The Standard Cosmological Model - Vatican … · The Standard Cosmological Model Achievements and...

The Standard Cosmological ModelAchievements and Issues

George Ellis

Department of MathematicsUniversity of Cape Town

May 11, 2017

Commemorating the Legacy of Fr. George LemaıtreSpecola Vaticana, Castel Gandolfo

Ellis (UCT) The Standard Cosmological Model May 11, 2017 1 / 43

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?


In memory of Fr W J Stoeger SJ


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The standard model

Static model (Einstein), taken for granted by all at the time, gives way toa dynamic geometric model (Friedmann, Lemaıtre, 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 physicscausation). But this does not determine topology


The standard model: Geometry

In comoving coordinates, the 4-velocity ua = dxa/dt of preferredfundamental observers is

ua = δa0, uaua = −1 (1)

and the metric tensor is

ds2 = −dt2 + a2(t)dσ2, (2)

where the 3-space metric dσ2 represents a 3-space of constant curvaturek . It can be represented by

dσ2 = dr2 + f 2(r)dΩ2, dΩ2 := dθ2 + sin2 θdφ2, (3)

where f (r) = sin r , r , sinh r if k = +1, 0,−1.


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

a

d2a

dt2= −4πG (ρ+ 3p) + Λ (5)

directly gives the deceleration due to matter and Λ; ρgrav = ρ+ 3p. Itsfirst integral is the Friedmann equation

3H2 = 8πGρ+ Λ− 3k

a2, (6)

where K = (3k/a2) is the curvature of the 3-spaces t = const Theevolution of a(t) is determined by any two, as any two imply the third.


The standard model

A geometric model becoming physical: Friedmann, Lemaıtre, 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 thesign of the spatial curvature. In particular k = +1 implies finite spatialsections and can have a bounce.


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× 1010years Dark energy or curvature

Table 1: The main epochs in the history of the Universe.


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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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Cosmological observationsDiscrete 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


Cosmological observationsBackground 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


Cosmological observationsCosmic 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 gravitationalinstability

Nucleosynthesis implies non-baryonic matter

Top-down effect from T (t) relation at that time, determined by theTolman solution (Friedmann equation with k = 0 and radiation)


The Copernican principlePhilosophy 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 frameimplies FLRW (Ehlers-Geren-Sachs)

Theorem(Mustapha-Hellaby-Ellis)

Can model any background model observations by an inhomogeneous LTBmodel with no Λ-term

Now CP can be observationally tested: good standard candles, timedrift of redshift, kinetic SZ effect: no major inhomogeneity (?)


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 thegeometry, 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)


Figure: (Left) Constraints on the cosmological density parameters obtained from theHubble diagram (direct measures), structure formation studies, and CMB measurementsbefore Planck (contextual). Credit: ESO. (Right) Fractions of the different energycomponents after Planck (plus nucleosnthesis). Credit:https://darkmatterdarkenergy.com/tag/dark-energy/.


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 governinghow universes operate. We have to therefore compare the oneuniverse with simulations of what might have been

Cosmic variance

Do variations from our model need explaining, or do we just accept themas statistical variation? e.g. CBR anomalies?


Particle and Visual Horizons

Largest scales: we can’t see beyond the horizon ...

Causal horizon: particle horizon (furthest matter we can haveinteracted with)

Rindler (1957): Concepts; Penrose (1963): conformal diagrams

Observational Horizon: Visual Horizon (furthest matter we can havereceived radiation from)

The furthest out matter visible by EM radiation is that we see on the LastScattering 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


Visual Horizons

Figure: Our past light cone C−(O) and visual horizon, based on the lastscattering surface Σt when the universe became transparent]. The particlehorizon is not visible: it depends on inflation (number of e-foldings).

.Ellis (UCT) The Standard Cosmological Model May 11, 2017 21 / 43

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.


A small Universe?

Is the universe closed on a scale smaller than the visual horizon? Then wecan 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 verycomputationally 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 ourown galaxy at different times in its history.


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 ingeneral (Weyl)

This is in particular true in the cosmological case: spatiallyhomogeneous, so boundary terms arising when integrating by parts donot necessarily vanish at infinity (MacCallum and Taub)

Maybe one of the innumerable possible spatial topologies will giveextra boundary terms in the field equations that may be of interestand give preferred dynamics


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The links between physics and cosmologyThe Synge Trick

Matter equation of state determines evolution (EFE) which thendetermines evolution of matter (Consn Eqns): Tab ⇒ Rab ⇒ gab ⇒ Tab.But can run the field equations backwards (The Synge Trick) to get anydesired evolution a(t) from some ”matter”: gab ⇒ Rab ⇒ Tab.

