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arXiv:astro-ph/0202006v1 1 Feb 2002
THE MEASURE OF COSMOLOGICAL PARAMETERS
W. L. FREEDMAN
Carnegie Observatories, 813 Santa Barbara St., Pasadena,
CA, 91101, USA
E-mail: wendy@ociw.edu
New, large, ground and space telescopes are contributing to an exciting and rapid
period of growth in observational cosmology. The subject is now far from its
earlier days of being data-starved and unconstrained, and new data are fueling a
healthy interplay between observations and experiment and theory. I report here
on the status of measurements of a number of quantities of interest in cosmology:
the expansion rate or Hubble constant, the total mass-energy density, the matter
density, the cosmological constant or dark energy component, and the total optical
background light.
1 General Background Cosmology and Assumptions
Based on the assumption that the universe is homogeneous and isotropic on
large scales, the framework for modern cosmology rests on Einstein’s general
theory of relativity, leading to the very successful hot big bang or Friedmann–
Robertson–Walker cosmological model. The dynamics of the expanding uni-
verse are described by the Friedmann equation, relating the expansion rate to
the density and curvature of the universe:
H2=8πGρm
3−k
R2+Λ
3
where R(t) is the scale factor, H= ˙
R
Ris the Hubble parameter (and H0is the
Hubble ‘constant’, the expansion rate at the present epoch), ρmis the aver-
age mass density, k is the curvature term, and Λ is the cosmological constant,
a term which represents the energy density of the vacuum. However, as de-
scribed in Section 3, the last term of the Friedmann equation may arise from a
different form of dark energy parameterized by ΩX, rather than a cosmological
constant. Generally, the matter density is expressed as Ωm= 8ΠGρm/3H2
0and
the vacuum energy density as ΩΛ= Λ/3H2
0. The values of these parameters
are not specified by the model, but must be determined experimentally.
Measurements of the cosmic microwave background (CMB) spectrum have
provided evidence of isotropy on large scales at the thousandths of a percent
level 1. But, until very recently, questions about the homogeneity of the uni-
verse on large scales have remained, primarily because of the discovery, in the
1980s, of unexpectedly large filaments, giant walls and voids – features having
1
dimensions up to the sizes of the survey volumes themselves (∼200 Mpc) 2.
However, in the most recent generation of galaxy surveys (LCRS
3, SDSS 4,
and 2DFGRS 5), which now extend six to more than an order of magnitude
times greater distances, out to redshifts of z∼0.3, the largest observed struc-
tures no longer rival the observed survey volumes. What were for Einstein two
convenient mathematical approximations, homogeneity and isotropy, are also
apparently very good approximations to the real universe.
Preliminary results from 2DFGRS 6,7and SDSS 8are beginning to provide
improved statistics on the shape of the galaxy power spectrum. The power
spectrum, P(k), is the Fourier transform of the correlation function (which
represents the probability of finding a pair of galaxies over a random distribu-
tion); k is the wave number. The power spectrum can be parameterized by
the shape parameter, Γ ∼Ωmh, which characterizes the horizon scale at the
time of matter and radiation equality. On small to intermediate scales, the ob-
served power spectrum is well represented by a power law, but then turns over
at larger scales. These recent, as well as earlier, surveys are consistent with a
value of Γ ∼0.15–0.20. An important advance resulting from these new large
surveys is that the scale range of the observed galaxy power spectrum is now
beginning to overlap that in the CMB anisotropy measurements, which will
offer an increasingly precise means of constraining a number of cosmological
parameters.
