May
25, 1999: In Julius Caesar William Shakespeare wrote
"But I am constant as the Northern Star, of whose true and
resting quality there is no fellow in the firmament." However,
had he known about cosmology, he'd have probably chosen a better
comparison than Polaris, which is actually a variable
star.
Stars like Polaris, called Cepheid variables, have a wonderful
property (wonderful to astronomers, that is). The rate at which
they pulsate is simply related to their luminosity. So, by observing
the apparent brightness of a Cepheid and its period of variability
it is straightforward to estimate its distance. Because astronomers
can measure the distance to Cepheids -- and, thus, to the galaxies
they inhabit -- these peculiar stars hold the key to many important
questions in Cosmology.
At today's
NASA Space Science Update, scientists announced that they
had used the Hubble Space Telescope to study Cepheid variables
and measure the expansion of the Universe with unprecedented
precision. To fully grasp the meaning of these new measurements
it may help to reflect on events 70 years ago, when a young American
astronomer named Edwin Hubble was using the great telescopes
of his day to study the large-scale structure of the Universe.
In 1929 as Hubble pursued his studies of distant galaxies, he
realized something extraordinary. Ten years earlier Shapley had
noticed that other galaxies appeared to be flying away from our
own Milky Way. Hubble had the insight to realize that not only
were these objects apparently speeding away, but the farther
away they were, the faster they appeared to be moving. Thus the
Hubble Constant, Ho, was born. Ho is a
number which relates a galaxy's apparent speed of recession to
its distance from the Milky Way, as shown in the graph below.
Scientists now know that the recessional velocites that we observe
are not actually distant galaxies flying through space, all away
from the Milky Way, but instead we are actually observing the
expansion of the Universe itself (and everything in it). This
expansion would look the same no matter what galaxy we actually
inhabited, and is one of the visible pieces of evidence that
points to a "Big Bang" origin for our Universe.
Left:
The Hubble Constant describes how fast objects appear to be moving
away from our galaxy as a function of distance. If you plot apparent
recessional velocity against distance, as in the figure above,
the Hubble Constant is simply the slope of a straight line through
the data.
The Hubble Constant is usually expressed in units of "kilometers
per second, per Megaparsec." One parsec is a unit of
distance equal to about 3.2 light years, and a Megaparsec is
a million times this, or about 3.2 million light years. So
what the Hubble Constant says is that for every 3.2 million light
years you look out into space, the objects there appear to be
receding from you at a rate of Ho kilometers per second. If
Ho is 100, then the objects appear to recede at 100
km/second for every 3.2 million light years you look out into
space. If Ho is 50, then you have to look about
6.4 million light years out into space for the same 100 km/second
recessional velocity.
But how do astronomers measure Ho
in the absence of a cosmic "radar gun" and mile-markers? There
are actually several ways, each with their own advantages, disadvantages,
and sources of uncertainty. Some involve the study of supernovae
with optical telescopes, others capitalize on a physical process
in distant clusters of galaxies called the Sunyaev-Zeldovich
effect that can be detected by combining X-ray images and
microwave astronomy measurements. Another method, used by scientists
working with the Hubble Space Telescope, for example, involves
looking at "Cepheid Variables" in distant galaxies.
Try This:
Your own model of Cosmological Expansion |
- Take a deflated balloon and
draw three dots on it, and label them 1, 2, and 3.
- Now begin to blow up the balloon.
- As you do this, look at the
movement of the three dots. They are all moving away from
each other as you inflate the balloon.
- Now pretend you are standing
on dot number 1. What is happening to dot 2 and 3? They
are moving away from dot 1. Is dot 1 in some preferred position
on the balloon?
- No, in fact the movement
of the other two dots is the same, regardless of whether you
put yourself on dot 1, dot 2, or dot 3.
- This is an excellent model of
the expansion of the universe. The dots are not moving on
the surface of the balloon, rather the fabric of the balloon
itself is expanding.
|
Cepheids (named as a group after the star delta-Cephei)
have been known for quite awhile, and they exhibit a very regular
fluctuation in their brightness on timescales of about 2 to 100
days. Polaris, the "North Star" is indeed
a member of this class of variables. It turns out that the frequency
with which they change their brightness is directly related to
the star's intrinsic luminosity. This so-called "Period-luminosity
relationship" was discovered by Henrietta Levitt, an astronomer
who worked at Harvard in the early decades of this century.
