Plasma & Cosmology The Cosmic Microwave Bacground Pt Said 2020 Amr El-Zant (CTP, BUE, Cairo) Useful refs: Liddle: Introduction to Cosmology (Newtonian) Ferreira: Lectures on General Relativity and Cosmology (simple intro with essentials) http://wwwastro.physics.ox.ac.uk/~pgf/B3..pdf Peacock: Cosmological Physics (Newtonian + GR).
Cosmic Distance Scale • Earth-Moon ~ 1 light second • Earth-Sun ~ 8 light minutes • Nearest stars >~ 5 light yrs
The Milky Way Galaxy Distance from sun to centre ~ 20 000 light years Farthest individual stars seen by naked eye ~ 1000 light years
Galactic Characteristics Nearest large galaxy > Million light years ~ Mpc Time scales ~ 100 million years; speeds ~ 100 km/s Mass scale ~ 10 7 to 10 13 solar masses (most apparently dark!) Average Density ~ 10 -24 kg/m 3 (larger near centre) Compare with 5000 for Earth and 1 kg/m 3 for air
Larger scales (and back in time) • Clusters of galaxies 1-10 Million light years • Large scale structure • > few 100 Million light years Cosmo molo logical l horizon ~13 Billion years
Gravity governs very weak Long time scales BUT ONLY ATTRACTIVE (NO POSITIVE AND NEGATIVE) Long range Wins on cosmic scales Makes and holds together stars and galaxies and determines the cosmological evolution
Newtonian Derivation of Cosmological Evolution Equations • Consider universe with uniform energy density • If scale large need GR as Newtonian gravity assumes instantaneous interaction • Take instead a patch that is small compared to the ‘ horizon ’ (distance light travels since ‘ beginning ’ ). • Because of homogeneity all patches same
Newton-Birkhoff theorem • Take said patch to be spherical (isotropy ) • Equation of motion
‘ Energy Integral ’ and interpretation • Integrate e. motion keeping enclosed mass constant 2 1 𝑒 𝑠 − 𝐻 𝑁 (<𝑠) = E 2 𝑒 𝑢 𝑠 • ‘ Energy ’ E universe forever expands (E>0) or eventually recontracts (E<0). • No equilibrium solutions (as in systems with random or rotational mean motion)! similar to a ball thrown vertically upward!
Dynamics of Different Universes Need to know: Rate of expansion as fn of cosmic history most directly via distance and ‘ redshift ’
Light from galaxies redshifted Nobel 2011 acceleration!
Expansion and its Acceleration: Dark Energy and Dark Matter e.g., Supernovae ‘ standard candles ’ Current acceleration Dark energy Past deceleration rate Dark matter
The Cosmic Microwave Background • Tells us of prior thermal equilibrium • Current temperature of spectrum: 2.728 Kelvin • Current energy density of CMB: • The average energy per photon (since distn ~ 𝐟 − 𝑭 ~ k T ~ h ν 𝒍𝑼 ) photon number density ~ Compare with < one proton per cubic meter!
Era of Tightly Coupled Plasma (exercise) • Currently interaction rate of CMB photons with matter negligible, but • As universe expands scale a increases energy per photon ~ h ν ~ 1/ λ ~1/a ~ k T Back in time higher density and temperature universe ionised Fraction of neutral atoms (~Hydrogen) suppre ressed by factor 𝑪𝒑𝒎𝒖𝒜𝒏𝒃𝒐𝒐 𝒈𝒃𝒅𝒖𝒑𝒔 𝒇 − 𝑪𝑰 ( 𝐶 𝐼 = 13.6 eV is Hydrogen ’ s binding energy ) 𝑼 9 9 photons per proton T rec ~ 14/ln 10 There are ~ 10 = 0.7 eV (used 10 9 𝑓 − 𝐶 𝐼 𝑈 ~1; proper calcgives 0.3) 3600 Kelvin a (rec ) =1/ 1300 z (rec.) = 1300 t (rec) / 13 ) ~ 300 000 yr yr for
Cosmic Plasma Coupling • Gas fully ionized strongly interacts with photons by Thompson scattering : • Electron placed in EM field oscillates • radiates back Crossection ~ power radiated / mean incident energy flux ~ Square of classical electron radius interaction rate (note relative vely ~ c = 1 here!)
Sound Waves in a Photon Fluid • Remember recombination? • Before that Baryons tightly coupled with photon gas. 1 • The latter behaves as ideal fluid with 𝑄 = 3 𝜍 (c = 1). 3 • It obeys a continuity equation for photon number ~ T ++ a momentum conservation equation. Wave Eq. for Temp. Perturbations: (with sol. Θ = Θ 0 Cos (c s k 𝑢) , if small Θ (0) ) 𝟐 ** Transformed wave equation obtained earlier acoustic waves travelling at 𝟒 c ** If system expanding , equation remains with t Valid in terms of this ‘ conformal time ’ (comoving dist light travels since t=0 )
Acou oustic Peaks • Photon fluid is permeated by sound waves • Frozen at recombination Temperature fluctuations: where and 𝑡 ∗ indicates recomb. era. From Wayne Hu pages 𝑜𝜌 ** Peaks indicate 𝑙 = (𝐥 𝐭 ∗ = 𝒐 𝝆) 𝑡 ∗ First peak 1 st Compression Second 1 st Rarefication Third 2 nd Compression more oscillations damping
Location on of the Peaks as s Stand ndard d Ruler • Object of commoving size 𝜇 a ppears of angular size 𝜄 at 𝑢𝑠𝑏𝑜𝑡𝑤𝑓𝑠𝑡𝑓 𝑑𝑝𝑛𝑝𝑤𝑗𝑜 𝑒𝑗𝑡𝑢𝑏𝑜𝑑𝑓 𝐸 : 𝜇 𝜌 2 𝑡 ∗ 𝑡 ∗ 𝜄 = So If 𝜄 = 𝐸 , and 𝑙 = 𝑜 𝑜 𝐸 Distance sound of photon wave travels can easily be calculated Given angle measured Euclidean distance D inferred! … .or angle calculated given assumptions about model! Given a = a (t) and knowing a ~ 1/T , current T and T (Rec)!
Curvatur ure and d Cosm smological Con onstant • In a closed universe objects appear closer peaks shifted to larger angles as where R is the radius of curvature In an open universe things will appear further away (the sin sinh) Also in a universe with Λ since
Baryon Loading • Perturbations exist even if there ’ s no initial temperature fluctuation (due to potential fluctuations from Inflation) • Adding baryons additional source (gravitational potential) term in wave Eq. compression larger than rarefaction (but no collapse as Jeans scale is of order of horizon for coupled baryons)
Radi diation n Driving and d DM fraction n 𝜲 𝑬𝑵 𝒃 𝒇𝒓 a
Summary of CMB Parameter Sensitivity Gives again ~70 % DE ~ 20 20 % DM
Fundamental Aspect of Gravity : Clustering Instability Gravity: i) always attractive ii) Long range I) ‘ Normal ’ sound wave Gravity against Pressure ii ) Gravi vity y beats pr pressure Collapse!
Fluctuations in the CMB seeding structure
Recommend
More recommend
