Download Statistical cosmollogy in retrospect

Cosmology
Statistical cosmol
Peter Coles looks at the past, present and future association of astronomy and cosmology with statistical theory.
Abstract
Over the past decade unprecedented
improvements in observational technology
have revolutionized astronomy. The effects
of this data explosion have been felt
particularly strongly in the field of
cosmology. Enormous new datasets, such
as those recently produced by the WMAP
satellite, have at last placed our
understanding of the universe on a firm
empirical footing. The new surveys have
also spawned sophisticated statistical
approaches that can do justice to the
quantity and quality of the information. In
this paper I look at some of the new
developments in statistical cosmology in
their historical context, and show that
they are just the latest example of a deep
connection between the fields of
astronomy and statistics that goes back at
least as far as Galileo.
I
t has become a bit of a cliché to refer to the
modern era of observational cosmology as
a “golden age”, but clichés are clichés
because they are often perfectly apt. When I
started my graduate studies in cosmology in
1985, the largest redshift survey in circulation
was the CfA slice (De Lapparent, Geller and
Huchra 1986) containing just over a thousand
galaxy positions. Now we have the AngloAustralian 2dF Galaxy Redshift Survey
(2dFGRS) in which the combination of widefield (two-degree) optics and multifibre spectroscopy has enabled redshifts of about a
quarter of a million galaxies to be obtained in
just a few years (Colless et al. 2001). Back in
1985, no experiment had been able to detect
variations in the temperature of the cosmic
microwave background (CMB) radiation across
the sky predicted by theories for the formation
of the structures seen in galaxy surveys. This
changed in 1992 with the discovery, by the
COBE team (Smoot et al. 1992), of large-scale
“ripples” in the CMB. This, in turn, triggered
immense theoretical and observational interest
in the potential use of finer-scale structure in the
CMB as a cosmological diagnostic. The stunning high-resolution maps produced by the
WMAP satellite (Bennett et al. 2003) are to
COBE what the 2dFGRS is to the CfA survey.
These huge new datasets, and the others that
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1: The huge improvement in angular resolution from degree to
arcminute scales
reveals unprecedented
detail in the temperature structure of the
CMB, but also poses
challenges. Millions
rather than thousands
of pixels mean that
even simple statistical
covariances require
more substantial
number-crunching for
WMAP (below) than
COBE (above). More
importantly, the high
signal-to-noise of
WMAP means that
systematic errors must
be understood even
more carefully than in
the COBE map.
will soon follow them, have led to a recognition
of the central role of statistics in the new era of
cosmology. The new maps may be pretty to look
at, but if they are to play a role in the empirical
science that cosmology aspires to be, they must
be analysed using objective techniques. This
requires a statistical approach. Patterns in the
distribution of galaxies and in the distribution
of hot and cold spots in the microwave sky must
be quantified using statistical descriptors that
can be compared with theories. In their search
for useful descriptors, cosmologists have borrowed ideas from diverse fields of mathematics,
such as graph theory and differential topology,
as well as adapting more traditional statistical
devices to meet the particular demands of cosmological data (Martinez and Saar 2002). The
power-spectrum, for example, a technique usually applied to the analysis of periodicities in
time series, has been deployed with great success
to the analysis of both 2dFGRS (Percival et al.
2001) and WMAP (Hinshaw et al. 2003). This
urgent demand for a statistical approach to cosmology is not new. It is merely the latest manifestation of a fascinating and deep connection
between the fields of astronomy and statistics.
Astronomy and the history of statistics
The attitude of many physicists to statistics is
summarized by two quotations: “There are lies,
damned lies, and statistics” (Benjamin Disraeli);
and “If your experiment needs statistics, you
ought to have done a better experiment”
(Ernest Rutherford). It was certainly my attitude when I was an undergraduate physics student that statistics was something done by
biologists and the like, not by real scientists. I
skipped all the undergraduate lectures on statistics that were offered, and only saw the error
of my ways when I started as a graduate student and realised how fundamentally important
it is to understand statistics if you want to do
science with observations that are necessarily
imperfect. Astronomy is about using data to
test hypotheses in the presence of error, it is
about making inferences in the face of uncertainty, and it is about extracting model parameters from noisy data. In short, it is about
statistics. This argument has even greater
strength when applied to cosmology, which has
all the peculiarities of astronomy and some of
its own. In mainstream astronomy one can survey populations of similar objects, count frequencies and employ arguments based on large
numbers. In cosmology one is trying to make
inferences about a unique system, the universe.
