Cosmological Parameters

Cosmological parameters
Linnea Hjalmarsdotter
December 9, 2005
Abstract
Cosmology of today has developed into a precision science with a well-established theoretical
framework including predictions and tests testable by astronomical observations. In this report we
outline the basics of the theory behind the standard model of present day cosmology and discuss the
impact of the mass and energy content of the Universe on its geometry as well as its historic and
future evolution. We describe the diﬀerent methods of observationally determining H0, ΩM and ΩΛ.
Finally, we discuss the latest results and their implications.
1
Introduction to curved spacetime
Our Universe, like any other homogeneous and isotropic four-dimensional curved space-time, can be
described by the metric
dr2
ds2 = dt2 − a2(t)
+ r2dθ2 + r2 sin2 θdφ2 ,
(1)
1 − kr2
where r is dimensionless, a(t) denotes the time-dependent scale factor, k is a curvature term that can
be chosen as +1, 0, or −1 depending on whether the constant curvature is positive, zero, or negative
respectively, and we set c = 1. This metric is called the Robertson-Walker line element, and we note
that for k = 0, it reduces to ﬂat Minkowski space. For k = +1, the Universe will be closed with a(t) as
the radius at time t. For k = −1, the Universe will be open and plausibly of inﬁnite extent.
The eﬀect of matter and energy on the curvature of space-time is given by Einstein’s equations of
general relativity
1
Rµν − g
2 µνR − Λgµν = 8πGTµν,
(2)
where Rµν is the Ricci tensor, R the Ricci scalar, Tµν the energy-momentum tensor, and Λ the so-
called cosmological constant. By choosing as the metric the Robertson-Walker line element above, and
calculating the metric connections we obtain from the 00 component
2
˙a
8πG
k
Λ
=
(ρ
+
(3)
a
3
m + ρrad) − a2
3
or by deﬁning the vacuum energy density
Λ
ρvac = ρΛ =
,
(4)
8πG
2
˙a
8πG
k
=
ρ
(5)
a
3
tot − a2
with ρtot = ρm + ρrad + ρvac. This is the Friedmann equation and describes the evolution of the scale
factor a depending on the contributions from matter and radiation as well as from the cosmological
constant.
Without the cosmological constant, Einstein’s equations do not permit static solutions, only expand-
ing or contracting. The concept of an expanding or contracting universe being too bold at the time -
it was generally taken for granted that the Universe would be static and eternal - this was considered
a failure which prompted Einstein to include the term. Later, when Hubble discovered the expansion
of the Universe, Einstein is said to have considered the introduction of the cosmological constant as his
biggest mistake. Today, however, it seems as there might be a need for a non-zero cosmological constant
after all. This will be further discussed in Section 3.5.
1

1.1
The expansion of the Universe
The predictions made by Friedmann (1922) and Lemaˆıtre (1927) of non-static solutions to Einsteins
equations were conﬁrmed with the discovery that distant galaxies are in fact moving away from us with
velocities proportional to their relative distances. The relationship v = H0d is known as Hubble’s law,
after its discoverer Edwin Hubble (1929). The Hubble constant H0 is one of the most fundamental
parameters of modern cosmology. The value of H0 deﬁnes the present observed value and is actually
more correctly expressed as H(t0), where t0 is the present time and H(t) for all times t determined by
˙a(t)
H(t) =
.
(6)
a(t)
As we can see from Friedmann’s equation, the evolution of the scale factor is governed by two additional
parameters, the mass density and the cosmological constant. The expansion of the Universe is an
expansion of space-time itself and thus does not involve any motion in coordinate space. If we assign
a set of coordinates (ri, θi, φi) to each galaxy i, these coordinates will not change when the Universe
evolves and the pattern of galaxies will stay the same. All the cosmological distances will, however, be
stretched by a factor a(t). A popular analogy is that of a ’raisin bread’, where the distances between
the raisins increase as the bread is raising. Nevertheless, using just ordinary Newtonian gravity, one can
show that a massive particle outside a spherical piece of the Universe expanding with a velocity v = H0d
will escape the gravitational attraction only if the density of space is
3H
ρ
02
≤ ρcrit =
.
