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Final Answers
© 2000-2005 Gérard P. Michon, Ph.D.

Cosmology 101

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Cosmos and Cosmology


(2002-07-30)   The Cosmological Principle

The Cosmological Principle states that, when viewed on a large enough scale, our physical Universe is essentially homogeneous and isotropic.  In other words, the distant Universe looks roughly the same in any broad direction from any typical point (technically, such "typical" points are comoving points in free fall).  Thus, the reason why the Earth cannot be at the "center" of the Universe, as once thought, is that there is no such "center" (alternately, the "center" is anywhere).  In that respect, the Universe resembles the surface of a perfect sphere:  All points are equivalent and no direction is special.

The idea that the Earth is not at the "center" of the Universe is an ancient one, but it was suppressed for a long time and its current prevalence is fairly recent...

In The Sand Reckoner (c.213 BC), Archimedes of Syracuse (287 BC-212 BC) reports that, according to Aristarchus of Samos (310 BC-c.230 BC), the Earth revolves around the Sun.  However, the opposite viewpoint advocated by Aristotle and Ptolemy later became official Church dogma and remained so for centuries.  The heliocentric idea was thus considered a dangerous heretical view when it was revived in 1514, by a Pole named Copernicus (1473-1543).  In spite of the courageous support of Kepler (1571-1630) and Galileo (1564-1642), Church coercion would not allow the "new" perspective to prevail easily (to say the least).  The Italian philosopher Giordano Bruno (1548-1600) was a noted early supporter of the Copernican heliocentric theory:  He was arrested in 1592, underwent a lengthy trial, refused to recant, and was burned at the stake in 1600.  This goes a long way toward explaining why Galileo "chose" to recant when he was similarly charged, in 1633.

However, once this viewpoint is adopted, it's only natural to think that the Earth should not occupy any special place whatsoever in the Universe as a whole.  After the discovery that the Milky Way (which harbors our Solar System) is only one of many similar galaxies, it seemed natural to assume that all such galaxies are essentially placed on an equal footing.  At a sufficiently large scale, the distant Universe should look essentially the same in any direction from any typical galaxy.

This statement was first called the Cosmological Principle by the British astrophysicist Edward Arthur Milne (1896-1950).  Before the Cosmological Principle was even known by that name, it had been used by the Russian cosmologist Alexander Aleksandrovich Friedmann (1888-1925), who should be given credit for the idea of an expanding Universe.

In 1922, Friedmann devised a model of an expanding Universe obeying both the Cosmological Principle and the equations of General Relativity, without the need for the so-called Cosmological Constant L (which Albert Einstein had introduced mostly to accomodate the [then] prevalent idea of a static Universe).  In 1929, the American astronomer Edwin Hubble independently discovered the first observational evidence of the expansion of the Universe...

There is now overwhelming observational evidence for the validity of the Cosmological Principle from careful measurements of the so-called Cosmic Microwave Background (CMB) which was discovered in 1964-65 by Arno A. Penzias and Robert W. Wilson.  The CMB has been found to be isotropic to a precision of about one part in 100 000.


(2002-07-24)   Redshift
How is redshift defined?  What's a cosmological redshift?

When some signal [sound, light, etc.] emitted at a frequency n is observed at frequency  n / (1+z) , the quantity z is called the redshift of the source for the observer.  In the case of visible light, a positive redshift does make the source appear redder (a negative redshift makes it look bluer and the opposite of a redshift is thus sometimes called a blueshift).

Redshift may have a number of combined causes, including the classical Doppler effect (which depends only on the radial velocity of the source), and the time dilation at the source due to its speed (Special Relativity) and/or surrounding gravity (General Relativity).  Finally, for very distant sources, there is also a cosmological redshift due to the fact that the wavelength of a traveling signal increases in the same proportion as the Universe expands.  In other words, light which was emitted from cosmic distances, when the Universe was (1+z) times smaller than now, is currently observed with a cosmic redshift equal to z.

