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Subject: [sci.astro] Cosmology (Astronomy Frequently Asked Questions) (9/9)
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------------------------------

Subject: Introduction

 sci.astro is a newsgroup devoted to the discussion of the science of
astronomy.  As such its content ranges from the Earth to the farthest
reaches of the Universe.

 However, certain questions tend to appear fairly regularly.  This
document attempts to summarize answers to these questions.

 This document is posted on the first and third Wednesdays of each
month to the newsgroup sci.astro.  It is available via anonymous ftp
from <URL:ftp://rtfm.mit.edu/pub/usenet/news.answers/astronomy/faq/>,
and it is on the World Wide Web at
<URL:http://sciastro.astronomy.net/> and
<URL:http://www.faqs.org/faqs/astronomy/faq/>.  A partial list of
worldwide mirrors (both ftp and Web) is maintained at
<URL:http://sciastro.astronomy.net/mirrors.html>.  (As a general note,
many other FAQs are also available from
<URL:ftp://rtfm.mit.edu/pub/usenet/news.answers/>.)

Questions/comments/flames should be directed to the FAQ maintainer,
Joseph Lazio ([e-mail]).

------------------------------

Subject: Table of Contents

[All entries last edited on 1998-02-28, unless otherwise noted.]

 I.01 What do we know about the properties of the Universe?
 I.02 Why do astronomers favor the Big Bang model of the Universe?
 I.03 Where is the center of the Universe?
 I.04 What do people mean by an "open," "flat," or "closed" Universe?
 I.05 If the Universe is expanding, what about me? or the Earth? or
      the Solar System?
 I.06 What is inflation?
 I.07 How can the Big Bang (or inflation) be right?  Doesn't it
    violate the idea that nothing can move faster than light?
    (Also, can objects expand away from us faster than the speed
    of light?)
 I.08 If the Universe is only 10 billion years old, how can we see
    objects that are now 30 billion light years away?  Why isn't
    the most distant object we can see only 5 billion light years
    away?
 I.09 How can the oldest stars in the Universe be older than the
    Universe?
 I.10 What is the Universe expanding into?
 I.11 Are galaxies really moving away from us or is space-time just
    expanding?
 I.12 How can the Universe be infinite if it was all concentrated into
    a point at the Big Bang?
 I.13 Why haven't the cosmic microwave background photons outrun the
    galaxies in the Big Bang?
 I.14 Can the cosmic microwave background be redshifted starlight?
 I.15 Why is the sky dark at night? (Olbers' paradox) [2001-10-02]
 I.16 What about objects with discordant redshifts?
 I.17 Since energy is conserved, where does the energy of redshifted
    photons go? [1998-12-03]
 I.18 There are different ways to measure distances in cosmology?
    [1999-07-06]

This section of the FAQ is largely extracted from Ned Wright's
Cosmology Tutorial,
<URL:http://www.astro.ucla.edu/%7Ewright/cosmolog.htm>, and was
written jointly by Ned Wright and Joseph Lazio, unless otherwise noted.

------------------------------

Subject:  I.01. What do we know about the properties of the Universe?

There are three key facts we know about the properties of the
Universe: galaxies recede, there's a faint microwave glow coming from
all directions in the sky, and the Universe is mostly hydrogen and
helium.

In 1929 Edwin Hubble published a claim that the radial velocities of
galaxies are proportional to their distance.  His claim was based on
the measurement of the galaxies' redshifts and estimates of their
distances.  The redshift is a measure of how much the wavelength of a
spectral line has been shifted from the value measured in
laboratories; if assumed to occur because of the Doppler effect, the
redshift of a galaxy is then a measure of its radial velocity.  His
estimates of the galaxies' distances was based on the brightness of a
particular kind of star (a pulsating star known as a Cepheid).

The constant of proportionality in Hubble's relationship (v = H * d,
where v is a velocity and d is a distance) is known as Hubble's
parameter or Hubble's constant.  Hubble's initial estimate was that
the Hubble parameter is 464 km/s/Mpc (in other words, a galaxy 1 Mpc =
3 million light years away would have a velocity of 464 km/s).  We
know now that Hubble didn't realize that there are two kinds of
Cepheid stars.  Various estimates of the Hubble parameter today are
between 50--100 km/s/Mpc.

