ASTROPHYSICS I: lecture notes, ETH Zurich 2018

9 covers in Choudhuri Section 9.3 and 9.6, and large fractions of Sections 10 and 11 notes provide a simplified treatment for book sections: 11.2 and 11.9. additional topic: the concordance model

9.1 Expansion of the local Doppler shifts, if objects move with radial velocity v relative to the observer – positive Doppler shift = z = motion away from observer – for v → c one needs to use the relativistic speed – the following is valid for v << c (say v < 0.1c)

∆λ vc v z = z}|{= λ c alternative description

λ λ + ∆λ ∆λ vc v obs = em = 1 + = 1 + z z}|{= 1 + λem λem λem c relativistic Doppler effect: v u λobs u1 + v/c = t = 1 + z λem 1 − v/c – (1 + β)/(1 − β) = (1 + β)(1 + β + β2 + β3 + ...) → (1 + β)2 for β → 0 – for v > 0.6c → z > 1 and ∆λ > λ

Hubble law

– Slipher (1914) noted that most galaxies are redshifted – Hubble (1929) proposed linear relationship between velocity and distance (Hubble law)

v = H0 d

– H0: Hubble constant (not a constant in time) – later studies confirmed this general expansion law

Interpretation: – the whole (observable) universe is expanding – no center can be recognized – we are part of a big explosion or Hubble flow

75 Determination of the Hubble constant

Task: distance determination for galaxies – absolute luminosities of galaxies are hard to determine accurately – must be calibrated in many steps, determining and using M for standard candles

  5 log10 d[pc] = m − M + 5 problems: – m can be affected by interstellar extinction – bright objects are rare and M is not well determined distance ladder

step method reachable calibration problems distance 1. trigonometric paral- 200 pc main sequence lumi- only few stellar clus- laxes nosity ters 2. main sequence fitting 100 kpc stellar clusters / LMC differencies in stellar (LMC) with Cepheids metallicities 3a. Cepheids P-L-relation 20 Mpc normal galaxies a lot of data required 3b. Cepheids P-L-relation 20 Mpc local Hubble law peculiar velocity of galaxies 3c. Cepheids P-L-relation 20 Mpc galaxies with SNIa SNIa are rare light curve 4. e.g. Tully-Fisher rel. 100 Mpc distant objects TF-rel. has large dis- persion 5. SNIa 2 Gpc deviations from linear best large distance Hubble law calibrator

Direct methods to determine Hubble law (alternative) gravitational lensing – time delays for individal images of lensed, variable quasar → distance to source and lens – problem: uncertainties in the gravitational potential of lensing object

Hubble constant: Problem for the determination of the Hubble constant: measured velocity: v = H0d + δv – peculiar velocities of galaxies δv ≈ ±1000 km/s – expected systematic effect H0 ≈ 50 − 100 km/s Mpc → systematic effects dominates only for d > 15 Mpc → for smaller distances the mean of a (large) sample is required

76 SLIDE: H0 from HST current value for the Hubble constant (HST-key program based on Cepheid distances)

−1 −1 H0 = 72 ± 8 km s Mpc radial velocity yields distance for δv  v v d = for v  c H0 or redshift → distance relation c d = z Mpc for v  c H0 Hubble time assuming that expansion was constant since the beginning of the Universe – time when all galaxies were at the same point

1 17 9 tH = = 4.4 · 10 s = 14 · 10 yr H0 → rough estimate for the

Redshift surveys galaxy distances follow from redshift for z > 0.01 (vr = cz > 3000 km/s) → 3-dim. galaxy distribution measured galaxy distribution shows: – filaments of enhanced galaxy densities – large empty bubbles (voids) – cluster of galaxies at intersections of filaments galaxy distribution on large scales: – isotropic – homogeneous

SLIDES: redshift surveys for galaxies

77 9.2 Newtonian Cosmology Cosmological principle: space is homogeneous and isotropic follows from: – galaxy distribution – 3-K cosmic micro-wave background (CMB)

Newtonian theory of gravity cannot describe universe: – g(r) = ∇Φ(r) = ∇(G R ρ/(r0 − r)dr0 – homogeneous space has a homogeneous gravitational potential Φ(r) = const – if ∇Φ = 0, then g = 0, → wrong → general relativity is required for description of the gravitation of the Universe

Simplification: Newtonian cosmology – important cosmological equation from Newtonian mechanics agree with general relativity → easy description of cosmology

Conceptual differences between Newtonian cosmology and relativistic cosmology:

Newtonian cosmology relativistic cosmology we are at the center of the Universe all points in space are equivalent galaxies are moving away in space space is expanding with galaxies redshift cause by Doppler effect due to re- redshift caused by stretching of light wave- cession of galaxies length due to expansion

Newtonian cosmology adopts 2 principles of the field theory from general relativity

– A. special relativity is valid like in electrodynamics (Maxwell’s equation) – information propagates with c (action at a distance is not instantaneous)

– B. Einsteins equivalence principle – a gravitational field is equivalent to an accelerated frame – gravity is considered as curvature of space time (gravity → spacetime is curved) – bodies are deflected from a rectilinear path in a gravitational field

9.3 The metric in general relativity definitions: metric: description of the curvature of spacetime geodesic: shortest path between two points

Examples A. no gravitational fields → space is flat and bodies move on straight lines B. with gravitational fields → space is curved and a body moves on curved path

78 Curvature for 2-dim. surfaces: A: flat (plane) surface – straight lines are geodesics – sum of the 3 angle in triangles α + β + γ = 180◦ B: curved surface – geodesic are curved – triangle α + β + γ 6= 180◦ – spherical surface: great circles are geodesics metric describes: 2 – distance squared ds between two nearby points (x1, x2, ...) and (x1 + dx1, x2 + dx2, ...)

Examples: – for a plane surface in polar coordinates (r, theta): ds2 = a2(dr2 + r2dθ2) – for a spherical surface with radius a (θ, φ): ds2 = a2 (dθ2 + sin2 θdφ2) = a2dΩ2

– general formulation (gα,β is the metric tensor)

2 ds = Σα,β gα,β dxαdxβ

9.4 Metric for the universe cosmological principle: homogeneous and isotropic geometry → metric has uniform (constant) curvature everywhere → there are only three cases under these conditions – zero curvature → flat (plane) geometry – positive curvature → spherical geometry – negative curvature → saddle-like geometry

SLIDE: spherical and saddle like surfaces

Overview on the mathematical formulation of metrics:

curvature zero positive negative plane (flat) spherical saddle-like 2-dimensional surface 2 2 2 2 2 2 2 2 2 2 2 2 2 ds = a (dx1 + x1dx2) a (dx1 + sin x1dx2) a (dx1 + sinh x1dx2) 3-dimensional space ds2 = a2 (dχ2 + χ2dΩ2) a2 (dχ2 + sin2 χdΩ2) a2 (dχ2 + sinh2 χdΩ2) 3-dimensional space (alternative formulation) χ = r sin χ = r sinhχ = r ds2 = a2 (dr2 + r2dΩ2) a2(dr2/(1 − r2) + r2dΩ2) a2(dr2/(1 + r2) + r2dΩ2) 3-dimensional space (compact formulation) k = 0 k = +1 k = −1 ds2 = ← a2((dr2/(1 − kr2) + r2dΩ2) →

79 Example: Volume of the positive curved Universe positive curved space metric has a finite volume volume element with length: – a dχ, a sin χdθ and a sin χ sin θdφ – yields volume element dV = a3 sin2χdχ sin θdθdφ and total volume (χ extending from 0 to π)

Z χ=π Z θ=π Z φ=2π V = a3 sin2 χdχ sin θdθ dφ = 2π2a3 χ=0 θ=0 φ=0 → comparable to a sphere which has a finite surface area without any edges

Spacetime: add a time part according to special relativity

ds2 = −c2dt2 + dx2 + dy2 + dz2 → Robertson-Walker metric = 4-dimensional spacetime metrics of the Universe:

dΩ  dr2 z }| { ds2 = −c2dt2 + a(t)2 + r2 (dθ2 + sin2 θdφ2) 1 − kr2 – only three values k = 0, +1, −1 for homogeneous and isotropic space – a(t) is the scale factor which can vary with time !

An expanding universe: – coordinates (r, θ, φ) defined with respect to a references (us) – co-moving coordinates (r, θ, φ) are constant in time for a galaxy (peculiar motion of galaxies are neglected) measure of distance d(t): Z r 1 d(t) = a(t) √ dr0 0 1 − kr02 speed of the expansion v(t): Z r 1 v(t) =a ˙(t) √ dr0 0 1 − kr02 → equivalent to the Hubble law v(t) a˙(t) H(t) = = d(t) a(t)

80 9.5 Friedmann equation Universe models based on Friedmann equation – old, outdated models – but give a simple picture on basic cosmological models

Assumptions: – spherical shell with radius a – located inside a larger volume with constant density:

SLIDE: sketch for spherical expansion potential energy per unit shell mass GM (4/3)πGρa3 4 E = − = = − πGρa2 pot a a 3 kinetic energy per unit shell mass 1 E = a˙ 2 kin 2 the total mechanical energy E (constant of motion) 1 4 E = a˙ 2 − πGρa2 2 3 is equal to the total energy described by the metric of the Universe (follows from Einstein equation): kc2 E = − metric 2 where k = 0, −1, +1 Friedmann equation: a˙ 2 kc2 8πG + = ρ a2 a2 3 Analogy with mechanics: – a projectile with positive total energy escapes to infinity – a projectile with negative total energy falls back due to the attraction of gravity k has opposite sign to energy E: – k = −1 → Universe will expand forever, E > 0 – k = +1 → Universe expansion stops and collapses, E < 0 – k = 0 → Universe is exactly on borderline, or expansion stops for a = ∞, E = 0

Borderline: – k = 0 defines the critical density of the Universe ρc 3H2 ρ = c 8πG

ρc for present day Hubble constant H0 is −26 −3 ρc,0 ≈ 1 · 10 kg m (corresponds to the mass of 6 protons per m3)

81 density parameter: ratio between measured ρ0 and critical ρc,0 ρ Ω = 0 ρc,0 Temporal evolution parameters evolve with time: – a(t),a ˙(t), H(t), ρ(t) and Ω(t) – values at present epoch: a0,a ˙ 0, H0, ρ0 and Ω0 – evolution of energy density: – energy density = matter density, for a matter dominated universe

a 3 ρc2 = ρ c2 0 0 a Solutions for the :

A. The critical solution (k = 0), 3 3 – ρ = ρc, Einstein-de Sitter model, (using ρ = ρ0a0/a ) s s 8πG 8πG a˙ = a ρ = a−1/2 ρ a3 3 3 0 0 Integration yields: s 3 8πG a3/2 = ρ a3 t 2 3 0 0 2 with the critical density ρ0 = 3H0 /8πG

a 3 2/3 = H0 t a0 2 for a Friedmann model with k = 0 the Universe expands ∝ t2/3: → expands for ever → expansion gets continuously slower

SLIDE: Friedmann models

B. The closed solution (k = +1)

– ρ > ρc – the Universe expands, reaches a maximal point and then collapses again – this Universe model has a finite volume – and a finite life time between and

C. The open solution (k = −1)

– ρ < ρc – the Universe expands, faster than for the k = 0 case – could be a good approximation for the early universe

82 Einstein equation

Einstein noticed (1917), that general relativity yields no steady state solution for ρ 6= 0: – the Hubble expansion was not known at that time – therefore he introduced a cosmological constant Λ, which allows for a steady state

a˙ 2 kc2 8πG Λ + = ρ + a a2 3 3

9.6 The cosmic micro-wave background (CMB) SLIDE: cosmic micro-wave background spectrum very important detection in – 1965 by Penzias and Wilson (radio engineers at Bell labs) – isotropic, black-body radiation of TCMB = 2.73 ± 0.06 K (mm-range = microwaves)

Interpretation: → redshifted (expanded) radiation from hot big bang → since then: expanding universe cooled – gas optically thick when ionized because of electron scattering – gas optically thin when neutral – transition happens for T ≈ 3000 K (Saha equation) – black-body radiation escapes and is still observable – wavelengths expand with Universe → TCMB ≈ 3 K today

Energy density of the background radiation (-radiation field):

2 4 ρRc = aBT

ρR: mass equivalent for energy density of photons aB = 4σ/c radiation constant relation between energy density and scale factor: a 4 ρ = ρ 0 R R,0 a

– ρR,0 current radiation density of CMB 3 – factor (a0/a) from the expansion of the universe – factor (a0/a) from the cosmological redshift of photons → evolution of the temperature of CMB 1 T ∝ CMB a Evolution of the radiation and mass (energy) density of the universe a 3 a 4 ρ = ρ 0 + ρ 0 M,0 a R,0 a

– ρR falls more rapidely ρM – ρR > ρM : radiation-dominated Universe at “early” phases

83 – ρR < ρM : matter-dominated Universe at later phases

Evolution of a radiation dominated Universe

Friedmann equation for k = 0 or flat universe: 4 but using ρR = ρR,0(a0/a) s 1 8πGρ a˙ = R,0 a2 a 3 0 Solution: a 32πGρ 1/4 = r,0 t1/2 a0 3

→ temperature evolution of the early universe (with a/a0 = T0/T there is):

