Fulvio Melia

*

# Initial Energy of a Spatially Flat Universe – a Hint of its Possible Origin

Fulvio Melia  Departments of Physics and Astronomy, and the Applied Math
Program, The University of Arizona, Tucson, Arizona, U.S.A.
<EMAIL_ADDRESS>
(27 October 2021)

###### Abstract

The evidence for a Big Bang origin of the Universe is truly compelling, though
its cause remains a complete mystery. As the cosmic spacetime is revealed to
us with ever improving detail, however, we are beginning to refine the range
of its possible initial conditions—at least within the framework of current
physical theories. The Universe, it seems, is spatially flat, and here we
discuss in clear, straightforward terms why this trait implies a cosmos with
zero kinetic plus gravitational energy, though apparently not zero total
energy. Such an outcome has far reaching consequences because of the
possibility that the Universe may have begun its existence as a quantum
fluctuation. Was this from ‘nothing,’ or perhaps a pre-existing vacuum? A non-
zero total energy would seemingly preclude the former scenario, but not
necessarily the latter, though this would then raise the question of how a
fluctuation with non-zero energy could have lived long enough, or
classicalized, for us to see it 13.5 billion years later. The high-precision
measurement of the Universe’s spatial curvature may thus constitute the first
tangible piece of evidence impacting a possible quantum beginning.

###### keywords:

Big Bang; cosmological parameters; cosmology: observations; cosmology: early
Universe; cosmology: theory; gravitation

††articletype: Comment

##  1 Introduction

Over the past half century, many attempts have been made at ‘calculating’ the
energy of the Universe using a variety of approaches with a diverse set of
underlying assumptions and definitions. With very few exceptions, these
studies have inevitably concluded that the Universe has zero total energy.
Given that the definition of what constitutes a viable contribution to this
energy has not been uniformly consistent, however, one may question whether
all of these treatments can be correct. In fact, with such ambiguity, how can
one be certain that any of them are physically meaningful? There is also the
suspicion that ‘zero energy’ is a fundamentally preferred answer because such
an initial condition lends considerable support to the notion that the
Universe may have started as a quantum event from ‘nothing’ or, at the very
least, from a pre-existing vacuum. These two scenarios are quite different, of
course, though the condition of zero energy seems to be a favoured requirement
in both cases.

The question of how much energy the Universe possesses first started emerging
via a topological argument due to P. Bergmann quoted by [Tryon (1973)], and
then elaborated upon by several other authors, including [Albrow (1973)],
[Guth (1981)], [Rosen (1994)], [Cooperstock (1994)], and several more since
then. All along, an important (though not exclusive) motivation for addressing
this question has been the proposal first made by [Tryon (1973)] that our
Universe is a fluctuation of the vacuum, to be understood in the context of
quantum field theory. Tryon conjectured that the appearance of the Universe
‘from nowhere’ would require it to have a net zero value for all its conserved
quantities, the most important of which is energy. He argued that the enormous
amount of mass energy in the cosmos is canceled by its (negative)
gravitational potential energy, and proceeded to show that these two
quantities, contained within the Hubble radius $R_{\rm h}\equiv c/H$, where
$H$ is the Hubble parameter, differ from each other only by a factor of order
unity.

Tryon made no mention of the positive energy due to the Universe’s expansion,
however, and envisaged the initial fluctuation as having taken place in the
vacuum of a larger space in which the cosmos is embedded. As such, the total
fluctuation energy $\Delta E$ need not have been zero, though one would then
have to explain, via the Heisenberg uncertainty principle, $\Delta E\Delta
t\sim\hbar/2$, how the Universe could have lived long enough for us to see it
$\Delta t\sim 13.5$ billion years later. Tryon’s paper was quite speculative,
but it was the seminal work that introduced a quantum origin as a possible
explanation for the Big Bang. He even went so far as to point out that a pre-
existing vacuum should be unstable against large-scale fluctuations in the
presence of a long-range, negative-energy, universal interaction, such as
gravity.

