4- Cosmology - II. introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 1

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1 4- Cosmology - II introduc)on to Astrophysics, C. Bertulni, Texs A&M-Commerce 1

2 4.1 - Solutions of Friedmnn Eqution As shown in Lecture 3, Friedmnn eqution is given by! H 2 = # " & % 2 = 8πG 3 ρ k 2 + Λ 3 (4.1) Using Λ= 8πGρ vc it cn lso be written s H 2 =! # " & % 2 = 8πG 3 ρ + ρ vc ( ) k 2 (4.2) Using Newton s 2 nd lw, the Grvittionl lw, nd p = γρ/3 for mtter γ = 0 or rdition γ = 1, one cn lso rewrite it s = 4πG 3 ( 1+γ)ρ + 8πG 3 ρ vc (4.3) In this lecture, we discuss the pplictions of these equtions to the structure nd dynmics of the Universe. introduc)on to Astrophysics, C. Bertulni, Texs A&M-Commerce 2

3 4.2 - Flt Universe In this cse, the Universe hs no curvture, k = 0 (nd using Λ = 0), nd Friedmnn eqution is! # " & % 2 = 8πG 3 ρ (4.4) As we sw in Eq. (3.56) q = 1/2. We consider the mtter-dominted nd the rdition-dominted Universes seprtely Mtter dominted Universe We hve p = 0 nd 3 ρ = const nd, using Eq. (4.4) (with Λ = 0), we get which leds to 2 2 = 8πG 3 ρ = 8πG 3 ρ 0! # " 0 & % 3 (4.5) 1/2 d = 2 3 3/2 + C = 8πGρ t (4.6) introduc)on to Astrophysics, C. Bertulni, Texs A&M-Commerce 3

4 Flt, mtter dominted Universe At the Big Bng, t = 0, = 0, so C = 0. Since the Universe is ssumed flt, k = 0, ρ 0 = ρ crit, we hve or = ( 6πGρ 0 ) 1/3 t 2/3 = ( 6πGρ crit ) 1/3 t 2/3 = 6πG 3H 2 0 # 0! = 3H 0 # 0 " 2 & % 2/3 t 2/3 (4.7) & 8πG % From this we compute the ge of the Universe t 0, which corresponds to the Hubble rte H 0 nd the scle fctor = 0, to be! " 1/3 t 2/3 t yers 9.1 eon (4.8) Notice tht we hd lredy obtined the correct power dependence of s function of t in Eq. (3.53). Now we lso obtined the correct multiplictive coefficient in Eq. (4.7). The model described in this subsection in known s the Einstein de Sitter model. introduc)on to Astrophysics, C. Bertulni, Texs A&M-Commerce 4

5 Flt, Rdition dominted Universe We hve p = ρ/3 nd 4 ρ = const nd, using Eq. (4.2) (with Λ = 0), we get d = C = 8πGρ (4.9) Agin, t the Big Bng, t = 0, = 0, thus C = 0, nd ρ 0 is equl to the criticl density ρ crit. Therefore, 3 t! = # 2π 0 3 Gρ crit " & % 1/4! t 1/2 = H 0 # " 2 & % 1/2 t 1/2 (4.10) For rdition-dominted Universe, the ge of the Universe would be much longer thn for mtter-dominted Universe: t 0 27 eon (4.11) Notice tht we hd lredy obtined the correct power dependence of s function of t in Eq. (3.52). Now we lso obtined the correct multiplictive coefficient in Eq. (4.10). introduc)on to Astrophysics, C. Bertulni, Texs A&M-Commerce 5

6 4.3 - Closed Universe In this cse, we hve k = 1 nd q > ½. Here we only present the results Mtter dominted We hve p = 0 nd 3 ρ = const. The solutions re obtined in prmetric form: 0 = q ( 1 cosη), with 2q 1 t t 0 = q 2q 1 η sinη ( ) where q is the deccelertion prmeter, given by Eq. (3.54) Rdition dominted (4.12) We hve p = ρ/3 nd 4 ρ = const. The solutions re obtined in prmetric form: 0 = 2q 2q 1 sinη, with t = t 0 2q 2q 1 1 cosη ( ) (4.5) (4.13) In both mtter- nd rdition-dominted closed Universes, the evolution is cycloidl the scle fctor grows t n ever-decresing rte until it reches point t which the expnsion is hlted nd reversed. The Universe then strts to compress nd it finlly collpses in the Big Crunch. (see next figure.) introduc)on to Astrophysics, C. Bertulni, Texs A&M-Commerce 6

