quantum mechanical stability of the Minkowski vacuum can be
guaranteed whatever the depth of the bare AdS vacuum.

The Kaloper-Westphal model also avoids the empty universe problem. This is
because we can bungee jump to the current vacuum from a parent de Sitter
vacuum with curvature $k_{+}^{2}\sim q/6$, where $q$ is not constrained to be
exponentially small in Planck units. Assuming sub-Planckian brane tension, we
might worry that the one condition we do have (92) is at odds with the weak
gravity conjecture for membranes (76). However, in the absence of kinetic
terms for the three-forms in the Kaloper-Westphal set-up, it is not
immediately clear how one should implement the weak gravity condition in the
first place. Of course, the absence of the kinetic terms would itself present
a challenge to string theory model builders.

###  6.4 More fun with irrationals

The key ingredients of the Kaloper-Westphal set-up can be realised in a simple
model of a single axion coupled to a four-form. In particular, we consider an
action of the form

$S=\int_{\mathcal{M}}d^{4}x\sqrt{|g|}\left[\frac{M_{pl}^{2}}{2}R-\frac{1}{2}\nabla_{\mu}\phi\nabla^{\mu}\phi-V(\phi)\right]\\\
+\int_{\mathcal{M}}\left[-\frac{1}{2}F\wedge\star F-\mu\phi
F\right]+S_{\text{boundary}}+S_{\text{membranes}}$ (94)

with $V(\phi)=V_{QFT}-m^{4}\cos(\phi/f)$. In terms of our generalised model
described earlier, this corresponds to the case where $i,j=1$ only and
$\omega^{ij}=1,Z_{ij}=1,\sigma_{i}=-\mu\phi$. The bulk constraints (13) to
(15) imply that

$\displaystyle 3M_{pl}^{2}k^{2}$ $\displaystyle=$ $\displaystyle
V_{QFT}-m^{4}\cos(\phi/f)+\frac{1}{2}c^{2}$ (95)
$\displaystyle\frac{m^{4}}{f}\sin(\phi/f)$ $\displaystyle=$ $\displaystyle\mu
c$ (96) $\displaystyle\chi$ $\displaystyle=$ $\displaystyle-\mu\phi-c.$ (97)

It follows that

$k^{2}=\frac{V_{QFT}-m^{4}\cos(\bar{\phi}(\chi)/f)+\frac{m^{8}}{2\mu^{2}f^{2}}\sin^{2}(\bar{\phi}(\chi)/f)}{3M_{pl}^{2}}\,,$
(98)

where $\bar{\phi}$ solves the equation

$\chi=-\mu\bar{\phi}-\frac{m^{4}}{\mu f}\sin(\bar{\phi}/f)\,.$ (99)

Nucleation of the membrane leads to a jump $\Delta\chi=-Q=-\eta q$, where $q$
is the fundamental membrane charge and $\eta=+1$ for branes and $\eta=-1$ for
anti-branes. This is consistent with the quantisation condition on $\chi=-Nq$
for some $N\in\mathbb{Z}$, with the membrane nucleation giving $\Delta
N=\eta$. We can rewrite (99) as

$N=\frac{\mu}{q}\bar{\phi}(N)+\frac{m^{4}}{q\mu f}\sin(\bar{\phi}(N)/f)\,,$
(100)

with the vacuum spectrum characterised by curvature

$k^{2}(N)=\frac{V_{QFT}-m^{4}\cos(\bar{\phi}(N)/f)+\frac{m^{8}}{2\mu^{2}f^{2}}\sin^{2}(\bar{\phi}(N)/f)}{3M_{pl}^{2}}\,.$
(101)

If the cosine potential were generated by non-perturbative corrections to the
axion $\phi$, and nothing else, we would expect the decay constant and
therefore the periodicity of the potential, to be set by the fundamental
membrane charge. This corresponds to the critical choice,
$f_{\text{crit}}=q/2\pi\mu$. The problem is that in such a set-up, the vacuum
curvature (101) is single valued, making it of no use in addressing the
cosmological constant problem. To see this note that if $f=f_{\text{crit}}$,
then (99) is invariant under $N\to N+n,\bar{\phi}\to\bar{\phi}+2\pi
nf_{\text{crit}}$. It follows that nucleation of a membrane simply triggers a
shift in $\bar{\phi}$ of size $\pm 2\pi f_{\text{crit}}$, leaving the
curvature unchanged.

