Is the present acceleration of the Universe caused by merging with other universes?

Abstract

We show that by allowing our Universe to merge with other universes one is lead to modified Friedmann equations that explain the present accelerated expansion of our Universe without the need of a cosmological constant.

Full text

JCAP12(2023)011
ournal of
Cosmology and Astroparticle Physics
An IOP and SISSA journal
J
Is the present acceleration of the
Universe caused by merging with other
universes?
J. Ambjørna,b and Y. Watabikic
aThe Niels Bohr Institute, Copenhagen University,
Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark
bInstitute for Mathematics, Astrophysics and Particle Physics (IMAPP),
Radbaud University Nijmegen,
Heyendaalseweg 135, 6525 AJ, Nijmegen, The Netherlands
cTokyo Institute of Technology, Dept. of Physics, High Energy Theory Group,
2-12-1 Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan
E-mail: ambjorn@nbi.dk,watabiki@th.phys.titech.ac.jp
Received August 30, 2023
Revised October 25, 2023
Accepted October 26, 2023
Published December 12, 2023
Abstract. We show that by allowing our Universe to merge with other universes one is lead to
modified Friedmann equations that explain the present accelerated expansion of our Universe
without the need of a cosmological constant.
Keywords: quantum cosmology, quantum gravity phenomenology
ArXiv ePrint: 2308.10924
c

2023 The Author(s). Published by IOP Publishing
Ltd on behalf of Sissa Medialab.
Original content
from
this work may be used under the terms of the Creative Commons
Attribution 4.0 licence. Any further
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maintain attribution to the author(s) and the title of the work,
journal citation and DOI.
https://doi.org/10.1088/1475-7516/2023/12/011

JCAP12(2023)011
Contents
1 Introduction 1
2 Expansion by merging with other universes 2
3 Including matter 5
4 Comparison of the different models 8
5 Discussion 13
A Analytic results for the three models 15
A.1 The fds-model 15
A.2 The fgcdt-model 16
A.3 The fmod-model 16
A.4 Duality in the fmod-model 17
1 Introduction
We have proposed a modified Friedmann equation [
1
–
3
] which changes the late time cosmology,
such that one does not need a cosmological constant to explain the present day acceleration of
our Universe. While it is our hope that this modified Friedmann equation can eventually be
derived from an underlying microscopic theory [
4
–
7
], we will here treat it as a phenomenological
model that we can obtain in a simple way from the standard Friedmann equation.1
Our starting point is the Hartle-Hawking minisuperspace action, which after the rotation
of the conformal factor can be written as
S=Zdt β˙v2
2Nv +λN v!.(1.1)
Before rotating to Euclidean spacetime and the rotation of the conformal factor, this action
is just the standard minisuperspace action
S=Zdt −β˙v2
2Nv −λN v!,(1.2)
written using the metric
ds2=−N2(t)dt2+a2(t)dΩd, dΩd=
d
X
i=1
dx2
i.(1.3)
In (1.1) and (1.2) we are using units where
c
=
ℏ
= 1 and the constant
β
= (
d−
1)
/d
, where
dis the dimension of space. We have used
v(t) = 1
κad(t), κ = 8πG, G =the gravitational constant. (1.4)
1
It should be noted that ideas somewhat related to the ones presented here and in [
1
–
7
] have also been
advocated in [8].
– 1 –

JCAP12(2023)011
as variable rather than the scale factor
a
(
t
).
v
(
t
)is proportional to the spatial
d
-volume at
time
t
and below we will just call it the
d
-volume. For the Hubble parameter
H
(
t
)we then
have
H(t)≡˙a
a=1
d
˙v
v.(1.5)
The reason we prefer to use
v
(
t
)instead of
a
(
t
)is that the minisuperspace action (1.1)–(1.2)
is then valid in all space dimensions
d >
1. Finally
λ
denotes the cosmological constant.
Presently we will ignore matter, only assume a cosmological constant term. Below we will
include matter in our considerations.
It should be noted that (1.1) also appears as the leading term in an effective action in
a model of four-dimensional quantum gravity known as Causal Dynamical Triangulations
(CDT). In that case one is not assuming a minisuperspace reduction, but performs the path
integral over all degrees of freedom except
v
(
t
). Thus (1.1) might be more general than
suggested by the minisuperspace reduction. This CDT result is a numerical result, obtained
via Monte Carlo Simulations of the CDT lattice gravity model where one identifies
G/ε2
and
λε4
with the corresponding dimensionless lattice coupling constants,
ε
denoting the length
of the lattice links, i.e. the UV lattice cut-off (see [
9
,
10
], and [
11
,
12
] for reviews). Quite
remarkably, for CDT in
d
= 1, (1.1) can be derived analytically (with
β
= 0), and it is
an effective action coming entirely from the path integral measure [
13
], since the classical
Einstein action is a topological invariant in the case of d= 1.
The classical Hamiltonian corresponding to (1.2) is2
H(v, p) = N v −p2
2β+λ!,(1.6)
where
p
denotes the momentum conjugate to
v
. In the following we will be interested in
d= 3, i.e. 2β= 4/3.
A classical solution corresponding to the action (1.2) is the de Sitter spacetime3
v(t) = v(t0) e√3λ(t−t0).(1.7)
We now want to go beyond this classical picture, but staying as close as possible to the
minisuperspace picture, by trying to include the possibility that our Universe can absorb
other universes, which we denote baby universes even if they are not necessarily small.
2 Expansion by merging with other universes
Since we will allow for other universes to merge with our Universe, we are really discussing a
multi-universe theory and like in a many-particle theory it is natural to introduce creation
and annihilation operators Ψ
†
(
v
)and Ψ(
v
)for single universes of spatial volume
v
. In a
full theory of four-dimensional quantum gravity the spatial volume alone will of course not
2
Since there is no time derivative of
N
(
t
)in (1.2), we should strictly speaking treat it as a constraint system
and we can choose as a Hamiltonian
Hu
=
H
+
u
(
t
)
PN
, where
PN
is the momentum conjugate to
N
, and impose
the constraint
PN
= 0 in phase space. This leads to
H
= 0 on the constraint surface as a consistent secondary
constraint and
˙
N
=
u
(
t
), expressing the invariance with respect to time reparametrization. In the following we
will use this invariance to choose Nconstant (N= 1), and then consider “on shell” solutions H= 0.
3
Note that the solution of
H
(
v, p
)=0corresponding to an expanding universe has
p
=
−√λ <
0. The
reason for the somewhat counter-intuitive negative values of
p
is the negative sign of the “kinetic” terms in (1.2)
and (1.6).
– 2 –

