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11institutetext: Maryland Center for Fundamental Physics, University of
Maryland, College Park, MD 20742, USA22institutetext: Department of Physics
and Astronomy, University of British Columbia,
6224 Agricultural Road, Vancouver, B.C. V6T 1Z1, Canada.33institutetext:
Brandeis University, Waltham, MA 02453, USA

# Accelerating cosmology from $\Lambda<0$ gravitational effective field theory

Stefano Antonini 2 Petar Simidzija 3 Brian Swingle 2 Mark Van Raamsdonk 2
Chris Waddell<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

A large class of $\Lambda<0$ cosmologies have big-bang / big crunch spacetimes
with time-symmetric backgrounds and asymptotically AdS Euclidean continuations
suggesting a possible holographic realization. We argue that these models
generically have time-dependent scalar fields, and these can lead to realistic
cosmologies at the level of the homogeneous background geometry, with an
accelerating phase prior to the turnaround and crunch. We first demonstrate
via explicit effective field theory examples that models with an
asymptotically AdS Euclidean continuation can also exhibit a period of
accelerated expansion without fine tuning. We then show that certain
significantly more tuned examples can give predictions arbitrarily close to a
$\Lambda$CDM model. Finally, we demonstrate via an explicit construction that
the potentials of interest can arise from a superpotential, thus suggesting
that these solutions may be compatible with an underlying supersymmetric
theory.

##  1 Introduction

In this paper, we consider the viability of models of cosmology based on
gravitational effective theories associated to a dual CFT via the AdS/CFT
correspondence. These EFTs have a negative extremum in their scalar potential
(the negative cosmological constant, $\Lambda<0$) with an associated stable
AdS solution. But they also have cosmological solutions of FRW form, including
flat big bang matter/radiation cosmologies. Naively, these wouldn’t be
realistic because of the negative cosmological constant. However, we will
argue via examples that in the solutions that are most likely to have a
holographic description, there will generically be time-dependent scalar
fields whose potential energy can give rise to an accelerating phase in the
cosmology without significant fine-tuning. We will show further that the
resulting scale factor evolution can be realistic.

The initial motivation to study cosmological models based on EFTs with
holographic duals is the goal of finding a complete quantum gravity for some
four-dimensional big bang cosmology, realistic or not. Since some solutions of
these $\Lambda<0$ effective theories have a known microscopic description via
holography, it is plausible that holographic tools may play a role in coming
up with a microscopic description of the cosmological solutions as well.111On
the other hand, it is presently much less clear how models with a positive
cosmological constant can be given a microscopic description; see Banks:2001px
; Strominger:2001pn ; Alishahiha:2004md ; Gorbenko:2018oov ; Coleman:2021nor ;
Freivogel2005 ; McFadden:2009fg ; Banerjee:2018qey ; Susskind:2021dfc  for
various approaches.

There is a strong hint for how holography might play a role: cosmological
solutions with $\Lambda<0$ and some combination of matter and radiation are
time-symmetric big-bang / big-crunch spacetimes whose analytic continuation
are real geometries with asymptotically AdS regions for Euclidean time
$\tau\to\pm\infty$; following Maldacena:2004rf  we will refer to these
geometries as wormholes (see McInnes:2004nx  for an early study of
accelerating cosmologies in similar setups). This suggests a picture where the
Euclidean theory is defined holographically via a pair of 3D CFTs and
observables in the cosmology are related by analytic continuation to
observables in this Euclidean theory. We have suggested a possible framework
for this description in Cooper:2018cmb ; Antonini2019 ; VanRaamsdonk:2021qgv ;
Antonini:2022blk ; Antonini:2022xzo , but we will not assume this in the
present paper.222Possible holographic constructions must address the
factorization puzzle Maldacena:2004rf ; suggested resolutions include
considering an ensemble of CFTs or a direct interaction between the CFTs
Maldacena:2004rf ; Betzios:2019rds ; VanRaamsdonk:2020tlr .

The existence of dual CFTs associated to the asymptotically AdS regions in the
Euclidean solution suggests that the solutions will often also have non-
trivial scalar fields Antonini:2022blk ; VanRaamsdonk:2022rts . Scalars with
$m^{2}<0$ will be present in the effective field theory in the fairly generic
situation that the CFTs have relevant scalar operators.333Recall that the
scalar masses are related to operator dimensions by
$m^{2}L_{\textnormal{AdS}}^{2}=\Delta(\Delta-3)$. These scalar fields vanish
at the asymptotically AdS boundary, but in the most generic asymptotically AdS
solutions, the $m^{2}<0$ scalars turn on as we move away from the boundary.
This is associated with RG flow in the dual CFT. The radial dependence of the
scalar field in the wormhole translates to a time-dependence of the scalar
field in the cosmology.

The main goal of this paper is to show that these scalar fields can naturally
give rise to an accelerating period in the cosmology, and that in some cases,
the acceleration can be realistic, with the cosmological scale factor matching
arbitrarily well with the scale factor of conventional models that provide a
good fit to observational data.444See Cardenas:2002np  for an early study of
accelerating negative $\Lambda$ cosmologies compatible with data.

Figure 1: Evolution of the scalar field in a typical potential in the
cosmology and its Euclidean continuation.

A typical potential for a scalar field associated with a relevant CFT operator
is shown in Figure 1 (left). The negative $m^{2}$ gives the quadratic decrease
in $V$ for small $\phi$, while the eventual increase is due to the interaction
terms that would typically be present; for example, a higher-order polynomial
potential should eventually be positive and growing if the EFT is stable.

In the Euclidean solution, the evolution of the scalar field from the middle
of the wormhole to the AdS boundary corresponds to the damped motion of a
particle in the potential $-V$ with damping constant $3a^{\prime}/a\equiv
3(da/d\tau)/a$. In order for this extremum to be associated with a stable AdS
solution, we must satisfy the Breitenlohner-Freedmann bound
$V^{\prime\prime}>-9/(4L_{AdS}^{2})$, which turns out to correspond to the
fact that the motion of the scalar field towards the extremum is overdamped.
Thus, the simplest evolution of the scalar in the wormhole is a simple
monotonic evolution from a lower value of the potential (i.e. a higher value
of $-V$) at the middle to the higher extremal value at the boundaries, as
shown in Figure 1 (right).

In the cosmology, the value of the scalar at the time-symmetric point is the
same as the value of the scalar at the middle of the Euclidean wormhole. The
evolution back towards the big bang or forwards towards the big crunch
corresponds to the anti-damped motion of the scalar in the potential $V$,
initially moving in the direction towards the positive part of the potential.
In many cases, the scalar will reach the positive region of the potential
before the scale factor goes to zero, so part of the cosmological evolution
has positive dark energy.

This opens up the possibility that some of these $\Lambda<0$ models might
exhibit an accelerating phase (as we have argued in Antonini:2022blk ;
Antonini:2022xzo ). To check this, we study various explicit examples in
Section 2. We first verify that solutions based on the scalar evolution shown
in Figure 1 can lead to periods of accelerated expansion in the cosmology
without excessive fine-tuning. We find examples of simple potentials that give
rise to acceleration for generic values of their parameters, meaning in a
codimension zero region of the parameter space.

Next, we check that by taking an appropriately chosen potential, we can find
models with acceleration where the scale factor is arbitrarily close to
$\Lambda$CDM or $w$CDM models for the past evolution. These models are finely
tuned, but we include them as a demonstration that the framework can in
principle give results consistent with observations. For realistic cosmology,
we want to match observations, not $\Lambda$CDM, and this requirement allows a
much broader class of potentials. We return to this point in the discussion.

If the effective field theory describing cosmology is related to some
underlying CFT, an interesting possibility is that this CFT and the
corresponding effective theory is supersymmetric,555It has been conjectured
that holographic CFTs with standard dual gravitational effective field
theories must be supersymmetric Ooguri:2016pdq , though this is not proven or
widely accepted. and that this supersymmetry is broken in the cosmology by the
time-dependent scalar field expectation value. The physics of such a model is
surely very complicated. If the underlying effective theory is some gauged
supergravity, the scalar expectation value would break gauge symmetry and
supersymmetry, and the low energy effective field theory relevant to the
cosmology would arise via RG flow in which various fields would become
strongly coupled (as in QCD). We will not attempt to understand any of these
things here, but ask instead a fairly naive question: can the scalar
potentials giving rise to realistic time-symmetric cosmologies with
asymptotically AdS Euclidean continuations arise from a superpotential in a
supersymmetric effective field theory? In Section 3, we find that the answer
is yes, providing an algorithm to construct a superpotential that yields the
right behavior of the scalar potential for the range of scalar field values
present in the cosmological solutions.

The results of this paper are explorations to see what is possible at the
level of effective field theory; we have not attempted to investigate whether
the detailed potentials giving rise to realistic cosmological backgrounds can
arise from microscopic constructions in string theory. But the fact that
$\Lambda<0$ effective theories are not immediately ruled out suggests that
cosmological models with asymptotically AdS continuations (and thus potential
holographic descriptions) have a chance of being realistic. We discuss various
directions for further investigation in Section 4.

##  2 Accelerating cosmologies from asymptotically AdS Euclidean geometries

Throughout this paper, we will consider spatially flat cosmological solutions

$ds^{2}=-dt^{2}+a(t)^{2}d\vec{x}^{2}$ (1)

of $\Lambda<0$ effective field theories. For simplicity, we will consider
models with a single scalar field with potential $V(\phi)$. Allowing matter
and radiation, the equations of motion are

$\ddot{\phi}+3H\dot{\phi}+\frac{dV}{d\phi}=0\>,\qquad H^{2}=\frac{8\pi
G}{3}\left[\frac{1}{2}\dot{\phi}^{2}+V(\phi)+\frac{\rho_{R}}{a^{4}}+\frac{\rho_{M}}{a^{3}}\right]\>,$
(2)

where $H=\dot{a}/a$.

