II Jose Plinio Baptista School of Cosmology March 014 CMB theory David Wands Institute of Cosmology and Gravitation University of Portsmouth
Part 1: overview introduction cosmological parameters a brief thermal history
Cosmic Microwave Background radiation discovered in 1965 by Arno Penzias and Robert Wilson relic thermal radiation from the hot big bang 3 Kelvin, just three degrees above absolute zero Many ground- and balloon-based experiments since e.g., South Pole Telescope, Atacama Cosmology Telescope
CoBE satellite launched by NASA in 1990 NASA.7 K in all directions +/- 3.3 mk Doppler shift due local motion (at 1 million miles per hour) +/- 18 µk intrinsic anisotropies
COBE launched 1990 WMAP launched 001, final data 01 Planck NASA launched 009, first data 013
Friedmann s dynamic cosmology slice up 4D spacetime into expanding 3D space with uniform matter density and spatial curvature scale factor a(t) Hubble rate H!a / a H 0 = 100h km s 1 Mpc 1 h 0.7 Friedmann equation from Einstein s energy constraint: H = 8πG 3 ρ + Λ 3 κ a 1= Ω m + Ω Λ + Ω κ
Precision cosmology from angular power spectrum background.uchicago.edu/~whu angular scale indicates flat space geometry, but also depends on nature of energy density in the universe
Planck - new standard model of primordial cosmology Cosmological parameters (Ade et al: Planck 013 results. XVI) h = 0.674 ± 0.014 = 0.686 ± 0.00 matter = 0.314 ± 0.00 ESA = 0.04 ± 0.05
cosmic pie NASA
500 Million Light Years A slice of the SDSS Credit: SDSS Same characteristic scale seen in distribution of galaxies today baryon acoustic oscillations Bob Nichol - ICG, Portsmouth
The isotropic CMB a brief thermal history
CMB = Black-body spectrum,t 0 =.75K Komatsu 011
Black-body <= thermal equilibrium Komatsu 011 Photon energy: Einstein-Boltzmann distribution: Number density: Energy density: f(p) = n = ( ) 3 c = ( ) 3 E = h = ~p 1 exp(~p/k B T ) 1 but not in thermal equilibrium with matter today Z Z 4 p f(p) dp '.4 4 p f(p) ~pdp= kb T ~c 3 15~ 3 c 3 (k BT ) 4
FLRW geometry Spatially flat metric: where conformal time = horizon size: Photon trajectory: 4-momentum: 3-momentum: ds = c dt + a ijdx i dx j = a d + ij dx i dx j Z cdt = a dx i d =ˆni = unit vector p i = pˆn i P µ = dxµ d p = g ij P i P j Wikipedia Geodesic equation: Hubble redshift: dp µ d + µ P P =0 ) 1 p 1+z p p 0 = a 0 a dp d = 1 a da d preserves Einstein-Boltzmann distribution: f(p) = 1 exp(~p/k B T ) 1 = 1 exp(~p 0 /k B T 0 ) 1 where temperature redshift: 1+z = T T 0 = a 0 a
natural units: for photons, distance = time, momentum = wavenumber = energy = temperature but by convention they have different units which we have to keep track of much easier to use natural units, such that Leaves only one dimensional constant = Newton s constant so only unit = Planck unit
8 G ds d a + (111) H ==a, dx dx. (11) 3 n 3 jo i ds = a d dx 4 G dx.constraint and evolution equation (111)for the The Einstein equations give + the ijfriedmann 4 G ound: FRW metric 0FLRW H = a ( + 3P ), (113) background Hbackground = (spatially a 3P ) (113) flat,( 0) FRW universe 3, dynamics n =+ o i j 3ds Friedmann =a d +. evolution equation (111) ein equations give the constraint the ij dx dx and for and energy conservation gives the continuity equation Dynamical equations 0 ij 8 G Spatially metric: spatially flat, = 0)flat FRW universe H ds = = cadt, + a ij dx i dxj = a d + ij dxi dxj (11) 3 Einstein equations give the Friedmann0 constraint and evolution equation for the = 3H ( + (114) 4 G P ), 0 und (spatially flat, = 0) FRW universe 8 G H = a ( + 3P ), (113) the continuity equation Einstein=energy constraint + Gij equations: a energy, G00density (11) 3 evolution where and P H w =are the total and the total pressure, a prime denotes 38 G 0 aand derivative with respect to4 G conformal the scale factor is a, and H (11) a0 /a is the energy conservation the =gives acontinuity, time,,equation 0 H 3 a ( + 3P ), H parameter. = (113) conformal Hubble 3 4 G 0 = 3H ( + P ), H0 = a ( + 3P ), (113) (114) Radiation domination: onservation gives the continuity 3equation where and P = w are the total energy density and the total pressure, a prime denotes Energy conservation rgy conservation gives the continuity equation 4 the scale factor 1/ a derivative with respect to conformal time,, is. a, and H a00 /a(115) is the 0 Pr = /3, / a, a / t / r r = 3H ( + P ), (114) conformal Hubble parameter. 