(2009-09-21) Olbers' Paradox
In an infinite and unchanging Universe, the night sky would be white!
Since the night sky isn't brightly lit, the obvious logical conclusion is that the
Universe is not infinite and unchanging.
We now know that it's finite and expanding.
Historically, the argument presented by Olbers helped clarify that viewpoint.
Arguably, the night sky is neither bright nor dark, it's just darker than
it used to and it's dimming steadily...
Nowadays, Heinrich Olbers (1758-1840) is often presented as an amateur
mathematician because he earned a living as a physician.
However, he must be credited with several important contributions to astronomy,
including an efficient method for computing cometary orbits (1779) and the
discovery of two of the three largest asteroids,
2 Pallas (28 March 1802) and
4 Vesta (29 March 1807).
Olbers' Paradox did not originate with Olbers but he
used his considerable reputation to popularize the argument.
In the cosmological view where outer space consists only of stars and empty
space, ... / ...
(2009-09-21) Galaxies
A short history of a mind-boggling idea.
The Milky Way
(Latin: Via Lactea) is the dim stripe of nebulous light
which can be seen on a clear night in the direction of Sagittarius.
Telescopes resolve that cloud into a system of
many stars to which our own Sun actually belongs.
It was a tremendous leap to imagine that all the fuzzy
spiral nebulae which can be observed through telescopes are
giant systems of stars similar to our own but at very great distances.
The famous catalog of 110
nebulae (including spiral nebulae) which
Charles Messier (1730-1817)
and
Pierre Méchain
(1744-1804)
published in 1777 was arguably little more than a list of nuisances
which might hinder the glamorized hunt for fleeting comets,
traveling through the solar system
The nature of those nebulae kept puzzling astronomers.
Because they have no measurable parallax,
it was clear early on that they are located at great distances from the
Solar system.
In spite of that, they don't look pointlike like stars do.
So, they must be much larger than stars.
Pierre-Simon Laplace
(1749-1827) advocated the dominant (misguided) opinion that the
spiral nebulae were rotating clouds forming new stars, according to the
nebular hypothesis
which had been formulated in 1734 by
Emanuel
Swedenborg (1688-1772) to explain the formation of the Sun itself.
Curiously, Swedenborg had also put forth another
unrelated seminal idea which would ultimately lead to the correct understanding
of the true nature of the spiral nebulae
(and the downfall of Laplace's views):
Swedenborg envisioned a definite order among visible stars, within a huge
local "starry sphere".
Although the details of his description are ambiguous and incompatible with modern views
(it's unclear whether he thought of the Via Lactea
as a polar axis or an equatorial ring for that sphere).
Swedenborg broke fantastic new grounds
when he suggested the mind-boggling possibility that there could be many
other such "starry spheres" at very large distances...
Around 1731, similar speculations were being made independently
by the gifted son of
a well-to-do carpenter and land owner from the hamlet of
Byers Green (6 miles
to the south of
Durham City)...
In 1750, having fulfilled his
unlikely destiny
of becoming an astronomer,
Thomas Wright
(1711-1786) published his proposal
that the distant starry formations envisioned by Swedenborg
(henceforth called galaxies )
might actually be visible to us in the form of nebulae.
Wright also explained the appearance of the Milky Way as
"an optical effect due to our immersion in
what locally approximates to a flat layer of stars."
As he wrote that, Wright was still thinking of this "layer" as part of
a large hollow spherical distribution
(actually, it's just a finite flat disk).
Both ideas were enthusiastically embraced by the philosopher
Immanuel Kant (1724-1804)
who credited Wright (but not Swedenborg) for them, although he candidly admitted
(in print)
that he had never read Wright's book !
Kant coined the popular term of Island Universes,
which he (correctly) envisioned as fairly thin rotating disks of very many stars
(thus discarding, possibly involuntarily, Wright's misconceptions about
hollow spherical distributions of stars).
Kant dubiously conjectured Sirius (the brightest
star in our Sky) to be located at the center of the Milky way
and he still called the Milky Way, the main plane of all motion.
When he published his
Universal
Natural History and the Theory of Heavens (1755)
Kant had no university degree yet, but he had this to say
against the views of Maupertuis on
nebulae in general and spiral nebulae in particular:
"It is much more natural and conceivable that these are not single stars
of such [huge] size, but systems of many stars, whose distance from us exhibits
them as being in so narrow a space that the light, which is imperceptible
from each one individually, becomes a uniform pale shimmering with
their immeasurable number."
