11

Aula 1/5Aula 1/5PrincípiosPrincípios

Curso de Cosmologia VII ECG, CBPF – Ago/2013Curso de Cosmologia VII ECG, CBPF – Ago/2013

Miguel QuartinMiguel QuartinInstituto de Física, UFRJInstituto de Física, UFRJ

Astrofísica, Relativ. e Cosmologia (ARCOS)Astrofísica, Relativ. e Cosmologia (ARCOS)

22

1. Sobre o Curso1. Sobre o Curso

Material didático:Material didático: ResumoResumo

Meus SlidesMeus Slides Hogg,Hogg, Distance measures in cosmology (astroph/9905116) Distance measures in cosmology (astroph/9905116) Roberto Trotta, Roberto Trotta, Bayes in the Sky, arXiv:0803.4089Bayes in the Sky, arXiv:0803.4089

Bibliografia suplementar nível básico:Bibliografia suplementar nível básico: Barbara Ryden, Barbara Ryden, Intro. to CosmologyIntro. to Cosmology

Bibliografia suplementar nBibliografia suplementar nível avançado:ível avançado: Amendola & Tsujikawa, Amendola & Tsujikawa, Dark EnergyDark Energy Mukhanov, Mukhanov, Physical Foundations of CosmologyPhysical Foundations of Cosmology

33

Sobre o Curso (2)Sobre o Curso (2)

Tópicos abordados:Tópicos abordados: [Aula 1] Fundamentos da Cosmologia[Aula 1] Fundamentos da Cosmologia [Aula 2] Dinâmica, modelos simples e [Aula 2] Dinâmica, modelos simples e ΛΛCDMCDM [Aula 3] Cosmologia Observacional[Aula 3] Cosmologia Observacional [Aula 4] [Aula 4] Estatística BayesianaEstatística Bayesiana [Aula 5] Análise de dados e métodos numéricos[Aula 5] Análise de dados e métodos numéricos

Tópicos não-abordados:Tópicos não-abordados: Cosmologia Relativística, Cosmologia Relativística, Formação de estruturasFormação de estruturas, , InflaçãoInflação,, Nucleossíntese Nucleossíntese primordial, CMB, Matéria Escura, Lentes Gravitacionais primordial, CMB, Matéria Escura, Lentes Gravitacionais etc.etc.

44

Unidades de PlanckUnidades de Planck

Usando as unidades de Planck, tem-se numericamente Usando as unidades de Planck, tem-se numericamente queque

55

Unidades de Planck (2)Unidades de Planck (2)

Usando as unidades de Planck, tem-se numericamente Usando as unidades de Planck, tem-se numericamente queque

Podemos “simplificar” nossas equações omitindo essas Podemos “simplificar” nossas equações omitindo essas constantesconstantes As constantes podem ser re-obtidas por análise As constantes podem ser re-obtidas por análise

dimensionaldimensional Exs:Exs:

66

Grandezas úteisGrandezas úteis

77

2. Fundamental Observations2. Fundamental Observations

Olber's ParadoxOlber's Paradox

Homogeneity and isotropyHomogeneity and isotropy

Hubble's (Lemaître's) LawHubble's (Lemaître's) Law

Cosmologist Particle BookCosmologist Particle Book

The CMBThe CMB

88

Olber's ParadoxOlber's Paradox

Why is the night Why is the night sky dark?sky dark? Infinite and static Infinite and static

universe bright →universe bright →sky!sky!

Solution 1: Solution 1: universe has universe has finite sizefinite size

Solution 2: Solution 2: universe has a universe has a finite agefinite age

99

Olber's Paradox (2)Olber's Paradox (2)

Parenthesis: Luminosity vs. Brightness vs. Intensity vs. FluxParenthesis: Luminosity vs. Brightness vs. Intensity vs. Flux

Luminosity (L)Luminosity (L) = total energy / time ; emitted or received = total energy / time ; emitted or received It is a It is a propertyproperty of the source; does of the source; does notnot depend on distance depend on distance

Intensity (I)Intensity (I) = = BrightnessBrightness = energy/(time = energy/(time xx det. area det. area xx solid ang.) solid ang.) It is a It is a propertyproperty of the source; does of the source; does notnot depend on distance depend on distance

Specific Intensity (ISpecific Intensity (Ivv) = I / (unit frequency)) = I / (unit frequency)

Flux (f)Flux (f) = Luminosity / (4 = Luminosity / (4ππ distance distance22)) depends on distancedepends on distance

1010

Olber's Paradox (3)Olber's Paradox (3)

Let's compute the sky brightnessLet's compute the sky brightness Let Let nn be the average # density be the average # density

of starsof stars Let Let LL be their average be their average

luminosity ( = energy/time)luminosity ( = energy/time)

The intensity differential dJ is:The intensity differential dJ is:

Where did we Where did we go wrong?go wrong?

1111

Olber's Paradox (4)Olber's Paradox (4)

Stars have finite (angular) size they obstruct stars →Stars have finite (angular) size they obstruct stars →behind thembehind them JJ no longer no longer ∞ ;∞ ; instead J ↔ average surface brightness of a star instead J ↔ average surface brightness of a star

L and/or n may depend on distanceL and/or n may depend on distance We would needWe would need

Universe could have finite size and/or ageUniverse could have finite size and/or age Cutoff in the integralCutoff in the integral

Flux might not go down as 1/rFlux might not go down as 1/r22

Due to non-euclidean geometryDue to non-euclidean geometry Due to redshift / expansionDue to redshift / expansion

1212

Homogeneity & IsotropyHomogeneity & Isotropy The universe is* homogeneous and isotropic on large scalesThe universe is* homogeneous and isotropic on large scales

Minimum scale ~ 100 MpcMinimum scale ~ 100 Mpc * – our observations are * – our observations are consistentconsistent with this hypotheses with this hypotheses

Isotropy & homogeneity independentIsotropy & homogeneity independent Isotropy around every point homogeneity→Isotropy around every point homogeneity→

Copernican Principle:Copernican Principle: we don't live in a special we don't live in a special location in the universelocation in the universe

Cosmological Principle:Cosmological Principle: on sufficiently large on sufficiently large scales, the properties of scales, the properties of the Universe are the the Universe are the same for all observerssame for all observers

1313

Homogeneity & Isotropy (2)Homogeneity & Isotropy (2)

Anisotropic & Anisotropic & HomogeneousHomogeneous

Isotropic & Isotropic & InhomogeneousInhomogeneous

Isotropy & homogeneity independent Isotropy around every point homogeneity→

1414

Homogeneity & Isotropy (3)Homogeneity & Isotropy (3)

1515

The Hubble's LawThe Hubble's Law

The Doppler allows us to measure radial velocities with The Doppler allows us to measure radial velocities with high precision;high precision;

Desvio parao azul

Desvio parao vermelho

1616

The Hubble's Law (2)The Hubble's Law (2) Lemaître (and later Hubble)* found out that galaxies are, Lemaître (and later Hubble)* found out that galaxies are,

in average, receding from us;in average, receding from us; The redshift is linear with distanceThe redshift is linear with distance The velocity is approx. also linear with distanceThe velocity is approx. also linear with distance

* Stigler's law of eponymy: "No scientific discovery is * Stigler's law of eponymy: "No scientific discovery is named after its original named after its original discoverer."discoverer."

