cosmology – Page 2 – Cosmic Reflections

Do Dark Matter and Dark Energy Exist?

Numerous searches for the particle or particles responsible for dark matter have so far come up empty. What if dark matter doesn’t really exist? Could there be alternative explanation for the phenomena attributed to dark matter?

In the November 10, 2017 issue of the Astrophysical Journal, Swiss astronomer André Maeder presents an intriguing hypothesis that non-baryonic dark matter need not exist, nor dark energy either. In “Dynamical Effects of the Scale Invariance of the Empty Space: The Fall of Dark Matter?” he suggests that scale invariance of empty space (i.e. very low density) over time could be causing the phenomena we attribute to dark matter and dark energy.

What is scale invariance? In the cosmological context, it means that empty space and its properties do not change following an expansion or contraction. Scales of length, time, mass, energy, and so on are defined by the presence of matter. In the presence of matter, space is not scale invariant. But take the matter away, and empty space may have some non-intuitive properties. The expanding universe may require adding a small acceleration term that opposes the force of gravity. In the earlier denser universe, this acceleration term was tiny in comparison to the rate at which the expansion was slowing down, but in the later emptier universe, the acceleration term dominates. Sound like dark energy, doesn’t it? But maybe it is an inherent property of empty space itself.

The existence of dark matter is primarily suggested by two observed dynamical anomalies:

Flat outer rotation curve of spiral galaxies (including the Milky Way)
Motions of galaxies within galaxy clusters

Many spiral galaxies have a well-known property that beyond a certain distance from their centers, their rotation rate (the orbital velocity of stars at that distance) stays nearly constant rather than decreasing as one would expect from Keplerian motion / Newtonian dynamics (think planets orbiting the Sun in our own solar system— the farther the planet is from the Sun, the slower it orbits). Only there seems to be evidence that the rotation curves of galaxies when they are young (as seen in the high-redshift universe) do have a Keplerian gradient, but in the present-day universe the rotation curve is flat. So, it appears, flat rotation curves could be an age effect. In other words, in the outer regions of spiral galaxies, stars may be orbiting at the same velocity as they did in the past when they were closer to the galactic center. Maeder writes:

…the relatively flat rotation curves of spiral galaxies is an age effect from the mechanical laws, which account for the scale invariant properties of the empty space at large scales. These laws predict that the circular velocities remain the same, while a very low expansion rate not far from the Hubble rate progressively extends the outer layers, increasing the radius of the Galaxy and decreasing its surface density like 1/t.

We need to study the rotation curves (as a function of galactocentric radius all the way out to the outermost reaches of the galaxy) of many more galaxies at different redshifts (and thus ages) to help us test the validity of the scale invariant vs. dark matter hypotheses. Maeder suggests a thorough rotation study of two massive and fast-rotating galaxies, UGC 2953 (a.k.a. IC 356; 50-68 Mly) and UGC 2487 (a.k.a. NGC 1167; 219-225 Mly), would be quite interesting.

The observed motions of galaxies within many galaxy clusters seems to indicate there is a substantial amount of unseen mass within these clusters, through application of the virial theorem. However, the motions within some galaxy clusters such as Coma (336 Mly) and Abell 2029 (1.1 Gly) may be explainable without the need to resort to “exotic” dark matter.

Then there’s the AVR (Age-Velocity Dispersion Relation) problem which, incidentally, has nothing to do with dark matter. But it may offer evidence for the scale invariant hypothesis. It is convenient to specify the motion of a star in a spiral galaxy such as the Milky Way in a galactocentric coordinate system.

U = component of velocity towards the galaxy center

V = component of velocity in the direction of galactic rotation

W = component of velocity orthogonal to the galactic plane

Maeder writes:

The AVR problem is that of explaining why the velocity dispersion, in particular for the W-component, considerably increases with the age of the stars considered … Continuous processes, such as spiral waves, collisions with giant molecular clouds, etc… are active in the disk plane and may effectively influence the stellar velocity distributions. However…vertical heating (the increase of the dispersion σ_W) is unexpected, since the stars spend most of their lifetime out of the galactic plane.

There may be more to “empty” space than meets the eye…

References
Maeder, A., 2017, ApJ, 849, 158
arXiv:1710.11425

Theory and Observation

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 F1: Philosophical choices necessarily underly cosmological theory.
Some cosmologists tend to ignore the philosophical choices underlying their theories; but simplistic or unexamined philosophical standpoints are still philosophical standpoints!

Cosmology, and indeed all human inquiry, is based on (at least) two unproven (though certainly reasonable) assumptions:

The Universe exists.
The human mind is at least to some degree capable of perceiving and understanding the Universe.

Any cosmological theory will have additional foundational unproven assumptions. These are called axioms. Ellis admonishes us to at least be aware of them, and to admit to them.