(7)
8πG
This means that to stop the expansion and make the Universe contract a mass density equal or higher
than this critical density is required. The value of the critical density is time-dependent, and the present
value is
ρ0,crit = 1.9 · 10−29h2g cm−3,
(8)
where
a(t)3
ρ0 = ρm
(9)
a30
is the matter density for the present scale factor (i.e. matter density now), and
H0
h =
.
(10)
100 km s−1Mpc−1
Today’s contribution from all kinds of matter and radiation to the critical density can be expressed as
ρ0
Ω0 =
.
(11)
ρcrit
By expanding the scale factor a(t) around the present time t0, we ﬁnd
1
a(t) = a(t0) 1 + H0(t − t0) − q0H2(t − t0)2 + ... ,
(12)
2
0
which deﬁnes the decceleration parameter
¨
a
q0 = −
,
(13)
aH20
or using the critical density,
1
3
q0 =
Ω0 +
Ω
2
2
iαi,
(14)
where α gives the equation of state as pi = αρi (for baryons α = 0, for radiation α = 1/3, and for
vacuum energy α = −1).
2
Cosmological models
The standard model of present day cosmology is built on the cosmological principle of a homogeneous
and isotropic Universe, the Robertson-Walker line element and the Friedmann equation, constituting
the Friedmann-Lemaitre-Robertsson-Walker model, or FLRW for short.
It is also customary to express the present day contributions to the critical energy density from
matter and the cosmological constant separately as
2

ρ
8πG
Ω
m
M =
=
ρ0,
(15)
ρcrit
3H20
ρΛ
Λ
ΩΛ =
=
.
(16)
ρcrit
3H20
Deﬁning
−k
ΩK =
,
(17)
a2H2
0
0
we can express the Friedmann equation as
ΩM + ΩΛ + ΩK = 1.
(18)
We note that there are only two independent contributions to the energy density. The curvature is
determined by the total density of matter and energy and vice versa. (In the early Universe, a contribu-
tion from the radiation density has to be included. This term vanishes for the present or future times,
reﬂecting the fact that radiation plays no major role today.)
The full solutions to the Friedmann equation for the scale factor a at all times t can be quite
complicated due to the diﬀerent a-dependence of the individual terms contributing to ρtot, but let us
look at a few simple cases and examine the long-time behaviour as t → ∞.
The simplest cases to consider are for Λ = 0. The Friedmann equation reduces to
ΩM + ΩK = 1.
(19)
The solution is
8πGρma3
˙a(t) =
0
− k.
(20)
3a(t)
For the (somewhat unrealistic) case of a Universe without matter, real value solutions for a(t) require
k = −1, an open geometry, and
a(t) = t.
(21)
This is the so-called Milne model and escribes a linearly expanding (or contracting) Universe.
Including the matter density term, the solutions for large t will depend on k. For k = 0,
2
t
3
a(t) = a0
.
(22)
t0
This is the so-called Einstein-de Sitter model. For k = −1, ˙a2 is always positive, which means an ever
increasing a(t). For k = +1, ˙a2 will be positive up to a critical largest value, given by
8πGρ
a
ma3
0
crit =
.
(23)
3
For larger t, the Universe will start to contract and eventually come to a ’Big Crunch’.
Including both matter and a non-zero cosmological constant we obtain a richer variety of cosmological
models. We have to solve
8πGρ
Λ a(t)2
˙a(t) =
ma3
0 − k +
.
(24)
3a(t)
3
If Λ < 0, a(t) can not become arbitrarily large to allow real solutions for ˙a. For the largest possible
value, ¨
a < 0, and we have an oscillating Universe.
For Λ > 0 and k = 0 or −1 we observe that for large a(t) the Universe will enter a period of
exponential expansion as the vacuum energy takes over. This situation can actually have occurred in
the past, during the period of inﬂation when the scale of the Universe was blown up exponentially during
a short period of time.
The energy densities and thus H0, Ω0, ΩM , and ΩΛ are measurable by observations and the focus
of today’s modern observational cosmology is to measure these parameters with ever better precision.
In the next section, we will outline some of the most important methods and present latest results. In
the last section, we will compare results from diﬀerent measurements and discuss the implications of the
current best estimates.