As we shall see, history has a lesson to teach us
on this matter too.
The connection between astronomy and statistics is by no means a one-way street. The
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ogy in retrospect
2: Great astronomers who also developed statistical methods.
Bernoulli
Bessel
Galileo
development of statistics from the 17th century
to the modern era is a fascinating story, the
details of which are described by Hald (1998).
To cut a long story short, it makes sense to
think of three revolutionary (but not necessarily chronological) stages. The first involved the
formulation of the laws of probability from
studies of gambling and games of chance. The
second involved applying these basic notions of
probability to topics in natural philosophy,
including astronomy (particularly astrometry
and celestial mechanics). The third, and most
recent stage, saw the rise of statistical thinking
in the life sciences, including sociology and
anthropology. It is the second of these stages
that bears the most telling witness to the crucial role astronomy has played in the development of mainstream statistics: an astonishing
number of the most basic concepts in modern
statistics were invented by astronomers. Here
are just a few highlights:
● Galileo Galilei (1632) was, as far as I am
aware, the first to find a systematic way of taking errors into account when fitting observations, i.e. assigning lower weights to
observations with larger errors. He used
absolute errors, rather than the squared error
favoured by modern approaches.
● Daniel Bernoulli (1735) performed a pioneering analysis of the inclinations of the planetary orbits, with the aim of determining
whether they were consistent with a random
distribution on the sky.
● John Michell (1767), although probably
more famous for his discussion of what are now
known as black holes, engaged in a statistical
analysis of the positions of the “fixed stars” on
the celestial sphere to see if they were consistent
with a random pattern.
● Laplace (1776) studied the problem of determining the mean inclination of planetary orbits.
● Carl Freidrich Gauss (1809) developed the
principles of least-squares in fitting observations.
● Laplace (1810) proved a special case of the
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Gauss
Jeffreys
Central Limit Theorem, namely that the distribution of the sum of a large number of independent variables drawn from some
distribution tends towards the Gaussian distribution regardless of the original distribution.
● Friedrich Wilhelm Bessel (1838), best known
for his functions, provided a general proof of
the Central Limit Theorem.
● John Herschel (1850) developed a theory of
errors based on the “normal”, i.e. Gaussian distribution. The phrase “normal distribution” is
now standard.
● Sir Harold Jeffreys (1939) published a book
on The Theory of Probability and so rekindled
a long-standing debate on the meaning of probability ignited by Laplace and Thomas Bayes to
which I shall return later on.
All these pioneering studies influenced modern-day statistical methods and terminology,
and all were done by scientists who were, first
and foremost, astronomers. It is only comparatively recently that there has been significant
traffic in the reverse direction. The Berkeley
statisticians Jerzy Newman and Elizabeth Scott
(1952), for example, made an important contribution to the study of the spatial distribution
of galaxy clustering.
A forensic connection
Before resuming the thread of my argument, I
can’t resist taking a detour in order to visit
another surprising point at which the history of
statistics meets that of astronomy. When I give
popular talks, I often try to draw the analogy
between cosmology and forensic science. We
have one universe, our one crime scene. We can
collect trace evidence, look for fingerprints,
establish alibis and so on. But we can’t study a
population of similar crime scenes, nor can we
perform a series of experimental crimes under
slightly different physical conditions. What we
have to do is make inferences and deductions
within the framework of a hypothesis that we
continually subject to empirical test. This
Laplace
Quételet
process carries on until reasonable doubt is
exhausted. Detectives call this “developing a
suspect”; cosmologists call it “constraining a
theory”. I admit that I have stretched this analogy to breaking point, but at least it provides
an excuse for discussing another famous
astronomer-come-statistician. Lambert Adolphe
Jacques Quételet (1796–1894) was a Belgian
astronomer whose main interest was in celestial
mechanics. He was also an expert in statistics.
As we have seen, this tendency was not unusual
for astronomers in the 19th century but
Quételet was a particularly notable example.
He has been called “the father of modern statistics” and, among other things, was responsible for organizing the first international
conference on statistics in 1853. His fame as a
statistician owed less to his astronomy, however,
than the fact that in 1835 he had written a book
called On Man. Quételet had been struck not
only by the regular motions displayed by heavenly bodies, but also by regularities in social
phenomena, such as the occurrence of suicides
and crime. His book was an attempt to apply
statistical methods to the development of man’s
physical and intellectual faculties. His later
work Anthropometry, or the Measurement of
Different Faculties in Man (1871) carried these
ideas further. This foray into “social physics”
was controversial, and led to a number of
unsavoury developments in pseudoscience such
as the eugenics movement, but it did inspire two
important developments in forensics.
