The Standard Cosmological Model
Achievements and Issues
George Ellis
Department of Mathematics
University of Cape Town
May 11, 2017
Commemorating the Legacy of Fr. George Lemaı̂tre
Specola Vaticana, Castel Gandolfo
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Table of Contents
1
The standard model
2
From philosophy to physics
3
The links between observations and cosmology
4
The links between physics and cosmology
5
Was there a start?
6
The multiverse and the Anthropic principle
7
From physics to philosophy?
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In memory of Fr W J Stoeger SJ
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Table of Contents
1
The standard model
2
From philosophy to physics
3
The links between observations and cosmology
4
The links between physics and cosmology
5
Was there a start?
6
The multiverse and the Anthropic principle
7
From physics to philosophy?
Ellis (UCT)
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The standard model
Static model (Einstein), taken for granted by all at the time, gives way to
a dynamic geometric model (Friedmann, Lemaı̂tre, Robertson, Walker):
RW Metric
Einstein Field Equations ⇒ Friedmann equation
Conservation equations
Matter description: dust, radiation, kinetic theory, scalar field
Perturbed FLRW Metric
Underlying view
Local physics everywhere determines global dynamics (bottom-up physics
causation). But this does not determine topology
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The standard model: Geometry
In comoving coordinates, the 4-velocity u a = dx a /dt of preferred
fundamental observers is
u a = δ0a , u a ua = −1
(1)
ds 2 = −dt 2 + a2 (t)dσ 2 ,
(2)
and the metric tensor is
where the 3-space metric dσ 2 represents a 3-space of constant curvature
k. It can be represented by
dσ 2 = dr 2 + f 2 (r )dΩ2 , dΩ2 := dθ2 + sin2 θdφ2 ,
(3)
where f (r ) = {sin r , r , sinh r } if k = {+1, 0, −1}.
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The standard model: Dynamics
Given an equation of state, the energy conservation equation
dρ/dt + 3H(ρ + p) = 0
(4)
relates the rate of change of the energy density ρ(t) to the pressure p(t);
ρinert = ρ + p. The Raychaudhuri equation
3 d 2a
= −4πG (ρ + 3p) + Λ
a dt 2
(5)
directly gives the deceleration due to matter and Λ; ρgrav = ρ + 3p. Its
first integral is the Friedmann equation
3H 2 = 8πG ρ + Λ −
3k
,
a2
(6)
where K = (3k/a2 ) is the curvature of the 3-spaces {t = const} The
evolution of a(t) is determined by any two, as any two imply the third.
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The standard model
A geometric model becoming physical: Friedmann, Lemaı̂tre, Robertson,
Tolman, Walker, MacVittie, Sandage, Lifschitz, Peebles, Guth
Backround dynamics
Cosmic epochs
Perturbation dynamics
Origin of structure
Inflation
Importance of spatial curvature
The models and their dynamics are completely different depending on the
sign of the spatial curvature. In particular k = +1 implies finite spatial
sections and can have a bounce.
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Epochs
Epoch
Start time
Matter source
Start
Ill defined
Unknown
Inflation
10−35 secs
Scalar field dominated
Hot Big Bang
10−3 secs
Radiation dominated
Visible universe
380, 000 years
Dark Matter dominated
Accelerating Universe
10 Gyrs
Dark energy dominated
Future
13.87 × 1010 years
Dark energy or curvature
Table 1: The main epochs in the history of the Universe.
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Table of Contents
1
The standard model
2
From philosophy to physics
3
The links between observations and cosmology
4
The links between physics and cosmology
5
Was there a start?
6
The multiverse and the Anthropic principle
7
From physics to philosophy?
Ellis (UCT)
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From philosophy to physics
Was once just regarded as philosophy (Rutherford)
Series of Unifications
Physics and cosmology
Gravity: apple, moon, and cosmos
Atomic physics, statistical physics, quantum theory: black body CMB
Nuclear Physics: nucleosynthesis
Particle physics: inflation??
Quantum Gravity: start of universe ??? (“Primordial atom”)
Transition from philosophy to physics
Included in the annual Particle Physics Data Group Report
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Table of Contents
1
The standard model
2
From philosophy to physics
3
The links between observations and cosmology
4
The links between physics and cosmology
5
Was there a start?
6
The multiverse and the Anthropic principle
7
From physics to philosophy?
Ellis (UCT)
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Cosmological observations
Discrete sources
Redshift 1 + z = a(tobs )/a(temit ),
Lookback time
Angular diameter distance r0 (z)
Luminosity distance (Reciprocity Theorem: DL = r0 (1 + z)).