The quantity   1+z   is the ratio of the observed wavelength to the emitted one and may be expressed as a simple product of several factors.  Each of these correspond to one of the 4 causes of redshift listed above:

1 + z     =    (1 + v/u)   
 1 
Vinculum
Vinculum
Ö 1-b2
  
 1 
Vinculum
Vinculum
Ö 1 - GM / Rc2
  
T 
Vinculum
 To
 
 Classical 
(Radial)
 Relativistic 
(Isotropic)
Gravitational  Cosmic 
Doppler Effect General Relativity

This "combined" formula is given mostly for educational purposes.  It's normally simplified in order to retain only the dominant effect(s) under consideration.
Symbols are used in the above formula with the following meanings:

  • v is the radial speed of the source.  For cosmological distances and/or curved signal propagation, this is defined in terms of the velocities of the observer and the source with respect to local comoving points at rest in the CMB:  The "radial" speed v is actually the difference between the projections of these velocities on the local tangents to the signal's "path". 
  • u is the celerity of the signal [its phase speed].  For light in a vacuum, u = c = 299792458 m/s.  For sound in dry air (20°C), u » 343.37 m/s. 
  • b is the ratio of the speed of the source to c (the speed of light).
    (There may be a nonzero transverse velocity, in which case b > v/c.) 
  • G is the gravitational constant.  M is the mass of some large nonrotating spherical body at a distance R from the source.  (The gravitational redshift factor given here as an example would, of course, be different for other mass distributions, but it's usually a good enough approximation whenever there's no rapidly rotating black hole in the vicinity...) 
  • The cosmic redshift factor T/To is the ratio of the "old" temperature T of the Cosmic Background at the source to the "newer" value seen by the observer ( To is currently about 2.728 K).  This factor is actually equal to Do / D, where D is any distance characterizing the whole Universe, like the average distance between major galaxies, or the wavelength of a typical background photon (which is indeed inversely proportional to T ).

Astronomers observe the redshift (z) directly by measuring the wavelengths of known lines in the atomic spectra of the light emitted by a distant source.  However, there is a dubious tradition to quote also the apparent recession speed of such distant sources (defined as the purely radial velocity of a nearby source with the same redshift, in the absence of General Relativistic effects).  This is obtained by retaining the first two factors of the above formula, (letting u = c and b = v/c), so that (1+z) 2 is  (1+b)/(1-b)  and we have:

v/c   =   b   =   [(1+z)2 -1] / [(1+z)2 +1]   =   z / [1 + z/2 + z2/(4+2z) ]

For example, a redshift  z = 1  corresponds to exactly 60% of the speed of light, whereas  z = 2  is 80% of the speed of light, and  z = 6  is (exactly) 96% of the speed of light...  Again, it's best to quote only z and ignore this "translation" which dates from an era when the expansion of the Universe was not yet understood...


(2002-12-09)   Hubble Law & Hubble Flow
What is Hubble's "constant"?

Arguably, modern cosmology originated in 1917 at the Lowell Observatory, when Vesto Melvin Slipher (1875-1969) observed that distant galaxies are all receding from our own Milky Way.  In 1929, Edwin P. Hubble (1889-1953) discovered (from sketchy observational data) that the recession speed (v) of a galaxy is roughly proportional to its distance from us (d).  The nonrelativistic coefficient of proportionality is now called Hubble's constant (H or H):

v   =   H d

Hubble's constant (H) actually describes the rate of expansion of the Universe, and its value evolves as the Universe ages.  Simple models of the Universe make the product of H and the Age of the Universe equal to a dimensionless number that depends on specific assumptions:  This product would be equal to 1 in a Universe of very low density [H would be the reciprocal of the Universe's age], but it would be 2/3 in a flat Universe (W = 1) dominated by ordinary matter, and only 1/2 in a radiation-driven expansion phase (the fireball conditions which prevailed for less than 56 000 years).  On the other hand, if some form of exotic stuff and/or a nonzero cosmological constant dominates the large-scale structure of the Universe (as modern data may indicate), the above product could be equal to or larger than one, so the Universe might be older than 1/H.

The actual value of H is difficult to determine experimentally, mostly because it's difficult to determine the distance to an object that's far enough to make its [unknown] proper motion a negligible factor in its observed redshift.  The latest estimates place H somewhere between 68 km/s/Mpc and 75 km/s/Mpc.

The reciprocal of H is sometimes called the Hubble time, and the Age of the Universe is commensurate with it.  One "s-Mpc/km" is 977 792 221 400 years, and the Hubble time corresponding to the above values of H is thus 75 or 68 times smaller than this, namely between 13 and 14.4 billion years...


