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17/11/2016

Cosmology

13.8 Gyrs of History

Gravity

Ruler of the

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 Strong Nuclear Force Responsible for holding particles together inside the nucleus. The nuclear strong force carrier particle is called the gluon. The nuclear strong interaction has a range of 10‐15 m (diameter of a proton).  Electromagnetic Force Responsible for electric and magnetic interactions, and determines structure of and molecules. The electromagnetic force carrier particle is the photon (quantum of ) The electromagnetic interaction range is infinite.  Weak Force Responsible for (beta) radioactivity. The weak force carrier particles are called weak gauge bosons (Z,W+,W‐). The nuclear weak interaction has a range of 10‐17 m (1% of proton diameter).  Responsible for the attraction between . Although the gravitational force carrier The hypothetical (carrier) particle is the graviton. The gravitational interaction range is infinite. By far the weakest force of .

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The weakest force, by far, rules the Universe … Gravity has dominated its evolution, and determines its fate …

Grand Unified Theories (GUT)

Grand Unified Theories

* describe how ∑ Strong ∑ Weak ∑ Electromagnetic Forces are manifestations of the same underlying GUT force …

* This implies the strength of the forces to diverge from their uniform GUT strength

* Interesting to see whether gravity at some very early instant unifies with these forces ???

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Newton’s

∑ In two thousand years of , no one ever guessed that the universe might be expanding.

∑ To ancient Greek astronomers and philosophers, the universe was seen as the embodiment of perfection, the heavens were truly heavenly: – unchanging, permanent, and geometrically perfect.

∑ In the early 1600s, developed his law of gravity, showing that in the heavens obeyed the same laws as motion on Earth.

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∑ However, Newton ran into trouble when he tried to apply his theory of gravity to the entire universe.

∑ Since gravity is always attractive, his law predicted that all the in the universe should eventually clump into one big ball.

∑ Newton knew this was not the case, and assumed that the universe had to be static

∑ So he conjectured that:

the Creator placed the such that they were

``at immense from one another.’’

In footsteps of Copernicus, Galilei & Kepler, Isaac Newton (1687) in his Principia formulated a comprehensive model of the world. Cosmologically, it meant

• absolute and uniform time • space & time independent of matter • : ‐ action at ‐ instantaneous • Universe edgeless, centerless & infinite • : Universe looks the same at every place in space, every moment in time • absolute, static & infinite space

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Einstein’s

Dynamic & Geometric Universe

In 1915, completed his General .

∑ is a “ theory’’: gravity is a manifestation of the geometry, , of space‐time.

∑ Revolutionized our thinking about the nature of space & time: ‐ no longer Newton’s static and rigid background, ‐ a dynamic medium, intimately coupled to the universe’s content of matter and .

∑ All phrased into perhaps the most beautiful and impressive scientific equation known to humankind, a triumph of human genius,

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… becomes a dynamic continuum, integral part of the structure of the cosmos … curved spacetime becomes force of gravity

… its geometry rules the world, the world rules its geometry…

Albert Einstein (1879‐1955; Ulm‐Princeton)

father of General Relativity (1915),

opening the way towards Physical

The supreme task of the physicist is to arrive at those universal elementary laws from which the cosmos can be built up by pure deduction.

(Albert Einstein, 1954)

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General Relativity:

Einstein Field Equations

Einstein Field Equation

18G RRgT 2 c4

Metric tensor: g  

Energy-Momentum tensor: T

p   TUUpg  2  c

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Einstein Field Equation

Einstein Tensor: 1 GR Rg 2

GT;;     0

Einstein Tensor only rank 2 tensor for which this holds:

GT

Einstein Field Equation

also: g;   0

Freedom to add a multiple of to Einstein tensor:

8G Gg  T c4

 :

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Einstein Field Equation

8G G g  T c4

Dark Energy

8G   GTT4   vac c

c4 Tg  vac 8G

Einstein Field Equation

18G RRgg T 2 c4

18G RRgTg 2 c4

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Curved Space:

Cosmological Principle Friedmann-Robertson Metric

A crucial aspect of any particular configuration is the geometry of spacetime: because Einstein’s General Relativity is a metric theory, knowledge of the geometry is essential.

Einstein Field Equations are notoriously complex, essentially 10 equations. Solving them for general situations is almost impossible.

However, there are some special circumstances that do allow a full solution. The simplest one is also the one that describes our Universe. It is encapsulated in the

Cosmological Principle

On the basis of this principle, we can constrain the geometry of the Universe and hence find its dynamical evolution.

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“God is an infinite sphere whose centre is everywhere and its circumference nowhere” Empedocles, 5th cent BC

Cosmological Principle: Describes the symmetries in global appearance of the Universe: The Universe is the same everywhere: ● Homogeneous - physical quantities (density, T,p,…) ● Isotropic The Universe looks the same in every direction

● Universality Physical Laws same everywhere The Universe “grows” with same rate in ● Uniformly Expanding - every direction - at every location

”all places in the Universe are alike’’ Einstein, 1931

Fundamental Tenet of (Non‐Euclidian = Riemannian) Geometry

There exist no more than THREE uniform spaces: 1) Euclidian (flat) Geometry Euclides 2) Hyperbolic Geometry Gauß, Lobachevski, Bolyai 3) Spherical Geometry Riemann

uniform= homogeneous & isotropic (cosmological principle)

