George F.R. Ellis – Page 2 – Cosmic Reflections

Constants of Nature

We continue our series of excerpts (and discussion) from the outstanding survey paper by George F. R. Ellis, Issues in the Philosophy of Cosmology.

The constants of nature are indeed invariant, with one possible exception: the fine structure constant, where there is claimed to be evidence of a very small change over astronomical timescales. That issue is still under investigation. Testing such invariance is fundamentally important, precisely because cosmology usually assumes as a ground rule that physics is the same everywhere in the universe. If this were not true, local physics would not guide us adequately as to the behaviour of matter elsewhere or at other times, and cosmology would become an arbitrary guessing game.

The fine structure constant (α) is a unitless number, approximately equal to 1/137, that characterizes the strength of the electromagnetic force between electrons. Its value is the same no matter what system of measurement one chooses. If the value of α were just a little smaller, molecular bonds would be less stable. If the value of α were just a little larger, carbon—which is essential to life—could no longer be produced inside of stars.

Do constants of nature, specifically dimensionless physical constants such as α, the fine structure constant, and μ, the proton-to-electron mass ratio¹, vary with time? This is an active topic of investigation. If constants of nature change at all, they change so slowly that it presents a formidable challenge to measure that change. But if they do indeed change, it would have profound implications for our understanding of the universe. A lot can happen in 13.8 billion years that might not be at all obvious in the infinitesimal interval of a human life or even human civilization.

“Despite the incessant change and dynamic of the visible world, there are aspects of the fabric of the universe which are mysterious in their unshakeable constancy. It is these mysterious unchanging things that make our universe what it is and distinguish it from other worlds we might imagine.” – J.D. Barrow, The Constants of Nature. (Vintage, 2003).

I’d like to conclude this discussion of constancy and change with a poem I wrote about the possibility of sentient life having a very different sense of time than we humans do.

Life On a Cold, Slow World

Life on a cold, slow world
On Europa, perhaps, or even Mars
On distant moons and planets of other stars.

A minute of time for some anti-freeze being
Might span a year for us human folk
(A greeting could take a week, if spoke.)

How fast our busy lives would seem to pass
Through watchful eyes we cannot see
Curious about our amative celerity.

The heartbeat of the universe runs slow and deep
We know only violent change, the sudden leap
But that which is most alive appears to sleep.

David Oesper

¹μ = m_p / m_e ≅ 1836

References
Barrow, J.D., Webb, J.K., 2005, Scientific American, 292, 6, 56-63

Ellis, G. F. R. 2006, Issues in the Philosophy of Cosmology, Philosophy of Physics (Handbook of the Philosophy of Science), Ed. J. Butterfield and J. Earman (Elsevier, 2006), 1183-1285.
[http://arxiv.org/abs/astro-ph/0602280]

The Beginning

We continue our series of excerpts (and discussion) from the outstanding survey paper by George F. R. Ellis, Issues in the Philosophy of Cosmology.

Thesis D1: An initial singularity may or may not have occurred.
A start to the universe may have occurred a finite time ago, but a variety of alternatives are conceivable: eternal universes, or universes where time as we know it came into existence in one or another way. We do not know which actually happened, although quantum gravity ideas suggest a singularity might be avoided.

If we imagine, for a moment, running the clock of the universe backwards to earlier and earlier times, its size gets smaller and its density gets larger until we reach a moment—even earlier than the putative inflationary era—when classical physics at the macroscopic level no longer applies and some (as yet unknown) quantum physics must apply to everything—even gravity. Therein lies the problem, because if you run the clock backwards just 5.39 x 10^-44 second from this time, you reach the purported moment of the Big Bang—the initial singularity. But whoa (or perhaps woe)! How can we say anything about the Big Bang—or even if it occurred at all—since the laws of known physics completely break down 5.39 x 10^-44 second (the Planck time) after the Big Bang! See the problem?

Perhaps the universe came into existence through a process analogous to radioactive decay where an alpha particle leaves a nucleus through quantum tunneling. Perhaps our universe “tunneled” into existence from somewhere else, and thus our beginning isn’t really the beginning. This is just one of many possibilities.