Hubble also measured the number of galaxies in different directions
and at different brightness in the sky.  He found approximately the
same number of faint galaxies in all directions (though there is a
large excess of bright galaxies in the northern sky).  When a
distribution is the same in all directions, it is isotropic.  When
Hubble looked for galaxies four times fainter than a particular
brightness, he found approximately 8 times more galaxies than he found
that were brighter than this cutoff.  A brightness 4 times smaller
implies a doubled distance.  In turn, doubling the distance means one
is looking into a volume that is 8 times larger.  This result
indicates that the Universe is close to homogeneous or it has a
uniform density on large scales.  (Of course, the Universe is not
really homogeneous and isotropic, because it contains dense regions
like the Earth.  However, if you take a large enough box, you will
find about the same number of galaxies in it, no matter where you
place the box.  So, it's a reasonable approximation to take the
Universe to be homogeneous and isotropic.)  Surveys of very large
regions confirm this tendency toward homogeneity and isotropy on the
scales larger than about 300 million light years.

The case for an isotropic and homogeneous Universe became much
stronger after Penzias & Wilson announced the discovery of the Cosmic
Microwave Background in 1965.  They observed an excess brightness at a
wavelength of 7.5 cm, equivalent to the radiation from a blackbody
with a temperature of 3.7+/-1 degrees Kelvin. (The Kelvin temperature
scale has degrees of the same size as the Celsius scale, but it is
referenced at absolute zero, so the freezing point of water is 273.15
K.)  A blackbody radiator is an object that absorbs any radiation that
hits it and has a constant temperature.  Since then, many astronomers
have measured the intensity of the CMB at different wavelengths.
Currently the best information on the spectrum of the CMB comes from
the FIRAS instrument on the COBE satellite.  The COBE data are
consistent with the radiation from a blackbody with T = 2.728 K.  (In
effect, we're sitting in an oven with a temperature of 2.728 K.)  The
temperature of the CMB is almost the same all over the sky.  Over the
distance from which the CMB travels to us, the Universe must be
exceedingly close to homogeneous and isotropic.  These observations
have been combined into the so-called Cosmological Principle: The
Universe is *homogeneous* and *isotropic*.

If the Universe is expanding---as the recession of galaxies
suggests---and it is at some temperature today, then in the past
galaxies would have been closer together and the Universe would have
been hotter.  If one continues to extrapolate backward in time, one
reaches a time when the temperature would be about that of a star's
interior (millions of degrees; galaxies at this time would have been
so close that they would not retain their form as we see them today).
If the temperature was about that of a star's interior, then fusion
should have been occurring.

The majority of the Universe is hydrogen and helium.  Using the
known rate of expansion of the Universe, one can figure out how long
fusion would have been occurred.  From that one predicts that,
starting with pure hydrogen, about 25% of it would have been fused to
form deuterium (heavy hydrogen), helium (both helium-4 and helium-3),
and lithium; the bulk of the fusion products would helium-4.
Observations of very old stars and very distant gas show that the
abundance of hydrogen and helium is about 75% to 25%.

------------------------------

Subject: I.02. Why do astronomers favor the Big Bang model of the
    Universe?

The fundamental properties of the Universe, summarized above, one can
develop a simple model for the evolution of the Universe.  This model
is called the Big Bang.

The essential description of the Big Bang model is that it predicts
the Universe was hotter and denser in the past.  For most of the 20th
century, astronomers argued about the best description of the
Universe.  Was the BB right? or was another model better?  Today, most
astronomers think that the BB is essentially correct, the Universe was
hotter and denser in the past.  Why?

When Einstein was working on his theory of gravity, around 1915, he
was horrified to discover that it predicted the Universe should either
be expanding or collapsing.  The prevailing scientific view at the
time was that the Universe was static, it always had been and always
would be.  He ended up modifying his theory, introducing a long-range
force that cancelled gravity so that his theory would describe a
static Universe.

When Hubble announced that galaxies were receding from us, astronomers
realized quickly that this was consistent with the notion that the
Universe is expanding.  If you could imagine "running the clock
backwards" and looking into the past, you would see galaxies getting
closer together.  In effect, the Universe would be getting denser.

