 |
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.
(2007-09-09)
Big Bang :
The vision of
l'abbé Lemaître (1927)
The term "Big Bang" was coined ironically by Fred Hoyle in 1956.
The idea that the entire Universe could have originated from
a single pointlike "primeval atom" was first formulated in 1927
by the Belgian mathematician, Abbé Georges
Lemaître (1894-1966) in a momentous article:
-
Un Univers homogène de masse constante et de rayon croissant rendant compte
de la vitesse radiale des nébuleuses extragalactiques
Annales de la Société scientifique de Bruxelles
A47, pp.49-59 (1927)
-
A homogeneous universe of constant mass and increasing radius accounting for the
radial velocity of extra-galactic nebulae
Notices of the Royal Astronomical Society
91, pp.483-490 (1931)
Lemaître had been ordained a catholic priest in 1923.
He then studied General Relativity at Cambridge under
Arthur Eddington and went on
to MIT. In 1925, he started lecturing at the
Université catholique de Louvain (UCL)
and accepted a full-time position there in 1927, as he obtained his Ph.D.
from MIT.
Georges Lemaître
presented his hypothesis of the primeval atom as describing
a day without
yesterday.
This idea is now universally called the Big Bang theory.
Curiously, the name "Big Bang" was originally a derogatory term
coined by Fred Hoyle
in 1956 to mock the concept...
Hoyle was then a leading proponent of a rival theory which is now all but forgotten.
The theory became universally accepted once the theoretical investigations
of Stephen Hawking and others
proved that General Relativity and the observed
expansion of the Universe do imply such a pointlike origin.
The viewpoint has even become part of the official Catholic doctrine
(Msgr. Lemaître was made an honorary prelate in 1960,
by Pope John XXIII).
Although Georges Lemaître himself
was not an official participant in the 1927 Solvay conference in Brussels
(on the foundations of Quantum Mechanics)
he was residing nearby and met with Einstein on that occasion.
Einstein was not impressed at the time; he was quoted as saying:
"Your calculations are correct, but your grasp of physics is abominable."
However, several years later (c.1935) Einstein would reportedly applaud Lemaître's
ideas as "the most beautiful and satisfactory explanation of creation to which I have ever listened".
What prompted that enthusiastic reaction from Einstein seems to have been
Lemaître's description of cosmic rays as possible left-overs from
the primeval explosion.
It turns out that this early insight does not really apply to high-energy cosmic rays
(as described by Robert Millikan)
but to the low-energy
photons of the cosmic microwave
background, discovered by
Penzias & Wilson in 1965, a few months before
Lemaître passed away...
(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 |
 |
| |
| Ö |
1-b2 |
|
|
| 1 |
|
| |
| Ö |
1 - GM / Rc2 |
|
|
| T |
|
| 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 dominant 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
or neutron star 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 Ho ):
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 the relativistic computation that we'll
outline next gives the same final result,
provided the density "r" is understood to
include the density of energy divided by c2
and a corrective term for what's now called dark energy.
First things first, here's the simple Newtonian argument:
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, whereby 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.
Although W is not
really "constant", the sign of W-1 is.
As advertised, those results remain valid within
General Relativity, whereby an homogeneous and
isotropic universe (as envisioned by
Friedmann in 1922)
is characterized by two parameters;
an increasing scale factor a(t)
(which is a length) and a dimensionless constant K
(the spatial Gaussian curvature
being K/a2 ).
The following relation is known as the
first
Friedmann equation :
H2 =
(a'/a)2 º
(8pGr + L) / 3
- K c2 / a2
Solving for r when
L=0 and
K=0 does give the aforementioned value of
ro .
Our previous Newtonian interpretation remains valid only if
Einstein's field equations hold
with a vanishing cosmological constant
(L=0).
Otherwise, a positive cosmological constant
can even entail an accelerating
expansion of the Universe,
which is compatible with the big picture that emerged from
observational data in 1998.
There's simply no Newtonian explanation for that !
More precisely, according to the above Friedmann equation,
a positive cosmological constant L
can be tallied as a supplemental density
L / 8pG
attributed to dark energy
(this is just a name)
which is added to the combined total from ordinary matter and energy
as well as dark matter.
Dark matter is distinct from dark energy
(which is uniform in space and repulsive on a cosmological scale)
because it can be lumped locally and features attractive gravitation
(as can be detected by gravitational lensing and/or the rotational speed
of neighboring visible matter).
Yet, dark matter is "stuff" which lacks any detectable
non-gravitational interaction with ordinary matter or radiation.
It cannot be made of any of the particles which Science has cataloged so far.
If the overall geometry of our universe wasn't very nearly flat
(like a Friedmann universe with K=0)
then it would either have collapsed
or thinned out beyond recognition a long time ago.
The current prevalent view is thus that
dark energy due to a positive cosmological constant
accounts for whatever is needed to balance the
Friedmann equation (assuming vanishing Gaussian curvature).
This would mean that empty space accounts for about
70% of the total tally (i.e, space itself and all its
matter-energy content, including
dark matter).
Mind-boggling, isn't it?
(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 » 0 | z |
z / H |
| z |
| (1+z) 2 -
1 | |
|
| (1+z) 2 + 1 |
|
| 2 |
|
[ |
1 -
|
1 |
|
] |
|
|
| 3H | (1+z)3/2 |
|
| 1 |
|
z |
|
|
| H | 1+z |
|
| 1 | 60% | 5619 | 6197 | 6518 | 7190 |
| 2 | 80% | 7019 | 7741 | 8691 | 9586 |
| 6 | 96% | 8222 | 9069 | 11175 | 12325 |
| ¥ | 100% | 8691 | 9586 |
13037 | 14379 |
| 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 seem to be slightly better
embodiments of comoving points, but such attempted refinements are vastly inferior to
the better characterization we shall now give...
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
(which indicates that the Earth and the Sun are not comoving)
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.
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...
George Smoot and John C. Mather (of NASA) have been jointly awarded the
2006 Nobel Prize in physics
"for their discovery of the black-body form and anisotropy of the cosmic microwave background radiation".
(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 & Dark Energy
The Sun and other stars have an orbital speed around the Galaxy which is much
larger than what it would be if gravitational forces were only due to all the
to the ordinary matter we can tally (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 found recently.
Although it's true that most ordinary matter
actually resides outside of galaxies,
all the evidence which has been gathered since 1998
seems to indicate that there's about five times more dark matter
than ordinary matter in the Universe.
Also, the fact that the expansion of the Universe is accelerating
indicates a positive cosmological constant in Einstein's
field equations.
This means that empty space itself has a nonzero energy density
of about 0.7 nPa
(this is roughly equivalent to the rest energy
of two hydrogen molecules—or one helium atom—per cubic meter of
empty space).
This type of energy is called dark energy
and it accounts for most (70%) of the following composition our Universe
[courtesy of NASA]:
On the trail of dark
energy by Eric Linder (CERN Courier, 2003-09-04)
(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 | |
| |
| 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 | |
| |
| h3 c3 ( e x
- 1 ) | | |
=
|
| 8p5 k4
T 4 | |
| |
| 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
| |
| |
| hn | | |
dn
=
|
| 16p z(3) k3
T 3 | |
| |
| 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 | |
| |
| 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
õ |
m
0 |
| x2 dx | |
| |
| ( e x
- 1 ) | | |
=
|
z(3) |
|
|
|