This is a key issue in terms of the nature of the universe: a space-time singularity is a dramatic affair, where the universe (space, time, matter) has a beginning and all of physics breaks down and so the ability to understand what happens on a scientific basis comes to an end. However eternal existence is also problematic, leading for instance to the idea of Poincaré’s eternal return: everything that ever happened will recur an infinite number of times in the future and has already occurred an infinite number of times in the past. This is typical of the problems associated with the idea of infinity. It is not clear in the end which is philosophically preferable: a singularity or eternal existence. That decision will depend on what criteria of desirability one uses.

While infinity is a highly useful mathematical device, one can make a strong argument that infinities do not exist in the physical universe (or even multiverse). Quantum physics already gives us a possible clue about the infinitely small: we appear not to be able to subdivide space or time any further than the Planck length (1.616 x 10^-35 meter) or the Planck time (5.39 x 10^-44 second). We would not be able to distinguish between two points less than a Planck length apart, nor two moments in time less than a Planck time apart. While harder to envision, might not there also be an upper limit to size? And time?

Thesis D2: Testable physics cannot explain the initial state and hence specific nature of the universe.
A choice between different contingent possibilities has somehow occurred; the fundamental issue is what underlies this choice. Why does the universe have one specific form rather than another, when other forms consistent with physical laws seem perfectly possible? The reasons underlying the choice between different contingent possibilities for the universe (why one occurred rather than another) cannot be explored scientifically. It is an issue to be examined through philosophy or metaphysics.

Metaphysics is the part of philosophy that deals with existence, space, time, cause and effect, and the like. Metaphysics begins where physics necessarily ends due to observational limitations.

Did anything exist before the Big Bang?

Was there a Big Bang?

What are the physical properties of the very early universe, when energy densities existed that are far beyond our ability to recreate in the laboratory?

What lies beyond our particle horizon?

Are there other universes?

Why does anything exist at all?

References
Ellis, G. F. R. 2006, Issues in the Philosophy of Cosmology, Philosophy of Physics (Handbook of the Philosophy of Science), Ed. J. Butterfield and J. Earman (Elsevier, 2006), 1183-1285.
[http://arxiv.org/abs/astro-ph/0602280]

Liddle, A.R. 2015, An Introduction to Modern Cosmology, 3rd ed., Wiley, ISBN: 978-1-118-50214-3.

Windows to the Earliest: Neutrinos and Gravitational Waves

We continue our series of excerpts (and discussion) from the outstanding survey paper by George F. R. Ellis, Issues in the Philosophy of Cosmology.

Thesis B7…
Neutrinos and gravitational waves will in principle allow us to peer back to much earlier times (the time of neutrino decoupling and the quantum gravity era respectively), but are much harder to observe at all, let alone in useful directional detail. Nevertheless the latter has the potential to open up to us access to eras quite unobservable in any other way. Maybe they will give us unexpected information on processes in the very early universe which would count as new features of physical cosmology.

The cosmic microwave background (CMB, T = 2.73 K) points us to a time 380,000 years after the Big Bang when the average temperature of the universe was around 3000 K. But there must also exist abundant low-energy neutrinos (cosmic neutrino background, CNB, CνB, relic neutrinos) that provide a window to our universe just one second after the Big Bang during the radiation dominated era. That’s when neutrinos decoupled from normal baryonic matter.

As the universe expanded, these relic neutrinos cooled from a temperature of 10¹⁰ K down to about 1.95 K in our present era, but such low-energy neutrinos almost never interact with normal matter. Even though the density of these relic neutrinos should be at least 340 neutrinos per cm³ (including 56 electron neutrinos per cm³ which will presumably be easier to detect), detecting them at all will be exceedingly difficult.

Neutrinos interact with matter only through the weak nuclear force (which has a very short range), and low-energy neutrinos are much more difficult to detect than higher-energy neutrinos—if they can be detected at all. If neutrinos have mass, then they will also interact gravitationally with other particles having mass, but this interaction is no doubt unmeasurable due to the neutrino’s tiny mass and the weakness of the gravitational force between subatomic particles.

The cosmic gravitational background (CGB) points us to the time of the Big Bang itself. Faessler, et al. (2016) state

The inflationary expansion of the Universe by about a factor 10²⁶ between roughly 10^-35 to 10^-33 seconds after the BB couples according to General Relativity to gravitational waves, which decouple after this time and their fluctuations are the seed for Galaxy Clusters and even Galaxies. These decoupled gravitational waves run since then with only very minor distortions through the Universe and contain a memory to the BB.