The solutions we are interested in are time-symmetric and arise via analytic
continuation from real Euclidean solutions

$ds^{2}=d\tau^{2}+a_{E}(\tau)^{2}d\vec{x}^{2}$ (3)

For the Euclidean solution, the equations of motion are

$\phi_{E}^{\prime\prime}+3H_{E}\phi_{E}^{\prime}-\frac{dV}{d\phi_{E}}=0\>,\qquad
H_{E}^{2}=\frac{8\pi
G}{3}\left[\frac{1}{2}(\phi_{E}^{\prime})^{2}-V(\phi_{E})-\frac{\rho_{R}}{a^{4}_{E}}-\frac{\rho_{M}}{a^{3}_{E}}\right]\>,$
(4)

where $a_{E}(\tau)=a(i\tau)$, $\phi_{E}(\tau)=\phi(i\tau)$, and
$H_{E}(\tau)=H(i\tau)$.

We would like to find potentials $V(\phi)$ and solutions $a(t),\phi(t)$ with
the following properties:

  * •

The scale factor evolution matches observational constraints, including a
recent phase of accelerated expansion

  * •

The solution is time-reversal symmetric about a recollapse point.

  * •

The analytic continuation of the solution (taking $t\to i\tau$ where $t=0$ is
the recollapse point) is asymptotically AdS for $\tau=\pm\infty$.

  * •

The scalar field satisfies the Breitenlohner-Freedman (BF) bound
$V^{\prime\prime}>-9/(4L_{AdS}^{2})$ at the two asymptotic AdS boundaries in
the Euclidean solution.

###  2.1 Constraints on the potential

There are a variety of constraints on the scalar potential $V$ coming from
these requirements.

First consider the solution in Lorentzian time (we will refer to this as the
“cosmology picture”). In order to have a time-symmetric solution, $a(t)$ and
$\phi(t)$ must be even functions (taking $t=0$ to be the recollapse time, so
that our present era corresponds to negative times) with $\dot{a}(0)=0$ and
$\dot{\phi}(0)=0$. The Friedmann equation shows that the potential must be
negative at this time-symmetric point, since $H=0$ here and $V$ must cancel
the remaining (positive) sources of energy density.

The scalar evolution equation is that of a particle in a potential with
damping $3H$ which is positive during the expanding phase. In order for the
scalar field to have zero time derivative at the time-symmetric point, this
particle must stop at $t=0$, and this is only possible if it is ascending the
potential just prior to $t=0$. Thus, going forward from the present time, the
scalar field must descend from positive to negative values of the potential
and then rise and stop at some $\phi_{0}$, passing through some minimum along
the way.

In the Euclidean continuation (we will refer to this as the “wormhole picture”
since the solutions have two asymptotically AdS boundaries connected by an
interior), the evolution equations are given in (4). Since $\ddot{a}(t=0)<0$,
we have $a_{E}^{\prime\prime}(\tau=0)>0$. For a solution that is
asymptotically AdS as $\tau\to\pm\infty$, $a$ must behave as
$e^{H_{AdS}|\tau|}$ for large $|\tau|$, so the simplest possibility is that
$a$ is monotonically increasing from $\tau=0$ towards each boundary. In this
case, the evolution of $\phi$ towards the boundary is the same as the damped
motion of a particle in potential $-V$. The scalar $\phi$ should approach a
constant as $\tau\to\pm\infty$, so the inverted potential $-V$ should have a
minimum here. Without loss of generality, we will take such a minimum to be
located at $\phi=0$.

Thus, the simplest possibility for the shape of the overall potential is that
shown in Figure 1 with the evolution in the cosmology picture and the wormhole
picture covering different regions of the potential as shown. We have argued
in the introduction that this shape is natural for effective field theories
associated with holographic CFTs.

A final constraint comes from requiring that the AdS solution which describes
the two asymptotic regions is stable to scalar perturbations. This requires
that the scalar field mass (the second derivative of the potential) satisfies
the BF bound

$\left.\frac{d^{2}V}{d\phi^{2}}\right|_{\phi=0}\geq\frac{9}{4}V(0),\;$ (5)

where we chose units such that $8\pi G/3=1$, in which $V(0)=-1/L_{AdS}^{2}$.
The condition (5) also ensures that the scaling dimensions of the
corresponding operator in the dual microscopic theory is real as required for
a unitary quantum field theory.

This bound turns out to have a simple implication for the scalar evolution
towards the extremum at $\phi=0$. First notice that, in order to ensure the
existence of the AdS asymptotic regions in the wormhole picture, the scalar
potential energy term must be the dominant contribution on the right-hand side
of the Euclidean Friedmann equation (4) as $\tau\to\pm\infty$. This implies
$H(\tau)\to H_{AdS}\equiv\sqrt{-V(0)}$ as $\tau\to\pm\infty$. By inspecting
the scalar field evolution equation in (4) it is then immediate to conclude
that the BF bound (5) is equivalent to the condition for the scalar field
motion to be overdamped in the vicinity of the extremum of the potential at
$\phi=0$. Therefore, moving from the center of the wormhole towards the
asymptotic boundaries, the scalar field will simply evolve from
$\phi=\phi_{0}$ to $\phi=0$, settling at the minimum of the inverted potential
$-V$ without any oscillation occurring in the asymptotic region.

To summarize, starting from the present value of the scalar potential, we
require that it decreases to negative values and then increases to an extremum
with a negative value, where the second derivative of the potential at the
extremum satisfies (5).

###  2.2 Existence and genericity of an accelerating phase

Starting with a potential satisfying these constraints, we can look for
solutions with the desired properties.

As initial conditions at the symmetric point $t=\tau=0$, we can take666In this
convention, $a$ will be less than one at the present time for examples
applicable to realistic cosmology, but we can rescale $a(t)\to a(t)/a(t_{0})$
to give the scale factor corresponding to the conventions of Sections 2.3 and
2.4.

$\phi(0)=\phi_{0},\qquad a(0)=1$ (6)

with vanishing derivatives
$\dot{a}(0)=a^{\prime}(0)=\dot{\phi}(0)=\phi^{\prime}(0)=0$. The matter and
radiation densities are constrained via the Friedmann equation at $t=0$ as

$\rho_{R}+\rho_{M}=-V(\phi_{0})\;.$ (7)

In the approximation where radiation can be neglected (valid for most of the
cosmological history and everywhere in the wormhole solution), the matter
density is fixed by $V(\phi_{0})$, so the solution is fully specified by
$\phi_{0}$, the scalar field value at $t=0$.

We would first like to check that it’s possible to have an accelerating phase
of cosmology before the recollapse without significant fine-tuning of the
potential. Evolving backward in time from the recollapse point in the
cosmology picture, the scalar field evolution corresponds to antidamped motion
in the potential $V$; it is natural that the scalar reaches positive values of
the potential, however this does not immediately imply an accelerating phase.
Such a phase further requires the positive scalar potential to dominate over
all other sources of energy density. This doesn’t always happen: it may be
that the growth of matter density, radiation density, or scalar kinetic energy
outpaces the increase in scalar potential as $a\to 0$ or that the scale factor
decreases to zero (i.e. we reach the big bang) before the potential energy
dominates. Thus, it is not true that we get an accelerating phase for all
potentials of the type shown in Figure 1 and all initial conditions.

On the other hand, it is not difficult to find examples of potentials and
initial conditions that do give rise to acceleration. Here we study a simple
example of a potential that satisfies all of our criteria and exhibits
acceleration over a significant fraction of the combined parameter space of
potential parameters and the initial condition $\phi_{0}$. For just this
subsection, we restrict to a simplified model with only radiation and scalar
field contributions to the stress tensor; we set the matter contribution to
zero for the purposes of this demonstration.

We consider a potential that is even about the AdS extremum $\phi=0$ with mass
$m^{2}=-9/4$ at the BF bound,

$V(\phi)=-1-\frac{9}{8}\phi^{2}+V_{\text{int}}(\phi)\;.$ (8)

In order to make the potential bounded from below and take positive values for
larger $\phi$, we include terms at higher order in $\phi^{2}$. We have found
that simply adding a $\phi^{4}$ term does not give solutions with an
accelerating phase. Instead, we consider a one-parameter family of even
interaction potentials

$V_{\text{int}}(\phi)=e^{g\phi^{2}}-g\phi^{2}-1$ (9)

where the last two terms are subtracted off to give an interaction potential
starting at order $\phi^{4}$.

We consider solutions for various model parameters $g$ and initial condition
$\phi_{0}$. These solutions are obtained by numerically integrating the
coupled scalar and Friedmann equations starting from the turning point where
$\phi=\phi_{0}$ and $H=\dot{\phi}=0$. We set $\rho_{M}=0$ and determine the
value of $\rho_{R}$ from $V(\phi_{0})$ using (2). The system is numerically
integrated back in time towards the big bang, then we numerically check if a
given solution has a numerically detectable period of accelerated expansion.

In Figure 2, we show the space of parameters $(g,\phi_{0})$ that leads to a
period of accelerated expansion in the cosmology picture before the
recollapse.

Figure 2: Space of model parameter $g$ and initial condition $\phi_{0}$ for
the exponential potential of (8) and (9). Solutions of the equations of motion
(2) and (4) with parameter $g$ and initial condition $\phi_{0}$ contained in
the red region exhibit an accelerating phase.

We see that a significant region of the space where the parameters are of
order 1 yields an accelerating phase. Thus, cosmological acceleration can be
obtained without significant fine-tuning of the model or initial conditions.

We can also consider how the mass of the field affects the allowed parameter
space. In Figure 3 we show the evolution of the parameter space exhibiting
accelerated expansion as $m^{2}$ becomes less negative. We see that
accelerated expansion becomes rapidly less generic as $m^{2}$ increases.
Conversely, we have also observed that if $m^{2}$ is allowed to go below the
BF bound, then a period of acceleration becomes more generic. Qualitatively,
this is because the slope of the potential near the turning point (which in
the present model is controlled mostly by the value of the mass) must be steep
enough for the anti-damped scalar field to quickly gain kinetic energy as we
evolve back towards the big bang (or forward towards the big crunch). This is
necessary for the scalar field to reach the positive region of the potential
before the scale factor vanishes.

Figure 3: Panel (A) is the same as Figure 2 with $m^{2}=-9/4$. Panels (B),
(C), and (D) correspond to masses $m^{2}=-8/4$, $m^{2}=-7/4$, and
$m^{2}=-6/4$, respectively. Clearly it becomes much less generic to have a
period of accelerated expansion as $m^{2}$ increases.