JCAP12(2023)011
provide a complete characterization of a state at a given time
t
. As mentioned we will here
make the drastic simplification to work in a minisuperspace approximation where the spatial
universe is characterized entirely by the spatial volume
v
. Thus, denote the quantum state
of a spatial universe with volume
v
by
|v⟩
. We consider now the multi-universe Fock space
constructed from such single universe states and denote the Fock vacuum state by
|
0
⟩
. Then
[Ψ(v),Ψ†(v′)] = δ(v−v′),Ψ†(v)|0⟩=|v⟩,Ψ(v)|0⟩= 0.(2.1)
In this way the (minisuperspace) quantum Hamiltonian that includes the creation and
destruction of universes can be written as [14]
ˆ
H=ˆ
H(0) −gZdv1Zdv2Ψ†(v1)Ψ†(v2) (v1+v2)Ψ(v1+v2)(2.2)
−gZdv1Zdv2Ψ†(v1+v2)v2Ψ(v2)v1Ψ(v1)−Zdv
vρ(v)Ψ†(v), ρ(v) = δ(v)
ˆ
H(0) =Z∞
0
dv
vΨ†(v)ˆ
H(0) vΨ(v),ˆ
H(0) =v −3
4
d2
dv2+λ!(2.3)
ˆ
H(0)
describes the quantum Hamiltonian corresponding to the action (1.1) (with
β
= 2
/
3)
and describes the propagation of a single universe, while the two cubic terms describe the
splitting of a universe into two and the merging of two universes into one, respectively. Finally,
the last term implies that a universe can be created from the Fock vacuum
|
0
⟩
provided the
spatial volume is zero. If it was not for this term
ˆ
H|
0
⟩
= 0, and the Fock vacuum would be
stable. Again, in our minisuperspace approximation we do not attempt to describe how such
a merging or splitting realistically takes place, the only thing which has our interest is how
the volume of space can be influenced by such merging or splitting processes, and for this the
minisuperspace model might give us some interesting hints.
Even the minisuperspace Hamiltonian
ˆ
H
is too complicated to be solved in general. A
universe can successively split in many, be joined by many and a part that splits off can
later rejoin, thereby changing the topology of spacetime. Since the Hamiltonian is essentially
dimension independent (all dimension dependence is absorbed in the coupling constants
κ
,
λ
and
g
), what we are describing is the so-called string field theory of two-dimensional CDT [
14
].
For this string field theory there exists a truncation that can be solved analytically [
15
],
4
called generalized CDT (GDCT), and that at the same time has our main interest from a
cosmological point of view. It follows the evolution of a universe (let us call it “our” Universe)
in time. During this evolution it can merge with other universes (denoted baby universes),
created at times
ti
with spatial volumes
vi
(
ti
)=0(see figure 1). We will assume these
universes have the same coupling constants as our Universe. The effective Hamiltonian,
the so-called inclusive Hamiltonian, first introduced in [
18
], governing the evolution of the
Universe, is obtained from the path integral by integrating over the times
ti
and summing over
the number of baby universes merging with our Universe. In the path integral, the various
baby universes, characterized at a given time
t
by spatial volumes
vi
(
t
), can themselves be
merged with other baby universes. One integrates over all possible
vi
(
t
), not necessarily
related to solutions of any classical equations.
Let us describe in some detail how such an inclusive Hamiltonian can be obtained from
ˆ
H
.
The simplest way to incorporate an absorption of baby universes into the propagation of our
4
Surprisingly, some CDT string field amplitudes can actually be calculated non-perturbatively to all order
in the genus expansion, see [16,17].
– 3 –

