Lecture 31: Cosmology

Don't Panic

Sections

1. Introduction
2. Cosmology, Human Concerns, and Philosophy
3. The Early History of Cosmology
4. The Expansion of the Universe
5. Hubble Time, Hubble Length, and the Observable Universe
6. Einstein, General Relativity, and the Einstein Universe
7. Friedmann-Lemaitre Models
8. The Accelerating Universe and the Friedmann-Lemaitre-Lambda Models
9. Dark Energy
10. The Concordance Model
11. The Fate of the Universe According to the Concordance Model
12. Big Bang Cosmology and the Constituents of Observable Universe
13. Inflation and Inflation Cosmology
14. Conclusion

---

Introduction

---

Modern cosmology is the science of the universe as a whole.

But, of course, it is not the science of everything.

Modern cosmology restricts itself to average behavior of the universe and, at just a smaller scale, to the large-scale structure of the universe: galaxies and galaxy clusters, superclusters, filaments, sheets, and voids.

To understand these things often entails understanding smaller things to some degree: e.g., stars, supernovae, supermassive black holes, quasars, atoms, nuclei, and quarks. It also requires macroscopic physics, thermodynamics, and quantum mechanics.

But there are things which are definitely excluded beyond the scope of cosmology: e.g., planets, biota, human beings, psychology, etc.

---

Cosmology, Human Concerns, and Philosophy

---

Cosmology in the broad sense has always been of importance to humans.

We are concerned about the meaning, purpose, and nature of our own existence.

Therefore about the meaning, purpose, and nature of the universe which sustains us and everything else.

Modern cosmology, of course, concerns itself with NATURE OF and leaves aside MEANING AND PURPOSE.

But MEANING AND PURPOSE probably hover just somewhere beyond the expressed concern many modern scientific cosmologists and are probably of interest to everyone interested in cosmology

If we ever did finally fathom the universe, it is easy to believe that that knowledge would have PHILOSOPHICAL IMPLICATIONS.

But modern scientific cosmologists are usually---but not always---reluctant to connect current thinking up with philosophical theories.

They are well aware that modern cosmological theories may well be wrong or superficial, and so drawing philosophical conclusions is premature---and, of course, we don't know how to draw them accurately anyway.

For example, Georges Lemaitre (1894--1966), who can justly be considered the grandparent of big bang cosmology, was a Jesuit et devot\'e a sa foi, naturalmente (No-525,530).

Georges Lemaitre (1894--1966).

But Lemaitre resisted identification of his PRIMEVAL ATOM THEORY (the theoretical ancestor of the big bang) with the creation of Genesis.

The former was speculative science; the latter, faith.

See Institut d'Astronomie et de Geophysique Georges Lemaitre. There is a good picture of Lemaitre with Robert Millikan and Einstein at Caltech on 1933jan10.

We will argue below big bang cosmology is NOT just speculative science any more: it is probably essentially right as far as it goes: at least it would be astonishing if it were just WRONG. (See section Big Bang Cosmology and the Constituents of Observable Universe.)

But it doesn't in itself tell us what happened before the big bang, what happened in other universe domains if they exist, and what the fate of the universe or our universe domain will be.

Universe domains illustrated.

So big bang cosmology is still superficial in deepest sense---or so most cosmologists would say, I think.

The issues outside of big bang cosmology are dealt with in broader theories, but those theories are more speculative and may well be just WRONG.

Of these broader theories inflation cosmology and the CYCLIC MODEL are the most favored: we will discuss inflation cosmology and briefly the CYCLIC MODEL below in the section Inflation and Inflation Cosmology.

Historically, cosmologists havn't been so circumspect about PHILOSOPHICAL IMPLICATIONS.

Myth-oriented cosmologists and philosophical cosmologists were or are often concerned with these implications.

Certainly, it's fun to play in PHILOSOPHY provided you don't take your own idiosyncratic ideas too seriously.

And who's to say when the next great advance in understanding in PHILOSOPHY will come and clarify things for us.

---

The Early History of Cosmology

---

All human societies, one supposes, have their own cosmological stories.

Most of these, one supposes---one hasn't done an actual count---have anthropomorphic dieties that order the universe.

But in even in these myths there is one supposes---one hasn't done an actual count---a regression to some simple early state with a very primitive matter and order perhaps somewhat personified as a very primitive god.

For example, the ancient Greek Hesiod's Theogony (origin of the gods) (Hesiod's Theogony translated by Hugh G. Evelyn-White 1914) posits CHAOS as the first substance or god and next GAIA (Mother Earth) and then other useful gods: Tartaros (Hell), Eros (Love), Erebos (Gloom), Nyx (Night), etc. ( The Structure of Hesiod's Theogony: Joe Farrell, University of Pennsylvania). Gods with more personality come in later generations.

So it seems that humans are predisposed to think that there must be a relatively simple beginning in time---or more abstractly outside of time.

The ancient Greek philosophers beginning in the 6th century BCE are the first persons recorded in history to try to develop rational theories about the universe: theories subject to argument and correction.

The Greeks were---compared to modern standards---weak on detailed observation and experimentation---they did practise them at times---and this limited their progress in cosmology.

One can characterize much of their theorizing as the making of RATIONAL MYTHS.

Some of their theories are very interesting.

The universe theory of the atomist philosophers LEUCIPPUS and DEMOCRITUS posited infinitely many worlds forming in vortices out of an infinite space of atoms in motion.

These worlds were always coming into existence and passing away.

This atomist theory bears are resemblance to ETERNAL INFLATION a modern theory---albeit one with a lot more math. We discuss ETERNAL INFLATION below in the section Inflation and Inflation Cosmology.

The cosmological theory that became dominant in Greco-Roman Antiquity and then in Medieval Islamic and European societies was that of Aristotle (384--322 BCE).

Aristotle, the Supreme Authority.

In his cosmology, the Earth was at the center of an eternal bounded spherical universe.

The boundary was a real physical Celestial Sphere on which the stars were pasted: the planets were closer and moved by gods or in monotheistic contexts by angels.

The Aristotelian Cosmos.

Beyond the Celestial Sphere was nothing: not even empty space---even in antiquity a lot of people found this ``not even empty space'' hard to accept.

Aristotle's theory actually bears a passing resemblance to the Einstein universe which we discuss below in the section Einstein, General Relativity, and the Einstein Universe.

[One begins to wonder if cosmology is an endless recycling of old ideas---albeit with a lot more math.]

The small Aristotelian universe was put in doubt to the astronomically-minded by Copernicus' HELIOCENTRIC SOLAR SYSTEM of 1543. First of all, by putting the Sun in the center of the SOLAR SYSTEM, of course.
Question: Was Copernicus the first proposer of the HELIOCENTRIC SOLAR SYSTEM?
1. Yes.
2. No. Some ancient Greek was first.
3. Maybe.




Answer 2 is right.

You are beginning to get the idea. Some Greek has thought of everything first.

Aristarchos of Samos (3rd century BCE): the first proposer of heliocentrism.

But Aristarchos only presented the bare idea of HELIOCENTRISM as far as we know although he probably knew some of the reasons why it was a good idea.

Copernicus is justly credited as the innovator since he presented a detailed argument that was somewhat convincing and was NOT ignorable.

This is often the way in the history of science.

The giver a convincing proof or argument gets more fame than the speculator---and this seems just.

HELIOCENTRISM also upset the Aristotelian universe by requiring the STARS to be extremely remote: this is the only reasonable way that the Earth could move and the stars not show PARALLAX to pre-modern observations.

But if the stars were very remote why should they be pasted on a big sphere?

Why not many stars spread throughout an infinite universe? In the context of heliocentrism, this idea was first put forward by Thomas Digges (circa 1546--1595) in 1576 (No-296).

As heliocentrism gained credence and the telescope revealed a quasi-infinity of new stars and that the Milky Way was a band of stars, the notion of an infinite or very large universe filled with stars became plausible.

Isaac Newton (1642/1643--1727) certainly thought in terms of infinite or very large universe filled with stars (No-375).

Newton of the Principia.

In unpublished work, Newton tried to construct a STATIC MODEL of such a universe---using Newtonian physics, of course (No-376).

His own theory of gravity suggested that matter would all collapse into a single clump at about??? the center of mass in any FINITE STATIC MODEL: it couldn't stay static.

Could extending the universe to infinity result in a balance of forces that allow the universe to stand up?

Newton did NOT come to a satisfactory answer.

In fact, Newtonian physics does NOT seem to allow one to construct an infinite universe model without new assumptions about physical laws (Bo-75,78).

Attempts in the 19th century to create a Newtonian theory of the whole universe foundered: they were all based on the idea that the universe as a whole had to be STATIC on average even though it was known that stars actually do move around (Bo-75).