2 The Total Matter/Energy Density
Currently, the determination of the total matter/energy density of the universe
is most accurately achieved from measurements of the angular power spectrum
of microwave background temperature anisotropies. Recent results from a
number of independent groups working with either balloon or high–altitude
mountain experiments are finding that the first acoustic peak in the angular
power spectrum is located at a multipole value of l∼210. This result is
consistent with a value of Ωtotal very near unity:
•Boomerang
9,10: 1.02 +0.06
−0.03
•DASI 11: 1.04 ±0.06
•MAXIMA 12: 0.9+0.18
−0.16
Under the assumption that the initial density fluctuations that gave rise to
these temperature fluctuations are Gaussian and adiabatic, these results pro-
vide the most compelling evidence to date that we live in a flat (Ωtotal = 1.0)
universe. The data are no longer restricted to the first peak alone; there is
solid evidence for additional peaks located at l∼540 and 840. CBI 13 has
provided coverage over the range l= 400 to 1500, showing a predicted sharp
2
decrease in power at higher multipoles. All of these experiments are consistent
with an adiabatic, cold dark matter (CDM) inflationary model with a power
law initial perturbation spectrum index, ns, close to 1. The next generation of
polarization experiments will test this hypothesis further and yield information
on the gravitational wave spectrum predicted by inflation.
3 The Cosmological Constant / Dark Energy
For several years, something has not been quite right in the state of cosmology.
Until a few years ago, the “standard” cold dark matter (sCDM) model with Ωm
= 1, h (= H0/ 100) = 0.5, ΩΛ= 0 was the favored model. However, this model
has been found to fall short in a number of very different ways. In numerical
simulations, sCDM fails to produce the large–scale distribution of galaxies,
producing too little power on large scales 14 . The expansion ages derived for
Hubble constants of 60 or more (most estimates in the recent literature) yield
ages of less than 11 Gyr for Ωm= 1; these ages are lower than most estimates
for the Galaxy15,16, which range from 12 to 15 Gyr. And finally, measurements
of the matter density (Section 4) have tended to yield values of only about 1/3
of the critical density.
All of these inconsistencies are removed by allowing a non-zero value for
the cosmological constant, or more precisely, by having a flat universe with
a contribution from some form of dark energy. Direct evidence for a current
acceleration of the universe has come with the observation that supernovae at
high redshifts are fainter than predicted by their redshifts, in comparison to
their local counterparts 17,18. Although systematic effects due dust or differing
chemical compositions could produce an observed difference between the high
and low redshift supernovae, several tests have failed to turn up any such
systematics. The resulting implication is that the universe is now undergoing
an acceleration, and the data are consistent with Ωm= 0.3, ΩX= 0.7, assuming
a flat universe, where ΩXrepresents some form of dark energy having a large,
negative pressure 19.
Although the CMB angular power spectrum contains information on ΩX,
a direct determination of ΩXis not possible from CMB measurements alone.
This is due to intrinsic degeneracies among the cosmological parameters which
allow models with the same matter content, but very different geometries 20 .
However, the position of the first peak in the CMB angular power spectrum,
combined with large–scale structure data, can be used to break these degen-
eracies and are also consistent with ΩX∼0.75 21.
In future, it may be possible to measure the expansion rate as a function of
redshift directly 22 by measuring the displacement of individual Lyman αlines
3
Figure 1: Predicted velocity shift in units of meters/second/century as a function of redshift
for different cosmologies. Based on the calculations of Loeb (dashed: Ωm= 1, dotted: Ωm
= 0.3, solid: Ωm= 0.3, ΩΛ= 0.7) for a value of H0of 70 km/sec/Mpc.
in quasar absorption spectra over a time baseline of a decade or so. The velocity
shift is predicted to be at the level of only ∼2 m/sec/century (see Figure 1),
beyond current technical reach, but of interest for the next generation of 30–
meter–class ground–based telescopes being planned for the next decade.
4 The Matter Density / Dark Matter
There are several completely independent routes to measuring the average
matter density of the universe. In the past decade, increasingly strong evidence
for dark matter has emerged, from a wide variety of independent studies. Only
about 5% of the universe appears to be made of ordinary baryons. However,
there is a strong consensus that the amount of dark matter falls well short of
the critical density.