Inspired by the Kaloper-Westphal model, we consider instead the case where
$f=wf_{\text{crit}}$ for some $w\notin\mathbb{Q}$. This could have arisen from
integrating out the heavier of two sectors in a doubled system with an
irrational ratio of membrane charges, reinforcing the connection to Kaloper
and Westphal’s original set-up. The curvature spectrum is now dense over a
finite range of values. To see this, we assume a rational approximation for
$w\approx p_{1}/p_{2}$, accurate to order $\epsilon$,

$w=\frac{p_{1}}{p_{2}}(1+\epsilon)$ (102)

for coprimes $p_{1}$ and $p_{2}$. If $\phi(N)$ solves (100), then we can show
that

$\phi(N+p_{1})\approx\bar{\phi}(N)+2\pi fp_{2}-\frac{2\pi
fp_{1}\epsilon}{\frac{2\pi m^{4}}{q\mu f}\cos(\bar{\phi}(N)/f)+w}\,.$ (103)

We now compare the vacuum curvature for the vacuum with flux number $N$ and
that with number $N+p_{1}$,

$\delta k^{2}=k^{2}(N+p_{1})-k^{2}(N)\approx-\left[\frac{2\pi
p_{1}m^{4}\sin(\bar{\phi}(N)/f)(1+\frac{m^{4}}{\mu^{2}f^{2}}\cos(\bar{\phi}(N)/f))}{3M_{pl}^{2}\left(\frac{2\pi
m^{4}}{q\mu f}\cos(\bar{\phi}(N)/f)+w\right)}\right]\epsilon.$ (104)

Since we can make $\epsilon$ arbitrarily small but non-zero with better and
better choices of $p_{1}$ and $p_{2}$, it is clear that the distance between
vacua can be made arbitrarily small and that the spectrum is dense. It turns
out that the curvature lies in a range
$k^{2}\in[k^{2}_{\text{min}},k^{2}_{\text{max}}]$, where

$k^{2}_{\text{min}}=\frac{1}{3M_{pl}^{2}}\left(V_{QFT}-m^{4}\right)$ (105)

and

$k^{2}_{\text{max}}=\frac{1}{3M_{pl}^{2}}\left[V_{QFT}+m^{4}+\frac{1}{2}m^{4}\left(\frac{m^{2}}{\mu
f}-\frac{\mu f}{m^{2}}\right)^{2}\theta\left(\frac{m^{2}}{\mu
f}-1\right)\right]$ (106)

where

$\theta\left(\frac{m^{2}}{\mu f}-1\right)=\begin{cases}1&m^{2}\geq\mu f\\\
0&m^{2}<\mu f\end{cases}.$ (107)

We now require that $V_{QFT}$, $\mu f$ and $m^{4}$ are such that the range of
curvature passes from large and positive, $k_{\max}^{2}\gg H_{0}^{2}>0$, to
large and negative values, $k_{\min}^{2}\ll-H_{0}^{2}<0$, with respect to the
current Hubble scale. Once this is guaranteed, the density of the spectrum
ensures that vacua exist whose curvature are sufficiently close to
observations.

Requiring the curvature of vacua to scan either side of the observed value
directly constrains the size of the QFT contribution to the vacuum energy. In
particular, we find that

$3M_{pl}^{2}H_{0}^{2}-m^{4}-\frac{1}{2}m^{4}\left(\frac{m^{2}}{\mu
f}-\frac{\mu f}{m^{2}}\right)^{2}\theta\ll
V_{QFT}\ll-3M_{pl}^{2}H_{0}^{2}+m^{4}.$ (108)

Assuming $m^{4}\gg M_{pl}^{2}H_{0}^{2}$, we can satisfy this just by imposing
an upper bound on the scale $|V_{QFT}|\lesssim m^{4}$. Typically we expect
$m^{4}\sim\Lambda_{\text{UV}}^{4}e^{-S}$, where $\Lambda_{\text{UV}}$ is some
UV scale666In QCD, this is set by the QCD scale, or more precisely, by the
scale of the quark condensates and the quark masses. In string
compactifications it is typically a few orders of magnitude below the Planck
scale, being suppressed by the volume of the Calabi-Yau.  and $S>1$ is the
action for the instanton that generates the potential. Although we cannot
extend the scale of the underlying vacuum energy to arbitrarily large values
in this model, we can easily accommodate scales just a few orders of magnitude
below the Planck scale.

