## Emergence

Physics is the fundamental science in that it describes the workings of the universe at all scales.  No other science is so comprehensive.

Will our knowledge of physics finally lead us to a “Theory of Everything”?  Perhaps, but the Theory of Everything alone will not be able to describe, predict, or explain its full expression upon/within the universe—no more so than our musical notation system can explain how a Brahms symphony was composed, nor its effect upon the listener.

Reductionism states that the whole is the sum of its parts, but emergence states that the whole is more than the sum of its parts.

There are many examples of emergent properties in the natural world, what one might call radical novelty.  Some examples:  crystal structure (e.g. a salt crystal or a snowflake), ripples in a sand dune, clouds, life itself.  Social organization (e.g. a school of fish or a city), consciousness.

John Archibald Wheeler (1911-2008) created a diagram that nicely illustrates an emergent property of the universe that is important to us.

Richard Wolfson writes,

At some level of complexity, emergent properties become so interesting that, although we understand that they come from particles that are held together by the laws of physics, we can’t understand or appreciate them through physics alone.

I like to think of emergence as an expression of creativity. Our universe is inherently creative, just as we humans express ourselves creatively through music, art, literature, architecture, and in so many other ways.

Creativity is the most natural process in the universe. It’s in our DNA.

But DNA alone can’t explain it.

References

Richard Wolfson, The Great Courses, Course No. 1280, “Physics and Our Universe: How It All Works”, Lecture 1: “The Fundamental Science”, 2011.

“And the end of all our exploring will be to arrive where we started and know the place for the first time.” – T. S. Eliot

## Counting Stars

Looking in all directions, how many stars are there brighter than a particular visual magnitude? Here’s an empirical formula that gives an approximation. It can be used over the range mv = +4.0 to +25.0.

$\textup{S} = 10^{-0.0003\,\textup{m}^{3} + 0.0019\,\textup{m}^{2} + 0.484\,\textup{m} + 0.795}$

where S is the approximate number of stars brighter than apparent visual magnitude m in the entire sky

Apparent Visual Magnitude# of Stars
4.0552
4.1618
4.2690
4.3772
4.4863
4.5964
4.61,077
4.71,204
4.81,345
4.91,503
5.01,679
5.11,875
5.22,094
5.32,338
5.42,611
5.52,914
5.63,253
5.73,631
5.84,051
5.94,520
6.05,042
6.15,623
6.26,271
6.36,992
6.47,794
6.58,687
6.69,681
6.710,786
6.812,015
6.913,382
7.014,900
7.116,588
7.218,464
7.320,547
7.422,860
7.525,428
7.628,278
7.731,441
7.834,949
7.938,839
8.043,152
8.147,932
8.253,229
8.359,096
8.465,592
8.572,784
8.680,743
8.789,549
8.899,287
8.9110,055
9.0121,955
9.1135,104
9.2149,627
9.3165,662
9.4183,362
9.5202,891
9.6224,431
9.7248,181
9.8274,358
9.9303,200
10.0334,965
10.1369,938
10.2408,426
10.3450,768
10.4497,330
10.5548,514
10.6604,755
10.7666,528
10.8734,349
10.9808,780
11.0890,430
11.1979,963
11.21,078,096
11.31,185,610
11.41,303,349
11.51,432,229
11.61,573,241
11.71,727,456
11.81,896,035
11.92,080,230
12.02,281,392
12.12,500,983
12.22,740,574
12.33,001,863
12.43,286,675
12.53,596,976
12.63,934,877
12.74,302,651
12.84,702,734
12.95,137,742
13.05,610,480
13.16,123,951
13.26,681,371
13.37,286,180
13.47,942,053
13.58,652,916
13.69,422,957
13.710,256,640
13.811,158,721
13.912,134,260
14.013,188,640
14.114,327,575
14.215,557,134
14.316,883,749
14.418,314,236
14.519,855,805
14.621,516,082
14.723,303,122
14.825,225,420
14.927,291,933
15.029,512,092
15.131,895,815
15.234,453,520
15.337,196,142
15.440,135,142
15.543,282,516
15.646,650,811
15.750,253,128
15.854,103,131
15.958,215,053
16.062,603,700
16.167,284,449
16.272,273,253
16.377,586,632
16.483,241,673
16.589,256,016
16.695,647,847
16.7102,435,879
16.8109,639,337
16.9117,277,932
17.0125,371,840
17.1133,941,667
17.2143,008,417
17.3152,593,453
17.4162,718,451
17.5173,405,353
17.6184,676,315
17.7196,553,644
17.8209,059,737
17.9222,217,010
18.0236,047,823
18.1250,574,401
18.2265,818,743
18.3281,802,538
18.4298,547,061
18.5316,073,074
18.6334,400,717
18.7353,549,396
18.8373,537,665
18.9394,383,103
19.0416,102,189
19.1438,710,168
19.2462,220,923
19.3486,646,831
19.4511,998,631
19.5538,285,275
19.6565,513,790
19.7593,689,134
19.8622,814,048
19.9652,888,922
20.0683,911,647
20.1715,877,479
20.2748,778,904
20.3782,605,508
20.4817,343,852
20.5852,977,352
20.6889,486,170
20.7926,847,110
20.8965,033,523
20.91,004,015,228
21.01,043,758,439
21.11,084,225,707
21.21,125,375,873
21.31,167,164,044
21.41,209,541,573
21.51,252,456,065
21.61,295,851,393
21.71,339,667,742
21.81,383,841,658
21.91,428,306,130
22.01,472,990,684
22.11,517,821,499
22.21,562,721,546
22.31,607,610,744
22.41,652,406,140
22.51,697,022,107
22.61,741,370,568
22.71,785,361,232
22.81,828,901,853
22.91,871,898,516
23.01,914,255,925
23.11,955,877,722
23.21,996,666,815
23.32,036,525,723
23.42,075,356,932
23.52,113,063,265
23.62,149,548,260
23.72,184,716,557
23.82,218,474,290
23.92,250,729,483
24.02,281,392,450
24.12,310,376,189
24.22,337,596,778
24.32,362,973,766
24.42,386,430,550
24.52,407,894,751
24.62,427,298,570
24.72,444,579,131
24.82,459,678,812
24.92,472,545,544
25.02,483,133,105