1717

109 anos-luz 2 x 109 anos-luz

1818

Variação de λ

1919

The Hubble's Law (3)The Hubble's Law (3)

Hubble's Law does not violate the copernican principle!Hubble's Law does not violate the copernican principle! Isotropic and homogeneous expansion produces Hubble's Isotropic and homogeneous expansion produces Hubble's

law for all observerslaw for all observers

2020

The Hubble's Law (4)The Hubble's Law (4)

We can describe such an expansion by a time-dependent We can describe such an expansion by a time-dependent scale factor a(t)scale factor a(t) Inhomogeneity a(t, r)→Inhomogeneity a(t, r)→ Anisotropy (shear) a(→Anisotropy (shear) a(→ t, t, θ, φθ, φ) or {a(t), b(t), c(t)}) or {a(t), b(t), c(t)}

2121

The Hubble's Law (5)The Hubble's Law (5)

If galaxies are receding from us, were we once together?If galaxies are receding from us, were we once together? Simplest first assumption: H(t) = const = HSimplest first assumption: H(t) = const = H00

This implies ALL galaxies were together at the SAME timeThis implies ALL galaxies were together at the SAME time

This is the base of the Big-Bang modelThis is the base of the Big-Bang model The above calculation ignores gravityThe above calculation ignores gravity

Gravity pulls galaxies in and slows expansion with timeGravity pulls galaxies in and slows expansion with time

H(t) > HH(t) > H0 0 in the past t→in the past t→ 00 < 13.8 Gyr < 13.8 Gyr

2222

Cosmologist Particle BookCosmologist Particle Book

4 types of particle are important is cosmology:4 types of particle are important is cosmology: Photons, “baryons” (protons+neutrons+electrons), neutrinos Photons, “baryons” (protons+neutrons+electrons), neutrinos

and dark matterand dark matter e- mass << proton masse- mass << proton mass Neutrinos are almost always “free-streaming”Neutrinos are almost always “free-streaming”

2323

Cosmologist Particle Book (2)Cosmologist Particle Book (2)

In thermal equilibrium the energy density of photons is In thermal equilibrium the energy density of photons is given by the given by the blackbodyblackbody spectrum spectrum

The total energy and The total energy and number density are:number density are:

2424

The The isotropic isotropic CMBCMB

We receive photons in the microwave spectrum from all We receive photons in the microwave spectrum from all directions isotropicallydirections isotropically Blackbody spectrum (the best one we have ever seen)Blackbody spectrum (the best one we have ever seen) Temperature well measured:Temperature well measured:

2525

2626

The isotropic CMB (2)The isotropic CMB (2)

T = 2,73 K

2727

Thermodynamic ConsiderationsThermodynamic Considerations

Blackbody CMB thermal equilibrium→Blackbody CMB thermal equilibrium→ We can use equilibrium thermodynamicsWe can use equilibrium thermodynamics Consider a co-moving (expanding like the universe) volume Consider a co-moving (expanding like the universe) volume

V V α α a(t) a(t)33..

Exerc!

2828

Thermodynamic Considerations (2)Thermodynamic Considerations (2)

2929

The isotropic CMB (3)The isotropic CMB (3)

Big-bang model in one line: universe starts very Big-bang model in one line: universe starts very hothot & & densedense; it expands; expansion makes it cool & empty.; it expands; expansion makes it cool & empty.

The CMB has a redshift z ~ 1100The CMB has a redshift z ~ 1100 We will see that 1 + z = a(tWe will see that 1 + z = a(t00) / a(t) / a(temem))

z ~ 1100 universe ~ 1100 smaller @ CMB epoch→z ~ 1100 universe ~ 1100 smaller @ CMB epoch→ → → T ~ 1100 higher @ CMB epochT ~ 1100 higher @ CMB epoch T(CMB) @ emission ~ 3000 K (near-infrared)T(CMB) @ emission ~ 3000 K (near-infrared)

Simple explanation by the Big-Bang modelSimple explanation by the Big-Bang model

EEmeanmean ~ 2.7 * 3000 / 11000 =~ 0.7 eV H ionization (13.6 eV)↔ ~ 2.7 * 3000 / 11000 =~ 0.7 eV H ionization (13.6 eV)↔

Discovery in the 60's: decisive evidence for that modelDiscovery in the 60's: decisive evidence for that model

3030

– – Fim da Aula 1 –Fim da Aula 1 –

3131

Aula 2/5Aula 2/5Dinâmica e Modelos SimplesDinâmica e Modelos Simples

Curso de Cosmologia VII ECG, CBPF – Ago/2013Curso de Cosmologia VII ECG, CBPF – Ago/2013

Miguel QuartinMiguel QuartinInstituto de Física, UFRJInstituto de Física, UFRJ

Astrofísica, Relativ. e Cosmologia (ARCOS)Astrofísica, Relativ. e Cosmologia (ARCOS)

3232

3. Newton vs. Einstein3. Newton vs. Einstein

Curvature in non-euclidean geometriesCurvature in non-euclidean geometries

The Friedmann-Lemaître-Robertson-Walker (FLRW) The Friedmann-Lemaître-Robertson-Walker (FLRW) metricmetric

Proper distanceProper distance

3333

Line ElementsLine Elements The line element The line element dsds tells one how to calculate the distance tells one how to calculate the distance

(interval) between 2 neighboring points(interval) between 2 neighboring points

3434

Line Elements (2)Line Elements (2)

3535

Line Elements (3)Line Elements (3)

““Circle” equidistant points of a given center→Circle” equidistant points of a given center→ e.g. lines of constant “latitude”e.g. lines of constant “latitude” On a sphere (constant positive curvature):On a sphere (constant positive curvature):

On a hyperboloid (constant negative curvature):On a hyperboloid (constant negative curvature):

3636

Line Elements (4)Line Elements (4)

In 3D we getIn 3D we get

These can be unified as:These can be unified as:

Or, using x = SOr, using x = Skk(r):(r):

3737

The Friedmann-Lemaître-Robertson-The Friedmann-Lemaître-Robertson-Walker (FLRW) metricWalker (FLRW) metric

Special relativity tells us how to compute separations in Special relativity tells us how to compute separations in 4-D space-times4-D space-times The separation The separation

depends on observer (spatial contraction)!depends on observer (spatial contraction)! We want to work with We want to work with invariantinvariant (observer-independent) (observer-independent)

quantities.quantities. c = const →c = const → Special relativity the ↔Special relativity the ↔ MinkowskiMinkowski metric metric

Line element:Line element: non-euclideannon-euclidean photons: ds^2 = 0 geodesics (photons: ds^2 = 0 geodesics (null-geodesicsnull-geodesics); ); matter: ds^2 < 0 geodesics (matter: ds^2 < 0 geodesics (timelike-geodesicstimelike-geodesics); );

3838

The FLRW metric (2)The FLRW metric (2)

We can generalize Minkowski to allow for spatial We can generalize Minkowski to allow for spatial expansion and/or contractionexpansion and/or contraction Assuming Assuming isotropyisotropy & & homogeneityhomogeneity, the most general metric , the most general metric

is the FLRW one, with line element:is the FLRW one, with line element:

The The metricmetric itself is a (rank 2) tensor itself is a (rank 2) tensor gg with components g with components gμνμν

3939

Proper DistanceProper Distance

For simplicity, let's consider instantaneous radial For simplicity, let's consider instantaneous radial separations: ds = a(t) drseparations: ds = a(t) dr The proper distance between two co-moving objects (one at The proper distance between two co-moving objects (one at

the origin) is:the origin) is:

The proper velocity is defined as:The proper velocity is defined as:

4040

Proper Distance (2)Proper Distance (2)

The proper velocity can be larger than The proper velocity can be larger than cc for faraway for faraway galaxiesgalaxies!!