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Distances in a uniformly curved spacetime is specified in terms of the Robertson‐Walker metric. The spacetime distance of a point at coordinate (r,q,f) is:

r 222 2 2 22 2 2 2 ds c dt a() t dr  Rck S d  sin d  Rc

r sin k  1 Rc where the function Sk(r/Rc) specifies the effect of curvature rr on the distances between Skk   0 points in spacetime RRcc r sinh k  1 Rc

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Friedmann-Robertson-Walker-Lemaitre (FRLW)

Universe

Einstein Field Equation

8G GT c4

 g,RW   RR  ,

p   TUUpg  2  c  diag c2 ,,, p p p

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Einstein Field Equation

8G GT c4

002222 8G GGRkcRc003/ 2  c

11222  8G GGRRRkcRp112/2 c

Friedmann-Robertson-Walker-Lemaitre Universe

 43Gp RRR 2  33c

8G  RRkcR 2222 33

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Cosmic Expansion Factor R()t at() R0

∏ Cosmic Expansion is a uniform expansion of space

Our Universe ?

Einstein‐ ?

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Friedmann-Robertson-Walker-Lemaitre Universe

43Gp   aaa 2  33c pressure cosmological density constant

curvature term 8Gkc2   22 2 aa 2 a 33R0

Because of General Relativity, the evolution of the Universe is fully determined by four factors:

∏ density ()t

∏ pressure p()t 22 k  0, 1, 1 ∏ curvature kc/ R0

R0: present curvature radius ∏ cosmological constant 

∏ Density & Pressure: ‐ in relativity, energy & momentum need to be seen as one physical quantity (four‐vector) ‐ pressure = momentum flux ∏ Curvature: ‐ gravity is a manifestation of geometry spacetime ∏ Cosmological Constant: ‐ free parameter in General Relativity ‐ Einstein’s “biggest blunder” ‐ mysteriously, since 1998 we know it dominates the Universe

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Relativistic Cosmology Newtonian Cosmology

43Gp 4G   aa   aaa 2  33c 3

2 8Gkc 228G aa22 a 2 aaE    2 3 33R0

22 Curvature Energy kc/ R0 E Cosmological  Constant

p Pressure

Cosmological Constant & FRW equations

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Friedmann-Robertson-Walker-Lemaitre Universe

43Gp   aaa 2  33c

8Gkc2   22 2 aa 2 a 33R0

Dark Energy & Energy Density

     8G

ppp  c2  p   8G

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Friedmann-Robertson-Walker-Lemaitre Universe

43Gp   aa  2 3 c

aG 2 8 kc22/ R  0 aa223

Cosmic Constituents

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The total energy content of Universe made up by various constituents, principal ones:

matter Wb baryonic matter Wm WDM radiation Wg photons W Wtot rad Wn neutrino’s

dark/ Wv vacuum energy

In addition, contributions by ‐ gravitational ‐ magnetic fields, ‐ cosmic rays … Poor constraints on their contribution: henceforth we will not take them into account !

LCDM Cosmology

• Concordance cosmology

- model that fits the majority of cosmological observations - universe dominated by Dark Matter and Dark Energy

LCDM composition today …

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Cosmic Energy Inventory

Fukugita & Peebles 2004

Cosmic Constituents:

Evolving Energy Density

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To infer the evolving energy density r(t) of each cosmic component, we refer to the cosmic energy equation. This equation can be directly inferred from the FRW equations pa   302  ca

The equation forms a direct expression of the adiabatic expansion of the Universe, ie.

2 Internal energy UcV  dU  pdV 3 Va Expanding volume

To infer r(t) from the energy equation, we need to know the pressure p(t) for that particular medium/ingredient of the Universe.

pa   302  ca

To infer p(t), we need to know the nature of the medium, which provides us with the equation of state,

pp (,) S

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3 • Matter: m ()tat ()

4 ∑ Radiation: rad ()tat ()

3(1w ) 2 ∑ Dark Energy: vv()tat () pwc w 1

 ()tcst .

Dark Energy:

Equation of State

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Einstein Field Equation curvature side

18G RRgg T 2 c4

18G RRgTg 2 c4

energy-momentum side

Equation of State

restframe c4 c4 Tg T   vac 8G vac 8G 00 1,ii  1

p   TUUpg  2  c

4 2 c 00 2 vacc  Tcvac  vac  8G    restframe: 4 ii  c Tp p  vac  8G

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Equation of State

c4  c2  vac 8G

c4 p  8G

2 pcvac vac

Dynamics

Relativistic Poisson Equation:

2 3p 4 G  2 c

3p  vac 20;  vaccG2 vac vac 8

2  0 Repulsion !!!

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Dark Energy & Cosmic Acceleration

Nature Dark Energy:

(Parameterized) Equation of State pwc()   2

Cosmic Acceleration: 43Gp   aa  2 3 c Gravitational Repulsion: 1 pwc 2 w a 0 3

Dark Energy & Cosmic Acceleration

DE equation of State

2 3(1w ) pwc()   ww()aaa (0 )

Cosmological Constant:

:1wcstw  .

