Hubble Constant H0 = H(T0)= S(T0)

Introduction to Cosmology

(essentials of the lectures)

F. Jegerlehner Humboldt University, Berlin

– summer 2011 –

0-0 General Setup

Cosmology = theory of universe at large ( > 1 Gigaparsec ) ∼ Content = pressure free “dilute gas of galaxies”, cosmic microwave background radiation, etc.: electromagnetically neutral stuff ➠ gravity only

Dynamics = Gravity ➠ Einstein’s General Relativity Theory

Cosmology’s metric space: assuming cosmological principle and Weyl’s postulate ➠

Comoving coordinates = distinguished cosmological frame of reference in which matter is at rest

F. Jegerlehner Humboldt University, Berlin 1 Robertson-Walker Metric the Metric of Cosmology

Cosmological principle: the cosmos looks homogeneous and isotropic spatial part must be 3–dimensional space of ⇒ constant curvature. Case of positive curvature can be viewed as a sphere in a 4–dimensional space:

2 2 2 2 x1 + x2 + x3 + x4 = 1/k = constant

x2 = k ~x 2 = 1/k r2 2x dx = 2rdr 4 − − ⇒ 4 4 − with r = ~x . In spherical coordinates: | | (x ,x ,x ) (r, θ, ϕ) yields 1 2 3 → 2 2 2 2 2 dℓ = dx1 + dx2 + dx3 + dx4 r2 dr2 = dr2 + r2 (dθ2 + sin2 θdϕ2) + 1/k r2 − dr2 = + r2 (dθ2 + sin2 θdϕ2) 1 kr2 − We can rescale this spatial line elements with the factor R2 and restrict k to values 0, 1: ± dr2 dℓ2 = R2 + r2 (dθ2 + sin2 θdϕ2) 1 kr2 − then Rr = physical radius, r = coordinate radius and the space has curvature k/R2.

F. Jegerlehner Humboldt University, Berlin 2 The parameter k characterizes the type of the geometry of the spaces of constant curvature

Type Geometry Curvature

k = +1 spherical positive k = 0 euclidean no (ﬂat) k = 1 hyperbolica negative − ❒ Cosmological principle + Weyl’s postulate (stuff in universe follows hydrodynamical ﬂow) ➠

the complete Robertson-Walker line element then may be written as

dr2 ds2 = c2dt2 R2(t) + r2 (dθ2 + sin2 θdϕ2) − 1 kr2 −

Determines all measurements of space and time in the universe!

to every point there exists a local rest frame: • co-moving system, co-moving coordinates

co-moving observer measures t as Eigentime • t is called the cosmic time • ρ(P ): total energy density • p(P ): isotropic pressure in the co-moving system only depend on t (cosmological principle, Weyl postulate)

F. Jegerlehner Humboldt University, Berlin 3 Propagation of light and spectral red shift: d2s =0

in radial direction: dθ =0; dϕ =0 cdt = dr R(t) √1 kr2 → − Spectral ratio: . ζ =1+ z = ν1 =1+ ∆λ ; ∆λ = c (∆t ∆t ) ν0 λ1 0 − 1 z is the spectral shift:

red shift ∆λ z > 0 z = λ1  z < 0 blue shift 

r1  t0 1 dr = dt c √1 k r2 S(t) r0=0 − t1 R R independent of t.

The r integral yields

r arcsin r ; k = 1 dr′ =  r ; k = 0 √ 2 1 kr′  Z0 −  arcsinh r ; k = 1 −   Red shift:

∆t0 ∆t1 ∆t0 S(t0) S(t0) = S(t1) hence ζ =1+ z = ∆t1 = S(t1)

Result: the time dependent scale factor S(t) implies a red shift

z = S(t0) 1 S(t1) −

F. Jegerlehner Humboldt University, Berlin 4 Distances and Hubble’s law:

The coordinate distance between an event (t1,r1) and the observer (t0,r0 = 0) is r1. The proper distance D can be deﬁned by setting dt =0: (τ = t t ) 1 0 − 1 r1 dr D1 = S(t0) √1 kr2 Z0 − r3 = S(t ) r + k 1 + 0 1 6 ··· t0 S(t ) = c dt 0 S(t) tZ1 τ τ 2 = cτ 1+ H + H2 (2 + q )+ 2 0 6 0 0 ···

Inverting this relation, we obtain D H D 2 H2 D 3 τ = 1 0 1 + 0 (1 q ) 1 + c − 2 c 6 − 0 c ··· and the red shift is the given by

2 2 z = H D1 + H0 (1 + q ) D1 + 0 c 2 0 c ···

Here, D1 is the present distance (observation time) of the galaxy.