The critical distance is called the Hubble distance dThe critical distance is called the Hubble distance dHH

The proper distance is The proper distance is unobservableunobservable (not in the past lightcone) (not in the past lightcone) we can only observe it we can only observe it indirectlyindirectly (assuming a model) (assuming a model)

However, the scale factor at emission a(tHowever, the scale factor at emission a(temem) ) isis observable observable

4141

Let's see what happens to a photon over large distancesLet's see what happens to a photon over large distances Without loss of generality, let's consider radial null-Without loss of generality, let's consider radial null-

geodesics separations: ds = 0 a(t) dr = →geodesics separations: ds = 0 a(t) dr = → ±± c dt c dt

Define: emission time (Define: emission time (ttee) and reception time () and reception time (tt00))

Proper Distance (3)Proper Distance (3)

We get this:We get this:If we subtract this:If we subtract this:

4242

Proper Distance (4)Proper Distance (4)

The integrand is effectively constant (The integrand is effectively constant (λλ/c ~ 10/c ~ 10-14 -14 s ~ 10s ~ 10-32-32 H H00-1-1))

4343

Measuring the curvatureMeasuring the curvature

In a homogenous & isotropic universe, all information is In a homogenous & isotropic universe, all information is

contained in: contained in: a(t)a(t), , κκ and and RR00 (if κ ≠ 0) (if κ ≠ 0)..

κ = +1 spherical (Riemann) geometry→κ = +1 spherical (Riemann) geometry→ κ = 0 flat geometry→κ = 0 flat geometry→ κ = –1 hyberbolical (Lobachevski) geometry→κ = –1 hyberbolical (Lobachevski) geometry→

Observations the radius of curvature R→Observations the radius of curvature R→ 00 must be much must be much

smaller than the Hubble radius (dsmaller than the Hubble radius (dHH))

Simple estimate if κ = 1: if we had RSimple estimate if κ = 1: if we had R00 <~ d <~ dHH we would see we would see

multiple images of galaxiesmultiple images of galaxies

4444

4545

4. Cosmic Dynamics4. Cosmic Dynamics

Einstein's equationsEinstein's equations The Friedmann equationThe Friedmann equation

Fluid equationsFluid equations

Equations of stateEquations of state

The cosmological constantThe cosmological constant

4646

Einstein's EquationsEinstein's Equations

The fundamental equations of general relativity are The fundamental equations of general relativity are Einstein's equations (10 second order partial diff. Eqs.)Einstein's equations (10 second order partial diff. Eqs.) Tensors are algebraic constructs in vector spaces, Tensors are algebraic constructs in vector spaces,

independent of coordinate systems (observers)independent of coordinate systems (observers) The The metricmetric is a function of the energy-momentum tensor is a function of the energy-momentum tensor TT

(symmetric)

(energy-momentum conservation)

4747

Einstein's Equations (2)Einstein's Equations (2)

It is often convenient to work on abstract-index notationIt is often convenient to work on abstract-index notation Tensors are represented by their coordinates in an Tensors are represented by their coordinates in an

undefined coordinate systemundefined coordinate system Sometimes implicitly assumed to be cartesianSometimes implicitly assumed to be cartesian

In a homogeneous and isotropic universe (FLRW metric), In a homogeneous and isotropic universe (FLRW metric), only 2only 2 of the above equations are non-zero & independent of the above equations are non-zero & independent

4848

Friedmann EquationFriedmann Equation

For a Newtonian derivation, see [Ryden]For a Newtonian derivation, see [Ryden] From Einstein's 00 (time-time) equation we have the so-From Einstein's 00 (time-time) equation we have the so-

called called Friedmann equationFriedmann equation

4949

Friedmann Equation (2)Friedmann Equation (2)

Since energy is positive, for Since energy is positive, for κ = – 1κ = – 1, there is a minimum , there is a minimum radius of curvature for the universeradius of curvature for the universe

For spatially flat universes, there is a critical density For spatially flat universes, there is a critical density related to the Hubble parameterrelated to the Hubble parameter

5050

Friedmann Equation (3)Friedmann Equation (3)

The universe has a The universe has a very lowvery low average density average density The critical density is roughly:The critical density is roughly:

1 hydrogen / 200 liters1 hydrogen / 200 liters 140 solar masses / kpc^3140 solar masses / kpc^3

5151

Friedmann Equation (4)Friedmann Equation (4)

In the Einstein Equation, the energy-momentum tensor In the Einstein Equation, the energy-momentum tensor TT is the total energy-momentum tensor, the is the total energy-momentum tensor, the sumsum of different of different TT's for all species (photons, baryons, neutrinos, dark 's for all species (photons, baryons, neutrinos, dark matter, etc.)matter, etc.)

It is convenient to define normalized density parametersIt is convenient to define normalized density parameters

5252

Friedmann Equation (5)Friedmann Equation (5)

The Friedmann equation is thus rewritten asThe Friedmann equation is thus rewritten as

We can also treat curvature as an We can also treat curvature as an effective fluideffective fluid, with , with associated energy density and pressureassociated energy density and pressure

In particular, at present we haveIn particular, at present we have

5353

Fluid EquationsFluid Equations

From Einstein's ij (space-space) equations we have the so-From Einstein's ij (space-space) equations we have the so-called called acceleration equationacceleration equation For a Newtonian derivation, again see [Ryden]For a Newtonian derivation, again see [Ryden]

Friedmann Eq. →

“acceleration equation”

5454

Fluid Equations (2)Fluid Equations (2)

We have derived 2 fundamental equations so farWe have derived 2 fundamental equations so far::

“acceleration equation”

“Friedmann equation”

We can combine them to derive the We can combine them to derive the conservation equationconservation equation (or simply (or simply fluid equationfluid equation))

Problem: we have 2 independent eqs. and 3 variablesProblem: we have 2 independent eqs. and 3 variables We need another equation!We need another equation!

Exerc!

5555

Equations of StateEquations of State

Normal fluids typically have a well-defined Normal fluids typically have a well-defined equation of equation of state (EoS)state (EoS), which relates its pressure with other thermo-, which relates its pressure with other thermo-dynamic quantities, like its energy dynamic quantities, like its energy EE or entropy or entropy SS

The equation of state can be non-linear and very The equation of state can be non-linear and very complicated in general (like in condensed matter)complicated in general (like in condensed matter)

In cosmology, we deal with dilute gases, with very simple In cosmology, we deal with dilute gases, with very simple EoSEoS

5656

Equations of State (2)Equations of State (2)

In cosmology typically very simple EoS→In cosmology typically very simple EoS→

Low density gas ideal gas→Low density gas ideal gas→ Consider an ideal & non-relativistic gasConsider an ideal & non-relativistic gas

Maxwell's velocity distribution →Maxwell's velocity distribution →

5757

Equations of State (3)Equations of State (3)

Fully relativistic matter has instead:Fully relativistic matter has instead:

Mildly relativistic matter is in-betweenMildly relativistic matter is in-between

Curvature has effectively:Curvature has effectively:

As we will see, a Cosmological Constant As we will see, a Cosmological Constant ΛΛ is described by is described by

Exerc!

