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 as W_k ≡ 1 - W_tot, where W_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 faster than 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 3.086×10¹⁶ 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 H₀, 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 e₀ (the electric force constant), m₀ (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.

SF Novels by John Cramer:  Printed editions of John's hard SF novels Twistor and Einstein's Bridge are available from Amazon at https://www.amazon.com/Twistor-John-Cramer/dp/048680450X and https://www.amazon.com/EINSTEINS-BRIDGE-H-John-Cramer/dp/0380975106 .  His new novel, Fermi's Question may be coming soon.

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).

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