3

3
Measuring the parameters
3.1
The Hubble parameter, H0
Hubble’s law states that distant galaxies are moving away from us with velocities proportional to their
relative distances, v = H0d. The value of H0 is found by measuring the recession velocities of galaxies,
whose distances are independently known by other measurements. The most reliable distance indicators
are obtained from the observed relation between the rotational velocity (distance independent) and the
apparent luminosity (related by the distance to intrinsic luminosity) of spiral galaxies. Another distance
indicator is the relation between observed and intrinsic luminosities of so-called astronomical standard
candles, objects with constant luminosity. One such example are type Ia supernovae, which will be
discussed more in detail in section 3.5. The present best estimate of H0 = 65 ± 5 km sec−1 Mpc−1,
combining diﬀerent types of measurements.
3.2
The total matter and energy density, Ω0
The cosmic microwave background radiation, or CMBR, provides a snapshot of the Universe at the so-
called last scattering surface, when photons scatter for the last time oﬀ free electrons, before the electrons
recombine with protons to form neutral matter (with a lot smaller cross section) at tls, about 300 000
years after the Big Bang. At that time, photons were still not decoupled from baryons. As the baryons
were falling into the potential wells of the density ﬂuctuations eventually making up the structure of the
Universe, the photons acted as a restoring force, which led to gravity-driven acoustic oscillations. These
oscillations show up in the CMBR as acoustic peaks in its power spectrum. The wavelength of the lowest
frequency acoustic mode λmax ∼ vstls and thus provides a standard ruler on the last scattering surface.
Both λmax and the distance to the last scattering surface depend on the curvature of the Universe and a
value of Ω0 can be derived from the location of the ﬁrst acoustic peak. Since this is a geometrical method
it assumes nothing about, and is insensitive to, the composition of matter and energy and is therefore
a measure of the curvature (in some sense rather ΩK than Ω0). CMBR anisotropy measurements have
now been carried out by more than twenty experiments, the most precise ones being COBE (1989) and
WMAP (2001). All the measurements have deﬁned the position of the ﬁrst acoustic peak at a value
consistent with Ω0 ≈ 1 (and Ωk = 0, implying either ΩM = 1 and ΩΛ = 0 or both > 0).
3.3
Baryonic matter, ΩB
The most precise determination of the baryon density in the Universe comes from a comparison with
measured primordal abundances of the light elements, D, 3He,4 He, and 7Li to predictions from Big
Bang nucleosynthesis. The agreement of measurements and theoretical predictions of the primordial
abundances is one of the strongest observational evidence for the Big Bang scenario, but the values
also give upper limits on ΩB, since the formation of the primordial elements depend strongly on the
baryon density. The best baryon density probe is deuterium since it can not be produced by any
known astrophysical processes, and the evolution of its abundance since Big Bang is therefore simple.
Measurements of the deuterium abundance from high-redshift gas-clouds, seen in absorption against
distant quasars, have yielded ΩBh2 = 0.019 ± 0.0012 or ΩB ≈ 0.05 with h = 0.65.
The results are consistent with those from two other measurements, based on entirely diﬀerent
physics. By comparing measurements of the opacity of the Ly-α forest toward high-redshift quasars
with high-resolution hydrodynamical simulations of structure formation, it is possible to obtain a lower
limit for ΩB through the intensity of the ionizing ﬁeld. Measurements indicate ΩBh2 ≥ 0.015.
The CMBR anisotropy also provides a determination of the baryon density from the amplitudes, or
more precisely from the diﬀerence between the amplitudes of the ﬁrst two acoustic peaks in the power
spectrum, described in the section above. The value from the most recent and accurate experiment
WMAP is consistent with ΩB ≈ 0.05.
3.4
Dark matter, ΩM − ΩB
The baryonic matter thus represents only a tiny fraction of the total mass density of the Universe
(assuming Ω0 = 1). It has, however, been known from other forms of mass measurements of for example
the rotational velocities of galaxies, that most of the matter in galaxies and galaxy clusters is in the form
of non-luminous matter. It is likely that the same situation applies to the Universe as a whole and that
the bulk of the contribution to the matter density should be searched for in the form of dark matter.