Number counts
Two point Correlation functions
Power spectra, BAO
Weak and strong gravitational lensing
The need for standard candles
Supernovae to the rescue
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Cosmological observations
Background Radiation
CMB (Gamow, Alpher, Herman, Peebles, Sachs and Wolfe).
Relic CMB radiation: Planck power spectrum (Reciprocity Theorem)
Relic CMB radiation: angular power spectrum
Relic CMB radiation: polarisation spectrum
SZ effect
But also all other wavelengths
Optical background
Radio Background
X-ray and γ-ray background
Related to sources and thermal history
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Cosmological observations
Cosmic relics
Remnants from near our past worldline
Elements from Nucleosynthesis
Baryosynthesis ??
Non-unified forces !
Classical not quantum state !!
Galaxies from quantum perturbations plus BAO plus gravitational
instability
Nucleosynthesis implies non-baryonic matter
Top-down effect from T (t) relation at that time, determined by the
Tolman solution (Friedmann equation with k = 0 and radiation)
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The Copernican principle
Philosophy to science
Is the universe in fact spatially homogeneous?
We can only see on one light cone from one point
Copernican Principle was a philosophical assumption
Spherical symmetry everywhere implies spatial homgeneity (Walker)
Spherical symmetry of free CMB in an expanding geodesic frame
implies FLRW (Ehlers-Geren-Sachs)
Theorem(Mustapha-Hellaby-Ellis)
Can model any background model observations by an inhomogeneous LTB
model with no Λ-term
Now CP can be observationally tested: good standard candles, time
drift of redshift, kinetic SZ effect: no major inhomogeneity (?)
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The standard model (other speakers)
Consensus model, converging lines of evidence
Cosmic parameters (other talks)
The best limits
The best limits on these parameters come not from direct testing of the
geometry, but from observations of structure formation outcomes: that is,
the top-down effect of cosmological parameters on local physics
Issues and tensions:
Inflaton?
Dark Matter?
Dark Energy?
Dark Halos??
Hubble constant??? (near and far values)
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Figure: (Left) Constraints on the cosmological density parameters obtained from the
Hubble diagram (direct measures), structure formation studies, and CMB measurements
before Planck (contextual). Credit: ESO. (Right) Fractions of the different energy
components after Planck (plus nucleosnthesis). Credit:
https://darkmatterdarkenergy.com/tag/dark-energy/.
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The uniqueness of the universe
There is only one universe
There is only one universe we see from one spacetime point
We only can observe our universe on one past light cone
We have to deduce 4-d spacetime structure from a 2-d image.
Distance estimations are therefore key
We can’t see many copies of the universe to deduce laws governing
how universes operate. We have to therefore compare the one
universe with simulations of what might have been
Cosmic variance
Do variations from our model need explaining, or do we just accept them
as statistical variation? e.g. CBR anomalies?
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Particle and Visual Horizons
Largest scales: we can’t see beyond the horizon ...
Causal horizon: particle horizon (furthest matter we can have
interacted with)
Rindler (1957): Concepts; Penrose (1963): conformal diagrams
Observational Horizon: Visual Horizon (furthest matter we can have
received radiation from)
The furthest out matter visible by EM radiation is that we see on the Last
Scattering Surface (COBE, WMAP, Planck, Bicep2): the visual horizon
(Ellis and Stoeger) lies inside particle horizon, unaffected by inflation
Absolute limits on observational verification
Different for different messengers
Will be larger in the future but we won’t be there
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Visual Horizons
Figure: Our past light cone C − (O) and visual horizon, based on the last
scattering surface Σt when the universe became transparent]. The particle
horizon is not visible: it depends on inflation (number of e-foldings).
.
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The furthest we can see
Figure: The particles that make up our visual horizon (WMAP, PLANCK)
Cant see any particles that exist further out by any form of EM radiation.
Similar horizons exist for neutrinos and gravitational waves.
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A small Universe?
Is the universe closed on a scale smaller than the visual horizon? Then we
can see round it since decoupling: multiple images of galaxies in the sky
Many possible spatial topologies for k = 0, k = +1, k = −1
Can test by direct observations but difficult
Can test by number counts but not easy
Can test by identical rings in CMB sky: very good test but very
computationally intensive
Small Universe
We probably don’t exist in small universe but case is not entirely closed
If we did it would be only case we could see all matter in the universe,
could actually predict the future from visible initial data, and see our
own galaxy at different times in its history.
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Fixing the spatial topology
What fixes the spatial topology?