As ever, nucleation of a membrane triggers a jump in one of the flux integers,
$N^{+}\to N^{-}=N^{+}\pm 1$, translating into a jump in vacuum curvature

$\Delta
k^{2}=-\frac{m^{4}\Delta[\cos(\bar{\phi}/f)]}{3M_{pl}^{2}}\left[1+\frac{m^{4}}{\mu^{2}f^{2}}\langle\cos(\bar{\phi}/f\rangle)\right]$
(109)

where

$\Delta[\cos(\bar{\phi}/f)]=\cos(\bar{\phi}(N^{+})/f)-\cos(\bar{\phi}(N^{+}\pm
1)/f)$ (110)

and

$\langle\cos(\bar{\phi}/f)\rangle=\frac{1}{2}\left[\cos(\bar{\phi}(N^{+})/f)+\cos(\bar{\phi}(N^{+}\pm
1)/f)\right]\,.$ (111)

This depends implicitly on where we are in the landscape through the
dependence on $N^{+}$. Descent from parent de Sitter vacua with curvature
$k_{+}^{2}>0$ is characterised by the value of $X=4M_{pl}^{4}\Delta
k^{2}/T^{2}>0$. If $\tau$ is the fundamental membrane tension, we can use the
fact that $\Delta[\cos(\bar{\phi}/f)]\geq-2$ and
$\langle\cos(\bar{\phi}/f)\rangle\leq 1$ to show that

$X\leq\frac{8M_{pl}^{2}m^{4}}{3\tau^{2}}\left[1+\frac{m^{4}}{\mu^{2}f^{2}}\right]=X_{\text{max}}\
.$ (112)

As in our analysis for the Bousso-Polchinski set-up, dangerous transitions
into AdS are kinematically forbidden if
$X_{\text{max}}<X_{0}(k_{+})=4M_{pl}^{2}k_{+}^{2}/\tau^{2}$. This introduces a
critical curvature
$k_{c}^{2}=\frac{2}{3}\frac{m^{4}}{M_{pl}^{2}}\left[1+\frac{m^{4}}{\mu^{2}f^{2}}\right]$
beyond which we cannnot descened into AdS through a single membrane
nucleation. For $k_{+}^{2}\lesssim k_{c}^{2}$, descent into AdS may be
possible. However, if we impose the modest constraint

$\frac{8M_{pl}^{2}m^{4}}{3\tau^{2}}\left[1+\frac{m^{4}}{\mu^{2}f^{2}}\right]<1$
(113)

we are guaranteed to have $0<X<1$ for all downward transitions. This ensures
that the lifetime of these vacua is determined by a tunnelling exponent $B\geq
B_{c}$ where

$B_{c}=\frac{3M_{pl}^{4}\Omega_{3}}{2m^{4}}\left[1+\frac{m^{4}}{\mu^{2}f^{2}}\right]^{-1}\,.$
(114)

By choosing parameters so that this is large we can ensure that the metastable
vacua are very long-lived. Furthermore, the condition (113) also ensures that
$|X_{M_{+}\to AdS_{-}}|<1$. As we saw previously, this guarantees the presence
of a pole in the transition rate as the parent vacuum approaches Minkowski,
rendering it arbitrarily long-lived.

