A Peek Outside Our Universe

Abstract

According to general relativity (GR), a universe with a cosmological constant Λ, like ours, is trapped inside an event horizon, r<3/Λ. What is outside? We show, using Israel (1967) junction conditions, that there could be a different universe outside. Our universe looks like a black hole for an outside observer. Outgoing radial null geodesics cannot escape our universe, but incoming photons can enter and leave an imprint on our CMB sky. We present a picture of such a fossil record from the analysis of CMB maps that agrees with the black hole universe predictions but challenges our understanding of the origin of the primordial universe.

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A peek outside our Universe
Enrique Gaztanaga, Pablo Fosalba
To cite this version:
Enrique Gaztanaga, Pablo Fosalba. A peek outside our Universe. Symmetry, MDPI, 2022, 14 (2),
pp.285. �hal-03196754v2�

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Citation: Gaztanaga, E.; Fosalba, P. A
Peek Outside Our Universe. Symmetry
2022,14, 285. https://
doi.org/
Academic Editor: Maxim Y. Khlopov
and Ignatios Antoniadis
Received: 23 December 2021
Accepted: 27 January 2022
Published: 31 January 2022
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Article
A Peek Outside Our Universe
Enrique Gaztanaga * 1,2 and Pablo Fosalba 1,2
1Institute of Space Sciences (ICE, CSIC), 08193 Barcelona, Spain
2Institut d Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain
*Correspondence: gaztanaga@darkcosmos.com
Abstract: According to general relativity (GR), a universe with a cosmological constant
Λ
, like ours,
is trapped inside an event horizon,
r<√3/Λ
. What is outside? We show, using Israel (1967) junction
conditions, that there could be a different universe outside. Our universe looks like a black hole
for an outside observer. Outgoing radial null geodesics cannot escape our universe, but incoming
photons can enter and leave an imprint on our CMB sky. We present a picture of such a fossil record
from the analysis of CMB maps that agrees with the black hole universe predictions but challenges
our understanding of the origin of the primordial universe.
Keywords: cosmology; dark energy; general relativity; black holes
1. Introduction
An event horizon (EH)
r∗
, or trapped surface, for a given observer can be defined
as the distance beyond which this observer will never see:
r<r∗
. The most famous EH
is, of course, that of a black hole (BH) of mass M. Dimensional analysis tells us that a
relativistic (
c
) gravitational (
G
) system of mass
M
has an associated EH:
r∗≃GM/c2
. The
same dimensional analysis indicates that a cosmological constant
Λ
is associated with a
relativistic EH,
r∗≃
1
/√Λc2
, which, in principle, is independent of
G
. We will show first
how these EHs appear as solutions to Einstein’s field equations and next how the different
EHs are related in our universe expansion.
The most general form for a metric with spherical symmetry in proper coordinates
xµ= (t,r,δ,θ)with c=1 can be written as [1]:
ds2=gµν dxµdxν=−A(t,r)dt2+B(t,r)dr2+r2dΩ2(1)
where we have introduced:
dΩ2=cos2δdθ2+dδ2
. Einstein’s field equations for that metric
in empty space ρ=p=Λ=0 result in the Schwarzschild (SW) metric:
ds2=−[1−2GM
r]dt2+dr2
1−2GM
r
+r2dΩ2(2)
where Mcan be interpreted as a singular point mass at r=0. As it is well known, the EH
at
r∗=
2
GM
prevents us from seeing such a naked singularity [
2
]. Outgoing null radial
geodesics cannot leave the interior of
r∗
, whereas incoming null radial geodesics can cross
inside r∗in proper time, even when, for an observer outside, this takes t=∞in her time.
1.1. deSitter Metric
The solution for Equation (1) with
ρ=p=M=
0 but
Λ̸=
0 is the deSitter (dS)
metric:
ds2=−[1−r2H2
Λ]dt2+dr2
1−r2H2
Λ
+r2dΩ2(3)
which is also static and has an EH, or trapped surface, at
r∗=
1
/HΛ
, where
H2
Λ≡
8
πGρΛ/
3
and
ρΛ=Λ/(
8
πG) + ρvac
. We include a constant
ρvac =V=V(ϕ) = −pvac
to account for
Symmetry 2022,14, 285. https://doi.org/10.3390/sym14020285 https://www.mdpi.com/journal/symmetry

