July 30, SLAC Summer Institute: Scott Dodelson

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2 Standard Cosmological Model: I Friedmann-Robertson-Walker (FRW) Metric / Friedmann Equation Constituents of the Universe Thermal History: Neutrino Abundance Thermal History: Weakly Interacting Massive Particles (WIMPs)

3 The Universe is Expanding

4 Scale Factor a quantifies expansion

5 Comoving Coordinates/Distances The coordinate differences on the grid are called comoving distances. They are the equivalent of longitude & latitude. To get a physical distance dl from a set of coordinate differences (dθ,dφ), use the metric. dl g θθ = = R R ( dθ + sin θdϕ ) ; g φφ = R sin θ

6 The scale factor a(t) is the key function in the Friedmann-Robertson-Walker metric In a flat universe (our universe) k=0, and the metric reduces to ds = dt + a ( t) dx or i dx i g = 1 g = δ a ( t) 00 ij ij ħ=c=k B =1

7 Comoving Coordinates/Distances Since we set a 0 =1, the comoving distance between objects is the physical distance one would get today if an infinitely long tape measure was placed between the objects. A tape measure placed between the same objects early on would find a physical distance of (a(t) x comoving distance)

8 The evolution of a is determined by Einstein s Equations 1 R Rg = 8 μν μν πgt μν Ricci Tensor Ricci Scalar Newton s Constant Energy Momentum Tensor

9 General Relativity in 1 Slide Metric inverse g μα g αν = δ μ ν Raise/lower indices with metric/inverse α T β = g αμ T μβ Christoffel Symbol Γ α μν = 1 g αβ g x βμ ν + g x βν μ g x μν β Ricci Tensor Ricci Scalar R μν α α = Γ μν, α Γ μα, ν R = g μν Rμν + Γ α βα Γ β μν Γ α βν Γ β μα

10 Example: Christoffel Symbol Γ 0 ij 1 = 1 = g 00 g 0α [ ] g + g g [ ] g + g g X 0i, j αi, j X 0 j, i αj, i ij,0 ij, α But the metric has no spatial dependence, so And g 00 =-1 and g ij =δ ij a, so Γ 0 ij 1 d = δ ij a = δijaa & dt

11 Time-Time Component of Einstein Equations R = R00 + R = 8πGT00 = 8πGρ Straightforward calculations lead to: a&& a R a&& = 6 a + a& a Plug in to get the Friedmann equation: a& a H = 8 π 3 G ρ

12 Space-Space Component of Einstein Equations a&& + a 1 a& a = 4πGP where P is the diagonal space-space component of the energy momentum tensor. Combine with the Friedmann equation to get: a&& a 4πG = ρ 3 ( + 3P) Deceleration unless ρ+3p is negative

13 That gravity typically decelerates is intuitive

14 Ordinary Matter would induce deceleration z 1 1 a

15 So we must invoke

16 Constituents of the Universe

17 Notation Cosmologists measure densities in units of the critical density, ρ cr =1.88 h x 10-9 g cm -3 where h defines the Hubble constant H 0 =100h km sec -1 Mpc -1 Two popular ways of writing the baryon density are: The total density in the universe is equal to the critical density Ω b = ρ b / ρ cr ω b = Ω b h h=0.73+/-0.03

18 Constituents of the Universe Redux Ω dark energy = 0.73±0.01 Ω cdm = 0.3±0.01 Ω b = 0.045± < Ω ν < 0.0 Ω γ = (5.0±0.)X10-5

19 Photon Energy Density The temperature of the cosmic microwave background (CMB) has been measured extremely well. Turn this into a measurement of the energy density. =.35x10-4 ev

20 0 Photon Energy Density Energy density of a gas of bosons in equilibrium: 3 d p 1 ρ = E 3 E ( ) / T π e 1 Spin states Bose-Einstein distribution Sum over phase space For massless particles, E=p, so 3 8π dpp ρ = 3 p / T (π ) e T dx x π 4 /15 = x π e 1 ρ γ = π 15 T 4

21 Evolution of the energy density This is a snapshot of energy budget today. What did this look like in the past?

22 Evolution of energy density Impose conservation of the energy-momentum tensor ρ ( a) 1 = ρ exp a [ + w a ] da 3 1 ( ) 0 a Energy density today Equation of state: w=p/ρ