5858

Equations of State (4)Equations of State (4)

A perturbation in the fluid generates sound wavesA perturbation in the fluid generates sound waves For adiabatic perturbations, the For adiabatic perturbations, the speed of soundspeed of sound is is

Causality w →Causality w → ≤≤ 1 1 w < 0 exponential perturbations→w < 0 exponential perturbations→

In general, In general, dark energy refers to any fluid with refers to any fluid with w < – 1/3

5959

– – Fim da Aula 2 –Fim da Aula 2 –

6060

Aula 3/5Aula 3/5O Modelo O Modelo ΛΛCDM e CDM e

Cosmologia ObservacionalCosmologia Observacional

Curso de Cosmologia VII ECG, CBPF – Ago/2013Curso de Cosmologia VII ECG, CBPF – Ago/2013

Miguel QuartinMiguel QuartinInstituto de Física, UFRJInstituto de Física, UFRJ

Astrofísica, Relativ. e Cosmologia (ARCOS)Astrofísica, Relativ. e Cosmologia (ARCOS)

6161

The Cosmological Constant The Cosmological Constant ΛΛ

Historically, Historically, ΛΛ was introduced by Einstein in 1917 to was introduced by Einstein in 1917 to produce a produce a static universestatic universe This was a fundamentally flawed approach due to This was a fundamentally flawed approach due to

dynamical instabilitydynamical instability Soon after, the expansion of the universe was discoveredSoon after, the expansion of the universe was discovered

In the past 2 decades, In the past 2 decades, ΛΛ was re-introduced with full force was re-introduced with full force as a simple model that explains the current acceleration of as a simple model that explains the current acceleration of the Universethe Universe Λ has a good Λ has a good side effectside effect: it makes the Universe : it makes the Universe olderolder Formally, the Einstein equations becomeFormally, the Einstein equations become

6262

The Cosmological Constant The Cosmological Constant Λ (2)Λ (2)

The modified fluid equations are:The modified fluid equations are:

The Λ terms can be absorbed into The Λ terms can be absorbed into εε & P by & P by identifyingidentifying

“Friedmann equation”

“acceleration equation”

“conservation equation”

6363

What could be behind Λ? Are there good physical What could be behind Λ? Are there good physical candidates with w = – 1?candidates with w = – 1? Answer: YES! vacuum (rest point) energy!→Answer: YES! vacuum (rest point) energy!→ Vacuum energy should Vacuum energy should notnot depend on the expansion of the depend on the expansion of the

universe constant →universe constant → εε Let's compute it from quantum field theoryLet's compute it from quantum field theory

A conservative cutoff at the LHC scale →A conservative cutoff at the LHC scale → See e.g. See e.g. S. WeS. Weinberg's book inberg's book Cosmology, p. 56Cosmology, p. 56

The Cosmological Constant The Cosmological Constant Λ (3)Λ (3)

6464

5. Single-Component Universes5. Single-Component Universes

Evolution of the energy densityEvolution of the energy density

Curvature–dominated universeCurvature–dominated universe

Spatially Flat UniversesSpatially Flat Universes Flat matter–dominated universeFlat matter–dominated universe Flat radiation–dominated universeFlat radiation–dominated universe Flat Flat Λ–Λ–dominated universedominated universe

6565

Evolution of the Energy DensityEvolution of the Energy Density

The total energy density (and pressure) is just the sum of The total energy density (and pressure) is just the sum of the individual energy density (and P) of each speciesthe individual energy density (and P) of each species

The conservation equation holds for each species The conservation equation holds for each species separatedly (neglecting interactions between species)separatedly (neglecting interactions between species)

This can be rewritten as

6666

Evolution of the Energy Density (2)Evolution of the Energy Density (2)

Integrating this equation we can get Integrating this equation we can get εε(a) for any species(a) for any species So far we are allowing So far we are allowing multiplemultiple species simultaneously species simultaneously

Assuming the different wAssuming the different wii to be (different) constants: to be (different) constants:

In particular for In particular for mmatter (w=0) and atter (w=0) and rradiation (w=1/3) we getadiation (w=1/3) we get

6767

6868

Evolution of the Energy Density (3)Evolution of the Energy Density (3)

Why do we use redshift all the time?Why do we use redshift all the time? It is directly measurable!It is directly measurable! a(t) [and thus z(t)] difficult to compute analyticallya(t) [and thus z(t)] difficult to compute analytically

From the Friedmann Equation we get:From the Friedmann Equation we get:

6969

(Spatially) Flat Universes(Spatially) Flat Universes

Observations universe is flat, or nearly flat→Observations universe is flat, or nearly flat→ Flatness should be (at least) a good approximationFlatness should be (at least) a good approximation Assuming flatness, equations are simplerAssuming flatness, equations are simpler

Let's consider a flat universe with one component (with Let's consider a flat universe with one component (with constantconstant EoS parameter EoS parameter ww) dominating over the others) dominating over the others

Exerc!

7070

(Spatially) Flat Universes (2)(Spatially) Flat Universes (2)

The relation z(t) is computed directly from a(t)The relation z(t) is computed directly from a(t)

The proper distance is then:The proper distance is then: The proper distance is then:The proper distance is then:

It is also simply to show that for It is also simply to show that for any wany w (except – 1): (except – 1):

7171

Matter-Dominated CaseMatter-Dominated Case

Let's now consider the caseLet's now consider the case This is called an This is called an Einstein-de-SitterEinstein-de-Sitter universe universe Non-relativistic matter w = 0→Non-relativistic matter w = 0→

Particular case of previous equationsParticular case of previous equations

7272

Radiation-Dominated CaseRadiation-Dominated Case

Let's now consider the caseLet's now consider the case This describes the early universe (pre-CMB): z >> 1000This describes the early universe (pre-CMB): z >> 1000

The mean energy of each photon is (blackbody)The mean energy of each photon is (blackbody)

7373

Radiation-Dominated Case (2)Radiation-Dominated Case (2)

The number density n(t) is then just:The number density n(t) is then just:

So both n(t) and So both n(t) and εε(t) formally diverge for t 0→(t) formally diverge for t 0→

Now for w = 1/3 we have:Now for w = 1/3 we have:

N(t) ~ 1 quantization →effects are crucial

Can only trust GR for t >> tP

7474

(Milne)

7575

6. Multiple-Component Universes6. Multiple-Component Universes

Matter + CurvatureMatter + Curvature

Matter + Matter + ΛΛ

Matter + Curvature + Matter + Curvature + ΛΛ

Radiation + MatterRadiation + Matter

The “standard model” (The “standard model” (ΛΛCDM)CDM)

7676

The Hubble ParameterThe Hubble Parameter

We will now deal with the full Friedmann equationWe will now deal with the full Friedmann equation

Dividing by HDividing by H0022::

Now, we saw thatNow, we saw that So:So:

7777

The Hubble Parameter (2)The Hubble Parameter (2)

In terms of redshift:In terms of redshift:

With the constraintWith the constraint

Multiplying by aMultiplying by a22::

7878

Matter + curvature + Matter + curvature + ΛΛ

This model has rich dynamicsThis model has rich dynamics We can have We can have re-collapsesre-collapses,, Big-Chills Big-Chills, , Big-BouncesBig-Bounces (no Big- (no Big-

Bang) and Bang) and loiteringloitering (almost static) universes (almost static) universes No general analytical solution of Friedmann eq.No general analytical solution of Friedmann eq. It is also a good description of the universe for z << 3600It is also a good description of the universe for z << 3600

It is a more complete It is a more complete a prioria priori model than just matter + model than just matter + ΛΛ Observations tell us flatness is a good approximationObservations tell us flatness is a good approximation

We need to understand the effects of curvature to interpret We need to understand the effects of curvature to interpret the observations in the first place!the observations in the first place!

7979

8080

Exerc!