If the Universe was denser in the past, then it was also hotter.  At
some point in the past, the conditions in the Universe would have
resembled the interior of a star.  If so, we should expect that
nuclear fusion would occur.  Detailed predictions of how much nuclear
fusion would have occurred in the early Universe were first undertaken
by George Gamow and his collaborators.  Since then, the calculations
have been refined, but the essential result is still the same.  After
nuclear fusion stopped, about 1000 seconds into the Universe's
history, there should be about one Helium-4 atom for every 10 Hydrogen
atoms, one Deuterium atom (heavy hydrogen) for every 10,000 H atoms,
one Helium-3 atom for every 50,000 H atoms, and one Lithium-7 atom for
every 10 billion H atoms.  These predicted abundances are in very good
agreement with the observed abundances.

As the Universe expanded and cooled, the radiation in it should have
also lost energy.  In 1965 Arlo Penzias and Robert Wilson were annoyed
to discover that no matter what direction they pointed a telescope,
they kept picking up faint glow.  Some physicists at Princeton
recognized that this faint glow was exactly what was expected from a
cooling Universe.  Since then, the COBE satellite has measured the
temperature of this radiation to be 2.728 +/- 0.002 K.

It is the combination of these excellent agreements between prediction
and observation that lead most astronomers to conclude that the Big
Bang is a good model for describing the Universe.

------------------------------

Subject:  I.03.  Where is the center of the Universe?

Often when people are told that galaxies are receding from us, they
assume that means we are at the center of the Universe.  However,
remember that the Universe is homogeneous and isotropic.  No matter
where one is, it looks the same in all directions.  Thus, all galaxies
see all other galaxies receding from them.  Hubble's relationship is
compatible with a Copernican view of the Universe: Our position is not
a special one.

So where is the center?  *There isn't one*.  Although apparently
nonsensical, consider the same question about the *surface* of a
sphere (note the *surface*).  Where's the center of a sphere's
surface?  Of course, there isn't one.  One cannot point to any point
on a sphere's surface and say that, here is the center.  Similarly,
because the Universe is homogeneous and isotropic, all we can say is
that, in the past, galaxies were closer together.  We cannot say that
galaxies started expanding from any particular point.

------------------------------

Subject:  I.04.  What do people mean by an "open," "flat," or "closed"
    Universe?

These different descriptions concern the future of the Universe,
particularly whether it will continue to expand forever.  The future
of the Universe hinges upon its density---the denser the Universe is,
the more powerful gravity is.  If the Universe is sufficiently dense,
at some point in the (distant) future, the Universe will cease to
expand and begin to contract.  This is termed a "closed" Universe.  In
this case the Universe is also finite in size, though unbounded.  (Its
geometry is, in fact, similar to the *surface* of a sphere.  One can
walk an infinite distance on a sphere's surface, yet the surface of a
sphere clearly has a finite area.)

If the Universe is not sufficiently dense, then the expansion will
continue forever.  This is termed an "open" Universe.  One often hears
that such a Universe is also infinite in spatial extent.  This is
possibly true; recent research suggests that it may be possible for
the Universe to have a finite volume, yet expand forever.

One can also imagine a Universe in which gravity and the expansion are
exactly equal.  The Universe stops expanding only after an infinite
amount of time.  This Universe is also (possibly) infinite in spatial
extent and is termed a "flat" Universe, because the sum of the
interior angles of a triangle sum to 180 degrees---just like in the
plane or "flat" geometry one learns in (US) high school.  For an open
Universe, the geometry is negatively curved so that the sum of the
interior angles of a triangle is less than 180 degrees; in a closed
Universe, the geometry is positively curved and the sum of the
interior angles of a triangle is more than 180 degrees.

The critical density that separates an open Universe from a closed
Universe is 1.0E-29 g/cm^3.  (This is an average density; there are
clearly places in the Universe more dense than this, e.g., you, the
reader with a density of about 1 g/cm^3, but this density is to be
interpreted as the density if all matter were spread uniformly
throughout the Universe.)  Current observational data are able to
account for about 10--30% of this value, suggesting that the Universe
is open.  However, motivated by inflationary theory, many theorists
predict that the actual density in the Universe is essentially equal
to the critical density and that observers have not yet found all of
the matter in the Universe.

------------------------------

Subject: I.05.  If the Universe is expanding, what about me? or the
    Earth? or the Solar System?

You, the reader, are not expanding, even though the Universe in which
you live is.  There are two ways to understand this.

