intro | why | general relativity | evolution | observations | plausibility | blunder | links & references
What are the observational constraints on the cosmological constant?
Observations provide various methods for constraining the value of the
cosmological constant in our universe because both the spatial
geometry and past evolution of the universe are affected by the
presence of a cosmological constant. Here various techniques and their
results (as of Feb. 1999) are presented.
High Redshift Objects
Bouncing universes can be ruled out due to the existence of high redshift (z > 5) objects and
the cosmic microwave background radiation
(CMB) at z ~ 1000. In order to have a bounce occur at a small enough scale factor to allow the redshift of the CMB our universe would have to have an impossibly small amount of matter (
< 10-9). Loitering universes produce a variety of strange observational effects and have been ruled out due to the existence of high redshift gravitationally lensed quasars (Gott 1985, 1987 & 1989).
"The Age Problem"
One of the more compelling evidence for the existence of a cosmological constant has been the fact that the age derived for a universe without a cosmological constant is younger that the age derived for the oldest stars. Some of the universe's oldest stars appear in globular clusters of stars. It is possible, through theoretical calculations of stellar evolution, to date the stars in globular clusters. Such calculations historically have yielded ages in the range of 15-18 billion years. If we assume the universe is purely made up of the matter we can detect (
0.3) then the universe is about 10-13 billion years old. The situation is worse if we require a flat, matter dominated universe ( = 1), then the universe is only 8-11 billion years old. However, a flat universe with matter at the level we observe ( 0.3) and a cosmological constant is about 12-16 billion years old, which is compatible with the age of the oldest stars.
Recently, though, the age problem has found another solution from results of the Hipparcos sattelite. Both the age of globular cluster and the age of the universe (through 
) depend on the distance scale used. Hipparcus has revised the distance scale by measuring parallaxes to some of the nearest Cepheid stars (variable stars which are extremely important in setting the distance scale.) This revision has adjusted the value of the Hubble constant to be around 60 km/s/Mpc, letting the open and flat matter dominated universes take the old extreme of the ages stated above (13 and 11 billion years respectively). The revision has also updated the ages of oldest globular clusters, bringing them down to 10-14 billion years old. Thus, the age problem may no longer exist, so a cosmological constant may not be needed to solve it. (See The Cosmological Implications of Hipparcus for a review of these results.)
Statistics of Gravitational Lensing
Light from high redshift objects may be lensed by
high mass concentrations such as galaxies and clusters of galaxies.
Lensing results in multiple images of the same object appearing on the
sky (see Gravitational
Lensing by Galaxy Clusters for a review of lensing). A
cosmological constant affects the geometry and evolution of the
universe and thus makes the statistics of lensing a powerful technique
in putting limits on the value of the cosmological constant in our
universe. Specifically one may include the observed number of lensed
sources, the redshift of the lens and source, the magnitude of the
source, and the separation of the two lensed images in a statistical
analysis. High redshift quasars provide a good sources for lensing
studies because they can be seen over cosmological distances, and high
mass elliptical galaxies provide good lenses. Although lensing is
sensitive to the value of the cosmological constant, there are several
uncertainties in determining the absolute number and specifics of
lensing expected for a given cosmological model. One must know the
distribution of lensing galaxies, their space density, lensing
potentials, and evolution with redshift in order to accurately
calculate the "optical depth" to lensing of a background source. There
are also observational selection effects that need to be considered
whenever a sample of sources is obtained (e.g. closer lenses are
easier to detect). These uncertainties may account for a factor of 2
error in the theoretical predictions of lensing. This is not so bad
however, since the relative predictions for different cosmological
models may differ by an order of magnitude (for flat universes, 
= 1 predicts 10 times as many lenses
as does = 0). All in all, since
higher values for predict higher
numbers of gravitational lenses, this technique offers a viable way of
putting upper bounds on the value of the cosmological constant.