Interestingly, this set-up seems to avoid any issues with the membrane weak
gravity conjecture. To see this, note that
$f=wf_{\text{crit}}=\frac{wq}{2\pi\mu}$, and so we can rewrite the condition
(113) as

$\frac{8M_{pl}^{2}m^{4}}{3\tau^{2}}\left[1+\frac{4\pi^{2}m^{4}}{w^{2}q^{2}}\right]<1$
(115)

which does not indicate any violation of (76). The last thing we need to do is
to check that we can indeed transition to a near Minkowski vacuum in a single
bungee jump and avoid the empty universe problem. Using equation (101), we see
that the flux integer for the near Minkowski vacuum satisfies

$\cos\left[\bar{\phi}\left(N^{\text{Mink}}\right)/f\right]=-\frac{\mu^{2}f^{2}}{m^{4}}\left(1\pm\upsilon\right)$
(116)

where

$\upsilon=\frac{1}{\mu
f}\sqrt{2\left(V_{QFT}+\frac{1}{2}\mu^{2}f^{2}+\frac{1}{2}\frac{m^{8}}{\mu^{2}f^{2}}\right)}\geq\sqrt{6}\frac{M_{pl}k_{\text{max}}}{\mu
f}$ (117)

where in the inequality we have used the formulae for $k_{\text{max}}$ for
both $\mu f\leq m^{2}$ and $\mu f>m^{2}$ given in equation (106). If Minkowski
is the daughter vacuum, it is generated by membrane nucleation in a parent de
Sitter vacuum of curvature

$k_{+}^{2}=\frac{m^{4}}{3M_{pl}^{2}}|\Delta[\cos(\bar{\phi}/f)]|\left(\upsilon-\frac{m^{4}}{2\mu^{2}f^{2}}|\Delta[\cos(\bar{\phi}/f)]|\right)$
(118)

where now

$\Delta[\cos(\bar{\phi}/f)]=\cos(\bar{\phi}(N^{\text{Mink}}\mp
1)/f)-\cos(\bar{\phi}(N^{\text{Mink}})/f)\ .$ (119)

Let us now assume that $\upsilon\gg m^{4}/M_{pl}\mu f>H_{0}$ and
$\omega=f/f_{\text{crit}}>1$. As we will show, these are a set of sufficient
conditions that allow us to avoid the empty universe problem. First up, note
that $\omega>1$ implies $|\Delta[\cos(\bar{\phi}/f)]|\sim\mathcal{O}(1)$777To
show that $|\Delta[\cos(\bar{\phi}/f)]|\sim\mathcal{O}(1)$ for $\omega>1$,
consider the case where $|\Delta[\cos(\bar{\phi}/f)]|\ll 1$. It follows that
$\bar{\phi}(N^{\text{Mink}}\mp
1)\approx\sigma\bar{\phi}(N^{\text{Mink}})+2n\pi f$ for some $n\in\mathbb{Z}$
and $\sigma=\pm 1$, which is consistent with the condition on the flux
integers (100) if and only if  $\mp 1\approx
n\omega+(\sigma-1)N^{\text{Mink}}$ (120) For $\sigma=1$, this implies
$\omega\approx 1/|n|$, contradicting the requirement that $\omega>1$. For
$\sigma=-1$, the condition (120) depends explicitly on $N^{\text{Mink}}$ and
is destabilised by radiative corrections to $V_{QFT}$. We therefore ignore
this possibility. . For $\upsilon\gg\frac{m^{4}}{\mu^{2}f^{2}}$ and
$|\Delta[\cos(\bar{\phi}/f)]|\sim\mathcal{O}(1)$ the curvature of the parent
vacuum is bounded from below as follows

$k_{+}^{2}\sim\frac{m^{4}\upsilon}{3M_{pl}^{2}}\geq\frac{m^{4}k_{\max}}{M_{pl}\mu
f}\,,$ (121)

where in the second inequality we have neglected $\mathcal{O}(1)$ factors and
used equation (117). As mentioned before, the absence of finely tuned
cancellations between the various contributions to the curvature requires
$k_{\max}\gg H_{0}$. If we further assume that the axion parameters satisfy
the inequality $m^{4}/M_{pl}\mu f>H_{0}$, as stated above, we find that

$k_{+}^{2}\gg\frac{m^{4}H_{0}}{M_{pl}\mu f}\gg H_{0}^{2}.$ (122)

Thus, the curvature of the parent vacuum vastly exceeds that of its offspring,
the current vacuum. It follows that we can bungee jump from a high scale
vacuum to the current vacuum with a single bubble nucleation and the empty
universe problem is avoided.