JCAP12(2023)011
t
Figure 1. Our Universe (blue), represented as one-dimensional circles, propagating in time, with baby
universes (red) merging and increasing the spatial volume.
Universe is to replace the quantum field Ψ(
v
), representing the disappearance of a universe
of spatial volume
v
when it is absorbed by our Universe, by a classical field
ψ
(
v
). Thus we
make the following replacement in the third term on the r.h.s. of eq. (2.2)
Ψ†(v1+v2)Ψ(v1)Ψ(v2)→Ψ†(v1+v2)ψ(v1)Ψ(v2)+Ψ†(v1+v2)Ψ(v1)ψ(v2)(2.4)
The terms on the r.h.s. of eq. (2.4) contribute to the propagation of our Universe since they
are of the form Ψ
†
(
v1
)
·
Ψ(
v2
), but contrary to the terms in
ˆ
H0
they are non-local in
v
. By
Taylor expanding Ψ
†
(
v
+
w
)around
v
, this non-locality can be expanded as a power series in
the non-local operator (d/dv)−1. After some algebra this leads to
Zdvdw Ψ†(v+w)wψ(w)vΨ(v) = ZdvΨ†(v)Fd
dv vΨ(v),(2.5)
where F(p)is the Laplace transform of ϕ(v) = vψ(v):
ϕ(v) = ϕ0+ϕ1v+ϕ2v2+··· , F (p) = Z∞
0
dv e−p v ϕ(v) = Γ(1)ϕ0
p+Γ(2)ϕ1
p2+··· (2.6)
Thus ˆ
H0from (2.3) will be replaced by the inclusive Hamiltonian
ˆ
Hinc =ˆ
H0−2gF d
dv v(2.7)
In GCDT
F
(
p
)is determined by the self-consistency requirement that our Universe, modified
by the impact
ϕ
(
v
)of baby universes, should be identical to the baby universes it absorbs.
We refer to [
15
] (or to the Lecture Notes [
19
,
20
]) for the details of this determination.
5
5The only difference compared the derivation in [15] is that the sign of gappearing in (2.9) and (2.8) will
be different from the sign appearing in [
15
], the reason being that we consider the absorption rather than the
emission of baby universes. In earlier articles discussing a modified Friedmann equation ([
1
,
2
]) we used the
notation from [
15
], but with negative
g
, which effectively meant that we were considering absorption rather
than emission of baby universes, like here.
– 4 –

JCAP12(2023)011
Finally, rotating back from the Hartle-Hawking metric to Lorentzian signature, as when going
from (1.1) to (1.2) and replacing
6
the
−id/dv
by the classical momentum
p
conjugate to
v
we
obtain the following semiclassical Hamiltonian
H=v−3
4p2+λ−2gF (p)=3
4v (p+α)r(p−α)2+2g
α!,(2.8)
where αsatisfies the equation7
α3−4
3λα −g= 0.(2.9)
It is seen that if
g
= 0 one obtains precisely the
H
in (1.6) (with
N
= 1). A solution to the
“on shell” Hamiltonian equations is
v(t) = v(t0) e 3
2Σ(t−t0),Σ = rα2+g
2α.(2.10)
Again, if we choose
g
= 0 we obtain the de Sitter solution (1.7). Increasing
g
will increase the
expansion rate of the universe, the intuition behind this being illustrated in figure 1, and we
can actually take the cosmological constant λ= 0 and still obtain an expanding universe:
v(t) = v(t0) ep27
8g1/3(t−t0).(2.11)
Thus we have the same classical de Sitter solution as (1.7) provided
g2/3=8
9λ, (2.12)
but the origin of this exponential expansion is now not a cosmological constant but instead
the “bombardment” of our Universe by baby universes.
3 Including matter
Until now we have considered our Universe, but without matter. We will now include matter
in the most simple way, as a matter density
ρm
(
v
)in the Hamiltonian (2.8).
8
In addition we
will only consider relatively late times in the evolution of our universe, namely the period after
the time of last scattering (
tLS
), where matter to a good approximation can be considered as
dust that exerts zero pressure. Thus the Hamiltonian has the form
H[v, p] = v(−f(p) + κρm(v) ), vρm(v) = v0ρm(v0),(3.1)
where
v0
and
ρm
(
v0
)denote the values at the present time
t0
and where we will later consider
three different choices of
f
(
p
). Before that, let us discuss the solution of the eoms for arbitrary
6
The rotation from the Hartle-Hawking metric to the Lorentzian metric involves a double analytic con-
tinuation, namely both in time and also a conformal factor rotation, which in the minisuperspace metric
parametrization becomes v→ −iv.
7
In the case of
g
= 0 we choose the solution
α
=
√λ >
0. In this case, as remarked in footnote 3,
p
=
−α <
0
and p(p−α)2=α−p. Since we consider g≥0, the solution α > 0will only increase with increasing gand
H= 0 leads to p=−α < 0.
8
The inclusion of matter in this way introduces an asymmetry between the baby universes and our Universe,
in the sense that we do not include such a matter term in the evolution of the baby universes.
– 5 –

JCAP12(2023)011
f(p)in (3.1). The eoms simplify since vρm(v)is constant.
˙v=∂H
∂p =−vf′(p),i.e.3˙a
a=˙v
v=−f′(p)(3.2)
˙p=−∂H
∂v =f(p),i.e. t −tLS =Zp
pLS
d p
f(p)(3.3)
By construction any solution to (3.2)–(3.3) will satisfy
H
=
const
, and we are interested in
the “on-shell” solutions H= 0, which by (3.1) implies that
f(p) = κρm(v) = κρm(v0)v0
v=f(p0)v0
v=f(p0)(1 + z)3,(3.4)
where p0denotes the value of pat present time t0and zdenotes the redshift at time t, i.e.
z(t) + 1 = a(t0)
a(t)=v(t0)
v(t)1/3
.(3.5)
Eq. (3.4) is the generalized Friedmann when eq. (3.2) is used to express pin terms of ˙a/a.
Using
p
as a parameter instead of
t
(the relation between the two parameters is defined by
eq. (3.3)) we can immediately write parametric expressions for a number of relevant functions
expressed in terms of f(p). Let us define these. The redshift z(t)or z(p)is
z=a(t0)
a(t)−1 = f(p)
f(p0)1
3−1.(3.6)
The Hubble parameter H(t)or H(p)is defined as (see also eq. (1.5))
H=˙a(t)
a(t)=−1
3f′(p).(3.7)
The formal density
ρf
(
t
)or
ρf
(
p
)related to the function
f
(
p
)is obtained by writing the
generalized Friedmann equation (3.4) as
˙a(t)
a(t)2
=κρm(v)
3+κρf(v)
3(3.8)
from which we deduce (using the eoms)
κρf(p) = 1
3f′(p)2−f(p).(3.9)
κdρf
dt =f(p)f′(p)2
3f′′(p)−1.(3.10)
We define the formal pressure Pfrelated to ρfby the energy conservation equation
d
dt(vρf) + Pf
d
dtv= 0.(3.11)
This leads to
Pf=f(p)2
3f′′(p)−1−ρf(v)(3.12)
– 6 –