The modern attempts to frame a physically consistent theory of the universe started with Einstein and his general relativistic model of 1917: this is the Einstein universe (No-520; Bo-97) which we will discuss below in the section Einstein, General Relativity, and the Einstein Universe.

---

The Expansion of the Universe

---

After the Copernican Revolution of the 16th and 17th centuries, the idea of space as quasi-infinite with a quasi-infinity of stars gained ground and Newton believed it as we mentioned above.

The history of the discovery of galaxies was discussed in IAWL 26: Discovery of Galaxies. This history is the essentially the history of observational cosmology.

Up until circa 1920, it was thought that the universe was essentially STATIC???: the stars and other galaxies (assuming they existed) were not moving on average even though star motions were known.

This belief is actually odd since the universe was obviously NOT thermodynamically static: i.e., it is NOT in thermodynamical equilibrium state:

Space is cold (as evidenced by our own planet Earth) and stars are hot.

Heat energy is steadily flowing out of stars in the form of electromagnetic radiation and not being returned.

Why should a system so obviously evolving thermodynamically be automatically assumed to be dynamically static on average? If evolving in one way, it could well be evolving in other ways?

Actually there was evidence in the early 1920s for large-scale motions.

Vesto Slipher (1875--1969) at Lowell Observatory in Flagstaff, Arizona starting in 1912 had been measuring the spectral shifts of spiral galaxies (No-522--523).

By 1925, Slipher had spectral shifts for 45 galaxies. Except for a few blueshifts, most were redshifts.

Interpreted as Doppler shifts---which in fact is NOT exactly right, though that is what was assumed originally by Slipher and, perhaps for awhile, by Edwin Hubble (1889--1953)---these results showed that most galaxies were moving away from ours and using the Doppler shift formula their RECESSION VELOCITIES were known.

The spectral shifts are actually a mixture of Doppler shifts due to relatively small peculiar motions and cosmological redshifts due to cosmological expansion.

For the blueshifted galaxies, the Doppler shift dominates: these are nearby galaxies that do not participate or not very much in a cosmological expansion relative to us.

At remoter distances, the cosmological redshift dominates.

The two effects are closely related and they coincidentally have the same 1st order formulae for relatively small shifts, and thus unclarity about which effect applied didn't wrong-track people much in the 1920s.

But they are distinct though many textbooks confuse the them.

Doppler shifts are caused by relative motions.

The cosmological redshift is caused by the growth of space under light as light travels through it. This growth stretches the wavelength of light: i.e., redshifts it.

Cosmological redshifts illustrated.
Question: Were the spiral nebulae known to be other galaxies by 1925?
1. Yes.
2. No.
3. Maybe.



Answer 1 is right.

By 1924, Hubble had shown that the Andromeda spiral nebula (M31) was another galaxy and by implication all other spirals were too (No-510).

Figuring out that ellipticals were other galaxies must have happened sometime soon thereafter.

But what was the FLOW PATTERN of the receding galaxies?

To know this you had to know, in addition to recession velocities, distances to the galaxies: i.e., where the galaxies were in space.

By 1929, Hubble had the distances by various means to 18 galaxies beyond the Milky Way (No-510).

[Recall Hubble with the 100-inch (i.e., 2.54 meter) Mt. Wilson Observatory telescope in southern California (before most of the smog and light pollution) had the best observing technology of his time (No-439).

This was essential to his discoveries.]

Using Slipher's or other spectral shifts, Hubble was able to find a remarkably simple relationship between distance and recession velocity (No-523).
He may have still been interpreting the shifts as Doppler shifts then, but maybe not.

However, as mentioned above, the 1st order Doppler shift formula, is coincidentally correct for the 1st order cosmological redshift.

This observational relationship first presented in 1929 is now called Hubble's law:

      v=Hd  ,

      where v is recession velocity nowadays measured in km/s,

      d is distance nowadays measured in Mpc,

      and H is a constant, now called the Hubble constant.

      Hubble's original value for H was

        540 (km/s)/Mpc (Bo-39).

      Hubble had considerable systematic distance errors, and so his
        value was rather badly wrong.

      The best modern value (which we will use throughout this lecture) is

        71 (+4/-3)  (km/s)/Mpc
	 (FK-653). 

Hubble extracted this relation from what we now call a Hubble diagram.

A schematic Hubble diagram. (This diagram has some kind of error I've never been able to fix.)

The line is the representation of Hubble's law and the slope of the line is Hubble's constant.

Hubble's law shows that there is a general expansion of distances between extragalactic objects when the redshift of remote objects is correctly interpreted as the cosmological redshift.

This general expansion is called the expansion of the universe.

Question: In the Hubble diagram there is a scatter about the straight line.
1. This is caused by observational error.

2. The gravitationally bound systems are not internally expanding and their components have rather random peculiar velocities which add Doppler shifts to the general expansion cosmological redshift to create net spectral shifts. The peculiar velocities cause the deviations from the straight line.

3. The gravitationally bound systems as wholes have their own rather random peculiar velocities which add Doppler shifts to the general expansion cosmological redshift to create net spectral shifts. The peculiar velocities cause the deviations from the straight line.

4. The general expansion is not exactly linear. There are large perturbations in it.





The first 3 answers are all partially right. Together they constitute what we believe to be the right answer.

So far observationally everything indicates the general expansion is exactly linear in the limit of small cosmological distances or look-back times (i.e., the travel time for light across those distances).

Remember when we look far away, we are looking to earlier epochs.

If look-back times are much less than the dynamical time scale of the universe (i.e., the time for significant motion or acceleration of the galaxies), general relativistic models of the universe predict Hubble's law for observations. We discuss these models below: see section Friedmann-Lemaitre Models and subsequent sections.

In fact, these models predict Hubble's law is exactly right for our universe domain for recession velocities and distances measured at exactly one instant in cosmic time (CL-14,47). (Such distances are called proper distances.) But this we CANNOT do this by direct observations.

These at-one-instant-in-cosmic-time recession velocities and distances must be determined by the cosmological model adopted, and so have that model's uncertainty.

Question: Hubble's law implies that:
1. the Milky Way is the center of cosmic expansion.
2. the expansion is a general scaling up of distances between non-gravitationally bound systems. All observers at located in bound systems see the same expansion behavior on average.




Well either answer could be right logically speaking.

But answer 2 is so overwhelmingly more acceptable that we must accept it as right.

There is ASSUMPTION in cosmology called Copernican principle: it states that we occupy no special place in the universe. This principle is a guiding simplifying principle in cosmology. We have no observational evidence or broadly accepted theoretical reason for thinking it is false. In fact, as far as we can tell it seems true.

The following diagram shows how to understand expansion without a center.

The universal expansion without a center.

General expansion without a center leads us into quandaries in Newtonian physics:

1. Newton postulated an absolute, infinite space. A finite system of galaxies can expand in this infinite space, but what does it mean to have an infinite system of galaxies expand? Is there an infinite system of galaxies?

2. There should be gravitational attraction between the galaxies leading to a deceleration of the expansion. We could postulate an unknown force that exactly cancels gravity and makes the expansion rate a constant. But such a force is totally ad hoc in the context of Newtonian physics:
   [In science ad hoc means just invented to solve in a non-compelling way (or to just avoid) a particular deep problem.]

   If there is a deceleration, then the systems moving with the expansion cannot be exactly inertial frames, except perhaps for one which would be very special system.

   Perhaps, it's our system. But that violates the Copernican principle. More concretely, we so see no evidence for universal non-inertial-frame effects in distant galaxies and galaxy clusters.

In fact, it seems impossible to construct pure Newtonian cosmological models that are consistent with the observable universe as we mentioned in the section The Early History of Cosmology.

SEMI-NEWTONIAN COSMOLOGICAL MODELS (with modified Newtonian physics) have been invented which are have some consistency with the observable universe (Bo-75), but since we know Newtonian physics is NOT fundamental, such models cannot be correct. They are, however, interesting historically and very useful pedagogically: but we won't pedagoge on them.

But we will remark that any physical theory of the expanding universe consistent with the Copernican principle must make inertial frames those frames participating in general expansion of the universe: i.e., the mean expansion aside from any peculiar motions.

We will take up the QUESTION of what if anything the universe or our universe domain is expanding into below in the section Friedmann-Lemaitre Models and subsequent sections.

The short answer to the QUESTION in big bang cosmology is the expansion is NOT into anything---but this may not be exactly right.

---

Hubble Time, Hubble Length, and the Observable Universe

---

Before going on to specific theories of expansion it is useful to consider certain characteristic quantities: the Hubble time and the Hubble length.

Hubble's law can be rewritten

     d=v(1/H) ,

from which one can see that 1/H must have the units of time.

If the recession velocities are CONSTANT in time, 1/H is the time since all the matter in the expanding universe was clumped together with infinite density.

We call 1/H the Hubble time t_H.

An epoch of infinite density is a singularity in which our laws of physics (including modern physis) fail. In fact the singularity is one meaning of the term big bang.