We conclude that this model of a single four-form scalar pair addresses the
cosmological constant problem in much the same way as the Kaloper-Westphal
model. However, there are some differences. For example, in the Kaloper-
Westphal model, the quantum mechanical stability of Minkowski is independent
of $|V_{QFT}|$, whereas here we find it can only be achieved for
$|V_{QFT}|<m^{4}$. This is not a significant limitation since we can easily
imagine $m$ lying just below the Planck scale. This new model also contains
certain features that are appealing from the perspective of fundamental
theory, including a standard kinetic term for the four-form without any
violation of the weak gravity conjecture for membranes.

Of course, the form of the axion potential may be further constrained by the
axion weak gravity conjecture Rudelius:2015xta . Naively, this would
correspond to the condition $fS<M_{pl}$ which is not especially restrictive
here. However, we think a more careful analysis in the context of a particular
UV completion of the model would be worthwhile.

Finally, it is worth noting that besides the vacuum transitions mediated by
the nucleation of fundamental membranes charged under the four-form, the
present model also allows for transitions between local minima while holding
the four-forms fixed. These correspond to the well known Coleman-de Luccia
instantons Coleman:1980aw , where local vacua are separated by domain walls,
and provide an extra channel for the relaxation of the cosmological constant.

##  7 Conclusions

Within the standard framework of quantum field theory, radiative corrections
to the vacuum energy are large, scaling like the fourth power of the cut-off.
If we assume naturalness, this scale should determine the overall size of the
renormalized vacuum energy. Corrections to the cosmological constant from
other sectors are now required to bring this result in line with observations,
bringing the corresponding vacuum curvature down to the present Hubble scale.
In an effective four-dimensional description of the universe in which gravity
is coupled to four-form field strengths, four-form flux provides a correction
to the cosmological constant. Nucleation of membranes charged under the
corresponding three-form potential, then allows the flux to jump by a single
quantised unit, changing the value of the cosmological constant. This yields a
landscape of possible vacua labelled by the quantised flux, scanned through
quantum tunnelling. To make contact with observation the landscape should be
sufficiently dense, with the curvature of vacua separated by no more than the
current Hubble scale. Furthermore, to avoid becoming lost in a cold and empty
universe, we must be able to reach the current vacuum from a high scale de
Sitter vacuum in a single nucleation event.

In Bousso and Polchinski’s seminal paper Bousso:2000xa , there are a large
number of four-forms, realising the required density of the landscape without
running into the empty universe problem. However, this is not enough to
address the cosmological constant problem. We also need to ask how the
observed vacuum is selected from the landscape of possibilities. Bousso and
Polchinski argue that the current vacuum should be selected using anthropic
arguments: if the vacuum curvature were too large and positive, structure
could not have formed due to the rapid exponential expansion; if it were too
large and negative, the universe would have started contracting before stars,
planets and complex life had ever come into existence. Anthropic arguments
have also been used in the context of the electroweak hierarchy problem where,
like in the cosmological constant case, naturalness is under pressure. For
example, a coupling between the Higgs-squared and one Giudice:2019iwl ;
Kaloper:2019xfj  or more four-forms Moretti:2022xlc  yields a landscape of
electroweak vacua, from which the current vacuum is selected through
anthropics.

In the recent history of string cosmology, anthropic ideas have indubitably
become the dominant explanation for the small value of the cosmological
constant seen in nature. We now seek to challenge that assertion, showing how
the current vacuum can often be selected on probabilistic grounds for a wide
class of string-like models of four-forms coupled to scalar fields, including
Bousso and Polchinski’s original set-up. With this new perspective, our world
exists independently of mankind or some other complex species. It is born from
probability.