Inspired by Quételet’s work on anthropometry,
Adolphe Bertillon hit upon the idea of solving
crime using a database of measurements of physical characteristics of convicted criminals (length
of head, width of head, length of fingers, and so
on). On their own, none of these could possibly
prove a given individual to be guilty, but it ought
to be possible using a large number of measurements to establish identity with a high probability. “Bertillonage”, as the system he invented
came to be known, was cumbersome but proved
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successful in several high profile criminal cases
in Paris. By 1892 Bertillon was famous, but
nowadays bertillonage is a word found only in
the more difficult Sunday newspaper crosswords. The reason for its demise was the development of fingerprinting. It’s a curious
coincidence that the first fingerprinting scheme
in practical use was implemented in India, by a
civil servant called William Herschel. But that
is another story.
The great debate
A particularly interesting aspect of modern statistical cosmology is the attention being given to
Bayesian methods. This has reopened a debate
that goes back as far as the early days of statistics, indeed to the first of the three revolutionary periods I referred to above. This debate
concerns the very meaning of the probabilistic
reasoning on which the field of statistics is based.
We make probabilistic statements of various
types in various contexts. “Arsenal will probably win the Premiership,” “There’s a one-in-six
chance of rolling a six,” and so on. But what
do we actually mean when we make such statements? Roughly speaking there are two competing views. Probably the most common is the
frequentist interpretation, which considers a
large ensemble of repeated trials. The probability of an event or outcome is then given by
the fraction of times that particular outcome
arises in the ensemble. In other words, probability is identified with some kind of proportion.
The alternative “Bayesian” view does not concern itself with ensembles and frequencies of
events, but is instead a kind of generalization of
the rules of logic. While ordinary logic, Boolean
algebra, relates to propositions that are either
true (with a value of 1) or false (value 0),
Bayesian probability represents the intermediate
state in which one does not have sufficient
information to determine truth or falsehood
with certainty. Notice that Bayesian probability
is applied to logical propositions, rather than
events or outcomes, and it represents the degree
of reasonable belief one can attach. Moreover,
all probabilities in this case are dependent on
the assumption of a particular model.
The name “Bayesian” derives from the Revd
Thomas Bayes (1702–61) who found a way of
“inverting” the probability involved in a particular problem involving the binomial distribution.
The general result of this procedure, now known
as Bayes’ theorem, was actually first obtained by
Laplace. The idea is relatively simple. Suppose I
have an urn containing N balls, n of which are
white and N–n of which are black. Given values
of n and N, one can write down the probability
of drawing m white balls in a sequence of M
draws. The problem Bayes solved was to find the
probability of n given a particular value of value
of m, thus inverting the probability P(m|n) into
P(n|m). This particular problem is only of acad3.18
emic interest, but the general result is important.
To give an illustration, suppose we have a
parameter A that we wish to extract from some
set of data D that we assume is noisy. A standard frequentist approach would proceed via
the construction of a likelihood function, which
gives the probability of the data D for each particular value of the parameter A and the property’s experimental noise P(D|A). The
maximum of this likelihood would be quoted
as the “best” value of the parameter: it is the
value of the parameter that makes the measured
data “most likely”. The alternative, Bayesian,
approach would be to “invert” the likelihood
P(D|A) to obtain P(A|D) using Bayes’ theorem.
The maximum of P(A|D) then gives the most
probable value of A. Clearly these two
approaches can be equivalent if P(A|D) and
P(D|A) are identical, but in order to invert the
likelihood correctly we need to multiply it by
P(A), a “prior probability” for A. If this probability is taken to be constant then the two
approaches are equivalent, but consistent
Bayesian reasoning can involve non-uniform
priors. To a frequentist, a prior probability
means model-dependence. To a Bayesian, any
statement about probability involves a model.
It has to be admitted, however, that the issue of
prior probabilities and how to assign them
remains a challenge for Bayesians.