Faessler, A., Hodák, R., Kovalenko, S., and Šimkovic, F. 2016
[https://arxiv.org/abs/1602.03347]

Small Universe

We continue our series of excerpts (and discussion) from the outstanding survey paper by George F. R. Ellis, Issues in the Philosophy of Cosmology.

4.3.1 Small universes
A Small Universe: a universe which closes up on itself spatially for topological reasons, and does so on such a small scale that we have seen right round the universe since the time of decoupling. Then we can see all the matter that exists, with multiple images of many objects occurring. This possibility is observationally testable by examining source statistics, by observation of low power in the large angle CBR anisotropies, and by detecting identical temperature variation on various circles in the CBR sky. There are weak hints in the observed CBR anisotropies (the lack of power on large angular scales) that this could actually be the case, but this is not solidly confirmed. Checking if the universe is a small universe or not is an important task; the nature of our observational relationship to the universe is fundamentally different if it is true.

In 1900, Karl Schwarzschild (1873-1916) was perhaps the first to suggest the idea of a small universe topology that would lead to multiple images of the same object at different points in the past. Though most cosmologists favor the idea of a very large universe with a simple topology, the possibility of a more complex small universe topology is still not out of the question. The universe might be measurably finite in some or all directions.

The smaller a finite topological region of space, the easier it should be to discover multiple images of the same object at different ages (except for CMB features which will all be the same age). The distribution of distant sources might show “patterns” that are related to more nearby sources. A comprehensive survey of sources at redshifts between about z=2 to z=6 is still needed before any conclusions can be drawn.

Another approach, of course, is to look at patterns in the CMB temperature (intensity) and polarization. Analyses of the most recent release of Planck satellite data, however, shows no evidence of a compact topology smaller than our visual horizon.

Luminet, J.-P. 2016, arXiv:1601.03884v2 [astro-ph.CO]

The Hidden Universe

We continue our series of excerpts (and discussion) from the outstanding survey paper by George F. R. Ellis, Issues in the Philosophy of Cosmology.

Thesis B6: Observational horizons limit our ability to observationally determine the very large scale geometry of the universe.
We can only see back to the time of decoupling of matter and radiation, and so have no direct information about earlier times; and unless we live in a “small universe”, most of the matter in the universe is hidden behind the visual horizon. Conjectures as to its geometry on larger scales cannot be observationally tested. The situation is completely different in the small universe case: then we can see everything there is in the universe, including our own galaxy at earlier times.

What an intriguing idea. If the entire universe (or the self-contained section we find ourselves in) is substantially smaller than the distance light has traveled since the universe became transparent to radiation (“decoupling”, about 380,000 years after the Big Bang), we might be able to see our Milky Way galaxy (and other galaxies) at various points in the past.

The key point here is that unless we live in a small universe, the universe itself is much bigger than the observable universe. There are many galaxies—perhaps an infinite number—at a greater distance than the horizon, that we cannot observe by any electromagnetic radiation. Furthermore, no causal influence can reach us from matter more distant than our particle horizon—the distance light can have travelled since the creation of the universe, so this is the furthest matter with which we can have had any causal connection. We can hope to obtain information on matter lying between the visual horizon and the particle horizon by neutrino or gravitational radiation observatories; but we can obtain no reliable information whatever about what lies beyond the particle horizon. We can in principle feel the gravitational effect of matter beyond the horizon because of the force it exerts (for example, matter beyond the horizon may influence velocities of matter within the horizon, even though we cannot see it). This is possible because of the constraint equations of general relativity theory, which are in effect instantaneous equations valid on spacelike surfaces. However we cannot uniquely decode that signal to determine what matter distribution outside the horizon caused it: a particular velocity field might be caused by a relatively small mass near the horizon, or a much larger mass much further away. Claims about what conditions are like on very large scales—that is, much bigger than the Hubble scale—are unverifiable, for we have no observational evidence as to what conditions are like far beyond the visual horizon. The situation is like that of an ant surveying the world from the top of a sand dune in the Sahara desert. Her world model will be a world composed only of sand dunes—despite the existence of cities, oceans, forests, tundra, mountains, and so on beyond her horizon.