-1/3 > w > -1: 3(1w ) decreases with time w aw10 Phantom Energy: 3(1w ) w aw10increases with time

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Dynamic Dark Energy

DE equation of State Dynamically evolving dark energy, parameterization: pwc()   2 wa() w0  (1 aw )a  w () a

a  1() wa   ww()aa (0 )exp 3 da   1 a 

Critical Density & Omega

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8Gkc2  22 aa 2 3 R0

Critical Density: 2 ‐ For a Universe with L=0 3H   ‐ Given a particular expansion rate H(t) crit 8G ‐ Density corresponding to a flat Universe (k=0)

In a FRW Universe, densities are in the order of the critical density,

3H 2  0 1.8791hgcm2293  10  crit 8G

29 2 3 0 1.8791 10 hgcm 11 2 3 2.78 10 hMMpc

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In a matter‐dominated Universe, the evolution and fate of the Universe entirely determined by the (energy) density in units of critical density:  8G   2 crit 3H

Arguably, W is the most important parameter of cosmology !!!

29 2 3 Present‐day 0 1.8791 10 hgcm Cosmic Density: 11 2 3 2.78 10 hMMpc

∏ The individual contributions to the energy density of the Universe can be figured into the W parameter:

42 4 ‐ radiation rad Tc/8 GT rad   22 crit crit 3Hc

‐ matter mdmb 

 ‐ dark energy/  cosmological constant  3H 2

rad  m 

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Critical Density

There is a 1-1 relation between the total energy content of the Universe and its curvature. From FRW equations: HR22 k  (1)  rad  m  c2

11k  Hyperbolic Open Universe

10k  Flat Critical Universe

11k  Spherical Close Universe

FRW Universe: Curvature

There is a 1-1 relation between the total energy content of the Universe and its curvature. From FRW equations: HR22 k  (1)  rad  m  c2

11k  Hyperbolic Open Universe

10k  Flat Critical Universe

11k  Spherical Close Universe

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Radiation, Matter & Dark Energy

∏ The individual contributions to the energy density of the Universe can be figured into the W parameter:

42 4 - radiation rad Tc/8 GT rad   22 crit crit 3Hc - matter mdmb 

 - dark energy/  cosmological constant  3H 2

rad  m 

Hubble Expansion

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∏ Cosmic Expansion is a uniform expansion of space

∏ Objects do not move themselves: they are like beacons tied to a uniformly expanding sheet:

 rt() atx () a Ht() a rt() atx () ax Htr () a a

∏ Cosmic Expansion is a uniform expansion of space

∏ Objects do not move themselves: they are like beacons tied to a uniformly expandingHubbleComoving Parameter:sheet: Position Comoving Position Hubble “constant”: ∫  H0 H(t=t0) rt() atx () a Ht() a rt() atx () ax Htr () a a

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∏ For a long time, the correct value of the Hubble constant H0 was a major unsettled issue:

‐1 ‐1 ‐1 ‐1 H0 = 50 km s Mpc H0 = 100 km s Mpc

∏ This meant distances and timescales in the Universe had to deal with uncertainties of a factor 2 !!!

∏ Following major programs, such as Hubble Key Project, the key projects and the WMAP CMB measurements,

2.6 1 1 HkmsMpc02.7 71.9

HhkmsMpc100 11 1 0 tH   H 1 thGyr0  9.78

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Just as the Hubble time sets a natural time scale for the universe, one may also infer a natural distance scale of the universe, the

Hubble Distance

c 1 RH 2997.9hMpc H0

Acceleration Parameter

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FRW Dynamics: Cosmic Acceleration Cosmic acceleration quantified by means of dimensionless q(t):

Examples: m  1; 0; q  rad  m  aa 2 q  0.5 q  2  0.3; 0.7; a m   q 0.65 q  m  2 

Dynamics

FRW Universe

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From the FRW equations:

2 Ht() rad,0 m ,0 1 0 243,0 2 H0 aa a

Expansion history at() Universe

a da Ht 0   0 rad,0 m ,0 a2 1  aa2 ,0 0

Age of a FRW universe at 1 Matter‐dominated Expansion factor a(t)

a da Hubble time Ht 1   t  0 rad m a2 1  H 21 2   t  aa 3 H

1 Matter‐dominated

21 t  3 H

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While general solutions to the FRW equations is only possible by numerical integration, analytical solutions may be found for particular classes of :

∏ Single‐component : ‐ empty Universe ‐ flat Universes, with only radiation, matter or dark energy

∏ Matter‐dominated Universes

∏ Matter+Dark Energy flat Universe

∏ Assume radiation contribution is negligible: 5 rad ,0 510 ∏ Zero cosmological constant:   0 ∏ Matter‐dominated, including curvature

m 1 m 1

m 1

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1 m k  0  0

2281G 8G FRW: aa  0 33a

2/3 t at()  t 0 Albert Einstein and discussing the Universe. In 1932 they published a paper together on the Einstein‐de Sitter universe, which is a model with flat 21 geometry containing matter as the only significant Age t0  substance. EdS Universe: 3 H0

0 m k 1  0 Empty space is curved

kc2 acst 2   . FRW: 2 R0

t at()  t0 1 Age t0  Empty Universe: H0

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0 m k  0  1

  2 H0  33H0  FRW: aa22 aHa 3 0

at() eHtt00()

Willem de Sitter (1872‐1934; Sneek‐Leiden) Age director Leiden Observatory De Sitter Universe: infinitely old alma mater: Groningen University

rad 1 0 k  0 In the very early Universe, the energy density is m completely dominated by radiation. The dynamics of the very early Universe is therefore fully determined by the  0 evolution of the radiation energy density: 1 2281G 8G0 FRW: aa  rad ()a  4 33a2 a 1/2 t at()  t0

Age 11 Radiation t0  Universe: 2 H0

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k  0 3(1w ) 2 vv()tat () pwc

FRW:

2 at() t33 w

Cosmological Transitions

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Dynamical Transitions

Because radiation, matter, dark energy (and curvature) of the Universe evolve differently as the Universe expands, at different epochs the energy density of the Universe is alternately dominated by these different ingredients.