7 Mtter Dominted Universe Evolu)on of the scle fctor (t) for the flt, closed nd open mher-dominted Friedmnn Universe. introduc)on to Astrophysics, C. Bertulni, Texs A&M-Commerce 7

8 4.4 - Open Universe In this cse, we hve k = - 1 nd q < ½. Here we only present the results Mtter dominted We hve p = 0 nd 3 ρ = const. The solutions re obtined in prmetric form: 0 = q ( coshη 1), with 2q Rdition dominted t t 0 = q 2q 1 sinhη 1 ( ) We hve p = ρ/3 nd 4 ρ = const. The solutions re obtined in prmetric form: 0 = 2q 1 2q sinhη, with t = t 0 2q 1 2q ( coshη 1) (4.14) (4.15) The previous figure summrizes the evolution of the scle fctor (t) for open, flt nd closed mtter-dominted Universes. In erlier times, one cn expnd the trigonometric nd hyperbolic functions to leding terms in powers of η, nd the nd t dependence on η for the different curvtures re given in the next tble. This shows tht t erly times the curvture of the Universe does not mtter. The singulr behvior t erly times is essentilly independent of the curvture of the Universe, or k. The Big Bng is mtter dominted singulrity. introduc)on to Astrophysics, C. Bertulni, Texs A&M-Commerce 8

9 Mtter Dominted Universe Solutions of Friedmnn eqution for mtter-dominted Universe. introduc)on to Astrophysics, C. Bertulni, Texs A&M-Commerce 9

(middle pnel) nd open (lower pnel) Friedmnn Universe.

10 Summry: Solutions of Friedmnn Eqution Evolu)on of the scle fctor (t) for the flt (upper pnel), closed (middle pnel) nd open (lower pnel) Friedmnn Universe. introduc)on to Astrophysics, C. Bertulni, Texs A&M-Commerce 10

11 Fltness problem According to Eq. (3.36), the density prmeter stisfies n eqution of the form Ω 1 = ρ(t) ρ c ρ c = 1 2 (t) The present dy vlue of Ω is known only roughly, 0.1 Ω 2. On the other hnd 1/(d/dt) 2 ~ 1/t 2 in the erly stges of the evolution of the Universe, so the quntity Ω 1 ws extremely smll. One cn show tht in order for Ω to lie in the rnge 0.1 Ω 2 now, the erly Universe must hve hd Ω 1 = ρ(t) ρ c ρ c (4.16) (4.17) This mens tht if the density of the Universe were initilly greter thn ρ c, sy by ρ c, it would be closed nd the Universe would hve collpsed long time go. If on the other hnd the initil density were ρ c less thn ρ c, the present energy density in the Universe would be vnishingly low nd the life could not exist. The question of why the energy density in the erly Universe ws so fntsticlly close to the criticl density is usully known s the fltness problem. introduc)on to Astrophysics, C. Bertulni, Texs A&M-Commerce 11

Horizon problem The number nd size of density fluctutions on both sides of the sky re similr, yet they re seprted by distnce tht is greter thn the speed of light times the ge of the Universe, i.e. they should hve no knowledge of ech other by specil reltivity (c, the speed of light is finite: c = 300,000 km/s).

