+ x 2 dω 2 = c 2 dt 2 +a(t) [ 2 dr 2 + S 1 κx 2 /R0

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1 Notes for Cosmology course, fll 2005 Cosmic Dynmics Prelude [ ds 2 = c 2 dt 2 +(t) 2 dx 2 ] + x 2 dω 2 = c 2 dt 2 +(t) [ 2 dr 2 + S 1 κx 2 /R0 2 κ (r) 2 dω 2] nd x = S κ (r) = r, R 0 sin(r/r 0 ), R 0 sinh(r/r 0 ). In homogeneous nd isotropic universe, everything is contined in κ, R 0 nd the scle fctor (t). If spce is curved, κ is non-zero nd R 0 is the rdius of curvture t t = t 0. The scle fctor (t) is dimensionless number, normlized to unity t t = t 0. The Friedmnn eqution The eqution tht describes how the scle fctor evolves with time is known s the Friedmnn eqution. Newtonin derivtion: The mss density in homogeneous, isotropic universe, ρ(t), depends on time, but not on position. Drw sphere bout the origin with constnt mss M S = 4π 3 ρ(t)r S(t) 3. We cn write the physicl rdius R S (t) in the form R S (t) = (t)r S, where (t) is the scle fctor of the universe nd r S is the rdius s mesured t t = t 0 (the comoving rdius). A test mss, m locted outside sphericlly symmetric object experiences the sme force it would feel if ll mss ws concentrted t the object s center.

2 Also, the test mss experiences no grvittionl force from the mtter outside the sphere. Thus, F = GM Sm R S (t) 2. The grvittionl potentil energy of the test mss (per unit mss) is E pot = GM S R S (t) = 4π 3 Gr2 S ρ(t)(t)2, nd the kinetic energy (per unit mss) E kin = dr S (t) = 1 dt 2 r2 Sȧ2. Since the sum of the grvittionl potentil energy nd the kinetic energy is constnt, U E kin + E pot (ȧ 1 2 r2 Sȧ2 = 4π 3 Gr2 S ρ(t)(t)2 + U, ) 2 = 8πG 3 ρ + 2U 1 rs 2 (t). 2 The future of the expnding sphere depends on the vlue of U: U > 0 ȧ 2 lwys positive eternl expnsion. U < 0 ȧ 2 zero t mx = (GM S )/(Ur S ) contrction. U = 0 ȧ 2 goes to zero in the infinite future. The exct form of the Friedmnn eqution, s derived by Friedmnn himself using GR, is (ȧ ) 2 = 8πG κc2 1 ǫ(t) 3c2 R The mss density ρ hs been replced by n energy density ǫ divided by c 2 (e.g., photons lso contributes to ǫ). The second substitution is 2U r 2 S = κc2 R 2 0 Tht is, if the energy U for test mss is zero, the universe is sptilly flt. If the test mss is unbound (U > 0), the universe is negtively curved. If the test mss is bound (U < 0), the universe is positively curved..

3 Since H(t) = ȧ/, we cn write H(t) 2 = 8πG 3c ǫ(t) κc2 2 R0(t). 2 2 The vlue of H(t) tody [mesured from v(t) = H 0 d(t)] is H 0 H(t 0 ) 70 kms 1 Mpc 1. H(t) is clled the Hubble prmeter nd the current vlue of the Hubble prmeter, H 0, is clled the Hubble constnt. For given vlue of the Hubble prmeter, there is criticl density, ǫ c (t) 3c2 8πG H(t)2. If the energy density of the universe is greter thn this vlue, the universe will be positively curved. If the energy density is less thn this vlue, the universe will be negtively curved. The current vlue of the criticl density is ǫ c,0 = 3c2 8πG H MeV m 3, or s n equivlent mss density, ρ c,0 ǫ c,0 /c kg m M Mpc 3, roughly equivlent to density of one hydrogen tom per 200 liters or one glxy per Mpc cube. Define dimensionless density prmeter s the rtio of the energy density nd the criticl density Ω(t) ǫ(t)/ǫ c (t), we cn write κc 2 1 Ω(t) = R0(t) 2 2 H(t). 2 Note tht since κ is constnt, 1 Ω(t) does not chnge sign. Directly mesuring the curvture by geometric methods is difficult. Consider

4 positively curved universe with circumference C 0 = 2πR 0. If C 0 ct 0 c/h 0, photons will hve hd time to circumnvigte the universe severl times (in the pst, C 0 ws even smller). Put C 0 to 10 million light yers. Looking towrd M31, which is 2 million light yers wy, we would see one imge 2 million light yers wy, showing M31 s it ws 2 million yers go, nother imge 12 million light yers wy, showing M31 s it ws 12 million yers go, nd so on (nd one imge in the opposite direction showing M31 s it ws 8 million yers go etc.). Since we don t see periodicities of this sort, we conclude tht if the universe is positively curved, R 0 > c/h Mpc. There is lower limit on the rdius of curvture lso for negtively curved universe. From the Friedmnn eqution s evluted t the present moment, we gin find tht R 0 (min = c/h 0 ). H 2 0 = 8πG 3c 2 ǫ 0 κc2 R 2 0 The Fluid nd Accelertion equtions Energy conservtion tells us tht dq = de + PdV, where dq is the het flow in or out of region, de is the chnge in the internl energy, P is the pressure nd dv is the chnge in volume of the region. In homogeneous universe, dq = 0 nd Ė + P V = 0. A sphere with rdius R S (t) = (t)r S hs nd internl energy V (t) = 4π 3 r3 S (t)3, E(t) = V (t)ǫ(t). The rte of chnge of the internl energy is Ė = V ǫ + V ǫ = V (, ) ǫ + 3ȧ,