As the Universe is dominated by either radiation, matter, curvature or dark energy, the cosmic expansion a(t) proceeds differently.

We therefore recognize the following epochs: ∏ radiation‐dominated era ∏ matter‐dominated era ∏ curvature‐dominated expansion ∏ dark energy dominated epoch

The different cosmic expansions at these eras have a huge effect on relevant physical processes

Dynamical Transitions

∏ Radiation Density Evolution 1 ()t  rada4 rad ,0

∏ Matter Density Evolution 1  ()t   mma3 ,0

22 ∏ Curvature Evolution kc11 kc 2222 1 0 Rt() a R0 a

∏ Dark Energy (Cosmological Constant)  ()tcst . 0 Evolution

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()t

crit,0

matter Density Evolution Cosmic Components radiation

dark energy

()t

crit,0

Matter‐Dark Energy Transition

matter Radiation‐Matter transition radiation

dark energy

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Radiation‐Matter Transition

∏ Radiation Density Evolution ∏ Matter Density Evolution 1 1 ()t   ()t   rada4 rad ,0 mma3 ,0

∏ Radiation energy density decreases more rapidly than matter density: this implies radiation to have had a higher energy density before a particular : Radiation aa rm dominance    a  rad ,0 mrad,0 ,0 rm  aa34 m,0 Matter aa rm dominance

Matter‐Dark Energy Transition

∏ Matter Density Evolution ∏ Dark Energy Density Evolution 1  ()t    ()tcst .  mma3 ,0  0

∏ While matter density decreases due to the expansion of the Universe, the cosmological constant represents a small, yet constant, energy density. As a result, it will represent a higher density after Matter aa m dominance

m,0 m,0 a  3   m a3 ,0 ,0 Dark energy aa m dominance

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Matter‐Dark Energy Transition

,0 0.27am 0.72 m,0 am  3 ,0 † mm,0 0.73a   0.57

Note: a more appropriate characteristic transition Flat is that at which the deceleration turns into Universe acceleration:

 m,0 † mm,0 ,0 a  3 am 33 m 22(1) 1m,0 ,0m ,0

Evolution Cosmological Density Parameter Limiting ourselves to a flat Universe (and discarding the contribution by and evolution of curvature):

to appreciate the dominance of radiation, matter and dark energy in the subsequent cosmological eras, it is most illuminating to look at the evolution of the cosmological density parameter of these cosmological components:

rad ()t m ()t  ()t

4 m,0a e.g. m ()t 4 rad,0 m ,0aa ,0

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m ()t

rad ()t

 ()t matter radiation Evolution Cosmic Density Parameter W radiation, matter, dark energy (in concordance Universe)

dark energy

m ()t

rad ()t

 ()t matter radiation

dark energy

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m ()t

rad ()t

 ()t matter radiation

Matter‐ Dark Energy Radiation‐Matter Transition transition

dark energy

Concordance Universe

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Concordance Universe Parameters

Hubble Parameter 11 H0 71.9 2.6 km s Mpc

Age of the Universe tGyr0 13.7 0.12

Temperature CMB TK0  2.725 0.001 Matter m  0.27 Baryonic Matter b 0.0456 0.0015 Dark Matter dm 0.228 0.013

5 Radiation rad 8.4  10 5 Photons (CMB)  510 5 Neutrinos (Cosmic)  3.4  10

 0.726 0.015 Dark Energy 

Total tot 1.0050 0.0061

LCDM Cosmology

• Concordance cosmology

- model that fits the majority of cosmological observations - universe dominated by Dark Matter and Dark Energy

LCDM composition today …

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 3/2 3  2 aa  Ht0 ln 1  31m,0 aamm   

transition epoch: m0 matter‐dominate to a  3 L dominated m 1 m0

amL~0.75

We can recognize two extreme regimes:

∏ aa m very early times

matter dominates the expansion, and m  1: Einstein‐de Sitter expansion,

2/3 3 at()m,0 Ht 0 2

∏ aa m very late times

matter has diluted to oblivion, and m  0 : de Sitter expansion driven by dark energy

1m00Ht at() am e

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aexp

aa m

1m00Ht at() am e

aexp expansion like aq 0; 0 De Sitter expansion acceleration past aa m future 2/3 3 at()m,0 Ht 0 2 expansion like today EdS universe aq 0; 0 deceleration

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Matter-Dark Energy Transition

,0 0.27am 0.72 m,0 am  3 ,0 † mm,0 0.73a   0.57

Note: a more appropriate characteristic transition Flat is that at which the deceleration turns into Universe acceleration:

 m,0 † mm,0 ,0 a  3 am 33 m 22(1) 1m,0 ,0m ,0

Key Epochs Concordance Universe

11 H0 71.9 2.6 km s Mpc

Radiation‐Matter 4 4 aeq tGyr02.813.7 10  0.12 tyreq 4.7 10 Equality

TK0  2.725

5 Recombination/ arec  1/1091 tyrsrec 3.77 0.03 10

Decoupling zrec 1090.88 0.72

Reionization  reion  0.084 0.016 90 6 Optical Depth tyrsreion 43267 10 z  10.9 1.4 reion Matter‐Dark Energy †† Transition Acceleration azmm0.60; 0.67 Energy azmm 0.75; 0.33 tGyrm  9.8

tGyreq  13.72 0.12 Today a0  1

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Observational Cosmology in FRW Universe

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In an (expanding) space with Robertson‐Walker metric,

r 222 2 2 22 2 2 2 ds c dt a() t dr  Rck S d  sin d  Rc

there are several definitions for distance, dependent on how you measure it.

