Introduction to Cosmology

Transcription

1 1 Introduction to Cosmology Mast Maula Centre for Theoretical Physics Jamia Millia Islamia New Delhi Collaborators: Nutty Professor, Free Ride Mast Maula (CTP, JMI) Introduction to Cosmology ICAC / 30

2 Outline 2 1 Homogeneous and Isotropic Model 2 Evolution Equation 3 Solutions of Evolution Equation 4 Standard Form of Evolution Equations Mast Maula (CTP, JMI) Introduction to Cosmology ICAC / 30

3 Outline 3 1 Homogeneous and Isotropic Model 2 Evolution Equation 3 Solutions of Evolution Equation 4 Standard Form of Evolution Equations Mast Maula (CTP, JMI) Introduction to Cosmology ICAC / 30

4 Homogeneous and Isotropic Model 4 Homogenity: Means that universe looks the same at each point. Isotropy: Means that universe looks the same in all directions. These are two important properties of space which are independent of each other. But isotropy at each point implies homogenity also. Cosmological principle: Universe is homogeneous and isotropic at any given cosmic time. The cosmological principle is supported by the observational evidence that the universe becomes smooth at large scales. The cosmological principle presents the idealized picture of the universe. The departure from homogenity and isotropy is extremely important which led to the structure formation in the universe. Mast Maula (CTP, JMI) Introduction to Cosmology ICAC / 30

5 Homogeneous and Isotropic Model 4 Homogenity: Means that universe looks the same at each point. Isotropy: Means that universe looks the same in all directions. These are two important properties of space which are independent of each other. But isotropy at each point implies homogenity also. Cosmological principle: Universe is homogeneous and isotropic at any given cosmic time. The cosmological principle is supported by the observational evidence that the universe becomes smooth at large scales. The cosmological principle presents the idealized picture of the universe. The departure from homogenity and isotropy is extremely important which led to the structure formation in the universe. Mast Maula (CTP, JMI) Introduction to Cosmology ICAC / 30

6 Hubble s law 5 We examine the motion of matter in a coordinate system in which it is at rest at the origin. We now ask for the velocity dstribution consistent with homgenity and isotropy. Hubbles law: v = H(t) r (1) 0 r Velocity field (1) is isotropic at O. Let us verify that (1) holds for any observer situated at a point A, The observer at A is in motion with respect to O. r = r r A. So, r A A r r v = v v A = H r H r A v = H( r r A ) v = H r 0 Therefore, velocity distribution (1) is homogeneous and isotropic. Mast Maula (CTP, JMI) Introduction to Cosmology ICAC / 30

7 Local expansion 6 The distance between two arbitrary points changes as Therefore, d r AB (t) dt = H(t) r AB ( t ) r AB (t) = r AB (t 0 ) exp H(t )dt t 0 Remark: The dynamics will be decided by H(t). If H(t) = const., then r AB (t) = r AB (t 0 )e (t t 0)H Mast Maula (CTP, JMI) Introduction to Cosmology ICAC / 30

8 Evolution of density 7 ρ = Differentiating w.r.t. time dρ(t) dt M 4π 3 R3 = 3M dr [ 4π 3 R4 ] dt But dr/dt = v = HR. Therefore, M R(t) dρ(t) dt = 3M [ 4π 3 R3 ] H = 3ρH dρ dt = 3ρH (2) Mast Maula (CTP, JMI) Introduction to Cosmology ICAC / 30

9 8 The last can also be obtained from continuity equation ρ - function of time alone: ρ t = (ρ v) ρ t = ρh r = 3ρH (3) ρ t = d dt ρ Homogeneity and isotropy is a preserved property in time. Mast Maula (CTP, JMI) Introduction to Cosmology ICAC / 30

10 Outline 9 1 Homogeneous and Isotropic Model 2 Evolution Equation 3 Solutions of Evolution Equation 4 Standard Form of Evolution Equations Mast Maula (CTP, JMI) Introduction to Cosmology ICAC / 30