There is no particular physical significance to the form of the potential we
have chosen; below, we will see that the exponentially increasing behavior of
the potential for large $\phi$ is not necessary to obtain acceleration.

###  2.3 An example matching flat $\Lambda$CDM

As a first example of a phenomenologically realistic cosmology based on the
framework introduced in Antonini:2022blk ; Antonini:2022xzo , we study
potentials able to reproduce to good accuracy the $\Lambda$CDM model. These
are potentials satisfying all the properties outlined above and having a
nearly flat positive region.

For our numerical analysis it is useful to work in units such that $8\pi
G/3=1$ and to introduce rescaled variables

$\tilde{t}=H_{0}t,\qquad\Omega_{i}=\frac{\rho^{now}_{i}}{H_{0}^{2}},\qquad\tilde{H}=\frac{H}{H_{0}},\qquad\tilde{V}(\phi)=\frac{V(\phi)}{H_{0}^{2}}\;,$
(10)

where $\rho_{i}^{now}$ are the energy densities at the present time. The
equations of motion then read

$\ddot{\phi}+3\tilde{H}\dot{\phi}+\frac{d\tilde{V}}{d\phi}=0\>,\qquad\tilde{H}^{2}=\frac{1}{2}\dot{\phi}^{2}+\tilde{V}(\phi)+\frac{\Omega_{R}}{a^{4}}+\frac{\Omega_{M}}{a^{3}}\>$
(11)

in the cosmology picture, and

$\phi_{E}^{\prime\prime}+3\tilde{H}_{E}\phi_{E}^{\prime}-\frac{d\tilde{V}}{d\phi_{E}}=0\>,\qquad\tilde{H}_{E}^{2}=\frac{1}{2}(\phi_{E}^{\prime})^{2}-\tilde{V}(\phi_{E})-\frac{\Omega_{R}}{a^{4}_{E}}-\frac{\Omega_{M}}{a^{3}_{E}}\>$
(12)

in the wormhole picture, where overdots and primes now represent derivatives
with respect to rescaled time variables. With this choice, both the Lorentzian
time $\tilde{t}=H_{0}t$ and the Euclidean time $\tilde{\tau}=H_{0}\tau$ are
measured in units of the Hubble time $1/H_{0}$. We also adopt the convention
$a(\tilde{t}_{0})=1$ where $\tilde{t}_{0}<0$ is the present time. This implies
that at the time-symmetric point the initial condition for the scale factor is
$a(0)=a_{0}>1$. The $\tilde{t}/\tilde{\tau}$ derivatives of $a$ and $\phi$
vanish at the symmetric point $\tilde{t}=\tilde{\tau}=0$. Given a value for
the density parameters $\Omega_{R}$ and $\Omega_{M}$ and a choice of $a_{0}$,
the initial condition for the scalar field is determined by the Friedmann
equation at $\tilde{t}=0$:

$\frac{\Omega_{R}}{a_{0}^{4}}+\frac{\Omega_{M}}{a_{0}^{3}}=-\tilde{V}(\phi_{0})\;.$
(13)

Figure 4: An example of the potential (14). $A$ fixes the value of
$\tilde{V}>0$ at the plateau, $B$ the value of $\tilde{V}(0)$, i.e. the
negative cosmological constant in the asymptotically AdS regions of the
wormhole solution. $C$, $X$, $\Delta$ determine the depth, location, and width
of the deep valleys present in the potential.

Let us now consider a five-parameter family of potentials given by

$\tilde{V}(\phi)=\tilde{V}_{0}(\phi)+\tilde{V}_{0}(-\phi)-B$ (14)

where

$\tilde{V}_{0}(\phi)=\frac{A-B}{2}\textnormal{erf}\left(\frac{\phi-X}{\Delta}\right)+C\exp\left[-\left(\frac{\phi-X}{\Delta}\right)^{2}\right]+\frac{A+B}{2}.$
(15)

and erf is the error function. An example of the potential (14) is plotted in
Figure 4. The parameter $A>0$ fixes the height of the plateau for large values
of $\phi$ and determines the value of the positive vacuum energy driving the
accelerated expansion in the cosmology picture, while $B<0$ gives the value of
the negative cosmological constant in the asymptotic AdS regions in the
wormhole picture. The parameters $C<0$, $X>0$ and $\Delta>0$ determine the
depth, location, and width of the deep valleys present in the potential,
respectively. In order to reproduce the $\Lambda$CDM model, the value of $A$
must match the value of the positive vacuum energy observed in our universe.

We can then study solutions of the cosmology and wormhole equations of motion
(11) and (12) with potential given by equations (14) and (15). We choose the
values for the cosmological parameters obtained by the Planck 2018
collaboration using TT, TE, EE + lowE + lensing + BAO data Planck:2018vyg :

$\Omega_{R}=9.18\times 10^{-5},\hskip 28.45274pt\Omega_{M}=0.311,\hskip
28.45274pt\Omega_{\Lambda}=0.6889$ (16)

where $\Omega_{\Lambda}$ is the density parameter associated with the vacuum
energy today. We further choose the initial condition $a_{0}=2$ for the scale
factor at the time-symmetric point, while the scalar field initial condition
$\phi_{0}$ is determined by equation (13) once the five parameters of the
potential are fixed.777We remind that the initial conditions for the cosmology
and wormhole pictures are the same. Note also that in general there are
multiple solutions of equation (13). As we have explained above, we are
interested in a solution such that $\phi_{0}$ is in between a minimum and a
maximum of the potential. In our numerical solutions depicted in Figures 5, 6,
and 8, they are taken to be

$A=0.6889,\hskip 19.91684ptB=-0.03,\hskip 19.91684ptC=-5,\hskip
19.91684ptX=1,\hskip 19.91684pt\Delta=0.0726775778709$ (17)

yielding $\phi_{0}=0.817319$.

#### Cosmological solution

By an appropriate fine-tuning of the potential’s parameters (see Appendix A.1)
leading to the values (17), the cosmological evolution reproduces to very good
accuracy the one predicted by the corresponding $\Lambda$CDM solution between
the early universe and the present day $\tilde{t}=\tilde{t}_{0}$, see Figure
5. For $\tilde{t}>\tilde{t}_{0}$, the potential energy eventually decreases
and becomes negative as the scalar field rolls down the potential, while the
kinetic energy $K_{\phi}$ increases. Finally, the scalar kinetic energy
decreases and vanishes as we reach the time-symmetric point $\tilde{t}=0$
(where we impose our initial conditions for both the cosmological and the
wormhole solutions). See Figure 6.

Figure 5: Scale factor evolution for the cosmological solution obtained using
the Planck cosmological parameters (16) (denoted by “CVac”). The potential
parameters are given in equation (17). The scale factor for the corresponding
$\Lambda$CDM solution is also depicted. The two scale factors are
indistinguishable until the present day $\tilde{t}=\tilde{t}_{0}=-1.16342$
(where $a(\tilde{t}_{0})=1$) — indicated by the black dashed line — and
beyond. Deviations in our solution from the $\Lambda$CDM behavior become
evident at late times as the universe approaches the turning point at
$\tilde{t}=0$. The contraction phase in our solution, not depicted here, can
be obtained by time-reversal. Figure 6: Scalar field kinetic and potential
energies, radiation contribution $\Omega_{R}/a^{4}$, matter contribution
$\Omega_{M}/a^{3}$, and scale factor $a(\tilde{t})$ as a function of time
$\tilde{t}$ for the expansion phase of our cosmological solution obtained
using the Planck cosmological parameters (16). The potential parameters are
given in equation (17). The contraction phase can be obtained by time-
reversal. The black dashed line indicates the present day
$\tilde{t}=\tilde{t}_{0}=-1.16342$ for which $a(\tilde{t}_{0})=1$. $K_{\phi}$
is negligible for most of the evolution until the present day. The universe
undergoes a radiation-dominated and a matter-dominated era before the current
potential energy-dominated era. In the future, the potential energy will
decrease and the kinetic energy increase and become dominant as the scalar
field rolls down the potential. Finally, the kinetic energy vanishes as the
universe reaches its turning point at $\tilde{t}=0$, where initial conditions
for our numerical solutions are imposed.

As a consistency check, we verified that the luminosity distance

$d_{L}(z)=\frac{1}{H_{0}a(t)}\int_{\tilde{t}}^{\tilde{t}_{0}}\frac{dt^{\prime}}{a(t^{\prime})}\>,\qquad
1+z=\frac{1}{a(t)}\>$ (18)

computed from our solution888For the solution obtained using the Planck
cosmological parameters (16) we can use the value of $H_{0}$ measured by
Planck, i.e. $H_{0}^{planck}=67.66\textrm{ km s}^{-1}\textrm{ Mpc}^{-1}$.
agrees with the Pantheon+SH0ES type Ia supernova (SN Ia) data Pantheondata .
In particular, we found that the luminosity distance in our solution is
indistinguishable from the one generated by the corresponding $\Lambda$CDM
solution in the range of redshifts $z$ covered by the SNIa data, see Figure
10. Therefore, our model fits the data as well as the $\Lambda$CDM model. Due
to the well-known Hubble tension issue DiValentino:2021izs , this also implies
that the solution obtained using the Planck 2018 parameters (16) is in tension
with supernovae data. We also studied a solution generated using cosmological
parameters obtained from supernovae observations Brout:2022vxf , which clearly
matches data much better. Whether specific realizations of our models could
help solve the Hubble tension is an interesting open question that we plan to
investigate in future work.

