# Rostock University, founded 1419

(on large scales > 100 Mpc) homogenous (translational invariant) and. isotropic (rotational invariant) = looks the same everywhere...

Rostock University, founded 1419

Hanse-Sail in August

Cosmology

Edwin Hubble 1929

Hubble Deep Field

Visible Matter Distribution

HST Deep Field

Sloan Digital Sky Survey (SDSS) – (1998-2006)

Edwin Hubble 1929

today: H0 ≈ 70 km/s/Mpc

Doppler Effect λObs = λSrc ⋅ γ (1 + β cos ϑ ) = λSrc ⋅ (1 + z ) 1+ β = λSrc ⋅ for ϑ = 0° 1− β

Redshift z = γ − 1 + βγ cos ϑ β = v/c for ϑ = 0 and β << 1 : z ≈ β

Expansion of the whole Universe

when expansion started = beginning of Universe (Big Bang) H0 = 70 km/s/Mpc 1/H0 = 14·109a

“Standard Candle” known absolute brightness (MB=–19.6) at maximum of supernova type Ia

known

Supernova Ia star 1: Red Giant star 2: Main Sequence Star star 1: White Dwarf star 2: Main Sequence Star star 1: White Dwarf mass increasing → Chandrasekhar-mass star 2: Red Giant mass flow to star 1 star 1: Supernova Ia

Distance Scale: Supernovae Type Ia

Universe is… (on large scales > 100 Mpc) homogenous (translational invariant) and isotropic (rotational invariant) =

looks the same everywhere

The Classical Friedmann Equation Sphere of mass M, radius R, density ρ expanding or contracting under gravity force

&R& = − GM = − 4πGρ R 3 R2

energy

&& R = − 43 πGρ (t ) R GM E 4πGρ 2 E 1 &2 R + + = 2 R = 3 R m m 2 & ⎛ R(t ) ⎞ 2E / m 8 ⎜⎜ ⎟⎟ = 3 πGρ (t ) + 2 ( ) ( ) R t R t ⎝ ⎠

(Newtonian Mechanics)

R

The Friedmann Equation 2 & ⎛ R (t ) ⎞ 2 E / m 2GM 2 E / m 8 ⎜⎜ ⎟⎟ = 3 πGρ (t ) + = + 2 3 2 R t ( ) R t R R ( ) ⎝ ⎠

• If E > 0 – right-hand side always positive – universe expands forever

• If E = 0 – RHS → 0 as t → ∞ – universe expands at ever-decreasing rate

• If E < 0 – ρ(t) ∝ R−3 – E-term ∝ R−2 – at R = GM /(–E/m) expansion reverses – universe headed for a Big Crunch

The fluid equation • Friedmann equation has two unknowns: ρ and R – need another equation – try thermodynamics: dQ = dE + P dV • energy in volume V is E / c2 = ρV; dE / c2 = Vdρ + ρdV • V ∝ R3 so dV / V = 3dR / R • dQ = 0 for expansion of universe

R& ⎛ P⎞ ρ& + 3 ⎜ ρ + 2 ⎟ = 0 R⎝ c ⎠ P⎞ ⎛ & ρ&R = −3R⎜ ρ + 2 ⎟ c ⎠ ⎝

the fluid equation

The acceleration equation • Friedmann eqn. multiplied by R2 2 2 2 8 & R(t ) = 3 πGρ (t ) R(t ) − kc ,

• differentiate:

k = E/m

&R ⎛ 8 π G 8 π G R P ⎞⎞ ⎛ 2 & & & & 2 RR = 2 RRρ + ρ&R = ⎜⎜ 2 ρ − 3⎜ ρ + 2 ⎟ ⎟⎟ 3 ⎝ 3 c ⎠⎠ ⎝

(

)

• using fluid equation:

P⎞ ⎛ & ρ&R = −3R⎜ ρ + 2 ⎟ c ⎠ ⎝ • simplify:

&& R 4πG ⎛ 3P ⎞ =− ⎜ρ + 2 ⎟ R 3 ⎝ c ⎠

Always deceleration unless pressure is negative!

Special Relativity • c = constant independent of (relative) motion all (inertial = not accelerated) systems are equivalent • transformation between systems in relative motion conserves linearity of distance and time

Lost of Simultaneousness flashlight (a) inside moving car: t(B) = t(F)

(b) outside observer: t(B′) < t(F′)

Photon-Clock (pendulum = photon) time dilation 2y t′ = c ( 2 y ) 2 + v 2t 2 t= c 2 2 ⎞ ⎛ v ( 2 y ) t 2 ⎜⎜1 − 2 ⎟⎟ = 2 = t ′2 c ⎝ c ⎠

y

vt

Lorentz contraction length = difference in coordinate at the same time if simultaneousness is lost, this is the distance of different spacetime-points

Special Relativity: Lorentz transformation = hyperbolic rotation

x'

= cosh y ⋅ x − sinh y ⋅ (ct )

(ct ' ) = − sinh y ⋅ x + cosh y ⋅ (ct ) v β = = tanh y c 1 γ= = cosh y 2 1− β

Minkowski-Rotation vs. Euklidian Rotation

Euklidean Rotation

Hyperbolic “Rotation” in Minkowski spacetime

Special Relativity: Lorentz transformation = hyperbolic rotation t′=const t=const

Relativistic Flight at β=0.90

scene illuminated by Planck radiation (2800K) geometrical effects (Lorentz-contraction, aberration geometry)

Relativistic Flight at β=0.90

…plus Doppler effect (blue shift)

Relativistic Flight at β=0.90

…plus flux change from aberration

Relativistic Flight at β=0.90

…plus flux change from aberration

The Lightcone

General Principle of Relativity physical laws are independent of the system of reference (coordinate system) special relativity: in inertial systems, with constant relative velocity general relativity: also with relative acceleration for mathematical description of physical laws: use local Cartesian systems but may be (sometimes must be) globally curved!

Equivalence Principle gravitational mass = inert mass → cannot distinguish acceleration m·a from gravity m·g locally! time dilation, Lorentz contraction → distortion of space-time by gravitation curved space!

Light Deflection at the Sun solar eclipse 1919

Gravitational Lensing

B1938+666 HST 1998

Gravitational Lensing Images

Light deflection near massive objects light paths

Observer

r = 2.5 rS

graphics: Uni Tübingen