FRW cosmology: an application of Einstein s equations to universe. 1. The metric of a FRW cosmology is given by (without proof)

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1 FRW cosmology: an application of Einstein s equations to universe 1. The metric of a FRW cosmology is given by (without proof) [ ] dr = d(ct) R(t) 1 kr + r (dθ + sin θdφ ),. For generalized coordinates q µ = (ct, r, θ, φ) (check this), 3. the above fixes the components of the metric g µν, which has no off-diagonal components. 4. One can see that this universe is spherically isotropic in spherical angles θ and φ. 5. and expanding with a scale factor R(t). 6. where k = 1,, 1 correspon to a cosmology that is open, flat or closed respectively. 7. k = 1 has a finite size, with the radial coordinate r < k = 1 cosmology is infinite without limits on the radial coordinate r. 9. Take a closed cosmology but consider the region where r be very very small, one essentially get the properties of a flat cosmology k =. 1. Calculate the Chistoffel symbol Γ µ µµ = 1/g µµ [g µµ,µ ] = forµ = t, θ, φ because these g µµ are independent of q µ in FRW metric (except that g rr depen on r if k = ±1). Calculate the Chistoffel symbol Γ a bc = 1 gaa [ g bc,a + g ac,b + g ab,c ] = if a b c thanks to vanishing cross terms g ab etc. So non-zero Γ requires that two of the three indices must be identical, e.g., Γ a ab, or Γa bb. Except that (try this) Γ i = 1 gii [ g,i + g i, + g i, ] = because no cross terms, and because g = 1 has vanishing derivative. 1

2 11. Calculate the Chistoffel symbol Γ i i for each i = 1,, 3. Γ 1 1 = 1/g11 [ g 1,1 + g 11, + g 1,1 ] [ ] d = 1/(1 kr )R(t) d(ct) (1 kr ) 1 R(t) = dr(t) Rd(ct) H(t)c 1 where H(t) is the Hubble parameter at time t. Likewise Γ i i = Γ θ tθ = Γ φ tφ = Γr tr = H(t)/c 1. Calculate the Chistoffel symbol Γ t ij. [ ] Γ ij = 1/ 1 dgij d(ct) 13. E.g., (try this) Γ = 1/g [ g, + g, + g, ] [ = 1/ 1 d ] d(ct) g = d[r(t) ] d(ct) r sin θ Γ i jk = 1 g ii [ g jk,i + g ik,j + g ij,k ] for i, j, k being three indices for spatial coordinates. The r.h.s. is independent of time t because the scale factor R(t) appears both in numerator and denominator. i, j, k must have an identical pair to be non-zero. 14. A possible Lagrangian is L(q µ, dq µ /) = = (d(ct)/) R [ (dr/) (1 r ) 1 + r (dθ/) + r sin θ(dφ/) ] The action is L here. We will show that q µ (s) = (s, cst, cst, cst) is a possible geodesics. 15. The geodesic equation for φ is d L (dφ/) = L φ = because L does not explicitly depend on φ. So we can integrate to find angular momentum R r sin θ(dφ/) = cst where we also used L = 1. Clearly dφ = is a possible solution,

3 16. The geodesic equation for θ is d R r dθ = L L θ = (L) 1 r The r.h.s. = if we assume dφ 17. The geodesic equation for r is (try this) d R dr L(1 kr ) = L r = (L) 1 ( ) dφ sin θ θ =. So a possible solution is dθ = q i =r,θ,φ ( dq i ) gii r The r.h.s. = if we assume dφ = dθ = and dr/ =. So a possible solution is dr =, i.e., fixed spatial coordinates r, θ and φ. 18. The geodesic equation for q 1 = ct is So (try this) d L (d(ct)/) = L ct ( ) d d(ct) L = dq i (L) 1 gii ct i=1,,3 The r.h.s = for fixed q i = (r, θ, φ). So a possible solution is d(ct) = 1, i.e., a stationary geodesic with the time coordinate t = s/c. 19. A set of geodesics has u µ dqµ = (1,,, ) c /c A universe with matter staying at fixed coordinates, which are called comoving coordinates.. The distance between two comoving observers at the same r and θ, but are separated by π angle in the φ direction, will expand dl = π g φφ dφ = R(t)[πr sin(θ)] 1. The spherical area of the same coordinate r expan as darea = π gθθ dθ π π g φφ dφ = R(t) rdθ [πr sin(θ)] = 4πR(t) r 3