Figure 7: Luminosity distance $d_{L}(z)$ computed for two solutions involving
rolling scalars (denoted by “CVac”) and their corresponding $\Lambda$CDM
solutions. For the Planck 2018 solution, the cosmological parameters are given
in equation (16), the potential parameters in equation (17), and we used
$H_{0}^{Planck}=67.66\textrm{ km s}^{-1}\textrm{ Mpc}^{-1}$ Planck:2018vyg .
For the SNeIa solution, the cosmological parameters are given by101010In the
late stages of this manuscript’s preparation a revised version of
Brout:2022vxf  was announced, with a slight correction of the best fits for
the cosmological parameters, yielding $\Omega_{M}=0.334\pm 0.018$ and
$\Omega_{\Lambda}=0.666\pm 0.018$. This difference is irrelevant for the
purposes of the present paper. $\Omega_{R}=9.96\times 10^{-5}$,
$\Omega_{M}=0.338$, $\Omega_{\Lambda}=0.662$, $H_{0}^{SN}=73.4\textrm{ km
s}^{-1}\textrm{ Mpc}^{-1}$ Brout:2022vxf , and the potential parameters are
$A=0.662$, $B=-0.03$, $C=-5$, $X=1$, $\Delta=0.0716914850735$; this yields
$\phi_{0}=0.824448$. The Pantheon+SH0ES experimental data are also depicted
Pantheondata . Our cosmological solutions and their corresponding $\Lambda$CDM
solutions are indistinguishable, meaning that our model matches supernovae
data as well as the $\Lambda$CDM model. Notice that the solutions generated
using Planck 2018 cosmological parameters are in tension with data, while the
ones generated using cosmological parameters derived from supernovae
observations agree with data: this is a manifestation of the Hubble tension.

#### Wormhole solution

First, we remark that the potential under consideration with the parameters
(17) used in our numerical analysis satisfies the BF bound (5). As a result of
the scalar field’s overdamped motion described above, the scale factor in the
wormhole solution $a_{E}(\tilde{\tau})$ increases away from the wormhole
center, the matter and radiation terms are suppressed as $\tilde{\tau}$
increases or decreases from $\tilde{\tau}=0$, and the scalar kinetic energy
$K_{\phi}$ becomes negligible as $\phi_{E}\to 0$ (see Figure 8, right panel).
Therefore, for $\tilde{\tau}\to\pm\infty$ the solution approaches pure AdS
with vacuum energy density parameter given by
$\Omega_{\Lambda}=3\tilde{V}(0)=3B$. As a consequence, the scale factor takes
the asymptotic form $a(\tilde{\tau})\to\exp(\sqrt{-B}\tilde{\tau})$ as
$\tau\to\pm\infty$ and the Hubble parameter approaches a constant value
$\tilde{H}(\tilde{\tau})\to\tilde{H}_{\infty}=\sqrt{-B}$ (see Figure 8, left
panel). These results confirm the existence, within the class of models
introduced in Antonini:2022blk ; Antonini:2022xzo , of well-defined wormhole
solutions associated with cosmological solutions able to reproduce to
arbitrary accuracy the predictions of the $\Lambda$CDM model.

Figure 8: Wormhole solution from the center of the wormhole towards the
asymptotic boundary at $\tilde{\tau}=\infty$, obtained using the Planck
cosmological parameters (16). The potential parameters are given in equation
(17). The evolution from the center of the wormhole towards the other
asymptotic boundary at $\tilde{\tau}=-\infty$ is obtained by reflection
symmetry around $\tilde{\tau}=0$. (Left) Hubble parameter as a function of the
coordinate $\tilde{\tau}$. The value of the Hubble parameter for the
corresponding pure AdS solution $\tilde{H}_{\infty}=\sqrt{B}$ is also
depicted. In the asymptotic region $\tilde{\tau}\to\infty$ the wormhole
solution approaches pure AdS. (Right) Scalar field kinetic and potential
energies, radiation contribution $\Omega_{R}/a_{E}^{4}$, and matter
contribution $\Omega_{M}/a_{E}^{3}$ as a function of the coordinate
$\tilde{\tau}$ along the wormhole direction. As we move towards the AdS
asymptotic boundary (i.e. as we increase $\tilde{\tau}$), the overdamped
scalar field approaches its value $\phi_{E}=0$ at the AdS boundary and its
kinetic energy vanishes. The radiation and matter contributions are suppressed
as the scale factor increases, and in the asymptotic region the negative
scalar potential energy $\tilde{V}(0)$ is dominant: the solution approaches an
AdS vacuum solution.

We would like to point out that, from a top-down point of view, there is
reason to be skeptical about the existence of the type of potentials studied
in this subsection on the grounds of Swampland conjectures. In particular, the
de Sitter conjecture Obied:2018sgi ; Garg:2018reu ; Ooguri:2018wrx  would
appear to rule out potentials with very flat regions such as the ones
introduced in equations (14) and (15). Nonetheless, this class of nearly-flat
potentials is a useful starting point for our discussion. Indeed, the present
analysis suggests that sufficient fine-tuning allows us to reproduce
arbitrarily well the predictions of the $\Lambda$CDM model between the early
universe and the present time while preserving the existence of a wormhole
solution with the scalar field satisfying the BF bound at the two asymptotic
AdS boundaries. A similar conclusion remains valid for a more general class of
potentials, as we will see in the next subsection.

###  2.4 An example matching flat $w$CDM

Though we have considered a flat potential to make direct contact with the
$\Lambda$CDM model, such tuning of the scalar potential is not at all
necessary, and we can consider a broader class of models consistent with
direct observations of $\Omega_{M}$ and the scale factor $a(t)$, which may
involve scalar evolution over a range of potential values $\Delta V/V=O(1)$.

For concreteness, we consider the spatially flat $w$CDM model,111111We neglect
radiation in this subsection. since direct constraints on the cosmological
parameters for these models from type Ia supernova observations, using the
Pantheon+SH0ES data set, are provided in Brout:2022vxf . We recall that this
model consists of modifying the dark energy contribution in the $\Lambda$CDM
model, by replacing the cosmological constant with a perfect fluid governed by
the equation of state $\rho=wp$, with $w$ constant. The corresponding Hubble
expansion is given by

$H(z)=H_{0}\sqrt{\Omega_{M}(1+z)^{3}+\Omega_{\Lambda}(1+z)^{3(1+w)}}\>,$ (19)

where $\Omega_{\Lambda}$ is the density parameter for the dark energy, so that
$\Omega_{\Lambda}=1-\Omega_{M}$ with the assumption of spatial flatness. The
constraints on the parameters of this model from Brout:2022vxf  are

$H_{0}=(73.5\pm
1.1)\>\textnormal{km}\cdot\textnormal{s}^{-1}\cdot\textnormal{Mpc}^{-1}\>,\quad\Omega_{M}=0.309^{+0.063}_{-0.069}\>,\quad
w=-0.90\pm 0.14\>.$ (20)

Figure 9: Rescaled potentials for the model $V(\phi)$ and reconstruction
$V_{w}(\phi)$, where the region of the latter probed by the scalar field in
the range $z\in(z_{\textnormal{min}},z_{\textnormal{max}})$ is shown. The
righthand plot is a close-up on this region. The precise form of the model
$V(\phi)$ can be found in Appendix A.2.

We can reproduce the evolution of the $w$CDM cosmology over an arbitrarily
large range of redshifts using our scalar field model. With the assumption
that only a single scalar field is relevant to the cosmological evolution, we
can actually deduce the potential associated with this scalar field given
$a(t)$ and $\Omega_{M}$. The kinetic energy can be expressed as

$K(t)\equiv{1\over 2}\dot{\phi}^{2}=-{1\over 8\pi G}\dot{H}-{3\over 16\pi
G}\Omega_{M}H_{0}^{2}{1\over a^{3}}\;,$ (21)

and we can take the square root and integrate, to obtain

$\phi(t)=\phi(t_{0})+\int_{t_{0}}^{t}d\hat{t}\sqrt{-{1\over 4\pi
G}\dot{H}(\hat{t})-{3\over 8\pi G}\Omega_{M}H_{0}^{2}{1\over
a^{3}(\hat{t})}}.$ (22)

Similarly, the potential energy is

$V(t)={1\over 8\pi G}\dot{H}+{3\over 8\pi G}H^{2}-{3\over 16\pi
G}\Omega_{M}H_{0}^{2}{1\over a^{3}}\;.$ (23)

We can alternatively express these in terms of the redshift $z$ and adopt the
rescaled quantities introduced in the previous subsection, obtaining

$\begin{split}\phi(z)&=\phi_{0}+\int_{0}^{z}dz^{\prime}\>\frac{\sqrt{\frac{1}{4\pi
G}(1+z^{\prime})\tilde{H}(z^{\prime})\tilde{H}^{\prime}(z^{\prime})-\frac{3}{8\pi
G}(1+z^{\prime})^{3}\Omega_{M}}}{(1+z^{\prime})\tilde{H}(z^{\prime})}\\\
\tilde{V}(z)&=-\frac{1}{8\pi
G}(1+z)\tilde{H}(z)\tilde{H}^{\prime}(z)+\frac{3}{8\pi
G}\tilde{H}(z)^{2}-\frac{3}{16\pi G}(1+z)^{3}\Omega_{M}\>.\end{split}$ (24)

We may then substitute the rescaled version of the Hubble expansion from
equation (19) into these expressions.

In the scalar field model, the effective equation of state parameter is given
by

$w=\frac{\frac{1}{2}\dot{\phi}^{2}-V}{\frac{1}{2}\dot{\phi}^{2}+V}\>.$ (25)

For a general scalar theory, $w$ will be redshift dependent, but with
appropriate choice of scalar potential, we may recover solutions for which $w$
is approximately constant over an arbitrarily large range of redshifts. We
denote by $V_{w}(\phi)$ the scalar potential reconstructing the $w$CDM model
for all $z>0$. For the best-fit parameters of $w$CDM listed above, we show
part of the reconstructed potential $\tilde{V}_{w}(\phi)=V_{w}(\phi)/H_{0}$ in
Figure 9.

Since the Pantheon+SH0ES data used to deduce the cosmological parameter
constraints probe only redshifts in the range
$z\in(z_{\textnormal{min}},z_{\textnormal{max}})$ with
$z_{\textnormal{min}}\approx 10^{-3}$ and $z_{\textnormal{max}}\approx 2.26$,
we are free to consider any scalar potential $V(\phi)$ which coincides with
$V_{w}(\phi)$ in the region over which the scalar solution evolves in this
range of redshifts. In Figure 9, we also plot a rescaled scalar potential
$\tilde{V}(\phi)=V(\phi)/H_{0}$ agreeing with $\tilde{V}_{w}(\phi)$ in the
appropriate range, but that satisfies the “UV constraints” necessary for the
existence of an analytic continuation of the cosmological solution to an
asymptotically AdS wormhole. The precise form of the potential
$\tilde{V}(\phi)$, as well as our method for constructing the potential and
the appropriate initial conditions for the cosmological solution, are given in
Appendix A.2. In the left panel of Figure 10, we plot the equation of state
evolution $w(z)$ extracted directly from the time-symmetric cosmological
solutions in the model with potential $\tilde{V}(\phi)$, confirming that they
reproduce that of the $w$CDM model with the desired precision, while in the
right panel of Figure 10 we confirm that the analytic continuation of these
solutions have AdS asymptotics.