23 Example 1: Non-relativistic matter ρ( a) 1 = ρ exp 3 a da a [ 1+ w( a ] 0 ) The pressure of non-relativistic matter is very small compared to the energy density (T<<m), so w=0. ρ ρ m m = a,0 3 Consistent with simple dilution by volume expansion

24 Example : Relativistic particles 1 ρ( a) = ρ exp 3 a da a [ 1+ w( a ] 0 ) To determine w, recall that the pressure is: P = g 3 d p (π ) 3 p 3E f ( E) For relativistic particles, E=p, so P=ρ/3, or w=1/3. ρ r ρ r = a,0 4 Volume dilution PLUS wavelength stretching T~1/a

25 Example 3: Cosmological Constant ρ( a) 1 = ρ exp 3 a da a [ 1+ w( a ] 0 ) One possibility is that the dark energy is a cosmological constant with w=-1. ρ = ρ Λ Λ,0 Empty space contains energy

26 Thermal History of the Universe The equation of state of dark energy is -1 to within 0% Structure begins to grow when the universe becomes matter dominated a EQ is extremely well-determined; pins down matter density

27 Neutrinos are produced in the early universe ν Alpher, Herman, & Gamow 1953 Assume there are no neutrinos initially when the tepmerature is much larger than m e. The rate for producing them via, e.g., e + is of order e Z ν nσ ~ T 3 α T 4 m Z

28 Compare this to the expansion rate H = 8πGρ ~ 3 T m Planck since the universe is radiation 3 Γ T dominated ν at early times The ratio of the neutrino production rate to the H 5MeV expansion rate is therefore ~ α T m 4 Z 5 m T Planck (1GeV ) (10 GeV ) T ~ 4 (100GeV ) 1GeV 3

29 Above 5 MeV, neutrinos are in equilibrium with the rest of the cosmic plasma Fermi-Dirac distribution with temperature T equal to the electron/photon temperature. After the neutrinos decouple from the rest of the plasma (T<MeV), they still maintain Fermi-Dirac distribution with T~1/a.

30 You might think neutrinos and photons have the same temperature today But photons gained energy from electron/positron annihilation when T~m e Use entropy conservation: sa 3 =constant Together with the fact that the neutrino temperature does scale as a -1 T s = 3 ν const

31 Compute this ratio before and after electron/positron annihilation photons electrons/positrons neutrinos Initially: T s = ct 3 ( + (7 / 8)[4 + 3 T 6]) c 3 ν = 43 4 Finally: T ( (7 / 8)6 ) 3 3 T + T s c γ ν 3 = = c γ ν ν Tν ( [ T / T ] 1/ 4) Equate the two to get: T T γ ν = /3

32 Calibrate off the well-known photon temperature to get the prediction n νν = 115N cm ν 3 Number of species of weakly interacting neutrinos There are ~ a hundred quadrillion cosmic neutrinos (flux of 115x3c = cm - sec -1 ) passing through this screen (~10 4 cm ) every second.

33 Unfortunately, we can t detect these because the cross-section is too small σ 63 Eν ~ 10 cm 4 10 ev So the detection rate is of order Γ ~ M 1kg detector yr 1

34 But these neutrinos do contribute to the energy density of the universe. The energy density of massive neutrinos is: Ω ν = ρ ρ ν = n ν m ν = 0.01 h ρ cr cr 1 m ev This could be as large as 10% of the total matter, so affects large scale structure. ν Cold Dark Matter Cold+ Hot Dark Matter

35 Weakly interacting stable massive particles could be the dark matter All particles start with density roughly equal to photon density When temperature drops beneath mass, annihilations deplete density until freeze-out

36 Annihilation rate compared to expansion rate Γ 1/ 3/ m / T T ~ nxσv ~ ( mt ) e σ m H ~ T m Planck

37 Freeze-out takes place when the two rates are equal e m/ T fo σ m ~ m 1 Planck After freeze-out, WIMP number scales as photon number density n n X γ ~ m T fo 3/ e m / T fo ~ 1 mσm Planck m T fo 3/

38 Multiply by mass to estimate the contribution to the energy density today Ω x ~ ρ cr T σ 3 0 m Planck m T fo 3/ Plug in numbers Ω x = 0.3h 3/ m 10 10T fo 37 cm σ Mild mass dependence, but mostly depends on cross-section only.

39 Easy to get correct dark matter abundance in suppersymmetric models De Austri, Trotta, & Roszkowski 006

40 Final Slide If you want to get your hands dirty check out Meet me at the bar tonight if you have a good idea about Models which simultaneously produce dark matter and dark energy