Kochanek (1996) has done a careful
analysis of the statistics of gravitational lensing, including the
number, redshifts, magnitudes, and separations of the lenses. He
investigated different lens models and included the statistical
uncertainties in the number of lenses, galaxies, quasars, and the
parameters relating galaxies luminosities to dynamical variables. He
finds an upper limit of < 0.66
at 95% confidence. For = 0
universes he finds > 0.2 at 90%
confidence. Click here to see plots of his
results.
Myungshin et al. (1997) use a
sample of seven lensed quasars to test different cosmological models.
They use the combined probabilities that the lens systems have the
observed image separation, source and lens redshifts, and lens
magnitudes to determine and . They find for a flat universe that
= 0.64 (+0.15, -0.26). They also
state that the = 1 universe is
excluded at the 97% confidence level, and that open, matter dominated
universes are less likely than flat universes with a non-zero
cosmological constant. Click here for a plot
of their results.
Chiba & Yoshii (1997) have done
calculations of lensing statistics using newly revised data on the
luminosity function and internal velocity dispersions of galaxies.
They compare their new theoretical predictions of the total number and
image separations of lenses to ones found from the Hubble Space Telescope Snapshot Lens Survey.
They find that the observations are in best agreement with a flat
universe with 0.8. Click here for plots of their data and results.
Chiba & Yoshii (1999) present new
limits on the cosmological constant based on a revised knowledge of the
luminosity function and internal dynamics of E/S0 galaxies. They
compare their models to existing lens surveys and find that a flat
universe with = 0.7 (+0.1, -0.2)
is most preferable.
High Redshift Supernovae
One of the effects of a cosmological constant is to change the
relations of physical distance to redshift. In principle, given a set
of objects with either a standard proper size or luminosity one could
determine the physical distance to the object. Along with a knowledge
of the redshifts of the objects the cosmological parameters of ,
and can be determined
unambiguously. In practice, however, it is very hard to find a set of
objects that is not subject to evolutionary effects, i.e. that are all
truly the same size or brightness at all redshifts. However, one set
of objects exists that seems to be free of evolutionary effects, they
are the type Ia supernovae. Type Ia supernovae exhibit a behavior
that allows the absolute magnitude of the supernovae (and thus their
actual distance from us) to be determined from the shape of their
light curve and their time varying spectra.
Perlmutter et al. (1997) have released
their findings from the first seven (of 28) supernovae from the high
redshift supernovae search of the Supernova Cosmology
Project. They find for a flat universe constaining both matter and
a cosmological constant that =
0.06 (+0.28, -0.34), or for a stringent upper limit < 0.51 (at the 95% confidence
level). For a purely matter dominated universe they find = 0.88 (+0.69, -0.60). Click here to see figures of their data and results.
Their results point to the absence of a cosmological constant in our
universe. The use of high redshift supernovae as a test of the
cosmological constant is a powerful technique, and the error bars on
their results should narrow with future analysis of more
supernovae.
Riess et al. (1998) use a set of 16
high redshift supernovae from the High-z
Supernova Search Team plus a set of 34 nearby supernovae to place
constraints on the Hubble constant, mass density, cosmological
constant, deceleration parameter, and the dynamical age of the
universe. For two different methods of fitting the supernova light
curves, they find for a flat universe = 0.68
0.10 and
= 0.840.09. Without requiring a flat universe they find
that > 0 at 98%
confidence.
Perlmutter et al. (1999) report
results from the analysis of 42 supernovae discovered by the Supernova Cosmology Project.
They find for a flat universe that = 0.71 (+0.08, -0.09). Without requiring a flat
universe they find > 0 at 99%
confidence.
Summary of Observational Constraints on the Cosmological
Constant
intro | why | general relativity | evolution | observations | plausibility | blunder | links & references
These pages were created by Eli Michael
(michaele@colorado.edu), with the inspiration of
Andrew Hamilton.
We are in the Department of
Astrophysical and Planetary Sciences at the University
of Colorado, Boulder.
2.15.99