Let us now define some terms that Ellis uses above.

visual horizon – the distance beyond which the universe was still opaque to photons due to high temperature and density

particle horizon – the distance beyond which light has not yet had time to reach us in all the time since the Big Bang; our particle horizon is, therefore, farther away than our visual horizon

spacelike surface – a three-dimensional surface in four-dimensional space-time where no event on the surface lies in the past or future of any other event on that surface; every point on the surface as it exists at one instant of time

Hubble scale – a cosmological distance unit equal to the reciprocal of the Hubble constant times the speed of light; see derivation below

A reasonable value for the Hubble constant H₀ is 70 km/s/Mpc. A galaxy one megaparsec distant has a cosmological recession velocity of 70 km/s, two megaparsecs distant 140 km/s, and so on.

You may notice that there are two units of distance in H₀: kilometers and megaparsecs. We can thus rewrite H₀in units of s^-1 (reciprocal seconds of time) as follows:

The Hubble time is defined as the inverse of the Hubble constant:

Converting this into more convenient units of years, we get

The Hubble scale is now simply the Hubble time multiplied by the speed of light.

Converting this into more convenient distance units of light years, and then parsecs, we get

As Ellis says, we are like ants in the Sahara desert that cannot see their Earth-universe beyond the sand dunes. Like the ant, is there a limit to our intellect as well?

Homogeneity and Isotropy

We continue our series of excerpts (and discussion) from the outstanding survey paper by George F. R. Ellis, Issues in the Philosophy of Cosmology.

4.2.2 Indirect determination: justifying a Friedmann-Lemaître geometry
Considered on a large enough angular scale, astronomical observations are very nearly isotropic about us, both as regards source observations and background radiation; indeed the latter is spectacularly isotropic, better than one part in 10⁴ after a dipole anisotropy, understood as resulting from our motion relative to the rest frame of the universe, has been removed.

No matter what direction we look, the universe looks statistically the same at a scale of hundreds of millions of light years. We call this property isotropy. Case in point: when compared one to the other, the Hubble Deep Fields look remarkably similar, even though they are about 135° apart in the sky.

Hubble eXtreme Deep Field in the constellation Fornax

Hubble Deep Field in the constellation Ursa Major

Taken individually, both of these deep fields also exhibit homogeneity, that is, they generally show a fairly uniform distribution of galaxies across the field.

Does the dipole anisotropy in the cosmic background radiation (CBR), due to our motion with respect the rest frame of the universe, indicate an absolute frame of reference? Not at all. Though the rest frame of the universe is the preferred frame for cosmology, it is not a particularly good frame of reference to use, for example, in describing the motion of the planets in our solar system. The laws of physics are the same in all inertial (unaccelerated) reference frames, so none of them can be “special”—or absolute. An absolute frame of reference would be one in which the laws of physics would be different—indeed simpler—but no such reference frame exists. And any non-inertial (accelerated) reference frame indicates there is an external force outside the system acting on the system, so it can never be used as an absolute frame of reference.

We’re moving toward Leo and away from Aquarius, relative to the cosmic background radiation

Top: CBR with nothing subtracted; Middle: CBR with dipole anisotropy subtracted; Bottom: CBR with both dipole anisotropy and galactic emission subtracted

Cosmic Background Radiation from the Planck spacecraft with anisotropies removed

If all observers see an isotropic universe, then spatial homogeneity follows; indeed homogeneity follows if only three spatially separated observers see isotropy. Now we cannot observe the universe from any other point, so we cannot observationally establish that far distant observers see an isotropic universe. Hence the standard argument is to assume a Copernican Principle: that we are not privileged observers. This is plausible in that all observable regions of the universe look alike: we see no major changes in conditions anywhere we look. Combined with the isotropy we see about ourselves, this implies that all observers see an isotropic universe.

The Copernican principle states that we are not privileged observers of the universe. Any observer elsewhere in the universe will see the same universe that we do. The laws of physics, chemistry, and biology are truly universal. The Copernican principle is a good example of the application of Occam’s razor: unless there is evidence to the contrary, the simplest explanation that fits all the known facts is probably the correct one.

Knowledge Limited: Deep Time, Deep Space

We continue our series of excerpts (and discussion) from the outstanding survey paper by George F. R. Ellis, Issues in the Philosophy of Cosmology.

Thesis A4: The concept of probability is problematic in the context of existence of only one object.
It is useful to distinguish between the experimental sciences—physics, chemistry, microbiology for example—on the one hand, and the historical and geographical sciences—astronomy, geology, evolutionary theory for example, on the other.

