If
you could halt the expansion of the universe and then travel outward in a
straight line, would you eventually come back to the place where you started?In other words, are all straight-line paths in the universe closed
circles?Or as
Euclid
would have put it, if you project two precisely parallel light beams out into
empty space, do they ever cross?A
recent analysis of the angular structure and lensing of the cosmic microwave
background radiation, as measured by the European Space Agency's Planck
Mission, suggests that the answer to these questions is "Yes".We will start by considering the curvature of space as represented in
general relativity (GR).

In
the words of general-relativity pioneer John A. Wheeler, "Space-time
grips mass, telling it how to move, while mass grips space-time, telling it how
to curve."In GR, the measure
of how much the space of the universe is curved by its average mass content is
specified by the curvature parameter W_{k}
(defined asW_{k }
≡
1 - W_{tot}, whereW_{tot} is the total mass density
parameter).If the curvature parameter W_{k} is precisely equal to 0 (because W_{tot} is exactly 1), then the universe is flat, and its space extends to
infinity in all directions.In such
a flat universe, the volume of an enlarging sphere increases exactly as the cube
of the sphere's radius.

If
the curvature parameter W_{k}
is greater than 0 (indicating a less-than-critical mass density and W_{tot} less than 1), then the universe is open and saddle shaped, and again
space extends to infinity in all directions.In such an open universe, the volume of an enlarging sphere increases
grows fasterthan the cube of the sphere's radius.

However,
if more-than-critical mass density is present and the curvature parameter W_{k} is less than 0, then the pull
of the mass-energy of the universe curves space into a closed hypersphere.A hypersphere is a sphere in four-dimensions, with a 3D "surface" in
which our familiar three-dimensional space resides.With W_{k} < 0, the universe is closed,
and space curves back on itself.In
such a closed universe, the volume of an enlarging sphere increases as less
than the cube of the sphere's radius, and parallel rays will eventually
cross and return to the place from which they started.

One
of the basic assumptions of the well established standard LCDM model (Cold Dark
Matter with cosmological constant L)
of cosmology is that W_{k }= 0 and the universe is flat.It
therefore has come as a significant surprise that the recent 2018 analysis of
Planck Mission data (PL18) gives the curvature parameter at
the 99% confidence level to be in
the range -0.007
> W_{k}>
-0.095.This suggests that W_{k}
is definitely negative and has a value of about -0.05.This result implies that the universe is closed rather than flat and has a radius of curvature on the order
of a billion parsecs or so. (Note:
a parsec is 3.26 light years or 3.086×10^{16}
meters.)

How
is this preference for a closed universe revealed by the Planck data?It comes from gravitational lensing, the tendency of light from a distant
source to be bent and focused by foreground mass.During its period of operation from 2009 to 2013, the ESA's Planck
Mission measured the temperature and polarization of the cosmic microwave
background on a very fine angular scale in order to study the baryonic
oscillations of the early universe and to determine the parameters of the
universe to unprecedented accuracy.At
the arc-minute angular scale, gravitational lensing in such data begins to show
up as additional correlations between adjacent measurements.There is more gravitational lensing in a closed universe than in a flat
one, resulting in an increase in such correlations.Consequently an analysis of lensing correlations in the PL18 data
provides a measure of W_{k}.We note that this enhanced lensing correlation also appeared a decade
earlier in the analysis of data from NASA's WMAP (Wilkinson Microwave
Anisotropy Probe)
Mission
, which made small-angle measurements of the cosmic microwave background when
the spacecraft operated from 2001 to 2010.However,
in the WMAP case the statistics and angular resolution were poorer than those of
Planck.While the WMAP analysis also
showed a tendency to negative values of W_{k},
it was also consistent with zero, and therefore inconclusive.

Up
to now the astrophysics community has been relatively satisfied with the LCDM
cosmological model, but it has not been without its problems.In my AV column in the March-April-2020 issue of Analog,
I discussed the problem of the disagreement between measurements of the Hubble
constant H_{0}, as
measured by Planck and as measured by more local (red-shift z<3) methods like
Type 1A supernovas, gravitational lensing, red giant stars, gravitational wave
emission, etc.Also, the LCDM
model invokes inflation, an exponential expansion of the universe at very early
times that somehow stops abruptly after the universe is nicely homogeneous and
flat.This inflation scenario is
essentially "put in by hand" to produce a flat homogeneous universe, with no
compelling reason why it should have taken place.Further, the value of the cosmological constant L of the LCDM
model deduced from Planck data is uncomfortably small, in that attempts to
predict it by theoretical models like string theory or quantum field theory give
absurdly large values.All in all,
there seems to be a growing body of evidence that as the accumulated
astrophysical data grows, the standard LCDM
model has increasing difficulty in accommodating it, perhaps pointing to a need
for radical revision of the model.

