3.1 Cosmological Parameters 1 Cosmological Parameters Cosmological models are typically defined through several handy key parameters: 2 Hubble Constant Defines the Scale of the Universe H0 = slope at t0 R0 0 t0 { / H = Hubble length 1 / H0 = Hubble time c 0 3 Cosmological Parameters 4 Cosmological Parameters 5 Cosmological Parameters H0 defines the spatial and temporal scale of the universe The other parameters (Ωx) determine the R0 shape of the R(t) curves 0 t0 { / H = Hubble length 1 / H0 = Hubble time c 0 6 Cosmological Parameters A few notes: The Hubble parameter is usually called the Hubble constant (even though it changes in time!) and it is often written as: -1 -1 -1 -1 h = H0 / (100 km s Mpc ), or h70 = H0 / (70 km s Mpc ) The current physical value of the critical density is -29 2 -3 r0,crit = 0.921 !10 h70 g cm The density parameter(s) can be written as: Wm + Wk + WL = 1 where Wk is a fictitious “curvature density” 7 Cosmological Parameters Recall the definitions of the cosmological parameters: If Wk = 0: The Friedmann Eqn. is now: 8 Cosmological constant implies that: A. There is an energy density that is constant in the comoving coordinates B. There is an energy density that is constant in the proper coordinates C. The expansion rate of the universe is constant D. The matter density is constant E. The mater density is equal to the critical 9 3.2 Solving the Friedmann Equation, and the Equation of State 10 Solving The Friedmann Equation In order to solve it, we also need to define the behavior of mass/energy density r(R) of any given mass/energy component, which is generally expressed through the equation of state, often written as a relation between pressure and density: P = w r w could be a constant, or a function of something 11 The EOS Parameter w • Defined by the equation p = w r • Often called by itself the “equation of state” – Note: this is not necessarily the best way to describe the matter/energy density; it implies a fluid of some kind… This may be OK for the matter and radiation we know, but maybe it is not an optimal description for the dark energy • Special values: w = 0 means p = 0, e.g., non-relativistic matter w = 1/3 is radiation or relativistic matter w = −1 looks just like a cosmological constant … but it can have in principle any value, and it can be changing in redshift 12 Equation of State (EOS) Some simple EOS we can consider: • “Matter”, “dust”, “galaxies”: p = 0 • Radiation: p = r c2 / 3 • “Cosmological Constant”: p = −r c2 In reality, the universe contains a mix of these components, and maybe others as well… Each will lead to a different evolution in redshift, and recall the basic GR paradigm: Density determines the expansion Expansion changes the density 13 Evolution of the Density Generally, r ~ R-3(w+1) • Matter dominated (w = 0): r ~ R-3 • Radiation dominated (w = 1/3): r ~ R-4 • Cosmological constant (w = –1): r = constant • Dark energy with w < –1 e.g., w = –2: r ~ R+3 – Energy density increases as is stretched out! – Eventually would dominate over even the energies holding atoms together! (“Big Rip”) In a mixed universe, different components will dominate the global dynamics at different times Note also that in principle, w could be a function of time, density, etc. 14 What is Dominant When? Matter dominated (w = 0): r ~ R-3 Radiation dominated (w = 1/3): r ~ R-4 Dark energy (w ~ –1): r ~ constant • Radiation density decreases the fastest with time – Must increase fastest on going back in time – Radiation must dominate early in the Universe • Dark energy with w ~ –1 dominates last; it is the dominant component now, and in the (infinite?) future Radiation Matter Dark energy domination domination domination 15 The Dark Energy dominates the expansion rate: A. In the early universe B. In the distant future C. Only if the matter density is less than critical D. Only if the total density is equal to critical E. At some point regardless of the matter density 16 3.3 Examples of Cosmological Models 17 Specific Models Consider several simple models: • k = 0, matter dominated, Einstein de Sitter • k = 0, radiation dominated • k < 0, r > 0 • k > 0 • L dominated 18 k = 0, matter dominated Einstein de Sitter Friedman Equation 2 1 2 8 ( a % 8 ⇒ a da = ± πGρ0 dt with k = 0 & # = πGρ ∫ 3 ∫ ' a $ 3 2 8 a 8 3 2 ' $ −3 ⇒ a = ±(3/ 2) πGρ0 t ⇒ % " = πGρ0a 3 & a # 3 8 ⇒ a 2 = πGρ a−1 3 0 2/3 ∂a 8 ⇒ = ± πGρ a−1 2 ⇒ a ∝ t ∂t 3 0 19 k = 0, radiation dominated Friedman Equation 2 ( a % 8 & # = πGρ with k = 0 ' a $ 3 2 ' a $ 8 −4 ⇒ % " = πGρ0a & a # 3 ⇒ a ∝ a−1 ⇒ ∫ ada ∝ t ⇒ a2 ∝ t 1/ 2 ⇒ a ∝ t 20 Matter Dominated Model 21 Positive Curvature Model: k = +1 Friedman Equation: 2 2 ' a $ 8 kc 2 8 2 2 % " = πGρ − 2 a a = πGρa − c & a # 3 a 3 −3 −4 2 −n if ρ ∝ a or ρ ∝ a then ρa ∝ a (n =1,2) decreases at some point a 2 = 0 a 4 & P # The acceleration equation is: = − πG$ ρ + 3 2 ! a 3 % c " And since all other quantities are positive, a!! < 0 Therefore, a collapse is inevitable 22 Cosmological Constant (L) Dominated 2 Friedman Equation: ' a $ 8 kc2 % " = πGρΛ − 2 & a # 3 a Which can be written as: 2 2 2 a = C0 ρa − kc Assuming that a is allowed to grow, then eventually 2 2 C0a dominates over − kc no matter what value of k If then ρΛ > 0 a! > 0 a Universe expands for ever (Note that if things get more complicated) ρΛ < 0 23 Models With Both Matter & Radiation Harder to solve for r (t) However, to a good approximation, now we can assume that k = 0 and either radiation or matter dominate Radiation Domination r g-dom m-dom Matter Domination log log a(t) ∝ t1/ 2 ∝ t 2/3 −3 −3/ 2 −2 ρm ∝ a ∝ t ∝ t −4 −2 −8/3 logt ργ ∝ a ∝ t ∝ t Generally, 8πGρ 2 −3 −4 = H 0 (ΩΛ,0 + Ωm,0a + Ωγ ,0a ) 3 24 Dynamics of the Universe R(t) ~ t 2/[3(w+1)] • Matter dominated (w = 0): R ~ t 2/3 § Decelerating • Radiation dominated (w = 1/3): R ~ t 1/2 § Decelerating • Cosmological constant (w = –1): R ~ e l t (special) § Accelerating • Where is the transition? § w > –1/3 decelerating § w < –1/3 accelerating 25 Pure L ! ¬High density 26 In the Einstein – de Sitter model: A. The matter density is equal to the critical B. The curvature is positive C. The universe expands first, but then collapses D. The universe is flat E. The expansion rate is proportional to the time 27 3.4 Distances in Cosmology 28 Distances in Cosmology A convenient unit is the Hubble distance, -1 28 -1 DH = c / H0 = 4.283 h70 Gpc = 1.322Î10 h70 cm and the corresponding Hubble time, -1 17 -1 tH = 1 / H0 = 13.98 h70 Gyr = 4.409Î10 h70 s At low z’s, distance D ≈ z DH . But more generally, the comoving distance to a redshift z is: where 29 Distances in Cosmology But the quantity really useful in computing the various physical quantities of interest is the “transverse comoving distance”, where we account for the curvature: where Wk is defined by: 30 Distances in Cosmology We can derive this for using the RW metric: To simplify, let’s put ourselves at the origin, then the light path is purely radial, and dq and df = 0, so: Taking the square root of both sides and integrating: 31 Distances in Cosmology In general this is non-analytic. In a special case of a L = 0 Universe, we have q0 = W0 / 2, and: For a non-zero L universe: Assuming Wk<0, if Wk >0 then the sinh becomes a sin and if Wk=0 then the sinh and the Wk drop out and all that’s left is the integral, which has to be evaluated numerically. 32 Comoving Distance Wm = 0.05, WL = 0 Wm = 0.2, WL = 0.8 Wm = 1, WL = 0 33 Luminosity Distance In relativistic cosmologies, observed flux (bolometric, or in a finite bandpass) is: f = L / [ (4p D2) (1+z)2 ] One factor of (1+z) is due to the energy loss of photons, and one is due to the time dialation of the photon rate. A luminosity distance is defined as DL = D (1+z), so that 2 f = L / (4p DL ). For a specific flux, however, (since Angstroms are also stretched by 1+z) 34 Luminosity Wm = 0.05, Distance WL = 0 Wm = 0.2, WL = 0.8 Wm = 1, WL = 0 35 Angular Diameter Distance Angular diameter of an object with a fixed comoving size X is by definition q = X / D However, an object with a fixed proper size X is (1+z) times larger than in the comoving coordinates, so its apparent angular diameter will be q = (1+z) X / D Thus, we define the angular diameter distance DA = D / (1+z) , so that the angular diameter of an object whose size is fixed in proper coordinates is q = X / DA 36 Angular W = 0.05, WL = 0 Diameter m Distance Wm = 0.2, WL = 0.8 Wm = 1, WL = 0 37 Volume Element This is useful, e.g., when computing the source counts.