Whether there was an actual singularity at the beginning of time is moot, but most astrophysicists probably think there was NOT.

They do think the big bang happened in the other meaning of the term: a time of very high density and temperature which was at the beginning of our universe domain's history.

Now the recession velocities are very probably NOT constant.

But for our universe domain 1/H should be a characteristic age of the universe in big bang cosmological models (which we discuss below in section Friedmann-Lemaitre Models and subsequent sections): i.e., 1/H should be order of the cosmic time since the big bang (in the 2nd meaning of the term).

We will in fact from here on always assume big bang cosmological models (which we discuss below in the section Friedmann-Lemaitre Models and subsequent sections) since there is very good evidence that they are close to true for the observable universe and the only models for the observable universe widely considered at present.

With the best modern value H=71 (km/s)/Mpc (FK-653), we find

      t_H = 1/H =               1                   3.085678*10**13 km
                     _________________________ *   _____________________
	                   71 (km/s)/Mpc                1 Mpc

           = 4.35*10**17 s

           = 13.8  Gyr .

Question: If we could freeze the expansion right now, from how far away could light reach us in a Hubble time?
1. 10 Mpc.
2. 4220 Mpc.
3. c/H.



Answer 3 and, assuming H=71 (km/s)/Mpc, answer 2 is right too.

c/H=c*(1/H) is just the speed of light times the Hubble time.

We call c/H the Hubble length L_H:

     L_H = c/H =  2.998*10**5 km/s
                 __________________
	           71 (km/s)/Mpc

         = 4220 Mpc .
	     

Given that the Hubble time is a characteristic age for the universe, the Hubble length is a characteristic size scale for the observable universe.

Signals from very much farther away than a Hubble length could NOT have reached us since the big bang.

The observable universe is the largest sphere centered on us that is CAUSALLY CONNECTED to us: i.e., sphere from within which a light signal could have reached us since the big bang.

No signals could reach us from outside.

This sphere is called the particle horizon (or cosmic light horizon) (CL-46--47) and, somewhat confusingly, its radius in proper distance is also called the particle horizon.

The particle horizon, of course, had a different size in the past when the signal started out which is one reason why the observable universe is a tricky concept.

[Actually, the observable universe defined this way is probably NOT entirely observable. The universe becomes opaque to most light as one looks back in time to the big bang.

We discuss this opaqueness below in the section Big Bang Cosmology and the Constituents of Observable Universe.

Perhaps some band of electromagnetic radiation or neutrinos or gravitational radiation will allow in principle observations to farther look-back times.

The universe is estimated to have become transparent to neutrinos at cosmic time of about 2 s after the hypothetical singularity. These neutrinos should have a temperature of about 2 K now and have a density of order 4*10**8 per m**3. But neutrinos are very hard to detect and this neutrino background has not yet been detected (FK-669).

Much of the observable universe is observable, but, in fact, its radius, the particle horizon, is not.

The particle horizon depends on exactly which big bang cosmological model one adopts.

But for those big bang cosmological models currently considered likely, the particle horizon is order of magnitude the same as the Hubble length (CL-47).

Thus the Hubble length (current best value L_H=4220 Mpc) is considered the characteristic size of the observable universe.

In the concordance model (which we discuss in the section The Concordance Model) the particle horizon is about 14000 Mpc.

The Hubble length and the observable universe.
Question: Actually, our definition of the Hubble length L_H=c/H raises a bit of a paradox.

By the Hubble law itself v=Hd (which as we argued above is exactly true for recession velocities and distances measured at one instant in cosmic time), a galaxy a Hubble length away should be moving RECESSION VELOCITY c.

Galaxies further away that a Hubble length should have RECESSION VELOCITIES greater than c.

But this result seems to violate special relativity.

1. Yes, it is a violation and something is wrong with our thinking.

2. The recession velocities are NOT ordinary velocities, but rates of growth of space between gravitationally bound systems. This growth rate can exceed the speed of light according to general relativity.





Answer 2 is right in relativistic theory, but you probably didn't know that---except that the question is rather leading.

---

Einstein, General Relativity, and the Einstein Universe

---

In 1905, Einstein presented his theory of special relativity (SR) which among other things gave the vacuum speed of light as the highest physical speed and showed that the flow of time was frame-dependent.

But SR does NOT explicitly deal with gravity.

In order to remedy this, Einstein went on a 10 year excursion into tensor calculus and differential geometry and emerged in 1915 with general relativity (GR) which we have already introduced in IAWL 25: Black Holes. (See also ( St. Andrews Mathematics Archives: Einstein biography.)

It was natural for Einstein to see if GR could be used to construct a model of the universe as whole.

[``Natural'' in the sense that one often tries to apply new theories to systems of great interest to which the new theories should be applicable.]

He came up with Einstein universe, as we now call it, first presented in 1917 (No-520; Bo-97)

The Einstein universe is in fact just about the first universe model developed from an exact mathematical physics theory and completely consistent with that physics.

[Recall pure Newtonian physics is not able it seems to yield a consistent physical model of the universe as whole (Bo-75,78).

Actually, Einstein earlier in 1917 presented an earlier GR universe model called the cylindrical model where space has the geometry of a 3-dimensional surface of a 4-dimensional cylinder (No-513). The cylindrical model never had much of a vogue though.]

In order to apply GR to the universe, Einstein made two major simplifying assumptions which are still usually used today at least for theories of our universe domain:
1. The Cosmological principle which is a glorified expression for the assumption that the universe looked at or averaged over sufficiently large scales is homogeneous (i.e., the same everywhere at one time) and isotropic (i.e, the same in all directions).
   [We believe the Cosmological principle is true for the observable universe, but you must look at and/or average over sizes of order 200 Mpc (CM-446; FK-596). Einstein did NOT know this in 1917, of course.]

2. The substance of the universe can be approximated a uniform perfect fluid of some kind: the simplest kind is pressureless (CL-33,40). We don't have to go into the exact specifications here.

Question: When starting out to model a complex system one:
1. assumes a highly simplified picture in order to capture the essential behavior, and then afterwards adds complex features as needed to match observation.

2. just experimentally determines how the complex system behaves and that constitutes a working model which tells you pretty completely how to manipulate the system.

3. a bit of both 1 and 2 is often a good strategy: the weighting of the mixture depends on cases.





There is NO right answer, of course.

Answer 2 is essentially how traditional technologists solved their problems: e.g., building the pyramids, building cathedrals, sailing the Pacific Ocean in outrigger canoes.

However, in dealing with the extremely advanced systems of the modern age answer 3 has usually been pursued.

But when you can't experiment, as in cosmology, answer 1 is about what you are stuck with.

You realize your first attempts may be too simple or just plain wrong, but you have to start WITHOUT complexities that you don't know how to deal with anyway: i.e., crawl before walking.

Einstein, deferring to contemporary 1917 astronomical opinion, thought the universe should be STATIC. He took this as an ASSUMPTION.

Why a good physicist like Einstein (to say the least) should defer to a bunch of astronomers is beyond me especially since the the obvious non-thermodynamic equilibrium state of the universe (which we discussed above in the section The Expansion of the Universe) pointed to an evolving universe.

[Einstein also wanted to conform to Mach's principle and that also led to a static universe (Bo-97--98; Fre-543). But we won't worry about this esoteric point.]

Einstein could NOT find a STATIC MODEL with GR has he had originally proposed it (No-520).

But by introducing a new term into his field equations---the cosmological constant Lambda---he could get a STATIC MODEL by fine-tuning the constant to provide a repulsion that exactly canceled the attraction of gravity.

The cosmological constant was an ad hoc device to obtain a STATIC MODEL, but at least it was just about the simplest modification of the field equations that one could imagine and it was sufficiently small that it had no significant effect on any other gravitational phenomena.

[Physicists have a word for a quantity one just inserts to get the answer one expects: a fudge factor because one cooks it up.

Students too are quite adept at creating fudge factors on tests.]

The STATIC MODEL is now called the Einstein universe. It is the 3-dimensional surface of a 4-dimensional hypersphere: i.e., it is finite, but unbounded, hyperspherical space (No-520; Bo-98).

The 4th spatial dimension has no physical meaning: there is nothing imagined off the 3-dimensional surface.

Now we have difficulty picturing curved 3-dimensional spaces, but the 2-dimensional analogs of curved spaces can pictured.

[I wonder how well we picture 3-dimensional flat space. Maybe we only picture it in the sense that experience tells us how things look in other directions from the one we are viewing an object from.]

2-dimensional space analogs of possible geometries of space in simple GR models of the universe.
[Omega is a density parameter that determines the curvature of the space in GR. We will discuss it below in the section Friedmann-Lemaitre Models and subsequent sections.]

The Einstein universe is analogous to the surface of an ordinary sphere in the diagram above.