We emphasize that, while we have only aimed here to reproduce the $w$CDM model
over the region $z\in(z_{\textnormal{min}},z_{\textnormal{max}})$ where it has
been constrained by SNe Ia data, we could recover the same model over an
arbitrarily large redshift interval through suitable choice of the scalar
potential.

Figure 10: (Left) Equation of state parameter $w$ versus redshift $z$ for
time-symmetric solutions to the model $\tilde{V}(\phi)$, with shaded region
given by the Pantheon+SH0ES constraint. (Right) Euclidean evolution of
$\tilde{H}(\tilde{\tau})$, with $\frac{1}{L}\equiv\sqrt{-V(0)}$ plotted for
reference. In the asymptotic region $\tilde{\tau}\to\infty$ the wormhole
solution approaches pure AdS, as expected. The precise form of the potential
$\tilde{V}(\phi)$ can be found in Appendix A.2.

##  3 Supersymmetric models

In our discussion so far, we have considered an effective gravitational field
theory involving scalar fields with dynamics controlled by an effective
potential $V$. We have shown that such a theory could describe the background
dynamics of our Universe while also admitting a Euclidean continuation
suggestive of a dual CFT description.

The most well-understood examples of gravitational theories which are dual to
CFTs are supersymmetric. It is thus interesting to ask whether there exist
supersymmetric theories whose effective low energy descriptions contain scalar
fields with potentials $V$, which give rise to effective cosmological
dynamics. In these examples, supersymmetry could be broken by the time-
dependent scalar field expectation value. Note that here we will not be
concerned with whether the supersymmetric theories under consideration have
explicitly known CFT duals. Instead we simply wish to understand whether
supersymmetry is consistent with the restricted class of scalar potentials
which we have considered so far.

We will consider $\mathcal{N}=1$ supergravity and restrict to considering only
the gravity and scalar sectors; presumably one can consistently set the
gravitino and chiral fermions to zero in solutions to the full equations of
motion. We follow the notation of deAlwis:2013jaa  and we set $8\pi
G_{\textnormal{N}}=\kappa^{2}=1$ in this section.

The $n_{c}$ scalar fields $\phi^{i}$ of $\mathcal{N}=1$ supergravity are
bottom components of chiral multiplets, such that the theory has bosonic
Lagrangian

$\mathcal{L}=\frac{1}{2}R-K_{i\bar{j}}\partial_{\mu}\phi^{i}\partial^{\mu}\bar{\phi}^{\bar{j}}-V_{W}(\phi,\bar{\phi})\>,$
(26)

where $K_{i\bar{j}}=\partial_{i}\bar{\partial}_{\bar{j}}K$ is the Kähler
metric arising from a Kähler potential $K$, and $V_{W}$ is the scalar
potential

$V_{W}=e^{K}\left(K^{i\bar{j}}D_{i}WD_{\bar{j}}\bar{W}-3|W|^{2}\right)\>,\qquad
D_{i}W=\partial_{i}W+K_{i}W\>,$ (27)

defined in terms of the superpotential $W$.

For simplicity we will consider the case of a single chiral multiplet,
$n_{c}=1$, and set the Kähler potential to the canonical form
$K(\phi,\bar{\phi})=\bar{\phi}\phi$. Theories with more general Kähler
potentials and more chiral multiplets may also be interesting to consider. In
our case the scalar potential is given by

$\displaystyle
V_{W}(\phi)=e^{\phi\bar{\phi}}\left(|W^{\prime}(\phi)+\bar{\phi}W(\phi)|^{2}-3|W(\phi)|^{2}\right).$
(28)

A specification of $W$ is part of the specification of the supersymmetric
theory, with the only restriction on $W$ being that it is a holomorphic
function of $\phi$. Thus we would like to know whether it is possible to
choose $W$ such that $V_{W}$ is a scalar potential like the ones considered in
previous sections, i.e. which gives rise to effective cosmological dynamics.

The scalar field $\phi$ arising from the supersymmetric theory is a complex
field, which we can decompose into two real fields as
$\phi=\phi_{R}+i\phi_{I}$. Thus $V_{W}$ is a (real) potential of two real
fields, $\phi_{R}$ and $\phi_{I}$, whose dynamics will in general be coupled —
the trajectory of $\phi$ in the complex plane will generally involve motion in
both the $\phi_{R}$ and $\phi_{I}$ directions.

However, although it is certainly possible that the low energy effective field
theory describing our Universe contains multiple real scalar fields, in the
previous sections we have shown that a single field with potential $V$ is
sufficient for the cosmology to be in quantitative agreement with measurements
of the scale factor, while still admitting an asymptotically AdS Euclidean
continuation. Thus we might hope that we can find a potential $V_{W}$ such
that the dynamics in the $\phi$ plane is one-dimensional, for example such
that $\phi$ rolls down the real axis, with $\phi_{I}=0$ for all times.
Moreover we would like $V_{W}(\phi_{R})=V(\phi_{R})$, so that the remaining
scalar degree of freedom $\phi_{R}$ provides us with the desired cosmological
evolution.

We will now show that, even with this restriction that $\phi_{I}$ effectively
decouples from the dynamics, given any cosmological potential $V(\phi_{R})$ it
is possible to find a superpotential $W(\phi)$ such that $V_{W}(\phi_{R})$ is,
to an arbitrarily good approximation, equal to $V(\phi_{R})$ within a given
fixed region. Furthermore, in this construction $W$ will be holomorphic
everywhere provided that $V$ is. Since the cosmological potentials which we
considered in previous sections are generically holomorphic everywhere, this
construction will produce holomorphic superpotentials $W$ which give rise to
realistic effective cosmological dynamics with Euclidean AdS asymptotics.

#### Constructing the superpotential

In our construction $W(\phi_{R})$ will be real, in which case Eq. (28) for
$\phi=\phi_{R}$ reads

$\displaystyle V_{W}(\phi_{R})$
$\displaystyle=e^{\phi_{R}^{2}}\left(\left[W^{\prime}(\phi_{R})+\phi_{R}W(\phi_{R})\right]^{2}-3W^{2}(\phi_{R})\right).$
(29)

This is a nonlinear ODE for $W$ and cannot be solved in closed form. To obtain
a linear equation, write

$\displaystyle V_{W}(\phi_{R})$
$\displaystyle=e^{\phi_{R}^{2}}\left(W^{\prime}+(\phi_{R}+\sqrt{3})W\right)\left(W^{\prime}+(\phi_{R}-\sqrt{3})W\right).$
(30)

Now let $f(\phi_{R})$ be some function, which we define shortly, and consider
the linear ODE

$\displaystyle
W^{\prime}(\phi_{R})+(\phi_{R}-\sqrt{3})W(\phi_{R})=f(\phi_{R}).$ (31)

The general solution is the sum of a homogeneous plus inhomogeneous piece,

$\displaystyle
W(\phi_{R})=CG(\phi_{R})+G(\phi_{R})\int_{0}^{\phi_{R}}du\frac{f(u)}{G(u)},$
(32)

where $C$ is an arbitrary integration constant, and the homogeneous piece is
given by

$\displaystyle
G(\phi_{R})=\exp\left(-\frac{1}{2}\phi_{R}^{2}+\sqrt{3}\phi_{R}\right).$ (33)

Substituting this solution for $W$ into (30) we obtain the scalar potential

$\displaystyle
V_{W}(\phi_{R})=\tilde{V}(\phi_{R})\left(1+\frac{\tilde{V}(\phi_{R})e^{-2\sqrt{3}\phi_{R}}}{12C^{2}}+\frac{1}{2\sqrt{3}C^{2}}\int_{0}^{\phi_{R}}du\,\tilde{V}(u)e^{-2\sqrt{3}u}\right).$
(34)

where we have defined

$\displaystyle\tilde{V}(\phi_{R})\equiv
2\sqrt{3}Ce^{\phi_{R}^{2}}f(\phi_{R})G(\phi_{R}).$ (35)

Because we are free to choose $f$, we are effectively free to choose
$\tilde{V}$. We can thus require $V_{W}(\phi_{R})=V(\phi_{R})$ and solve (34)
for $\tilde{V}$. Although this cannot be done in closed form, we can obtain a
series approximation for $\tilde{V}$ in powers of $1/C^{2}$. Notice that $C$
is an arbitrary constant, and so it can be chosen large enough to be a good
expansion parameter; we will discuss this in more detail below. Namely,
writing

$\displaystyle\tilde{V}(\phi_{R})=\tilde{V}_{0}(\phi_{R})+\frac{1}{C^{2}}\tilde{V}_{2}(\phi_{R})+\frac{1}{C^{4}}\tilde{V}_{4}(\phi_{R})+\dots,$
(36)

it is possible to recursively solve the equation $V_{W}(\phi_{R})=V(\phi_{R})$
for the $\tilde{V}_{n}$. The first two terms in the expansion are

$\displaystyle\tilde{V}_{0}(\phi_{R})$ $\displaystyle=V(\phi_{R}),$ (37)
$\displaystyle\tilde{V}_{2}(\phi_{R})$
$\displaystyle=-\frac{V(\phi_{R})}{2\sqrt{3}}\left(\frac{e^{-2\sqrt{3}\phi_{R}}V(\phi_{R})}{2\sqrt{3}}+\int_{0}^{\phi_{R}}du\,V(\phi_{R})e^{-2\sqrt{3}u}\right),$
(38)

and expressions for higher order terms can be straightforwardly obtained by
recursion. Substituting (35) into (32) we obtain the superpotential in terms
of $\tilde{V}$,