In 1910,
Vesto
Melvin Slipher (1875-1969) tried to argue Laplace's case but ended up
showing that the other view was probably correct!
The mystery was finally solved when it was confirmed that the nebulae
were extremely large objects at huge distances.
The proof came in 1923 when Edwin Hubble turned his attention to the largest of them,
the Andromeda Nebula,
which is visible to the naked eye and was first described as a nebulous smear
in AD 964 by
Azophi (AD 903-986).
Using the 100-inch
Hooker telescope
on Mount Wilson,
(in Los Angeles County)
Hubble managed to identify a Cepheid variable star (now called V1) in the Andromeda nebula.
He measured the period of V1 to be 31.4 days from which its absolute luminosity
can be deduced. The faintness of V1 meant that it was more than a million light-years away.
So are the other stars of the nebula which is actually composed of nearly a trillion
stars. Andromeda was thus revealed to be one of the island universes,
envisioned by Kant and our view of the cosmos was forever changed!
The key tool for Hubble's momentous discovery was the
relation between
the period of a Cepheid variable and its absolute luminosity, which had been discovered
by Henrietta Leavitt (1868-1921)
by observing many nearby Cepheids, whose distances
in parsecs (pc) could be obtained directly
by triangulation
from an observation base narrower than 2 au.
What Hubble did for Andromeda, Leavitt had done 20 years earlier for the Magellanic clouds
(where she cataloged 1777 variable stars between 1904 and 1908).
The work of Hubble was a bit flawed because of the existence of two classes of Cepheid variables,
depending on which population they belong to,
a distinction which Walter Baade (1893-1960)
introduced in 1944.
Working at the
Mount Wilson Observatory (MWO),
from 1931 to 1958, Baade took advantage of the low light pollution in 1944
(resulting from the blackout of Los Angeles during WWII) and could resolve individual stars in the
center of the Andromeda Galaxy, which helped with the aforementioned distinction. As a result,
Baade rescaled the results of Hubble by a factor of two in 1952.
The most recent estimate of our distance to the center of the Andromeda Galaxy
is 2.537 million light-years.
The Cosmological Principle says that, when viewed on a large enough scale,
our physical Universe is mostly homogeneous and isotropic.
In other words, the distant Universe looks roughly the same
in any broad direction from any typical point (technically, such points
are comoving points in free fall).
Thus, the reason why the Earth isn't at the center of the Universe, as once thought,
is that there's 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 ancient,
but it was suppressed for a long time and its current prevalence is fairly recent...
In his Sand Reckoner masterpiece (c.213 BC),
Archimedes of Syracuse (287 BC-212 BC) reports that
Aristarchus of Samos (310 BC-c.230 BC) was already
arguing that the Earth revolves around the Sun. However, the opposite viewpoint advocated by
Aristotle (384-322 BC) and
Ptolemy of Alexandria (c. AD 87-165)
remained official Church dogma for centuries.
The heliocentric idea was thus considered a dangerous heretical view when it was finally
revived privately, 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
(in part for political reasons unrelated to his cosmology).
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 shouldn't
occupy any special place whatsoever in the Universe as a whole.
After the discovery that the Milky Way
(which harbors our own Solar System) is only
one of many similar galaxies, it seemed fairly obvious 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.
That statement was first called theCosmological 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 accommodate
the [then] prevalent idea of a static Universe).
In 1929, the American astronomer Edwin Hubble (1889-1953)
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
(Nobel 1978).
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 in jest 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, AbbéGeorges
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 was obtaining 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...
Using 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 the above, the definite integral of
x3/( e x - 1 )
was obtained as the sum
(for n = 1 to ¥)
of all definite integrals of x3e -nx :
As the n-th one is 6/n4 the whole sum is
p4/15
(the reciprocals of fourth powers add up to
p4/90).
The total energy density of blackbody radiation
is thus proportional to the fourth power of its absolute temperature T.
In terms of the
Stefan-Boltzmann constant
(s)
the above is equal to (4s/c) T 4.
Well before Planck, that fact was known as the
Stefan-Boltzmann law, which was discovered
experimentally by Jozef Stefan (1835-1893) the year
Einstein was born (1879).
