adventures with the one-point distribution function

Adventures with the One-Point Distribution

Peter Coles

Castiglioncello, Italy1st September 2015

Lecture 1Probability

“The Essence of Cosmology is Statistics”

George McVittie

1 May 2023

SAY “PRECISION COSMOLOGY” ONE MORE TIME…

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.”

Questionable Aspects of the Standard Cosmology

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

Ingredients of the Standard Cosmology

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

1 May 2023

Direct versus Inverse Reasoning

Theory (, H0…)

Observations

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

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

Beware the Prosecutor’s Fallacy!

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..

Edgeworth Expansion

This is useful for many things, but does not guarantee that the result is a proper probability distribution!

Extreme Value Statistics (exact)

Given the distribution of X, what is the distribution of Xmax?

1

21max

1max

)()()(

)(

)Pr(.....)Pr()Pr()Pr(

).....sup(:}{

n

n

n

ni

zFznfzg

zF

zXzXzXzX

XXXX

Extreme Value Statistics(asymptotic)

For any distribution of exponential type, in the sense that

Then there is a stable asymptotic distribution

0)(

)(1lim

xfxF

dxd

x

n

n

abzzG

xaszX

expexp)(

)Pr( max

PDF

Waizman, Ettori & Moscardini, 2011

arXiv:1105.4099See also: Colombi et al.

2011; Davis et al. 2011 etc

These use alternative parametrisations

Conclusions

• The success of the standard cosmology is very constraining…

• But while we may have less freedom than we did 30 years ago, we do have one thing that we didn’t have then: DATA!

• Might there be things lurking in data we already have?