This is not a sterile debate about the meaning
of words. The interpretation of probability is
important for many reasons, not least because
of the implications it holds for the scientific
method. The frequentist interpretation, the one
favoured by many of the classical statistical
texts, sits most comfortably with an approach
to the philosophy of science like that of Karl
Popper. In such a view, data are there to refute
theories, and repeated failure to refute a theory
provides no reason for stronger belief in it. On
the other hand, Bayesians advocate “inductive”
reasoning. Data can either support a model,
making it more probable, or contradict it, making it less probable. Falsifiability is no longer the
sole criterion by which theories are judged.
There is not space here to go over the pros and
cons of these two competing approaches, but it
is interesting to remark that the early historical
development of mathematical statistics owes
much more to the Bayesian approach than frequentist ones but these fell into disrepute later.
Most standard statistical texts, such as the compendious three-volume Kendall and Stuart
(1977), are rigorously frequentist. When Harold
Jeffreys (1939) advocated a Bayesian position he
was regarded as something of a heretic. The book
by Ed Jaynes (2003), published posthumously, is
a must for anyone interested in these issues.
The Bayesian interpretation holds many
attractions for cosmology. One is that the universe is, by definition, unique. It is not impossible to apply frequentist arguments in this
arena by constructing an ensemble of universes,
but one has to worry about whether such an
ensemble is meant to be real or simply a construct to allow probabilities to become proportions. Bayesians, on the other hand, can deal
with unique entities quite straightforwardly.
Another reason for cosmologists to be interested in Bayesian reasoning is that they are quite
used to the idea that things are modeldependent. For example, one of the important
tasks set during the era of precision cosmology
has been the determination of a host of cosmological parameters that can’t be predicted a
priori in the standard Big Bang model. These
parameters include the expansion rate (Hubble
constant) H0 , the spatial curvature k, the mean
cosmic density in various forms of material
Ωm (cold dark matter), Ων (neutrinos), Ωb
(baryons), and so on. In all, it is possible to infer
joint values of 13 or more such parameters from
measurements of the fluctuations of the cosmic
microwave background. But such parameters
only make sense within the overall model
within which they are defined. This massive
inference problem also demonstrates another
virtue of the Bayesian approach. The result
(posterior probability) of one analysis can be
fed into another as a prior probability.
Probable worlds
As well as its practical implications for data
analysis, a Bayesian approach also leads to useful insights into more esoteric topics; see
Garrett and Coles (1993). Among these is the
Anthropic Principle which is, roughly speaking,
the assertion that there is a connection between
the existence of life in the universe and the fundamental physics that governs large-scale cosmological behaviour. This expression was first
used by Carter (1974) who suggested adding the
word “Anthropic” to the usual Cosmological
Principle to stress that our universe is “special”,
at least to the extent that it has permitted intelligent life to evolve. Barrow and Tipler (1986)
give a complete discussion.
There are many otherwise viable cosmological models that are not compatible with the
observation that human observers exist. For
example, heavy elements such as carbon and
oxygen are vital to the complex chemistry
required for terrestrial life to have developed.
And it takes around 10 billion years of stellar
evolution for generations of stars to synthesize
significant quantities of these elements from the
primordial gas of hydrogen and helium that
exists in the early stages of a Big Bang model.
Therefore, we could not inhabit a universe
younger than about 10 billion years old. Since
the size of the universe is related to its age if it
is expanding, this line of reasoning, originallly
due to Dicke (1961) sheds some light on the
question of why the universe is as big as it is. It
has to be big, because it has to be old if there
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has been time for us to evolve within it. Should
we be surprised that the universe is so big? No,
its size is not improbable if one takes into
account the conditioning information.
This form of reasoning is usually called the
Weak Anthropic Principle (WAP), and is essentially a corrective to the Copernican Principle
that we do not inhabit a special place in the universe. According to WAP, we should remember
that we can only inhabit those parts of spacetime compatible with human life, away from the
centre of a massive black hole, for example. By
the argument given above, we just could not
exist at a much earlier epoch than we do. This
kind of argument is relatively uncontroversial,
and can lead to useful insights. From a Bayesian
perspective, what WAP means is that our existence in the universe in itself contains information about the universe and it must be included
when conditional probabilities are formed.