8.1 Criteria for theories
As regards criteria for a good scientific theory, typical would be the following four areas of assessment: (1) Satisfactory structure: (a) internal consistency, (b) simplicity (Ockham’s razor), and (c) aesthetic appeal (‘beauty’ or ‘elegance’); (2) Intrinsic explanatory power: (a) logical tightness, (b) scope of the theory—the ability to unify otherwise separate phenomena, and (c) probability of the theory or model with respect to some well-defined measure; (3) Extrinsic explanatory power, or relatedness: (a) connectedness to the rest of science, (b) extendability—providing a basis for further development; (4) Observational and experimental support, in terms of (a) testability: the ability to make quantitative as well as qualitative predications that can be tested; and (b) confirmation: the extent to which the theory is supported by such tests as have been made.

As you can see, a theory is not an opinion. It must be well-supported by facts. It must be internally consistent. It must have explanatory power. The Russian physicist A. I. Kitaĭgorodskiĭ (1914-1985) put it succinctly: “A first-rate theory predicts; a second-rate theory forbids, and a
third-rate theory explains after the event.” Einstein’s special and general relativity are spectacular examples of first-rate theories. In over 100 years of increasingly rigorous and sophisticated experiments and observations, relativity has never been proven to be incorrect.

Ellis emphasizes the importance of observational and experimental support in any scientific theory.

It is particularly the latter that characterizes a scientific theory, in contrast to other types of theories claiming to explain features of the universe and why things happen as they do. It should be noted that these criteria are philosophical in nature in that they themselves cannot be proven to be correct by any experiment. Rather their choice is based on past experience combined with philosophical reflection. One could attempt to formulate criteria for good criteria for scientific theories, but of course these too would need to be philosophically justified. The enterprise will end in infinite regress unless it is ended at some stage by a simple acceptance of a specific set of criteria.

So, even our criteria about what makes a good scientific theory rest upon axioms that cannot be proven. But unlike religion, scientific theories never posit the existence of any supernatural entity.

Thesis F3: Conflicts will inevitably arise in applying criteria for satisfactory cosmological theories.
The thrust of much recent development has been away from observational tests toward strongly theoretically based proposals, indeed sometimes almost discounting observational tests. At present this is being corrected by a healthy move to detailed observational analysis of the consequences of the proposed theories, marking a maturity of the subject. However because of all the limitations in terms of observations and testing, in the cosmological context we still have to rely heavily on other criteria, and some criteria that are important in most of science may not really make sense.

String theory? Cosmic inflation? Multiverse? If a theory is currently neither testable nor directly supported by observations, is it science, or something else?

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]

Emergence of Complexity

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

7.3 Emergence of complexity
As the universe evolves an increase of complexity takes place in local systems as new kinds of objects come into being that did not exist before—nuclei, atoms, stars and galaxies, planets, life, consciousness, and products of the mind such as books and computers. New kinds of physical states come into being at late times such as Bose-Einstein condensates, that plausibly cannot exist without the intervention of intelligent beings.

The first atoms formed about 400 thousand years after the Big Bang. The first stars, at about 100 million years. The emergence of atoms, stars, planets, life, intelligence, humans, morality, a Brahms symphony, etc. are a natural consequence of all the physical laws that existed at the moment of the Big Bang, 13.8 billion years ago. There is nothing supernatural about the unfolding of the universe, remarkable as it is. It is a completely natural process. The only possibility of anything supernatural, I believe, is the cause of the Big Bang itself. And, without scientific evidence…

We may never know or be able to understand the Big Bang, but the parturient possibilities contained in that creative moment are truly mind boggling: all that we see around us, all that was and is yet to be, existed then in a nascent state. The universe as it evolves is not merely moving the furniture around, but it is creating entirely new structures and entities that never existed before.

Through the emergence of intelligence across billions of years, the universe has, at last, become self-aware. Our consciousness is its consciousness.

What Is and What Might Have Been

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 E2: We cannot take the nature of the laws of physics for granted.
One cannot take the existence and nature of the laws of physics (and hence of chemistry) as unquestionable in cosmology—which seems to be the usual habit in biological discussions on the origin and evolution of life. This is in stark contrast to the rest of science, where we are content to take the existence and nature of the laws describing the fundamental behaviour of matter as given and unchangeable. Cosmological investigation is interested in the properties of hypothetical universes with different physical behaviour. Consideration of ‘what might have been’ is a useful cosmological speculation that may help throw light on what actually is; this is a statement of the usefulness of ‘Gedanken experiments‘ in cosmology.

Practical science, engineering, and technology are prescriptive. If we do a, we know from experience that b will occur. Using the laws of physics, we can predict the location of the Moon as a function of time, put a spacecraft in orbit around Saturn, or build a light bulb that will illuminate. Though we may be curious, we are not required to know why or how these laws exist—or how they might have been different—only that they do work, time and time again.