In the experimental sciences, one is usually free to both observe and experiment. For example, we can observe at what temperature water freezes with different concentrations of various salts such as sodium chloride. But in the historical and geographical sciences, one can observe but we are seldom, if ever, able to experiment. We can observe the properties of the Sun and other stars, but we cannot create a star nor modify its properties to see how that alters its development. We must infer how individual stars or classes of stars change with time by observing many stars of different masses at various points along their continuum of existence. And what of objects that are unique or that happened only once? The evolution of life on Earth, the Grand Canyon, and the Universe itself? The greatest challenge in the historical and geographical sciences besides not being able to run experiments is the enormous amount of time it takes for measurable changes to occur. How can we humans—who seldom live more than a century—begin to comprehend changes that occur over a million, let alone a billion, years?

Thesis B1: Astronomical observations are confined to the past null cone, fading with distance.
Uncertainty grows with distance and time. The vast scale of the universe implies we can effectively only view it from one spacetime event (“here and now”).

Cosmology is both a geographic and a historical science combined into one: we see distant sources at an earlier epoch, when their properties may have been different. As we are looking back in the past, source evolution must be taken into account; their properties at the time they emitted the light may be quite different from their properties now. We can only determine the distances of objects if we understand this evolution; but in practice it is one of the unknowns we have to try to determine.

Distant sources appear very small and very faint, both because of their physical distance, and because their light is highly redshifted (due to the expansion of the universe). Simply detecting them, let alone determining their characteristics, becomes rapidly more difficult with distance. Furthermore absorption by intervening matter can interfere with light from distant objects. The further back we look, the worse these problems become; thus our reliable knowledge of the universe decreases rapidly with distance.

Another name for the “null cone” Ellis mentions above is light cone. A light cone is a two-dimensional model of our three spatial dimensions, plus time. We build up the cone using a series of circles along the time dimension.

First, let’s consider that you, the observer, as experimenter, produce an isotropic flash of light sometime this year at a particular location. The flash of light will move outward in all directions at the speed of light. The concentric circles below show the location of the wavefront from your flash in the year 2027, 2037, and 2047 when it is 10 light years, 20 light years, and 30 light years from Earth, respectively, and so on. If we add a time axis that is perpendicular to the plane of our two-dimensional “Flatland” and points away from you, we see that we can build up a cone from the ever-expanding circular wavefront at every instant of time. This is the future light cone.

Similarly, when you look out into the depths of space on a clear night you are also inexorably looking back in time. Light from a star 10 light years away left on its journey to Earth in 2007. If the star is 20 light years away, the light began its journey in 1997. If 30 light years away, in 1987, and so on. Again, if we add a time axis perpendicular to our two spatial dimensions, now pointing towards you (coming from the past), we see that we can build up a cone from the incoming wavefront’s location at each moment of time in the past. This is the past light cone.

Now, if we put the past and future light cones together we get the full view of our location in spacetime, as shown below. The two cones meet at the “here and now”. Keep in mind that the diagram below is a two-dimensional representation of a 3D object (two spatial dimensions and one time dimension), but in reality, this should be a four-dimensional object (three spatial dimensions and one time dimension).

So, our view from the “here and now” is small and provincial. Instead of obtaining a panoramic snapshot of our universe as it currently exists today, we are being served up old photos instead. But quite useful, nonetheless.

Beginnings, Quantum Gravity, and Inflation

We continue our series on the outstanding survey paper by George F. R. Ellis, Issues in the Philosophy of Cosmology.

2.6 Inflation
Particle horizons in inflationary FL models will be much larger than in the standard models with ordinary matter, allowing causal connection of matter on scales larger than the visual horizon, and inflation also will sweep topological defects outside the visible domain.

The particle horizon is the distance beyond which light would have not yet had time to reach us in all the time since the Big Bang. The visual horizon is the distance beyond which the universe was still opaque to photons due to high temperature and density. The visual horizon, therefore, is not as far away as the particle horizon. FL stands for Friedmann-Lemaître, the standard models of a flat, open, or closed universe.

What is inflation? At the moment of the Big Bang, the expansion of the universe accelerated exponentially for a very short period of time. This caused portions of space that had been close enough together to be causally connected to become causally disconnected. While inflation does a very good job of explaining many observed features of our universe, such as its uniformity in all directions, at this point it is an untestable hypothesis (unlike special and general relativity), and the underlying physical principles are completely unknown.