How many stars are there in our Milky Way galaxy? Between 100 and 400 billion stars. Many stars are not very luminous, and can only be seen in the immediate solar neighborhood. That is one source of uncertainty.

How many galaxies are there in the observable universe? Something like two trillion (2 × 1012).

How many stars are in the observable universe? Something like a septillion (1024). A trillion trillion!

And, just so you know, our universe is probably much larger than the volume that we can observe.

How does the Universe love thee? Let us count the stars…

References

“How many stars are in the sky?”, Space Math, NASA Goddard Space Flight Center, accessed February 29, 2020, https://spacemath.gsfc.nasa.gov/weekly/6Page103.pdf.

Wikipedia contributors, “Galaxy,” Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/w/index.php?title=Galaxy&oldid=942479372 (accessed February 29, 2020).

Wikipedia contributors, “Milky Way,” Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/w/index.php?title=Milky_Way&oldid=942977760 (accessed February 29, 2020).

## Multiverse

George F. R. Ellis writes in Issues in the Philosophy of Cosmology:

9.2 Issue H: The possible existence of multiverses
If there is a large enough ensemble of numerous universes with varying properties, it may be claimed that it becomes virtually certain that some of them will just happen to get things right, so that life can exist; and this can help explain the fine-tuned nature of many parameters whose values are otherwise unconstrained by physics.  As discussed in the previous section, there are a number of ways in which, theoretically, multiverses could be realized.  They provide a way of applying probability to the universe (because they deny the uniqueness of the universe).  However, there are a number of problems with this concept.  Besides, this proposal is observationally and experimentally untestable; thus its scientific status is debatable.

My 100-year-old uncle—a lifelong teacher and voracious reader who is still intellectually active—recently sent me Max Tegmark’s book Our Mathematical Universe: My Quest for the Ultimate Nature of Reality, published by Vintage Books in 2014. I could not have had a more engaging introduction to the concept of the Multiverse. Tegmark presents four levels of multiverses that might exist. They are

Level I Multiverse: Distant regions of space with the same laws of physics that are currently but not necessarily forever unobservable.

Level II Multiverse: Distant regions of space that may have different laws of physics and are forever unobservable.

Level III Multiverse: Quantum events at any location in space and in time cause reality to split and diverge along parallel storylines.

Level IV Multiverse: Space, time, and the Level I, II, and III multiverses all exist within mathematical structures that describe all physical existence at the most fundamental level.

There seems little question that our universe is very much larger than the part that we can observe. The vast majority of our universe is so far away that light has not yet had time to reach us from those regions. Whether we choose to call the totality of these regions the universe or a Level I multiverse is a matter of semantics.