11 The acceleration of a particle with mass m due to the gravitational force of M is given by: 10 d 2 R(t) dt 2 = GM R 2 (t) M R(t) m d 2 R(t) dt 2 = d dr (HR) = H dt dt +RdH dt dh dt = GM R 3 H2 = H 2 R+R dh dt dh dt = H 2 4π 3 Gρ (4) Remark: If H = const, eqn (4) is inconsistent. Infact H = const (with dh dt = R R H2 ) would imply R(t) > 0, which cannot come from eqn (4). Mast Maula (CTP, JMI) Introduction to Cosmology ICAC / 30

12 Friedman Equation: Multiply eqn (3) by dr(t)/dt: ( dr d 2 ) R dt dt 2 = GM dr R 2 dt ( ) 1 d dr 2 = d ( ) GM 2 dt dt dt R [ ( ) ] d 1 dr 2 GM = 0 dt 2 dt R ( ) dr 2 GM dt R M = 4 3 πρr3 = A = const (5) Determining the constant A {t 0, ρ 0 } present epoch and select a value of R = R 0 at t = t 0 for the sphere. 4 Mast Maula (CTP, JMI) Introduction to Cosmology 2 ICAC / 30 1 ( dr ) 2

13 General character of the solution of (6) 12 At present, dr dt > 0 implies that R was smaller in the past, but 8πG 3 ρr2 was larger, so dr dt was larger in the past: Explanation R(t 0 ) = 0, dr dt = + t = t 0 t=t0 R 0 by definition, R(t) < 0, ρ > 0 Hence R(t) was smaller and smaller as we go into past deeper and deeper, and dr/dt becomes larger and larger. Consequently, there was an epoch, say t = 0, when dr R(t = 0) = R(0) = 0 dt = t=0 Mast Maula (CTP, JMI) Introduction to Cosmology ICAC / 30

14 Critical density 13 The prediction of the future depends upon the sign of [ρ 0 3H 0 /8πG] or how the present density compares with the critical density ρ c : ρ c = 3H2 0 8πG We also defined the dimensionless density parameter Ω 0 : R(t) Ω 0 ρ 0 ρ = 8πGρ 0 3H t 0 t Mast Maula (CTP, JMI) Introduction to Cosmology ICAC / 30

15 Classification of the solution 14 1 I ρ 0 > ρ c The second term in eqn (6) is positive. As R increases, the first term decreases and eventually becomes equal to the second term at a particular time. The RHS of eqn (6) then vanishes and expansion ceases, and contraction begins. 2 II ρ 0 < ρ c RHS of equation (6) is positive, leading to expansion forever. As t, R dr dt = t= [ ] 8πG 1/2 3 R2 0(ρ c ρ) R(t) ρ < ρ o c ρ = ρ o c ρ > ρ o c t max 3 III ρ 0 = ρ c Expansion continues without bound. Mast Maula (CTP, JMI) Introduction to Cosmology ICAC / 30

16 Classification of the solution 14 1 I ρ 0 > ρ c The second term in eqn (6) is positive. As R increases, the first term decreases and eventually becomes equal to the second term at a particular time. The RHS of eqn (6) then vanishes and expansion ceases, and contraction begins. 2 II ρ 0 < ρ c RHS of equation (6) is positive, leading to expansion forever. As t, R dr dt = t= [ ] 8πG 1/2 3 R2 0(ρ c ρ) R(t) ρ < ρ o c ρ = ρ o c ρ > ρ o c t max 3 III ρ 0 = ρ c Expansion continues without bound. Mast Maula (CTP, JMI) Introduction to Cosmology ICAC / 30