12 Horizon problem The number nd size of density fluctutions on both sides of the sky re similr, yet they re seprted by distnce tht is greter thn the speed of light times the ge of the Universe, i.e. they should hve no knowledge of ech other by specil reltivity (c, the speed of light is finite: c = 300,000 km/s). us 13 billion yers 13 billion yers At some time in the erly Universe, ll prts of spcetime were cuslly connected, this must hve hppened fter the spcetime fom er, nd before the time where thermliztion of mtter occurred. This is known s the horizon problem. introduc)on to Astrophysics, C. Bertulni, Texs A&M-Commerce 12

13 Infltion Infltion occurs when the vcuum energy contribution domintes the ordinry density nd curvture terms in Friedmnn Eqution (4.1). Assuming these re negligible nd substituting Λ = constnt, one gets The solution is! # 2 " % & t = H 2 ( t) = + Λ 3 ( ) = exp " # Λ 3 t % & ' (4.18) (4.19) When the Universe is dominted by cosmologicl constnt, the expnsion rte grows exponentilly. introduc)on to Astrophysics, C. Bertulni, Texs A&M-Commerce 13

Infltion phse Prticle physics predicts tht the interctions mong prticles in the erly Universe re controlled by forces (or force fields (sf)), which cn hve negtive pressure (p sf = ρ sf, where ρ sf is

14 Infltion phse Prticle physics predicts tht the interctions mong prticles in the erly Universe re controlled by forces (or force fields (sf)), which cn hve negtive pressure (p sf = ρ sf, where ρ sf is the energy density of the fields.) If ρ sf >> ρ rd, Friedmnn eqution reds: d 2 dt 2 = 4πG 3 "# ρ sf + 3p sf % = 8πG 3 ρ sf (4.20) The solution is: t ( ) = 0 exp! " # t t int % & ; t int = 3 8πGρ sf (4.21) The model predicts t inf ~ s. Infltion lsts s long s the energy density ssocited with the force fields is trnsformed into expnsion of the Universe. If this lsts from t 1 = s to t 2 = s, then the Universe expnds by fctor exp(δt/t inf ) ~ e 100 ~ After this exponentil increse, the Universe develops s we discussed before. introduc)on to Astrophysics, C. Bertulni, Texs A&M-Commerce 14

kt (temperture) united forces big bng time ~ 10 2 GeV electromgnetic + wek ~ 10 13 GeV electromgnetic + wek ~ 10

15 Grnd Unifiction Theory (GUT) of forces The seprtion of forces hppens in phse trnsitions with spontneous symmetry breking. NOTE: we will discuss more bout forces nd pr)cles in forthcoming lecture. kt (temperture) united forces big bng time ~ 10 2 GeV electromgnetic + wek ~ GeV electromgnetic + wek ~ s + strong ~ GeV electromgnetic + wek ~ s + strong + grvity Plnck time introduc)on to Astrophysics, C. Bertulni, Texs A&M-Commerce 15

16 Infltion Reding mteril There re number of other observtions tht re suggestive of the need for cosmologicl constnt. For exmple, if the cosmologicl constnt tody comprises most of the energy density of the universe, then the extrpolted ge of the universe is much lrger thn it would be without such term, which helps void the dilemm tht the extrpolted ge of the universe is younger thn some of the oldest strs we observe! A cosmologicl constnt term dded to the infltionry model, n extension of the Big Bng theory, leds to model tht ppers to be consistent with the observed lrge-scle distribution of glxies nd clusters, with COBE's mesurements of cosmic microwve bckground fluctutions, nd with the observed properties of X-ry clusters. The Infltion Theory proposes period of extremely rpid (exponentil) expnsion of the universe shortly fter the Big Bng. While the Big Bng theory successfully explins the shpe of the cosmic microwve bckground spectrum nd the origin of the light elements, it leves open number of importnt questions: Why is the universe so uniform on the lrgest length scles? Why is the physicl scle of the universe so much lrger thn the fundmentl scle of grvity, the Plnck length, which is one billionth of one trillionth of the size of n tomic nucleus? Why re there so mny photons in the universe? Wht physicl process produced the initil fluctutions in the density of mtter? The Infltion Theory, developed by Aln Guth, Andrei Linde, Pul Steinhrdt, nd Andy Albrecht, offers nswers to these questions nd severl other open questions in cosmology. It proposes period of extremely rpid (exponentil) expnsion of the universe leding to the Big Bng expnsion, during which time the energy density of the universe ws dominted by cosmologicl constnt term tht lter decyed to produce the mtter nd rdition tht fill the universe tody. The Infltion Theory links importnt ides in modern physics, such s symmetry breking nd phse trnsitions, to cosmology. The Infltion Theory mkes number of importnt predictions: () the density of the universe is close to the criticl density, nd thus the geometry of the universe is flt. (b) the fluctutions in the primordil density in the erly universe hd the sme mplitude on ll physicl scles. (c) there should be, on verge, equl numbers of hot nd cold spots in the fluctutions of the cosmic microwve bckground temperture. introduc)on to Astrophysics, C. Bertulni, Texs A&M-Commerce 16