To illustrate how broadly this discourse has meandered over the years, it has
even been suggested that a pre-existing vacuum would have permitted an
enormous range of fluctuations (Šekrst, 2020), but intelligent beings would
see only a Universe whose outcome is suitable for them to exist. This
philosophically interesting argument, known as the anthropic principle (Barrow
Tipler, 1988), is quite forgiving because only a Universe capable of
supporting life would be seen, no matter how statistically improbable it would
be compared to other possibilities. But notice that implicit in this argument
is the requirement that there be a distribution of possible quantum
fluctuations, among which the one that produced our highly improbable Universe
was just a representative of many (perhaps an infinite number of) others.

This was also an issue considered in some detail by Ellis (2006), who argued
that a choice between various contingent initial conditions somehow occurred
at the Big Bang, but it is not clear how that selection came about, at least
not in the absence of an anthropic principle. The questions one may ask in
this context seem endless, even beyond scientific exploration. For example,
why did the Universe’s expansion as a bubble from that pre-existing vacuum
start when it did, rather than at some previous time in a presumably pre-
existent eternity? Attempts at avoiding such difficult or unanswerable queries
often avoid a beginning by subscribing to cyclic phases, as in Tolman’s series
of expansion and collapse cycles (Tolman, 1931), or Linde’s eternal chaotic
inflation (Linde, 2007) and Khoury et al.’s ekpyrotic universe (Khoury .,
2001).

Several alternative proposals have also been made for initiation events in the
absence of space and time, though the distinction between such models, in
which literally ‘nothing’ existed prior to the Big Bang and those with a pre-
existing vacuum, is sometimes blurred. The obvious corollary to the former is
that both space and time would have been created at the Big Bang along with
the Universe itself. This class of cosmological origins includes the ‘Universe
from Nothing’ concept promoted by several workers, such as Krauss (2012),
Vilenkin (1982), Zeldovich Starobinskij (1984) and, more recently, He .
(2014). But this approach tends to face greater scrutiny than the pre-existing
vacuum idea, for the simple reason that an ‘ontic seed’ must have been
necessary to actually cause the fluctuation, as has been argued by Isham
(1994). In addition, this class of models is distinguished from the pre-
existing vacuum idea by the absolute requirement of zero energy throughout the
Universe’s history.

For such reasons, the ‘Universe from Nothing’ proposals certainly receive more
vigorous criticism than their pre-existing vacuum counterparts. For example,
Kohli (2016) describes several flaws in the fundamental basis of these models,
amplifying the point made earlier by Isham. On the one hand, he argues, these
models assume that the Universe originated from nothing, yet they require all
the complex machinery of variational principles, differential and pseudo-
Riemannian geometry, topology, general relativity and quantum field theory,
without ever addressing the question of where these come from. Indeed, even
the theory used to describe the quantum event itself often requires an
underlying superspace, which is absent by default.

Having said this, our goal in this paper is not so much to argue for or
against either class of models, but merely to motivate our exploration of what
the recent cosmological observations are telling us about the most critical
ingredient in all of these scenarios, i.e., the total energy ($E$) of the
Universe. As we saw above, the first attempts (by Bergmann and Tryon) were
rather simple, and probably unrealistic. For example, Bergmann’s closed
universe would not be consistent with the spatial flatness we are measuring
today. And, as noted earlier, Tryon did not actually demonstrate that $E$ is
exactly zero, partly because he did not include all possible contributions to
it. But these were only the first forays; there have been many others since
then.

Nevertheless, attempts at calculating the energy of the Universe prior to the
2000’s were largely based on the use of pseudotensors, rendering those
approaches highly questionable (Faraoni Cooperstock, 2003). Pseudotensors may
be useful tools in studies of bounded systems that are asymptotically
Minkowskian, but the Universe is not bounded for Friedmann-Lemaître-Robertson-
Walker (FLRW) metrics with a spatial curvature constant $k\leq 0$ (see Eq. 1
below). A more reasonable proposal for calculating $E$ with a full general
relativistic treatment was introduced by Faraoni Cooperstock (2003), who
showed that $E$ is constant for $k\leq 0$. They then argued that these
universes are asymptotically Minkowskian and must therefore have a vanishing
energy, concluding that $E=0$ from the beginning.