The form of the relation exhibits the two key parameters H0 and q0 and

D1 z = H0 c

is the Hubble law.

The latter would be valid as an exact relation if the expansion would be

F. Jegerlehner Humboldt University, Berlin 5 non-accelerated: q0 =0 such that S(t)= S(t ) (1+(t t ) H ) ; q =0 0 − 0 0 0

Depending on the value of q0 we have:

q > 0 the expansion is decelerated • 0 q < 0 the expansion is accelerated • 0 Observed: ﬁrst by Hubble 1929:

Present status: H = 74.2 3.6 (km/s)/Mpc [HST 2009] 0 ±

In 1968, the ﬁrst good estimate of H0, 75 km/s/Mpc, was published by Allan Sandage, but it would be decades before a consensus was achieved. Proved universal expansion!!!

Measuring S˙(t0) as well as S¨(t0):

F. Jegerlehner Humboldt University, Berlin 6 S˙(t0) Hubble constant H0 = H(t0)= S(t0)

S¨(t0) S(t0) q0 = q(t0)= 2 deceleration parameter − (S˙(t0)) A closer look: deviation from linear relation, from observations one presently extracts:

H0 = 74.2 ± 3.6 km/sMpc(HST) 71 ± 4 km/sMpc (WMAP)

q0 = −0.60 ± 0.02

Type Ia super nova reveal an accelerated expansion of our universe

Requires positive cosmological constant (Λ > 0)! Proves accelerated expansion!!!

Supported by power spectrum of CMB ﬂuctuations!

Note: Cosmological constant not so important for evolution of universe in the

F. Jegerlehner Humboldt University, Berlin 7 past, but determines its future! (see Freidmann’s equations)

For q0 =0 the age of the universe is

1 9 ● H− = (13.7 0.2) 10 years [Hubble age] 0 ± · the time back a Big Bang must have been taking place.

The corresponding horizon of the universe, where the escape velocity reaches the speed of light c is ● D = c 3.34 Gpc max H0 ≃ about 10.9 109 light years. · Note: observable Cosmology commences at > 1 Gpc and ends at the horizon 3.34 Gpc. ∼

F. Jegerlehner Humboldt University, Berlin 8 Friedmann equation

Homogeneous, isotropic matter distribution: Einstein’s ﬁeld equations Friedmann’s solution ⇒

2 ˙ 2 2 R 8πGρtot c H = = k 2 R! 3 − R

G Gravitational constant

ρtot Total energy-matter density R(t) Radius of the universe R˙ (t) Expansion rate (R˙ dR/dt) ≡ k 0, 1 Type of geometry ± Cosmological constant Λ: Λ ρ + ρ ρ ; ρ = tot|Λ=0 Vacuum→ tot Vacuum 8πG For simplicity we take: Λ = 0 (was not relevant in past, just starts to be relevant) Observations require Λ > 0 (dark energy)!

F. Jegerlehner Humboldt University, Berlin 9 Friedmann’s equation: derivation

Energy balance: per unit mass (mass in spherical shell = 4πR2dR ρ)

Kinetic energy + potential energy = constant 1 G M(R) v2 = constant 2 − R 4π M(R)= R3 ρ, v = dR/dt R˙ 3 ≡ yields, after division by R2 and multiplication with 2, 2 R˙ 8πG ρ c2 = k 2 R! − 3 − R

F. Jegerlehner Humboldt University, Berlin 10 Basic dynamical equations of Cosmology

R˙ 2 = Λ R2 + 8πG ρR2 kc2 : Friedmann equation 3 3 − d (ρR3) = 3p R2 : Energy balance dR − p = p(ρ) : Equation of state

The equation of state of matter and radiation is determined by the thermodynamic properties of the energy distribution: gas, liquid, solid, dust, radiation. For the state of matter electromagnetic, weak and strong interaction forces play the key role.