The simple way to understand the reason you're not expanding is that
you are held together by electromagnetic forces.  These
electromagnetic forces are strong enough to overpower the expansion of
the Universe.  So you do not expand.  Similarly, the Earth is held
together by a combination of electromagnetic and gravitational forces,
which again are strong enough to overpower the Universe's expansion.
On even larger scales---those of the Solar System, the Milky Way, even
the Local Supercluster of galaxies (also known as the Virgo
Supercluster)---gravity alone is still strong enough hold these
objects together and prevent the expansion.  Only on the very largest
scales does gravity become weak enough that the expansion can win
(though, if there's enough gravity in the Universe, the expansion will
eventually be halted).

A second way to understand this is to appreciate the assumption of
homogeneity.  A key assumption of the Big Bang is that the Universe is
homogeneous or relatively uniform.  Only on large enough scales will
the Universe be sufficiently uniform that the expansion occurs.  You,
the reader, are clearly not uniform---inside your body the density is
about that of water, outside is air.  Similarly, the Earth and its
surroundings are not of uniform density, nor for the Solar System or
the Milky Way.

This latter way of looking at the expansion of the Universe is similar
to common assumptions in modelling air or water (or other fluids).  In
order to describe air flowing over an airplane wing or water flowing
through a pipe, it is generally not necessary to consider air or water
to consist of molecules.  Of course, on very small scales, this
assumption breaks down, and one must consider air or water to consist
of molecules.  In a similar manner, galaxies are often described as
the "atoms" of the Universe---on small scales, they are important, but
to describe the Universe as a whole, it is not necessary to consider
it as being composed of galaxies.

Also note that the definitions of length and time are not changing in
the standard model.  The second is still 9192631770 cycles of a Cesium
atomic clock and the meter is still the distance light travels in
9192631770/299792458 cycles of a Cesium atomic clock.

------------------------------

Subject:  I.06.  What is inflation?

The "inflationary scenario," developed by Starobinsky and by Guth,
offers a solution to two apparent problems with the Big Bang.  These
problems are known as the flatness-oldness problem and the horizon
problem.

The flatness problem has to do with the fact that density of the
Universe appears to be roughly 10% of the critical density (see
previous question).  This seems rather fortuitous; why is it so close
to the critical density?  We can imagine that the density might be
0.0000001% of the critical value or 100000000% of it.  Why is it so
close to 100%?

The horizon problem relates to the smoothness of the CMB.  The CMB is
exceedingly smooth (if one corrects for the effects caused by the
Earth and Sun's motions).  Two points separated by more than 1 degree
or so have the same temperature to within 0.001%.  However, two points
this far apart today would not have been in causal contact at very
early times in the Universe.  In other words, the distance separating
them was greater than the distance light could travel in the age of
the Universe.  There was no way for two such widely separated points
to communicate and equalize their temperatures.

The inflationary scenario proposes that during a brief period early in
the history of the Universe, the scale size of the Universe expanded
rapidly.  The scale factor of the Universe would have grown
exponentially, a(t) = exp(H(t-t0)), where H is the Hubble parameter,
t0 is the time at the start of inflation, and t is the time at the end
of inflation.  If the inflationary epoch lasts long enough, the
exponential function gets very large.  This makes a(t) very large, and
thus makes the radius of curvature of the Universe very large.

Inflation, thus, solves the flatness problem rather neatly.  Our
horizon would be only a very small portion of the whole Universe.
Just like a football field on the Earth's surface can appear flat,
even though the Earth itself is certainly curved, the portion of the
Universe we can see might appear flat, even though the Universe as a
whole would not be.

Inflation also proposes a solution for the horizon problem.  If the
rapid expansion occurs for a long enough period of time, two points in
the Universe that were initially quite close together could wind up
very far apart.  Thus, one small region that was at a uniform
temperature could have expanded to become the visible Universe we see
today, with its nearly constant temperature CMB.

The onset of inflation might have been caused by a "phase change."  A
common example of a phase change (that also produces a large increase
in volume) is the change from liquid water to steam.  If one was to
take a heat-resistant, extremely flexible balloon filled with water
and boil the water, the balloon would expand tremendously as the water
changed to steam.  In a similar fashion, astronomers and physicists
have proposed various ways in which the cooling of the Universe could
have led to a sudden, rapid expansion.

It is worth noting that the inflationary scenario is not the same as
the Big Bang.  The Big Bang predicts that the Universe was hotter and
denser in the past; inflation predicts that as a result of the physics
in the expanding Universe, it suddenly underwent a rapid expansion.
Thus, inflation assumes that the Big Bang theory is correct, but the
Big Bang theory does not require inflation.