Symmetry 2022,14, 285 2 of 6
vacuum energy (or the potential of a trapped scalar field), which is physically degenerate
with
Λ
. The inside of
r∗=
1
/HΛ
is causally disconnected and corresponds to an EH: radial
null events (
ds2=
0) connecting
(
0,
r0)
with
(t
,
r)
take
t=∞
to reach
r=r∗
from any
point inside.
1.2. The FLRW Metric and Λ
The Friedmann–Lemaitre–Robertson–Walker (FLRW) flat (
k=
0) metric in spherical
comoving coordinates
(τ
,
χ
,
δ
,
θ)
, corresponds to a homogeneous and isotropic space-time:
ds2=−dτ2+a(τ)2hdχ2+χ2dΩ2i(4)
where the scale factor
a(τ)
describes the expansion/contraction as a function of time. For a
comoving observer and a perfect fluid, the field equations reduce to:
3H2≡3˙
a
a2
=8πG(ρma−3+ρRa−4+ρΛ)(5)
where
ρm
and
ρR
are the matter and radiation density today (
a=
1) and
ρΛ=−pΛ
is
the effective cosmological constant density introduced in the dS metric. Given
ρ
and
p
at
some time, we can find
a=a(τ)
and determine the metric in Equation (4). Observations
show that the expansion rate today is dominated by
ρΛ
. This indicates that the FLRW
metric describes the interior of a trapped surface of size
r∗=
1
/HΛ
, like the dS metric. In
fact, both metrics are equivalent in that regime [
3
,
4
]. They also reproduce the steady-state
cosmological principle [5].
Interpreting
Λ
as a boundary term to the GR equations and imposing the so-called
zero action principle,
r∗=
1
/HΛ
can be given in terms of ordinary contributions to the
energy density, ρΛ=<ρm/2 +ρR>[6,7].
1.3. SW–FLRW Perturbation
The Schwarzschild (SW) metric is commonly used to describe the outside of BHs or
stars and it should be understood as a perturbation inside a larger background. An example
of this is the Schwarzschild–deSitter (SW–dS) metric:
ds2=−[1−2GM
r−r2H2
Λ]dt2+dr2
1−2GM
r−r2H2
Λ
+r2dΩ2(6)
which corresponds to an exact solution with both non-zero
Λ
and
M
. More generally,
the FLRW metric is the background to the SW solution [
8
], e.g., it replaces
r=a(τ)χ
in
Equation (2). For large
r
, we recover the dS metric, as in Equation (6) above, which is
equivalent to the FLRW metric dominated by
Λ
[
3
]. Close to the BH at
r∗<r<
1
/HΛ
, we
recover the SW metric.
2. Outside Our FLRW Universe
In proper coordinates
r=aχ
, the FLRW metric with
H=H(τ)
is also trapped inside
the same EH as the dS metric,
r∗=
1
/HΛ
, because
H(τ)>HΛ
. We can see this by
considering outgoing radial null geodesics in the FLRW metric (Equation (4)):
rout =a(τ)Z∞
τ
dτ′
a(τ′)=aZ∞
a
dln a′
a′H(a′)<1
HΛ
=r∗(7)
which shows that signals cannot escape from the inside to the outside of the EH. However,
incoming radial null geodesics a(τ)Rτ
0dτ
a(τ)could be larger than r∗if we look back in time
long enough. This shows that observers living in the interior are trapped inside the EH,
but they can, in principle, observe what happened outside.