JCAP12(2023)011
and the formal equation of state parameter wfis defined (for ρf= 0) by
wf=Pf
ρf
=f(p)2
3f′′(p)−1
1
3f′(p)2−f(p)−1(3.13)
The definitions of
ρf
and
Pf
ensure that our eoms can be written in the standard GR form,
expressed in terms of a,ρm,ρfand P.
Finally, we will need the standard definitions of some cosmological distances.
DH
is
defined as the inverse Hubble parameter
DH=1
H=−3
f′(p)(3.14)
while
DM=Zz
0
dz′
H(z′)=Zt0
t
dt′a(t0)
a(t)=Zp
p0
dp
f(p0)f(p0)
f(p)2/3
(3.15)
and
DV=3
qzDHD2
M,(3.16)
represents a kind of average of the various distances and the so-called angular diameter is
often defined as
θ=rs
DV
,(3.17)
where
rs
is the co-moving sound horizon at
tdrag
.
θ
is an important variable, e.g. in the
observations of baryon acoustic oscillations (BAO). Since the physics associated with these
oscillations takes place before
tLS
and thus
tdrag < tLS
,
rs
will be independent of the functions
fwe consider, for reasons to be discussed below.
In these formulas the coupling constants
κ
,
λ
and
g
are so far arbitrary, and so are
t0
and
tLS
(or equivalently, via (3.3))
p0
and
pLS
). If we from observations are given
H0
, the
Hubble parameter at present time
t0
, and
zLS
,
9
the redshift at the time of last scattering
tLS
,
we can determine p0and pLS:
f′(p0) = −3H0, f(pLS) = f(p0)(1 + zLS)3.(3.18)
This finally leads to the determination of
t0−tLS
by eq. (3.3). We need more (experimental)
input to determine the coupling constants that enter in
f
(
p
). We will discuss this in the
next section.
Let us now consider the three choices of
f
(
p
)in (3.1), where
f
(
p
)is given by (2.8). The
first choice corresponds to g= 0,λ > 0:
fds(p) = 3
4p2−λ. (3.19)
This is of course just the
f
(
p
)associated with the standard de Sitter Hamiltonian (1.6). We
include this choice in order to compare the late time cosmology of the standard ΛCDM model
9
Of course, what is given by observations is the temperature
T
(
t0
)of the CMB.
T
(
tLS
)can be calculated
by atomic physics and is to a large extent independent of the cosmological model, as is also the statement that
T(tLS)/T (t0) = a(t0)/a(tLS ) = 1 + zLS.
– 7 –

JCAP12(2023)011
with the results from our new cosmological models. The second choice corresponds to
λ
= 0
and g > 0. Thus
fgcdt(p) = 3
4(p+α)q(p−α)2+ 2α2, α =g1/3.(3.20)
This is our model where baby universes of any size
v
can merge with our Universe. The third
choice is motivated by expanding fgcdt(p)in powers of α/p:
fgcdt(p) = 3
4 p2+α2 2α
p+O α2
p2!!! (3.21)
At early times,
v
(
t
)becomes small and according to (3.1)
ρ
(
v
(
t
)) will be large. Thus
H
(
v, p
) = 0
implies that
|p|
is large at early times (i.e. times larger than but close to
tLS
). In addition
we expect that in our Universe
g
is small according to (2.12). Thus we expect that keeping
only the first term in the expansion (3.21) is a good approximation except at quite late times
where the exponential expansion will dominate and where
|p|
is close to
g1/3
, and this was one
of the reasons we in earlier work considered this approximation, which when inserted in (3.4)
led to what we denoted the modified Friedmann equation. We thus define this
f
(
p
)as follows
fmod(p) = 3
4p2+2g
p.(3.22)
The last term in (3.22) is effectively a time dependent cosmological constant. At very late
time, when the matter term
ρ
(
v
)
∝
1
/v
plays no role, the modified Friedmann equation
implies that
p
=
−
(2
g
)
1/3
and the term acts precisely as a cosmological constant, as already
shown in (2.11), and in analogy with the
−λ
term in (3.19). However, for smaller
t
,
|p|
will
increase, as discussed above, and the last term in (3.22) will be less important.
The three functions
fds
,
fmod
and
fmod
are so simple that even the integrals appearing
in (3.3) and (3.15) can be expressed as known analytic functions and in this sense the models
are fully solvable. In an appendix we present some details and point out a curious symmetry
of the fmod-model.
Returning to the expansion (2.6) that defined the inclusive Hamiltonian, it is seen that
the approximation (3.22) corresponds to
ϕ
(
v
) =
ϕ0
+
O
(
v
), i.e. to the absorption of baby
universes of infinitesimal size (which is one reason we introduced the word “baby universe”
for the universes absorbed by our Universe). The absorption of such small baby universes can
only result in a change of the spatial volume if infinitely many are absorbed per unit time
and unit spatial volume.
4 Comparison of the different models
In the three models we introduced above there are two coupling constants. The gravitational
coupling constant
κ
is fixed by local experiments and we will not discuss it any further.
fds
(
p
)
has the cosmological coupling constant
λ
as a parameter, while
fmod
(
p
)and
fgcdt
(
p
)have the
universe-merging coupling constant
g
as parameter.
λ
or
g
are actually determined already
from the values
H0
and
zLS
mentioned above provided we can assume they play no role for
t<tLS
. We know this is true for
λ
in the ΛCDM model and from (2.12), which is of course
not exactly true when we add matter to the model, we nevertheless expect that it is also true
for gin the other models. This implies that
f(p)≈3
4p2for 0 < t < tLS.(4.1)
– 8 –