One of the most fundamental outstanding questions in cosmology is the
nature of the dark matter. No one simple explanation is sufficient to explain all
of the available data - that is, there appears to be more than one type of dark
matter. There is dark matter in galaxies thought to be in the form of ordinary
baryons whose form has not yet been identified, perhaps warm (100,000◦K)
gas or faint stellar remnants; there is cold, dark matter in clusters whose form
has also not been identified, but which must be non–baryonic or else it would
4
violate big bang nucleosynthesis constraints; atmospheric neutrino experiments
yield evidence for hot, non–baryonic dark matter in the form of neutrinos with
a total mass density about equal to that in luminous stars (∼
<1% of the critical
density); perhaps there is warm or self–interacting dark matter that might
explain some of the observed properties of galaxies on small (galaxy) scales
And there is also the evidence that 70% of the total mass/energy density of
the universe is in a form of dark (vacuum) energy (Section 3).
Most recent estimates of the global matter density have used clusters of
galaxies as probes of the matter distribution, assuming that clusters are large
enough that they are representative of the overall average mass density. A
number of independent techniques have been used for Ωmestimates: cluster
mass–to–light ratios 23 , the baryon density in clusters both from x-ray 24 and
Sunyaev-Zeldovich 25 measurements, the distortion of background galaxies be-
hind clusters or weak lensing 26 (also on supercluster scales of ∼3 h−1Mpc 27 ),
and the existence of very massive clusters at high redshift 28 . A widespread
consensus has emerged that the apparent matter density appears to fall in
the range Ωm∼0.2-0.4, at least on scales up to about 2h−1Mpc; i.e., only
∼20-40% of the critical density required for a flat, Ωtotal = 1 universe.
5 The Hubble Constant
The Hubble constant, the current expansion rate of the universe, is one of
the most critical parameters in big bang cosmology. Together with the energy
density of the Universe, it sets the age, t, and the size of the observable Universe
(Robs = ct). The square of the Hubble constant relates the total energy density
of the Universe to its geometry. The density of light elements (H, D, 3He, 4He
and Li) synthesized after the Big Bang also depends on the expansion rate.
Determinations of mass, luminosity, energy density and other properties of
galaxies and quasars require knowledge of the Hubble constant. In addition,
the Hubble constant defines the critical density of the Universe (ρcrit =3H2
8πG ).
The critical density (and therefore H) further determines the epoch in the
Universe at which the density of matter and radiation were equal. Hence, the
growth of structure in the Universe is also dependent on the expansion rate.
During the radiation era, growth of matter on small scales is suppressed – the
turnover in the power spectrum corresponds to the point at which the Universe
changes from radiation to matter dominated. This feature, set by the critical
density, is used to normalize cosmic structure formation models 29,30.
For several decades, over a factor of two discrepancy has persisted in
measurements of the Hubble constant spanning a range of about 40 to 100
km/sec/Mpc. Why has the measurement of H0been so difficult? It requires
5
Figure 2: Velocity versus distance for galaxies within 400 Mpc calibrated by the Cepheid
distance scale. Distances for five secondary methods are plotted: the Tully–Fisher relation
for spiral galaxies(filled circles), type Ia supernovae (filled squares), the fundamental plane for
elliptical galaxies (filled triangles), type II supernovae (open squares) and surface brightness
fluctuations (filled diamonds). A correction for metallicity has been applied to the Cepheid
calibration. A fit to the slope yields a value of H0= 72 km/sec/Mpc. One-sigma error
bars are also indicated. The bottom panel plots H0versus distance and the horizontal line
corresponds to H0= 72 km/sec/Mpc.
measurements of recession velocities and the distances to galaxies at large
distances (where deviations from the smooth Hubble expansion are small).
Progress in measuring H0has been limited by the fact that measuring accu-
rate distances presents an enormous challenge. Primarily as a result of new
instrumentation at ground-based telescopes, and with the availability of the
Hubble Space Telescope (HST), the precision with which H0can be measured
has evolved at a rapid pace.