Is our universe or the Level I multiverse infinite? Most likely not. That infinity is a useful mathematical construct is indisputable. That infinite space or infinite time exists is doubtful. Both Ellis and Tegmark agree on this and present cogent arguments as to why infinity cannot be associated with physical reality. Very, very large, or very, very small, yes, but not infinitely large or infinitely small.

Does a Level II, III, and IV multiverse exist? Tegmark thinks so, but Ellis raises several objections, noted above and elsewhere. The multiverse idea remains quite controversial, but as Tegmark writes,

Even those of my colleagues who dislike the multiverse idea now tend to grudgingly acknowledge that the basic arguments for it are reasonable. The main critique has shifted from “This makes no sense and I hate it” to “I hate it.”

I will not delve into the details of the Level II, III, and IV multiverses here. Read Tegmark’s book as he adroitly takes you through the details of eternal inflation, quantum mechanics and wave functions and the genius and tragic story of Hugh Everett III, the touching tribute to John Archibald Wheeler, and more, leading into a description of each multiverse level in detail.

I’d like to end this article with a quote from Max Tegmark from Mathematical Universe. It’s about when you think you’re the first person ever to discover something, only to find that someone else has made that discovery or had that idea before.

Gradually, I’ve come to totally change my feelings about getting scooped. First of all, the main reason I’m doing science is that I delight in discovering things, and it’s every bit as exciting to rediscover something as it is to be the first to discover it—because at the time of the discovery, you don’t know which is the case. Second, since I believe that there are other more advanced civilizations out there—in parallel universes if not our own—everything we come up with here on our particular planet is a rediscovery, and that fact clearly doesn’t spoil the fun. Third, when you discover something yourself, you probably understand it more deeply and you certainly appreciate it more. From studying history, I’ve also come to realize that a large fraction of all breakthroughs in science were repeatedly rediscovered—when the right questions are floating around and the tools to tackle them are available, many people will naturally find the same answers independently.

References
Ellis, G.F.R., 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]

Tegmark, Max. Our mathematical universe : my quest for the ultimate nature of reality. New York: Alfred A. Knopf, 2014.

“You passed your exam in many parallel universes—but not in this one.”

## The Laws of Physics and the Existence of Life

George F. R. Ellis writes in Issues in the Philosophy of Cosmology:

The first requirement is the existence of laws of physics that guarantee the kind of regularities that can underlie the existence of life.  These laws as we know them are based on variational and symmetry principles; we do not know if other kinds of laws could produce complexity.  If the laws are in broad terms what we presently take them to be, the following inter alia need to be right, for life of the general kind we know to exist:

• Quantization that stabilizes matter and allows chemistry to exist through the Pauli exclusion principle.

• The neutron-proton mass differential must be highly constrained.  If the neutron mass were just a little less than it is, proton decay could have taken place so that by now no atoms would be left at all.

• Electron-proton charge equality is required to prevent massive electrostatic forces overwhelming the weaker electromagnetic forces that govern chemistry.

• The strong nuclear force must be strong enough that stable nuclei exist; indeed complex matter exists only if the properties of the nuclear strong force lies in a tightly constrained domain relative to the electromagnetic force.

• The chemistry on which the human body depends involves intricate folding and bonding patterns that would be destroyed if the fine structure constant (which controls the nature of chemical bonding) were a little bit different.

• The number D of large spatial dimensions must be just 3 for complexity to exist.

It should not be too surprising that we find ourselves in a universe whose laws of physics are conducive to the existence of semi-intelligent life.  After all, we are here.  What we do not know—and will probably never know: Is this the only universe that exists?  This is an important question, because if there are many universes with different laws of physics, our existence in one of them may be inevitable.  If, on the other hand, this is the only universe, then the fantastic claims of the theists, or at least the deists, become more plausible.

You may wonder why I call the human race semi-intelligent.  Rest assured, I am not being sarcastic or sardonic.  I say “semi-intelligent” to call attention to humanity’s remarkable technological and scientific achievements while also noting our incredible ineptness at eradicating war, violence, greed, and poverty from the world.  What is wrong with us?

References
G.F.R. Ellis, 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 Anthropic Question

George F. R. Ellis writes in Issues in the Philosophy of Cosmology:

9.1 Issue G: The anthropic question: Fine tuning for life
One of the most profound fundamental issues in cosmology is the Anthropic question: why does the Universe have the very special nature required in order that life can exist?  The point is that a great deal of “fine tuning” is required in order that life be possible.  There are many relationships embedded in physical laws that are not explained by physics, but are required for life to be possible; in particular various fundamental constants are highly constrained in their values if life as we know it is to exist:

Ellis goes on to quote Martin Rees.

