(Lecures 7-8) Liddle, Chaper 5 Simple cosmological models (i) Hubble s Law revisied Self-similar srech of he universe All universe models have his characerisic v r ; v = Hr since only his conserves homogeneiy and isoropy Recall r = a( ) x where r s he proper (physical) disance, commoving disance r is he srech facor, and x he = a ( ) x (since x = 0 by definiion) r ( ) = a ( ) a( ) a ( ) x r( ) where r ( ) is he velociy, a( ) x he disance, and hence a ( ) is Hubble s consan Velociy disance law is a snapsho of he universe a a given ime However, noe ha as we look ou in disance, we look back in ime ( c ) Hence we would expec he Hubble law o change a large disances (earlier imes), when hings were moving faser, and he universe was expanding a a differen rae (Picure 6) We will reurn o his laer Redshif and expansion of he universe (revisied) Redshif z = obs emied emied This implies ha 1 + z = obs emied z is due o he expansion of space iself; i is no a Doppler shif (somehing moving hrough space) Phoons don have binding energy, herefore don resis he expansion of he universe (problems 1, quesion 15) 1+ z = obs = obs emi a( emi ) This can be proved rigorously in GR We ofen se a( 0 ) = 1, ie now 1 1+ z = a( emi ) eg if z = 4, hen emi = 1 5, so he universe was 1 5 is curren size when he ligh was emied This is regardless of he deails of he cosmological model Cosmology ells us how long ago his was Solve Friedmann Equaion
We need o undersand he properies of consiuens In paricular, he equaion of sae ; how P and are relaed (as in hermodynamics) (i) (ii) Maer We can always consider his as a gas (even whole galaxies), since he universe is so big (Can be baryonic, or Dark Maer) We know ha E oal = 0 c, or E oal = m 0 c + ( 1)m 0 c res mass A all imes of relevance, maer is non-relaivisic E k << m 0 c (moves so slowly), so pressure is (roughly) 0 (pressure is E k / Volume, ie energy densiy erm) Therefore i can be ignored Radiaion Phoons (+ neurinos) Radiaion has no res mass, bu los of energy due o moion Therefore does have pressure A sandard resul for relaivisic maer is: P = rad c where he comes from he dimensions This is very similar o he resul in sandard kineic heory, P gas = 1 gas U, where U is he mean E k square speed In general, we have boh = aer + radiaion = + r Consider he effecs one a a ime Fluid equaion wih aer 0 (maer only universe) + a a ' = 0 = ( a a = a 0,0 a ' using some inegraion,,0 is he curren densiy of maer NB: if a is no wrien as, where he 0 denoes ha i is now Since a 0 = a( 0 ) 1;, i should be a 0 0 =,0 a obvious Boxes shrink as hey go back in ime Volume goes down by a Now subsiue ino he Friedmann equaion, wih k = 0 for simpliciy (and since observaion srongly favours k = 0 )
a a a a = 8'G( = 8'G a = 8'G ( m,0 a ( m,0 a ) a * 1 a 1 The rae of change of he scale facor indicaes ha he universe is deceleraing as graviy is pulling i back o he origin Time variaion? Inegrae he equaion Through separaion of variables, we ge: = ie = 0 0 8'G( m,0 Therefore since =,0 a = 0 m,0 ' 1, we ge Differeniae he previous equaion wr : 1 a ( ) = a ( 0 ) 1 = 1 = H ( ) A classic cosmological soluion Radiaion-only universe P rad = rad c Proceed vihe Fluid equaion as before (Liddle) A physical approach is more insrucive Run he universe backwards, ie conrac i The volumes conrac by he phoons change by shorer ) So he energy densiy rad = r,0 a( ) 4 = 1 where 0, and also 1 So change energy by a( ) 1 (higher in he pas; 4
Going back o he Friedmann equaion, wih k = 0 again: = 8'G ( r,0 4 Wihou worrying abou consan facors; 1 a da d a a 1 (which is differen o he maer case) or; = 0 0 1 and again, ( r ) = 0 r,0 ' (which is he same as he maer case) This is anoher classic cosmological soluion Noe ha he case when radiaion dominaes implies ha expansion rae is slower 1 dominaes han when he maer relaivisic effec: energy has mass, herefore has graviy Pressure ( E k erms) does no blow up he universe ; i is exacly he opposie Mixures: = aer + rad A general soluion of he Friedmann equaion is complicaed However, one erm dominaes in regions of mos ineres, ie now and he fuure (maer erm), or a early imes (radiaion erm) 1 Consider when radiaion dominaes, herefore ses he scale facor change wih ime 1 rad (as above) So he smaller consiuen aer 1 a( ) Therefore aer falls off more slowly han radiaion as universe expands So evenually, maer will come o dominae Consider when maer dominaes aer
he smaller consiuen, radiaion 1 a( ) 8 4 Therefore aer falls off more slowly han rad (again) Once maer dominaes, i says dominan When he wo lines cross, Z ~ 1000 ; each bi of he universe is 1000 imes smaller han i is now in scale Radiaion dominaes earlier, due o phoons having higher energy a early imes a( ) Radiaion dominaed 1 a rad a 4 8 Maer dominaes As we go from radiaion dominaed o maer dominaed, he expansion rae speeds up 1 Evoluion including he k-erms
1 k = 0 k < 0 (ie U > 0, E k dominaes) FE = a a = 8'G kc a his is always posiive, and hence expands forever A lae imes, kc a dominaes since 1 a bes a erm a a = ' kc a a lae imes; a ( E k dominaing) velociy ends o become consan Free expansion k > 0, ie U negaive, so E p wins Terms on he RHS of he Friedmann equaion cancel Evenually, he RHS becomes negaive, ie posiive curvaure, ie graviy dominaes universe grinds o a hal, and hen re-collapses Big crunch
(picures 8, 9, 0) Observable Parameers Hubble s consan (now): H 0 70 ± 10kms 1 Mpc 1 Uncerainy arises from he difficuly in measuring he disances o beer han 10 Use various forms of sandard lenghs, and have o calibrae Ofen parameerize he uncerainy H 0 = 100h km s 1 Mpc 1 H 0 larger universe expanding faser The Densiy Parameer Friedmann equaion wihou k (universe is fla): a a = H ( ) = 8'G ( Criical densiy cri ( ) = H 8G Pu H ino SI unis: cri ( 0 ) = 188h 10 6 kg m This is very small, ie 11h hydrogen aoms per meer cubed