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Adventures with the One-Point Distribution
Peter Coles
Castiglioncello, Italy1st September 2015
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Lecture 1Probability
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“The Essence of Cosmology is Statistics”
George McVittie
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1 May 2023
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SAY “PRECISION COSMOLOGY” ONE MORE TIME…
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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.”
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Questionable Aspects of the Standard Cosmology
•General Relativity •Cold Dark Matter•Cosmological Constant•Cosmological Principle•Primordial Gaussian fluctuations•Inflation•Baryons•Neutrinos•Radiation…
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Ingredients of the Standard Cosmology
•General Relativity •Cold Dark Matter•Cosmological Constant•Cosmological Principle•Primordial Gaussian fluctuations•Inflation•Baryons•Neutrinos•Radiation…
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1 May 2023
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Direct versus Inverse Reasoning
Theory (, H0…)
Observations
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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!
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“If tortured sufficiently, data will confess to almost
anything”
Fred Menger
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A)!|P(MM)|P(A
Beware the Prosecutor’s Fallacy!
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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..
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Edgeworth Expansion
This is useful for many things, but does not guarantee that the result is a proper probability distribution!
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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
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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
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Waizman, Ettori & Moscardini, 2011
arXiv:1105.4099See also: Colombi et al.
2011; Davis et al. 2011 etc
These use alternative parametrisations
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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?