The classical approach to weighing the dark matter involves the use of mass-to-light ratios. The idea
is to extrapolate ratios found for galaxies or galactic clusters to the Universe as a whole and divide by
the critical mass-to-light ratio to obtain ΩM . Best estimates suggest ΩM = 0.20 ± 0.04. The problem
4

with this method, however, is that the extrapolation from a structure such as a cluster of galaxies to
the entire Universe might not hold, especially since the luminosity density of the Universe itself evolves
strongly with redshift.
Another method, involving less assumptions, is based on determining the ratio of baryon to total mass
density in galactic clusters. Most of the baryons in a cluster reside in the hot, x-ray emitting intracluster
gas. The mass of this gas can be determined by either measuring its X-ray ﬂux, and/or mapping the
Sunyaev-Zel’dovich CMBR distortion caused by CMBR photons scattering oﬀ hot electrons in the gas.
The total mass of the cluster can be measured by its dynamics or by mapping using gravitational lensing.
Assuming that the baryon density of the Universe is the same fraction of the total mass as the baryons
in the gas of the clusters to the total cluster masses, one can then derive a value by making use of the
baryon density from section above. Best estimates suggest ΩM = 0.4 ± 0.1, signiﬁcantly larger than
the values derived from calculating mass-to-light ratios, and are supported by several other diﬀerent
methods as well as theoretical considerations for structure formation. A value of ΩM
ΩB is required
for the evolution of the structure that we see today. If the mass had been in form of baryons only
density ﬂuctuations seen in the CMBR today would have begun to grow only at the time of decoupling
(z = 1000), and would not have had time to evolve and produce all the structure we see today. There
are therefore no viable models for structure formation without a signiﬁcant amount of dark matter.
Such considerations also rules out a signiﬁcant fraction of the dark matter as being in the form
of dark baryons. We know that only about 10 percent of baryons are in form of luminous matter or
stars. The reminding 90 percent is made up of dark baryonic matter most of which probably resides in
intracluster gas as described above but also in ’dark stars’ such as white dwarfs, neutron stars, black
holes or objects of mass below the hydrogen burning limit. But the fact that there seems to be a lot
more matter than there are baryons, including this dark fraction, require most of the dark matter to
be in some other exotic form of particles. The currently most favoured candidates are relic elementary
particles left over from the Big Bang, heavy stable particles with very weak interactions, such as e.g.
the neutralino. No such particle has however yet been detected, neither in space nor in particle physics
experiments on Earth.
3.5
Dark energy, ΩΛ
Accepting the notion of most of the matter in the Universe being in the form of some unknown exotic
particle or particles, there is still a discrepancy between ΩM = 0.4 and the curvature measurements
implying that the total matter/energy density of the Universe is equal to the critical one, i.e. Ω0 = 1.
There is obviously a need for an additional component, some kind of ’dark energy’. This component has
to be smoothly distributed not to have been detected in measurements of matter density. Its properties
are also severely constrained by structure formation, the age of the Universe, and the CMBR anisotropy.
In order not to have interferred with matter domination during the period from matter-radiation equality
until very recently, and thus hampering structure formation, the dark-energy component must have been
much less important in the past than it is today. The simplest example of such a component would be
vacuum energy described by Einstein’s cosmological constant. A property of vacuum energy is that
its equation of state is p = αρ with α = −1 and thus a contribution to the energy density by the
vacuum energy gives a negative pressure. Inserting in the expression for the decceleration parameter
q0, we observe that a consequence of ΩΛ > 0 is a negative decceleration parameter an thus accelerated
expansion.
An accelerated expansion should be measurable and in 1998 two groups (Perlmutter et al. and
Reiss et al.) presented the ﬁrst results favouring a non-zero ΩΛ, based on measurements of apparent
magnitudes vs. redshifts for type Ia supernovae (SNeIa). The results are shown in Fig.1. The method is
based on the notion of SNe Ia as so-called standard candles. Since they are all the result of an explosion of
a white dwarf exceeding the Chandrasekhar mass of approximately 1.4M , they are thought to have the
same absolute luminosities. The diﬀerence in apparent (observed) luminosities is then only a function of
distance which is related to redshift via Hubbles law, v0 = H0d. If the expansion rate was constant, the
measured luminosities vs. redshift should give a perfectly linear relation. The fact that distant galaxies
have lower redshift and thus are moving slower than predicted by Hubble’s law means that the expansion
is speeding up. The ﬁrst break-trough results were followed up by even more precise measurements of
even more distant supernovae and the best ﬁt to the current data (Supernova Cosmology Project, Knop
et al. 2003) corresponds to a ﬂat universe with ΩM =0.25 and ΩΛ=0.75.