Compute the boundary lines

between Bounce/Chill

and Chill/Crunch

8181

The The ΛΛCDM ModelCDM Model

Very good description of the universe at all* timesVery good description of the universe at all* times * - after inflation (or for z << 10^10)* - after inflation (or for z << 10^10) Describes correctly all current observations: CMB, Describes correctly all current observations: CMB,

Supernovae, BAO, Big-Bang Nucleosynthesis (BBN), Supernovae, BAO, Big-Bang Nucleosynthesis (BBN), gravitational lensing, etc.gravitational lensing, etc.

Planck numbers (2013):

8282

8383

8484

8585

Chapter 7 Chapter 7 Measuring Cosmological ParametersMeasuring Cosmological Parameters

Cosmological DistancesCosmological Distances Comoving dist.Comoving dist. Luminosity dist.Luminosity dist. Angular-diameter dist.Angular-diameter dist.

Standard CandlesStandard Candles Standard RulersStandard Rulers

Supplement BibliographySupplement Bibliography Hogg - Hogg - Distance measures in cosmology Distance measures in cosmology (astroph/9905116)(astroph/9905116)

8686

Distances in CosmologyDistances in Cosmology

Inside the solar system Laser Ranging→Inside the solar system Laser Ranging→ Shoot a strong laser at a planet and measure the time it Shoot a strong laser at a planet and measure the time it

takes to be reflected back to ustakes to be reflected back to us Inside the galaxy stellar parallax→Inside the galaxy stellar parallax→

Requires precise astrometry.Requires precise astrometry. Maximum distance measured: 500 pc (1600 ly), by the Maximum distance measured: 500 pc (1600 ly), by the

Hipparcos satellite (1989–1993)Hipparcos satellite (1989–1993) 2013 launch of Gaia satellite (2013 – 2019) parallax up → →2013 launch of Gaia satellite (2013 – 2019) parallax up → →

to ~50 kpcto ~50 kpc Compare with:Compare with:

Milky Way ~15 kpc radius→Milky Way ~15 kpc radius→ Andromeda ~1 Mpc→Andromeda ~1 Mpc→

8787

Luminosity DistanceLuminosity Distance

We can We can definedefine the luminosity distance d the luminosity distance dLL by by analogyanalogy with with the euclidean distance given by the measured flux of a the euclidean distance given by the measured flux of a source of known intrinsic luminosity (i.e., a source of known intrinsic luminosity (i.e., a standard candlestandard candle))

In FLRW, the area of a sphere is given byIn FLRW, the area of a sphere is given by

8888

Luminosity Distance (2)Luminosity Distance (2)

Apart from the area distortion due to curvature, Apart from the area distortion due to curvature, expansion introduces a (1+z)expansion introduces a (1+z)22 correction: correction: Expansion Doppler we measure → →Expansion Doppler we measure → → larger wavelengthslarger wavelengths → →

energy drops by 1+z energy drops by 1+z Expansion The →Expansion The → raterate of photons arriving are also smaller of photons arriving are also smaller

than the rate of photons emitted also by 1+zthan the rate of photons emitted also by 1+z

In particular, in flat-spaces we getIn particular, in flat-spaces we get

8989

Angular Diameter DistanceAngular Diameter Distance

For an object of known physical size For an object of known physical size ℓℓ (i.e. a standard (i.e. a standard ruler), the distance is related to its angular size by (for ruler), the distance is related to its angular size by (for small angles)small angles)

ℓdA

9090

9191

9292

Summary of DistancesSummary of Distances

Based on HoggBased on Hogg (astroph/9905116)(astroph/9905116) Big H and small h →Big H and small h → The Hubble Distance:The Hubble Distance:

The auxiliary function E(z):The auxiliary function E(z):

Remember that for radial geodesics:Remember that for radial geodesics:

9393

Summary of Distances (2)Summary of Distances (2)

So we define the So we define the line-of-sight comovingline-of-sight comoving distance as the distance as the distance constant for objects in the Hubble flow:distance constant for objects in the Hubble flow:

AllAll other distances can be defined in terms of d other distances can be defined in terms of dCC

We define the We define the transverse comovingtransverse comoving distance as the distance distance as the distance that when multiplied by that when multiplied by δθδθ gives the comoving distance gives the comoving distance between 2 objects at the same z & separated by between 2 objects at the same z & separated by δθδθ::

9494

Summary of Distances (3)Summary of Distances (3)

The The angular diameterangular diameter distance is given simply by: distance is given simply by:

When discussing gravitational lensing effects, one When discussing gravitational lensing effects, one naturally need to compute dnaturally need to compute dAA between two objects, one at between two objects, one at zz11, the other at z, the other at z22. The d. The dAA's do 's do notnot sum! sum!

E.g.: for E.g.: for κ κ < 0 (< 0 (ΩΩκκ00 > 0), we have > 0), we have

9595

Summary of Distances (4)Summary of Distances (4)

As we have shown, the As we have shown, the luminosity distanceluminosity distance is related to is related to the angular diameter distance in a simple way:the angular diameter distance in a simple way:

The The distance modulusdistance modulus relates d relates dLL with the astronomer's with the astronomer's beloved beloved magnitudemagnitude (negative log) system (negative log) system Ancient greeks stars visible at night were classified in 6 →Ancient greeks stars visible at night were classified in 6 →

different magnitude (m) cathegoriesdifferent magnitude (m) cathegories m=1 the brightest; m=6 the fainter→ →m=1 the brightest; m=6 the fainter→ →

9696

– – Fim da Aula 3/5 –Fim da Aula 3/5 –

9797

Aula 4/5Aula 4/5Supernovas, BAO e Supernovas, BAO e Estatística BayesianaEstatística Bayesiana

Curso de Cosmologia VII ECG, CBPF – Ago/2013Curso de Cosmologia VII ECG, CBPF – Ago/2013

Miguel QuartinMiguel QuartinInstituto de Física, UFRJInstituto de Física, UFRJ

Astrofísica, Relativ. e Cosmologia (ARCOS)Astrofísica, Relativ. e Cosmologia (ARCOS)

http://tinyurl.com/mylycfw

9898

Standard CandlesStandard Candles

In order to use the luminosity distance we need good In order to use the luminosity distance we need good standard candles (known intrinsic L)standard candles (known intrinsic L)

There are 2 classic standard (rigorously, There are 2 classic standard (rigorously, standardiziblestandardizible) ) candles in cosmology:candles in cosmology: Cepheid variable starsCepheid variable stars Type Ia SupernovaeType Ia Supernovae

Both classes have Both classes have intrinsic variabilityintrinsic variability, but there are , but there are empirical relations that allow us to calibrate and empirical relations that allow us to calibrate and standardizestandardize them them

9999

CepheidsCepheids

Cepheid variable stars are very luminous (L ~ 400 – 40000 Cepheid variable stars are very luminous (L ~ 400 – 40000 LLsunsun) stars which oscillate with period T ~ 1 – 60 days) stars which oscillate with period T ~ 1 – 60 days Henrietta Leavitt discovered in the 1910's that there is a Henrietta Leavitt discovered in the 1910's that there is a

strong correlation between T and Lstrong correlation between T and L Longer T higher L↔Longer T higher L↔ By looking at the Magellanic Clouds only, she knew their By looking at the Magellanic Clouds only, she knew their

distance was ~ similardistance was ~ similar Milky Way Cepheids can be calibrated with parallaxMilky Way Cepheids can be calibrated with parallax

Measuring T gives L and dMeasuring T gives L and dLL, up to some scatter, up to some scatter Scatter in distance modulus DM is ~ 0.2 magScatter in distance modulus DM is ~ 0.2 mag arXiv:1103.2976 measurement of 600 Cepheids with →arXiv:1103.2976 measurement of 600 Cepheids with →