In such a space traveling in a straight line (a line that seems to be straight at every locality) should bring you back to where you started and if you looked long enough you should see the back of your head.

[The Einstein universe is actually unstable (Bo-118; No-527). Any perturbation will start it on runaway expansion or contraction.

Here is a mechanical system that illustrates system instability.

Stability of a mechanical system.

Exactly how the many local perturbations that exist in the real universe could have affected a real Einstein universe is hard to say. But for awhile a universe evolving away from a static Einstein universe was a well considered model of the expanding universe (No-527).]

After 1929 and the observational discovery of expansion of the universe (No-523) and discussion with Hubble in 1930 (No-526), Einstein abandoned his model and subsequently said (probably pretty often, but certainly to George Gamow) that inventing the cosmological constant was the biggest blunder of his life.
[He certainly meant his scientific life, not his private life. Einstein wasn't a good family man, in fact: late in life he commented that he had failed at marriage twice and that was his greatest shame.

Question: What Einstein meant about his biggest blunder was that:
1. the cosmological constant was a simple math error.
2. he had avoided trusting his own original field equations and thereby missed his chance to predict the expansion of the universe.




Answer 2 is right.

The cosmological constant was a mistake in its original use, but it didn't go away.

It continued to be useful in for other cosmological fix-ups---``the refuge of scoundrel cosmologists''---and in fact it has come back in a new function with a vengeance as we'll see below the section The Accelerating Universe and the Friedmann-Lemaitre-Lambda Models.

But though Einstein blundered, others did not: expansion of universe was predicted from GR models before it was observationally discovered.

---

Friedmann-Lemaitre Models

---

Aleksandr Aleksandrovich Friedmann (1888--1925) in Russia in the early 1920s and Georges Lemaitre (1894--1966) in Belgium just a bit later and independently developed NON-STATIC GR MODELS OF THE UNIVERSE with the cosmological constant Lambda set to zero, but otherwise following Einstein's original assumptions which we discussed above in section Einstein, General Relativity, and the Einstein Universe (see also No-524--527): i.e., the Cosmological principle and the representation of the mass-energy of the universe by a homogeneous fluid.

Because the cosmological constant Lambda set to zero, the Friedmann-Lemaitre models are pure GR models.

In these models, the universe begins from a singularity which was once called the POINT ORIGIN (Bo-85,181), but is now called the big bang.

[The term big bang was first used derisively by Fred Hoyle (1915--2001) in a BBC radio program in 1950 (No-532): Hoyle was a coinventor of the steady-state universe (Bo-140ff,152ff) which was a serious rival of big bang cosmology up to the early 1960s.]

The term big bang also means the short time after the singularity in which the light elements of the universe are now believed to have been synthesized in big bang cosmology.

We should emphasize that the big bang is NOT a pressure explosion, but an initial condition of the models (CL-36).

The big bang in these models starts an initial expansion of the universe that is DECELERATED at all times by the mutual gravity of the mass-energy of the universe.

The fate of these models is determined by Omega.

[Standard html does NOT have a Greek font. Thus we use the English names Omega and Lambda (which is cosmological constant introduced in the section Einstein, General Relativity, and the Einstein Universe) for the Greek letters that usually stand for the quantities.

The Greek Alphabet: alpha, beta, gamma, ...]

Omega is the ratio of the universal mass-energy density to a critical mass-energy density which is a natural time-dependent parameter of the models:

    critical density =  3H**2    =  9.5*10**-27 kg/m**3   ,
                       ________
                        8*pi*G

      where H is the Hubble constant and G is the gravitational constant
      (FK-646).

   Note the present-day critical density is a very small value. 
     The density of water is 1000 kg/m**3 recall.

   The ratio is 

     Omega =    density
            ________________  ,
            critical density

   Omega, density, and critical density are all time dependent, but
     Omega 

         if greater than 1, stays greater than 1 for all time,
         if less than 1, stays less than 1 for all time,
         if exactly 1, stays exactly 1 for all time,

    in pure Friedmann-Lemaitre models with Lambda=0 if there are no
    perturbations sufficient to change things. 

Omega (which is a sort of unitless density) decides the geometry of space and---if the models can be extended to all space---whether the universe is finite or infinite.

[Remember in GR mass-energy determines the distortion of spacetime and spacetime tells mass-energy how to move.]

Many people do NOT believe that the models can be extended to all space and so the latter decision is moot.

These people believe the models can only be extended to our universe domain.

2-dimensional space analogs of possible geometries of space in simple GR models of the universe.

The evolution of the Friedmann-Lemaitre models can be represented compactly by the time evolution of the universal scale factor a(t).

a(t) does not have to be assigned any specific length itself for our purposes.

[Actually a(t) can be assigned a specific value that has geometrical significance in non-flat spaces (CL-11,12). But we won't worry about that.]

It is just a value proportional to the proper distance between any two points that participate in the expansion of the universe.

a(t) thus just gives the relative scaling up of the universe.

The ``t'' in a(t) is the cosmic time: the time in a frame that is expanding with the mean expansion of the universe. All these frames have the same time flowing at the same rate. This is just a condition of the Friedmann-Lemaitre models and similar models.

There is a convenient choice for a(t) for plotting purposes at least.

Universal scale factor a(t)

A diagram illustrates how a(t), and thus how the expansion evolves with cosmic time in the three qualitatively distinct versions of the Friedmann-Lemaitre models.

Friedman-Lemaitre models.

The slope of the curves in the diagram is the rate of change of of a(t): i.e., recession velocity of distance a(t). Since the slope always decreases with cosmic time the expansion continuously decelerates for the Friedmann-Lemaitre models.

The Omega<1 and Omega=1 versions expand forever (although always at a decreasing rate because of the deceleration) and the universe (or universe domain) will end in a BIG CHILL which we will discuss below in the section The Fate of the Universe According to the Concordance Model.

If Omega>1, then the universe (or universe domain) will eventually recollapse and there will be a BIG CRUNCH. Since the BIG CRUNCH is itself a singularity, we don't really know what happens then or later.

Some have imagined an OSCILLATING UNIVERSE where the BIG CRUNCH is the BIG BANG of a subsequent epoch. This idea as originally suggested has not lasted, but in the guise of the CYCLIC MODEL it has made a bit of a comeback. We briefly discuss the CYCLIC MODEL in the section Inflation and Inflation Cosmology.

A KEY POINT is that the Friedmann-Lemaitre models predict either an expansion or a contraction of the universe: i.e., a(t) is never constant, but always changing.

And this was done before Hubble observationally discovered Hubble's law and Hubble was aware of the prediction??? and it is hard not to believe that it guided his thinking (No-524).

Hubble's law itself is consequence of the Friedmann-Lemaitre models which Lemaitre himself had explicitly shown before about 1928 without making a big point of it (No-524--526).

This is, of course, the theoretical Hubble's law which is exact for recession velocities and distances measured at one instant in cosmic time.

But the observational law agrees with the theoretical law for cosmologically short distances or cosmologically short look-back times as we have already noted above.

Also, as we mentioned above, the cosmological redshift due to the universal expansion is not actually at Doppler shift (FK-636--637).

Doppler shifts are caused by relative motions.

The cosmological redshift is caused by space expanding under the light as it propagates through expanding space.

The formulas for the two kinds of spectral shift are different, but they agree to 1st order in low velocity and thus many books loosely speak of the cosmological redshift as Doppler shift.

The Friedmann-Lemaitre models in themselves do NOT tells us everything. In particular they do NOT tell us the value of the Hubble constant or of Omega. So they are certainly incomplete cosmological models.

In principle, the Hubble constant and Omega can be determined by observations, of course. In fact, they have.

How Hubble constant is determined we have already discussed. How Omega is determined we will briefly mention below in the section The Accelerating Universe and the Friedmann-Lemaitre-Lambda Models.

But the fact that the Friedmann-Lemaitre models predicted the expansion of the universe and Hubble's law before they were discovered is very impressive.

It certainly did suggest that they were right as far as they went.

Question: Which version of the Friedmann-Lemaitre models corresponds to the actual universe (or universe domain)?
1. The Omega > 1, hyperspherical model.
2. The Omega = 1, flat model
3. The Omega < 1, hyperbolical model.
4. None of them.




Answer 4 is right according to the best modern observations.

All the Friedmann-Lemaitre models are decelerating at all times after the big bang singularity.

There is good evidence now that the expansion of the observable universe is accelerating.

The accelerating universe is the subject of the next section The Accelerating Universe and the Friedmann-Lemaitre-Lambda Models

But our discussion of the Friedmann-Lemaitre models has not been a waste of time as we will also see in the next section.

---

The Accelerating Universe and the Friedmann-Lemaitre-Lambda Models

---

How we got where we are now in modern cosmology with lots of omissions:

1. 1917: The static, hyperspherical Einstein universe with non-zero cosmological constant Lambda which was invented to make the universe static (No-520; Bo-97).