JCAP12(2023)011
That being the case, we can calculate
tLS
as well as the relation between
tLS
and
pLS
without
reference to the specific models. For the purpose of illustration, assume (incorrectly) that we
can ignore the radiation density ρr(t)all the way down to t≈0. Eq. (3.3) then leads to
tLS =ZpLS
−∞
dp
3
4p2=−4
3
1
pLS
.(4.2)
Eqs. (3.18) determine our coupling constant since the first equation determines
p0
and the
last equation (the generalized Friedmann equation at tLS ) can be written as
f(p0) = 3p2
LS
4 (1 + zLS)3(4.3)
So given
tLS
, and thus
pLS
, we can determine either
λ
from
fds
(
p
), or
g
from
fmod
(
p
)or
fgcdt
(
p
).
The
tLS
used in cosmology is obtained by also including the radiation density. In this
case we obtain an additional term in eq. (3.3) from ∂H/∂v and
tLS =ZpLS
−∞
dp
3
4p2+1
3κρr(v(p)),3
4p2=κρm(v) + κρr(v).(4.4)
This
tLS
is smaller than the one given in (4.2) and leads to a somewhat different
pLS
, which
we should use in (4.3) to determine the coupling constants. We can effectively compensate for
the smaller tLS by shifting the origin of time.
Having determined the coupling constants we can now compare the different models
for given
H0
and
zLS
. While
zLS
is fixed (almost) model independent to 1090, there are
presently two values of
H0
that do not agree within 5
σ
(a fact that is denoted the
H0
tension).
One value is deduced from “local” measurements, using various space candle techniques [
21
].
We denote it
HSC
0
. This value (
HSC
0
= 73
.
04
±
1
.
04 km/s/Mpc) is almost independent of
cosmological models. The other value is deduced from the CMB data created at
tLS
. It is
using cosmological models in a number of ways, among those to extrapolate to present time
t0
.
This value [
22
] (
HCMB
0
= 67
.
4
±
0
.
5km/s/Mpc), which we denote
HCMB
0
, is model dependent
and the value usually referred to is based on the ΛCDM model.
When comparing our two models with the ΛCDM model we will only use the CMB data
to determine
T0
, the temperature of the microwave background radiation at the present time.
We have not attempted to use the wealth of information contained in the CMB temperature
fluctuations since it requires the use of metric perturbations and it is presently not clear
how to incorporate such perturbations in our extended models. We will simply present some
of the variables we listed in the previous section, namely
H
(
z
),
DV
(
z
)and
wf
(
z
), and
t0
,
calculated for the three cosmological models, in two cases, namely for (
HSC
0, zLS
)(case A)
and for (
HCMB
0, zLS
)(case B). For
H
(
z
)and
DV
(
z
)we can also compare with measurements
from which one can extract these observables for relatively low values of z.
In figure 2we have shown
a
(
z
)
H
(
z
) =
H
(
z
)
/
(1 +
z
)in case A and B for the three models.
It is seen that in case A the
fgcdt
and the
fmod
models fit the data better than the
fds
model
(the ΛCDM model), while in case B the opposite is the case. Of course we do not know yet
whether
HSC
0
or
HCMB
0
is the correct
H0
value, but if
HSC
0
turns out to be the correct value
the
fgcdt
and the
fmod
models seems a better choice for the data presented here. Of course
there are many other data to take into account if one wants to make a multi-parameter fit,
but as already remarked most these data should be matched to the cosmological model using
perturbation expansions not yet available for the
fgcdt
and
fmod
models. In table 1we have
– 9 –

JCAP12(2023)011
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
z
55
60
65
70
75
80
aH
Figure 2.
a
(
z
)
H
(
z
) =
H
(
z
)
/
(1 +
z
)shown for the three models for
H0
= 73
.
0(case A). We have
included in red the error bars around this point, corresponding to
z
= 0, but the curves all start at
73.0. The orange curve corresponds to
fds
, the red curve to
fgcdt
and the blue curve to
fmod
. In case
B the three models all start at
H0
= 67
.
4, the value suggested by the Planck Collaboration. The green
curve corresponds to
fds
, the dashed red curve to
fgcdt
and the dashed blue curve to
fmod
. Inserted in
red are data from 0
< z <
3: the first three points from the baryon acoustic oscillation data [
24
], the
next from quasars [25] and the last two data points from Ly-αmeasuments [26,27].
χ2
red(ds) χ2
red(gcdt) χ2
red(mod)
case A 3.5 1.5 1.8
case B 1.2 1.7 5.6
Table 1. The reduced χ2values obtained from figure 2.
χ2
red(ds) χ2
red(gcdt) χ2
red(mod)
case A 4.7 2.1 1.0
case B 1.6 4.1 13.7
Table 2. The reduced χ2values obtained from figure 3.
listed the reduced
χ2
values just calculated from the data and the corresponding error bars
shown in figure 2in Case A and B.
Figure 3shows
DV
(
z
)
/rs
for the
fds
,
fgcdt
and
fmod
models, normalized by
(
DV
(
z
)
/rs
)
CMB
, i.e. the
DV
(
z
)
/rs
for the
fds
model in case B. We see here the same trend as
in figure 2: the
fgcdt
and the
fmod
models fit the data better in case A, while the
fds
-model
fits better in case B. In table 2we list the reduced χ2values obtained from figure 3.
In figure 4we display
wf
(
z
)for the three models. Of course
wfds
(
z
) =
−
1. The two other
models have negative
wf
(
z
)
<−
1. In standard cosmology this is a sign that some unphysical
– 10 –