For nearby Cepheids that can easily be resolved in ground-based
telescopes, it is fairly easy to measure both the brightness
and the period of the star. With these measurements
in hand, by using the period-luminosity relationship, one can
find the luminosity of the star. And since the luminosity
and brightness are related by an inverse-square law, the distance
to the star can be determined with a bit of simple 7th or 8th-grade
algebra. This works great for stars within our
galaxy, and in some nearby galaxies like M31 (Andromeda) and
M33. Indeed Hubble himself put the question of "the
realm of the nebulae" to bed in 1923 by measuring Cepheids
in these galaxies, and deducing a distance of about 900,000 light
years. This distance was at least a factor of 20 or more
farther than any object within our galaxy, showing that these
nebulae were indeed galaxies in their own right, rather than
collections of stars, gas, and dust within our own Milky Way.
But on cosmological scales, 900,000 light years is the equivalent
of a local phone call. At these nearby distances, the cosmological
expansion proceeds very slowly (two dots near each other on the
surface of the balloon experiment above hardly move with respect
to one another) with velocities equal to or less than typical
random motions of stars within the galaxy, more like the velocity
of the Earth around the Sun, about 30 km/second or less.
Going deeper into space, where the expansion proceeds more rapidly,
to look at individual Cepheids was a difficult task for most
ground based telescopes. So until the Hubble Telescope came
along, with its ability to resolve Cepheids in extremely distant
galaxies, astronomers had to be a bit more resourceful in getting
at the Hubble Constant.
And resourceful they were. Using a variety of methods,
many fraught with uncertainty and large systemtic errors, the
answers have traditionally come out somewhere between 50 km/s/Mpc
and 100 km/s/Mpc. And therein lies the rub. In this
difference between 50 and 100 km/s/Mpc lies the fate of the entire
Universe.
Ten+1 measurements of the Hubble Constant
Method Used |
Citation |
Value (km/second/Megaparsec) |
Cepheid variables in distant galaxies |
W. Freedman et al (1999) |
70 +/- 7 |
M101 group velocity and distance |
Sandage and Tammann (1974) |
55.5 +/- 8.7 |
Virgo Cluster |
Peebles (1977) |
42 - 77 |
Globular Clusters |
Hanes (1979) |
80 +/- 11 |
Virgo Sc HII luminosities |
Kennicutt (1981) |
55 |
Type I supernovae |
Branch (1979) |
56 +/- 15 |
Type I supernovae |
Sandage and Tammann (1982) |
50 +/- 7 |
Infrared Tully-Fisher relation |
Aaronson and Mould (1983) |
82 +/- 10 |
SN-Ia and Cepheids |
Sandage, et al. (1994) |
55 +/- 8 |
Cepheids in Virgo (M100) |
Freedman, et al. (1994) |
80 +/- 17 |
Surface Brightness Fluctuation |
Tully (1993) |
90 +/- 10 |
The fate of the Universe actually depends
on four numbers, which have been described as the "Holy
Grail of Cosmology." Three of these numbers are independent,
meaning if you find three, the fourth one can be computed from
the three you know. Among these numbers is the Hubble
Constant.
The first number is called the
``deceleration parameter,'' and it is a measure of how fast the
cosmological expansion is speeding up or slowing down. Or,
colloquially, does the Universe have it's foot on the gas or
on the brake? Written by scientists as qo ("q-naught"),
a positive value of the deceleration parameter means the universe
is slowing down (foot on the brake), a value of 0 means the universe
is expanding at a constant speed (coasting), and a negative value
means that the universe is accelerating (foot on the gas).
The second number is the ``density
parameter.'' Called either `omega' or `sigma,' depending
on the units one uses, this number describes how much "stuff"
is in the universe. The larger the density parameter, the more
stuff there is. Issues related to "missing mass" are
often tied up in measurements of the density parameter that turn
out to be smaller than one needs to theoretically halt the expansion,
for example through the self-gravitation of all objects in the
Universe.