But it is not clear that this argument works for an FLRW spacetime with
$k\not=0$, since it is not asymptotically Minkowskian when expressed in terms
of comoving coordinates, though one can demonstrate that it is equivalent to
Minkowski space with an appropriate gauge transformation (Melia, 2012). Thus,
one may conclude for $k\not=0$ that $E$ is constant, but not that it is also
zero. For example, Kohli (2016) points out that any spatially homogeneous and
non-static universe, i.e., one that does not contain a global timelike Killing
vector, is necessarily not asymptotically flat. On the other hand, their
result for $k=0$ appears to agree with one of our conclusions in this paper,
and we shall return to this in § IV below. It should be pointed out, though,
that Faraoni and Cooperstock did not include all possible contributions to
$E$. They based their approach on the use of a Lagrangian, from which they
derived the Hamiltonian, so they certainly considered both the ‘kinetic’ and
‘potential’ energy components, but did not include the cosmic fluid’s energy
density $\rho$ itself in the overall energy budget.

In this paper, we shall take a different—much more pedagogical—approach to
interpret the value of $E$, based on the observational determination that our
Universe is spatially flat (i.e., $k=0$). We shall focus our attention on the
most reliable ‘energy conservation’ equation we have for the FLRW spacetime,
i.e., the Friedmann equation (see Eq. 2 below). We shall interpret its meaning
using a Newtonian approach, though strongly motivated by the Birkhoff-Jebsen
theorem (Birkhoff Langer, 1923) and its corollary. And we shall clearly
distinguish between the different forms of energy contributing to $E$.

##  2 The Cosmic Spacetime

Theoretical cosmology is founded on the Cosmological Principle (CP),
comprising two essential symmetries: homogeneity and isotropy, at least on
spatial scales (Yadav ., 2010) larger than $\sim 300$ Mpc. An important
consequence of this principle is the highly simplified form of the (FLRW)
metric used to describe the cosmic spacetime (Melia, 2020), normally expressed
as

$ds^{2}=c^{2}dt^{2}-a^{2}(t)[dr^{2}(1-kr^{2})^{-1}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})]\;.$ (1)

This form is written in terms of comoving coordinates, in which $t$ is the
proper (sometimes called ‘cosmic’) time, $r$ is the radius, $\theta$ is the
poloidal angle and $\phi$ is the azimuth. Unless an object in the Universe
exhibits so-called ‘peculiar’ motion, its comoving coordinates remain fixed.
Proper motion and the universal expansion itself are instead given in terms of
the proper radius, $R=a(t)r$, in which the expansion factor $a(t)$ is solely a
function of time, not position. The most important element in this equation
for this paper is the spatial curvature constant $k$, whose value is $+1$ for
a closed universe, $0$ for a flat, open universe, and $-1$ for an open
universe. In reality, $k$ could be any real number, though one typically
renormalizes the radius $r$ in any given cosmological model in order to reduce
it to one of these three distinct integer values.

If one accepts the FLRW metric as the basis of our cosmology, most of the work
in building a framework for the origin and evolution of the Universe revolves
around the behavior of $a(t)$. Each of the models we have today assumes some
combination of matter and energy, predicting its own distinct time dependence
of the expansion factor. And in the majority of cases, the underlying
dynamical equations for $a(t)$ are obtained by introducing the metric in
Equation (1) into Einstein’s field equations of general relativity (Melia,
2020). The one most relevant to us here is the Friedmann equation,

$H^{2}\equiv\left({\dot{a}\over a}\right)^{2}={8\pi G\over
3c^{2}}\rho-{kc^{2}\over a^{2}}\;,$ (2)

in which an overdot denotes a derivative with respect to $t$, $\rho$
represents the total energy density in the cosmos, and $H\equiv\dot{a}/a$ is
the Hubble parameter. In preparation for our discussion in § IV, we emphasize
here that the Friedmann equation is exact. No approximations were made in
deriving it directly from the full expression of Einstein’s equations in
general relativity.