The main idea was inspired by the recent proposal of KaloperKaloper:2022oqv
and Kaloper and Westphal Kaloper:2022jpv , although we have shown how it can
be readily extended. In the wide class of models under consideration, we have
identified a parameter that controls the lifetime of low scale vacua, similar
to the current vacuum we observe. If this parameter can be suitably bounded, a
pole appears in the corresponding bounce actions as the curvature of the
parent vacuum goes to zero. This guarantees the quantum stability of the
Minkowski vacuum, and the longevity of those vacua close to Minkowski in
Planck units. In the Kaloper-Westphal model, this bound corresponds to an
absolute bound on the membrane charge in terms of its tension. In the Bousso-
Polchinski model, the bound is also sensitive to the scale of the renormalised
vacuum energy. However, the sensitivity is not too severe: one can easily
accommodate a Planckian energy density for the QFT vacuum, with the membrane
tension and charge lying just an order of magnitude or two below the Planck
scale. That said, in both the Kaloper-Westphal and the Bousso-Polchinski set-
ups, these probabilistic arguments seem to require a mild violation of the
membrane weak gravity conjecture. With this in mind, we have presented a new
model where the same probabilistic methods are used but where there is no such
violation.

This new approach offers some hope for addressing the cosmological constant
problem using probabilistic methods. Notwithstanding some concerns associated
with various swampland considerations, this could be relevant to a wide class
of four-dimensional effective field theories whose underlying structures are
compatible with string compactifications.

#### Acknowledgements

We would like to thank Alex Westphal for useful discussions. AP was supported
by STFC consolidated grant number ST/T000732/1 and YL by an STFC studentship.
For the purpose of open access, the authors have applied a CC BY public
copyright licence to any Author Accepted Manuscript version arising.

##  Appendix A Tunnelling rates for general boundary conditions

For a general choice of boundary condition on the three-forms, the tunnelling
exponent computed from the geometry of the bounce is given by

$B=\lambda B_{N}+(1-\lambda)B_{D}$ (123)

where $B_{N}$ is the tunnelling exponent computed for Neumann boundary
conditions ($\lambda=1$), given by (38), and $B_{D}$ for Dirichlet boundary
conditions, given by

$B_{D}=B_{N}-\Omega_{3}\sum_{i}Q_{i}A^{i}(0)\\\
+\Omega_{3}\Delta\left\\{\frac{\sum_{i}\chi_{i}c^{i}}{3k^{4}}\left[3\rho^{\prime}(r)-\rho^{\prime}(r)^{3}\right]^{0}_{r_{\text{min}}}\right\\}.$
(124)

Recall that for Neumann boundary conditions ($\lambda=1$), configurations with
$k_{+}^{2}\leq 0$ and $\epsilon_{+}=-1$ are always infinitely suppressed; for
Dirichlet boundary conditions ($\lambda=0$), they will also be infinitely
suppressed provided $(2V+\sum_{i}\sigma_{i}c^{i})_{+}<0$. If this is not the
case, there will be a catastrophic instability.

For configurations of physical interest, the tunnelling exponent simplifies.
Indeed, $B_{N}$ is now given by (39) for Neumann boundary conditions on the
three-forms, and

$\displaystyle B_{D}=$ $\displaystyle
B_{N}+\Omega_{3}\left[\frac{M_{pl}^{2}}{3T}\rho(0)^{3}\left(v\Delta
k^{2}-\frac{T^{2}}{2M_{pl}^{4}}u\right)-\sum_{i}Q_{i}A^{i}(0)\right]$ (125)
$\displaystyle+\frac{2\Omega_{3}}{3k_{+}^{2}k_{-}^{2}}\Bigg{\\{}(u\Delta
k^{2}-v\langle k^{2}\rangle)\left(1-\frac{M_{pl}^{2}}{T}\rho(0)\Delta
k^{2}\right)\left.+\frac{T}{4M_{pl}^{2}}\rho(0)\left[v\Delta k^{2}-2u\langle
k^{2}\rangle\right]\right\\}$

for Dirichlet boundary conditions, where
$u=-6M_{pl}^{2}+\langle(2V+\sum_{i}\sigma_{i}c^{i})/k^{2}\rangle$ and
$v=\Delta\left[(2V+\sum_{i}\sigma_{i}c^{i})/k^{2}\right]$.

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