 3c2 1/4 T = t−1/2 32πGaB or inserting values: 1.52 · 1010 1.31 T [K] = q or T [MeV ] = q t [s] t [s]

9.7 The concordance model in cosmology – description of the evolution and energy density of the Universe – the currently accepted and most commonly used model – easy comparison of data and models – many things are still not proven

Hubble constant: H0 = 71 km/(s Mpc)

Energy density of the universe: luminous mass – luminous stars producing UV-optical-IR continuum light): –ΩLum = ρLum/ρc ≈ 0.01 baryonic matter (protons, nucleons and electrons) – stars (including M-dwarfs and white dwarfs), interstellar and intergalactic matter –Ωb ≈ 0.05 (weakly interacting massive particles WIMPs) – as measured by rotation curves, virial theorem of clusters and gravitational lensing –Ωd ≈ 0.25 – nature of these particles is unknown – neutrinos contribute only a few % to dark matter – an apparent acceleration of the cosmological expansion – based on SNIa distances (SNIa are too faint for a decelerating expansion) – “repulsive pressure” which increases with decreasing density ?

84 – energy density Λ of this unexplained component is very high –ΩΛ = 0.7

SLIDE: SNIa Hubble diagram

Energy density: – baryonic matter Ωb ≈ 0.05, dark matter Ωd ≈ 0.25, dark energy Ωb ≈ 0.70 – universe is (close to) flat Ωb + Ωc + ΩΛ ≈ 1 Current values for densities: −26 3 – critical density: ρc,0 ≈ 1 · 10 kg m −27 3 – dark matter density: ρd,0 ≈ 2.5 · 10 kg m −28 3 – baryon mass density: ρb,0 ≈ nH mH = 5 · 10 kg m 4 2 −31 3 – radiation density: ρR,0 = aT /c = 5 · 10 kg m

Particle densities −3 – hydrogen nH ≈ 0.2 m 8 −3 – photons density nγ,0 ≈ 4 · 10 m 8 −3 – neutrino density nν,0 ≈ 4 · 10 m

Equation for the concordance model:

a˙ 2 kc2 8πG h a 3 a 4 i h a 3 a 4 i + = ρ 0 + ρ 0 + ρ = H2 Ω 0 + Ω 0 + Ω a2 a2 3 M,0 a R,0 a Λ 0 M,0 a R,0 a Λ

– matter density: ρM,0 = ρb,0 + ρd,0 – ρΛ = Λ/8πG 2 – 8πG/3 = H0 /ρc,0

Interpretation: – radiation-dominated Universe very early – matter-dominated Universe latter up to about now – in future the Λ-term dominates – the curvature seems to be close to flat – the expansion of the universe is accelerating (except for the very beginning)

Big questions: – Are the weakly interacting massive particles the dark matter? – What is dark energy ?

85 9.8 The thermal history of the Universe The Universe started very hot and dense and expands since then → steady decrease of T → steady decrease of ρ

Reactions in thermodynamic equilibrium A + B + ... ↔ L + M + ...

– reaction rates Γ (1/Γ is characteristic time for a reaction) reaction system is in equilibrium, if – rates for forward and backward reactions are equal – reaction rate Γ is faster than the expansion rate H =a/a ˙ – characteristic time 1/Γ for one reaction

Γ  H reaction is not in equilibrium for Γ  H – number of stable particles stay constant, if Γ  H – “freeze-out” condition for Γ ≈ H

Example: pair production and annihilation γ + γ ↔ x +x ¯

– photon pairs can create matter x, anti-matterx ¯ particle pairs 2 – requirement: Eγ = hν > m0,xc , photon energies exceeds rest mass energies of particle – all particle conservation laws are fulfilled (charge, lepton-number, etc.)

SLIDE: thermal history of the Universe

Important events in the early Universe epoch – time: < 10−6 s – quark-gluon plasma – energies ∼> GeV are to high to form hadrons – the Large Hadron Colliders at CERN investigates quark-gluon plasmas – nγ ≈ nq (number density of photons nγ = nq for ) hadron epoch (hadrons - particles held together by the strong force) – time: 10−6 − 1 s – quarks are bound into hadrons – hν < GeV (< rest mass of p and n), no pair-production of nucleons, e.g. p andp ¯ – hadrons and anti-hadrons eliminate each other (annihilation)

h + h¯ → γ + γ

86 – only the stable hadrons proton and neutron survive – small asymmetry, which originates from earlier phases, but process is not known

n − n¯ ∆n h h ≈ b ≈ 10−8 nh + nh¯ nγ – : matter only exists because of this matter-anti-matter asymmetry 8 – still today: nγ ≈ 10 nb after annihilation (see particle densities in concordance model) end of the hadron area – remaining stable particles: p, n, e−, e+, ν’s, and dark matter neutrinos decouple, because of low density they do not interact with “matter” anymore → there must be a neutrino background in the Universe (Tν ≈ 2 K) lepton epoch – time: 1 s – 10 s – leptons dominated the “interacting” matter of the Universe