2. 1917: The de Sitter universe developed by Willem de Sitter (1872--1934) (No-518,521; CL-28--29,159). This is zero-density, exponentially expanding universe.

   
             a(t) = a_0 exp(t/t_0) , where
   
                 exp is exponential function:  e=2.71828...
                 raised to the power t/t_0.
   
                 The exponential function is not at all
                 esoteric:  it describes the growth of
                 a bank account with compound interest
                 and uncontrolled population growth;
   
                 a_0 is some fiducial scale distance;
   
                 t_0 is some fiducial cosmic time.
    
           

   Historically, the de Sitter universe is of interest because it was the first model to predict expansion, but, of course, it wasn't right because it had no mass-energy. But nowadays it has been revived as a component of inflation cosmology which has an exponential growth phase.

   We discuss inflation cosmology below in the section Inflation and Inflation Cosmology.

3. 1917 to Mid-1960s: Various universe models including Friedmann-Lemaitre models had vogues for periods of time.

4. Mid-1960s to 1998: In this period Friedmann-Lemaitre models which were the standard cosmological models. They predicted the expansion of the universe and their early dense phase allowed for the constituents of the observable universe to develop. The latter is a subject discuss below in the section Big Bang Cosmology and the Constituents of Observable Universe.

5. 1998 on: In 1998 the first evidence appeared that the universal expansion was accelerating: to be more exact, positively accelerating unlike the Friedmann-Lemaitre models where the expansion is always negatively accelerating (i.e., decelerating).

The acceleration was first discovered by studying Type Ia supernovae (SNe Ia).

These are very bright objects that can be seen using the modern giant telescopes to beyond 2500 Mpc (FK-649). Recall the current value for the Hubble length is 4220 Mpc, and so SNe Ia can be seen to cosmologically large distances.

Their maximum luminosities are known reasonably well, and thus one can determine their luminosity distances from the inverse-square law for light: luminosity distances are not the same as proper distances (except in a static universe), but they can be MEASURED.

 
                        L
           Recall F=___________
                     4*pi*r**2

           implies

           r =sqrt[ L/(4*pi*F) ] .

           But this calculation neglects the expansion of space during
           the photon flight time and assumes flat geometry.

           Consequently, the distance r so determined is a funny distance
           that we call luminosity distance.  

           It can be used, nonetheless, to determine cosmological 
           parameters in a way we will NOT go into. 

    

The cosmological redshifts of the parent galaxies of the SNe Ia can be measured too.

Thu, one can determine a Hubble diagram for SNe Ia.

Such diagrams extend to great distances with pretty high accuracy, and thus allowed a more sensitive test of Hubble's law and the nature of the expansion of the universe than before.

A schematic Hubble diagram.

The 1998 Hubble diagram for SNe Ia showed deviations from Hubble's law that could reasonably be fitted only if an ACCELERATION of the expansion were assumed.

[The deviations were NOT due to peculiar motions.]

When the ACCELERATION was first announced, people were somewhat skeptical. The data had a lot of random UNCERTAINTY---the plots looked like scatter diagrams to me---and there could have been many systematic errors.

But since 1998 the data for SNe Ia has continued to firm up and in addition 2 other independent evidences for ACCELERATION have appeared.

To summarize without giving any details about how one knows:

1. The SN Ia Hubble diagram directly points to acceleration.

2. Analysis of the cosmic microwave background (CMB) (which we discuss below in the section Big Bang Cosmology and the Constituents of Observable Universe) strongly suggests that Omega=1.00+/-0.02, but gravitational mass in galaxy clusters indicates Omega_matter=0.27+/-0.04 (FK-650,653).

   The missing mass-energy can be interpreted as some kind of dark energy that is powering acceleration.

3. The integrated Sachs-Wolfe effect (CL-375) (which we won't go into) also points to acceleration.

So ACCELERATION is rather well established: it still might be proven non-existent, but that would take a lot of counter-evidence now.

How does one accommodate the ACCELERATION theoretically?

Well the possibilities quasi-endless.

But the simplest way is to fetch Einstein's cosmological constant Lambda back from the storeroom of discarded theories and put it back in the GR field equations, but now tune it to give the measured ACCELERATION instead of a static Einstein universe.

One can then derive what one can call Friedmann-Lemaitre-Lambda models which are just the Friedmann-Lemaitre models with the cosmological constant Lambda as an extra free parameter: i.e., an extra controlling variable whose value is not determined by the model and must be determined by observational or other means.

The universal scale factor a(t) for an appropriate Friedmann-Lemaitre-Lambda model evolves schematically as in the following diagram.

Accelerating universe based on CM-455.

The increasing slope of the accelerating a(t) curve is the signature of ACCELERATION.

Note that the accelerating model starts in a decelerating phase and then makes a transition to ACCELERATION about 5 Gyr ago.

The transition to ACCELERATION is less certain than the ACCELERATION itself, but recent data for SNe Ia suggest it (FK-650--651).

---

Dark Energy

---

The ACCELERATION in the Friedmann-Lemaitre-Lambda models is simply turned on by setting the cosmological constant Lambda to a sufficiently large positive value (CM-454).

But what does Lambda mean physically?

In Einstein's field equations Lambda just appears as a modification to how gravity affects spacetime. It is a ``just so'' modification.

In modern physics, there is a strong preference to interpret the Lambda as representing a NEGATIVE PRESSURE (GR-277--278) which has an associated dark energy. We call this energy dark energy because we don't see it so far in any other way than through its effect on the universal expansion.

The NEGATIVE PRESSURE itself causes an inward sucking instead of an outward push like ordinary pressure. But this pressure effect is weak and cancels completely if the NEGATIVE PRESSURE is uniform in space which is the case when assuming it is governed by the cosmological constant (GR-520).

What is not canceled is the cumulative effect of the gravity of dark energy which becomes stronger over larger separations (GR-279). And this gravity effect is REPULSIVE---antigravity of a sort at last it seems (GR-278).

Einstein used the REPULSIVE GRAVITY---a very ugly kind of gravity---to exactly counter ordinary gravity and make the static Einstein universe. But the cosmological constant can be turned up to drive the ACCELERATION of the universe.

Now COSMOLOGICAL-CONSTANT DARK ENERGY has a constant energy density in time and space. It is about the simplest kind of dark energy imaginable.

That the COSMOLOGICAL-CONSTANT DARK ENERGY DENSITY is constant in time is UNUSUAL. For example, matter density must decrease because of the expansion of the universe. In fact, as universal scale factor a(t) increases, volumes increase by a(t)**3, and matter density falls as a(t)**(-3).

The COSMOLOGICAL-CONSTANT DARK ENERGY DENSITY can be treated as a contribution to Omega: we denote this contribution by Omega_Lambda.

Question: Omega is:
1. the cosmological constant by an other name.
2. the last letter of the Greek alphabet.
3. the ratio of the mass-energy density of the universe (or universe domain) to the critical mass-energy density. It is the parameter that determines the geometry of space in the Friedmann-Lemaitre models and the Friedmann-Lemaitre-Lambda models.



Answer 2 and 3 are right. But answer 3 is best context.

From the analysis of the CMB data, galaxy cluster data, and SNe Ia ASSUMING the dark energy is a COSMOLOGICAL-CONSTANT DARK ENERGY, one finds Omega_Lambda=0.73+/-0.04 (FK-653).

But the dark energy need not have constant density in either time or space and it may interact with other forms of mass-energy in ways we do not know.

So far nothing in the observations tells us to go beyond the simple theory of COSMOLOGICAL-CONSTANT DARK ENERGY.

But it would NOT be surprising if eventually we had to.

Is there any reason for believing there could be just COSMOLOGICAL-CONSTANT DARK ENERGY from physical theory?

Yes, quantum field theory suggests there could be COSMOLOGICAL-CONSTANT DARK ENERGY, but alas predicts its density to be 10**120 times bigger than needed to fit the observed ACCELERATION (e.g., Carroll, S. 2003, p. 3, Why is the Universe Accelerating?).

This remarkable OVERESTIMATE suggests that the dark energy is move complex than the simple COSMOLOGICAL-CONSTANT DARK ENERGY with a density which is constant in time and space.

It is also strange that the COSMOLOGICAL-CONSTANT DARK ENERGY DENSITY from the analysis of the CMB data is Omega_Lambda=0.73+/-0.04, which is COMPARABLE to the matter density Omega_matter=0.27+/-0.04 (FK-653).

There may be some deep reason why the dark energy and matter should be connected in which case the dark energy CANNOT be simply a COSMOLOGICAL-CONSTANT DARK ENERGY.

[Possibly the COMPARABILITY is explained by the Anthropic principle.

The are an infinity or a quasi-infinity of universe domains with different structures.

Those somewhat like our own universe domain may NOT be able to support life if there was no COMPARABILITY.