17 Plnck mss, time nd length Reding mteril Before t = 10 3 s, the Universe ws dominted by quntum mechnics. A mesure for this point in time is the Compton wvelength of the Universe R C = Mc where M is ll the mss in the Universe, ħ = h/ 2π, with h = x MeV.s. the Plnck constnt. The Schwrzschild rdius is defined s the boundry between spcetime regions round n object of mss M tht cn nd cnnot communicte with the externl Universe. It is given by R S = 2GM c 2 (4.23) (4.24) Equting the Compton wvelength nd the Schwrtzschild rdius leds to the so-clled Plnck mss,! M Pl = c # & " G % 1/2 = GeV / c 2 (4.25) which is equivlent to temperture T = K, using M Pl c 2 ~ kt. Vi the uncertinty reltion from quntum mechnics, ΔEΔt ~ ħ/2, or ΔxΔp ~ ħ/2, one cn get Plnck time nd length, i.e. t Pl = s nd l Pl = cm. At these lengths nd times, time itself becomes undefined. In typicl models of infltion the phse trnsition tkes plce t tempertures round the Grnd Unifiction Temperture (GUT) when the Universe ws bout s old. Suppose tht the Universe styed in the infltionry stte for s. This my pper to be short time, but in fct infltion lsted for 100 hundred times the ge of the Universe t the time infltion strted. Consider smll region with rdius round, sy, cm before infltion. After infltion the volume of the region hs incresed by fctor of (e 100 ) 3 = This huge increse solves some of the problems previously mentioned: () fltness nd (b) horizon problems. introduc)on to Astrophysics, C. Bertulni, Texs A&M-Commerce 17

18 Infltion solves the horizon problem The CMB is found to very smooth (ΔT/T ~ 10 5 ). This suggests tht the different prts of our universe communicted before they equilibrted. Light cn only trvel finite distnce during the finite ge of Universe nd photons of the CMB, coming from opposite directions, nd observed tody cn hve no informtion from ech other. However, infltion llows every prt of the Universe to hve been in close proximity before the infltion phse. After the end of infltion the vcuum energy of the field driving infltion ws trnsferred to ordinry prticles, so reheting of the Universe took plce. During infltion the Universe itself supercooled with T ~ e Ht. The period of infltion nd reheting is strongly non-dibtic, since there is n enormous genertion of entropy t reheting. After the end of infltion, the Universe restrts in n dibtic phse with the stndrd conservtion of the energy. In fct the Universe restrts from very specil initil conditions tht the horizon nd fltness problems re voided. introduc)on to Astrophysics, C. Bertulni, Texs A&M-Commerce 18

19 Infltion solves the fltness problem "#%&'()#&*+!,-)./! 8'&#1&*1!0)1,%! 2&1(34!)! )54,*6&5%,! 7#(6,*4,! "#%&'()#&*+! 0)1,%! -*,4,#'! In the cosmologicl models the k/ 2 term, with k = ±1, could hve been importnt. However, with the sudden infltion the scle fctor hs incresed by ~ nd the vlue of k/ 2 hs been reduced by So, even if we strted from lrge curvture term there is no other mechnism necessry to rrive t the nerly flt Universe. During infltion the energy density of the Universe is constnt, wheres the scle fctor increses exponentilly, s described by Eqs. (4.19) of (4.21). This mens tht Ω must hve been exponentilly close to unity, Ω = 1 to n ccurcy of mny deciml plces. introduc)on to Astrophysics, C. Bertulni, Texs A&M-Commerce 19

4 The dynamical FRW universe

4 The dynamical FRW universe 4 The dynmicl FRW universe 4.1 The Einstein equtions Einstein s equtions G µν = T µν (7) relte the expnsion rte (t) to energy distribution in the universe. On the left hnd side is the Einstein tensor which