JCAP12(2023)011
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
z
0.90
0.95
1.00
1.05
DV/rs
DV/rsCMB
Figure 3. The three curves starting at value 1 for
z
= 0 are curves of (
Df
V
(
z
)
/rs
)
/
(
Dfds
V
(
z
)
/rs
)
CMB
for
the models with
f
=
fgcdt
(red, dotted),
fmod
(blue, dotted) and
fds
(green), all for the chosen value
H0
=
HCMB
0
(case B). The last curve is of course 1, except for the error bars reflecting the uncertainty
of
HCMB
0
. The three curves starting at value 0.94 for
z
= 0 are the curves for
f
=
fgcdt
(red),
fmod
(blue) and
fds
(orange), all for
H0
=
HSC
0
(case A), and again normalized by (
Dfds
V
(
z
)
/rs
)
CMB
. The
data points and error bars are obtained from various BAO surveys (see [3] for a detailed table).
degrees of freedom have been added to the system. However, here one cannot conclude that,
since merging with baby universe seems more like having a time dependent cosmological
constant, but without the problem that a time dependent cosmological constant will break
the invariance of the model under time-reparametrization. Allowing a time dependence of the
cosmological “constant”: λ→λ(t), changes wfds:
wfds (z) = −1→w˜
fds =−1−1
3H(t)
˙
λ(t)
λ(t).(4.5)
Thus, if
λ
(
t
)is growing with time, assuming the universe is expanding (i.e.
H
(
t
)
>
0), it
follows that
w˜
fds
(
t
)
<−
1. If we consider the
fmod
model then
λ
(
t
)is replaced by
−
3
g/
2
p
,
and effectively it acts in the same way: for small
t→
0
p
(
t
)
→ −∞
while for
t→ ∞
p
(
t
)
→ −
(2
g
)
1/3
. In fact using (3.13) it follows immediately that
wfmod
(
z
)goes monotonically
from −3/2at z=∞(t= 0), to −1for z=−1(t=∞), as also illustrated in figure 4.
The Planck data, combined with BAO as well as supernovae type Ia data, is consistent
with the ΛCDM model with a value
H0
= 67
.
4
±
0
.
5. The consistency is corroborated by
treating
w
in the dark energy equation of state
p
=
wρ
as a parameter, the value of which is
then determined to be
w
=
−
1
.
03
±
0
.
03 [
22
]. However,
w
is not well determined by the Planck
data alone (
w
=
−
1
.
56
±
0
.
5, ([
22
], table 4)). If one allows for a more general time dependent
w
(
z
)extension of the ΛCDM model, the picture is the same, but even including BAO as well
as supernovae type Ia data the best fit will produce a curve which is both consistent with the
w
(
z
)curve for our gcdt-model as well as consistent with the pure ΛCDM model within error
bars (figure 5, [
23
]). One trend to observe in the analysis presented in [
23
] (see figure 3, 4
– 11 –

JCAP12(2023)011
-0.2 0.2 0.4 0.6 0.8 1.0
z
-1.5
-1.4
-1.3
-1.2
-1.1
-1.0
w
Figure 4.
wf
(
z
)for the three models. The orange curve is
wfds
=
−
1, which is true for any value of
H0
. The red curves are for the
fgcdt
-model, dotted curve for
HSC
0
(case A) and full curve for
HCMB
0
(case B). The blue curves are for the
fmod
-model, dotted curve for
HSC
0
(case A) and full curve for
HCMB
0(case B). For z→ −1(t→ ∞)wf(z)→ −1.
t0(ds) t0(gcdt) t0(mod)
case A 13.3 Gyr 13.5 Gyr 13.9 Gyr
case B 13.8 Gyr 14.0 Gyr 14.4 Gyr
Table 3. The values of t0for the three models in cases A and B.
and 6) is that larger values of
H0
correlate with smaller values of
w
(
z
). If
H0
is in the range
72–74 km/s/Mpc, as is the case for the best fits to our models, one obtains in the Planck
collaboration analysis of the extended ΛCDM models, values of
w
(
z
)that are typically in the
range predicted by our models. In this sense our w(z)is compatible with the Planck w(z).
Finally, in table 3we list the various values of
t0
in the six cases discussed. In case
A
t0
(
ds
) = 13
.
3
Gyr
and
t0
(
gcdt
) = 13
.
5
Gyr
seem uncomfortable short and in case B
t0
(
mod
) = 14
.
4
Gyr
is probably too long. However, had we started out with a
H0
in between
HSC
0
and
HCMB
0
we can obtain values of both
t0
(
mod
)and
t0
(
gcdt
)that are comfortable in
agreement with the age of the universe as determined from the oldest stars. In particular
for the
fgcdt
-model, just looking at the data in figures 2and 3it is clear that if we tried to
determine the best value of
H0
for the model from the data (including now the observed
HSC
0
as a data point), the optimal value of
H0
would precisely be between
HSC
0
and
HCMB
0
. In
table 4we have listed values of
H0
obtained for the three model by minimizing the
χ2
for the
data shown in figure 2, together with the corresponding (minimal) value of the reduced
χ2
,
gt3
0
and
g/
Λ
3/2
, where we for Λhave used the value corresponding to the
fds
-model with the
H0
listed in table 4. The
fgcdt
- and
fmod
-models are doing slightly better than the
fds
-model,
but the error bars in figure 2are too large to obtain any precision determination of
H0
in this
way. The same is even more true if one had used the data in figure 3to determine the “best
value” of
H0
for the three models. In figure 5we have shown the equivalent of figure 3, using
the values of H0from table 4.
While we have some choice for different models for absorption of baby universes, once
H0
is fixed the values of observables like
H
(
z
),
DV
(
z
)and
fm
(
z
)
σ8
(
z
)are fixed for each such
model. If, for some reason we decide that a specific value of
H0
is trustworthy the values
– 12 –