Hubble (HST) gives HHubble (HST) gives H00 with to 3% precision with to 3% precision

100100

Type Ia SupernovaeType Ia Supernovae

Supernovae (SNe) are Supernovae (SNe) are very brightvery bright explosions of stars explosions of stars There are 2 major kinds of SNeThere are 2 major kinds of SNe

Core-collapse (massive stars which run out of H and He)Core-collapse (massive stars which run out of H and He) Collapse by mass accretion in binary systems (Collapse by mass accretion in binary systems (type Iatype Ia))

White dwarf + red giant companion (single degenerate)White dwarf + red giant companion (single degenerate) White dwarf + White dwarf (double degenerate)White dwarf + White dwarf (double degenerate) Type Ia SNe explode with a more standard energy releaseType Ia SNe explode with a more standard energy release

Chandrasekar limit on white dwarf mass: MChandrasekar limit on white dwarf mass: Mmaxmax = 1.44 M = 1.44 Msunsun

Beyond this instability explosion→ →Beyond this instability explosion→ → Besides having less intrinsic scatter, it was discovered by Besides having less intrinsic scatter, it was discovered by

Phillips in '93 that there is a strong correlation between the Phillips in '93 that there is a strong correlation between the brightness and duration of a supernovaebrightness and duration of a supernovae

101101

102102

SupernovaeSupernovae

103103

104104

Type Ia Supernovae (2)Type Ia Supernovae (2)

105105

Type Ia Supernovae (3)Type Ia Supernovae (3)

106106

Type Ia Supernovae (2)Type Ia Supernovae (2)

After taking the stretch – luminosity correlation into After taking the stretch – luminosity correlation into account scatter in distance modulus DM ~ 0.2 – 0.3 mag→account scatter in distance modulus DM ~ 0.2 – 0.3 mag→ Current and near-future scatter down to ~ 0.15 mag→Current and near-future scatter down to ~ 0.15 mag→

What is the fundamental limit? 0.12 mag? 0.1 mag?What is the fundamental limit? 0.12 mag? 0.1 mag? Supernovae can be seen Supernovae can be seen very farvery far

Farthest type-Ia supernova yet: Farthest type-Ia supernova yet: z = 1.914z = 1.914 (arXiv:1304.0768) (arXiv:1304.0768) Nearby SNe can be callibrated with Cepheids (1103.2976)Nearby SNe can be callibrated with Cepheids (1103.2976) Allows measurent of dAllows measurent of dLL to high z to high z

Allows constraints on cosmologyAllows constraints on cosmology Allows Allows 2011 Nobel Prize 2011 Nobel Prize

107107

Hubble diagramHubble diagram ddLL(z)(z)

108108

Baryon Acoustic OscillationsBaryon Acoustic Oscillations

109109

Baryon Acoustic Oscillations (2)Baryon Acoustic Oscillations (2)

110110 (dAng

2

dComov

)1/3 (Mpc / h)

Baryon Acoustic Oscillations (3)Baryon Acoustic Oscillations (3)

SDSS (2005)

111111

Baryon Acoustic Oscillations (4)Baryon Acoustic Oscillations (4)

WiggleZ (2011)

(dAng

2

dComov

)1/3 (Mpc / h)

112112

113113

Estatística BayesianaEstatística Bayesiana

[R. Trotta 0803.4089] [R. Trotta 0803.4089] [Amendola & Tsujikawa cap. 13][Amendola & Tsujikawa cap. 13]

114114

TopicsTopics

Short review of probability theoryShort review of probability theory

Bayes' TheoremBayes' Theorem

The Likelihood methodThe Likelihood method

Model SelectionModel Selection

[optional] Fisher Matrix[optional] Fisher Matrix

115115

ProbabilitiesProbabilities

Classical interpretation of probability: infinite realization Classical interpretation of probability: infinite realization limit of relative frequencieslimit of relative frequencies Probability: “the number of times the event occurs over the Probability: “the number of times the event occurs over the

total number of trials, in the limit of an infinite series of total number of trials, in the limit of an infinite series of equiprobable repetitions.”equiprobable repetitions.”

Let's define 2 random (stochastic) variables x and y (e.g. Let's define 2 random (stochastic) variables x and y (e.g. numbers on a die roll).numbers on a die roll). p(X) is the probability of getting the result x = Xp(X) is the probability of getting the result x = X p(X, Y) or p(X p(X, Y) or p(X ∩∩ Y) prob of getting results x = X AND y = Y→ Y) prob of getting results x = X AND y = Y→ p(X | Y) or p(X ; Y) prob of x = X given the fact that y = Y→p(X | Y) or p(X ; Y) prob of x = X given the fact that y = Y→ p(X p(X ∪∪ Y) prob of getting results x = X OR y = Y→ Y) prob of getting results x = X OR y = Y→

116116

Probabilities (2)Probabilities (2)

Some properties:Some properties: Joint probabilities are symmetricJoint probabilities are symmetric

Joint prob of Joint prob of independentindependent events events

Joint prob of Joint prob of dependentdependent events events

Disjoint prob of Disjoint prob of mutually exclusivemutually exclusive events events

In particularIn particular

117117

Probabilities (3)Probabilities (3)

Let's discuss the conditional probability property:Let's discuss the conditional probability property:

Suppose A refers to “person that studies physics” and B to Suppose A refers to “person that studies physics” and B to “person that plays piano”“person that plays piano”

Suppose also that we know that: p(B) = 1/100 and p(A, B) = Suppose also that we know that: p(B) = 1/100 and p(A, B) = 1/ 10001/ 1000 In other words, out of 1000 random people, 10 will play the In other words, out of 1000 random people, 10 will play the

piano and 1 will play the piano AND be a physicistpiano and 1 will play the piano AND be a physicist So, if someone plays piano, he has 1/10 chance of being also So, if someone plays piano, he has 1/10 chance of being also

a physicista physicist

118118

Bayes' TheoremBayes' Theorem

Note thatNote that The probability of A given B is The probability of A given B is notnot the prob of B given A. the prob of B given A. E.g.: The probability of winning the lottery given that you E.g.: The probability of winning the lottery given that you

played twice in your life is played twice in your life is notnot the same as the probability the same as the probability that you played twice in your life given that you won the that you played twice in your life given that you won the lottery!lottery!

From the symmetry of the joint probabilities we getFrom the symmetry of the joint probabilities we get

This is the This is the Bayes TheoremBayes Theorem of conditional probabilities of conditional probabilities

119119

InterpretationInterpretation

Classical “Classical “frequentistfrequentist” interpretation of probability (infinite ” interpretation of probability (infinite realization limit of relative frequencies) has limitationsrealization limit of relative frequencies) has limitations It is It is circularcircular assumes that the repeated trials have same → assumes that the repeated trials have same →

probability of outcomesprobability of outcomes Cannot deal with unrepeatable situations [e.g. (i) probability I Cannot deal with unrepeatable situations [e.g. (i) probability I

will die in a car accident; (ii) prob the Big-Bang happened the will die in a car accident; (ii) prob the Big-Bang happened the way it did]way it did] ““what is the probability that it rained in Manaus during what is the probability that it rained in Manaus during

D. Pedro II 43D. Pedro II 43rdrd birthday?” birthday?” How to correct for finite realizations? How many realizations How to correct for finite realizations? How many realizations

are needed for the frequencies to be approx. the probabilities? are needed for the frequencies to be approx. the probabilities? This approximation is to which % accuracy?This approximation is to which % accuracy?