4 . The volume of a k = 1 universe is (try this) 1 1 R(t) V = grr drarea = 1 r dr4πr(t) r = 4πR(t) 3 which is finite and expanding with R(t) 3. 1 r 1 r dr 3. This FRW cosmology correspon to one with uniform distribution of matter energy momentum. 4. The energy momentum tensor of a uniform media is defined by 5. where T µν ;ν =. T µν = u µ u ν (ρ + P/c ) Pg µν 6. and the matter particles are stationary in the co-moving coordinates u µ = (1,,, ) c. 7. The contract form of the stress tensor T µ µ = g µν T µν = g µν u µ u ν (ρ + P/c ) Pg µν g µν T µ µ = 1cc(ρ + P/c ) + 15 P( ) = ρc 3P 8. Starting from Einstein s equation R µν g µν R/ + Λg µν = κt µν 9. Let s omit the cosmological constant term, and rewrite the Einstein s equation as (prove it yourself) Namely (lower T αβ by g αµ g βν ) R µν = κ(t µν 1/g µν T α α ) R µν = κ [ g αµ g βν u α u β (ρ + P/c ) Pg µν (ρc 3P)/g µν ] R µν = κ [ δ µ δ ν cc(ρ + P/c ) (ρc P)/g µν ] 3. (Check) R = κ (ρc + 3P) R 11 = κ (ρc P) R(t) /(1 kr ) R = κ (ρc P) R(t) r R 33 = κ (ρc P) R(t) r sin θ 4

5 31. Now compute the Ricci tensor from the previous computed Chistoffel symbols. R = Rα α = Γ α,α Γα α, + Γ α Γβ αβ Γα β Γβ α we get (try cancel certain Christoffel symbols yourself) 3. R = 3 d 3 d(ct) Γi i + ( ) Γ i i i=1 So Einstein equation R term becomes (try doing sum of identical terms with i = 1,, 3): R = 3 d [ ] [ ] dr dr 3 = FriedmannEq = 3 R d(ct) Rd(ct) Rd(ct) Rc = + 3P) κ(ρc i=1 R 11 = R α 1α1 = Γα 11,α Γα 1α,1 + Γα 11 Γβ αβ Γα 1β Γβ 1α (Lengthy) reductions of Chistoffel symbols give Einstein equation R ii component R ii = k + R R/c + Ṙ /c R g ii = κ (ρc P) These combine to give the Ricci scalar R = g R + 3 ] i=1 gii R ii = [ 6 R/c R 6Ṙ /c R 6k R = κ(3p ρc ). E.g., for a static closed universe R rr = /R(t) g rr = /R(t) /(1 r ), and Ricci scalar R = 6/R(t), consistent with the curvature of a sphere of radius R(t). Clearly Eqs. for R 11, R and R 33 are redundant, because of a spatially homogenous and isotropic density. Combining R µν gives (try it) G R Rg / = FriedmannEq1 = 3Ṙ /c + k R = κρc 33. Physics of Friedmann Eq.: Universe deaccelerates R < if there is only normal matter with ρc > P >. An accelerating universe must include a positive cosmological constant term (check this), simply replaces ρ ρ + ρ Λ, P P ρ Λ c, ρ Λ c Λ 8πG 34. Physics of Friedmann Eq1.: the flatness of the universe is decided by the balance of the density and expansion rate ) 3k (Ṙ R + 3 c = κ(ρ + ρ Λ )c, κ 8πG R c 4 which contains a dark energy ρ Λ c term from the cosmological constant. g ii 5