A key project of the Hubble Space Telescope (HST) was the measurement
of distances to a sample of nearby galaxies using Cepheid variables. These
intrinsically bright stars follow a well–defined relation between luminosity and
period of variation, from which, given a measurement of apparent luminosity
and period, their distances can be established by applying the inverse square
law. The Cepheid distances may then be used to provide a calibration of
several independent methods for measuring relative distances which can be
extended further than Cepheids: for example, type Ia supernovae, the Tully–
Fisher relation (a relation between the luminosity of a spiral galaxy and its
rotational velocity), or the fundamental plane for elliptical galaxies (relating
luminosity to velocity dispersion). A summary of the final results for the
key project 31,32 yields a value of H0= 72 ±3 (statistical) ±7 (systematic)
6
km/sec/Mpc based on 5 different methods (see Figure 2).
These measurements are consistent with other recent measurements of the
Hubble constant from the measurement of the Sunyaev–Zeldovich effect and
time delays for gravitational lenses. These methods, currently with systematics
at the ± ∼20–25% level are yielding values of H0∼60 km/sec/Mpc 33,34 for
cosmologies in which Ωm= 0.3, ΩΛ= 0.7.
6 The Optical Extragalactic Background Light (EBL)
The luminosity of stars falls off as 1
r2, but the surface area intercepted increases
as r2. In an infinite universe, the cancelling r2terms ought to result in a
constant surface brightness, and the sky should be as bright as the surface
of a star (Olber’s paradox). This paradox was resolved with the discovery of
the expansion of the universe, and the finite lifetimes (and the universe itself).
But the measurement of the actual value of the optical night sky brightness
has remained elusive simply because of just how dark the sky actually is. And
because the optical background light is swamped by both by the foreground
airglow of the Earth as well as the foreground zodiacal light in the ecliptic plane
(at the level of two orders of magnitude), to date it has only been possible to
place limits on the EBL contribution. With HST it has been possible for the
first time to make a direct measurement of the total optical background light
from extragalactic sources 35,36,37 .
The total star formation history of the universe is recorded in the extra-
galactic background light. The EBL is a record of the baryonic mass processed
in stars, and the formation of elements heavier than lithium (the metal pro-
duction). The intensity of the EBL can be expressed:
IEBL =c
4πZt0
tf
ρbol (t)
1+zdt
where tfand t0represent the formation and current epochs respectively, ρbol
is the total bolometric luminosity, and the factor (1+z) accounts for the ex-
pansion of the universe 38,39 . The measured EBL also includes a contribution
from active galactic nuclei (AGN) and accreting black holes in quasars; recent
estimates suggest that this contribution could amount to about 15% 41 .
An important lower bound to the optical luminosity in extragalactic sources
can be obtained by integrating the luminosities from individually detected
galaxies (for example, the Hubble Deep Field (HDF)40,38 ). However, the total
amount of light in galaxies cannot yet be determined directly from individual
galaxy counts because cosmological surface–brightness dimming (which goes as
(1+z)4) is sufficiently severe that even intrinsically bright galaxies of L∗(Milky
7
Figure 3: Ultraviolet images of M51 and M101 as they would appear at successive redshifts
from Kuchinski et al. . Top panel: (Left) Ground-based U-band (3500 Angstrom) image of
M51. (Center) M51 artificially redshifted to z=0.6, corresponding to the HST WFPC2 606W
filter. (Right) M51 artificially redshifted to z= 1.2, corresponding to the HST WFPC2 814W
filter. Bottom panel: (Left) UIT far-ultraviolet (1500 Angstrom) image of M101. (Right)
Simulated images of M101 at redshifts where the rest-frame UIT filter bandpass would
coincide with the four HST WFPC2 filters used to image the Hubble Deep Field.
Way) luminosities can be missed at high redshift in deep surveys, and the outer
lower surface–brightness regions in galaxies can easily escape detection even in
relatively nearby galaxies 35,37. Figure 3, from Kuchinski et al. 42 illustrates
directly how the well–known Messier objects, M51 and M101 would appear
when observed at successively higher redshifts. Corrections for redshift and
surface–brightness dimming are applied to rest–frame ultraviolet (U–band and
1500 Angstrom) images. Only the highest surface–brightness features remain
visible at high redshift (see also the discussion by Colley et al. 43), and these
reasonably bright (as well as all fainter) galaxies will be missed in deep galaxy
surveys.