Cosmology, though firmly rooted in science, is different. We are passive observers in a very large and very old universe, and there is no absolute guarantee that the laws of physics that work for us so well in the here and now apply to all places and at all times. We must attempt to understand the laws of physics in a larger context that does involve some well-reasoned and reasonable speculation.

“Not only does God … play dice, but He sometimes confuses us by throwing them where they can’t be seen.” – Stephen Hawking

“Sometimes attaining the deepest familiarity with a question is our best substitute for actually having the answer.” – Brian Greene

In politics, governance, sociology, and philosophy, too, I would submit to you that consideration of “what might have been” is useful in helping us to understand what actually is. Such reflection, en masse, might even lead to substantive change.

“Why is it that here in the United States we have such difficulty even imagining a different sort of society from the one whose dysfunctions and inequalities trouble us so? We appear to have lost the capacity to question the present, much less offer alternatives to it. Why is it so beyond us to conceive of a different set of arrangements to our common advantage?” – Tony Judt

Getting back to cosmology, however, for the moment…

Indeed if one wants to investigate issues such as why life exists in the universe, consideration of this larger framework—in essence, a hypothetical ensemble of universes with many varied properties—is essential (this is of course not the same as assuming an ensemble of such universes actually exists). However, we need to be very cautious about using any claimed statistics of universes in such a hypothetical ensemble of all possible or all conceivable universes. This is usually not well defined, and in any case is only relevant to physical processes if either the ensemble actually exists, rather than being a hypothetical one, or if it is the outcome of processes that produce well-defined probabilities—an untestable proposal. We can learn from such considerations the nature of possible alternatives, but not necessarily the probability with which they might occur (if that concept has any real meaning).

It is easy to imagine a universe without life. But we obviously do not live in such a universe. There may be other universes devoid of life.

For the more thoughtful among us, it is easy to imagine a civilization without war, guns, violence, extrinsic suffering¹ caused by others, or deprivation. Obviously, we do not live in such a society. But how can we say it is impossible, or even improbable? It would be easy to find many millions of people in the world even today that would never fight in a war, would never own or use a gun, who would never resort to violence, who would never cause others to suffer, and who would make eliminating deprivation and poverty a top priority. The question for the scientists is: what is wrong with the rest of us?

^{¹Extrinsic suffering is suffering caused by others or circumstances completely outside of one’s control. Intrinsic suffering, on the other hand, is self-inflicted—through our own failings, poor judgement, or mistakes that we make.}

Growing Older

As we grow older,
That which is older grows upon us.
Time accelerates,
And the world seems a smaller place.

The years go by like months,
The months go by like weeks,
The weeks go by like days,
The days go by like hours,
And the hours go by like minutes.

And our world which in our youth was all that we knew
Slowly reveals itself to be a surprisingly alien place,
Full of centuries of hard work, unlikely events, and compromise:
The world could be a very different (and better) place,
Even within the confines of human nature.

Taken to its natural conclusion
Were we each to live for millennia, perhaps longer
We would find eternity in an instant
And infinity at the door.

David Oesper

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?

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]

The Sachs-Wolfe Effect

The cosmic microwave background (CMB) peaks at a wavelength of 1.9 mm and frequency 160.23 GHz, if spectral radiance is defined in terms of frequency. If spectral radiance is defined in terms of wavelength, then the CMB peaks at wavelength 1.1 mm. This radiation comes from all directions, and the curve of intensity as a function of wavelength very closely approximates a perfect black body having temperature 2.725 Kelvin. Since the Big Bang 13.8 billion years ago, the universe has expanded and cooled so that today its temperature is 2.725 K.

About 380,000 years after the Big Bang, the universe had expanded and cooled enough so that for the first time it became transparent to electromagnetic radiation. Thus when we accurately map the exact spectrum of the cosmic microwave background in different directions, we can construct a “baby picture” of the universe when it was only 380,000 years old.

Our baby picture is not smooth but has features. At that early time, the universe had already developed into denser regions, and less dense ones. Now, it is important to note that cosmic microwave background photons that left a denser part of the universe have been gravitationally redshifted to slightly longer wavelengths (and lower frequencies) to a greater extent than elsewhere. This is called the Sachs-Wolfe effect.

Lots of exciting cosmological information is coming out of mapping the tiny differences in the CMB spectrum as we look in different directions. I’m wondering, though, if anyone has seen temporal variations in the CMB? In other words, if you stay pointed in a particular direction and carefully measure the CMB spectrum over time, does it change or fluctuate at all (after all sources of noise have been removed)? Even though our current understanding of cosmology might lead us to believe that the CMB would not change fast enough for us to measure, has anybody looked?

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?