##  3 The Flatness Problem

Over the past several decades, precision measurements (Spergel ., 2003; Planck
Collaboration ., 2016) of the cosmic microwave background (CMB) have shown
that the Universe is very nearly flat, perhaps completely flat, with an
inferred spatial curvature constant $k\approx 0$. Therefore, the density
$\rho$ appears to be at (or very near) the so-called ‘critical’ density

$\rho_{c}\equiv{3c^{2}H^{2}\over 8\pi G}\;,$ (3)

obtained by setting $k=0$ in Equation (2).

A density close to $\rho_{c}$ can be difficult to explain, however, unless $k$
has in fact always been exactly zero. To see why, let us define

$\Omega\equiv\rho/\rho_{c}\;,$ (4)

and re-write Equation (2) as follows:

$1=\Omega(t)-{kc^{2}\over{\dot{a}^{2}}}\;.$ (5)

Using subscript ‘0’ to denote quantities pertaining to the present time
$t_{0}$, we therefore find that

$\Omega(t)-1=\left({\dot{a}_{0}\over\dot{a}}\right)^{2}\left(\Omega_{0}-1\right)\;.$
(6)

This equation may not look very peculiar, but let’s focus on the
multiplicative factor $(\dot{a}_{0}/\dot{a})^{2}$ on the right-hand side.
Today, we’ve all become accustomed to the idea that the Universe may be
accelerating, but this was not the situation at early times; it was actually
decelerating at an enormous rate. Let’s say for simplicity that $\rho$ was
dominated by radiation. This would certainly have been the case during the
first few hundred thousand years, though not afterwards. One may then put
$\rho\propto a^{-4}$ in Equation (2), since the energy density of a radiation
field scales inversely with the spatial volume ($\propto a^{3}$), and
inversely with the wavelength ($\propto a$), due to redshifting as the
Universe expands. One can therefore easily see from the solution to Equation
(2) that $a(t)\propto t^{1/2}$, as long as $k$ is small. (Incidentally, had we
assumed $\rho$ was dominated by matter instead, we would have found that
$a(t)\propto t^{2/3}$, but such differences have no impact on this argument.)
The prefactor in Equation (6) would thus be $({\dot{a}_{0}/\dot{a}})^{2}\sim
t/t_{0}$. So comparing the quantity $\Omega(t)-1$ at the GUT (Grand
Unification Theory) time ($t_{\rm GUT}\sim 10^{-35}$ seconds; see below) with
its counterpart 13.5 billion years later, we see that it must have been $\sim
10^{48}$ times smaller compared to its value today. Any indication that
$\Omega_{0}\approx 1$ today therefore suggests that $\Omega(t)$ must have been
fine tuned to nearly exactly $1$ as $t\rightarrow 0$.

This fine-tuning is the origin of the so-called ‘flatness’ problem in
cosmology, in the sense that it seems to require yet one more special initial
condition without any physical explanation (Lightman, 1990; Hawking, 1974).
But this clearly stems from the view that the Universe could have been born
with any value of $k$, so an initial condition $\Omega(t)=1$ with $k\not=0$
would appear to be highly unlikely. To address this issue, several different
approaches have been taken to explain why the Universe appears to be flat (or
nearly flat) today, in spite of it possibly having begun with an arbitrary
spatial curvature.

One possible solution is to again invoke the anthropic principle (Collins
Hawking, 1973), arguing that only a Universe with the correct density to form
galaxies and stars would give rise to intelligent observers who would then
inquire about the initial conditions. It should then not be surprising that we
do not live in other types of universes with unsuitable values of $\Omega(t)$
at the beginning.