------------------------------

Subject: I.07.  How can the Big Bang (or inflation) be right?  Doesn't
    it violate the idea that nothing can move faster than light?
     (Also, can objects expand away from us faster than the speed
    of light?)

In the Big Bang model the *distance* between galaxies increases, but
the galaxies don't move.  Since nothing's moving, there is no
violation of the restriction that nothing can move faster than light.
Hence, it is quite possible that the distance between two objects is
so great that the distance between them expands faster than the speed
of light.

What does it mean for the distance between galaxies to increase
without them moving?  Consider two galaxies in a one-dimensional Big
Bang model:
                          *-|-|-|-*
                          0 1 2 3 4

There are four distance units between the two galaxies.  Over time the
distance between the two galaxies increases:

                       * - | - | - | - *
                       0   1   2   3   4

However, they remain in the same position, namely one galaxy remains
at "0" and the other remains at "4."  They haven't moved.

(Astronomers typically divide the distance between two galaxies into
two parts, D = a(t)*R.  The function a(t) describes how the size of
the Universe increases, while the distance R is independent of any
changes in the size of the Universe.  The coordinates based on R are
called "co-moving coordinates.")

------------------------------

Subject:  I.08. If the Universe is only 10 billion years old, how can
    we see objects that are now 30 billion light years away?  Why
    isn't the most distant object we can see only 5 billion light
    years away?

When talking about the distance of a moving object, we mean the
spatial separation NOW, with the positions of both objects specified
at the current time. In an expanding Universe this distance NOW is
larger than the speed of light times the light travel time due to the
increase of separations between objects as the Universe expands. This
is not due to any change in the units of space and time, but just
caused by things being farther apart now than they used to be.

What is the distance NOW to the most distant thing we can see?  Let's
take the age of the Universe to be 10 billion years. In that time
light travels 10 billion light years, and some people stop here.  But
the distance has grown since the light traveled.  Half way along the
light's journey was 5 billion years ago.  For the critical density
case (i.e., flat Universe), the scale factor for the Universe is
proportional to the 2/3 power of the time since the Big Bang, so the
Universe has grown by a factor of 22/3 = 1.59 since the midpoint of
the light's trip.  But the size of the Universe changes continuously,
so we should divide the light's trip into short intervals.  First take
two intervals: 5 billion years at an average time 7.5 billion years
after the Big Bang, which gives 5 billion light years that have grown
by a factor of 1/(0.75)2/3 = 1.21, plus another 5 billion light years
at an average time 2.5 billion years after the Big Bang, which has
grown by a factor of 42/3 = 2.52.  Thus with 1 interval we get 1.59*10
= 15.9 billion light years, while with two intervals we get
5*(1.21+2.52) = 18.7 billion light years.  With 8192 intervals we get
29.3 billion light years.  In the limit of very many time intervals we
get 30 billion light years.

If the Universe does not have the critical density then the distance
is different, and for the low densities that are more likely the
distance NOW to the most distant object we can see is bigger than 3
times the speed of light times the age of the Universe.

------------------------------

Subject: I.09. How can the oldest stars in the Universe be older than
    the Universe?

Obviously, the Universe has to be older than the oldest stars in
it. So this question basically asks, which estimate is wrong:

   * The age of the Universe?
   * The age of the oldest stars? or
   * Both?

The age of the Universe is determined from its expansion rate: the
Hubble constant, which is the ratio of the radial velocity of a
distant galaxy to its distance. The radial velocity is easy to
measure, but the distances are not. Thus there is currently a 15%
uncertainty in the Hubble constant.

Determining the age of the oldest stars requires a knowledge of their
luminosity, which depends on their distance. This leads to a 25%
uncertainty in the ages of the oldest stars due to the difficulty in
determining distances.

Thus the discrepancy between the age of the oldest things in the
Universe and the age inferred from the expansion rate is within the
current margin of error.

------------------------------

Subject: I.10. What is the Universe expanding into?

This question is based on the ever popular misconception that the
Universe is some curved object embedded in a higher dimensional space,
and that the Universe is expanding into this space.  This
misconception is probably fostered by the balloon analogy that shows a
2-D spherical model of the Universe expanding in a 3-D space.