A universe hospitable to life—what we might call a biophilic universe—has to be special in many ways … Many recipes would lead to stillborn universes with no atoms, no chemistry, and no planets; or to universes too short lived or too empty to evolve beyond sterile uniformity.

Physics does not tell us anything (yet) about why the fundamental constants and other parameters have the values they do.  These parameters include, for example, the speed of light, the Planck constant, the four fundamental forces and their relative strengths, the mass ratio of the proton and the electron, the fine-structure constant, the cosmological density parameter, Ωtot, relative to the critical density, and so on.  And, why are there four fundamental forces?  Why not five?  Or three?

Also, why do we live in a universe with three spatial dimensions and one time dimension?  Others are possible—even universes with two or more time dimensions.

But it appears that only three spatial dimensions and one time dimension is conducive to life (at least life as we know it), as shown in the diagram above (Whittle 2008).

In fact, altering almost any of the parameters would lead to a sterile universe and we could not exist.  Is the universe fine-tuned for our existence?

Let’s assume for the moment it is.  Where does that lead us?

1. As our understanding of physics advances, we will eventually understand why these parameters must have the values that they do. -or-
2. We will eventually learn that some of these parameters could have been different, and still support the existence of life. -or-
3. God created the universe in such a way that life could exist -or-
4. We’re overthinking the problem.  We live in a life-supporting universe, so of course we find the parameters are specially tuned to allow life. -or-
5. There exist many universes with different parameters and we just happen to find ourselves in one that is conducive to life. (The multiverse idea.)

#4 is the anthropic explanation, but a deeper scientific understanding will occur if we find either #1, #2, or #5 to be true.  #3 is problematic for a couple of reasons.  First of all, how was God created?  Also, deism has a long history of explaining phenomena we don’t understand (“God of the gaps”), but in time we are able to understand each phenomenon in turn as science progresses.

The anthropic explanation itself is not controversial.  What is controversial is deciding to what degree fine tuning has occurred and how to explain it.

In recent years, the multiverse idea has become more popular because, for example, if there were a billion big bangs and therefore a billion different universes created, then it should not be at all surprising that we find ourselves in  one with just the right set of parameters to allow our existence.  However, there is one big problem with the multiverse idea.  Not only do we have no physical evidence that a multiverse exists, but we may never be able to obtain evidence that a multiverse exists, due to the cosmological horizon problem1.  If physical evidence of a multiverse is not forthcoming, then in that sense it is not any better than the deistic explanation.

To decide whether or not there is only one combination of parameters that can lead to life we need to rule out all the other combinations, and that is a tall order.  Recent work in this field suggests that there is more than one combination of parameters that could create a universe that is hospitable to life (Hossenfelder 2018).

Thinking now about why our universe is here at all, it seems there are just two possibilities:

(1)  Our universe has a supernatural origin.

(2)  Our universe has a natural origin.

If our universe has a supernatural origin, then what is the origin of the supernatural entity (e.g. God)?  If, on the other hand, our universe had a natural origin (e.g. something was created out of nothing), didn’t something have to exist (laws of physics or whatever) before the universe came into existence?  If so, what created those pre-conditions?

In either case, we are facing an infinite regression.  However, we could avoid the infinite regression by stating that something has to exist outside of time, that is to say, it has no beginning and no ending.  But isn’t this just replacing one infinity with another?

Perhaps there’s another possibility.  Just as a chimpanzee cannot possibly understand quantum mechanics, could it be that human intellect is also fundamentally limited?  Are the questions in the previous two paragraphs meaningless or nonsensical in the context of some higher intelligence?

1We appear to live in a universe that is finite but very much larger than the region that is visible to us now, or ever.

References
G.F.R. Ellis, 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]

Sabine Hossenfelder, Lost in Math: How Beauty Leads Physics Astray (Basic Books, 2018).