Of interest for cosmology: Radiation: 1 2 dominates at very high temperatures p = 3 ρc Matter pressure-free: p = 0 after cooling down and dilution by expansion

Vacuum energy: p = ρc2 after cooling in future becomes dominating term −

F. Jegerlehner Humboldt University, Berlin 11 Observable quantities:

Curvature : K(t) = k/R2(t) Hubble constant : H(t) = R˙ (t)/R(t) Deceleration parameter : q(t) = (R¨(t) R(t))/R˙ 2(t) − Friedmann equation:

2 κ H (t) = 2 K(t) 2 (ρ(t) + p(t)) c (q(t)+1) − 2 κ H (t) λ = (ρ(t) + 3p(t)) 3 2 q(t) 2 − c

2 8πG ( , 2 ) λ = Λ/c κ = 3c

F. Jegerlehner Humboldt University, Berlin 12 Matter dominated universe

Universe as a gas cloud of galaxies

Non-relativistic gas, kinetic energy negligible relative to rest mass: dust

p 0 ∼ Density of non-relativistic particles is proportional to 1/R3

3 3 ρ(t) R (t) = ρ0 R0 = constant R˙ 2 κ C = ρ R2 k = mat k c2 3 − R − κ C ρ R3 = constant mat ≡ 3 elementary integrable k = 1: R(t)= Cmat (1 cos η), ct = Cmat (η sin η) 2 − 2 − 9 1/3 2/3 k = 0: R(t)=( 4 Cmat) (ct) k = 1: R(t)= Cmat (cosh η 1), ct = Cmat (sinh η η) − 2 − 2 − Behavior for small t: 9 R(t) ( C )1/3 (ct)2/3 , k = 0, 1 ≃ 4 mat ± independent of k ! all three cases: Big-Bang solutions Cold “Big-Bang” !!!

F. Jegerlehner Humboldt University, Berlin 13 Behavior for large t:

k = 1: cyclic, cycle ctc = πCmat 9 1/3 2/3 k = 0: R(t)=( 4 Cmat) (ct) k = 1: R(t) ct − ≃

The product M = ρR3 is constant and corresponds to the total mass. Galaxies, clusters of galaxies, dark matter (v c) ≪

F. Jegerlehner Humboldt University, Berlin 14 Einstein - de Sitter Universe

k = 0 is the limiting case in between closed and open universe No curvature : K(t) 0 ≡ Deceleration parameter : qEdS(t) 1/2 ≡ 2 3 H (t) Matter density : 2 ρEdS(t) = κ c κ C ρ R3 mat ≡ 3 κ 3 H2(t) = R3 3 κ c2 κ 3 H2(t) 9 = C (ct)2 3 κ c2 4 mat Hence cosmic time is: 2 −1 t = 3 H (t) Today: H 0.52 10−28 cm−1 c 0 ∼ × × ρ 4.6 10−30gr/cm3 c2 0EdS ≃ × × − t = 2 H 1 13 109 years 0EdS 3 0 ≃ × Age of the universe if it would be ﬂat! ⇒ (CMB says: universe is ﬂat!) Generally:

ρ>ρEdS if k = +1 ρ(t) = ρEdS(t) 2q(t) · ⇒ ( ρ<ρEdS if k = 1 −

F. Jegerlehner Humboldt University, Berlin 15 Critical density: ρ ρ c ≡ EdS Density parameter in units of the critical density:

Ω0 = ρ0/ρc

Ω = 1 Critical density Space just remains open, infinite, flat. ⇒ Ω > 1 Lots of matter Space closes under gravitational attraction! ⇒ Gravity wins and stops expansion. The universe collapses into itself, universe dies from heat. Ω < 1 Little matter Gravity not sufficient to stop expansion, universe ⇒ expands forever, space open, universe dies from cold.

F. Jegerlehner Humboldt University, Berlin 16 Radiation dominated universe

the universe as a ﬁreball

1 2 p = 3 ρc

Since E R−1 we have γ ∼ energy density of relativistic particles is proportional to 1/R4

4 4 ρ(t) R (t) = ρ0 R0 = constant R˙ 2 κ C2 = ρ R2 k = rad k c2 3 − R2 − κ C2 ρ R4 = constant rad ≡ 3 elementary integrable k = 1: R(t) = C2 (C ct)2 rad − rad − k = 0: R(pt) = √2 Crad ct k = 1: R(t) = (C + ct)2 C2 − rad − rad Behaviorp at small t: R(t) 2 C ct, k = 0, 1 ≃ rad ± independent of k !p all three cases: Big-Bang solutions Hot Big-Bang !!! HOT BIG BANG

F. Jegerlehner Humboldt University, Berlin 17 k = 1: cyclic, cycle ctc = 2Crad

Behavior at large t: k = 0: R(t) = √2 Crad ct k = 1: R(t) ct − ≃

The product S = ρR4 is constant and corresponds to the total entropy. Radiation, photons, neutrinos, highly relativistic particles (v c) ∼

F. Jegerlehner Humboldt University, Berlin 18 For radiation in thermal equilibrium (black body radiation) the Stefan-Boltzmann law:

ρ = aT 4 c2 , a = 8.4 10−36gr/cm3(◦K)4 radiation × with ρR4 = constant follows 1 ρ T 4 ∝ ∝ R4 or

R0 T = T0 R

Isotropic microwave radiation = cosmic black body radiation

T 2.725◦K 0 ≃ Big-Bang radiation !