While it is possible to think of the Universe this way, it is not
necessary, and---more importantly---there is nothing whatsoever that
we have measured or can measure that will show us anything about the
larger space.  Everything that we measure is within the Universe, and
we see no edge or boundary or center of expansion.  Thus the Universe
is not expanding into anything that we can see, and this is not a
profitable thing to think about.  Just as Dali's Crucifixion is just a
2-D picture of a 3-D object that represents the surface of a 4-D cube,
remember that the balloon analogy is just a 2-D picture of a 3-D
situation that is supposed to help you think about a curved 3-D space,
but it does not mean that there is really a 4-D space that the
Universe is expanding into.

------------------------------

Subject: I.11. Are galaxies really moving away from us or is
    space-time just expanding?

This depends on how you measure things, or your choice of coordinates.
In one view, the spatial positions of galaxies are changing, and this
causes the redshift.  In another view, the galaxies are at fixed
coordinates, but the distance between fixed points increases with
time, and this causes the redshift.  General relativity explains how
to transform from one view to the other, and the observable effects
like the redshift are the same in both views.

------------------------------

Subject:  I.12. How can the Universe be infinite if it was all
    concentrated into a point at the Big Bang?

Only the *observable* Universe was concentrated into a point at the
time of the Big Bang, not the entire Universe.  The distinction
between the whole Universe and the part of it that we can see is
important.

We can see out into the Universe roughly a distance c*t, where c is
the speed of light and t is the age of the Universe.  Clearly, as t
becomes smaller and smaller (going backward in time toward the Big
Bang), the distance to which we can see becomes smaller and smaller.
This places no constraint on the size of the entire Universe, though.

------------------------------

Subject:  I.13. Why haven't the CMB photons outrun the galaxies in
    the Big Bang?

Once again, this question assumes that the Big Bang was an explosion
from a central point.  The Big Bang was not an explosion from a single
point, with a center and an edge.  The Big Bang occurred everywhere.
Hence, no matter in what direction we look, we will eventually see to
the point where the CMB photons were being formed.  (The CMB photons
didn't actually form in the Big Bang, they formed later when the
Universe had cooled enough for atoms to form.)

------------------------------

Subject:  I.14.  Can the CMB be redshifted starlight?

No!  The CMB radiation is such a perfect fit to a blackbody that it
cannot be made by stars.  There are two reasons for this.

First, stars themselves are at best only pretty good blackbodies, and
the usual absorption lines and band edges make them pretty bad
blackbodies.  In order for a star to radiate at all, the outer layers
of the star must have a temperature gradient, with the outermost
layers of the star being the coolest and the temperature increasing
with depth inside the star.  Because of this temperature gradient, the
light we see is a mixture of radiation from the hotter lower levels
(blue) and the cooler outer levels (red).  When blackbodies with these
temperatures are mixed, the result is close to, but not exactly equal
to a blackbody.  The absorption lines in a star's spectrum further
distort its spectrum from a blackbody.

One might imagine that by having stars visible from different
redshifts that the absorption lines could become smoothed out.
However, these stars will be, in general, different temperature
blackbodies, and we've already seen from above that it is the mixing
of different apparent temperatures that causes the deviation from a
blackbody.  Hence more mixing will make things worse.

How does the Big Bang produce a nearly perfect blackbody CMB?  In the
Big Bang model there are no temperature gradients because the Universe
is homogeneous.  While the temperature varies with time, this
variation is exactly canceled by the redshift.  The apparent
temperature of radiation from redshift z is given by T(z)/(1+z), which
is equal to the CMB temperature T(CMB) for all redshifts that
contribute to the CMB.

------------------------------

Subject:  I.15. Why is the sky dark at night? (Olbers' paradox)

If the Universe were infinitely old, infinite in extent, and filled
with stars, then every direction you looked would eventually end on
the surface of a star, and the whole sky would be as bright as the
surface of the Sun.  This is known as Olbers' Paradox after Heinrich
Wilhelm Olbers (1757--1840) who wrote about it in 1823--1826 (though
it had been discussed earlier).  A common suggestion for resolving the
paradox is to consider interstellar dust, which blocks light by
absorping it.  However, absorption by interstellar dust does not
circumvent this paradox, as dust reradiates whatever radiation it
absorbs within a few minutes, which is much less than the age of the
Universe.