JCAP12(2023)011
fds-model fgcdt-model fmod-model
H071.2 72.2 73.9
χ2
red 2.4 1.2 1.4
t013.5 Gyr 13.6 Gyr 13.8 Gyr
gt3
02.983 1.224
g/Λ3/20.971 0.380
Table 4. The best values of H0for the three models together with the corresponding parameters.
0.0 0.5 1.0 1.5 2.0 2.5 3.0
z
55
60
65
70
75
80
aH
Figure 5. The graphs for
H
(
z
)
/
(1+
z
)for the three models using the values of
H0
from table 4. Orange
graph corresponds to fds-model, blue graph to the fmod -model and red graph to the fgcdt -model.
of these observables can then be viewed as predictions for a given model. It was in this
spirit we presented the data from the observed
H
(
z
)and
DV
(
z
)for case A and B in figure 2
and figure 3and the corresponding tables 1and 2. Another “local” variable is the density
fluctuations of matter for
z <
2. Presently the error bars on these data are too large for being
used in a simple analysis like done above in figure 2and figure 3(see [
3
] for an attempt).
However, this will most likely change dramatically with the new observations to be obtained
by the Euclid satellite.
5 Discussion
Let us start by emphasizing the extreme simplicity of the models considered. It is based on
the Hamiltonian
H=Nv−f(p) + κρm(v), ρm(v)v=ρm(v0)v0,(5.1)
and the corresponding eoms are
f(p) = κρm(v),˙v
v=−f′(p),˙p=f(p).(5.2)
– 13 –

JCAP12(2023)011
Thus, (for suitable
f
), if the universe starts out with a small spatial volume
v
(
t
)for small
t
,
ρm
(
v
(
t
)) is large and the first equation implies that
−p
is large. The second equation then
implies that the expansion is fast, and the last equation that −pincreases, i.e. |p|decreases,
which again implies that the rate of expansion per unit volume
˙v/v
decreases. Eventually,
if
v
(
t
)continues to grow to infinity,
ρm
(
v
)goes to zero. Thus
f
(
p
)goes to zero, i.e.
p
goes
to a constant. The middle equation then implies that
v
(
t
)approaches an pure exponential
expansion. As we have argued, assuming the absorption of baby universes this exponential
expansion is very natural if the chance of such an absorption per unit time is proportional to
the spatial volume. Therefore, there is no need to introduce a cosmological constant (“negative
gravity”) by hand: the gravitational force indeed wants to limit the expansion of the universe,
but this is counteracted by the “bombardment” of our Universe by baby universes.
The model we have suggested is of course quite primitive and unrealistic, but taking two
limits, one where only universes with infinitesimal spatial volumes are absorbed, the
fmod
model, and one where universes of any size can be absorbed and where our Universe and the
other universes are on equal footing, the
fgcdt
model, show that the models are reasonable
insensitive to the detailed distribution of baby universe sizes. One can therefore hope that
they reflect the results one would obtain from more realistic models.
We have presented our model as a “late time” cosmological model and in the present
formulation it has nothing to say about the universe for
t<tLS
. Viewed as such a late time
cosmological model it is interesting that it favors the local measured value
HSC
0
somewhat
compared to the value
HCMB
0
. However, in view of the simplicity of the model, we will not
press this as an important point. Let us rather discuss if this multi-universe scenario has a
chance to answer some of the early-time questions in cosmology.
We have nothing to add about inflation as it is presented in various models. However, the
fact that the universe has expanded from, say, a Planckian size to 10
−5
m in a very short time,
invites the suggestion that this expansion was caused by a collision with a larger universe,
i.e. that it was really our Universe which was absorbed in another “parent” universe. Since
we have presently no detailed description of the absorption process, it is difficult to judge if
such a scenario could take place in a way that would actually solve the problems inflation
was designed to solve, but one interesting aspect of such a scenario is that there is no need
for an inflaton field.
While a continuous absorption of microscopic baby universes probably can be accom-
modated in a non-disruptive way in our Universe, it is less clear what happens if the “baby”
universe is not small, since we have not suggested an actual mechanism for such absorption.
Maybe the least disruptive situation would be one where the absorption happened inside a
black hole. The unknown mechanism of absorption could maybe favor such a scenario when
the sizes of the baby universes are not infinitesimal. Recall that a Reissner-Nordström black
hole actually connects to different universes. We are not seriously suggesting such a black hole
scenario, but we mention it to point out that there is room for a lot of interesting considerations.
Ultimately, any realistic model should be specific about how the absorption occurs.
The present value of the cosmological constant
λ
in the ΛCDM model is according to
some viewpoints embarrassingly small. According to (2.12),
λ
being small when expressed in
Planck units (
κ
= 1) also implies that
g
is small. Historically, before observations pointed to
a small
λ
in the ΛCDM model, many people favored
λ
= 0 as a result of some underlying
mechanism not yet fully understood, e.g. Coleman’s mechanism [
31
]. We might still need
such a mechanism to explain why a
λ
coming from the zero-point fluctuations of quantum
– 14 –