Not given by the Einstein field equations per se
Maybe, via variational principle (non-local)
Making a variation and imposing symmetries do not commute in
general (Weyl)
This is in particular true in the cosmological case: spatially
homogeneous, so boundary terms arising when integrating by parts do
not necessarily vanish at infinity (MacCallum and Taub)
Maybe one of the innumerable possible spatial topologies will give
extra boundary terms in the field equations that may be of interest
and give preferred dynamics
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Table of Contents
1
The standard model
2
From philosophy to physics
3
The links between observations and cosmology
4
The links between physics and cosmology
5
Was there a start?
6
The multiverse and the Anthropic principle
7
From physics to philosophy?
Ellis (UCT)
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The links between physics and cosmology
The Synge Trick
Matter equation of state determines evolution (EFE) which then
determines evolution of matter (Consn Eqns): Tab ⇒ Rab ⇒ gab ⇒ Tab .
But can run the field equations backwards (The Synge Trick) to get any
desired evolution a(t) from some ”matter”: gab ⇒ Rab ⇒ Tab .
Theorem
Can choose V (φ) to get any almost desired result a(t) because KG
equation and conservation eqns imply each other (Madsen and Ellis)
The inflaton
Dark energy. Unphysical if ρ + p < 0.
The fact we can determine such a potential does not guarantee there is a
corresponding field out there, unless we experimentally confirm it. Same
applies to any hypothesized Lagrangian.
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The links between physics and cosmology
The Physics Horizon
Our ability to experimentally test the physics relevant to cosmology dies
away as we try to understand the earlier times
Atomic physics Ok: CMB is black body
Nucleosynthesis Ok
Baryosynthesis not yet
Inflation not except if inflaton is Higgs
Pre-inflation (quantum gravity?) is not
Unknown Physics: The Physics Horizon
We don’t know what the physics of the really early universe is
McCrea: uncertainty principle for cosmology (largest scales) as companion
to uncertainty principle for quantum theory (smallest scales)
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Table of Contents
1
The standard model
2
From philosophy to physics
3
The links between observations and cosmology
4
The links between physics and cosmology
5
Was there a start?
6
The multiverse and the Anthropic principle
7
From physics to philosophy?
Ellis (UCT)
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Was there a start?
Singularity Theorems
Did the universe have a start? (Lemaı̂tre)
FLRW singularities if ρ + 3p < 0, Λ = 0
Raychaudhuri equation and anisotropies: shear makes no difference
Hawking-Penrose singularity Theorems: refocussing of past light cone,
plus energy conditions implies geodesic incompleteness
Existence of CMB is sufficient to give the required refocussing
(Hawking-Ellis)
A crisis for physics (Wheeler)
A start to space, time, matter/fields, physics itself. Physics loses its ability
to predict to earlier times because it did not apply
Not taken seriously sometimes
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Was there a start?
John Wheeler emphasized that existence of spacetime singularities - an
edge to spacetime, where not just space, time, and matter cease to exist,
but even the laws of physics themselves no longer apply - is a major crisis
for physics:
The existence of spacetime singularities represents an end to the
principle of sufficient causation and to so the predictability
gained by science. How could physics lead to a violation of itself
– to no physics?
It is unclear how to deal with this in physics or in philosophical terms.
We can proceed by extrapolation from the known to the unknown. No
guarantee whatever this is correct.
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Was there a start?
Energy conditions
But what about scalar fields?
Inflation implies existence of scalar fields that violate the energy
conditions: ρ + 3p < 0
Alleged inflationary singularity theorems - exclude exponential and
bouncing models by fiat (alhtough k = 0 de Sitter is indeed singular)
Emergent models (Eddington-Lemaı̂tre)- but stability
Bouncing models - but inhomogeneity (Penrose)
Quantum gravity and quantum cosmology?
But is an eternal model in fact preferable? What underlies its existence
and its nature? Basic issue unresoived
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The start had to be very special
Inflation: motivation and achievements (Linde, Penrose)
Flatness, Monopole, Homogeneity
Almost scale-free quantum seeds of structure
but
calculations almost always in almost-FLRW models
no unique physically well motivated inflaton
the key issue of how classical perturbations arise is unsolved. We live
in a unique universe with a specific classical outcome.
Top-down effects in cosmology
The local arrow of time is a consequence of global initial conditions (a very
low energy initial state), which inter alia is required to allow inflation to
start: suppression of gravitational degrees of freedom (Penrose)
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Table of Contents
1
The standard model
2
From philosophy to physics
3
The links between observations and cosmology
4
The links between physics and cosmology
5
Was there a start?