The resolution of Olbers' paradox comes by recognizing that the
Universe is not infinitely old and it is expanding.  The latter effect
reduces the accumulated energy radiated by distant stars. Either one
of these effects acting alone would solve Olbers' Paradox, but they
both act at once.

------------------------------

Subject:  I.16. What about objects with discordant redshifts?

A common objection to the Big Bang model is that redshifts do not
measure distance.  The logic is that if redshifts do not measure
distance, then maybe the Hubble relation between velocity and distance
is all wrong.  If it is wrong, then one of the three pillars of
observational evidence for the Big Bang model collapses.

One way to show that redshifts do not measure distance is to find two
(or more) objects that are close together on the sky, but with vastly
different redshifts.  One immediately obvious problem with this
approach is that in a large Universe, it is inevitable that some very
distant objects will just happen to lie behind some closer objects.

A way around this problem is to look for "connections"---for instance,
a bridge of gas---between two objects with different redshifts.
Another approach is to look for a statistical "connection"---if high
redshift objects tend to cluster about low redshift objects that might
suggest a connection.  Various astronomers have claimed to find one or
the other kind of connection.  However, their statistical analyses
have been shown to be flawed, or the nature of the apparent "bridge"
or "connection" has been widely disputed.

At this time, there's no unambiguous illustration of a "connection" of
any kind between objects of much different redshifts.

------------------------------

Subject: I.17 Since energy is conserved, where does the energy of
    redshifted photons go?
Author: Peter Newman <[e-mail]>

The energy of a photon is given by E = hc/lambda, where h is Planck's
constant, c is the speed of light, and lambda is its wavelength.  The
cosmological redshift indicates that the wavelength of a photon
increases as it travels over cosmological distances in the Universe.
Thus, its energy decreases.

One of the basic conservation laws is that energy is conserved.  The
decrease in the energy of redshifted photons seems to violate that
law.  However, this argument is flawed.  Specifically, there is a flaw
in assuming Newtonian conservation laws in general relativistic
situations.  To quote Peebles (_Principles of Physical Cosmology_,
1995, p. 139):

      Where does the lost energy go? ... The resolution of this
      apparent paradox is that while energy conservation is a good
      local concept ... and can be defined more generally in the
      special case of an isolated system in asymptotically flat space,
      there is not a general global energy conservation law in general
      relativity theory.

In other words, on small scales, say the size of a cluster of
galaxies, the notion of energy conservation is a good one.  However,
on the size scales of the Universe, one can no longer define a
quantity E_total, much less a quantity that is conserved.

------------------------------

Subject: There are different ways to measure distances in cosmology?
Author: Joseph Lazio <[e-mail]>

Yes!

There are at least three ways one can measure the distance to objects:

 * parallax;
 * angular size; or
 * brightness.

The parallaxes of cosmologically-distant objects are so small that
they will remain impossible to measure in the foreseeable future (with
the possible exception of some gravitationally-lensed quasars).

Suppose there exists an object (or even better a class of objects)
whose intrinsic length is known.  That is, the object can be treated
as a ruler because its length known to be exactly L (e.g., 1 m, 100
light years, 10 kiloparsecs, etc.).  When we look at it, it has an
*angular diameter* of H.  Using basic geometry, we can then derive its
distance to be
                       L
                D_L = ---
                       H

Suppose there exists an object (or even better a class of objects)
whose intrinsic brightness is known.  That is, the object can be
treated as a lightbulb because the amount of energy it is radiating is
known to be F (e.g., 100 Watts, 1 solar luminosity, etc.).  When we
look at it, we measure an *apparent* flux of f.  The distance to the
object is then
                              F
                D_F =sqrt( ------ )
                           4*pi*f

In general, D_L *is not equal to* D_F!

For more details, see "Distance Measures in Cosmology" by David Hogg,
<URL:http://xxx.lanl.gov/abs/astro-ph/9905116>, and references within.
Plots showing how to convert redshifts to various distance measures
are included in this paper, and the author will provide C code to do
the conversion as well.  Even more details are provided in "A General
and Practical Method for Calculating Cosmological Distances" by Kayser
et al., <URL:http://xxx.lanl.gov/abs/astro-ph/9603028> or <URL:
http://multivac.jb.man.ac.uk:8000/helbig/Research/Publications/info/angsiz.html>.
Fortran code for calculating these distances is provided by the second
set of authors.

------------------------------

Subject: Copyright

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