They all involve the central distance function, the RW Distance Measure,

r Dr() RSck Rc

Imagine an object of proper size d, at redshift z, its angular size Dq is given by

r dz1  d datRS()    ck D D Rc A

Angular Diameter distance:

D D  A 1 z

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Imagine an object of L(ne), at redshift z, its flux density at observed no is L  LL e S bol bol S  o  2 bol 2 2 2 41 Dz 41 Dz 4 DL

Luminosity distance:

DDL  1 z

Observing in a FRW Universe, we locate in terms of their redshift z. To connect this to their true physical distance, we need to know what the coordinate distance r of an object with redshift z, c Rdr dz 0 Hz()

In a FRW Universe, the dependence of the Hubble expansion rate H(z) at any redshift z depends on the content of matter, dark energy and radiation, as well ss its curvature. This leads to the following explicit expression for the redshift‐distance relation,

c 2341/2 R dr11   z   1  z  1  z dz 0 0  ,0mrad ,0  ,0  H0

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Observing in a FRW Universe, we locate galaxies in terms of their redshift z. To connectin this a matter-dominated to their true physical Universe, distance, wethe need redshift-distance to know what the relation coordinate is distance r of an object with redshift z,

c 231/2 R dr11 z 1 z dz 000 H0

In a FRW Universe, the dependence of the Hubble expansion rate H(z) at any redshiftfrom z depends which on one the maycontent find of thatmatter, dark energy and radiation, as well ss its curvature. This leads to the following explicit expression for the redshift‐distance relation, cdzz  Rr 0  H0 0 11zz 0 

The integral expression cdzz  Rr 0  H0 0 11zz 0 

k  1 can be evaluated by using the substitution: u2  0 0 1 z

This leads to Mattig’s formula:

rc2 00zz21  0  1 Dz() RSck 2 RHc 001 z

This is one of the very most important and most useful equations in .

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rc2 00zz21  0  1 Dz() RSck 2 RHc 001 z

In a low-density Universe, it is better to use the following version:

rcz 110 z Dz() RSck RHzc 0 1 1100zz /2

For a Universe with a cosmological constant, there is not an easily tractable analytical expression (a Mattig’s formula). The comoving Distance r has to be found through a numerical evaluation of the fundamental dr/dz expression.

For all general FRW Universe, the second-order distance-redshift relation is identical,

only depending on the deceleration parameter q0:

rc1 2 Dz() RSck  z 1 q0 z RHc 0 2

q0 can be related to W0 once the equation of state is known.

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Dr1  DRSAck  11zz Rc

In a matter‐dominated Universe, the angular diameter distance as function of redshift is given by: 121rc DzAck() RS00 z 2 1 0 z 1 1 zRH2 2  c 0 0 1 z

  ()z  DA The angular size q(z) of an object of physical size l at a redshift z displays an interesting behaviour. In most FRW universes is has a minimum at a medium range redshift –

z=1.25 in an Wm=1 EdS universe –and increases again at higher .

In a matter‐dominated Universe, the angular diameter distance as function of redshift is given by: 121rc DzAck() RS00 z 2 1 0 z 1 1 zRH2 2  c 0 0 1 z

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r DDLck(1 z ) (1 zRS )  Rc

In a matter‐dominated Universe, the as function of redshift is given by: rc2 Dz() (1 zRS ) z 2 1 z 1 Lck 2 00 0 RHc 00

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FRW

Thermodynamics

43Gp  To find solutions a(t) for the aaa 2  33c expansion history of the Universe, for a particular FRW Universe ,

8Gkc2  one needs to know how the 22 2density r(t) and pressure p(t) aa 2 aevolve as function of a(t) 33R0

FRW equations are implicitly equivalent to a third Einstein equation, the energy equation, pa   302  ca

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Important observation: the energy equation, pa   302  ca is equivalent to stating that the change in internal energy UcV  2 of a specific co‐expanding volume V(t) of the Universe, is due to work by pressure: dU  p dV

Friedmann‐Robertson‐Walker‐Lemaitre expansion of the Universe is Adiabatic Expansion

Adiabatic Expansion of the Universe:

∑ Implication for Thermal History ∑ Evolution of cosmic components

For a medium with adiabatic index g: TV 1  cst

4 T Radiation (Photons)   T  0 3 a

5 T0 Monatomic Gas   T  2 () 3 a

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Adiabatic Expansion of the Universe:

∑ Implication for Thermal History ∑ Temperature Evolution of cosmic components

For a medium with adiabatic index g: TV 1  cst

4 T Radiation (Photons)   T  0 3 a

5 T0 Monatomic Gas   T  2 (hydrogen) 3 a

Radiation & Matter

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The Universe is filled with thermal radiation, the photons that were created in The Big Bang and that we now observe as the Cosmic Microwave Background (CMB).

The CMB photons represent the most abundant species in the Universe, by far !