But the explanation that currently enjoys the most widespread acceptance (if
not outright belief) is inflation, a period of exponential growth in the
Universe only $\sim 10^{-35}$ seconds after the Big Bang, possibly associated
with the separation of the strong and electroweak forces at the aforementioned
GUT scale. In his paper, Guth (1981) proposed that such an early phase of very
rapid expansion would solve several puzzles in cosmology, including the
flatness problem. A requirement of inflation is that the density $\rho$ remain
constant for a brief, though critical, period, and it is not difficult to see
from Equation (2) that, under such circumstances, $a(t)$ would grow
exponentially (again ignoring any small value of $k$). Thus, any value of
$\Omega(t)-1$ at the beginning would rapidly shrink to near zero by the end of
inflation, due to the enormously large factor $\dot{a}\sim e^{Ht}$ in the
r.h.s. denominator of Equation (6). And the Universe would emerge back into
the standard hot Big Bang configuration to resume its expansion resulting in
$\Omega_{0}\sim 1$ today. In other words, inflation could in principle wash
out any initial condition associated with $\Omega(t)$ resulting from an
arbitrary value of $k$.

Inflationary theory is not universally accepted, however, and recent
observations are starting to raise doubt about whether it can actually explain
the various horizon problems plaguing the standard model (Melia, 2018), and
the fluctuation spectrum seen in the CMB (Liu Melia, 2020). There are still
too many gaps in the theory, and the continued refinement of cosmological
measurements may eventually disprove it. For example, even after four decades
of development, inflationary cosmology still lacks specificity regarding the
field that is driving it (Bird ., 2008). The various versions of the theory
spawned by this ambiguity contain parameters and initial conditions that
themselves require fine-tuning (Ijjas ., 2014), analogously to how ‘unnatural’
the initial density would be in the absence of inflation (see Eq. 6).

There are therefore good reasons to explore other explanations for the
flatness of the Universe, even more so today because the ever improving
observations are telling us that $k$ is not only close to zero, but is
probably exactly zero. So the ‘flatness’ problem may be moot. It may not be a
problem at all, if it turns out that there is a physical reason why the
Universe had to have perfect spatial flatness as one of its initial
conditions.

##  4 Cosmic Energy

Today, we have a better understanding of what $k$ stands for. Spatial
curvature is not merely a geometric property of the spacetime—it results from
a well defined physical attribute, which can be understood with a more
thorough inspection of the Friedmann Equation (2). Though this expression
derives from Einstein’s field equations, it is not difficult to find a simple
heuristic argument for its physical meaning, which includes an explanation for
the spatial curvature constant $k$. This is most easily done by invoking the
corollary to the Birkhoff-Jebsen theorem (Birkhoff Langer, 1923), which allows
us to understand why the influence of any isotropic, external source of
gravity completely cancels within a spherical cavity. This theorem is a
relativistic generalization of Newton’s theory—that the gravitational field
outside a spherically symmetric body is indistinguishable from that of the
same mass concentrated at its center. It was first pointed out by Weinberg
(1972) that, due to this symmetry, a limited use of Newtonian gravity is
permitted for some cosmological applications, such as the one we are concerned
with here.

A consequence of isotropy is that every observer in the cosmos experiences net
zero acceleration due to the surrounding mass. But in fact the relative
acceleration between any two points is not zero; it depends on the mass/energy
content between them. This is the reason why the corollary to the Birkhoff-
Jebsen theorem is so germane to our discussion, because it describes the field
inside an empty spherical cavity at the center of an isotropic distribution,
which may be used to calculate the relative acceleration locally. The metric
inside such a cavity is Minkowskian, a situation not unlike we find in
electromagnetism, where we would calculate the electric field inside a
spherical cavity embedded within an otherwise uniform charge distribution to
be zero. So even in the classical limit, it is safe to argue that the medium
exterior to a sphere can be represented as a sequence of shells with an ever
increasing radius, each of which produces zero effect within the cavity.

Figure 1: The Birkhoff-Jebsen theorem (Birkhoff Langer, 1923) and its
corollary tell us that in an isotropic Universe, the mass-energy everywhere
outside (gray) of a spherical region (blue) of radius $r_{s}$ (i.e., at all
$r>r_{s}$) has no gravitational influence at $r\leq r_{s}$. Thus, the
gravitational acceleration of any point on the shell at $r_{s}$ relative to an
observer at the origin depends solely on the mass-energy contained within this
(blue) sphere.