The optical HST EBL measurements are shown in Figure 4, from Bern-
8
Figure 4: Background light in units of νIνfrom 0.1 to 1000 µm. The optical (UVI)
EBL measurements are from Bernstein, Freedman & Madore (BFM) 37, along with EBL
measurements and limits at near– and far–infrared, and submillimeter wavelengths; solid
lines are theoretical predictions (see references cited in BFM).
stein, Freedman & Madore 37 , in addition to measurements at longer wave-
lengths. The grey shaded band corresponds to the model of Dwek et al. 44
scaled to fit the HST data. This model is based on a star formation model
including corrections for dust extinction and reradiation. From this figure, it
can be seen that approximately 30% of the optical radiation (that produced in
stars) is absorbed and then reradiated by dust at infrared wavelengths. The
total EBL contribution from 0.l to 1000 µm amounts to 100 ±20 nW/m2/sr.
The measured optical EBL exceeds the light computed from counts in
the Hubble Deep Field by a factor (depending on bandpass) ranging from
two to three 35,37 . There are two major components to the missing light,
the first being from the outer parts of detected galaxies (beyond the aperture
used for photometry measurements), and the second being from galaxies below
the surface brightness threshold of the HDF. Roughly 30% of the missing
light comes from the outer parts of galaxies and 60% from galaxies below the
detection threshold. This background light measurement is consistent with a
star formation history of the universe characterized by a steep increase of star
formation between redshifts of 0 and 1 (as observed in the CFRS survey45 ) and
a flat or slightly increasing star formation rate beyond in the redshift range 1-
4, consistent with observations of star–forming galaxies at these redshifts (the
9
Lyman break galaxies) 46. Thus, much of the contribution to the background
light may be due to normal galaxies at redshifts less than 4 that are missed
because of cosmological surface–brightness dimming (and K-corrections due to
band shifting with redshift). However, an earlier generation of stars, not yet
detected, may also be contributing to the total background light 47. The value
measured for the optical EBL implies that the mass processed through stars
contributes about 1% of the critical density (for h = 0.7); i.e., a small overall
contribution, but a factor of two higher than previous estimates.
7 Concluding Remarks
At the current time, there is an impressive convergence on a new standard
cosmological model: a flat Ωtotal = 1 model with Ωm∼0.3, ΩX∼0.7, h =
0.7, and an age of about 14 billion years. As the number of independent deter-
minations of these parameters increases, so will our confidence in this overall
picture. Importantly, this convergence does not depend on the results from a
single experiment. For example, a non-zero value for a dark energy component,
ΩX, is implied not solely by the type Ia supernovae, but also by the combined
CMB anisotropy and large–scale structure surveys. The position of the first
acoustic CMB peak at a scale of about 1 degree strongly favors a flat uni-
verse, whereas several direct estimates of Ωmyield low values, again pointing
to a missing energy component. Values of the Hubble constant exceeding 60
km/sec/Mpc lead to a conflict with the ages of the Milky Way globular clusters
in an ΩX= 0 universe. Many of the parameter measures would have to move
beyond their current 2–σerror bounds for the cosmological model to change.
Given the difficulty in eliminating systematic errors, such a possibility cannot
be ruled out, but as more accurate data (from MAP, Planck, SDSS, Chandra,
SIM, GAIA, LSST, GSMT and other ongoing or planned experiments) become
available, the prospects for robustly constraining the cosmological parameters,
ultimately leading to an understanding the physical basis of the underlying
cosmological model, are looking extremely promising.
Acknowledgments
I gratefully acknowledge support by NASA through grant GO-2227-87A from
STScI for HST observations, and the ASTRO-2 Guest Investigator Program
through NAG8-1051 for the UIT observations. It was a pleasure to participate
in the inauguration of the Michigan Center for Theoretical Physics, and I thank
the organizers for a very enjoyable meeting.
10
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