The CMB radiation field is PERFECTLY thermalized, with their energy distribution representing the most perfect blackbody spectrum we know in nature. The energy

density un(T) is therefore given by the spectral distribution,

81h 3 uT()  ce3/hkT 1

At present, the temperature T of the cosmic radiation field is known to impressive precision,

TK0  2.725 0.001

With the energy density un(T) of CMB photons with energy hn given, we know the number density nn(T) of such photons:

uT() 81 2 nT()  hce 3/hkT 1

The total number density ng(T) of photons in the Universe can be assessed by integrating the number density nn(T) of photons with frequency n over all ,

 nT() nTd () TK 2.725  0  2 3 81 kT nT( ) 412 cm3 d 60.4   3/hkT  0 ce1  hc

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Having determined the number density of photons, we may compare this with the

number density of baryons, nb(T). That is, we wish to know the PHOTON‐BARYON ratio,

n B Bcrit   nb  mm nB pp

The baryon number density is inferred from the baryon density. here, for simplicity, we have assumed that baryons (protons and neutrons) have ‐24 the same mass, the proton mass mp ~ 1.672 â 10 g. At present we therefore find

52 3 nhgcmbb1.12 10 We know that Wb~0.044 and h~0.72:

n 9 n 1  1.60  10  3.65  1073g cm 0 0 2 nb nhBb

From simple thermodynamic arguments, we find that the number of photons is vastly larger than that of baryons in the Universe.

n 9 0 1.60  10 nb

In this, the Universe is a unique physical system, with tremendous repercussions for the thermal history of the Universe. We may in fact easily find that the cosmic photon‐baryon ratio remains constant during the expansion of the Universe,

n nt() b,0 b a3 nt() n,0   0 ntbb() n,0 1 n nt() Tt ()3 nt ()  0 aa33

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The photon‐baryon ratio in the Universe remains constant during the expansion of the Universe, and has the large value of

nt() n,0 9 0 1.60 10 ntbb() n,0

This quantity is one of the key parameters of the Big Bang. The baryon‐photon ratio quantifies the of the Universe, and it remains to be explained why the Universe has produced such a system of extremely large entropy !!!!!

The key to this lies in the very earliest instants of our Universe !

Hot Big Bang

Key Observations

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  Olber’s paradox: the night sky is dark finite age Universe (13.7 Gyr)

 Hubble Expansion uniform expansion, with expansion ~ distance: v = H r

 Explanation Helium Abundance 24%: light chemical elements formed (H, He, Li, …) after ~3 minutes …

 The Cosmic Microwave Background Radiation: the 2.725K radiation blanket, remnant left over hot ionized plasma neutral universe (379,000 years after Big Bang)

 Distant, deep Universe indeed looks different …

In an infinitely large, old and unchanging Universe each line of sight would hit a :

Sky would be as bright as surface of star:

Night sky as bright as Solar Surface, yet the night sky is dark

finite age of Universe (13.7 Gyr)

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In an infinitely large, old and unchanging Universe each line of sight would hit a star:

Sky would be as bright as surface of star:

Night sky as bright as Solar Surface, yet the night sky is dark

finite age of Universe (13.8 Gyr)

Hubble Diagram:

• Hubble 1929: Universe expands !!!! • Supernova Cosmic Expansion Projects (1998) is accelerating

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Mass Fraction Light Elements

24% 4He nuclei traces D, 3He, 7Li nuclei 75% H nuclei (protons)

Between 1‐200 seconds after Big Bang, temperature dropped to 109 K:

Fusion protons & neutrons into light atomic nuclei

4. Cosmic Microwave Background

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And there was light...

4. Cosmic Microwave Background

Thermal Background Radiation Field

T=2.725 K

∏Discovery Penzias & Wilson (1965) Nobelprize 1978

∏ Echo of the Big Bang: perfect thermal nature can only be understood when Universe went through very hot and dense phase:

∑ Ultimate proof Hot Big Bang !!!!! T ~ 3000 K

zdec=1089 (Δzdec=195); tdec=379.000 yrs

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Recombination &

waterstofatomen protonen & electronen

lichtdeeltjes/fotonen

Cosmic Light (CMB): the facts

 Ontdekt in 1965 door Penzias & Wilson, Nobelprijs 1978 !!!!!

 Kosmisch Licht dat het gehele heelal uniform vult

 Temperatuur: Tγ=2.725 K  Fotonen veruit het meest voorkomende deeltje in de natuur:

-3 nγ ~ 415 cm 9  Per atoom in het Heelal: nγ/n B ~ 1.9 x 10

 Ultieme Bewijs van de Big Bang !!!!!!!!!!!!!!!!!!!

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CMB Radiation Field Blackbody Radiation

∑ COBE‐DIRBE: temperature, blackbody 21h 3 BT()  ce2/hkT 1 •T = 2.725 K

• John Mather Nobelprize physics 2006

∑ Most accurately measured Spectrum Ever !!!!!

Note: far from being an exotic faraway phenomenon, realize that the CMB nowadays is counting for approximately 1% of the noise on your tv set … Courtesy: W. Hu

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Cosmic Light (CMB): most abundant species

By far, the most abundant particle species in the Universe

nγ/n B ~ 1.9 billion

The appearance of the Universe does change when looking deeper into the Universe:

Depth=Time

Galaxies in Hubble Ultra Deep Field

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The appearance of the Universe does change when looking deeper into the Universe:

Depth=Time

Quasars (very high z)

How Much ?