(2002-07-24)   W
What is meant by "critical density"?  What's the omega (W) constant?

Following Steven Weinberg (The First Three Minutes, 1977), we'll introduce the notion of critical density in the framework of Newtonian mechanics.  It turns out that a relativistic computation would give the same final result  (provided the density "r" is understood to include the density of energy divided by c):

Consider a sphere of radius R much smaller than the whole universe, but large enough to apply the Cosmological Principle.  If r is the average mass density of the Universe, such a sphere is roughly homegeneous and its total mass M is equal to r times its volume, namely:

M   =   4pR3r/3

The (Newtonian) potential energy of a galaxy of mass m near the surface of the sphere is -mMG/R (where G is Newton's Universal Constant of Gravitation):

- mMG/R   =   - 4pmR2rG/3

On the other hand, this galaxy has a (purely radial) speed V = HR given by Hubble's Law (H = H(t) being the value of Hubble's constant at the present time t) and its kinetic energy is therefore:

½ m V2   =   ½ m H2 R2

The total energy of the galaxy is the sum of the above two terms and remains constant as the Universe expands:

m R2  [ H2/2  -  4prG/3 ]

If this total energy is positive, the galaxy will eventually escape to infinity with some kinetic energy left over.  If it's negative, this won't happen and, in fact, the Universe's expansion will eventually stop and reverse (the Universe will then collapse).  Between these two alternatives is the critical case where the bracket in the above expression is precisely zero, and the Universe keeps on expanding forever, but just barely so  (the relative speed of two typical galaxies eventually approaches zero but their distance still keeps growing to infinity).  The above expression shows that this happens precisely when the density r of the Universe is equal to the following quantity ro, which is called the critical density :

ro   =   3 H2 / 8pG           W = r / ro

The ratio (W) of the actual density r to the critical density is the famous omega "constant", which determines the ultimate fate of the Universe:  If W is less than or equal to 1, the Universe will expand forever, otherwise it will eventually collapse.


(2003-07-16)   Look-Back Time
How is the "look-back time" of distant objects determined?

The redshift of a very remote object is observed directly, but all other indicators of its distance are deduced from some cosmological model of the Universe...  In particular, the look-back time of a distant source is defined as the time ellapsed since the light reaching us was emitted.  Because of the Universe's expansion, such a distant source is always farther away than what would be naively estimated by multiplying its look-back time by the speed of light.

Our estimate of look-back time depends on which model of the Universe we rely on.  Currently, one popular view is that the total energy content of the Universe is zero (negative gravitational energy balances matter and other forms of positive energy) and that the existence of some sort of exotic "dark" stuff and/or a nonzero Cosmological Constant makes the Universe behave gravitationally, on a large scale, as if it was effectively empty (its expansion does not slow down).  This viewpoint is illustrated by the last two columns of the following table:

Look-Back Times (Millions of Years) for 2 Cosmic Models & 2 Values of  H
Cosmic
Redshift
( z )
"Apparent"
Recession
( b )
Matter-Dominated
( W = 1 )
Zero Total Energy
( Effectively, W = 0 )
75 km/s/Mpc 68 km/s/Mpc 75 km/s/Mpc 68 km/s/Mpc
z » 0z z / H
z
 (1+z) 2 - 1 
vinculum
(1+z) 2 + 1
 2      é
ë
 
1 -  
 
1   ù
û
vinculum vinculum
3H(1+z)3/2
 1      é
ë
  z   ù
û
vinculum vinculum
H1+z
160%5619619765187190
280%7019774186919586
696%822290691117512325
¥100%86919586 1303714379
If the Universe was indeed dominated by ordinary matter, it would be younger than the oldest stars in it !

(2002-12-09)   Distance
What is distance in a cosmological context?

Astronomers estimate distance in many different ways.  It's not at all obvious that all such methods end up measuring the same thing.  In fact, they don't.

In an observational cosmological context, the distance to a distant object is [probably] best defined as the distance its light has traveled before reaching the observer.  This definition would mean distance and look-back time are simply proportional (the coefficient of proportionality being Einstein's constant).  Thus the relation between distance and redshift would depend, as discussed above, on how the Universe has expanded between the emission and the reception of light.