Planck’s spectrum: energy density (ω =2πν) 1 ω2 1 ~ω u = ǫ(T ) , ǫ(T )= ~ω + ω π2 c3 2 exp ~ω/kT 1 −

F. Jegerlehner Humboldt University, Berlin 19 General covariant formulation:

Four-velocity ﬁeld (liquid, gas): uµ = dxµ/ds

in the co-moving system uµ = (1,~0)

particle current: . jµ(x) = nuµ

Energy-Momentum-Tensor: . T µν (x) =(ρ + p) uµuν pgµν −

2 8πG Einstein equation: ( , 2 ) λ = Λ/c κ = 3c 1 R g R λg = κT µν − 2 µν − µν µν

For RW-metric ➠ Friedmann’s equation!

F. Jegerlehner Humboldt University, Berlin 20 Thermodynamics

T Temperature k Boltzmann constant kT Thermal energy ǫ = ρc2 Energy density n Particle density p Pressure m particle mass g Spin factor (number of degrees of freedom) µ Chemical potential a + b c + d ↔ µa + µb = µc + µd Particle number in thermal equilibrium:

2 2 2 2 g ∞ E /c m c E dE n Fermions = 2 3 − 2 Bosons 2π ~ 2 E µ c mc pexp − 1 Z kT ± 2 2 2 2 g ∞ E /c m c E dE ǫ Fermions = 2 3 E − 2 Bosons 2π ~ 2 E µ c mc pexp − 1 Z kT ± 2 2 2 2 g ∞ E /c m c E dE p Fermions = 2 3 − 2 Bosons 2π ~ 2 E µ c mc pexp − 1 Z kT ±

F. Jegerlehner Humboldt University, Berlin 21 Explanations: distribution functions

Density of states in momentum space (Quantum Mechanics): d3p d3p p2dpdΩ = = h3 (2π~)3 (2π~)3 Integrated over angles in case of isotropic distribution: p2dpd p2dp dΩ = (2π~)3 2π2~3 Z Since p = E2/c2 m2c2 we have pdp = EdE/c2 and − therefore p 1 E dE E2/c2 m2c2 2π2~3 − c2 is the density of statesp in the energy distribution , while 1 exp E−µ 1 kT ± is the statistical occupation probability (Quantum statistics). Fermions can occupy a state 0- or 1-times (Pauli-Principle), − −1 leads to weight factor E µ exp kT + 1 Bosons can occupy any state arbitrarily often, leads to the − −1 weight factor exp E µ 1 kT −

F. Jegerlehner Humboldt University, Berlin 22 Limiting cases

In the relativistic limit (kT mc2) one has ≫ 4 ζ(3) g kT 3 = = 2 nBosons 3 nFermions π ~c 2 3 = 8 = π g kT ǫBosons 7 ǫFermions 30 ~c kT 2 3 In the = 8 = 2π g kT sBosons 7 sFermions 45 ~c k 1 p Fermions = ǫ Fermion Bosons 3 Bosons non-relativistic limit (kT mc2) the difference between ≪ Fermions and Bosons disappears:

kT 3 mc2 3 mc2 n = g exp ~c 2πkT − kT s mc2n s = , ǫ = nmc2 , p = nkT T Energy density of the relativistic components: Stefan-Boltzmann law π ~c 3 ρc2 = g∗ kT 30 kT with g∗ the effective number of relativistic degrees of freedom = sum of all contributions of the relativistic particle species.

F. Jegerlehner Humboldt University, Berlin 23 g-Factor

Degrees of freedom, particle species for kT 300GeV ≫ all particles of the Standard Model are in thermodynamic equilibrium free quarks and gluons all masses negligible all particles at the same temperature Fermions: Leptons : e±(2), µ±(2), τ ±(2), 3 ν(1), 3ν ¯(1) Quarks : 3(uu,¯ dd,¯ ss,¯ cc,¯ b¯b, tt¯)(2) Bosons: W ±(3), Z0(3), γ(2),Higgs(1), 8 Gluons(2) Fermions : 90 7 = 315 · 8 4 Bosons : 28

Total : 427 g = 4 = 106.75 With decreasing temperature also g decreases, Particles freeze out as a consequence of their mass.

