The Linear Universe Matt Johnson Perimeter Institute/York - PowerPoint PPT Presentation

The Linear Universe Matt Johnson Perimeter Institute/York University Thursday, 4 July, 13

Modelling the Universe • Most of cosmology is described by General Relativity and Relativistic Hydrodynamics. Thursday, 4 July, 13

Modelling the Universe • Most of cosmology is described by General Relativity and Relativistic Hydrodynamics. • The fundamental variables: ρ i ( x, t ) = u µ ( x, t ) = g µ ν ( x, t ) = metric fluid densities fluid velocities Thursday, 4 July, 13

Modelling the Universe • Most of cosmology is described by General Relativity and Relativistic Hydrodynamics. • The fundamental variables: ρ i ( x, t ) = u µ ( x, t ) = g µ ν ( x, t ) = metric fluid densities fluid velocities • The laws: Einstein and continuity equations r µ T µ ν = 0 X G µ ν = 8 π GT µ ν T µ ν = ( ρ i + p i ) u iµ u i ν + p i g µ ν i Thursday, 4 July, 13

Modelling the Universe • Most of cosmology is described by General Relativity and Relativistic Hydrodynamics. • The fundamental variables: ρ i ( x, t ) = u µ ( x, t ) = g µ ν ( x, t ) = metric fluid densities fluid velocities • The laws: Einstein and continuity equations r µ T µ ν = 0 X G µ ν = 8 π GT µ ν T µ ν = ( ρ i + p i ) u iµ u i ν + p i g µ ν i Set of coupled PDE’s -- need initial conditions! Thursday, 4 July, 13

The Linear Universe • For much of the history of the Universe: g µ ν ( x, t ) = ¯ g µ ν ( t ) + δ g µ ν ( x, t ) δ = small ρ i ( x, t ) = ¯ ρ i ( t ) + δρ i ( x, t ) u µ + � u µ ( x, t ) u µ ( x, t ) = ¯ Thursday, 4 July, 13

The Linear Universe • For much of the history of the Universe: g µ ν ( x, t ) = ¯ g µ ν ( t ) + δ g µ ν ( x, t ) δ = small ρ i ( x, t ) = ¯ ρ i ( t ) + δρ i ( x, t ) u µ + � u µ ( x, t ) u µ ( x, t ) = ¯ • Why was the universe so nearly homogeneous? Thursday, 4 July, 13

The Linear Universe • For much of the history of the Universe: g µ ν ( x, t ) = ¯ g µ ν ( t ) + δ g µ ν ( x, t ) δ = small ρ i ( x, t ) = ¯ ρ i ( t ) + δρ i ( x, t ) u µ + � u µ ( x, t ) u µ ( x, t ) = ¯ • Why was the universe so nearly homogeneous? • For today: this is an extraordinary convenience! Thursday, 4 July, 13

The Linear Universe • For much of the history of the Universe: g µ ν ( x, t ) = ¯ g µ ν ( t ) + δ g µ ν ( x, t ) δ = small ρ i ( x, t ) = ¯ ρ i ( t ) + δρ i ( x, t ) u µ + � u µ ( x, t ) u µ ( x, t ) = ¯ • Why was the universe so nearly homogeneous? • For today: this is an extraordinary convenience! Homogeneous Background Universe G µ ⌫ = 8 ⇡ G ¯ ¯ T µ ⌫ G µ ν = 8 π GT µ ν Fluctuations � G µ ⌫ = 8 ⇡ G � T µ ⌫ Thursday, 4 July, 13

Initial Conditions • The homogeneous Universe: p i = w i ρ i ρ i ( t 0 ) ¯ Densities today Types of fluids Thursday, 4 July, 13

Initial Conditions • The homogeneous Universe: p i = w i ρ i ρ i ( t 0 ) ¯ Densities today Types of fluids • The Linear Universe: Characterize statistics of inhomogeneities! P [ g µ ⌫ ( x, t = 0)] ⇥ ⇤ P [ ⇢ i ( x, t = 0)] u iµ ( x, t = 0) P Assume our Universe is typical. Thursday, 4 July, 13