2.7 The very early universe
Quantum gravity processes are presumed to have dominated the very earliest times, preceding inflation. There are many theories of the quantum origin of the universe, but none has attained dominance. The problem is that we do not have a good theory of quantum gravity, so all these attempts are essentially different proposals for extrapolating known physics into the unknown. A key issue is whether quantum effects can remove the initial singularity and make possible universes without a beginning. Preliminary results suggest that this may be so.

We currently live in a universe where the density may be too low to observe how gravity behaves at the quantum level. Though we may never be able to build a particle accelerator with energies high enough to explore quantum gravity, quantum gravity might possibly play a detectable role in high-density stars such as white dwarfs, neutron stars, or black holes. At the time of the Big Bang, however, the density of the universe was so high that quantum gravity certainly must have played a role in the subsequent development of our universe.

Do we live in the universe that had no beginning and will have no end? A universe that is supratemporal—existing outside of time—because it has always existed and always will exist? Admittedly, this is an idea that appeals to me, but at present it is little more than conjecture, or, perhaps, even wishful thinking.

2.7.1 Is there a quantum gravity epoch?
A key issue is whether the start of the universe was very special or generic.

Will science ever be able to answer this question? I sincerely hope so.

2.8.1 Some misunderstandings
Two distantly separated fundamental observers in a surface {t = const} can have a relative velocity greater than c if their spatial separation is large enough. No violation of special relativity is implied, as this is not a local velocity difference, and no information is transferred between distant galaxies moving apart at these speeds. For example, there is presently a sphere around us of matter receding from us at the speed of light; matter beyond this sphere is moving away from us at a speed greater than the speed of light. The matter that emitted the CBR was moving away from us at a speed of about 61c when it did so.

Thus, there are (many) places in our universe that are receding from us so fast that light will never have a chance to reach us from there. Indeed, the cosmic background radiation that pervades our universe today was emitted by matter that was receding from us at 61 times the speed of light at that time. That matter never was nor ever will be visible to us, but the electromagnetic radiation it emitted then, at the time of decoupling, is everywhere around us. Think of it as an afterglow.

A Small, Big, or Really Big Universe?

George F. R. Ellis writes in section 2.4.2 of his outstanding survey paper, Issues in the Philosophy of Cosmology:

Clearly we cannot obtain any observational data on what is happening beyond the particle horizon; indeed we cannot even see that far because the universe was opaque before decoupling. Our view of the universe is limited by the visual horizon, comprised of the worldlines of furthest matter we can observe—namely, the matter that emitted the CBR at the time of last scattering.

The picture we obtain of the LSS by measuring the CBR from satellites such as COBE and WMAP is just a view of the matter comprising the visual horizon, viewed by us at the time in the far distant past when it decoupled from radiation.

Visual horizons do indeed exist, unless we live in a small universe, spatially closed with the closure scale so small that we can have seen right around the universe since decoupling.

The major consequence of the existence of visual horizons is that many present-day speculations about the super-horizon structure of the universe—e.g. the chaotic inflationary theory—are not observationally testable, because one can obtain no definite information whatever about what lies beyond the visual horizon. This is one of the major limits to be taken into account in our attempts to test the veracity of cosmological models.

Let’s start by defining a few of the terms that Ellis uses above.

particle horizon – the distance beyond which light has not yet had time to reach us in all the time since the Big Bang

decoupling – the time after the Big Bang when the Universe had expanded and cooled enough that it was no longer a completely ionized opaque plasma; atoms could form and photons began traveling great distances without being absorbed

worldlines – the path of a photon (or any particle or object) in 4-dimensional spacetime: its location at each and every moment in time

CBR – cosmic background radiation

LSS – last scattering surface

COBE – Cosmic Background Explorer

WMAP – Wilkinson Microwave Anisotropy Probe

(And, Planck should be added now, too)

Now the question. Do we live in a small, big, or really big universe? The best answer we can give now (or, perhaps, even in the future) is that we live in a really big universe, though it is unlikely to be infinite. Ellis himself provides a cogent argument in section 9.3.2 of the paper referenced here that infinity, while a very useful mathematical tool, does not ever exist in physical reality. We shall investigate this topic in a future posting.