Too small a COSMOLOGICAL-CONSTANT DARK ENERGY DENSITY and the universe may not have formed the right kind of galaxies and stars.

Too large a COSMOLOGICAL-CONSTANT DARK ENERGY DENSITY and the universe would have expanded too quickly ever to form galaxies

It is very hard to prove an argument based on the an Anthropic principle.

But such an argument could be falsified if the dark energy density and matter density were fine-tuned beyond the needs (so far as we can tell) of making the universe domain suitable for us to be here.

For example the ratio is now of dark energy density to matter density is approximately 3 to 1.

If the ratio were exactly 3 to 1, then that is more exactness than is needed for a universe domain approximately like ours, and strongly suggests a deep connection between dark energy and matter.]

---

The Concordance Model

---

Assuming that the Friedmann-Lemaitre-Lambda model is correct, one adjusts its free parameters to fit the modern data from SNe Ia, the CMB (which we will discuss below in the section Big Bang Cosmology and the Constituents of Observable Universe), and galaxy cluster observations.

The resulting model is called the concordance model.

______________________________________________________________________________

Concordance Model Parameters
________________________________________________________________________________

Quantity               Value                    Short explanation

________________________________________________________________________________

Hubble constant H      71(+4/-3) (km/s)/Mpc     present day expansion rate

Hubble time 1/H        13.8 Gyr                 characteristic universe age 

Age of universe        13.7+/-0.2 Gyr           time since the big bang

Hubble length c/H      4220 Mpc                 characteristic radius of 
                                                  the observable universe

particle horizon       14000 Mpc                radius of particle horizon
                                                  or observable universe 

Age of universe        379,000+/-8,000 years    time when the protons 
  at recombination                                and electrons combined
                                                  to make neutral hydrogen 

Omega                  1.02+/-0.02              total of all mass-energy density
 (density parameter)                              of universe divided by
                                                  critical density.
                                                  Omega determines the geometry
                                                  of space.

Omega_dark_energy      0.73+/-0.04              dark energy contribution

Omega_exotic           0.23+/-0.04              exotic dark matter

Omega_ordinary_matter  0.044+/-0.04             ordinary matter:  dark and
                                                  luminous 

Omega_luminous         0.007                    luminous matter:  stars,
                                                  H I gas, H_2 gas,
                                                  hot H II gas in X-ray galaxy 
                                                  clusters. 

________________________________________________________________________________

References:
1. FK-653
2. Davis, T. M., & Lineweaver, C. H. 2003, Publications of the Astronomical Society of Australia, astro-ph/0310808, Expanding Confusion: common misconceptions of cosmological horizons and the superluminal expansion of the Universe
3. Spergel, D. N. et al. 2003, ApJ, astro-ph/0302209, First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameters
4. Cen, R., & Ostriker, J. P. 1999, ApJ, 514, 1, astro-ph/980628, Where are the Baryons?
5. Fields, B. D., & Sarkar, S. 2003, astro-ph/0406663, Big Bang Nucleosynthesis in The Review of Particle Properties (2004)

         
________________________________________________________________________________

Some comments about the concordance model are in order:

1. The many of the observations have gone into determining concordance model parameters are thought to be very accurate. But the uncertainties could be larger than is thought.

2. If the Friedmann-Lemaitre-Lambda models (of which the concordance model is a special case0 are essentially wrong then derived quantities like the age of the universe and Omega might be wrong.

   If a model is wrong, then observations need to be interpreted in a different way.

3. And the model may indeed be wrong. As we have discussed above a COSMOLOGICAL-CONSTANT DARK ENERGY is the simplest theory of the ACCELERATION, and so far this theory is adequate to explain the observations.

   Future observations, particularly of the past history of the universal expansion using SNe Ia, may show that COSMOLOGICAL-CONSTANT DARK ENERGY is wrong.

   This would NOT be surprising.

   The simplest adequate theory is often the first theory one should investigate---crawl before running---but astronomers are always finding things more complicated than they first thought: it's sort of expected.

4. The exotic dark matter may well be some strange, new particle that interacts weakly with matter, except through gravity: a WIMP (weakly interacting massive particle).

   There are many ideas about this particle and some tentative claims of possible detections of something in the laboratory or astronomically, but the possibilities are still wide open.

   The requirement for exotic dark matter is an inference as we now explain.

   Big bang nucleosynthesis (which discuss below the section Big Bang Cosmology and the Constituents of Observable Universe) predicts Omega_ordinary_matter equal to about 0.044, and velocity studies of galaxies and galaxy clusters demand Omega_matter at about 0.27+/-0.04.

   Big bang nucleosynthesis is itself a very robust theory, and so we are forced to believe exotic dark matter is likely.

   If we ever discover exotic dark matter, it will have profound implications for cosmology and fundamental physics.

   [Note if MOND (for MOdified Newtonian Dynamics) turns out to be at all right, then the need for exotic dark matter may vanish and all of cosmology would be affected in radical ways.]

5. Most of the ordinary matter is dark. From big bang nucleosynthesis, Omega_ordinary_matter equal to about 0.044.

   But adding up all the luminous matter gives only 0.007 (e.g., Cen, R., & Ostriker, J. P. 1999, ApJ, 514, 1, astro-ph/980628, Where are the Baryons?).

   Ordinary dark matter is 5 to 6 times more abundant than luminous matter.

   Some of this ordinary dark matter may be in the form of brown dwarfs, dim white dwarfs, dim neutron stars, and black holes.

   But perhaps not much of it. The idea that MACHOs (MAssive Compact Halo Objects) (brown dwarfs, dim white dwarfs, dim neutron stars, and black holes) may make up a lot of the dark matter is at present in doubt. Re-analysis of the MACHO data suggests there may be few or almost no MACHOs (e.g., Evans, N. W. & Belokurov V. 2004, astro-ph/0411222, RIP: The MACHO Era [1974--2004]). But the issue is very controversial right now.

   So what is the ordinary dark matter?

   At present, the favored idea seems to be that it is diffuse, warm/hot intergalactic gas (i.e., ionized H and He with temperatures in range 10**5--10**7) and slightly warmer and colder gas makes up most or maybe almost all of the ordinary dark matter (e.g., Cen, R., & Ostriker, J. P. 1999, ApJ, 514, 1, astro-ph/980628, Where are the Baryons? [hereafter CO], CO-3).

   This gas is almost invisible because it emits low energy X-rays (which are mostly drowned out by Milky Way X-ray emission) and extreme ultraviolet light to which the neutral Milky Way hydrogen is opaque (CO-3).

   We may be on the verge of detecting it????.

   The gas is still warm/hot from gravitational collapse from the initial gas of the universe and, secondarily, from feedback from formed galaxies (CO-4). The gas cools very slowly.

   Eventually, maybe in several Hubble times, some of the warm/hot gas will be cooled enough to collapse into new galaxies (CO-5). This would keep star formation going in the universe for some time. But the continued and accelerated expansion of the universe might prevent all of it from collapsing????.

---

The Fate of the Universe According to the Concordance Model

---

Fire and Ice

Some say the world will end in fire,
Some say in ice.
From what I've tasted of desire
I hold with those who favor fire.
But if it had to perish twice,
I think I know enough of hate
To know that for destruction ice
Is also great
And would suffice.


Robert Frost (1874--1963). First published 1923. Download site: Representative Poetry Online, Department of English, University of Toronto.

What is the fate of the universe (or our universe domain) if the concordance model is taken as absolutely correct?

Well a highly speculative sketch---which gets more speculative as it goes along---is as follows (HI-477):

1. Current, cosmic time is 13.7*10**9 years.

2. At cosmic time of order 10**13 years, all star formation will have ended and the universe will consist of mainly of exotic dark matter (probably), compact remnants (white dwarfs, neutron stars, and black holes), and, perhaps, some uncollapsable H and He gas.

3. It is possible that protons decay (and this entails the elimination of neutrons too) into electrons, positrons, neutrinos, and photons with a half-life of something in excess of 10**32 years. Thus on some cosmic time in excess of 10**32 years, the protons and neutrons will have all decayed. This will leave a dilute gas of electrons, positrons, neutrinos, and photons, and exotic dark matter particles, and black holes.

4. Black holes are theorized to evaporate by Hawking radiation.

   By cosmic time of order 10**100 years the black holes may have evaporated and the vastly expanded universe could be only a very dim, dilute gas of electrons, positrons, neutrinos, and photons, and maybe exotic dark matter particles.

   This is the end of the story---but there's no reason to put much faith in it: it's a very speculative story.

---

Big Bang Cosmology and the Constituents of the Observable Universe

---

Now the Friedmann-Lemaitre-Lambda models are big bang models in the sense that they begin from a big bang singularity, and so we've been discussing big bang cosmology for some time.

But big bang cosmology is more than the Friedmann-Lemaitre-Lambda models.