17 Classification of the solution 14 1 I ρ 0 > ρ c The second term in eqn (6) is positive. As R increases, the first term decreases and eventually becomes equal to the second term at a particular time. The RHS of eqn (6) then vanishes and expansion ceases, and contraction begins. 2 II ρ 0 < ρ c RHS of equation (6) is positive, leading to expansion forever. As t, R dr dt = t= [ ] 8πG 1/2 3 R2 0(ρ c ρ) R(t) ρ < ρ o c ρ = ρ o c ρ > ρ o c t max 3 III ρ 0 = ρ c Expansion continues without bound. Mast Maula (CTP, JMI) Introduction to Cosmology ICAC / 30

18 Maximum Age Estimate 15 t = 0, R = 0. Suppose the expansion rate is constant and given by the present value of Hubble parameter, ( ) dr R(t 0 ) R 0 = t 0 = H 0 R 0 t 0 dt t=0 t 0 = 1 H 0 T 0 h years H 0 100kms 1 mpc 1 h t h 1 years 0.37 < H 0 t 0 < 1.47 Mast Maula (CTP, JMI) Introduction to Cosmology ICAC / 30

19 Outline 16 1 Homogeneous and Isotropic Model 2 Evolution Equation 3 Solutions of Evolution Equation 4 Standard Form of Evolution Equations Mast Maula (CTP, JMI) Introduction to Cosmology ICAC / 30

20 Solutions of Evolution Equation 17 ρ 0 = ρ c or Ω 0 = 1 ( ) dr 2 = 8πG dt 3 ρr2 ρ(t) = ρ 0 R 3 R3 0 ( dr dt ) 2 = 8πG ρ 0 R R R(t) = R(t = 0)(t/t 0 ) 2/3 drr 1/2 = const.dt ( ) t 2/3 R 0 = R(t) t 0 R(t) t 2/3 Ṙ(t) 2 3 t 1/3 Ṙ(t) R = t Mast Maula (CTP, JMI) Introduction to Cosmology ICAC / 30

21 18 Therefore, t 0 = H 0. ρ(t) = ρ 0R 3 0 R 3 (t) = ρ 0R 3 0 R 3 0 (t/t 0) 2 = 3H2 0 8πGt 2 t2 0 = 3H πGt H 2 0 = 1 6πGt 2 ρ(t) = 1 6πGt 2 Mast Maula (CTP, JMI) Introduction to Cosmology ICAC / 30

22 Pressure corrections: Relativistic effects 19 Consistency check: Differentiating (9), we get Ṙ = HR lead to ρ (ρ t + 3H + Pc ) 2 = 0 (7) R R = 4πG (ρ + Pc ) 3 (8) Ṙ 2 2 4πG 3 R2 ρ = A (9) Ṙ R 4πG 3 R2 ρ 4πG 3 2 ρṙr = 0 R R = 4πG (ρ + Pc ) 3 2 System (7), (8) and (9) are consistent. Mast Maula (CTP, JMI) Introduction to Cosmology ICAC / 30

23 Pressure corrections: Relativistic effects 19 Consistency check: Differentiating (9), we get Ṙ = HR lead to ρ (ρ t + 3H + Pc ) 2 = 0 (7) R R = 4πG (ρ + Pc ) 3 (8) Ṙ 2 2 4πG 3 R2 ρ = A (9) Ṙ R 4πG 3 R2 ρ 4πG 3 2 ρṙr = 0 R R = 4πG (ρ + Pc ) 3 2 System (7), (8) and (9) are consistent. Mast Maula (CTP, JMI) Introduction to Cosmology ICAC / 30

24 Case of Radiation Domination 20 Equation of State: Out of the eqs (7), (8) and (9), only two are independtent. Eq. (9) can be obtained from (7) and (8). These eqs can be solved and ρ(t), R(t) can be uniquely determined provided the equation of state (= relation between ρ and P) is given. This relation, in simple cases, can be written as P = ωρc 2, ω = 0 Dust 1 3 Radiation 1 Cosmological constant ω = 1 3, ρ 0 = ρ c ρ t + 3H (ρ + Pc 2 ) = 0 ρ t + 4Hρ = 0 ρ(t) = ρ 0 R 4 0 R 4 (t) Mast Maula (CTP, JMI) Introduction to Cosmology ICAC / 30