F. Jegerlehner Humboldt University, Berlin 24 “Planck Epoch”

Gravitation, relativity (electromagnetism) and quantum theory each are characterized by a typical fundamental constant ⇒

Basic constants of physics: G, c, ~ (Planck 1913)

They may be combined into

“smallest” distance: Planck length: ~G 33 ℓ = 3 = 1.616252(81) 10− cm Pl c · q“biggest” mass: Planck mass: ~/c 5 m = = 2.2 10− gr Pl G · shortestq time: Planck time: 44 t = ℓ /c = 5.4 10− sec Pl Pl ·

F. Jegerlehner Humboldt University, Berlin 25 Using E = mc2 we obtain: ~ highest energy m = c = 1.22 1019 GeV Pl G · q Planck scale

Using Ethermal = kBT we obtain: highest temperature 2 5 mPlc ~c 32 ◦K k = Gk2 = 1.416786(71) 10 B B · (Temperatureq of the Big Bang)

Classical cosmology based on GRT stops to be valid

for t

At the Planck scale: uniﬁcation of gravity with the other forces quantum gravity expected. No compelling answers, no working theory.

F. Jegerlehner Humboldt University, Berlin 26 History of the Universe

The further we look back to the past, the universe appears to be compressed more and more. We therefore expect the young universe was very dense and hot:

Density

Radiation ւ Matter ւ

Time Radiation Matter ↑ dominates dominates TODAY

At Start a Light-Flash: ➽ Big-Bang (ﬁreball) Light quanta very energetic, all matter totally ionized, all nuclei disintegrated. Elementary particles only!: γ, e+, e−,p, p,¯ ···

+ Particle– Processes: 2γ e + e− 2γ ↔ p¯ + p Antiparticle . ↔ Symmetry! .

☛ Digression into high energy physics: example LEP

F. Jegerlehner Humboldt University, Berlin 27 + e e− Annihilation at LEP

45.5 GeV 45.5 GeV

Electron Positron (Matter) ✷(Antimatter)

2 E =2meﬀc

Mini Big Bang !

F. Jegerlehner Humboldt University, Berlin 28 What happens in Electron Positron Collisions? •

Matter and Antimatter annihilate to pure light or heavy light (Z’s), which re-materializes in new forms of matter (mostly showers of unstable particle).

Matter: Electron e− J. J. Thomson 1897 Proton p E. Rutherford 1911 Neutron n J. Chadwick 1932 Relativity Theory A. Einstein 1905

+ W. Heisenberg, Quantum Theory E. Schrödinger 1925 P. Dirac, M. Born

Antimatter: Positron e+ P. Dirac 1928, C.D. Anderson 1932

Antiproton p¯ E. Segré, O. Chamberlain 1955

Antineutron n¯ B. Cork et al. 1956/58

Science Fiction as Reality!

Theory: Matter Antimatter Symmetry ↔ To every particle there exists an antiparticle of opposite charge Nature: in our universe: antimatter is missing!

F. Jegerlehner Humboldt University, Berlin 29 Energy versus temperature correspondence:

5 1◦K 8.6 10− eV ≡ × (Boltzmann constant kB)

↓ temperature of an event T 1.8 1011 T ∼ × × ⊙ In nature such temperatures only existed in the very early universe: 2 t = 2.4 1MeV sec. kBT √g∗(T ) ↓ 10 t 0.3 10− sec. after B.B. ∼ × early universe

0◦K 273.15◦ C absolute zero temperature ≡−

◦ T⊙ 5700 K surface temperature of the Sun ≃ g (T ) number of highly relativistic degrees of freedom at given T ∗

F. Jegerlehner Humboldt University, Berlin 30 History of the Universe (continuation)

10−43 sec. quantum-chaos . . ??? · 2 10−6 sec. nucleon–antinucleon annihilation  × ·   matter  −9 ·  Relict: nB/nγ 10 visible  ⇒ ≃ ·  today ·  2 sec. electron–positron annihilation   ·   200 sec. Helium synthesis  yields neutrinos ! have tiny masses

may contribute substantially to → matter density of the universe

500’000 years from now on matter dominates

700’000 years Hydrogen recombination ·  ⇒ universe transparent ·  there are no free charges anymore  photons decouple

and cool down further

neutral matter (atoms)

galaxies, clusters of galaxies, ··· 13 Billion years today

F. Jegerlehner Humboldt University, Berlin 31 “Annihilation Drama of Matter”

X q e ··· − ν 35 10− sec. ??? γ X¯ e+ ··· q¯ ν¯

¯ q :q ¯ = 1,000 000 001:1 X, X–Decay: + ⇒ { e− : e = 1,000 000 001:1 W q e− ν 30 10− sec. γ W¯ q¯ e+ ν¯

10 0.3 10− sec. LEP events ×

qq¯ γγ:

→ q e− ν 4 10− sec. γ e+ ν¯ + e e− γγ: → q e− ν 1 sec. γ ν¯

F. Jegerlehner Humboldt University, Berlin 32 “Nucleo-Synthesis”

quark-gluon-plasma❊ ❊ ❊

❊ q ❊

q ❊ nucleons (proton , neutron )

❊ ❊❊ ❊ N p n ❊ N q

❊❊ ❊ → p Some protons p and neutrons n form Helium nuclei n n p

After 700 000 years: light atoms (electrically neutral)

• •

p p n n p

Helium Hydrogen •

H : He 3:1 together 98% of baryonic∼ matter ! Where are the γ’s ?

Cosmic Microwave Background (T 2.725◦ K) ⇒ ≃

F. Jegerlehner Humboldt University, Berlin 33 ✩ CMB Discovery

Penzias & Wilson 1965, Nobel Prize 1978

to a full proof that the isotropic Cosmic Microwave Background indeed follows a Planck distribution COBE satellite

(Mather 1990) ✫ T = (2.725 0.02)◦K 0γ ±

F. Jegerlehner Humboldt University, Berlin 34 A compilation of CMB black-body spectral data

✩ CMB the Proof and the Noise

(Smoot 1990)

❏ δT/T 10−5 ∼ G.F. Smoot, J.C. Mather, Nobel Prize 2006

F. Jegerlehner Humboldt University, Berlin 35 ΩB from Big Bang nucleosynthesis provides strong limits for the existence of new light particles (not neutrinos only) and also yields e.g. a bound for the τ neutrino mass:

< mντ ∼ 0.5MeV ! excludes range 0,5 MeV to 25 MeV ⇒ (25 MeV is the present limit from laboratory experiments)

Nucleosynthesis of the light elements is very sensitive to details. Only works if standard cosmology describes correctly the history of the universe back to

t ∼> 0.01 sec. after B.B. i.e. up to temperatures equivalent to

T ∼< 10 MeV.

F. Jegerlehner Humboldt University, Berlin 36 Dark Matter

Galaxy rotation curves! missing mass!

F. Jegerlehner Humboldt University, Berlin 37

10: 1

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invisible visible

“halo” galaxy

F. Jegerlehner Humboldt University, Berlin 38 Cosmic Microwave Background: COBE, WMAP

CMB temperature ﬂuctuations ❏ δT/T 10−5 ∼ Spherical harmonics analysis of temperature ﬁeld ∆T (ˆn) power ⇒ spectrum ℓ (ℓ + 1) Cℓ

F. Jegerlehner Humboldt University, Berlin 39 Acoustic peaks: baryons oscillating in ripples of dark matter; ◦ ◦ angular size φ 180 ℓ 180 ∼ ℓ ⇒ ∼ φ ❒ oscillations damped by Silk damping at large ℓ

❒ low ℓ 5 obscured by cosmic variance ≤ ❒ low ℓ> 5 Sachs-Wolfe effect: gravitational ﬂuctuations red shift partly eaten up by simultaneous change in Hubble rate

● 1st main peak horizon at last scattering total energy/mass content ⇔ ● 2nd peak total baryon density (in agreement with result from Big Bang nucleosynthesis of light elements)

● 3rd peak gives total dark matter density

speciﬁc information in height of the peaks!

F. Jegerlehner Humboldt University, Berlin 40 ● dark energy 70% in conjunction with constraint from deceleration parameter in SN Ia redshifts ● dark matter (as required independently by many other observations): 25%

● baryonic matter: 4.5% ∼ ● neutrinos 0.3%

Remark: ❒ Astronomers estimate Ω =0.2 0.1 0 ± Resulting model: ΛCDM–model

F. Jegerlehner Humboldt University, Berlin 41 Inﬂationary world model: Ω0 = 1

ΩB = Ω(Nucleons)<0.05 ∼

??? ΩX 0.95 ??? ∼

??? What are to other 95% of the universe made of ??? New in recent years: Einstein’s GRT must be modiﬁed to include cosmological constant Λ ⇒ interpreted as dark energy

A model for dark energy is the SM Higgs mechanism!

2 mH ∆Tµν = gµν VHiggs(v) = gµν 8√2 GF

F. Jegerlehner Humboldt University, Berlin 42 V (H) the Higgs potential (in unitary gauge) and v =< H > Higgs vacuum expectation value. Equation of state of Higgs condensate p = ρ c2. H − H Precisely the right pattern but 60 orders of magnitude too big!

The cosmological constant question is not

why is there a cosmological constant,

rather why is it so small?