It is also an explanation and history of the constituents of the universe (or of our universe domain at least) beginning from an early hot, dense phase of the universe which is the second meaning of the term big bang.

[We go beyond the perfect fluid picture for the matter in the universe that the simple Friedmann-Lemaitre-Lambda models assume.]

The ancestor of the big bang theory for the constituents was Georges Lemaitre's (1894--1966) PRIMEVAL ATOM THEORY---but we won't discuss that historically interesting, but long discarded, theory: it was essentially pre-nuclear physics (No-530).

Georges Lemaitre (1894--1966).

In the 1940s (by which time nuclear physics was somewhat elucidated), George Gamow (1904--1968), Ralph Alpher, and Robert Hermann worked out an early version of big bang nucleosynthesis (BBN) (No-531ff, 559ff): the theory that the elements were synthesized by nuclear fusion from hydrogen nuclei in an early hot, dense phase of the universal expansion: i.e., in the big bang in the 2nd meaning of the term.

Originally, Gamow et al. tried to show that all the nuclei could have been formed in this early phase (Bo-58), but later this turned out to be impossible it seems (No-560). The heavier nuclei are accounted for by nucleosynthesis in stars followed by ejection by stellar winds and supernovae (No-540).

But stars cannot account for certain light elements (or more precisely nuclei): helium (He-4 and He-3), deuterons (D or H-2), lithium (Li-7 and Li-6), and, perhaps, some beryllium (Be-7).

The case for He is particularly acute: there seems too much to have been produced in stars.

Recall the universal abundances by mass are about 71 % H, 27 % He, and 2 % metals: these numbers, in particular the last one, vary a bit from source to source (e.g., Cox-28).

The light elements can be accounted for by BBN.

The idea is to start cosmic time at some early hot, dense phase of the universe with some simple primordial constituents and then run the clock forward synthesizing the nuclei as space expands and cools. The gas expands with space and this cools it by a commonplace physical effect: adiabatic cooling---which we won't go into, but its everywhere including everyday life.

We can't start the clock at TIME ZERO. At infinite density, our physical concepts break down.

In fact we can't start earlier than the Planck time:

            t_Planck = 10**-43 s 

which is a time earlier than which the density is so high that quantum effects on gravity must have been important by general quantum mechanical principles even though we don't know much about those effects (CL-122).

GR must fail it is thought before the Planck time which is a good reason for not believing in the infinite density singularity.

The very earliest times before a second or so are in a very speculative realm where the matter is believed to be so hot and dense that only QUARKS and LEPTONS and their antiparticles exist and in which matter and antimatter are about equal in abundance (FK-668).

[There are probably also exotic dark matter particles about, but what they are doing is uncertain.]

Quarks make up protons and neutrons.

Free QUARKS exist only under super-dense conditions. If you try to pull apart composite particles (e.g., protons) made up of quarks under less dense conditions, new quarks come into existence to complete make new composite particles.

The energy from the pulling apart goes into making the new composite particles.

Leptons are electrons, positrons (antielectrons), neutrinos, and some less common species.

Matter and antimatter mutually annihilate to produce photons.

The mutual annihilation destroys the antimatter and leaves a trace of matter.

It is thought in theories of particles that there is some asymmetry in properties between matter and antimatter that slightly favors matter (FK-668).

To just give a simple sketch of the early universe a sequence of snapshots is useful.

We assume that the early universe is very homogeneous: i.e., among other things has nearly constant temperature, density, and composition at any given cosmic time. The continuous expansion causes the temperature and density to fall steadily.

There are small density fluctuations that will be the seeds of the large-scale structure that will form in of order the first billion years. Gravitational runaways will start from the seeds.

Early universe 1: t= about 10**-35 s.

In this early phase, the strong nuclear, weak nuclear, and electromagnetic forces may have been united: i.e., acted in the same way (FK-664). There may be exotic particles around too. This epoch may have been just before inflation: see just below and FK-661.

Early universe 2: t = about 2 s.

Early universe 3: t = about 3--15 minutes

Early universe 4 (recombination epoch): t = about 380,000 years.

Early universe 5: t = about 1 Gyr.

After snapshot 5, the universe (or universe domain) continues to evolve to our present epoch.

The recombination epoch is when the electrons and nuclei combine to form NEUTRAL ATOMS: mainly hydrogen and helium, of course.

The NEUTRAL ATOMS have much lower cross sections for interactions with photons of the temperature of recombination epoch of about 3000 K (FK-670).

So much so that after about the recombination epoch the matter in the universe domain becomes largely transparent to the primordial photons after the

Before the recombination epoch, the photons interacted strongly with matter and thus matter and photons were held a the same temperature. At recombination itself this temperature was about 3000 K as noted above (FK-670). The photons then had a blackbody spectrum of about 3000 K.

After recombination epoch the primordial photons stream off through space only slightly interacting with matter again.

They do interact a little, of course: they can scatter off free electrons in space, run into stars and planet, be affected by gravitational effects, and other lesser interactions.

The primordial photons cool by expansion of the universe. Their wavelengths scale with the universal scale factor and their density decreases as the volumes scale up.

In fact, it can be shown that primordial photon distribution remains blackbody-like with a constantly decreasing temperature due to expansion.

In 1949, Alpher and Hermann predicted the present-day temperature of the primordial photons would be about 5 K. (No-559).

Using WIEN'S LAW

                   2900 micron-K
     lambda_peak = _____________  = about 600 microns = 0.06 cm  
                        5 K

which by common definition is long wavelength infrared (HZ-54; FK-94). But the microwave band is redward of 0.1 cm where much of the primordial photon spectrum is.

Thus, this relic primordial photon gas is called the cosmic microwave background (CMB).

In 1965, Arno Penzias and Robert Wilson working with a Bell Laboratories radio telescope in Holmdel, New Jersey fortuitously discovered the CMB at 7.3 cm: a nearly uniform radiation from all directions (No-561ff).

[It is de rigueur in this context to mention that to be sure that they weren't just detecting instrumental noise, they had to clean history's most famous pigeon guano out of their antenna.]

Penzias and Wilson got the 1978 Nobel prize for their discovery; Alpher and Hermann did NOT get a Nobel prize.
[A Swedish friend once commented to me: ``The question is not why the Swedes decide the Nobel prizes; the question is how do they decide the Nobel prizes.'' Apparently, it's a mystery even in Sweden.]

The CMB is best measured from space, because the atmosphere is opaque to much of the spectrum of the CMB.

The first highly accurate CMB measurement over a broad wavelength range was reported from the COBE probe circa 1990 (FK-640).

---

Cosmic Microwave Background (CMB) from COBE and other detectors.

The plot is logarithmic on all three axes.

The microwave band by one convention is redward of 0.1 cm; blueward is infrared (HZ-54).

The data is excellently fit by a blackbody spectrum has a temperature of T=2.726+/-.001 K.

Credit: CMB Astrophysics Research Program: COBE site This is a Lawrence Berkeley Lab site reporting on George Smoot's group.

---

The CMB has a large-scale variation caused by the Earth's peculiar motion with respect to the MEAN FRAME participating in the expansion of the universe (FK-640--641).

Question: The effect that actually causes the large-scale variation is:
1. the cosmological redshift.
2. Doppler shift.
3. the cosmological blueshift.




Answer 2 is right.

There is a slight blueshift in the direction of the Earth's motion and a slight redshift in the opposite direction.

Answer 1 is the reason the CMB has cooled down from 3000 K to about 3 K since the recombination epoch.

The solar system's motion relative to the MEAN FRAME is in fact best known from the large-scale CMB variation.

The motion is 371 km/s in the direction of Leo and away from Aquarius (FK-640--641).

We can deduce that the Local Group is moving at 620 km/s relative to the mean expansion in the direction the Hydra-Centaurus supercluster (FK-640--641).

If the large-scale variation is removed, there remain small-scale random fluctuations in CMB temperature of order 200 micro-Kelvins or in relative terms of 1 part in 10**4.

[These fluctuations were first measured to low accuracy by the COBE satellite in the early 1990s. People were relieved that that were finally found. They had been expecting them for some time.

Since then the measurements of the fluctuations have been considerably improved particularly by the WMAP satellite that has been active since 2001 (NASA's Wilkinson Microwave Anisotropy Probe (WMAP)).]

These CMB temperature fluctuations are believed to correspond to primordial density fluctuations that were the seeds for the gravitational collapses that led to the formation of the galaxies and the large-scale structure (see IAWL Lecture 28: Galaxies).

---

Cosmic Microwave Background (CMB) all-sky map from WMAP circa 2003.

Why is there no scale?

The average temperature is 2.725+/-0.001 K.

The colors code deviations in micro-Kelvins: dark blue (-200), green (0), red (+200).

Credit: NASA/WMAP Science Team.

---

To return to the cosmic abundances of the elements and BBN.