6 Parameter Model of the Universe Λ CDM 3 parameters 2 parameters 1 parameter (for CMB) P [ δ g µ ν ( x, t = 0)] τ Thursday, 4 July, 13

Modelling the universe Initially small fluctuations collapse to form galaxies, stars, etc. That’s it! The rest is details. Thursday, 4 July, 13

Giving Thanks • The non-linear Universe • GR is highly non-linear - inferring the state of the early universe would be like asking for the weather 100 million years ago based on the weather today. • No general classification of metrics - how to characterize initial conditions? • Shock waves, singularities, oh my! Thursday, 4 July, 13

Giving Thanks • The linear Universe • Simple evolution allows initial conditions to be inferred. • Background evolution and growth of structure can be analyzed separately. • Simple classification of initial conditions and metric degrees of freedom. • Physics on different scales evolves independently (Fourier modes independent). Thursday, 4 July, 13

The rest Now for some details.... Thursday, 4 July, 13

The homogeneous universe Thursday, 4 July, 13

The homogeneous universe Cool 13.7 Billion Years: the present. Diffuse 9.1 Billion Years: our sun ignites. 100 million years galaxies and first stars form. 380,000 years: neutral atoms form. 1 second: atomic nuclei form. 10 − 6 seconds: protons and neutrons form. Hot ?Big Bang? Dense Thursday, 4 July, 13

The homogeneous universe • The metric in a flat, homogeneous, isotropic universe: ds 2 = � dt 2 + a 2 ( t ) δ ij dx i dx j comoving coordinates Thursday, 4 July, 13

The homogeneous universe • The metric in a flat, homogeneous, isotropic universe: ds 2 = � dt 2 + a 2 ( t ) δ ij dx i dx j comoving coordinates time ∆ s ∆ x constant comoving distance = growing physical distance Thursday, 4 July, 13

The homogeneous universe • Conformal time: dt Z ds 2 = a 2 ( η ) � d η 2 + δ ij dx i dx j ⇤ ⇥ ⌘ = a ( t ) Thursday, 4 July, 13

The homogeneous universe • Conformal time: dt Z ds 2 = a 2 ( η ) � d η 2 + δ ij dx i dx j ⇤ ⇥ ⌘ = a ( t ) ⌘ � ⌘ 0 = ± ( x � x 0 ) time space The big bang: when a(t)=0 Thursday, 4 July, 13

The homogeneous universe • Conformal time: dt Z ds 2 = a 2 ( η ) � d η 2 + δ ij dx i dx j ⇤ ⇥ ⌘ = a ( t ) particle horizon Z a 0 =1 d ln( a ) dt Z ∆ x = a ( t ) = aH time a =0 space comoving horizon The big bang: when a(t)=0 Thursday, 4 July, 13

The homogeneous universe • Equations of motion in a homogeneous universe: ✓ ˙ ◆ = 8 π G ρ a H 2 ⌘ G µ ν = 8 π GT µ ν 3 a Thursday, 4 July, 13

The homogeneous universe • Equations of motion in a homogeneous universe: ✓ ˙ ◆ = 8 π G ρ a H 2 ⌘ G µ ν = 8 π GT µ ν 3 a r µ T µ ν = 0 ρ = � 3 H ( ρ + p ) ˙ p = w ρ Thursday, 4 July, 13

The homogeneous universe • Equations of motion in a homogeneous universe: ✓ ˙ ◆ = 8 π G ρ a H 2 ⌘ G µ ν = 8 π GT µ ν 3 a r µ T µ ν = 0 ρ = � 3 H ( ρ + p ) ˙ p = w ρ • Solutions: 2 ρ = ρ 0 a − 3(1+ w ) a ( t ) = a 0 t 3(1+ w ) different fluids gravitate differently! Thursday, 4 July, 13

