- Cosmology relies heavily on statistics and probability to analyze astronomical data and test theories of the universe.
- Bayesian probability provides a rigorous way to assign probabilities to hypotheses based on prior knowledge and new data, and update beliefs.
- The universe appears "fine tuned" for life with parameter values that allow complexity; Bayesian reasoning can help assess if these are surprising coincidences.
- The concordance model of cosmology posits that initial fluctuations in the early universe formed a Gaussian random field, but some anomalies in cosmic microwave background data could indicate "weirdness" beyond this simple picture.
21. Precision Cosmology
“…as we know, there are known knowns;
there are things we know we know. We also
know there are known unknowns; that is to
say we know there are some things we do not
know. But there are also unknown unknowns
-- the ones we don't know we don't know.”
25. Fine Tuning
• In the standard model of cosmology the
free parameters are fixed by observations
• But are these values surprising?
• Even microscopic physics seems to have
“unnecessary” features that allow
complexity to arise
• Are these coincidences? Are they
significant?
• These are matters of probability…
26. What is a Probability?
• It’s a number between 0 (impossible) and 1
(certain)
• Probabilities can be manipulated using simple
rules (“sum” for OR and “product” for “AND”).
• But what do they mean?
• Standard interpretation is frequentist (proportions
in an ensemble)
27. Bayesian Probability
• Probability is a measure of the “strength of
belief” that it is reasonable to hold.
• It is the unique way to generalize
deductive logic (Boolean Algebra)
• Represents insufficiency of knowledge to
make a statement with certainty
• All probabilities are conditional on stated
assumptions or known facts, e.g. P(A|B)
• Often called “subjective”, but at least the
subjectivity is on the table!
28. Balls
• Two urns A and B.
• A has 999 white balls and 1 black one; B
has 1 white balls and 999 black ones.
• P(white| urn A) = .999, etc.
• Now shuffle the two urns, and pull out a
ball from one of them. Suppose it is white.
What is the probability it came from urn
A?
• P(Urn A| white) requires “inverse”
reasoning: Bayes’ Theorem
29. Urn A Urn B
999 white
1 black
999 black
1 white
P(white ball | urn is A)=0.999, etc
30. Bayes’ Theorem: Inverse
reasoning
• Rev. Thomas Bayes
(1702-1761)
• Never published
any mathematical
papers during his
lifetime
• The general form
of Bayes’ theorem
was actually given
later (by Laplace).
31. Bayes’ Theorem
• In the toy example, X is “the urn is A” and Y is
“the ball is white”.
• Everything is calculable, and the required
posterior probability is 0.999
I)|P(Y
I)X,|I)P(Y|P(X
=I)Y,|P(X
32. Probable Theories
I)|P(D
I)H,|I)P(D|P(H
=I)D,|P(H
• Bayes’ Theorem allows us to assign probabilities
to hypotheses (H) based on (assumed)
knowledge (I), which can be updated when data
(D) become available
• P(D|H,I) – likelihood
• P(H|I) – prior probability
• P(H|D,I) – posterior probability
• The best theory is the most probable!
33. Why does this help?
• Rigorous Form of Ockham’s Razor: the hypothesis
with fewest free parameters becomes most
probable.
• Can be applied to one-off events (e.g. Big Bang)
• It’s mathematically consistent!
• It can even make sense of the Anthropic
Principle…
34. Null Hypotheses
• The frequentist approach to statistical
hypothesis testing involves the idea of a
null hypothesis H0,which is the model
you are prepared to accept unless there
is evidence to the contrary.
• Under the null hypothesis one then
constructs the sampling distribution of
some statistic Q, called f(Q).
• If the measured value of Q is unlikely
on the basis of H0 then the null
hypothesis is rejected.
35. Type I and Type II Errors
• There are two ways of making an error in this
kind of test.
• Type I is to reject the null when it is actually
true. The probability of this happening is called
the significance level (or p-value or “size”),
usually called α. It is usually chosen to be 5%
or 1%.
• The other possibility is to fail to reject the null
when it is wrong. If the probability of this
happening is β then (1-β) is called the power.
36. Bayesian Hypothesis Testing
Two of the advantages of this is that it
doesn’t put one hypothesis in a special
position (the null), and it doesn’t
separate estimation and testing.
Suppose Dr A has a theory that makes a
direct prediction while Professor B has
one that has a free parameter, say λ.
Suppose the likelihoods for a given set of
data are P(D|A) and P(D|B,λ)
39. Is there anything wrong with
Frequentism?
• The laws for manipulating probabilities are
no different
• What is different is the interpretation.
• OK to imagine an ensemble, but there is
no need to assert that it is real! (mind
projection fallacy)
• The idea of a prior is worrying for many,
but is the only way to make this reasoning
consistent
40. Prior and Prejudice
• Priors are essential.
• You usually know more than you
think..
• Flat priors usually don’t make much
sense.
• Maximum entropy, etc, give useful
insights within a well-defined theory:
“objective Bayesian”
• “Theory” priors are hard to assign,
especially when there isn’t a theory…
41. Why is the Universe
(nearly) flat?
• Assume the
Universe is one of
the Friedman
family
• Q: What should we
expect, given only
this assumption?
• Ω=1 is a fixed
point (so is Ω=0)..
• The Universe is
walking a
tightrope..
42. ˙a
2
=
8πGρ
3
a
2
−kc
2
The Friedman Models
The simplest relativistic cosmological models are
remarkably similar (although the more general
ones have additional options…)
¨a=−
4πGρ
3
a
Solutions of these are complicated, except when
k=0 (flat Universe). This special case is called
the Einstein de Sitter universe.
Notice that
ρ ∝
1
a3
For non-relativistic
particles (“dust”)
Curvature
45. The Cosmic Tightrope
• We know the Universe doesn’t have either
a very large or a very small one, or we
wouldn’t be around.
• We exist and this fact is an observation
about the Universe
• The most probable value of is therefore
very close to unity
• Still leaves the mystery of what trained
the Universe to walk the tightrope
(inflation?)
54. Cosmology is an exercise in data compression
Cosmology is a massive
exercise in data
compression...
….but it is worth looking at
the information that has
been thrown away to check
that it makes sense!
56. How Weird is the Universe?
• The (zero-th order) starting point is
FLRW.
• The concordance cosmology is a “first-
order” perturbation to this
• In it (and other “first-order” models),
the initial fluctuations were a
statistically homogeneous and isotropic
Gaussian Random Field (GRF)
• These are the “maximum entropy”
initial conditions having “random
phases” motivated by inflation.
• Anything else would be weird….
60. Types of CMB Anomalies
• Type I – obvious problems with data
(e.g. foregrounds)
• Type II – anisotropies (North-South,
Axis of Evil..)
• Type III – localized features, e.g. “The
Cold Spot”
• Type IV – Something else (even/odd
multipoles, magnetic fields, ?)
65. Weirdness in Phases
ΔT (θ,φ )
T
=∑∑ al,m Ylm(θ,φ)
| | [ ]ml,ml,ml, ia=a φexp
For a homogeneous and isotropic Gaussian
random field (on the sphere) the phases are
independent and uniformly distributed. Non-
random phases therefore indicate weirdness..