5 giving the fluid eqution. ǫ + 3ȧ (ǫ + P) = 0, The Friedmnn nd fluid equtions cn be combined to derive the ccelertion eqution: ä = 4πG (ǫ + 3P). 3c2 The energy density ǫ slows down the expnsion. A component with pressure P < ǫ/3 will cuse the expnsion of the universe to speed up. A cosmologicl constnt hs P = ǫ. (Negtive pressure is the sme s tension.) Lerning to love Lmbd Einstein first published GR in 1915 when the universe ws thought to be sttic. (In fct, it ws by no mens settled tht glxies besides our own ctully existed.) The only permissible sttic universe is totlly empty universe. Einstein introduced new term, the cosmologicl constnt Λ, into the Friedmnn eqution (ȧ ) 2 = 8πG 3c 2 ǫ κc2 R Λ 3. The fluid eqution is unchnged nd the ccelertion eqution tkes the form ä = 4πG 3c 2 (ǫ + 3P) + Λ 3. Introducing Λ is equivlent to introducing n energy density ǫ Λ = c2 8πG Λ. For ǫ Λ to remin constnt with time, the ssocited pressure must be P Λ = ǫ Λ = c2 8πG Λ.

6 Now, setting Λ = 4πGρ, the Friedmnn eqution reduces to 0 = 4πG 3 ρ κc2 R 2 0 i.e, positively curved, sttic universe. However, the solution is unstble nd Einstein ws eger to bndon Λ s he ws presented with Hubble s 1929 evidence for n expnding universe. Note however tht Hubble s vlue of H kms 1 Mpc 1 leds to Hubble ge of H0 1 2 Gyr, less thn hlf the ge of the Erth. If the vlue of Λ is lrge enough to mke ä > 0, then ȧ ws smller in the pst thn it is now, nd consequently the universe is older thn the Hubble time. In fct, you cn mke the universe rbitrrily old by crnking up the vlue of Λ. More recently, Λ hs been employed to ccelerte the expnsion rte of the universe. Wht is the cosmologicl constnt? The Heisenberg uncertinty principle E t h permits prticle-ntiprticle pirs to spontneously pper nd then nnihilte in n otherwise empty vcuum. The energy density ǫ vc is ssocited with the density of the virtul prticles nd ntiprticles ǫ vc E P l 3 P,, ev/ m 3. This is 124 orders of mgnitude lrger thn the criticl density for our universe spectculrly poor mtch between theory nd observtion. Equtions of Stte Since we hve two independent equtions which describe how the universe expnds nd three unknowns the functions (t), ǫ(t), nd P(t) we need n eqution of stte, relting the pressure P to the energy density ǫ. For ll substnces of cosmologicl importnce, the eqution of stte cn be written in simple liner form: P = wǫ, where w is dimensionless constnt. A low-density gs of non-reltivistic prticles hs w <v2 > 3c 2 1.

7 For gs of rdition (either photons or highly reltivistic mssive prticles) w = 1 3. (Smll perturbtions in the pressure will trvel t the speed of sound c s = c dp 1/2 = w c. dǫ To preserve cuslity, w is restricted to the rnge 1 w 1.) The prticulrly interesting vlues of w re: w = 0 (cold, non-reltivistic gs or dust) w = 1/3 (rdition) w = 1 (cosmologicl constnt) (A network of cosmic strings hs w = 1/3 nd tht network of domin wlls hs w = 2/3.) Summry The Friedmnn eqution is given by (ȧ The fluid eqution is given by ) 2 = 8πG 3c ǫ(t) κc2 2 R0(t). 2 2 ǫ + 3ȧ (ǫ + P) = 0. The ccelertion eqution is given by ä = 4πGq (ǫ + 3P). 3c2 Given n eqution of stte for the different components nd the current energy densities, we cn solve for (t) nd ǫ(t).

A5682: Introduction to Cosmology Course Notes. 4. Cosmic Dynamics: The Friedmann Equation. = GM s

A5682: Introduction to Cosmology Course Notes. 4. Cosmic Dynamics: The Friedmann Equation. = GM s 4. Cosmic Dynmics: The Friedmnn Eqution Reding: Chpter 4 Newtonin Derivtion of the Friedmnn Eqution Consider n isolted sphere of rdius R s nd mss M s, in uniform, isotropic expnsion (Hubble flow). The