One example of a useful insight gleaned in this
way relates to the Dirac Cosmology. Dirac
(1937) was perplexed by a number of apparent
coincidences between large dimensionless ratios
of physical constants. He found no way to
explain these coincidences using standard theories, so he decided that they had to be a consequence of a deep underlying principle. He
therefore constructed an entire theoretical edifice
of time-varying fundamental constants on the socalled Large Number Hypothesis. The simple
argument by Dicke outlined above, however, dispenses with the need to explain these coincidences in this way. For example, the ratio
between the present size of the cosmological
horizon and the radius of an electron is roughly
the same as the ratio between the strengths of the
gravitational and electromagnetic forces binding
protons and electrons. (Both ratios are huge: of
order 1040). This does indeed seem like a coincidence, but remember that the size of the horizon
depends on the time: it gets bigger as time goes
on. And the lifetime of a star is determined by
the interplay between electromagnetic and gravitational effects. It turns out that both these ratios
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bil
3: The 2dFGRS cone
showing more than
200 000 galaxies with
recession velocities
up to 100 000 km/s,
selected from the
APM survey. The CfA
redshift survey, state
of the art in 1986,
contained only two
thousand galaxies
with top recession
velocity 15 000 km/s.
The 2dFGRS is the
closest to a fair
sample of the galaxy
distribution that has
ever been produced.
23h
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reduce to the same value precisely because they
both depend on the lifetime of stellar evolution:
the former through our existence as observers,
and the latter through the fundamental physics
describing the structure of a star. The “coincidence” is therefore not as improbable as one
might have thought when one takes into account
all the available conditioning information.
Some cosmologists, however, have sought to
extend the Anthropic Principle into deeper
waters. While the weak version applies to physical properties of our universe such as its age,
density or temperature, the “Strong” Anthropic
Principle (SAP) is an argument about the laws
of physics according to which these properties
evolve. It appears that these fundamental laws
are finely tuned to permit complex chemistry,
which, in turn, permits the development of biology and ultimately human life. If the laws of
electromagnetism and nuclear physics were only
slightly different, chemistry and biology would
be impossible. On the face of it, the fact that the
laws of Nature do appear to be tuned in this
way seems to be a coincidence, in that there is
nothing in our present understanding of fundamental physics that requires the laws to be conducive to life in this way. This is therefore
something we should seek to explain.
In some versions of the SAP, the reasoning is
essentially teleological (i.e. an argument from
design): the laws of physics are as they are
because they must be like that for life to develop.
This is tantamount to requiring that the existence of life is itself a law of Nature, and the
more familiar laws of physics are subordinate to
it. This kind of reasoning appeals to some with
a religious frame of mind but its status among
scientists is rightly controversial, as it suggests
that the universe was designed specifically in
order to accommodate human life. An alternative construction of the SAP involves the idea
that our universe may consist of an ensemble of
mini-universes, each one having different laws of
physics to the others. We can only have evolved
in one of the mini-universes compatible with the
development of organic chemistry and biology,
so we should not be surprised to be in one where
the underlying laws of physics appear to have
special properties. This provides some kind of
explanation for the apparently surprising properties of the laws of Nature mentioned above.
This latter form of the SAP is not an argument
from design, since the laws of physics could
vary haphazardly from mini-universe to miniuniverse, and is in some aspects logically similar to the WAP, except with a frequentist slant.
Fingerprinting the universe
I hope I have demonstrated that the relationship
between statistics and cosmology is an intimate
and longstanding one. This does not mean that
the modern era does not pose new challenges,
some of them resulting from the sheer scale of the
data flows being generated. To understand some
of these new problems it is perhaps helpful to step
back about 20 years and look at the sort of methods that could be applied to cosmological data
and the sort of questions that were being asked.
In the 1980s there was no universally accepted
“standard” cosmological model. The curvature
and expansion rate of the universe had not yet
been measured, the average density of matter
was uncertain by a factor of at least 10. Some
basic ideas of the origin of cosmic structure had
been established, such as that it formed by gravitational condensation from small initial fluctuations perhaps generated during an
inflationary epoch in the early universe, but the
form of the initial fluctuation spectrum was
uncertain. The largest redshift surveys covered
a volume less than 100 Mpc deep and contained
on the order of a thousand galaxies.