This observed cosmological distance is thus not a simple concept and it's fairly useless in theoretical speculations, where the distance of an object to the [arbitrary] origin is best defined as the value of a space-coordinate when the time-coordinate is the same [we're talking curvilinear coordinates in curved space, here].  In an expanding universe, this latter flavor of distance is greater than the former one.  [The source has "moved away" after emitting its light.]

The straightforward parallax method, based on Euclidean trigonometry, may not be valid for very large distances and/or when strong gravity is present; the three angles of a large physical triangle may not quite add up to 180°.  Although the parallax angles of galaxies are actually far too small to be measured, we may wonder how trigonometry could be used in principle to measure intergalactical distances...  The very concept of distance is worth questioning under at least three types of extreme conditions:

  • Extremely small scales:  The Planck length (1.6´10-35 m) is the characteristic unit of a scale at which physical space itself is thought to lack any kind of smoothness.  Geometry breaks down when we "look" this close.  This is the not-yet-understood domain of quantum gravity. 
  • Extreme curvature:  Around black holes, our Euclidean intuition fails.  It's best to avoid considering the "distance to the center of a black hole", because this distance would turn out to be infinite under most definitions. 
  • Extremely large scales:  As the Universe expands, so does the distance between two objects sufficiently far apart.  The expansion of the Universe may thus introduce a significant delay in the light signals that go from one object to the other.  It becomes important to state precisely what is meant by "distance" in such a context, as discussed above.


(2002-07-24)   Comoving Points & CMB Anisotropy
What are "comoving points" ?

In the Euclidean space of classical geometry, motion is actually considered relative to some immobile framework of fixed points.  This viewpoint is not a practical proposition within our expanding physical Universe considered as a whole.  Instead, the cosmological approach is to introduce reference points whose relative motions are entirely due to the general expansion of space itself, whatever that may be.  By definition, such points are said to be comoving.

The relative motions of galaxies are not entirely due to the expansion of the Universe (nearby galaxies attract each other) and their centers of mass are thus not strictly comoving.  However, descriptions of our expanding Universe will often discard the distinction for the sake of simplifying the presentation.  (The centers of fairly large clusters of galaxies could be slightly better embodiments of comoving points, but even such refinements are vastly inferior to the better characterization we give next.)

The most practical viewpoint is to characterize a comoving point as a point which is at rest with respect to the Cosmic Microwave Background (CMB).  The Sun is not comoving (relative to the CMB, its speed is about 370 km/s ).  Neither is the center of mass of our Local Group of galaxies, which moves at about 600 km/s with respect to the CMB (three dozen galaxies are thus not a large enough chunk of matter to estimate the value of our local Hubble flow).

The dipolar anisotropy of the CMB from our local viewpoint indicates that the Earth and the Sun are not comoving.  This was first precisely determined in 1977 by the so-called U2 Anisotropy Experiment which was flown aboard the NASA Ames U2 jet aircraft by a UC Berkeley group.  The results were later confirmed from outer space, by the "Cosmic Background Explorer" (COBE) launched on November 18, 1989.  George Smoot masterminded both projects.

The Universe after 379 000 years,
according to WMAP (Feb. 2003)...
Click for larger picture from NASA. Knowing our own speed in the CMB is just the beginning.  The tiny irregularities in the CMB offer a baby picture of the Universe at the age of about 379 000 years, when it first became transparent.  (A COBE picture made headlines in April 1992.)

On June 30, 2001, NASA launched its "Microwave Anisotropy Probe" (MAP) at a cost of $145 000 000.  It is 45 times more sensitive than COBE and its angular resolution is 33 times better.  MAP was renamed in honor of David T. Wilkinson, who died on September 2, 2002 (WMAP = "Wilkinson Microwave Anisotropy Probe").  It arrived at L2 on Oct. 1, 2001 (the second Lagrange point "L2" is a semi-stable orbital position on the Earth-Sun line, 1.5 million km further from the Sun than the Earth).  A first sky scan was completed in April 2002 and the WMAP results were finally released on February 11, 2003...


(2002-07-30)   The Anthropic Principle

The Anthropic Principle is simply the statement that the Universe we observe must allow intelligent life to evolve, or else we would not be here to observe it.