It was established theoretically five years later (1884) by his
star studentLudwig Boltzmann (1844-1906).
Planck used the bold new idea of quantization to justify his radiation formula,
which unifies asymptotically two different regimes:
The classical
Rayleigh-Jeans
law (1900,1905) for low frequencies and the
Wien distribution (1896)
which fails for low frequencies.
Incidentally, Planck expressed the Stefan constant
s in term of the
new constant h he introduced for the occasion:
To a temperature T = 2.72548(57) K
corresponds an energy density of:
7.565723(18) 10-16 T4 =
4.1747(36) 10-14 Pa
( 1 Pa = 1 J / m3 )
That's about 261 electronvolts per liter;
261 eV/L or 0.26056(23) eV/cc.
Number of Photons in Blackbody Radiation:
Since each photon has energy hn,
the density of photons per unit volume is the following integral
(whose value involves
Apéry's constant
z(3), defined as
the sum of the reciprocals of all perfect cubes ) :
ó õ
¥ 0
un
hn
dn
=
16pz(3) k3
T 3
h3 c3
= ( 20.286826(36) 106 ) T3
For the CMB, that's 0.41072(26) 109 photons per m3 (411 photons per cc).
Average Energy of a Thermal Photon:
It's the ratio of the total energy density to the 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)
Thermal Photon at the Peak of the Frequency Spectrum:
When the spectral density of the blackbody energy is plotted against frequency
(as above) it's proportional to x3 / (exp(x)-1)
and reaches a maximum when exp(-x) = 1-x/3.
That's to say that, in the most energetic interval of frequencies,
a thermal photon has the following energy:
h n = ( 2.82143937212...) kT
Thermal Photon at the Peak of the Wavelength Spectrum:
In the nineteenth century, the spectral energy per wavelength interval
(instead of frequency interval) was commonly plotted.
The maximum of that diagram is reached at a totally different point,
corresponding to the solution of exp(-x) = 1-x/5. Namely:
h n = ( 4.965114231744...) kT
HINT: Maximize
ul defined by
ul dl
=
un dn ,
knowing l n = c.
(2018-05-20) Primordial Nuclear Soup
Mostly hydrogen and helium. Some deuterium and a little lithium.
In the very first second of the Universe, the overabundance of neutrinos and thermal
photons helped maintain an equilibrium with roughly the same number of neutrons and protons.
The intense radiation overwhelmed any nuclear bonds which might otherwise have formed.
At a slightly lower temperature, protons become significantly more abundant than
neutrons, on account of their slightly lower mass.
An equilibrium was established well before nuclear fusion started for good
(when the Universe was about 10 seconds old) with roughly only one neutron to six protons.
During the next 20 minutes or so, some lone neutrons did decay before they fused into atomic
nuclei, which lowered the neutrons-to-protons ratio from 1:6 to 1:7.
However, most of them formed stable deuterium nuclei by fusing with protons.
Most of those deuterium nuclei fused with each other to form Helium-4
(alpha particles). Some fused with protons to form stable Helium-3 directly or
with remaining neutrons to form radioactive tritium
which would be completely transformed to Helium-3 by beta-decay in a few short centuries
(the half-life of tritium is 12.3 years ).
The 1:7 ratio of neutrons to protons at the end of the so-called BBN (Big-Bang nucleosynthesis)
doesn't depend much on reaction details.
It matches the often-quoted primordial abundance of helium
(25% by mass). In the main, with 14 protons and 2 neutrons we form 12 hydrogen nuclei and
one helium-4 nucleus. The latter thus accounts for a quarter of the total mass.
On the other hand, the relatively low abundance of unfused deuterium nuclei (deuterons)
at that time depends very much on the dynamics of the primordial reactions
and is therefore a good test of any model thereof.
The lack of stable or semi-stable nuclei containing 5 nucleons means that
very few stable heavier nuclei emerged from the primordial nucleosynthesis.
By far, the most abundant of those was
Lithium-7,
with a primordial abundance of roughly one atom in ten billion.
Lithium-6 is a distant second.
In 2011, investigating the tiny primordial concentration of
Beryllium-9 was
proposed as a probe into the details of the BBN (Big-Bang nucleosynthesis).
As the Universe evolved into galaxies
harboring several generations of stars,
primordial deuterium has been steadily depleted in all stellar cores faster than regular hydrogen.