25 21 (Ṙ R ) 2 = 8πG 3 ρ Ṙ(t) 1 R(t) H(t) = 1 2t ( ) t 1/2 R(t) = R 0 t 0 t 0 = 1 2H 0 ρ(t) = πG t 2 Mast Maula (CTP, JMI) Introduction to Cosmology ICAC / 30

26 Outline 22 1 Homogeneous and Isotropic Model 2 Evolution Equation 3 Solutions of Evolution Equation 4 Standard Form of Evolution Equations Mast Maula (CTP, JMI) Introduction to Cosmology ICAC / 30

27 Comoving Coordinates: 23 These coordinates are are carried along with the expansion. Since the expansion is uniform, the relation between the Physical and Comoving Coordinates is given by r(t) = a(t) x a(t) scale factor The uniformity of expansion is encoded in the scale factor. ( ) dr 2 = 8πG dt 3 ρr2 8πG 3 R2 0[ρ 0 ρ c ] Put R(t) = a(t)x (ȧ a ) 2 = 8πG 3 ρ H x 2 a 2 Mast Maula (CTP, JMI) Introduction to Cosmology ICAC / 30

28 Let Kc 2 = A x 2 or K = Also, from (8) Putting c = 1 A x 2 c 2. Dimensionally [K] = L2 T 2 1 (ȧ a L 2 L 2 T 2 = L 2 ) 2 = 8πG 3 ρ Kc2 a 2 ä a = 4πG ( ρ + 3P ) 3 c 2 ρ + 3H(ρ + P) = 0 t (ȧ ) 2 = 8πG a 3 ρ K a 2 24 ä a = 4πG (ρ + 3P) 3 Mast Maula (CTP, JMI) Introduction to Cosmology ICAC / 30

29 Thermal History 25 For K = 0, ρ r (t) = 3 32πGt 2 ρ r (t) = αt 4 T = ( ) 3 1/4 t 1/2 32πGα T K = t 1/2 sec ρ r (t) 1 ( a0 ) a 4 T(a) = T 0 a Mast Maula (CTP, JMI) Introduction to Cosmology ICAC / 30

30 Decoupling 26 Decoupling takes place at a temperature equal to the binding energy of Hydrogen atom. 3k B T d = 13.6eV T d = 13.6eV 3k B K (1eV 10 4 K) However, temperature in reality is much smaller than this T d 3000K a 0 = a d T 0 Since T K = T 1/2, decoupling time t d = sec yrs. Mast Maula (CTP, JMI) Introduction to Cosmology ICAC / 30

31 27 Remark: Ω r 1 Ω m a = Ω0 r 1 Ω 0 m a Ω 0 m a a eq Ω 0 t eq Ω 3/2 0 years Mast Maula (CTP, JMI) Introduction to Cosmology ICAC / 30

32 Cosmological Constant 28 Einstein introduced the cosmological constant to make the universe static (later described by him as his biggest blunder ). Λ > 0 and ρ: H = 0 ä > 0 ρ Λ = H 2 = 8πG 3 ρ K a 2 + Λ 3 Λ 8πG a(t) e Λ 3 t P Λ = Λ 8πG ä(t) a(t) = 4πG 3 (ρ Λ + 3P Λ ) = + 8πG 3 ρ Λ accelerated expansion. Inflation Mast Maula (CTP, JMI) Introduction to Cosmology ICAC / 30

33 29 H 2 = 8πG 3 ρ K a 2 + Λ 3 1 = Ω m + Ω Λ K a 2 H 2 Ω m + Ω Λ 1 = K a 2 H 2 Mast Maula (CTP, JMI) Introduction to Cosmology ICAC / 30

34 For Further Reading 30 J. V. Narlikar An Introduction to Cosmology (Cambridge University Press, 2002) Steven Weinberg Cosmology (Oxford University Press) Mast Maula (CTP, JMI) Introduction to Cosmology ICAC / 30