$\displaystyle
W(\phi_{R})=CG(\phi_{R})\left(1+\frac{1}{2\sqrt{3}C^{2}}\int_{0}^{\phi_{R}}du\,\tilde{V}(u)e^{-2\sqrt{3}u}\right),$
(39)

and using the series expansion for $\tilde{V}$ in terms of $V$, we can,
perturbatively in powers of $1/C^{2}$, write the superpotential $W$ in terms
of the cosmological potential $V$:

$\displaystyle
W(\phi_{R})=CG(\phi_{R})\left[1+\frac{1}{2\sqrt{3}C^{2}}\int_{0}^{\phi_{R}}du\,V(u)e^{-2\sqrt{3}u}+\mathcal{O}\left(C^{-4}\right)\right].$
(40)

#### Truncating the expansion

If we use the full expression for $W(\phi_{R})$ derived above, including all
corrections in powers of $1/C^{2}$, then we would expect to find an exact
equality $V_{W}(\phi_{R})=V(\phi_{R})$ for all values of $\phi_{R}$. Of
course, it is not possible to work with the infinite series of terms. Instead
we can truncate the series expansion of $W$ at $n$ terms, obtaining the
truncated superpotential, which we denote $W_{n}$. For example, $W_{2}$ is
obtained by keeping two terms in the expansion,

$\displaystyle
W_{2}(\phi_{R})=CG(\phi_{R})\left(1+\frac{1}{2\sqrt{3}C^{2}}\int_{0}^{\phi_{R}}du\,V(u)e^{-2\sqrt{3}u}\right),$
(41)

which results in the scalar potential

$\displaystyle
V_{W_{2}}(\phi_{R})=V(\phi_{R})\left(1+\frac{V(\phi_{R})e^{-2\sqrt{3}\phi_{R}}}{12C^{2}}+\frac{1}{2\sqrt{3}C^{2}}\int_{0}^{\phi_{R}}du\,V(u)e^{-2\sqrt{3}u}\right).$
(42)

Recall that $C$ appeared simply as an undetermined integration constant, and
we are free to set it to any value whatsoever. Therefore we have the somewhat
unusual freedom to set our expansion parameter $1/C$, and thus the terms
truncated in going from $W$ to $W_{2}$, as small as we like. In terms of the
potentials, given any value of $\phi_{R}$ we can make $V_{W_{2}}(\phi_{R})$
arbitrarily close to the cosmological potential $V(\phi_{R})$. In other words,
as we take $C$ large $V_{W_{2}}$ converges pointwise to $V(\phi_{R})$. If we
restrict to values of $\phi_{R}$ in any compact subset of the real line, then
the convergence is uniform within this subset. Thus we have successfully
managed to embed any scalar potential $V$ — including cosmological potentials
— into a supersymmetric theory, to an arbitrary degree of accuracy.

In Fig. 11 we illustrate with a simple example the deviation of $V_{W_{2}}$
from $V$ for various values of $C$. Notice that $V_{W_{2}}(\phi_{R})$ is in
excellent agreement with $V(\phi_{R})$ for $\phi_{R}>0$, but for a fixed value
of $C$ the agreement breaks down for negative enough values of $\phi_{R}$.
This is because the exponential factor in the integrand in (41) suppresses the
contribution of the subleading term for positive values of $\phi_{R}$, but
enhances this same contribution for negative $\phi_{R}$. This effect can be
traced back to us choosing to set the second bracketed factor in (30) equal to
$f$. Had we chosen to set the first factor equal to $f$, the same analysis
would result in an approximation $V_{W_{2}}(\phi_{R})$ of $V(\phi_{R})$ which
is accurate for all $\phi_{R}<0$, but breaks down for $\phi_{R}$ positive
enough. Since throughout this paper we have worked with the convention that
the scalar field rolls along the positive real axis, it makes sense to
consider, as we have done, the construction which gives an accurate
approximation $V_{W_{2}}(\phi_{R})\approx V(\phi_{R})$ for positive
$\phi_{R}$. Nevertheless, by taking $C$ large enough, we can make
$V_{W_{2}}(\phi_{R})$ arbitrarily close to $V(\phi_{R})$ for $\phi_{R}$
greater than any real $\phi_{R}^{*}$.

Figure 11: Comparison of $V(\phi_{R})=-1-3\phi_{R}^{2}+\phi_{R}^{4}$ and the
potential $V_{W_{2}}(\phi_{R})$ given by equation (42), obtained from the
truncated superpotential $W_{2}$ given by equation (41).

#### Maximum at $\phi_{R}=0$

One potential issue with the truncated potential $V_{W_{2}}(\phi_{R})$ is that
it is not symmetric with respect to $\phi_{R}\rightarrow-\phi_{R}$; see Fig.
11. Despite this, from (42) we find

$\displaystyle
V_{W_{2}}^{\prime}(\phi_{R})=V^{\prime}(\phi_{R})\left(1+\frac{2V(\phi_{R})e^{-2\sqrt{3}\phi_{R}}}{12C^{2}}+\frac{1}{2\sqrt{3}C^{2}}\int_{0}^{\phi_{R}}du\,V(u)e^{-2\sqrt{3}u}\right),$
(43)

and hence $V_{W_{2}}^{\prime}(0)=0$ if $V^{\prime}(0)=0$. Thus, although
$V_{W_{2}}(\phi_{R})$ is not symmetric, it does have a local maximum at
$\phi_{R}=0$, allowing for the asymptotically AdS solution $\phi_{R}=0$ in the
Euclidean continuation. Therefore it is not a problem that
$V_{W_{2}}(\phi_{R})$ is not symmetric.

#### Dynamics of a complex $\phi$

So far we have restricted the complex field $\phi=\phi_{R}+i\phi_{I}$ to the
real line, i.e. we have set $\phi_{I}=0$. This has allowed us to explicitly
construct the superpotential $W_{2}(\phi_{R})$, which gives a scalar potential
$V_{W_{2}}(\phi_{R})$ that is arbitrarily close to the cosmological potential
$V(\phi_{R})$. Analytically continuing away from the real axis, we obtain the
superpotential $W_{2}(\phi)$ as a function of complex $\phi$. Notice from (41)
that if $V(\phi)$—the analytic continuation of the cosmological potential
$V(\phi_{R})$—is analytic in some simply connected region of the complex plane
containing $\phi=0$, then $W_{2}(\phi)$ is analytic in this same region. The
cosmological potentials considered in previous sections were analytic
everywhere, and so the corresponding superpotential is also analytic
everywhere.

The superpotential $W_{2}(\phi)$ defines a scalar potential $V_{W_{2}}(\phi)$
for complex values of $\phi$ via Eq. (28),

$\displaystyle
V_{W_{2}}(\phi)=e^{\phi\bar{\phi}}\left(|W_{2}^{\prime}(\phi)+\bar{\phi}W_{2}(\phi)|^{2}-3|W_{2}(\phi)|^{2}\right).$
(44)

Notice that $V_{W_{2}}(\phi)$ is not holomorphic even if $W_{2}(\phi)$ is. In
fact, $V_{W_{2}}(\phi)$ is real for all $\phi$, and so it cannot be
holomorphic. Thus instead of thinking of $V_{W_{2}}(\phi)$ as a function of a
single complex variable $\phi$, it makes more sense to think of it as a (real)
function of two real variables, $\phi_{R}$ and $\phi_{I}$.

By construction we have ensured that $V_{W_{2}}(\phi)$ restricted to the real
axis is approximately equal to the cosmological potential $V(\phi_{R})$. An
important question is: if at some initial time the values of $\phi$ and
$\dot{\phi}$ are both real (i.e. $\phi_{I}=0=\dot{\phi}_{I}$), then does
$\phi$ remain real (i.e. does $\phi_{I}$ remain zero) throughout its dynamical
evolution? In other words we are asking whether the imaginary component of
$\phi$ decouples, leaving only a single real scalar degree of freedom to
source the cosmological dynamics.

To answer this, suppose that $\phi$ is spatially homogeneous and consider the
equation of motion for the fields $\phi_{R}$ and $\phi_{I}$ with potential
$V_{W_{2}}(\phi_{I},\phi_{R})$,

$\displaystyle\ddot{\phi}_{i}+3H\dot{\phi}_{i}+\partial_{i}V_{W_{2}}=0.$ (45)

Since $W_{2}(\phi)$ is analytic in the complex plane and real on the real
axis, it can be shown that
$\overline{W_{2}(\phi)}=W_{2}(\bar{\phi})$.121212Proof: Since $W_{2}(\phi)$ is
analytic
$W_{2}(\phi)=W_{2}(\phi_{R}+i\phi_{I})=\sum_{n=0}^{\infty}\frac{1}{n!}W_{2}^{(n)}(\phi_{R})(\phi-\phi_{R})^{n}$.
Since $W_{2}(\phi_{R})$ is real
$\overline{W_{2}^{(n)}(\phi_{R})}=W_{2}^{(n)}(\phi_{R})$. Hence
$\overline{W_{2}(\phi)}=\sum_{n=0}^{\infty}\frac{1}{n!}W_{2}^{(n)}(\phi_{R})(\bar{\phi}-\phi_{R})^{n}=W_{2}(\bar{\phi})$.
Equation (44) then implies that $V_{W_{2}}(\bar{\phi})=V_{W_{2}}(\phi)$ which
implies that $\partial V_{W_{2}}/\partial\phi_{I}$ is identically zero on the
real axis. Therefore if $\phi_{I}$ and $\dot{\phi}_{I}$ vanish at some initial
time, they vanish for all times; if $\phi$ starts rolling along the real axis,
it stays on the real axis. Hence we are able to construct any effective single
field cosmological model, such as those discussed in previous sections, by
starting with a supersymmetric Lagrangian.

#### Stability

Having shown that $\phi_{I}=0=\dot{\phi}_{I}$ initially implies vanishing
$\phi_{I}$ for all times, i.e. $\phi_{I}$ decouples from $\phi_{R}$, it is
important to ask whether the dynamics of $\phi_{R}$ is stable against small
perturbations which make $\phi_{I}$ or $\dot{\phi}_{I}$ nonzero. Such
perturbations will inevitably spontaneously occur due to quantum fluctuations,
so it is crucial to consider their effects.