In any universe with features that rule out intelligent life, there would not be anybody around to wonder why such features exist...  Yet, there is a general feeling that the Anthropic Principle by itself provides a poor sort of explanation.  Indeed, if we were to assume that there's only one possible universe, it seems that there should always be a reason for what we observe, other than our own existence.  Thus, cosmologists often find the Anthropic Principle somewhat repugnant and will invoke it only as a last resort...

The alternative, however, is that there could very well "be" (in some obscure sense) many universes.  Some have intelligent observers in them and some don't.  The Anthropic Principle simply states that our own Universe can only be of the former type.  In fact, André Linde's "chaotic inflation" theories do predict that the creation of a universe like ours is best explained as part of a process which creates a large multiplicity of universes in which the fundamental constants of nature may have different values.  If that viewpoint is correct, there would not be any ultimate explanation for the values of the fundamental physical constants, except that their range should be compatible with the Anthropic Principle...

Now, the tricky part is that the dubious "existence" of other universes is entirely irrelevant, by definition, to the physics of our own Universe.  As an irrelevant assumption does not change anything, we may conclude that the Anthropic Principle (which may or may not be ultimately needed ) is fully justified even if we leave open the "existence" of anything outside of our own Universe.


(2002-10-28)   Dark Matter

In our own Galaxy, the Sun and other stars have an orbital speed which is much larger than what it would be if gravitational forces were only due to to the visible matter we've tallied (stars and interstellar gas).  The same observation can be made in other galaxies as well.  Galaxies have massive dark halos which consist of some strange stuff, called dark matter.

Although the early evidence for the existence of dark matter came from galactic rotation curves, the relative speeds of galaxies in some clusters also imply the existence of intergalactic dark matter to hold clusters together (at the large speeds observed, the galaxies would otherwise have flown apart a long time ago).  More localized evidence has also been recently found.

 Come back later, we're
 still working on this one...


(2003-07-14)   The Cosmic Microwave Background (CMB)
What's the energy density of the Cosmic Microwave Background today?

The Cosmic Microwave Background is a gas of photons with a blackbody spectrum, whose current temperature has been measured to be T = 2.728(2) K.  The energy density [energy per unit of volume] contributed by the photons whose frequencies are between n and n+dn is therefore given by Planck's formula:

 
un dn   =   
 
8p hn3 dn
vinculum
c3 ( exp( hn / kT ) - 1 )

Introducing the variable   x = hn / kT , the energy density of all photons is thus:

ó¥
õ
0  
 
un dn   =   
 
ó¥
õ
0    
8p k4 T 4 x3 dx
vinculum
h3 c3 ( e x - 1 )
 
    =    
 
8p5 k4 T 4
vinculum
15 h3 c3

In this computation, the definite integral of   x3/( e x - 1 ) may be obtained as the sum (for n = 1 to ¥) of the definite integrals of  x3 e -nx...  The n-th term is 6/n4 and they all add up to p4/15 (the reciprocals of fourth powers add up to p4/90).

In terms of the Stefan-Boltzmann constant (s) the above is equal to  (4s/c) T 4.  The total energy density of blackbody radiation is thus proportional to the fourth power of the absolute temperature T:  To a CMB temperature of 2.728(2) K corresponds an energy density of 4.190(13) 10-14 Pa  (1 Pa equals one joule per cubic meter) which is about 260 electronvolts per liter, or 0.26 eV/cc.

Number of Photons in Blackbody Radiation:

Since each has energy hn, the density of the photons is the following integral, whose value involves Apéry's number z(3), the sum of the reciprocal cubes :

ó¥
õ
0    
un
vinculum
hn
 
  dn     =    
 
16p z(3) k3 T 3
vinculum
h3 c3
For the CMB, this is about 410 000 photons per liter (410 photons per cc).

Average Energy of a Thermal Photon:

It's the ratio of the total energy density to the above density of photons, namely:

p4
vinculum
30 z(3)
 
  kT     =     ( 2.70117803291906389613472623...)   kT
 

Median Energy of a Thermal Photon:

m kT   »   2.35676305705  kT
(where m is given by the relation at right)          
ó m
õ0    
x2 dx
vinculum
( e x - 1 )
 
    =    
 
z(3)
 

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