Even though general relativity shows us how matter defines the geometry of our observable universe, it tells us nothing about the topology of our universe, in other words, its global properties. Do we live in a wrap-around universe where if we set off in one direction and traveled long enough, we’d eventually return to the same point in spacetime? Is the topology of our universe finite or infinite? At the moment it appears that we are not able to observe enough of the universe to discern its topology. If that is true, we may never be able to understand what type of universe we live in. But observational cosmologists will continue to search for the imprint of topology on our visible universe at the largest scales.

Liddle, A.R. 2015, An Introduction to Modern Cosmology, 3rd ed., Wiley, ISBN: 978-1-118-50214-3.

Cosmologically Distant Objects Appear Magnified

George F. R. Ellis writes in section 2.3.3 of his outstanding survey paper, Issues in the Philosophy of Cosmology:

…there is a minimum apparent size for objects of ﬁxed physical size at some redshift z_c = z_∗ depending on the density parameter and the cosmological constant. The past light cone of the observer attains a maximum area at z_∗; the entire universe acts as a gravitational lens for further oﬀ objects, magnifying their apparent size so that very distant objects can appear to have the same angular size as nearby ones. For the Einstein-de Sitter universe, the minimum angular diameter is at z_∗ = 1.25; in low density universes, it occurs at higher redshifts.

Electromagnetic radiation such as visible light that we observe from a source that is in motion relative to us (the observer) experiences a change in wavelength that is given by

This is called redshift and is positive for a source that is moving away from us and negative for a source that is moving towards us. The higher the relative speed toward or away from us, the greater the magnitude of the redshift. Superimposed upon the kinematic velocities of individual galaxies relative to our Milky Way galaxy, since 1929 we have known that there is a cosmological redshift (called the Hubble flow) that is always positive and increasing in magnitude with increasing distance between any two galaxies. In the nearby universe, the redshift (or blueshift) from kinematic velocities (often referred to as “peculiar velocities”) swamp the contribution from the Hubble flow, so some galaxies are actually approaching each other. A good example of this is M31 and the Milky Way galaxy. For more distant galaxies, however, the cosmological redshift swamps any contribution from the kinematic velocities. Thus, redshift becomes a useful proxy for distance at cosmological distances.

From our everyday experience, we know that the further away an object is, the smaller is its angular size. However, there comes a point where the angular size of an object reaches a minimum, and at even greater distances, its angular size increases! As George Ellis states above, the entire universe acts as a gravitational lens to magnify distant objects.

Michael Richmond presents an equation for angular size as a function of redshift (based on some classical assumptions about the structure of the universe). In his equation, the angular size of an object also depends upon the value we choose for H₀, the Hubble constant, the matter density parameter, Ω_M, and, of course, the physical size of the object of interest.

Let’s work through an example using this equation. The visible part of the Andromeda Galaxy is estimated to be about 220,000 light years across. In megaparsecs, that is 0.0675. This is the value we will use for S.

For the Hubble constant, H₀, let use a recent result: 71.9 km/s/Mpc.

And, for the matter density parameter, Ω_M, let’s use 1.0. This indicates that we live in a universe that has just enough matter for the universe to eventually recollapse, were it not for dark energy. Though Richmond’s equation above only applies to a matter-dominated universe where the dark energy density parameter Ω_Λ is zero, as George Ellis indicates above, a minimum angular diameter is still reached in a universe with dark-energy (i.e. low density universe), only this occurs at a higher redshift than that presented here.

I have not been able to find or derive a more general equation for angular size as a function of redshift that will work for a dark-energy-dominated universe (perhaps a knowledgeable reader will post a comment here providing some insight into this issue), but it will be a useful exercise to continue with the calculation assuming the matter-dominated Einstein-de Sitter universe.

Casting Michael Richmond’s equation into the following SAS program, I was able to find that the Andromeda galaxy would reach a minimum angular size of 11.3 arcseconds at z = 1.25, as shown below.

In principle, measuring the angular size of a “standard” object at various redshifts could allow us to determine what kind of universe we live in. But there’s a problem. As we look further out into space we are also looking further back in time, so there is no guarantee that a “standard” object in today’s universe (say, a spiral galaxy such as M31) would have looked the same or even existed billions of years ago.

Richmond, Michael, Two classic cosmological tests
[https://web.archive.org/web/20180909221238/http://spiff.rit.edu/classes/phys443/lectures/classic/classic.html]