To determine the effects of such small perturbations away from real values of
$\phi$, we must evaluate $\partial^{2}V_{W}/\partial\phi_{I}^{2}$ for
$\phi_{I}=0$. If this is positive, the potential $V_{W_{2}}(\phi)$ curves
upwards as $\phi$ moves away from the real axis, and thus the motion along the
real axis is stable to small perturbations.

Equation (44) gives $V_{W_{2}}(\phi)$ for all complex values of $\phi$.
Setting $\phi=\phi_{R}+i\phi_{I}$ and differentiating, we find

$\displaystyle\frac{\partial^{2}V_{W_{2}}}{\partial\phi_{I}^{2}}\Bigg{|}_{\phi_{I}=0}=8e^{2\sqrt{3}\phi_{R}}C^{2}+F_{0}(\phi_{R})+\frac{F_{2}(\phi_{R})}{C^{2}},$
(46)

where $F_{0}(\phi)$ and $F_{2}(\phi)$ are analytic functions (which depend on
$V$) and are thus bounded on any compact set $I\subset\mathbb{R}$. Therefore,
since the leading term in this expansion is positive, it is always possible to
take $C$ large enough so that
$\partial^{2}V_{W_{2}}/\partial\phi_{I}^{2}|_{\phi_{I}=0}>0$ for any
$\phi_{R}\in I$.131313Recall that taking $C$ large is the same limit which
ensures $V_{W_{2}}(\phi_{R})\approx V(\phi_{R})$ in the first place. In this
case the dynamics of $\phi_{R}$ along the real axis is guaranteed to be stable
under fluctuations in the imaginary directions, and so even at the level of
perturbations the field $\phi_{I}$ decouples from the dynamics of
$\phi_{R}$.141414Note that the dynamics is also stable in the double
analytically continued (Lorentzian wormhole) picture Antonini:2022blk ,
because $\partial^{2}V_{W_{2}}/\partial\phi_{I}^{2}>0$ trivially satisfies the
BF bound, while $\partial^{2}V_{W_{2}}/\partial\phi_{R}^{2}$ satisfies the BF
bound by construction. Thus we truly have a single field cosmological model,
starting from a supersymmetric Lagrangian.

##  4 Outlook

In this paper, we have argued that accelerated expansion is generic (i.e.
arises without fine tuning) in $\Lambda<0$ cosmologies arising from
asymptotically AdS Euclidean wormholes, for simple scalar potentials with a
form that is natural in theories with a dual CFT. We found that there exist
models of this type which can match $\Lambda$CDM to high accuracy at the level
of background geometry, so the framework can give realistic cosmologies. We
found that the scalar potential in these examples can arise from a
superpotential, so the effective field theory can potentially have a
supersymmetric point for $\phi=0$. There are many directions for further
study.

#### Direct reconstruction of the potential

The class of models considered in this paper make at least two generic
predictions: that one should have a dynamical scalar field contributing to the
energy density of the universe and that the universe should eventually begin
to decelerate. With this in mind, it is interesting to consider direct
measurements of the scale factor from supernova observations and search for
evidence of an evolving scalar field. In principle, the scalar potential in a
single field model can be reconstructed directly from knowledge of the scale
factor $a(t)$ and $\Omega_{M}$ when radiation can be ignored. The explicit
expressions given in equations (22) and (23) define $V(\phi)$ parametrically.
However, $\Omega_{M}$ (the present day matter contribution to the energy
density) is not well constrained by direct observation, and the reconstruction
of $V(\phi)$ for given $\Omega_{M}$ is quite sensitive to small changes in
$a(t)$. As a result, there is actually considerable freedom in the form of the
potential that can be consistent with data, and thus considerable room for the
types of models considered here.151515One example of this flexibility is
highlighted in Figure 9 which shows in red a potential with significant
variation that provides a good fit to the low redshift measurements of the
scale factor. Plausibly, the type of fine-tuning that we required to match
$\Lambda$CDM precisely will not be required if the goal is to provide a good
fit with direct obserations of the scale factor. We will discuss these issues
along with a more complete analysis of the supernova data in a forthcoming
paper dataPaper . We note that it is particularly interesting at present to
consider models that go beyond $\Lambda$CDM given the Hubble tension
di2021realm .

#### Including fluctuations in a realistic effective field theory model

In this paper, we have focused entirely on the evolution of the homogeneous
background. We found that there are effective theories which are compatible
both with the cosmological history obtained from $\Lambda$CDM or $w$CDM and
with having a stable asymptotically AdS analytic continuation. It is very
interesting to understand whether these effective theories are also capable of
reproducing the observed spectrum of fluctuations in the CMB. One question we
would particularly like to understand is whether primordial inflation is
required or if the existence of the analytic continuation is already enough to
give a sensible spectrum of fluctuations.

We also want to understand if the scalar potentials we are considering could
have other phenomenological effects. For example, considering fluctuations
around a very flat potential raises the prospect of a new long-ranged scalar-
mediated force. One can also ask if scalar particles would be produced in any
appreciable quantity and, if so, could such particles form a component of dark
matter? These are issues that must be addressed to claim a fully realistic
model, quite apart from objections to flat potentials arising from Swampland
conjectures.

#### Towards a microscopic construction

Here we have focused exclusively on effective theories, but it remains an
important open question to find concrete microscopic realizations of these big
bang / big crunch cosmologies with a holographic dual. In the search for such
constructions, it makes sense to again back off of demanding a fully realistic
model. Instead, we can ask if relatively simple models can be found, perhaps
corresponding to a pure radiation scenario or a scenario with a relatively
simple scalar potential exhibiting some period of acceleration.

It is also interesting to further explore the issue of supersymmetric
effective theories. Here we showed that a superpotential can be constructed to
match any potential we devise, but presumably not all such superpotentials
arise from a UV-complete and supersymmetric microscopic theory.

Acknowledgements: We acknowledge support from the U.S. Department of Energy
grant DE-SC0009986 (B.G.S.), the U.S. Department of Energy, Office of Science,
Office of Advanced Scientific Computing Research, Accelerated Research for
Quantum Computing program “FAR-QC” (S.A.), the National Science and
Engineering Research Council of Canada (NSERC) and the Simons foundation via a
Simons Investigator Award and the “It From Qubit” collaboration grant.

##  Appendix A Details of the potential examples

###  A.1 Fine-tuning potentials with a flat region

In this appendix we explain how the parameters of potentials with a flat
region of the form (14)-(15) must be chosen and fine-tuned to reproduce the
$\Lambda$CDM cosmological evolution between the early universe and the present
day.

In order to have a phase of accelerated expansion in the cosmological
evolution, the positive potential energy at the plateau (which is fixed by
observational data to be $A=\Omega_{\Lambda}$) must be dominant over every
other term on the right-hand side of the Friedmann equation. This implies that
the scalar field’s kinetic energy $K_{\phi}$ must be very small when the field
reaches the plateau. Evolving backwards in time from the time-symmetric point
$\tilde{t}=0$ in the cosmology picture, the scalar field behaves like an anti-
damped particle. Therefore, intuitively, an accelerated expansion can take
place only if $|B|\ll|A|$ — where $B$ sets the value of $V(0)$. In this way,
for an appropriately fine-tuned potential, the scalar field just barely makes
it up to the plateau and the kinetic energy does not become dominant until
very close to the Big Bang (or Big Crunch, if we evolve forward in time from
the time-symmetric point).

Next, we must require that the BF bound (5) is satisfied. This can be achieved
by choosing the value of $X$ to be large enough that the deep valleys are not
too close to the maximum at $\phi=0$, so that
$(d^{2}\tilde{V}/d\phi^{2})|_{\phi=0}$ is suppressed. However, $X$ must not be
too large, otherwise the cosmological evolution will reach the Big Bang
singularity before the scalar field can reach the valleys (and then the
plateau): in this case, no accelerated expansion phase occurs.

Moreover, the (negative) parameter $C$ — which controls the depth of the
valleys — must be large enough that the scalar field acquires sufficient
kinetic energy in its anti-damped motion to reach the positive plateau. But it
cannot be too large, or the kinetic energy will be large when the scalar
reaches the plateau and no accelerated phase can take place. We remark that no
particular fine-tuning has been carried out so far on the parameters: we just
identified the general criteria to choose the parameters $B$, $C$ and $X$ in
order for an accelerated expansion to be possible and for the BF bound to be
satisfied (we remind that $A$ is fixed by data). The specific choice of their
values is arbitrary and irrelevant for the sake of reproducing the
$\Lambda$CDM evolution.

Finally, we fine-tune the value of $\Delta$161616Note that we could have
instead fixed a value of $\Delta$ and fine-tuned the parameter $C$. to allow
the scalar field to reach the plateau with a very small kinetic energy. This
yields a phase of accelerated expansion, and will allow us to reproduce the
$\Lambda$CDM cosmological evolution between the early universe and the present
day.

Note that, as we evolve backwards in time towards the Big Bang, the scalar
field’s kinetic energy will eventually become dominant. Our goal is to
reproduce as best as possible the results of the $\Lambda$CDM model, in which
there is no such term in the Friedmann equation, between the early universe
and the present day. More precisely, an important criterion for the
feasibility of these models is that we should not have a significant departure
from $\Lambda$CDM between the present time and the time of Big Bang
Nucleosynthesis (BBN) (which roughly occurs in the range of temperatures
$T\sim 10^{9}-10^{10}$ K), otherwise the abundances of various elements could
vary appreciably from those experimentally observed. These temperatures
correspond to scale factors which are between $10^{-9}$ and $10^{-10}$ times
smaller than the current value of the scale factor $a(\tilde{t}_{0})=1$. We
can therefore ask whether it is possible to tune our model with a rolling
scalar such that the scalar field’s kinetic energy is a subdominant
contribution in the cosmological Friedmann equation for $10^{-10}\lesssim
a(\tilde{t})\leq 1$. In other words, we would like to determine whether the
kinetic energy dominance can be pushed sufficiently early in time that it
would not impact the predictions of BBN.