M. J. Rees, Our Cosmic Habitat (Princeton and Oxford, 2003).

Mark Whittle, “Fine Tuning and Anthropic Arguments”, Lecture 34, Course No. 1830.  Cosmology: The History and Nature of Our Universe.  The Great Courses, 2008.  DVD.
[https://www.thegreatcourses.com/courses/cosmology-the-history-and-nature-of-our-universe.html]

## Where Cosmology Meets Philosophy

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 physical explanatory power of inflation in terms of structure formation, supported by the observational data on the fluctuation spectra, is spectacular.  For most physicists, this trumps the lack of identification and experimental verification of the underlying physics.  Inflation provides a causal model that brings a wider range of phenomena into what can be explained by cosmology, rather than just assuming the initial data had a specific restricted form.  Explaining flatness (Ω0 ≅ 1 as predicted by inflation) and homogeneity reinforces the case, even though these are philosophical rather than physical problems (they do not contradict any physical law; things could just have been that way).  However claims on the basis of this model as to what happens very far outside the visual horizon (as in the chaotic inflationary theory) results from prioritizing theory over the possibility of observational and experimental testing.  It will never be possible to prove these claims are correct.

Inflation is one compelling approach to explaining the structure we see in the universe today.  It is not necessarily the only one, but it currently has the most support.  Basically, a tiny fraction of a second after the Big Bang, the universe expanded dramatically.  Around 10-36 seconds after the Big Bang the universe had a diameter on the order of 1.2 × 10-27 meters.  To put that size in perspective, the diameter of a proton is between 0.84-0.87 × 10−15 meters.  So, when inflation began, the entire universe had a diameter almost a trillion times smaller than a single proton!  10-34 seconds later when the inflationary period was coming to an end, the size of the universe was a little over half the distance to Alpha Centauri!

The basic underlying cosmological questions are:
(1)  Why do the laws of physics have the form they do?  Issues arise such as what makes particular laws work?  For example, what guarantees the behaviour of a proton, the pull of gravity?  What makes one set of physical laws ‘fly’ rather than another?  If for example one bases a theory of cosmology on string theory, then who or what decided that quantum gravity would have a nature well described by string theory?  If one considers all possibilities, considering string theory alone amounts to a considerable restriction.
(2)  Why do boundary conditions have the form they do?  The key point here is, how are specific contingent choices made between the various possibilities, for example whether there was an origin to the universe or not.
(3)  Why do any laws of physics at all exist?  This relates to unsolved issues concerning the nature of the laws of physics: are they descriptive or prescriptive?  Is the nature of matter really mathematically based in some sense, or does it just happen that its behaviour can be described in a mathematical way?
(4)  Why does anything exist?  This profound existential question is a mystery whatever approach we take.

The answer to such questions may be beyond the limits of experimental science, or even beyond the limits of our intellect.  Maybe, even, these questions are as meaningless as “What lies north of the north pole?1because of our limited intellect.  Many would claim that because there appears to be limits to what science or human intellect can presently explain, that this constitutes evidence for the existence of God.  It does not.  Let’s just leave it as we don’t know.

Finally, the adventurous also include in these questions the more profound forms of the contentious Anthropic question:
(5)  Why does the universe allow the existence of intelligent life?
This is of somewhat different character than the others and largely rests on them but is important enough to generate considerable debate in its own right.

Well, a seemingly flippant answer to this question is we wouldn’t be here if it didn’t, but that begs the question.  Perhaps intelligent life is the mechanism by which the universe becomes self-aware, or is this just wishful thinking?  In the end, I am willing to admit that there may be some higher power in the universe—in the scientific pantheist and humanist sense—but I will stop short of calling that “God” in any usual sense of the term.

The status of all these questions is philosophical rather than scientific, for they cannot be resolved purely scientifically.  How many of them—if any—should we consider in our construction of and assessments of cosmological theories?

Perhaps the limitations of science (and, therefore, cosmology) is more a manifestation of the limitations of our human intellect than any constraint on the universe itself.

One option is to decide to treat cosmology in a strictly scientific way, excluding all the above questions, because they cannot be solved scientifically.  One ends up with a solid technical subject that by definition excludes such philosophical issues.  This is a consistent and logically viable option.  This logically unassailable position however has little explanatory power; thus most tend to reject it.

Let’s call this physical cosmology.

The second option is to decide that these questions are of such interest and importance that one will tackle some or all of them, even if that leads one outside the strictly scientific arena.  If we try to explain the origin of the universe itself, these philosophical choices become dominant precisely because the experimental and observational limits on the theory are weak; this can be seen by viewing the variety of such proposals that are at present on the market.

And let’s call this metaphysical cosmology.

1Attributed to Stephen Hawking

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]

Ryden, Barbara. 2003.  Introduction to Cosmology. San Francisco: Addison Wesley.

## 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:

1. Flat outer rotation curve of spiral galaxies (including the Milky Way)
2. 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 σW) 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

## 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

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 = mp / me ≅ 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 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 H0 in 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?

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]

## 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.

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.

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.

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]