Modern BBN depends on the ratio baryons (protons and neutrons in this context [e.g., En-470]) to primordial photons.

The ratio is an adjustable free PARAMETER of the calculations.

The WMAP measurements of the CMB and observed primordial deuteron abundance actually give a value for this ratio of (6.13+/-0.25)*10**-10. (Spergel, D. N. et al. 2003, ApJ, astro-ph/0302209, First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameters; Mathews, G. J., et al. 2004, Phys. Rev. D, submitted, astro-ph/0408523, Big Bang Nucleosynthesis with a New Neutron Lifetime).

With the ratio parameter set to this value, the predictions of calculations of BBN can be compared with measured light elements corrected for stellar nucleosynthesis effects where possible.

_____________________________________________________________________________

BBN Predictions and Observed Elemental Abundances
Corrected where Possible for Stellar Nucleosynthesis Effects 
_____________________________________________________________________________

Element        BBN           Observed                   Quantity

_____________________________________________________________________________

He             0.246          0.245+/-.001              mass fraction
D (H-2)        2.5            2.5 to 3                  D/H * 10**-5  
He-3           1              none available            He-3/H * 10**-5
Li-7           4.5            1.5+/0.5                  Li-7/H * 10**-10
_____________________________________________________________________________

Reference: Mathews, G. J., et al. 2004, Phys. Rev. D, submitted, astro-ph/0408523 Big Bang Nucleosynthesis with a New Neutron Lifetime

_____________________________________________________________________________

The agreement is actually excellent over 10 orders of magnitude, but remember the deuteron abundance was fitted????.

The Li-7 does not agree within error, but Li-7 can both be created and destroyed in stars and correcting observed abundance to primordial abundance is very uncertain. At the moment, there is no reason to believe this discrepancy is a significant problem.

---

Let us summarize the strongest evidence for big bang cosmology.

1. The observed expansion of the universe is predicted. This was done before it was observed.

2. The observed CMB is predicted naturally as the relic of the hot early universe (or universe domain). This was also done before it was observed.

3. The observed light element abundances are well predicted.

4. The observed large-scale structure can be explained as having been seeded by the primordial density fluctuations that manifest themselves in CMB fluctuations.

   Calculations starting from the primordial fluctuations and using many assumptions especially about the dark matter do seem to be reproducing the observed large-scale structure though a lot of uncertainty remains (FK-671ff).

5. The oldest stars whose age we can well determine by modeling are in Milky Way globular clusters. The modeling is subject to some uncertainties, but has improved greatly since it began in the 1950s. The ages are of order 12.5+/-1.0 Gyr (FK-638).

   These ages are less than the 13.7 Gyr age of the universe concordance model (FK-653).

   At present, there is no problem with contents of the universe being older than the big bang cosmology predicted age of the universe which in the past has occasionally been an embarrassment (Bo-39,51--52).

At present, there are no viable, attractive rivals to big bang cosmology.

And this has really been so since the the 1960s despite the attempts of mavericks like Fred Hoyle (1915--2001) to present viable alternatives.

The alternatives have always had many ad hoc and/or complicating assumptions.

These assumptions are mostly fix-ups to try to explain things that big bang cosmology explains in natural way.

Big bang cosmology is a very robust theory nowadays.

It would be astonishing if it turned out to be just plain WRONG.

It is probably right as far as it goes.

But big bang cosmology does NOT go everywhere.

1. It has free parameters (e.g., the Hubble constant, Omega, and the baryon to photon ratio) that have to be set by observations. An ideal theory should tell us everything without observational input.

2. It does NOT explain exotic dark matter nor dark energy. Since it doesn't explain dark energy we do NOT understand the ACCELERATION or how long it will last.

   Perhaps MOND (for MOdified Newtonian Dynamics) will upset things more or clarify things if it turns out to be at all correct.

   [Recall MOND requires both Newtonian gravity and general relativity to be modified for low accelerations.

   Such modifications could radically change much of our cosmological theory.]

3. It does NOT explain what happened before the big bang singularity if that existed at all, or what happened before the earliest time that we can track backward in time using known physics: i.e., the Planck time.

4. It does NOT tell us if our universe domain is the whole universe or not.

   Many intuit that our universe domain is NOT the whole universe.

5. There are what are called the horizon problem and the flatness problem that have no natural explanation in the opinion of many in big bang cosmology.

   We will take these problems up just below in the section Inflation and Inflation Cosmology.

---

Inflation and Inflation Cosmology

---

``I could be bounded in a nut shell and count myself a king of infinite space, were it not that I have bad dreams.''

Hamlet, Act II, Scene 2. at MIT's The Complete Works of William Shakespeare.

The relevance of the quote will emerge below.

Inflation is the name for a super-rapid expansion phase that may have happened in the early universe. The expansion is much more rapid than in the conventional Friedmann-Lemaitre-Lambda models.

After the inflation the cosmic evolution is supposed to track into a Friedmann-Lemaitre-Lambda model or something similar.

The idea of inflation was developed independently by Alexei Starobinsky in Russia in the late 1970s and Alan Guth in the US in late 1979 who also coined the term inflation (Ov-240,242,245).

Since then the concept of inflation has spawned a quasi-infinity of inflation theories and has evolved somewhat.

It was and is considered a good idea just because it offers explanations for three problems.

[In science ``problem'' often means a perplexing feature of observations or theory that needs to be explained in order to exorcise perplexity.]

The three problems are:
1. MAGNETIC MONOPOLE PROBLEM: Guth's original reason for studying inflation was to solve the MAGNETIC MONOPOLE PROBLEM. Magnetic monopoles are isolated magnetic poles: we ordinarily always see magnetic dipoles: there is always and north and south pole.

   The problem is one that PARTICLE PHYSICISTS created for themselves. Grand Unified Theories (GUTs), which unite the strong nuclear, weak nuclear, and electromagnetic forces, seem to predict that MAGNETIC MONOPOLES should be created in the early universe and be as common as protons and be much more massive (Ov-239--240). But none are observed and they havn't caused a rapid recollapse of the universe.

   A phase of inflation would decrease the MAGNETIC MONOPOLE DENSITY to practically unobservable: Guth originally estimated about one MAGNETIC MONOPOLE in the observable universe (Ov-245).

   Problem solved---if it ever really existed.

2. HORIZON PROBLEM: The temperature of CMB is extremely uniform. It shows fluctuations of only about 1 in 10,000 (FK-645).

   This implies that the whole early universe was very homogeneous and in very nearly in exact thermodynamic equilibrium (i.e., at nearly the same temperature).

   But in Friedmann-Lemaitre-Lambda models, points on opposite sides of sky from which CMB flux originated were never CAUSALLY CONNECTED (except perhaps at that the physically indeterminate big bang singularity itself).

   Those points are not within each other's particle horizons.

   How could the early universe have such a uniform temperature if it never had a chance to thermally equilibrate?

   More generally how could the early universe be so homogeneous?

   This is the HORIZON PROBLEM.

   One could solve the problem by just saying the universe was created ex nihilo with uniform conditions at the big bang singularity or at the PLANCK TIME.

   But why then was it not created exactly uniform?

   And anyway physicists do not like to accept at blank wall at the PLANCK TIME. They want to know what happened before.

   Inflation solves the HORIZON PROBLEM by saying all the observable universe and more started from a minute region of space that was CAUSALLY-CONNECTED and in thermodynamic equilibrium and inflationary expansion blew it up to sizes that the expansion of Friedmann-Lemaitre-Lambda models could not achieve.

   This solves the HORIZON PROBLEM and also indicates that beyond the region of inflation there are other universe domains.

3. FLATNESS PROBLEM: The observable universe has Omega very close to 1: the best available measurements give Omega=1.02+/-0.02 for the present epoch (FK-653).

   Friedmann-Lemaitre-Lambda models with Lambda=0, Omega(t) (i.e., Omega as a function of cosmic time) always diverges from 1, unless it is exactly 1. In other words, Omega=1 is an unstable state.

   The divergence of Omega.

   In order to as close to 1 as it is today (i.e., for the universe domain to be as flat as it is today i.e., 1.02+/-0.02 [FK-653"]), Omega at the Planck time must satisfy

   
                |Omega(Planck time)-1| = about 10**-60
   
          

   (CL-154).

   The universe must have had Omega fine-tuned to very nearly 1 at early times.

   Perhaps this is just a fundamental initial condition, but as with the HORIZON PROBLEM physicists don't like to that idea.

   They prefer to regard the present day closeness of Omega to 1 as a problem to be solved: i.e., the FLATNESS PROBLEM.

   Now the dark energy or non-zero cosmological constant Lambda must complicate the analysis---but no text says the FLATNESS PROBLEM goes away.

   Inflation solves the FLATNESS PROBLEM by ``stretching'' space flat: this also means the requisite energy density of space for flatness must also be created.