MaxEnt 201613th July, Ghent

Cosmology: A Bayesian Perspective

Peter Coles (@telescoper)

Lecture 1Probability

“The Essence of Cosmology is Statistics”

George McVittie

Direct versus Inverse Reasoning

Theory (, H0…)

Observations

3 May 2023

Picture: R. Trotta

Urn A Urn B

999 white 1 black

999 black 1 white

P(white ball | urn is A)=0.999, etc

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

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(YI)X,|I)P(Y|P(X=I)Y,|P(X

The Expanding Universe

Picture: R. Trotta

3 May 2023

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…

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)

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!

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).

Probable Theories

I)|P(DI)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!

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…

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..

a2=8πGρ

3 a2−kc 2

The Friedman ModelsThe 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∝1a3

For non-relativistic particles (“dust”)

Curvature

Cosmological Parameters

We do not know how to set the initial conditions for the expanding Universe, nor do we know precisely what forms of matter and energy fill the Universe. What we have to do is make models and see if they fit the observations. A “model” is a solution of the Friedmann equation and is usually written in terms of a set of parameters

Km

akcG

aa

kcaaGa

133

833

8

2

2

2

2

22

22

TG

Gravity Stuff

TG ?

?TG

Whatever it is, Dark Energy is a terrible name

for it…

• What is important is not so much the energy, but the pressure…

• Dark Energy has to act like something with negative pressure (or tension)

3 May 2023

Other Observations

• Supernovae (Type Ia)• Large-scale structure measurements (BAO)• Gravitational Lensing (Weak and Strong)• CMB lensing• Peculiar Velocities

fainter

EUCLID

SAY “PRECISION COSMOLOGY” ONE MORE TIME…

Bayesian Hypothesis TestingTwo 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 .Suppose the likelihoods for a given set of data are P(D|A) and P(D|B,)

Occam’s Razor

λ)B,|(DB)|(λdλA)|(D

(B)(A)

λ)B,|(Dλ)(B,dλA)|(D(A)

D)|λ(B,dλD)|(A=

D)|(BD)|(A

PrPrPr

PrPr

PrPrPrPr

PrPr

PrPr

Occam factor

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…

Bayesian estimation

aI)d,aa|xI)p(x|ap(a=K

I),aa|xI)p(x|aKp(a=I),xx|ap(a

mmnm

mnmnm

.......

............

1111

11111

This involves finding the posterior distribution of the parameters given the data and any prior information.

Evidence!

A)!|P(MM)|P(A

Beware the Prosecutor’s Fallacy!

The “Maximally Boring Universe”?

• There are many unanswered theoretical questions!

• So far the questions we’ve asked have been the “easy” ones

• Now that this “boring” stuff is out of the way, cosmology will start to get interesting!

• Because we now have a better idea what to ask!

“CONCORDANCE”

Ingredients of the Standard Cosmology

•General Relativity •Cold Dark Matter•Cosmological Constant•Cosmological Principle•Primordial Gaussian fluctuations•Inflation•Baryons•Neutrinos•Radiation…

Questionable Aspects of the Standard Cosmology

•General Relativity •Cold Dark Matter•Cosmological Constant•Cosmological Principle•Primordial Gaussian fluctuations•Inflation•Baryons•Neutrinos•Radiation…

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!

“If tortured sufficiently, data will confess to almost

anything”

Fred Menger

Theories

Observations

FrequentistBayesian

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.”

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….

Beyond the Power Spectrum

• So far what we have discovered is largely based on second-order statistics…

• This is fine as long as we don’t throw away important clues…

• ..ie if the fluctuations are statistically homogeneous and istropic, and Gaussian..

Weirdness in PhasesΔT (θ,φ )

T=∑∑ a l,mY lm (θ,φ )

ml,ml,ml, ia=a exp

For a homogeneous and isotropic Gaussian random field (on the sphere) the phases are independent and uniformly distributed. Non-random phases therefore indicate weirdness..

The Prosecutor’s Fallacy!

P(A|M)P(M|A)

CMB Anomalies•Type I – obvious problems with data (e.g. foregrounds)

•Type II – anisotropies and alignments (North-South, Axis of Evil..)

•Type III – localized features, e.g. “The Cold Spot”

•Type IV – Something else (even/odd multipoles, magnetic fields, ?)

Low Quadrupole?

Parity Violation?

(from Copi et al. 2005)

(from Hansen et al. 2004)