3.6
Cosmic concordance
The results of the best values and conﬁdence intervals of the cosmological parameters as measured by
the diﬀerent methods are shown graﬁcally in Fig 1, right panel. We ﬁnd that the values of the total
Ω0 deﬁne a concordance region in the ΩM -ΩΛ plane with ΩΛ from the supernova measurements that is
5

Figure 1: Figure 1. left Figure from the Supernova Cosmology Project (Knop et al. 2003) showing the
relation between apparent magnitude and redshifts for distant supernovae. The observations favour a
ﬂat Universe with a non-zero cosmological constant and must be interpreted as an accelerated expan-
sion. right Concordance in ΩM -ΩΛ space between results from diﬀerent methods (from the Supernoova
Cosmology Project).
further constrained by the results for ΩM . This means that not only are we now able to measure these
cosmological parameters with reasonable error estimates, the results from totally diﬀerent experiments
also seem to agree with each other. The values of ΩM +ΩΛ(or ΩK) of approximately 1.0, ΩB ≈ 0.05, ΩM ≈
0.3, and ΩΛ≈ 0.75 seem to imply that we live in a ﬂat universe with a non-zero comological constant
with the energy density made up of about 5 % baryonic matter, 25 % non-baryonic or dark matter and
75 % vacuum energy.
The results, however, have some strange implications. The cosmological parameters are strong func-
tions of time. How can it be that we today happen to live in a Universe which is so close to the critical
density? And what is the meaning of ΩΛ= 1? In natural units, the vacuum energy density for Ω0 = 1
equals ±ρvac ≈ 10−46 GeV4. This is about a factor of 10122 smaller than what one could expect to
emerge from a quantum theory of gravity using the Planck mass as a mass scale. The smallness of the
cosmological constant is a problem to both cosmology and theoretical physics. Some particle physicists
believe that when the problem is understood, the answer will be exactly zero. Others have tried to
resolve the issue by invoking a dynamical cosmological constant, evolving with time and the scale factor.
In such a scenario, even if the true vacuum is zero, it is possible that not all ﬁelds have evolved to
their state of minimum energy, but are still ’rolling’ towards it. An evolving cosmological constant is
sometimes referred to as ’quintessence’.
4
Summary
To summarize, recent experiments, especially perhaps the most recent measurements of the CMBR
anisotropy together with the results from the distance vs. redshift measurements of distant supernovae,
have provided accurate determinations of the cosmological parameters H0, Ω0, ΩM , and ΩΛ. The
results between diﬀerent methods of measurements also seem to coincide very well with each other thus
conﬁning the values to a small parameter space in the ΩM -ΩΛ plane. Not only do measured values
provide observational evidence for the correctness of the Big Bang scenario. They are now also accurate
enough to provide constrains on the evolution of the expansion and the ultimate fate of our Universe.
Future results from the second year of the WMAP project as well as the launch of the Planck mission
(in 2007), will further reﬁne our understanding of these cosmological parameters governing the evolution
of space-time.
References
[1] L. Bergstr¨om & A. Goobar, Cosmology and Particle Astrophysics, Springer, 2004
6

[2] R. A. Knop et al., New constrains on ΩM , ΩΛ and ω from an Independent Set of 11 High-Redshift
Supernovae Observed with the Hubble Space Telescope, Astrophysical Journal, 2003, 598, 102
[3] J. Silk, The Big Bang, W. H. Freeman and company, 2001
[4] M. S. Turner, Cosmological Parameters, astro-ph/9904051, 1999
[5] High Redshift Supernova Search Supernova Cosmology Project (Berkeley lab),
http://www-supernova.lbl.gov/
7