Let us imagine ‘carving’ a sphere of radius $R_{\rm s}(t)=a(t)r_{\rm s}$ out
of an otherwise homogeneous, isotropic universal medium (fig. 1). The rest of
the Universe has zero gravitational influence inside this cavity. We then fill
the sphere with density $\rho$ all the way up to an infinitesimal distance
away from $R_{\rm s}(t)$. According to the Cosmological Principle, this
density needs to be a function solely of $t$. In addition, the FLRW metric
tells us that every pair of points within and outside the sphere recede from
each other at a rate proportional to the expansion factor $a(t)$, which itself
is the same everywhere. But according to the Birkhoff-Jebsen theorem and its
corollary, we only need to consider the source of gravity within $R_{s}$ to
determine the local dynamics of this region (i.e., the behavior of $a$)
extending out to this radius.

We place an observer at the center of this sphere, where they calculate the
kinetic energy of a shell with thickness $dR$ at $R$ to be

$dK=4\pi R^{2}\,dR\,{\rho(t)\dot{R}^{2}\over 2c^{2}}\;,$ (7)

where the factor $c^{2}$ converts the energy density to a mass density. One
must be aware, of course, that there are limitations to how far one can take
this classical, heuristic approach. Inferring a mass density using this method
for the purpose of calculating a "kinetic" energy is reasonable for a matter
dominated Universe, perhaps even a matter plus radiation Universe, as long as
the radiation in the latter is ‘trapped’ within the matter, allowing one to
use the $E=mc^{2}$ conversion to infer the gravitational mass. This approach
would not be sensible for a purely radiation-dominated Universe, however. If
the observer now integrates this quantity from $r=0$ (their location) to
$r=r_{\rm s}$ (the edge of the sphere), they find that the total kinetic
energy of the sphere relative to them is

$K={2\pi\over 5}{\rho(t)\over c^{2}}a^{3}{\dot{a}}^{2}\,{r_{\rm s}}^{5}\;.$
(8)

This sphere also has gravitational energy (remember this is a classical
derivation). The observer finds that the corresponding potential energy of a
shell at $R$ is

$dU=-4\pi R^{2}\,dR\,{GM(R)\rho(t)\over c^{2}R}\;,$ (9)

where

$M(R)={4\pi\over 3}\,R^{3}\,{\rho(t)\over c^{2}}$ (10)

is the total mass enclosed within this radius. Thus, after integrating this
quantity from $r=0$ to $r_{\rm s}$, they find that the total potential energy
of the sphere (relative to them at the origin) is

$U=-{16\pi^{2}G\over 15}{\rho(t)^{2}\over c^{4}}a^{5}{r_{\rm s}}^{5}\;.$ (11)

Since they are doing this calculation classically, they therefore conclude
that the sphere expanding away from them has a net kinetic plus gravitational
energy

$E={2\pi\over 5}{\rho(t)\over c^{2}}a^{3}{\dot{a}}^{2}\,{r_{\rm
s}}^{5}-{16\pi^{2}G\over 15}{\rho(t)^{2}\over c^{4}}a^{5}{r_{\rm s}}^{5}\;,$
(12)

and if they re-arrange this slightly, a very recognizable form appears:

$\left({\dot{a}\over a}\right)^{2}={8\pi G\over 3c^{2}}\rho(t)+{5c^{2}E\over
2\pi\rho(t)\,a^{5}\,{r_{\rm s}}^{5}}\;.$ (13)

The local conservation of energy relative to the observer at the origin is
thus simply the Friedmann Equation (2), so long as one interprets the spatial
curvature constant to be

$k\equiv-{10\over 3\,{r_{\rm s}}^{2}}\left({\epsilon\over\rho}\right)\;,$ (14)

written in terms of the local kinetic plus gravitational energy density

$\epsilon\equiv{3E\over 4\pi\,{R_{\rm s}}^{3}}\;,$ (15)

and the energy density $\rho$ in the cosmic fluid. But note that $\epsilon$
does not include $\rho$ itself. Incidentally, calling the quantity $k$ defined
in Equation (14) a constant would be justified only as long as $\epsilon$ and
$\rho$ scale in proportion to each other. In a classical context, this simply
means that the sum of kinetic plus gravitational energy (i.e., $E$) is
conserved, which is what one would expect without an energy non-conserving
influence affecting the cosmic expansion. One must be careful in general
relativity, however, since the energy measured by an observer can change in a
time-dependent spacetime. As we shall discuss shortly, however, even this
distinction may not matter for our interpretation of $k$ if $\epsilon$ is
strictly zero, because then $\epsilon$ would always be zero in both the
classical and relativistic contexts.