6 35. The pressure satisfies the conservation law. which reduces to (without proof) = T µ ;µ = [ (ρ + P/c )u µ u Pg µ] ;µ d(ρc R 3 ) = Pd(R 3 ) i.e., energy in an expanding volume R(t) R(t) R(t) is lost by a positive pressure doing work. One can invent a w-parameter w P ρc = d(ρc R 3 ) ρc d(r 3 ) 36. Cosmological constant term can act as a positive density and negative pressure, i.e., w = 1, It satisfies energy conservation. d(ρ Λ c R 3 ) = ( ρ Λ c )d(r 3 ) 37. Ordinary matter has negligible positive pressure w, so ρ R 3 (check). Radiation (photons) have significant positive pressure, w = 1/3, so ρ R 4 would satisfy the conservation equation. So in an expanding box of R 3 the number of photons is fixed, but the total energy inside ρc R 3 drops as R 1, i.e., the thermal energy per photon drops as R 1 with the expansion (check). The universe had lots of short-wavelength photons when R was small. Neutrinos are in between ordinary matter and photon since they have a small rest mass, which makes the photon-like in the early universe when the thermal energy is high, and matter-like in late universe when the thermal energy has dropped. 38. Solve the Friedmann equation for a flat universe dominated by an energy ρ = BR n, we get ) 3 (Ṙ c = κbc R n H (dr/r/dt) R n R We can solve this (try it) and get an expansion history of the universe: R(t) t n So the cosmos expan with R(t) t.5 in the early universe where photon energy was dominant with n = 4, and R(t) t.66 later on when matter dominates with n = 3. The current best explanation of the cosmic data also requires that presently we are dominated by an n dark energy or a cosmological constant, this implies a very rapid expansion t, actually an exponential growth. 6

7 39. In current cosmology literature, one often see the same equation, but written with these notations (try it) 3H 8πG = 3H [ Ωpho (1 + z) 4 + Ω mat (1 + z) 3 + Ω Λ + Ω openness (1 + z) ] 8πG where H is the present Hubble parameter, and 1 + z R /R(t), where R is the present size, and Ω is the present fraction of each energy term, and the last term is due to curvature; Ω openness (1 Ω mat Ω Λ Ω pho ) We get an open universe if there is too little energy density, i.e., if Ω < 1 presently. 4. Current observations favor a nearly flat cosmology with Ω pho =.5, Ω mat.3, Ω Λ.7. The nearly flat cosmology requires some kind of inflation theory to explain. The difference between Ω mat.3 and the amount of ordinary baryons Ω bary.4 in the universe suggests a missing matter (dark matter). The coincidence of Ω Λ being comparable to Ω mat is a mystery of physics. 41. Einstein s equation can also be modified while keeping its co-variant tensor form. E.g., Rb a δb a F(Re)/ e = 8πG(T e e ) where f(r) is an algebraic function of the contacted Ricci tensor, and the modified gravitational constant G(T) is an algebraic function of the contracted stress tensor. More careful designs (starting from the action principle) can yield Einstein-like equations, preserving the conservation of energy. It remains to be seen if these modified theories can explain the observations more naturally without dark matter or dark energy. 4. If we use a new coordinate χ such that c 4 T a b r sin χ, dr = cosχdχ, 1 r = cos χ The metric of a closed FRW cosmology can be rewritten as = d(ct) R(t) [ dχ + sin χ(dθ + sin θdφ ) ], k = 1 The χ coordinate can be thought as the polar angle in the 4D sphere where the pole is the extra dimension. Photon propagates with = = d(ct) R(t) (dχ) we can solve it (try it) and calculate the horizon size t d(ct) χ = dχ = R(t) 7

8 In normal matter models the horizon size is finite because R t /3, but in inflation models (where R t ) early on, the horizon χ is infinite (try it), so everywhere was connected once by inflation. Likewise for open cosmology = d(ct) R(t) [ dχ + sinh χ(dθ + sin θdφ ) ], k = 1 if use r = sinhχ, and dr/ 1 + r = dχ (try it). And finally for flat cosmology = d(ct) R(t) [ dχ + χ (dθ + sin θdφ ) ], k = 43. Alternatively, one sometimes see in research papers the conformal gauge = R(η) { dη [ dχ + sin χ(dθ + sin θdφ ) ]}, k = 1 for a closed universe. Here one defines a dimensionless time coordinate η by R(η)dη ct In these conformal coordinate, metric is especially simple for a flat cosmology, where = R(η) { dη [ dx + dy + dz ]}, k = where we used cartessian instead of spherical coordinates. So flat FRW cosmology is Minkowski metric dilated by the R(η) scale factor in all four dimensions. 8