It's the only form of nuclear fuel available to brown dwarfs
which are, by definition, too small for gravitational collapse to produce the temperatures required
to ignite hydrogen fusion.
Estimates for the abundance of primordial deuterium must be derived
from measurements of various present-day environments.
Our best estimates are that the early Universe had 27 atoms of
deuterium per million of hydrogen atoms.
Although it was thought that about a third of the primordial deuterium
had been destroyed over time, some hydrogen clouds in the Milky Way
still show a deuterium concentration as high as 23 ppm
(just 15% below the estimated primordial abundance).
The abundance of Deuterium currently observed in the Earth's oceans
is higher: About 156 ppm or, more precisely, one atom
of deuterium (D) for about 6420 atoms of protium (H).
Most is in the form of semi-heavy water molecules (HDO)
which are about 3210 times more common in seawater than "pure"
heavy water molecules
(deuterium oxide, D2O).
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 makes 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 four
causes of redshift listed above
(some authors quote only three such causes by viewing
both types of Doppler shifts as a single aspect of the same phenomenon;
I beg to differ):
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
Rarely, if ever, are the quantified effects of the four causes of redshift
explicitly unified in this way.
Almost always, that grand formula is reduced down to only one or two dominant
redshift factors.
The symbols have 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 immediate 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, in a cosmological context,
it's best to quote only z and ignore this dubious "translation".)
In April 2009, a cosmological redshift of
8.2 was observed for the gamma-ray burst identified as
GRB 090423
(which lasted for a few seconds, during the violent implosion of a short-lived
massive star into a black hole). At the time,
that object was the oldest and most distant ever seen.
The light we saw from it had been emitted when the Universe was
9.2 times smaller than
today (the temperature of the CMB was then about 25 K).
In October 2010, one galaxy with a redshift of 8.55
(HUDF.YD3)
was found among the 10000 galaxies of the
Hubble
Ultra Deep Field.
The oldest light we can observe is that of the CMB itself.
It was emitted when the Universe became transparent to electromagnetic
radiation, at a temperature of about 3000 K.
This corresponds to a redshift of about 1100.
Botched in
Yahoo! Answers (2010-11-18) Multiple Choice Exam
In Einstein's General Theory of Relativity,
the redshift of galaxies in the Universe is correctly interpreted as:
A Doppler-shift due to the motions of the galaxies through expanding space.
An "aging" of the light.
Space itself is expanding with time;
the wavelengths of photons are stretched while they travel through the expanding space.
The difference in temperatures of distant and nearby galaxies.
Clueless students who systematically pick the longest answers would
enjoy an unfair advantage here: The only correct answer is (c).
Answer (a) was once used by Edwin Hubble and others outside
of the framework of General Relativity
(it still appears occasionally in misguided essays).
However, this is not acceptable in the proper context of General Relativity
where typical galaxies occupy locations that are as motionless as
the expansion of the Universe can possibly allow
(i.e., comoving points with fixed coordinates).
As discussed above,
the Doppler effect
is another cause of redshift
which is unrelated to the cosmological redshift.
Answer (b) conjures up other deprecated viewpoints which are
not compatible with General Relativity,
unless you redefine "aging" of light as the matching of the
wavelength of traveling photons to the changing scale of
the Universe described by the correct answer (c)
Answer (d) is wrong but barely so.
It would be correct if we could interpret "temperature of a
galaxy" as the temperature of the Cosmic Background
around that galaxy at the time when the light we see was emitted
from it.
(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) took Vesto Slipher's velocity data
(incidentally, witout giving Slipher proper credit) and
tried to correlate them with his own distance measurements
(obtained with the method inaugurated by Henrietta Leavitt
(1868-1921) in 1908). What Hubble discovered (from sketchy data)
is 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.
Various models
of the Universe make the product of H into the Age of the Universe
equal to a dimensionless number which 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].
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).
However, if some form of exotic stuff and/or a nonzero cosmological constant
dominates the large-scale structure of the Universe
(as modern data
indicates), 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 precisely 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 homogeneous 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 negative (it's a binding energy). It's equal to:
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 mechanical 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.
W is not
really constant but the sign of W-1 is
(in this model at least).