The third number is simply the
age of the universe. How old is this place? We can
actually get a decent handle on lower-limits to this number simply
by recognizing that the universe has to be older than the oldest
objects in it. Radioactive isotope studies yield ages of
at least nine to sixteen billion years for our Galaxy, and the
ages of the oldest star clusters put the limit at about 14 billion
years. Give or take a billion years, a reasonable lower
limit to the age of the Universe turns out to be about 15 billion
years.
The fourth number is our Hubble
Constant Ho. Two other numbers that are often
discussed in cosmology, the "cosmological constant,"
which Einstein referred to as one of his greatest blunders, and
the "curvature of the Universe," are determined by
the density parameter and the deceleration parameter.
The way these numbers play together mathematically allows
one to write the product of Ho and the age of the
universe as some function of the density parameter and the deceleration
parameter.
(Age of the Universe) x (Hubble
Constant) = A complicated function of the density and deceleration
parameters
In other words, given our fairly robust limit of the age of
the Universe, and a reliable measurement of the Hubble Constant,
their product restricts our Universe to only a very limited set
of combinations of density parameter and deceleration parameter,
many of which can be further ruled out by other independent observations.
Conversely, if one assumes a cosmological model (i.e. values
for sigmao and qo), a measure of the Hubble
Constant can give you a measure of the age of the universe.
Regardless of the approach, by getting a reliable measurement
of the Hubble Constant, we therefore have a much greater understanding
of the nature and eventual fate of the Universe. If the
Hubble Constant is large enough, for example, for a given age
of the Universe, the deceleration parameter must be negative,
meaning that the Universe is not slowing down, but instead is
accelerating its expansion.
One can actually draw a "map" of where our Universe
might be within the realm of all possibilities, to see these
implications of a measurement of Ho. Using the
`sigma-q' notation, it looks something like the accompanying
figure.
The density parameter `sigmao' is plotted on the horizontal
axis, and the deceleration parameter `qo' is plotted
on the vertical axis. High-density, rapidly decelerating
universes are represented at the upper right part of the diagram,
and rapidly accelerating, low-density universes are at the lower
left. The diagonal dot-dashed line that runs
from the origin (0,0) upwards to the right represents universes
where the cosmological constant (lambda) is zero (sigmao
= qo). Below this line, the cosmological constant
is positive, above it, it's negative. The second diagonal
line, dashed-only, delineates the universes that have flat topology
(k = 3 sigmao - qo - 1 = 0).
The Einstein-deSitter Universe |
The dot located at sigmao=qo=0.5
in the accompanying diagram is the so-called "Einstein de
Sitter" universe. It is unique in the sense that it
is the only universe that possesses flat topology, and has a
zero cosmological constant.
Mathematically, it is much easier
to treat as well, as many of the equations can be done by hand
in this unique case. The model has been a favorite of scientists
for aesthetic, theoretical, and other reasons.
If one assumes that this model
is indeed the one that describes our Universe, large measurements
of the Hubble Constant can result in an age of the Universe that
is younger than the objects we know are in it, an obvious impossibility.
In such a case, either the model
is not the correct one, or the measurement of Ho is
in error. |
The curves from upper left to lower right are the locus
of points where the Hubble Constant times the age of the Universe
equals some number, indicated on the line. For example, the Einstein-deSitter
model located at the bold dot (sigmao = qo
= 1/2) has a value of Ho times the age of the universe
equal to 2/3. By measuring the Hubble Constant and multiplying
by the age of the Universe, we are restricted to live on one
of these curves.
We are somewhere on this diagram...but where?
A value of Ho = 70 km/second/Mpc in an Einstein-de
Sitter Universe results in an age of the universe slightly less
than 10 billion years, and with the uncertainty in Ho,
a value of about 12 billion years is likely not out of the question
for this model.
With an age of 15 billion years, consistent with the ages
of the oldest star clusters [ref],
and a Hubble Constant measurement of 70 km/s/Mpc, one obtains
a product of 1.07 (when the units are accounted correctly). This,
combined with a strong theoretical impetus for a flat topology
(k = 0), would confine our position to a narrow area in the lower
left of the diagram, with a positive cosmological constant.
It is indeed possible that our latest measurements of the
Hubble Constant may have shown us that our universe is accelerating,
and will continue to do so forever. Nature appears to have its
foot on the cosmological gas pedal - and we're along for the
ride. |