120120

Interpretation (2)Interpretation (2)

The The BayesianBayesian interpretation is based on Bayes' Theorem interpretation is based on Bayes' Theorem Re-interpret the theorem not in terms of regular random Re-interpret the theorem not in terms of regular random

variables but in terms of data (D) and theory (T)variables but in terms of data (D) and theory (T) Inverse statistical problem: what is the probability that Inverse statistical problem: what is the probability that

theory T is correct given we measured the data D?theory T is correct given we measured the data D?

The “theory” might be a model (such as The “theory” might be a model (such as ΛΛCDM or DGP) of CDM or DGP) of just the parameter values of an assumed model (such as just the parameter values of an assumed model (such as ΩΩm0m0 and and ΩΩΛ0Λ0 , assuming , assuming ΛΛCDM).CDM).

121121

Interpretation (3)Interpretation (3)

Bayesian analysis has some philosophical implicationsBayesian analysis has some philosophical implications The The best theorybest theory will be the will be the most probablemost probable theory theory Bayesian analysis carry a mathematically precise Bayesian analysis carry a mathematically precise

formulation of Occam's Razor: “if 2 hypotheses are equally formulation of Occam's Razor: “if 2 hypotheses are equally likely, the hypothesis with the fewest assumptions should likely, the hypothesis with the fewest assumptions should be selected”.be selected”. Only strong reason before the 18Only strong reason before the 18thth century to choose century to choose

Copernicus' model over Ptolomy'sCopernicus' model over Ptolomy's Karl Popper: we prefer simpler theories to more complex Karl Popper: we prefer simpler theories to more complex

ones “because their empirical content is greater; and ones “because their empirical content is greater; and because they are better testable” simple theories are →because they are better testable” simple theories are → more more easily falsifiableeasily falsifiable

122122

The Likelihood MethodThe Likelihood Method

Let's define the data as a vector Let's define the data as a vector xx and the parameters as a and the parameters as a vector vector θθ

We write Bayes' theorem asWe write Bayes' theorem as L →L → posteriorposterior probability probability p(p(θθ) prior probability→) prior probability→ f →f → likelihoodlikelihood function function g(g(xx) as we will see usually just a normalization factor→ →) as we will see usually just a normalization factor→ →

We are interested in the posteriorWe are interested in the posterior In the literature people sometimes refer to the posterior In the literature people sometimes refer to the posterior

also as the “likelihood” (hence the letter L)also as the “likelihood” (hence the letter L)

123123

The Likelihood Method (2)The Likelihood Method (2)

The posterior is a probability, so it has to be normalized to The posterior is a probability, so it has to be normalized to unityunity

This integral is called the evidenceThis integral is called the evidence g(x) does not depend on the parameters, so useless for g(x) does not depend on the parameters, so useless for

parameter determinationparameter determination But very useful to choose between modelsBut very useful to choose between models

124124

– – Fim da Aula 4/5 –Fim da Aula 4/5 –

125125

Aula 5/5Aula 5/5Estatística BayesianaEstatística Bayesianae Métodos Numéricose Métodos Numéricos

Curso de Cosmologia VII ECG, CBPF – Ago/2013Curso de Cosmologia VII ECG, CBPF – Ago/2013

Miguel QuartinMiguel QuartinInstituto de Física, UFRJInstituto de Física, UFRJ

Astrofísica, Relativ. e Cosmologia (ARCOS)Astrofísica, Relativ. e Cosmologia (ARCOS)

http://tinyurl.com/mylycfw

126126

Prior and PrejudicePrior and Prejudice

Priors are inevitable in the likelihood (posterior) methodPriors are inevitable in the likelihood (posterior) method Frequentist don't like it subjective prior knowledge→Frequentist don't like it subjective prior knowledge→ Bayesianists have to learn to love it after all, we →Bayesianists have to learn to love it after all, we → alwaysalways

know something before the analysisknow something before the analysis E.g.: we can use p(E.g.: we can use p(ΩΩm0 m0 < 0) = 0 as it does not make sense to < 0) = 0 as it does not make sense to

have negative matter densityhave negative matter density We can add information from previous experiment. E.g. We can add information from previous experiment. E.g.

experiment A measured h = 0.72 experiment A measured h = 0.72 ±± 0.08, so we can use, say, 0.08, so we can use, say, the gaussian priorthe gaussian prior

127127

Prior and PrejudicePrior and Prejudice

You are free to choose your prior but choice must be →You are free to choose your prior but choice must be →explicitexplicit

You HAVE to choose a prior p(→You HAVE to choose a prior p(→ θθ) = 1 *is* a particular ) = 1 *is* a particular prior, which under a parameter change will no longer be prior, which under a parameter change will no longer be constant. constant. E.g.: p(t) = 1 E.g.: p(t) = 1 ≠≠ p(z) = 1 p(z) = 1 ≠ p(log t) = 1 …≠ p(log t) = 1 … E.g. 2: a measurement of ΩE.g. 2: a measurement of ΩΛΛ00 assumes the strong prior that assumes the strong prior that

the model is the model is ΛΛCDMCDM Priors may be subjective, but analysis is objectivePriors may be subjective, but analysis is objective Priors are an Priors are an advantageadvantage of Bayes no inference can be made → of Bayes no inference can be made →

without assumptionswithout assumptions Data can show that the priors were “wrong”Data can show that the priors were “wrong”

128128

The Likelihood Method (3)The Likelihood Method (3)

If we are not interested in model selection we can neglect If we are not interested in model selection we can neglect the function g(x)the function g(x) The posterior L must then be normalizedThe posterior L must then be normalized The best-fit parameter values are the ones that maximize LThe best-fit parameter values are the ones that maximize L

The The n%n% confidence region confidence region RR of the parameters are the region of the parameters are the region around the best fit for whicharound the best fit for which

The confidence region in general is not symmetricThe confidence region in general is not symmetric

129129

The Likelihood Method (4)The Likelihood Method (4)

If the likelihood (i.e. the posterior) has many parameters, If the likelihood (i.e. the posterior) has many parameters, it is interesting to know what information it has in each it is interesting to know what information it has in each parameter (or each pair) independently of the othersparameter (or each pair) independently of the others We must do a weighted sum on the other parametersWe must do a weighted sum on the other parameters

This is referred to as marginalization over a parameterThis is referred to as marginalization over a parameter

130130

The Likelihood Method (5)The Likelihood Method (5)

It is customary to use the following confidence regions: It is customary to use the following confidence regions: 68.3%, 95.4% and 99.73%. The reason is that for gaussian 68.3%, 95.4% and 99.73%. The reason is that for gaussian posteriors, these are the 1, 2 and 3 standard deviations.posteriors, these are the 1, 2 and 3 standard deviations. We therefore often refer to these regions, for simplicity, as We therefore often refer to these regions, for simplicity, as

just the 1just the 1σσ, 2, 2σσ and 3 and 3σσ regions regions

Say, if for Say, if for ΩΩm0 m0 the best fit is 0.3 the best fit is 0.3 and and

the 68% confidence region (1the 68% confidence region (1σσ) is [0.1, 0.4] ) is [0.1, 0.4] we writewe write

Note that here the 2Note that here the 2σ region will σ region will notnot be be

131131

The Likelihood Method (6)The Likelihood Method (6)

1σ

2σ

best fit

132132

How to Build the Likelihood?How to Build the Likelihood?