The homogeneous universe Log H Ρ L radiation ⇢ / a − 4 w = 1 / 3 , matter ⇢ / a − 3 w = 0 , dark energy w = � 1 , , ⇢ / const . Log H time L ρ = ρ 0 a − 3(1+ w ) Thursday, 4 July, 13

The homogeneous universe • Evolution of the scale factor: a 1.5 dark energy 1.0 matter 0.5 t 0.2 0.4 0.6 0.8 1.0 1.2 1.4 t 0 radiation Thursday, 4 July, 13

The homogeneous universe • Energy budget: ✓ H ◆ 2 X X Ω i a − 3(1+ w i ) Ω i = 1 = H 0 i i Thursday, 4 July, 13

The homogeneous universe • Energy budget: ✓ H ◆ 2 X X Ω i a − 3(1+ w i ) Ω i = 1 = H 0 i i Ω r ⇠ 10 − 4 � � Thursday, 4 July, 13

The homogeneous universe • Energy budget: ✓ H ◆ 2 X X Ω i a − 3(1+ w i ) Ω i = 1 = H 0 i i H 0 = 100 h km s Mpc h = Ω r ⇠ 10 − 4 � � 3000 Mpc 1 Mpc = 3 × 10 23 meters 1 Mpc = 3 . 3 × 10 6 Lyr Thursday, 4 July, 13

The homogeneous universe 1 z = λ obs − λ em • Redshift: a = 1 + z λ obs a = 1 , z = 0 Thursday, 4 July, 13

The homogeneous universe 1 z = λ obs − λ em • Redshift: a = 1 + z λ obs a = 1 , z = 0 z eq = 3400 , z ∗ = 1090 , z re ⇠ 11 , z gal ⇠ 11 � 12 , z surveys < ⇠ 1 , z Λ = . 28 , z Virgo = . 003 Thursday, 4 July, 13

The Inhomogeneous Universe • There is structure in the Universe: • In Fourier space: ρ ( x, t ) = 1 ρ ( k, t ) e i ~ X k · ~ x V ρ ( k, t ) e ρ ( t ) f ¯ 3000 ρ ( k, Thursday, 4 July, 13

The Inhomogeneous Universe • There is structure in the Universe: • In Fourier space: ρ ( x, t ) = 1 ρ ( k, t ) e i ~ X k · ~ x V ρ ( k, t ) e ρ ( t ) f ¯ 3000 ρ ( k, Thursday, 4 July, 13

The Inhomogeneous Universe • There is structure in the Universe: • In Fourier space: ρ ( x, t ) = 1 ρ ( k, t ) e i ~ X k · ~ x V ρ ( k, t ) e ρ ( t ) f ¯ 3000 ρ ( k, Thursday, 4 July, 13

The Inhomogeneous Universe • There is structure in the Universe: • In Fourier space: ρ ( x, t ) = 1 ρ ( k, t ) e i ~ X k · ~ x V large on small ρ ( k, t ) e scales ρ ( t ) f ¯ 3000 ρ ( k, Thursday, 4 July, 13

The Inhomogeneous Universe • There is structure in the Universe: O (10 6 ) galaxies Structure: δρ > O (10 3 ) ⇠ 1 clusters ¯ ρ superclusters O (1) Thursday, 4 July, 13

The Inhomogeneous Universe • There is structure in the Universe: O (10 6 ) galaxies Structure: δρ > O (10 3 ) ⇠ 1 clusters ¯ ρ superclusters O (1) • There is structure on all scales which have had a chance to undergo gravitational collapse. • The largest structures in the Universe define a scale above which the fluctuations in density are linear: k = . 1 � . 01Mpc − 1 λ ∼ 1% of observable Universe wave number Thursday, 4 July, 13

The Inhomogeneous Universe • There is structure in the Universe: Luminous: Baryons Photons Semi-Luminous: Neutrinos Thursday, 4 July, 13