We thus see that a Universe with positive spatial curvature ($k>0$) contains
negative net energy ($\epsilon<0$) and is therefore bounded—for which the more
common terminology is that the Universe is ‘closed’. Negative spatial
curvature arises when this energy is positive ($\epsilon>0$), which
characterizes an unbounded (or ‘open’) Universe. A Universe with zero net
energy ($\epsilon=0$) is spatially flat ($k=0$), but also ‘open’. To put this
in context, let us consider the following simple analogy. Imagine throwing a
ball vertically into the air, ignoring any resistance. Its motion will depend
on how its initial speed ($v_{\rm init}$) compares to the value it needs
($v_{\rm esc}\equiv\sqrt{2GM_{\rm Earth}/R_{\rm Earth}}$) to escape from
Earth. If $v_{\rm init}<v_{\rm esc}$, the ball slows down, stops and then
falls back to its point of origin. The ball is bounded to Earth because its
total energy is negative. If $v_{\rm init}>v_{\rm esc}$, the ball has positive
energy and is unbounded. It will continue receding to infinity with a speed
always greater than zero. The critical initial condition is $v_{\rm
init}=v_{\rm esc}$, for which the ball rises indefinitely, though with an ever
decreasing speed, coming to rest at infinity.

A spatially flat Universe therefore has zero local energy density $\epsilon$.
Normally, attempts at quantitatively handling gravitational energy in general
relativity face the daunting problem of localization (Landau, 1962). Here,
however, the $\epsilon=0$ condition holds at every point in the Universe, so
it follows that the total kinetic plus gravitational energy must also be zero,
no matter how one chooses to do the integration. This simplified heuristic
argument therefore teaches us that spatial flatness is a unique, fundamental
physical property of the Universe, which apparently contains equal portions of
positive kinetic (expansion) energy and negative gravitational energy.
Moreover, $k$ is constant, so this perfect balance must have been present from
the beginning, constituting an important initial condition at the time of the
Big Bang.

##  5 Discussion

It is tempting to think that perhaps our Newtonian approach may lack certain
nuances manifested solely via the full general relativistic treatment. But
remember that our calculation of $\epsilon$ has been carried out strictly on a
local basis, with full justification from the Birkhoff-Jensen theorem. In
fact, the proper radius of the sphere used in this construction, i.e.,
$a(t)r_{s}$, may be chosen arbitrarily small compared to the Hubble radius
$R_{\rm h}=c/H$, because we are interested merely in the local value of
$\epsilon$, point by point. And once $\epsilon$ is known for a given observer,
every spacetime point in the FLRW cosmos has precisely the same value, at
least according to the Cosmological principle. There is therefore nothing
missing from Equation (13) that would otherwise emerge from an analogous
calculation using Einstein’s equations. In particular, notice that this
equation matches precisely the actual Friedmann equation (2) derived from
general relativity.

The key result of this derivation is a clear, unambiguous identification of
the energy $E$ in Equation (13), which appears to be directly proportional to
the spatial curvature constant $k$. It includes only kinetic plus
gravitational energy, but not a contribution from $\rho$ itself. Our result
therefore agrees completely with that of Faraoni Cooperstock (2003) in the
case of $k=0$, who used an entirely different method of identifying the
kinetic and potential energy of the cosmic expansion. In both cases, the
conclusion is that $E$ is zero, but it only represents the kinetic and
gravitational energy. Thus, if the latest observations are telling us that
$k=0$, we cannot avoid the conclusion that the Universe has net positive
energy, when the integrated value of $\rho$ is taken into account.

