Academic Training Lectures Rocky Kolb Fermilab, University of Chicago, & CERN

1. Cosmology and the origin of structure Academic Training Lectures Rocky Kolb Fermilab, University of Chicago, & CERN Rocky I: The universe observed Rocky II: The growth of cosmological structure Rocky III: Inflation and the origin of perturbations Rocky IV: Dark matter and dark energy

2. Rocky II: Growth of structure • Linear regime: quantative analysis • Jeans analysis • Sub-Hubble-radius perturbations (Newtonian) • Super-Hubble-radius perturbations (GR) • Harrison-Zel’dovich spectrum • Dissipative processes • The transfer function • Linear evolution • Non-linear regime: word calculus • Comparison to observations • A few clouds on the horizon

3. Growth of small perturbations • Today (12 Gyr AB) • radiation and matter decoupled • Before recombination (300 kyr AB) • radiation and matter decoupled

4. Seeds of structure time

5. Simulation Oxford English Dictionary

6. Power spectrum • Assume there is an average density • Expand density contrast in Fourier modes • Autocorrelation function defines power spectrum

7. Jeans analysis Jeans analysis in a non-expanding fluid: matter density pressure velocity field gravitational potential Perturb about solution*

8. Jeans analysis real: perturbations oscillate as sound waves imaginary: exponentially growing (or decaying) modes Jeans wavenumber Jeans mass solutions of the form gravitational pressure vs. thermal pressure

9. Sub-Hubble-radius (RH=H-1) Jeans analysis in an expanding fluid: scale factor a(t) describes expansion, unperturbed solution:* • Solution is some sort of Bessel function: • growth or oscillation depends on Jeans criterion • In matter-dominated era • For wavenumbers less than Jeans

10. Super-Hubble-radius (RH=H-1) • complete analysis not for the faint of heart • interested in “scalar” perturbations • fourth-order differential equation • only two solutions “physical” • other two solutions are “gauge modes” • which can be removed by a coordinate • transformation on the unperturbed metric

11. Bardeen 1980 Reference spacetime: flat FRW

13. Bardeen 1980 Reference spacetime: flat FRW Perturbed spacetime (10 degrees of freedom):

14. reference flat spatial hypersurfaces actual curved spatial hypersurfaces

15. scalar, vector, tensor decomposition 1 2+1 2+1+2+1 10 evolution of scalar, vector, and tensor perturbations decoupled

16. Vector Perturbations: • are not sourced by stress tensor • decay rapidly in expansion Tensor Perturbations: • perturbations of transverse, traceless component of the • metric: gravitational waves • do not couple to stress tensor Scalar Perturbations • couple to stress tensor • density perturbations!

17. Super-Hubble-radius in synchronous gauge and uniform Hubble flow gauge • in matter-dominated era • in radiation-dominated era

18. sub Hubble-radius super Hubble-radius log d log d P(k) ~ kn log k log k kH • Harrison-Zel’dovich in radiation-dominated era in matter-dominated era P(k) ~ kn kH

19. log d log d log d P(k) ~ k1 log k log k log k kH • Harrison-Zel’dovich “flat” spectrumD2(k)=k3P(k)~const in radiation-dominated era kH kH

20. log d logP(k) P(k) ~ k1 log k log k kH P(k) ~ k1 P(k) ~ k-3 • Harrison-Zel’dovich “flat” spectrum D2(k)=k3P(k)~const • in radiation-dominated era • no growth sub-Hubble radius • growth as t super-Hubble radius • in matter-dominated era • power spectrum grows as • t2/3 on all scales kH

21. logP(k) log k P(k) ~ k1 P(k) ~ k-3 • Power spectrum for CDM matter-radiation equality k1 k-3

22. Dissipative processes • Collisionless phase mixing – free streaming If dark matter is relativistic or semi-relativistic particles can stream out of overdense regions and smooth out inhomogeneities. The faster the particle the longer its free-streaming length. Quintessential example: eV-range neutrinos

23. The evolved spectrum

24. Dissipative processes • Collisionless phase mixing – free streaming If dark matter is relativistic or semi-relativistic particles can stream out of overdense regions and smooth out inhomogeneities. The faster the particle the longer its free-streaming length. Quintessential example: eV-range neutrinos • Collisional damping – Silk damping As baryons decouple from photons, the photonmean-free path becomes large. As photons escape from dense regions, they can drag baryons along, erasing baryon perturbations on small scales. Baryon-photon fluid suffers damped oscillations.

25. The evolved spectrum

26. Linear evolution today z=10 z=100 z=1000

27. Linear evolution today z=10 z=100 z=1000

28. Life ain’t linear! • Many scales become nonlinear at about the same time • Mergers from many smaller objects while larger • scales form • N-body simulations for dissipation-less dark matter • Hydro needed for baryons • Power spectrum well fit if G = Wh ~ 0.2 • There is more to life than the power spectrum Alex Szalay

29. Large- scale structure fits well

30. Small-scale structure-cusps Moore et al.

31. Small-scale structure-satellites Moore et al.

32. Rocky II: Growth of structure • Linear regime: quantative analysis • Jeans analysis • Sub-Hubble-radius perturbations (Newtonian) • Super-Hubble-radius perturbations (GR) • Harrison-Zel’dovich spectrum • Dissipative processes • The transfer function • Linear evolution • Non-linear regime: word calculus • Comparison to observations • A few clouds on the horizon

33. Cosmology and the origin of structure Academic Training Lectures Rocky Kolb Fermilab, University of Chicago, & CERN Rocky I: The observed universe Rocky II: The growth of cosmological structure Rocky III: Inflation and the origin of perturbations Rocky IV: Dark matter and dark energy