JCAP12(2023)011
fields will not create a large
λ
.
10
However, it might be easier to explain why
λ
is exactly zero
than to explain why it is unnaturally small. If
λ
= 0 can be proven, we could then still have
an exponentially expanding universe caused by baby universe absorption.
Like
λ
,
g
appears in our model just as a coupling constant, reflecting in some way the
“density” of the baby universes “surrounding” our Universe. By comparing our model to
observations
g
has to be small. However, it is a coupling constant in a larger multiverse theory
that we have not yet been able to solve. This leaves the hope that a consistent solution of
this larger theory might determine
g
. We have started the program to unveil this theory [
4
–
7
],
but it is still work in progress.
Acknowledgments
JA thanks the Perimeter Institute for hospitality while this work was completed and the
research was supported in part by the Perimeter Institute for Theoretical Physics. Research
at the Perimeter Institute is supported by the Government of Canada throuh the Department
of Innovation, Science and Economic Development and by the Province of Ontario through
the Ministry of Colleges and Universities.
YW acknowledges the support from JSPS KAKENHI Grant Number JP18K03612.
A Analytic results for the three models
In this appendix we list some of the analytical results for the three models defined by
fds
,
fgcdt
and
fmod
, respectively. As remarked above we have already analytic parametric representations
of all the observables considered, using
κp
as parameter. Here we will provide the explicit
expressions and also provide some of them as analytic functions of time t.
A.1 The fds-model
fds(p)is given (3.19). From (3.3) we can find tas a function of p:
t=Zp
−∞
dp′
fds(p′)=−2
√3λtanh−1 2√λ
√3
1
p!(A.1)
p=−2√λ
√3coth √3λ
2t!.(A.2)
fds(p) = λ
sinh2√3λ
2t,i.ev(t) = κρm0v0
λsinh2 √3λ
2t!(A.3)
The only non-trivial function is DMdefined by eq. (3.15). The integral can be expressed by
hypergeometric functions and one representation is
DM=6
√3λλ
κρm0 1/3
sinh1/3 √3λ
2t′!2F1"1
6,1
2;7
6;−sinh2 √3λ
2t′!#
t0
t
(A.4)
=6
√3λλ
κρm0 1/3λ
fds(p′)1/6
2F11
6,1
2;7
6;−λ
fds(p′)
p0
p
(A.5)
10
There are other viewpoints where the value of
λ
is claimed to be natural (or at least not embarrassingly
small) in a quantum field context when one uses renormalization group arguments [
28
,
29
] (see [
30
] for a review).
– 15 –

JCAP12(2023)011
where
λ
κρm0
=λ
fds(p0)= sinh2 √3λ
2t0!.(A.6)
A.2 The fgcdt-model
fgcdt(p)is given by (3.20). From (3.3) we can find tas a function of p:
t=Zp
−∞
dp′
fgcdt(p′)=4
3√6α
tanh−1
q2
3(−p+ 2α)
p(p−α)2+ 2α2
−tanh−1 r2
3!
(A.7)
Definining tcby
3√6α
4tc= tanh−1 r2
3!= cosh−1√3 = sinh−1√2,(A.8)
we can invert (A.7) to find:
p=−αsinh 3√6α
4(t+tc)+2√2
sinh 3√6α
4(t+tc)−√2(A.9)
fgcdt(p) = 9√3α2
2
cosh 3√6α
4(t+tc)
sinh 3√6α
4(t+tc)−√22(A.10)
Again the only non-trivial function is
DM
defined by eq. (3.15). In this case the integral in
eq. (3.15) is a generalized hypergeometric functions, a so-called Appell-F1function:
DM=1
fgcdt(p0)1/3p′+α
6α21/3
F11
3,1
3,1
3,4
3;x+(p′), x−(p′)
p0
p
(A.11)
where
fgcdt(p0) = κρm0 , x±(p) = 2±i√2
3α(p+α).(A.12)
A.3 The fmod-model
fmod(p)is given by (3.21). From (3.3) we can find tas a function of p:
t=Zp
−∞
dp′
fmod(p′)=4
3√3a arctan 2p−a
√3a+1
2√3log (p−a)2+ap
(p+a)2+π
2!(A.13)
where
a= 21/3α, t →0 for p→ −∞, t → ∞ for p→ −a. (A.14)
Again we can express
DM
, defined by the integrals in eq. (3.15), in terms of hypergeometric
functions. One representation is:
DM=3
52
32/31
α2f(p0)1/3(p′)5/32F1 5
9,2
3;14
9;−(p′)3
2α3!
p0
p
.(A.15)
– 16 –

JCAP12(2023)011
A.4 Duality in the fmod-model
We end by observing that we have a kind of duality in the
fmod
-model. Define a (modified)
Hubble parameter by
h
(
t
) =
˙v/v
= 3
˙a/a
= 3
H
(
t
), where
H
(
t
)is the usual Hubble parameter.
We then have the equations:
3
4 p2+2α3
p!=κρm(v), h(v) = −3
4 2p−2α3
p2!.(A.16)
These equations are invariant under the replacement
p←→ −γ
α
α3
p, κρm←→ α
γh, γ = 21/3.(A.17)
The mapping p→ −γα2/p has some similarity with a T-duality map in string theory.
References
[1] J. Ambjørn and Y. Watabiki, A modified Friedmann equation,Mod. Phys. Lett. A 32 (2017)
1750224 [arXiv:1709.06497] [INSPIRE].
[2]
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