Nevertheless, one must still contend with several possible caveats to this
result. First, it may turn out that $k$ is close to, but not exactly, zero, in
which case there may still be some wiggle room to keep the overall sum of
energies zero. We note, in this regard, that the most precise measurement of
the spatial curvature to date, from the Planck mission (Planck Collaboration
., 2020), is quoted as $\Omega_{k}=0.001\pm 0.002$, where $\Omega_{k}$ is
defined as the last term in Equation (2) divided by $H^{2}$. There is no
question this parameter is fully consistent with zero, but it may still allow
for a very slight positive or negative bias. This condition is the essential
caveat with this work, for if it turns out that $k$ is in fact not exactly
zero, then the conclusions we draw here become moot. We note, in this regard,
that observations other than those pertaining to the CMB also indicate a
likely flat Universe, consistent with Planck (Wei Melia, 20202, 20201), though
the precision of these alternative measurements is inferior to that of the CMB
satellite missions. Nevertheless, one must keep in mind that any discussion
relating the total energy of the Universe to its spatial flatness becomes much
more difficult to resolve if $k$ is not zero, given in part by the
complications discussed earlier of handling the energy in a general
relativistic framework when the spacetime curvature changes with time.

Second, inflation may have happened after all, in which case $\Omega_{0}-1$ in
Equation (6) is close to zero today only because of the significant dilution
that took place during the inflationary expansion at $t\sim 10^{-35}$ sec.
Having said this, it is also possible that $k=0$ was an initial condition
regardless of whether inflation happened or not, in which case our inference
on the total energy would be valid nonetheless. It is beyond the scope of the
present paper to discuss inflation at greater length, but it might be helpful
to point out that the observational evidence in favor of such an event having
occurred in the early Universe is rather slim. As noted earlier, the latest
analysis of the Planck data suggests that a slow-roll inflaton potential could
not simultaneously have solved both the CMB temperature horizon problem and
provided a mechanism for seeding the primordial fluctuation spectrum Liu Melia
(2020). And at a more fundamental level, it is even debatable whether these
quantum fluctuations could have classicalized to produce the classical large-
scale structure we see today Melia (2021).

Third, in addition to inflation, there are other possible reasons why the
Universe may appear close to spatial flatness today, even if $k=0$ was not an
initial condition. For example, a key factor in many ‘cyclic’ models of the
Universe is a so-called ekpyrotic phase, in which the contraction preceding
the Big Bang evolves ultra-slowly (Raveendran Sriramkumar, 2019; Ijjas
Steinhardt, 2019). In such models, it is this ekpyrotic phase that can explain
the smoothness and flatness of the Universe on large scales, and it can also
generate a distinctive primordial spectrum of scalar and tensor fluctuations.

So there are still many unknowns, and while our conclusions in this paper are
suggestive, we would not call them ‘final’. We are beginning to get some
guidance from the data concerning the nature of the Big Bang, but there is
still much work to do before we can identify its true physical origin.

##  6 Conclusion

In a way, this result completes the derivation begun by Tryon in his seminal
paper (Tryon, 1973), in which he attempted to balance the gravitational energy
by the rest mass energy in the cosmic fluid. As we now understand it, the
kinetic energy of expansion exactly balances its gravitational counterpart,
leaving $\rho$ as an ‘excess’ positive contribution to $E$.

If these results withstand the test of time, it would not be possible to
reconcile the presently ‘measured’ non-zero energy of the Universe with the
requirements of a quantum birth from ‘nothing’. Perhaps a quantum fluctuation
in a pre-existing vacuum might still be consistent, but one would need to
understand how its energy $E$—in the context of our Universe’s measured value
of the Planck constant, $\hbar$—could have allowed it to grow and classicalize
and live long enough for us to see it today. Would this be an indication that
$\hbar$ had to be different prior to the Big Bang?

### Author contributions

Fulvio Melia is the sole author of this paper.

### Financial disclosure

None reported.

### Conflict of interest

The authors declare no potential conflict of interests.

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## Author Biography

Fulvio Melia is an internationally recognized leader in astrophysics and
cosmology, having previously published seven books in physics and astronomy,
and approximately 400 refereed publications in the these fields. He has also
served as associate editor of the Astrophysical Journal and Astrophysical
Journal Letters.