Cosmic Curvature

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FRW Universe: Curvature

There is a 1-1 relation between the total energy content of the Universe and its curvature. From FRW equations: HR22 k  (1)  rad  m  c2

11k  Hyperbolic Open Universe

10k  Flat Critical Universe

11k  Spherical Close Universe

Cosmic Microwave Background

Map of the Universe at Recombination Epoch (WMAP, 2003): ∑ 379,000 years after Big Bang ∑ Subhorizon perturbations: primordial waves ∑ ∆T/T < 10-5

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Measuring Curvature

Measuring the Geometry of the Universe:

∑ Object with known physical size, at large cosmological distance

● Measure angular extent on sky

● Comparison yields light path, and from this the curvature of space

W. Hu

In a FRW Universe: lightpaths described by Robertson-Walker metric

Geometry of 222 2 2 22r 2 2 2 ds c dt a() t dr  Rck S d  sin d  R   Space c 

Measuring Curvature

∑ Object with known physical size, at large cosmological distance:

∑ Sound Waves in the Early Universe !!!!

W. Hu

In a FRW Universe: lightpaths described by Robertson-Walker metric Temperature Fluctuations  r  ds222 c dt a() t 2 dr 2  R 22 S d 2  sin 2 d 2 CMB  ck  Rc 

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Music of the Spheres

● small ripples in primordial matter & photon distribution ● gravity: - compression primordial photon gas - photon pressure resists

● compressions and rarefactions in photon gas: sound waves

● sound waves not heard, but seen: - compressions: (photon) T higher - rarefactions: lower

● fundamental mode sound spectrum - size of “instrument”: - (sound) horizon size last

● Observed, angular size: θ~1º - exact scale maximum compression, the “cosmic fundamental mode of music” W. Hu

The Cosmic Tonal Ladder

The WMAP CMB temperature power spectrum Cosmic sound horizon

The Cosmic Microwave Background Temperature Anisotropies: Universe is almost perfectly flat

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The WMAP CMB temperature power spectrum

Angular CMB temperature fluctuations

CMB: Universe almost perfectly Flat !

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Cosmic Constraints

Wm vs. WL

 q m 2  HR22 k (1) c2 m 

SCP Union2 constraints (2010) on values of matter density Wm dark energy density WL

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Standard Big Bang:

what it cannot explain

  the Universe is remarkably flat, and was even (much) flatter in the past

 Horizon Problem the Universe is nearly perfectly isotropic and homogeneous, much more so in the past

 Monopole Problem: There are hardly any magnetic monopoles in our Universe

 Fluctuations, seeds of structure Structure in the Universe: origin

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Flatness Problem

Flatness Problem

FRW Dynamical Evolution:

Going back in time, we find that the Universe was much flatter than it is at the present.

Reversely, that means that any small deviation from flatness in the early Universe would have been strongly amplified nowadays …

We would therefore expect to live in a Universe that would either be almost W=0 or W~¶;

Yet, we find ourselves to live in a Universe that is almost perfectly flat … Wtot~1

How can this be ?

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Evolution W

From the FRW equations, one can infer that the evolution of W goes like (for simplicity, assume matter‐dominated Universe),

11 0 (1 z ) 1()1at   ()z 0 10 z

These equations directly show that HR22 a  0 1 k  (1) c2

implying that the early Universe was very nearly flat …

Flatness Evolution

11 1()1at  0

∏ At radiation‐matter equiv. 4 1210rm  

∏ ‐8 anuc~3.6ä10 14 1310nucl  

∏ Planck time 60 1110P  

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Horizon Problem

Cosmic Horizons

Light travel in an expanding Universe:

∑ Robertson‐Walker metric: ds222 c dt a() t 22 dr 2 ∑ Light: ds  0

t cdt t cdt d  Rat () Hor  Hor  0 at() 0 at()

Horizon distance in comoving space Horizon distance in physical space

Horizon of the Universe: distance that light travelled since the Big Bang

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Cosmic Horizons

t cdt RatHor  ()  RctHor  3 0 at() In an Einstein‐de Sitter Universe Horizon distance in physical space

Horizon of the Universe: distance that light travelled since the Big Bang

Cosmic Horizons

t cdt RatHor  ()  RctHor  3 0 at() In an Einstein‐de Sitter Universe Horizon distance in physical space

The horizon distance at recombination/decoupling 1/2 (CMB), z  1.74 1/2 dec angular size on the sky: Hor 0  1100

Horizon of the Universe: distance that light travelled since the Big Bang

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Cosmic Horizons

The horizon distance at 1/2 recombination/decoupling  1/2 zdec (CMB), Hor 1.74 0  angular size on the sky: 1100

 Large angular scales:  1 NOT in physical contact

 Small angular scales:  1 In physical (thus, also thermal) contact

Horizon of the Universe: distance that light travelled since the Big Bang

Size Horizon Recombination

COBE measured fluctuations: > 7o Size Horizon at Recombination spans ~ 1o

How can it be that regions totally out of thermal contact have the same temperature ?

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Size Horizon Recombination

COBE measured fluctuations: > 7o Size Horizon at Recombination spans angle ~ 1o

COBE proved that superhorizon fluctuations do exist: prediction !!!!!