6
The multiverse and the Anthropic principle
7
From physics to philosophy?
Ellis (UCT)
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The multiverse and the Anthropic principle
Does a multiverse exist?
Multiverses: the 9 claimed types (Brian Greene, Tegmark)
Implied by physics? - Eternal chaotic inflation. But physics not tested
Anthropic explanation? - The issue of vacuum energy
But unimodular gravity can also solve it
Claimed measures a problematic issue: infinities, non-uniqueness,
cannot be tested
Observable predictions: issue of underdetermination
Claimed conflation of Everett with other forms?
Are any of them scientifically proven outcomes???
It is good hypothesis making - but where is the confirmation?
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Claimed infinities
Particularly problematic are claimed infinities. Infinity is not a big number.
It is a quantity that is unattainable no matter how long you wait or what
you do Thus it does not occur in the real physical universe. As stated
eloquently by David Hilbert, while the concept of infinity is needed for
various mathematical purposes,
”The infinite is nowhere to be found in reality, no matter what
experiences, observations, and knowledge are appealed to”.
This is in stark contrast to the various claims it exists in physical reality.
Claimed infinities in cosmology are not science
There is no way to prove they exist. The claim is completely untestable.
They may be what the universe tends to in the far distant future.
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Table of Contents
1
The standard model
2
From philosophy to physics
3
The links between observations and cosmology
4
The links between physics and cosmology
5
Was there a start?
6
The multiverse and the Anthropic principle
7
From physics to philosophy?
Ellis (UCT)
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From physics to philosophy?
Some workers are claiming we need to change the criteria for what is a
scientific theory because of this problem of proof as regards multiverses
(and similar for string theory): our theory is so good you have to believe it.
Criteria for a scientific theory?
Test theories observationally and recognise role of auxiliary
hypotheses (Lakatos)
Limits of possible verification in cosmology
Limits of certainty in cosmology
Do we go with non-empirical theory confirmation? (Dawid) or
weakened empirical requirements (Susskind, Carroll)?
Retrogressive steps?
Beyond science
Some multiverse suggestions. e.g. simulation hypothesis, are pure fantasy
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CERN Researchers Apologize For Destruction Of 5 Parallel
Universes In Recent Experiment
GENEVA—Expressing deep regret over the catastrophic incident that
occurred within the Large Hadron Collider, officials from the European
Organization for Nuclear Research, also known as CERN, held a press
conference Monday to apologize for the destruction of five parallel
universes in a recent experiment.
“We are sorry to report that in conducting research involving high-powered
proton-proton collisions, we inadvertently caused the implosion of five
universes nearly identical to our own,” said CERN Director-General Fabiola
Gianotti, adding that billions of people worldwide might have experienced
momentary vertigo around 9:45 a.m. as a result of several of their
alternate identities being wiped from existence.
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CERN Researchers Apologize For Destruction Of 5 Parallel
Universes In Recent Experiment
“I’d like to emphasize that there is no need to worry, as we were able to
contain the damage before our own time stream disintegrated into oblivion
like the others. Furthermore, in order to perform an investigation, the LHC
will be shut down for the remainder of the afternoon.”
At press time, a team of CERN researchers in a parallel universe was
preparing to perform the exact same experiment
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Closing notes
A solid model
The standard model is well established from the Hot Big Bang era on,
with some puzzles unresolved: issues that are under investigation
How does this relate to life?
Universe provides galaxies made of stars
Heavy elements to create planets
Planets hospitable to life
Laws of physics that allow life to exist
A bio-friendly universe: not just in terms of the laws of phyiscs, but also in
its initial conditions (or its ongoing cyclic state, if that is the case)
The deep issue: possibility spaces for physics and life
The deep issues are philosophical, not scientific
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Lemaı̂tre again
A real vision
Lemaı̂tre on The Primeval Atom:
“The evolution of the world can be compared to a display of
fireworks that has just ended : some few red wisps, ashes and
smoke. Standing on a well-chilled cinder, we see the slow fading
of suns, and we try to recall the vanishing brilliance of the origin
of the worlds...”
This has been truly vindicated with time.
He was the first real cosmologist.
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Issues
Always consider all curvatures
Consider alternative topologies
Avoid infinities
Avoid ρ + p < 0
Running EFE backwards is quasi-physics
Test theories observationally and recognise role of auxiliary
hypotheses (Lakatos)
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The two conflicting Hubble
parameters come from direct
measures on the one hand via
SN1a, and from indirect
estimation from structure
formation studies via Planck
and BAO on the other.
Maybe that’s a clue.
Figure: Credit: ESO.
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