Symmetry 2022,14, 285 3 of 6
What is outside
r∗=
1
/HΛ
in the FLRW metric? The FLRW comoving coordinates
(τ
,
χ)
can be matched to the SW proper coordinates
(t
,
r)
. The joint metric is what [
4
,
6
] call
a BHU solution. The particular case where the inside is dS (and the outside SW) is called
BH.fv (where fv stands for false vacuum). The BHU metric is also a solution to Einstein’s
field equations. To prove this, we simply need to show that the junction follows the Israel
matching conditions [
9
]. The two metrics can be matched on a timelike hypersurface
Σ
of
constant χ:
ds2
Σ=habdyadyb=−dτ2+a2(τ)χ2
§dΩ2(8)
and the extrinsic curvature
K
at
Σ
is the same in both sides. The matching conditions
h−=h+and K+=K−reduce to [4]
r=R(τ) = a(τ)χ§;˙
R2=R2H2=R∗
R(9)
where
R∗≡
2
GM
. Staring from small
a
, as we increase
τ
, both
R
and
˙
R=HR
grow until
we reach
HR =c=
1, which corresponds to the event horizon
R∗=
2
GM =
1
/HΛ
. It
takes
t=∞
in SW time to asymptotically reach
R∗
. This proves that the joint BHU metric
is also a solution to Einstein’s field equations with no surface terms in the junction. This is
equivalent to stating that the
Λ
term corresponds to a trapped surface
R∗=
1
/HΛ
in the
FLRW metric which matches the EH of a BH. Generalization to
k̸=
1 is straightforward
(see §12.5.1 in [1,4,10]).
Recall that the SW metric is a perturbation of a larger FLRW metric, i.e, a BH-like
metric embedded in a background described by the FLRW metric. This means that outside
r∗=
1
/HΛ
, we have another FLRW metric, like in a Matryoshka doll. From the outside, the
inner FLRW metric looks like a BH. There could be many other BHUs inside and outside
r∗=1/HΛ, so the structure could be better described by a fractal.
2.1. Causal Structure
In the FLRW universe, the Hubble horizon
rH
is defined as
rH=c/H
. Scales larger
than
rH
cannot evolve because the time a perturbation takes to travel that distance is larger
than the expansion time. This means that
r>rH
scales are "frozen out" (structure cannot
evolve) and are causally disconnected from the rest. Thus,
c/H
represents a dynamical
causal horizon that is evolving (blue line in Figure 1).
We can sketch the evolution of our universe in Figure 1. A primordial field
ψ
settles or
fluctuates into a false (or slow-rolling) vacuum which will create a BH.fv with a junction
Σ
in
Equation (8), where the causal boundary is fixed in comoving coordinates and corresponds
to the particle horizon during inflation,
χ§=c/(aiHi)
, or the Hubble horizon when
inflation begins. The size
R=a(τ)χ§
of this vacuum grows and asymptotically tends to
R∗=c/H
following Equation (9) with
H=Hi
. The inside of this BH will be expanding
exponentially,
a=eτHi
, while the Hubble horizon is fixed at 1
/Hi
. When this inflation ends
[
11
–
14
], vacuum energy excess converts into matter and radiation (reheating). This results
in BHU, where the infinitesimal Hubble horizon starts to grow following the standard Big
Bang evolution. The observable universe (or particle horizon) after inflation, χO, is:
χO=χO(a) = Za
ae
dln a′
a′H(a′)=χO(1)−¯
χ(a), (10)
where
ae
is the scale factor when inflation ends. For
ΩΛ≃
0.7, the particle horizon today is
χO(
1
)≃
3.26
c/H0
and
¯
χ(a) = R1
adln a/(aH)
is the radial lookback time, which, for a flat
universe, agrees with the comoving angular diameter distance,
dA=¯
χ
. The observable
universe becomes larger than
R∗
when
a>
1, as shown in Figure 1(compare dotted and
dashed lines). This shows that observers like us, living in the interior of the BH universe, are
trapped inside
R∗
but can nevertheless observe what happened outside. We can estimate
χ§
from
ρΛ=<ρm/
2
+ρR>
(see §I.A), where the average is in the lightcone inside
χ§
.
For
ΩΛ≃
0.7, [
6
] found that
χ§≃
3.34
c/H0
, which is close to
χO
today. Imagine that
ΩΛ