By appropriately fine-tuning the parameter $\Delta$ we were able to push the
crossover between the early kinetic-energy-dominated era and the radiation-
dominated era to $\tilde{t}=\tilde{t}^{*}$ with
$\tilde{t}^{*}-\tilde{t}_{BB}=\mathcal{O}(10^{-13})$ (where $\tilde{t}_{BB}$
corresponds to the Big Bang) in our numerical solutions, corresponding to
$a(\tilde{t}^{*})=\mathcal{O}(10^{-7})$. We were not able to reach the desired
value $a(\tilde{t}^{*})=\mathcal{O}(10^{-10})$ because of numerical precision
issues related to the excessively small time steps needed, but there is no a
priori obstacle in pushing the kinetic-energy-dominated era arbitrarily close
to the cosmological singularity.

By implementing this fine-tuning procedure after choosing the Planck 2018
cosmological parameters (16) and the initial condition $a_{0}=2$, we obtained
the set of parameters (17) used in our numerical analysis. The same procedure
was also used to fine-tune the potential for the solution employing SNIa-
derived cosmological parameters depicted in Figure 10.

###  A.2 Scalar potential for $w$CDM model

For completeness, we provide some details related to the analysis in Section
2.4.

To construct a scalar potential $V(\phi)$ agreeing with $V_{w}(\phi)$ in a
suitable region, permitting cosmological solutions which satisfy observational
constraints from SNe Ia and have asymptotically AdS Euclidean continuations,
we proceed as follows:

  1. 1.

Determine the redshift dependence of the scalar field $\phi_{w}(z)$ and
potential $V_{w}(z)$ for the given $w$CDM model,171717The Hubble expansion
$H(z)$ for the $w$CDM model is given in equation (19), using the parameters
recorded in Section 2.4. The redshift dependence of the scalar $\phi(z)$ and
potential $V(z)$ may then be computed from $H(z)$ using equation (48). in the
region $z\in(z_{\textnormal{min}},z_{\textnormal{max}})$ where type Ia
supernova data have been used to constrain cosmological parameters. We have
the freedom to choose the value $\phi_{w}(0)\equiv\phi_{0}$ and the sign of
$\phi_{w}^{\prime}(0)$; we will take both to be positive. Note however that
the present scalar kinetic energy is fixed to

$\frac{1}{2}\dot{\phi}_{w}(z=0)^{2}=\frac{1}{8\pi
G}H_{0}H^{\prime}(0)-\frac{3}{16\pi G}H_{0}^{2}\Omega_{M}\equiv K_{0}\>.$ (47)

  2. 2.

Evaluate $\phi_{w}(z)$ and $V_{w}(z)$ for a large sample $\\{z_{i}\\}$ of
redshifts in the interval $(z_{\textnormal{min}},z_{\textnormal{max}})$, to
obtain an array of pairs $\\{(\phi_{w}^{(i)},V_{w}^{(i)})\\}$.

  3. 3.

Choose a model $V(\phi;\\{p\\},q)$, with parameters $\\{p\\}$ and $q$,
satisfying:

     * •

$V(0;\\{p\\},q)<0$

     * •

$-\frac{9}{4}<\frac{V^{\prime\prime}(0;\\{p\\},q)}{|V(0;\\{p\\},q)|}<0$

     * •

$V(\phi;\\{p\\},q)$ behaves as a polynomial for $\phi\gtrsim\phi(z=0)$.

In our case, the model has a Gaussian trough between $\phi=0$ and
$\phi=\phi_{0}$, with a height controlled by the parameter $q$.

  4. 4.

Apply the “shooting method” to determine parameters $\\{p\\}$ and $q$ for
which the model has a time-symmetric solution with the correct scalar kinetic
energy $K_{0}$ at $z=0$:

     * •

For various choices of $q$, use the array $\\{(\phi_{w}^{(i)},V_{w}^{(i)})\\}$
to determine the best-fit parameters $\\{p_{\textnormal{fit}}(q)\\}$.

     * •

Solve the equations of motion for $a(t),\phi(t)$ for the best-fit models
$V(\phi;\\{p_{\textnormal{fit}}(q)\\},q)$ with each choice of $q$ to determine
for which value $q_{*}$ there exists a future time $t$ for which
$\dot{a}(t)=\dot{\phi}(t)=0$; this determines the model for which the solution
has a moment of time symmetry. We use initial conditions
$\phi(t_{0})=\phi_{0}$, $\dot{\phi}(t_{0})=-\sqrt{2K_{0}}$, and $a(t_{0})=1$.

  5. 5.

As a check, evaluate the solution to the model
$V(\phi;\\{p_{\textnormal{fit}}(q_{*})\\},q_{*})$ with time-symmetric initial
conditions, and verify that the equation of state $w(z)$ behaves appropriately
for $z\in(z_{\textnormal{min}},z_{\textnormal{max}})$, and that the Euclidean
continuation has AdS asymptotics.

It will frequently be useful to work with quantities rescaled by the Hubble
parameter $H_{0}$, as defined in Section 2.3.

We observe that the procedure outlined above has significant freedom in the
choice of model $V(\phi)$; since our goal is merely to demonstrate the
existence of a scalar potential satisfying the list of criteria outlined in
the main text, it suffices to construct a specific example.

#### Constructing an example

We first deduce the redshift dependence of the scalar field $\phi_{w}(z)$ and
the rescaled potential $\tilde{V}_{w}(z)$ from the rescaled Hubble expansion
$\tilde{H}(z)$ for the $w$CDM model, using

$\begin{split}\phi_{w}(z)&=\phi_{0}+\int_{0}^{z}dz^{\prime}\>\frac{\sqrt{\frac{1}{4\pi
G}(1+z^{\prime})\tilde{H}(z^{\prime})\tilde{H}^{\prime}(z^{\prime})-\frac{3}{8\pi
G}(1+z^{\prime})^{3}\Omega_{M}}}{(1+z^{\prime})\tilde{H}(z^{\prime})}\\\
\tilde{V}_{w}(z)&=-\frac{1}{8\pi
G}(1+z)\tilde{H}(z)\tilde{H}^{\prime}(z)+\frac{3}{8\pi
G}\tilde{H}(z)^{2}-\frac{3}{16\pi G}(1+z)^{3}\Omega_{M}\>.\end{split}$ (48)

We also deduce the present time-derivative $\dot{\phi}(\tilde{t}_{0})$, where
the overdot denotes a derivative with respect to the rescaled time coordinate;
in units $\frac{8\pi G}{3}=1$, we find for the $w$CDM model

$\dot{\phi}_{w}(\tilde{t}_{0})=-0.2628687886\>.$ (49)

Next, we choose as our model function

$\begin{split}\tilde{V}(\phi;A,B,C,\Delta,X,\mathbf{V})&=\tilde{V}_{0}(\phi;\ldots)+\tilde{V}_{0}(-\phi;\ldots)\\\
\tilde{V}_{0}(\phi;\ldots)&=\frac{A-B}{2}\textnormal{erf}\left(\frac{\phi-X}{\Delta}\right)+C\exp\left(-\frac{(\phi-X)^{2}}{\Delta^{2}}\right)\\\
&\qquad+\left(\sum_{k=1}^{4}\frac{V_{k}}{k!}(\phi-X)^{k}\right)\left(1+\textnormal{erf}\left(\frac{\phi-X}{\Delta}\right)\right)+\frac{A}{2}\>.\end{split}$
(50)

This is similar to the asymptotically flat potential introduced in Section 2.3
to reproduce the $\Lambda$CDM model, though we have effectively replaced the
flat region with a polynomial.

We will choose to situate the present time at $\phi_{0}=0.7$, the trough in
the potential at roughly $X=0.5$, the width $\Delta=0.06$, and to take
$B=-0.08$, which fixes the potential at $\phi=0$ to this value; we choose
these values for concreteness, though one could proceed analogously for many
suitable choices of these quantities. Our choices are motivated by the facts
that (1) having the trough join suitably to the polynomial piece of the
potential requires $(\phi_{0}-X)$ to be a few multiples of $\Delta$, and (2)
$B$ should be sufficiently large in absolute value that we avoid recollapse in
the wormhole picture. The parameter $C$, controlling the depth of the trough,
is the parameter that we will vary for the shooting method.

With these choices, we proceed to apply the shooting method, solving the
equation (11) with initial conditions
$a(t_{0})=0,\phi(t_{0})=\phi_{0},\dot{\phi}(t_{0})=-0.2628687886$, for various
choices of $C$ and best-fit $(A,V_{1},V_{2},V_{3},V_{4})$. Our procedure
yields, to our working precision, the parameters $C=-7.80623080636$ and

$\begin{split}&A=1.13258420902349\>,\quad V_{1}=-4.21938633468535\>,\quad
V_{2}=50.3190576443637\>,\\\ &\qquad\qquad\qquad
V_{3}=-370.514499837809\>,\quad V_{4}=1408.97547041725\>.\end{split}$ (51)

We plot the corresponding potential $V(\phi)$, as well as $V_{w}(\phi)$, in
Figure 9 of the main text. We find that the moment of time symmetry occurs at
rescaled time $\tilde{t}$ with

$\tilde{t}-\tilde{t}_{0}=0.5712956836\>,$ (52)

at which time the scale factor and scalar field are given by

$a(\tilde{t})=1.41641139203233\>,\quad\phi(\tilde{t})=0.358665688585761\>.$
(53)

To check our results, we can solve both the Lorentzian and Euclidean equations
of motion (11) and (12) for the model with scalar potential $V(\phi)$,
assuming time-symmetric initial conditions at

$\phi=0.358665688585761\>,$ (54)

the location suggested by equation (53). Assuming that the present time
$\tilde{t}_{0}$ satisfies $a(\tilde{t}_{0})=1$, we can extract the redshift
dependence of the equation of state parameter $w(z)$ over the range
$z\in(z_{\textnormal{min}},z_{\textnormal{max}})$ using equation (25), and
verify that it is approximately constant and equal to the Pantheon+SH0ES value
$w=-0.90\pm 0.14$; we show this in Figure 10 in the main text. We can also
verify that the Euclidean solution is asymptotically AdS by plotting the
rescaled “Euclidean Hubble expansion” $H(\tau)$ and observing that it
approaches a constant value, as also shown in Figure 10 in the main text.

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