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Energetic Tail of Photon Bath

Energetic Tail of Photon Bath. hv. # hardest photons. ~ # baryons. Evolution of Sound Speed. Expand a box of fluid. Radiation. Matter. Show C 2 s = c 2 /3 /(1+Q) , Q = (3 ρ m ) /(4 ρ r ) ,  Cs drops from c/sqrt(3) at radiation-dominated era

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Energetic Tail of Photon Bath

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  1. Energetic Tail of Photon Bath hv # hardest photons ~ # baryons

  2. Evolution of Sound Speed Expand a box of fluid

  3. Radiation Matter Show C2s = c2/3 /(1+Q) , Q = (3 ρm) /(4 ρr) ,  Cs drops • from c/sqrt(3) at radiation-dominated era • to c/sqrt(5.25) at matter-radiation equality Matter number density Random motion energy Non-Relativistic IDEAL GAS

  4. hv hv Compton-scatter Keep electrons hot Te ~ Tr until redshift z After decoupling (z<500), Cs ~ 6 (1+z) m/s because T Te R Until reionization z ~ 10 by stars quasars Cs ~ 6 (1+z) m/s dP dP dX dX

  5. Growth of Density Perturbations and peculiar velocity

  6. Peculiar Motion • The motion of a galaxy has two parts: Proper length vector Uniform expansion vo Peculiar motion v

  7. Damping of peculiar motion (in the absence of overdensity) • Generally peculiar velocity drops with expansion.

  8. Non-linear Collapse of an Overdense Sphere • An overdense sphere is a very useful non linear model as it behaves in exactly the same way as a closed sub-universe. • The density perturbations need not be a uniform sphere: any spherically symmetric perturbation will clearly evolve at a given radius in the same way as a uniform sphere containing the same amount of mass.

  9. R, R1 log Rmax t-2 Rmax/2 virialize t Background density changes this way logt

  10. Gradual Growth of perturbation

  11. Equations governing Fluid Motion

  12. Let • We define the Fractional Density Perturbation:

  13. Motion driven by gravity: due to an overdensity: • Gravity and overdensity by Poissons equation: • Continuity equation: Peculiar motion and peculiar gravity both scale with δ and are in the same direction. The over density will rise if there is an inflow of matter

  14. the equation for linear growth • At high z>>1 & matter domination • In the equation Gravity has the tendency to make the density perturbation grow exponentially. Pressure makes it oscillate

  15. Nearly Empty Pressure-less Universe

  16. The Jeans Instability • Case 1- no expansion • Assume the density contrast  has a wave-like form • Assume no expansion •  the dispersion relation Pressure support gravity

  17. At the (proper) JEANS LENGTH scale we switch from • standing sound waves for shorter wavelengths to • the exponential growth of perturbations for long wavelength modes • <J, 2>0  oscillation of the perturbation. • J, 20exponential growth/decay

  18. Timescale: • Application: Collapse of clouds, star formation.

  19. Jeans Instability • Case 2: on very large scale >>J of Expanding universe • Neglect Pressure (restoring force) term

  20. Einstein de Sitter • Generally M=1 log Log R/R0

  21. Case III: Relativistic Fluid • equation governing the growth of perturbations being:

  22. Jeans Mass Depends on the Species of the Fluid that dominates • If Photon dominates: • If DarkMatt dominates & decoupled from photon: cst=distance travelled since big bang

  23. Jeans Mass past and now Flattens out at time of equality. Galaxy can form afterwards

  24. Dark Matter Overdensity Growth Condition • GROW Possible only if • During matter-domination (t > teq) or • during radiation domination, but on proper length scales larger than • sound horizon ( > cs t) & • free-streaming length of relativistic dark matter ( > c tfs )

  25. Theory of CMB Fluctuations • Linear theory of structure growth predicts that the perturbations: will follow the following coupled equations. Or

  26. Where  is the perturbation in the gravitational potential, with Gravitational Coupling

  27. F m • This is similar to a spring with a restoring force: • Frestoring=-m2x Term due to friction (Displacement for Harmonic Oscillator) x t

  28. The solution of the Harmonic Oscillator equation is: • Amplitude is sinusoidal function of k cs t • if k=constant and oscillate with t • or t=constant and oscillate with k. For B or R

  29. We don’t observe directly-what we actually observe is temperature fluctuations. • The driving force is due to dark matter over densities. • The observed temperature is: Effect due to having to climb out of graviatational well

  30. The observed temperature also depends on how fast the Baryon Fluid is moving. Doppler Term

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