Symmetry 2022,14, 285 4 of 6
is caused by some dark energy and has nothing to do with
χ§
. We still have that
χ§≲χO
,
because otherwise
χ§
would have crossed
RH =
1 early on, resulting in a smaller
χO
than
measured (see Figure 1).
Figure 1. Proper coordinate:
R=a(τ)χ
in units of
c/H0
as a function of cosmic time
a
. The Hubble
horizon
c/H
(blue continuous line) is compared to the observable universe
aχO
after inflation (dashed
line) and the primordial causal boundary
χ§=c/(aiHi)
(dot-dashed red line). Larger scales (green
shading) are causally disconnected and smaller scales (yellow shading) are dynamically frozen. After
inflation,
c/H
grows again. At
a≃
1, the Hubble horizon reaches our event horizon
R∗=c/HΛ
. At
the CMB last scattering, we can observe both frozen and causally disconnected perturbations.
Thus, at the time of the CMB last scattering (when
dA≃χO
),
χ§
corresponds to an
angle
θ=χ§/dA≲
1 rad
≃
60 deg. Therefore, we can actually observe scales larger than
χ§
, scales that are not causally connected! This could be related to the so-called CMB
anomalies (i.e, apparent deviations with respect to simple predictions from
Λ
CDM, see
[
15
,
16
] and references therein) or the parameter tensions in measurements from vastly
different cosmic scales or times [17–20]).
2.2. A Peek Outside
A recent analysis of the Planck temperature anisotropy data, Ref. [
21
], shows that the
distribution of best-fit dark energy density
ΩΛ
exhibits three distinct regions across the
CMB sky (marked by the three large grey circles in Figure 2). These regions have radii
ranging from 40 to 70 degrees. The sizes of these structures are in agreement with the
scale of the causal boundary
χ§
for
Λ
dominated universes. As shown in [
21
], the size of
each of these regions is correlated with the mean value of
ΩΛ
over that portion of the sky,
in good agreement with the BHU prediction. The same large-scale anisotropic patterns
are observed for the distribution of other basic
Λ
CDM cosmological parameters. This
represents a very significant break-down of the main hypothesis of the Big Bang model:
the assumption that the universe is isotropic on a large scale. The observed anisotropy has
a tiny probability (of order 10
−9
) of being a Gaussian statistical fluctuation of an otherwise
isotropic universe [21].
In summary, Figure 2shows that: (a) regions with a given value of
ΩΛ
have a cor-
responding angular size
θ
that agrees with the BHU prediction (see Figure 31 in [
21
]),
(b) causally disconnected regions of the sky (larger than
χ§
) fit the same physical model of
acoustic oscillations very well, and (c) the background is similar but with parameters that
are significantly different across disconnected regions. This suggests that the underlying
physical mechanism sourcing the observed anisotropy encompasses scales beyond our
causal universe. This is in apparent tension with simple models of inflation (as sources of
perturbation for the largest scales) and opens the door to revisiting our basic understanding
of the origin of the primordial universe [22].

Symmetry 2022,14, 285 5 of 6
Figure 2. Map of the best-fit values of the dark energy density
ΩΛ
across the celestial sphere,
estimated from partial sky (discs) measurements of the Planck CMB maps. The large grey circles
delimit areas across the sky with significantly different values of ΩΛ.
Acknowledgments: This work has been supported by Spanish MINECO grants PGC2018-102021
and ESP2017-89838-C3-1-R and EU grants LACEGAL 734374 and EWC 776247 with ERDF funds and
grant 2017-SGR-885 of the Generalitat de Catalunya.
Conflicts of Interest: The authors declare no conflict of interest.
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