The likelihood is a function of the data functional form →The likelihood is a function of the data functional form →depends on the instrument used to collect the datadepends on the instrument used to collect the data Usually instruments have (approximately) either Gaussian Usually instruments have (approximately) either Gaussian

or Poisson noise. or Poisson noise. It is common to assume by default a Gaussian noiseIt is common to assume by default a Gaussian noise

If the likelihood is Gaussian in the data and the data are If the likelihood is Gaussian in the data and the data are independent (uncorrelated errors) we haveindependent (uncorrelated errors) we have

133133

Model SelectionModel Selection

We now want to address the more general problem: how We now want to address the more general problem: how to tell which of 2 competitive theories are statistically to tell which of 2 competitive theories are statistically better given some data?better given some data?

Frequentist approach: compare the reduced Frequentist approach: compare the reduced χχ22 (i.e. the (i.e. the χχ22 per degree of freedom – d.o.f.) of the data in the 2 theoriesper degree of freedom – d.o.f.) of the data in the 2 theories The The χχ22–distribution with –distribution with kk degrees of freedom is the degrees of freedom is the

distribution of a sum of the squares of distribution of a sum of the squares of kk independent independent standard normal (i.e. gaussian) random variables.standard normal (i.e. gaussian) random variables.

The p.d.f. is given by (although we do not use it explicitly)The p.d.f. is given by (although we do not use it explicitly)

134134

Model Selection (2)Model Selection (2)

This is the distribution if the likelihood of the data was This is the distribution if the likelihood of the data was exactly given by exactly given by

In a nutshell, it is the sum of squares of the “distance, in In a nutshell, it is the sum of squares of the “distance, in units of standard deviations, between data points and units of standard deviations, between data points and theoretical curve”theoretical curve” We refer to the We refer to the total total χχ22 as as the sum the sum

Frequentist mantra: good models haveFrequentist mantra: good models have

135135

Model Selection (3)Model Selection (3)

The bayesian equivalent to The bayesian equivalent to χχ22 comparison is the comparison is the Bayes Bayes ratioratio ratio of → ratio of → evidencesevidences of models “1” and “2” of models “1” and “2” For a model “M” the evidence isFor a model “M” the evidence is

BB1212 > 1 model 1 is favored by the data (and vice-versa)→ > 1 model 1 is favored by the data (and vice-versa)→

If you have an a priori reason to favor a model generalize →If you have an a priori reason to favor a model generalize →the above to include model priorsthe above to include model priors

The Bayes factor between 2 models is justThe Bayes factor between 2 models is just

136136

Model Selection (4)Model Selection (4)

The Bayes factor has several advantages over simple The Bayes factor has several advantages over simple χχ22

If the If the data is poordata is poor and a particular parameter of one model and a particular parameter of one model is unconstrained by it, the model is is unconstrained by it, the model is notnot penalized penalized E.g.: a given dark energy model has a parameter related E.g.: a given dark energy model has a parameter related

to, say, cluster abundance at z = 2, for which data is poor. to, say, cluster abundance at z = 2, for which data is poor. This is good, because poor data This is good, because poor data ≠≠ poor model! poor model!

Mathematically the posterior is approx. flat on this →Mathematically the posterior is approx. flat on this →parameter assuming (as usual) that the priors are →parameter assuming (as usual) that the priors are →independent we have that:independent we have that:

137137

Model Selection (5)Model Selection (5) To get a better intuition, we can study the simple case of To get a better intuition, we can study the simple case of

gaussian likelihoods + gaussian priors analytical E(x)→gaussian likelihoods + gaussian priors analytical E(x)→ Assuming uncorrelated parameters, the posterior is then Assuming uncorrelated parameters, the posterior is then

(integrating over the data):(integrating over the data):

138138

Model Selection (6)Model Selection (6)

The evidence is then given byThe evidence is then given by

Let's analyze the 3 distinct terms aboveLet's analyze the 3 distinct terms above fmax is the max likelihood how well the model fits the data→fmax is the max likelihood how well the model fits the data→ is always < 1 penalizes extra parameters →is always < 1 penalizes extra parameters →

constrained by the data Ockham's Razor factor→constrained by the data Ockham's Razor factor→ exp[ … ] penalizes cases where prior best fit is very →exp[ … ] penalizes cases where prior best fit is very →

different than posterior best fitdifferent than posterior best fit

139139

Jeffrey's ScaleJeffrey's Scale

As we have seen: BAs we have seen: B1212 > 1 model 1 is favored by the data → > 1 model 1 is favored by the data →

(and vice-versa)(and vice-versa) There is no absolute rule of how big must BThere is no absolute rule of how big must B1212 be to conclude be to conclude

whether one model must be replaced by anotherwhether one model must be replaced by another A simple rule-of-thumb though is just to use a simple scale A simple rule-of-thumb though is just to use a simple scale

to guide the discussion. to guide the discussion. Jeffrey's scaleJeffrey's scale is often used: is often used:

140140

141141

Fisher MatrixFisher Matrix

In a nutshell the Fisher Matrix method is an →In a nutshell the Fisher Matrix method is an →approximationapproximation for the computation of the posterior under for the computation of the posterior under the assumption that it is Gaussian the assumption that it is Gaussian in the parametersin the parameters Advantages: Advantages:

very fast to compute (either analytically or numerically)very fast to compute (either analytically or numerically) gives directly the (elliptical) confidence-level contoursgives directly the (elliptical) confidence-level contours

Disadvantages: Disadvantages: gives wrong results when non-gaussianity is stronggives wrong results when non-gaussianity is strong no intrinsic flags to warn you when non-gaussianity is strongno intrinsic flags to warn you when non-gaussianity is strong numerical derivatives can be noisynumerical derivatives can be noisy

For a 4-page quick-start guide, see: arXiv:0906.4123For a 4-page quick-start guide, see: arXiv:0906.4123 For more detail, see Amendola & Tsujikawa, Sect. 13.3For more detail, see Amendola & Tsujikawa, Sect. 13.3

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Fisher Matrix (2)Fisher Matrix (2)

We write the posterior as a multivariate gaussianWe write the posterior as a multivariate gaussian

The matrix The matrix FF is called the Fisher (or information) matrix is called the Fisher (or information) matrix To compute To compute FF, we Taylor expand the posterior near its , we Taylor expand the posterior near its

peak – the maximum likelihood (ML) pointpeak – the maximum likelihood (ML) point We need to compute first this point but this is simple:We need to compute first this point but this is simple:

When doing forecasts for future experiments, we know the When doing forecasts for future experiments, we know the ML beforehand (it is our fiducial model)ML beforehand (it is our fiducial model)

For real data multi-dim. minimization algorithms are fast→For real data multi-dim. minimization algorithms are fast→

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Properties of the Fisher MatrixProperties of the Fisher Matrix

Once we have Once we have FF, the , the covariance matrixcovariance matrix is simply its is simply its inverseinverse For 2 parameters:For 2 parameters:

The ellipses axis lengths (The ellipses axis lengths (α α aa and and α bα b) and rotation angle ) and rotation angle are given by the eigenvalues and eigenvectors of are given by the eigenvalues and eigenvectors of CC::

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Properties of the Fisher Matrix (2)Properties of the Fisher Matrix (2) MarginalizationMarginalization over a parameter simply remove the → over a parameter simply remove the →

line & column of that parameter from line & column of that parameter from CC = = FF–1–1 and invert and invert the new, reduced the new, reduced CC

FixingFixing a parameter to its best fit simply remove the line → a parameter to its best fit simply remove the line →& column of that parameter from & column of that parameter from FF

Adding datasets simply add →Adding datasets simply add → FFtottot = = FF11 + + FF22

Changing variables simple jacobian transformation→Changing variables simple jacobian transformation→