The only forms of statistical analysis that
could be performed at this time had to be simple in order to cope with the sparseness of the
data. The most commonly used statistical analysis of galaxy clustering observations involved the
two-point correlation (e.g. Peebles 1980). This
measures the statistical tendency for galaxies to
occur in pairs rather than individually. The correlation function, usually written ξ(r), measures
the excess number of pairs of galaxies found at
a separation r compared to how many such pairs
would be found if the distribution of galaxies in
space were completely random. In other words,
ξ(r) measures the excess probability of finding a
galaxy at a distance r from a given galaxy, compared with what one would expect if galaxies
were distributed independently throughout
space. A positive value of ξ(r) indicates that
there are more pairs of galaxies with a separation r; galaxies are then said to be clustered on
the scale r. A negative value indicates that galaxies tend to avoid each other, so they are anticlustered. A completely random distribution,
usually called a Poisson distribution, has ξ(r) = 0
for all r. Estimates of the correlation function
of galaxies indicate that ξ(r) is a power-law
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5: The CMB angular power
spectrum for WMAP, COBE,
ACBAR and CBI data. Note
that the WMAP results are
consistent with the other
experiments and that the
combined WMAP and CBI
points constrain the angular
power spectrum of our sky
with great accuracy and
precision. The agreement
between the measured
spectrum and the model
shown not only suggests
the basic cosmological
framework is correct, but
also allows parameters of
the model to be estimated
with unprecedented
accuracy. The polarization
spectrum shown below is
less certain, but does show
a perhaps surprising peak
at very low spherical
harmonics usually attributed
to the astrophysical effect
of cosmic reionization.
(WMAP Science Team
images.)
1
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WMAP
CBI
ACBAR
4000
3000
2000
1000
0
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(l+1) Cl /2π(µK2)
4: The 2dFGRS
power-spectrum.
Note the small error
bars resulting fron
the large sample
size. There is some
suggestion of
“wiggles” in the
spectrum that may
be the remnants of
the acoustic peaks
in the WMAP angular
spectrum (figure 5).
The power spectrum
is a useful diagnostic
of clustering pattern;
a complete
statistical description of the filaments
and voids in figure 3
will require more
sophisticated
measures. (2dFGRS
Team/Will Percival.)
power spectrum ∆2(k)
function of the form ξ(r) ≈ (r/r0)–1.8, where the
constant r0 is usually called the correlation
length. The value of r0 depends on the type of
galaxy used, but is around 5 Mpc for bright
galaxies. On larger scales than this, however, ξ(r)
becomes negative, indicating the presence of
large voids. The correlation function is mathematically related to the power spectrum P(k) by
a Fourier transformation; the power-spectrum is
in many ways a better descriptor of clustering on
large-scales than the correlation function and
has played a central role in the analysis of the
2dFGRS I referred to earlier. Since they form a
transform pair, ξ(r) and P(k) contain the same
information about clustering pattern. A slightly
different version of the power-spectrum not
based on Fourier modes but on spherical harmonics is the principal tool for the analysis of
CMB fluctuations especially for the purpose of
extracting cosmological parameters.
The correlation function and power-spectrum
are examples of what are called second-order
clustering statistics (the first-order statistic is
simply the mean density). Their success in pinning down cosmological parameters relies on the
fact that according to standard cosmological
models, the initial fluctuations giving rise to
large-scale are Gaussian in character, meaning
that second-order quantities furnish a complete
description of their spatial statistics. The
microwave background provides a snapshot of
these initial fluctuations, so the spherical harmonic spectrum is essentially a complete
description of the statistics of the primordial universe. The universe of galaxies is not quite such
a pristine reflection of the initial cosmic fluctuations, but the power-spectrum is such a simple
and robust measure of clumping that one can
correct for nonlinear evolution, bias and distortions produced by peculiar motions in a relatively straightforward manner, at least on large
scales (Percival et al. 2001). This is why secondorder statistics have been so fruitful in the analysis of the latest surveys and in establishing the
basic parameters of the standard cosmology.
One has to bear in mind, however, that
second-order statistics are relatively blunt instruments. There is much more information in
2dFGRS and WMAP than is revealed by their
power-spectra, but this information is not so
much about cosmological parameters as about
deviations from the standard model. The broadbrush description afforded by a handful of numbers will have to be replaced by measurements
of the fine details left by complex astrophysical
processes involved in structure formation and
possible signatures of unknown physics. The
strength of the second-order statistics is that they
are insensitive to these properties, so their detection will require higher-order approaches tuned
to probe particular departures from the standard
model. Does the CMB display primordial nonGaussianity (Komatsu et al. 2003), for example?
reionization
TE cross power
spectrum
2
1
0
–1
0
What can one learn about galaxy formation
from the clustering of galaxies of different types
in different environments (Madgwick et al.
2003)? The recent stunning discoveries are not
the end of the line for statistical cosmology, but
are signposts pointing in new directions. ●
Peter Coles is Professor of Astrophysics at the
University of Nottingham.
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