0 = 3H ( + P ), (114) total energy density and the total pressure, a prime denotes onformal time,, the scale factor is a, and H a /a is the = 3H ( + P ), (114) Matter domination: P = w are the total energy density and the total pressure, a prime denotes Radiation domination: and respect P = w to areconformal the total energy density and the total is pressure, a prime 0 denotes /3 and H a /a is the with time, Pm = 0,, the m scale / a 3 factor, a / ta, /. (116) 0 tive with respect to conformal time,, the scale and H a /a is the 4factor is a, 1/ bble parameter. P = /3, /a, a / t /. (115) al Hubble parameter. Vacuum domination Matter domination: omination: ion domination: Pv = v c v = constant1/, a / eht / 1/( 1 4 Pm = 0, 4 m / a 3, 1/ a / t/3 /. /3, /a a /at/ t 1///.., P =P =/3, /3, / a,4, / a Matter-radiation equality Vacuum domination mination: domination: Pv = a/t /3 /3 (115) (115) /.! m,0 a0 3 m h 1 +zeq = = 3.4 10 Ht v c av eq= constant a / e 0.14 / 1/( 1 r,0, 3 3 ) ) (117) (116) (115) (118) (117)
Temperature-time relation: At su ciently high temperatures (k B T mc, the rest mass energy of the particle) we expect all types of elementary particles to become relativistic. If they interact with the photons they will share the same equilibrium temperature and so all types of relativistic particles will have a density proportional to T 4. We can write the density as = g e 30 (k B T ) 4 h 3 c 5 (58) where g e is a sum over the e ective number of degrees of freedom of particles. g is for photons alone, corresponding to the two di erent polarizations. All other relativistic bosons (with integer spin) counts one per degree of freedom, while for fermions (halfinteger spins, e.g., neutrinos) it is 7/8 per degree of freedom due to their Fermi statistics rather than Bose-Einstein statistics [see Eq.(69)]. Remember, however that the energy density in a K = 0 radiation dominated model can also be written in terms of the expansion rate, and thus the time since the big bang: Thus we have a relation between temperature and time = 3H 8 G = 3 3 G(t t ) (59) t t = 1 H = v u t 3 3 G 30 h 3 c 5 g e 1 (k B T ), (60) which can be written as t t 1sec 1 p ge 1MeV. (61) k B T
electron-photon scattering: Scattering processes double (radiative) Compton scattering: changes photon number Compton scattering: (relativistic electrons) Thomson scattering: (non-relativistic electrons -> elastic) e + $ e + + e + $ e + e + $ e + energy
Spectral distortions: Full thermalization e + $ e + + double Compton scattering maintains black-body spectrum above redshift z th = 10 6 b h 0.0 /5 below this Compton scattering redistributes energy, but conserves photon number µ-distortion Compton scattering maintains statistical equilibrium above redshift y-distortion e + $ e + f(p) = 1 exp((~p µ)/k B T ) 1 z µ =5 10 4 b h 0.0 1/ low frequency electrons can gain energy from energetic electrons (inverse Compton) leads to deficit in intensity in low energy (Rayleigh-Jeans) region of spectrum T T ~p kb T = y e.g., thermal Sunyaev-Zel dovich from hot cluster gas along line of sight
Mechanisms leading to spectral distortions: particle decay or annihilation? but electron-positron annihilation, for example, occurs at z>>z th evaporation of primordial black holes? dissipation of (large?) density perturbations on small scales thermal Sunyaev-Zel dovich (y-distortion) from hot cluster gas along line of sight e.g., South Pole Telescope (009) Planck SZ cluster survey All-sky upper bounds from COBE/FIRAS y < 1.5 x 10-5 µ < 9 x 10-5