Horizon Problem

The horizon distance at 1/2 recombination/decoupling  1/2 zdec (CMB), Hor 1.74 0  angular size on the sky: 1100

 Angular scales: Hor 1

How can it be that regions that were never in thermal still have almost exactly the same temperature T~2.725 K

Horizon of the Universe: distance that light travelled since the Big Bang

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Structure Problem

Millennium Simulation

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Millennium Simulation

The Universe should be Uniform: homogeneous & isotropic

Where does all structure in the Universe come from ?

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Monopole Problem

Inflationary Universe

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10-36 sec after Big Bang:

Inflation of the Universe

As time evolves and the Universe expands, Essential due to the decreasing temperature, the Ingredient/Extension potential V(f) of the Universe goes down Standard Cosmology and culminates in a broken symmetry

Inflationary Universe

Phase transition Early Universe

‐ GUT transition : t ~ 10‐36 sec ??

‐ (false) vacuum potential induces exponential (de Sitter) expansion

Universe blows up by factor N > 1060

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Cosmic Inflation

~ 10-36 sec. after Big Bang:

Universe expands exponentially: factor 1060 in 10-34 sec

Size current visible Heelal:

begin inflation: 10-15 size atoom end inflation: diameter dime

Essential Ingredient/Extension Standard Cosmology

Inflationary Universe

Phase transition Early Universe

‐ GUT transition : t ~ 10‐36 sec ??

‐ (false) vacuum potential induces exponential (de Sitter) expansion Inflation explains the horizon problem in a natural fashion: a region that initially was much smaller than the pre‐inflation horizon, Universe blows up by factor rapidly blew to a superhorizon size: any region on the CMB sky was in N > 1060 thermal contact before inflation !

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Essential Ingredient/Extension Standard Cosmology

Inflationary Universe

Phase transition Early Universe

‐ GUT transition : t ~ 10‐36 sec ??

‐ (false) vacuum potential Fluctuation Generation during Inflation: induces exponential Quantum fluctuations around the potential get magnified to (de Sitter) expansion macroscopic scales.

Universe blows up by factor N > 1060

Propelling Inflation:

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Inflatie &

Inflationary Universe

Explains:  Horizon Problem  Flatness Problem  Monopole Problem

And …  Origin of Structure

Towards the end of inflation, the Universe converts the tremendous amount of vacuum energy that drives inflation into a surge of newly created radiation and particles (“latent heat”). This is the Reheating Phase of inflation.

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Cosmic Future & Cosmic Horizons

Cosmic Horizons

Particle Horizon of the Universe: distance that light travelled since the Big Bang

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Cosmic Horizons

Fundamental Concept for our understanding of the physics of the Universe:

∏ Physical processes are limited to the region of space with which we are or have ever been in physical contact.

∏ What is the region of space with which we are in contact ? Region with whom we have been able to exchange photons (photons: fastest moving particles)

∏ From which distance have we received light.

∏ Complication: ‐ light is moving in an expanding and ‐ fighting its way against an expanding background

∏ This is called the

Horizon of the Universe

Cosmic Particle Horizon

Light travel in an expanding Universe:

∑ Robertson-Walker metric: ds222 c dt a() t 22 dr 2 ∑ Light: ds  0

t cdt t cdt d  Rat () Hor  Hor  0 at() 0 at()

Horizon distance in comoving space Horizon distance in physical space

Particle Horizon of the Universe: distance that light travelled since the Big Bang

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Cosmic Particle Horizon

Particle Horizon of the Universe: distance that light travelled since the Big Bang

t cdt t cdt d  Rat () Hor  Hor   0 at() 0 at()

Horizon distance in comoving space Horizon distance in physical space

In a spatially flat Universe, the horizon distance has a finite value for w>-1/3

2 2 33 w at() t dtHor ()00 ct 13 w

Cosmic Particle Horizon

Particle Horizon of the Universe: distance that light travelled since the Big Bang

In a spatially flat Universe, the horizon distance has a finite value for w>-1/3

2 2 33 w dt() ct at() t Hor 0013 w

dtHor ()00 3 ct flat, matter-dominated universe

flat, radiation-dominated universe dtHor ()00 2 ct

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Infinite Particle Horizon

Particle Horizon of the Universe: distance that light travelled since the Big Bang

In a spatially flat Universe, the horizon distance is infinite for w<-11/3

2 2 33 w dt() ct at() t Hor 0013 w

In such a universe, all of space is causally connected to observer:

In such a universe, you could see every point in space.

Cosmic

Light travel in an expanding Universe:

∑ Robertson-Walker metric: ds222 c dt a() t 22 dr 2 ∑ Light: ds  0

 cdt  cdt d  Rat () event  event  t at() t at()

Event Horizon distance in Event Horizon distance in comoving space physical space

Event Horizon of the Universe: the distance over which one may still communicate …

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Cosmic Event Horizon

Event Horizon of the Universe: distance light may still travel in Universe.

 cdt  cdt d  Rat () EHor  EHor   t at() t at()

Event Horizon distance comoving space Event Horizon distance physical space

In a spatially flat Universe, the event horizon is:

 2  13 w  33 w 33 w at() t dtEHor ()0   t  t

Cosmic Event Horizon

Event Horizon of the Universe: distance light may still travel in Universe.

 2  13 w  dt() t33 w at() t33 w EHor 0    t

In a spatially flat Universe, the event horizon is: 13 w wd1/3   33 w EHor dttEHor ()0 

wd1/ 3  EHor finite shrinking event horizon:

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Cosmic Fate

100 Gigayears: the end of Cosmology

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