8 ne− ≈ ne+ ≈ nγ ≈ 10 nb – they are first in thermal equilibrium, e.g. e− + e+ ↔ γ + γ – then the temperature falls below the threshold and most leptons annihilate e− + e+ → γ + γ

− −8 – only n(e ) ≈ n(p) ≈ 10 nγ survive – time: 10 s – 1000 s – proton - neutron equilibrium was established before the lepton epoch

p + e− → n + ν and n + e+ → p +ν ¯ neutron/proton ratio – in equilibrium, like for the Boltzmann level population law – proton is ground state, neutron is excited state

n h (m − m )c2 i n = exp − n p np kBT

– equilibrium ratio np/nn ≈ 1 for high temperature (hadron epoch) – reaction stops for T ≈ 0.8 MeV when ratio is about 5:1 – but free-neutrons decay in about 10 minutes: n → p + e− +ν ¯ helium synthesis – if all neutrons are used to form He, then the resulting H/He particle ratio is n n − 2n n H = p n = 2 p − 4 nHe nn/2 nn

– this yields for np/nn = 5 → nH/nHe = 6 – observed value nH/nHe ≈ 10

87 – because neutron decays the “average” ratio is np/nn ≈ 7 : 1 for this phase reactions for the nucleosynthesis of He – are only possible, if 2H can survive (radiation field must be soft) – processes – p + n → 2H + γ, deuterium production – 2H + 2H → 3He + n or 3H + p – 2H + 3H → 4He + n – 2H + 3He → 4He + p – some reactions creating 7Li or 7Be – no heavy element are formed, because no stable nuclei with mass 5 or 8 exist

SLIDE: Nucleosynthesis

– the exact outcome, depends on the barion to photon number ratio η = nB/nγ – especially the 2H abundance – observations are in good agreement with the models – primordial mass fraction of He: ≈ 0.28 – primordial mass fraction of 2H: ≈ 10−5 photon epoch – time: 10 s – 380’000 yr – temp: 109 − 4000 K – Universe is a plasma, the interacting particles are p+, 4He+2, e− – in thermodynamic equilibrium with the radiation field – and non-interacting neutrinos and dark matter particles – average photon energy → eV – transition from radiation dominated to matter dominated Universe – when energy of ≈ 108 photons of 10 eV ≈ 1 baryon of 1 GeV end of photon epoch: – (re)-combination = atoms form: e.g. p+ + e− → H0 + γ – universe becomes transparent and radiation escapes – this event produced the CMB dark ages – time: 380’000 yr – 108 yr – temp: 4000 − 30 K – cold neutral H0 and He0 gas: only HI 21 cm emission expected (not detected yet) end of dark ages – first stars form in proto-galaxies

88 9.9 after the first minutes the Universe was composed of – plasma with p+, 4He+2, e− – radiation – neutrinos: hot = relativistic (because low mass), weakly interacting particles – WIMPS: cold = non-relativistic, weakly interacting massive particles which component forms the first structures – WIMPS - because they feel only gravity and there is no counteracting gas pressure – not plasma or gas, because contraction increases T and gas pressure counteracts – not relativistic particles (ν, γ), because they are too fast to cluster by gravitation

SLIDE: structure formation

Steps for the structure formation:

Before the re-combination phase – dark matter fluctuations from big bang – density fluctuation δρ/ρ grow, in a matter dominated Universe – plasma is attracted by the dark matter gravity – but this enhances gas pressure and plasma bounces back – this creates sound waves (accustic or pressure waves) → the CMB shows these pressure waves with δT/T ≈ 10−5 → CMB provides a map of the dark matter distribution

During the dark ages: – dark matter accumulates in growing potentials – the gas cools and “falls” later into the dark matter potentials → first stars in proto-galaxies are formed → proto-galaxies and first quasars start to re-ionize the Universe → bright sources are tracers of the dominant dark matter distribution

Many complex baryonic processes on small scales → define galaxy structures → define super-massive black-hole evolution → define intergalactic gas properties → define chemical evolution (nucleosynthesis) of the Universe

SLIDE: high-redshift observations

89