Theorem

Can choose V (φ) to get any almost desired result a(t) because KGequation 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 acorresponding field out there, unless we experimentally confirm it. Sameapplies to any hypothesized Lagrangian.


The links between physics and cosmologyThe Physics Horizon

Our ability to experimentally test the physics relevant to cosmology diesaway 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 companionto uncertainty principle for quantum theory (smallest scales)


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Was there a start?Singularity Theorems

Did the universe have a start? (Lemaıtre)

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 abilityto predict to earlier times because it did not apply

Not taken seriously sometimes


Was there a start?

John Wheeler emphasized that existence of spacetime singularities - anedge 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 crisisfor physics:

The existence of spacetime singularities represents an end to theprinciple of sufficient causation and to so the predictabilitygained 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. Noguarantee whatever this is correct.


Was there a start?Energy conditions

But what about scalar fields?

Inflation implies existence of scalar fields that violate the energyconditions: ρ+ 3p < 0

Alleged inflationary singularity theorems - exclude exponential andbouncing models by fiat (alhtough k = 0 de Sitter is indeed singular)

Emergent models (Eddington-Lemaıtre)- but stability

Bouncing models - but inhomogeneity (Penrose)

Quantum gravity and quantum cosmology?

But is an eternal model in fact preferable? What underlies its existenceand its nature? Basic issue unresoived


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 livein 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 verylow energy initial state), which inter alia is required to allow inflation tostart: suppression of gravitational degrees of freedom (Penrose)


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The multiverse and the Anthropic principleDoes 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?


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 whatyou do Thus it does not occur in the real physical universe. As statedeloquently by David Hilbert, while the concept of infinity is needed forvarious mathematical purposes,

”The infinite is nowhere to be found in reality, no matter whatexperiences, 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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From physics to philosophy?

Some workers are claiming we need to change the criteria for what is ascientific 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 auxiliaryhypotheses (Lakatos)

Limits of possible verification in cosmology

Limits of certainty in cosmology

Do we go with non-empirical theory confirmation? (Dawid) orweakened empirical requirements (Susskind, Carroll)?

Retrogressive steps?

Beyond science

Some multiverse suggestions. e.g. simulation hypothesis, are pure fantasy


CERN Researchers Apologize For Destruction Of 5 ParallelUniverses In Recent Experiment

GENEVA—Expressing deep regret over the catastrophic incident thatoccurred within the Large Hadron Collider, officials from the EuropeanOrganization for Nuclear Research, also known as CERN, held a pressconference Monday to apologize for the destruction of five paralleluniverses in a recent experiment.

“We are sorry to report that in conducting research involving high-poweredproton-proton collisions, we inadvertently caused the implosion of fiveuniverses nearly identical to our own,” said CERN Director-General FabiolaGianotti, adding that billions of people worldwide might have experiencedmomentary vertigo around 9:45 a.m. as a result of several of theiralternate identities being wiped from existence.


CERN Researchers Apologize For Destruction Of 5 ParallelUniverses In Recent Experiment

“I’d like to emphasize that there is no need to worry, as we were able tocontain the damage before our own time stream disintegrated into oblivionlike the others. Furthermore, in order to perform an investigation, the LHCwill be shut down for the remainder of the afternoon.”

At press time, a team of CERN researchers in a parallel universe waspreparing to perform the exact same experiment


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 inits 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


Lemaıtre againA real vision

Lemaıtre on The Primeval Atom:

“The evolution of the world can be compared to a display offireworks that has just ended : some few red wisps, ashes andsmoke. Standing on a well-chilled cinder, we see the slow fadingof suns, and we try to recall the vanishing brilliance of the originof the worlds...”

This has been truly vindicated with time.

He was the first real cosmologist.


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 auxiliaryhypotheses (Lakatos)


Figure: Credit: ESO.

The two conflicting Hubbleparameters come from directmeasures on the one hand viaSN1a, and from indirectestimation from structureformation studies via Planckand BAO on the other.Maybe that’s a clue.


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