As advertised, the results of the previous section
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 factora(t)
(which is a length) and a dimensionless constant K
(the spatial Gaussian curvature
being K/a2 ).
Under those assumptions, the temporal (00) component of
Einstein's field equations yield the following relation:
First Friedmann Equation (1922)
H2 º
(a'/a)2 =
(8pGr + L) / 3
- K c2 / a2
Solving for r when
L=0 and
K=0 gives 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 First 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.
unless right-handed neutrinos are involved.
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 is due to a positive cosmological constant
that accounts for whatever is needed to balance the above
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?
As the Universe expands, its density (r>0)
and pressure (p>0) decrease steadily. So does the above negative bracket.
If the cosmological constant (L) is positive,
the right-hand-side becomes positive after a while,
which means that the expansion of the Universe
accelerates after a certain point.
Since 1998, it's known that we passed that point about
5 billion years ago.
This shows that L is
positive (which Einstein's equations don't require).
(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.
All other indicators of its distance depend on some
cosmological model of the Universe.
In particular, the look-back time of a distant source is defined as
the time elapsed 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.
Well, this "naive" estimate is
one possible definition of cosmological distance, which
may be called the distance "to us".
The distance "from us" (which could be defined, in this context, as the time it
would take for photons we send now to reach the distant object)
is certainly larger than that in a rapidly expanding Universe, but it's not
infinite unless the expansion is accelerating out of control...
Our estimate of look-back time depends on which model of the Universe
we rely on.
Currently, the most popular such model is the
Lambda-CDM_model
which allows for the existence of dark matter and a positive
Cosmological Constant which accelerates the expansion of the Universe.
The table below presents two (obsolete) versions of the semi-Newtonian model.
The first one is utterly incompatible with modern observations.
The second one (represented by the last two columns)
is surprisingly good for a rough estimate.
It assumes that the Universe behaves gravitatonally as if it was effectively empty.
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
t =
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
A Newtonian Universe
dominated by ordinary matter 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 intergalactic 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.
On Oct. 1, 2001, It arrived at the second Lagrange point
L2 (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.
MAP was renamed in honor of David T. Wilkinson, who died on September 2, 2002
(it's now called WMAP = Wilkinson Microwave Anisotropy Probe)
and the first WMAP results
were released on February 11, 2003.
George Smoot and John C. Mather (of NASA) were jointly awarded the
2006 Nobel Prize in physics
"for their discovery of the black-body form and anisotropy of the cosmic microwave background radiation".
(2021-01-18) Worm on a rubber band
Even a slow worm will eventually reach the end of a fast-stretching band.
The Anthropic Principle is the statement that the Universe we
observe must allow intelligent life to evolve, or else we wouldn't
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, Andrei 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
I would like to learn that Newton's laws must be modified
in order to correctly describe gravitational interactions at large distances.
That's more appealing than a universe filled with a new kind of
particle. Vera Rubin (1928-2016)
Dark Matter & the Pull of Galaxies:
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
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 most ordinary matter
actually resides outside of galaxies,
all the evidence that has been gathered since 1998
seems to indicate that there's about five times more dark matter
than ordinary matter in the Universe.
Dark Energy & the Accelerating Expansion of the Universe:
In 1998, the study of distant Type-1A supernovae
(Nobel 2011) indicated
that the expansion of the Universe is accelerating.
Such an acceleration can be accounted for by a small positive cosmological constant
(L) in Einstein's
field equations.
This would mean that
empty space itself has a nonzero energy density
of about 0.5 nPa or 3 keV/cc
(that's roughly equivalent to the rest energy
of 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 of our Universe [courtesy of NASA]:
Ryan Landfield (2009-04-10;
UA931)
The Pioneer Anomaly
A residual (decaying) sunward acceleration of
8.74(45) 10 -10 m/s2
This is commensurate with the Hubble acceleration
(i.e., the product of the Hubble constant
H by the speed of light c).
H c =
6.9 10 -10 m/s2
In July 2012, Slava G. Turyshev et al. apparently nailed
that coffin with a full quantitative explanation of
the "anomalous" acceleration in term of the recoil of infrared photons
emitted by the onboard nuclear power and bouncing
off the back of the parabolic communication antenna.
This would explain not only the magnitude of the observed effect
but also its exponential decrease with time at a
rate similar to the decay
of the plutonium which powers the Pioneer probe
(behind its own antenna).