Is
it possible to accommodate W_{k}<0
without doing too much violence to the standard LCDM model of cosmology?The answer is yes, but with some fine-tuning discomfort.Let me explain about "fine tuning" in theoretical physics.In physics theories there are always adjustable parameters.For example,
Newton's 2^{nd} Law (F = ma) there is an implicit parameter associated with the
question of how much acceleration a
does a given mass m
receive from a given force F.
In this case, the parameter is "1", and it is buried in the units of the
variables.In Maxwell's
electrodynamics there are three such adjustable constants.They are e_{0}
(the electric force constant), m_{0} (the magnetic force constant),
and c
(the speed of light).However, these
constants are not arbitrary because they can be measured to excellent accuracy
in the laboratory.

In
theoretical models that are more removed from the laboratory, there are
adjustable parameters that cannot be directly measured.Fine-tuning describes theoretical situations in which the parameters of a
model must be adjusted to very precise values in order to fit observations,
without any explicit mechanism that produces the needed values.The need for fine tuning is considered a negative aspect."Naturalness", defined as the absence of the need for fine tuning, is
considered a positive aspect of a theory.

We
note in this context that it has been argued by science writer John Gribbin and
astrophysicist Martin Rees that our universe is fine-tuned to an extraordinary
degree to produce the conditions that permit the existence of life and
intelligence.They argue that if any
one of a half-dozen parameters describing our universe was even slightly
different, the development of life as we know it would have been impossible.

In
modern cosmology, the cosmological constant L already requires fine tuning in all
models that can predict it to accommodate its very small value.It is also possible to construct non-flat models of primordial inflation
that lead to a value of W_{k}
that is less than zero, but these also require fine tuning.Perhaps a more serious problem is that almost all of the local
(red-shift z<3) observables
of cosmology suggest flatness and strongly disfavor a closed universe.

The
discrepancy in the Hubble constant, as discussed in my AV column in the
March-April-2020 Analog,
is that the universe seems to have had a significantly lower expansion rate when
the cosmic microwave background was first formed than its expansion rate at
later times.One theoretical
speculation discussed in that AV column was that by adding an "early dark
energy" to Big Bang cosmology one could have an early universe that expanded
at a slower rate than that of the later universe.The scenario involves conjuring up this extra early mass-energy that
somehow vanishes and ceases its gravitational pull at later times.

Now
with curvature we have a similar problem: preliminary evidence that the universe
had enough mass to be closed at early times, but at later times it had just the
right critical mass density to become flat and open.Could the early-dark-energy scenario also be applied to the curvature
problem?In particular, could the
presence of this hypothetical early dark energy have increased the mass density
of the universe by enough to produce the closure observed by PL18, but then
somehow vanished at later times, leaving behind just enough mass density to
produce the flatness observed by the local measurements?That would seem to violate the cherished law of conservation of
mass-energy, but perhaps that law doesn't apply to entire universes.Further, I am not sure that it makes any geometrical sense to have a closed
and finite universe that later becomes flat and infinite.Does GR allow that?In any
case, the cosmological picture that is emerging with improvements in
cosmological data seems to be telling us that our universe is even stranger than
we had imagined.

The
alternative is that there is something wrong with these data sets that are
telling us two different stories about our universe.Future improved cosmological measurements are planned, and new ideas and
new theoretical models are emerging.Somewhere
the must be a solution to this most fundamental of problems.

John
G. Cramer's 2016 nonfiction book (Amazon gives it 5 stars) describing his transactional interpretation of
quantum mechanics, The
Quantum Handshake - Entanglement, Nonlocality, and Transactions,
(Springer, January-2016) is available online as a hardcover or eBook at:http://www.springer.com/gp/book/9783319246406
or https://www.amazon.com/dp/3319246402.

Alternate
View Columns Online: Electronic reprints of 212 or more "The Alternate
View" columns by John G. Cramer published in Analog
between 1984 and the present are currently available online at: http://www.npl.washington.edu/av
.

References:

The
Planck 2018 Analysis: "Planck 2018 Results VI.Cosmological
Parameters", Aghanim, N., et al.
(The Planck Collaboration); ArXiv:1807.06209
[astro-ph.CO] (2018).

Evidence
for a Closed Universe: "Planck evidence for a closed universe and a possible crisis for
cosmology", , Alessandro Melchiorri, and Joseph Silk, Nature
Astronomy (2019) doi:10.1038/s41550-019-0906-9; ArXiv:1911.02087
[astro-ph.CO] (2019).

Fine
Tuning: Cosmic Coincidences - Dark Matter,
Mankind, and Anthropic Cosmology, John Gribbin and Martin Rees, ReAnimus
Press, Golden, CO (2015), ISBN-13: 978-1511915816.

Early
Dark Energy: "Early Dark Energy Can Resolve the Hubble Tension", Vivian
Poulin, Tristan L. Smith, Tanvi Karwal, and Marc Kamionkowski, Phys. Rev. Lett. 122,
221301 (2019).