Other sources, quantum vacuum ﬂuctuations, quark condensates, other spontaneous symmetry

breaking at higher scales etc. Also VHiggs(0) = 0 assumed above, no justiﬁcation!

What about the missing dark matter?

F. Jegerlehner Humboldt University, Berlin 43 ➜ WIMPS ? (Weakly Interacting Massive Particles)

WIMPs determine the type of structures one observes in the universe: distribution of galaxies, clusters of galaxies, voids, ... WIMPs ⇒ weak=weak force ➥ decouple earlier ➥ can build structures earlier (clumping matter) HDM: Hot Dark Matter (m 50 eV) ≈ Candidates: Light Massive Neutrinos

(neutrinos have tiny masses, provide established form of DM!) Contribution too small, only can account for tiny fraction of DM CDM: Cold Dark Matter (m 105 1012 eV) ∼ − Candidates: NEUTRALINO in SUSY-theories with conserved R-parity

F. Jegerlehner Humboldt University, Berlin 44 Comparison

COBE/WMAP data (large scales)

m Data on distribution of galaxies (small scales)

Ωtot = 1.006(6)

ΩΛ = 0.74(3)

ΩM = 0.26(2)

ΩCDM = 0.21(2)

ΩB = 0.044(4) 5 Ω = 4.48(4) 10− γ · Ω = 0.0009 0.048 ν − Ω = Ω Ω Ω CDM M − B − ν ΩHDM = Ων( ν mν<1 eV) ∼ P PDG 2010

F. Jegerlehner Humboldt University, Berlin 45 Problem of ﬁne-tuning

Time evolution of the density function: 1 Ω(t)= 1 x(t) −

3k/R2 x(t)= 8πGρ

In order that the density today has a value 10 times higher or 10 times lower than the critical value one has to assume that at Planck times the density was coinciding to about 60 digits with the critical value 1.

43 < 60 Ω(10− sec.) 1 ∼ 10− ! | − | Solution by Inﬂation

Cure: add an inﬂation term, which blows up the early universe, such that it looks ﬂat today: 8π 1 V (φ)+ φ˙2 3m2 2 P l to be added to the right hand side of Friedmann’s equation (must be the dominating for small times)

F. Jegerlehner Humboldt University, Berlin 46 “Baryogenesis”

n /n 0 ; n /n 10−9 p¯ p ≃ p γ ≃ Conditions for the possibility of the baryogenesis: (A. Sacharov 1967)

① Baryon number violating processes must exist (B–L violation!)

② CP violation ! Cronin, Fitch 1964 (NP 1980) (Violation of time-reversal symmetry) ﬁrst seen in neutral Kaon decays 0.3% effect, more recently established in B-meson physics 100% effect ∼ CP violation precisely as predicted by SM of elementary B-factories: BaBar and Belle ⇒ ③ Thermal far from equilibrium ( non-stationary time evolution (as predicted by Friedmann’s equations [GRT])) is true as the universe is expanding

Standard Model of elementary particle physics cannot explain these facts: ➜ NEW PHYSICSREQUIRED likely at energies not yet accessible by experiment (except LHC may reveal new physics and hopefully does)! GUT’S, SUSY, STRING’S ??? in GUT’s: X–bosons and leptoquarks! ATLAS, CMS and LHCB experiments at the LHC⇒

F. Jegerlehner Humboldt University, Berlin 47 “Nucleosynthesis”

After baryogenesis and e+e−-annihilation: n /n 0 ; n /n 10−9 p¯ p ≃ p γ ≃ At few seconds after B.B. and temperature T 1010 ◦K ∼ dropping fast

ne = np and nB = nP + nn conserved neutrons, protons in thermal equilibrium n + ν ⇌ p + e− , n + e+ ⇌ p +ν, ¯ n ⇌ p + e− +ν ¯

① Ratio nn/np is given by the Boltzmann equation: n 2 n =e−(mp−mn) c /kB T np which at T 1010 ◦K gives n /n = 0.223. ∼ n p ② Below T 1010 ◦K , no new neutrons are formed and ∼ the ratio is ﬁxed. Yet it is too hot for deuterium to form, so protons and neutrons remain free. The free neutrons decay into protons via β–decay (n p + e− + ν , half → e life 617 seconds). About 4 minutes after B.B. temperature has dropped to T 109 ◦K , and deuterium ∼ can form. At this point, neutron decay has rebalanced

the neutron to proton ratio to nn/np = 0.164. ③ Now time is ready for nucleosynthesis. At T 109 ◦K , ∼ deuterium will survive. So nuclear reactions will form deuterium, tritium, and helium:

F. Jegerlehner Humboldt University, Berlin 48 p + n Deuterium + n Tritium + p He4 → → → p + n Deuterium + p He3 + n He4 → → →

④ Now per proper volume: nHe helium, nH hydrogen

nuclei: close to all neutrons are in helium: nHe = nn/2, n = n 2 n = n n fractional abundance H p − He p − n ⇒ by weight of helium Y = 4nHe/(4nHe + nH ) = 2n /(2n + n n ) = 2n /(n + n ) = 2x/(1 + x), n n p − n n p n with xn /n = 0.164 at T 109 ◦K we have n p ∼ Y = 0.282. Observation close to 25%.

⑤ D is a steep function of baryon number n B ⇒ Baryometer ΩB = 0.02

⑥ Cosmic concordance: one value of nB predicts light element abundances 4He, D, 3He, 7Li rather well and in agreement with value form CMB!

F. Jegerlehner Humboldt University, Berlin 49 F. Jegerlehner Humboldt University, Berlin 50 Under conditions after the B.B. (low density of baryons after baryon antibaryon annihilation) only adding up nucleon by nucleon could be successful. Start of nucleosynthesis hindered by small binding energy of Deuteron [bottleneck]. Formation of D requires low enough temperature (at about T =0.1 MeV) but in the meantime neutrons density decreases by neutrons decaying. At the end most of the remaining neutrons survive bounded in the most sable of the light elements 4He.

The lack of stable elements 9Be of masses 5 and 8 makes it

very difﬁcult for BBN to 7Li 6Li progress beyond Lithium

4He and even Helium. Need No stable 3 He nuclei conditions found in stars for 2H formation of the heavy elements. 1H

The Bottlenecks in BBN 8B 10B 11B 12B

7Be 8Be 9Be

6Li 7Li 8Li

3He 4He 5He

Elements of order ➄ and ➇ missing • 1H 2H 3H (totally unstable) Barrier for production • 1n of heavier elements in the early universe heavy elements must have been produced in stars • (extreme temperature and pressure)

F. Jegerlehner Humboldt University, Berlin 51 The Standard Cosmological Model

● Standard ΛCDM cosmological model is an exceedingly successful phenomenological model

● Rests on three pillars ❒ Inﬂation: sources all structure ❒ Cold Dark Matter: causes growth from gravitational instability ❒ Cosmological Constant: drives acceleration of expansion that are poorly understood from fundamental physics

● ΛCDM and its generalizations to dark energy and slow-roll inﬂationary models is highly predictive and hence highly falsiﬁable.

Wayne Hu’s summary

F. Jegerlehner Humboldt University, Berlin 52 ❇ ❇ ❇ The hot Big Bang is supported very well by many facts. The picture of the universe is one of the outstanding intellectual achievements of the 20th century. It bridges physics at smallest scales with physics at largest scales - the cosmic bridge. It involves mysteries about its beginning and leads us into a uncertain future. ❇ ❇ ❇

Last but not least: Standard Model of particle physics cannot explain

Dark matter, • Baryogenesis [matter vs antimatter asymmetry] • (missing B violation, missing amount of CP violation),

Does not explain why cosmological constant is so small • Why universe is ﬂat [Inﬂation is beyond the SM physics] • One of today’s motivations and a great challenge for high energy particle physics Find what’s beyond the SM ● dark matter ● B-violating interactions ● and all that ....

F. Jegerlehner Humboldt University, Berlin 53 6 9 10−12 s 20 µs 100 s 10 y 15 10 y 1020 ×

baryon antibaryon 16 annihilation 10 electron radiation matter positron dominated dominated annihilation decoupling of W and Z K) ◦ 12 bosons 10

epoch of recombination QCD 8 10 phase neutrino barrier the present transition epoch log (Temperature

104 epoch of nucleosynthesis photon barrier galaxies/quasars 2.7 ◦K 1 10−18 10−16 10−14 10−12 10−10 10−8 10−6 10−4 10−2 1 log (Scale Factor)

Thermal History

Dark Ages after last scattering (epoch of recombination): between 150 million to 1 billion years reionization of atoms takes place. The ﬁrst stars and quasars form from gravitational collapse. The intense radiation they emit reionizes the intergalactic gas (hydrogen) of the surrounding universe. From this point on, most of the universe is composed of plasma. Stars cannot be seen until reionization becomes negligible due to ongoing expansion and decreasing matter density (dilution).

F. Jegerlehner Humboldt University, Berlin 54 As far as we can see.

F. Jegerlehner Humboldt University, Berlin 55