3.1 Cosmological Parameters

Transcription

1 3.1 Cosmological Parameters 1

2 Cosmological Parameters Cosmological models are typically defined through several handy key parameters:

3 Hubble Constant Defines the Scale of the Universe R 0 H 0 = slope at t 0 0 { t 0 1 / H 0 = Hubble time c / H 0 = Hubble length 3

4 Cosmological Parameters 4

5 Cosmological Parameters 5

6 Cosmological Parameters H 0 defines the spatial and temporal scale of the universe R 0 The other parameters (Ω x ) determine the shape of the R(t) curves 0 { t 0 1 / H 0 = Hubble time c / H 0 = Hubble length 6

7 A few notes: Cosmological Parameters The Hubble parameter is usually called the Hubble constant (even though it changes in time!) and it is often written as: h = H 0 / (100 km s -1 Mpc -1 ), or h 70 = H 0 / (70 km s -1 Mpc -1 ) The current physical value of the critical density is ρ 0,crit = h 70 g cm -3 The density parameter(s) can be written as: Ω m + Ω k + Ω Λ = 1 where Ω k is a fictitious curvature density 7

8 Cosmological Parameters Recall the definitions of the cosmological parameters: If Ω k = 0: The Friedmann Eqn. is now: 8

9 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

10 3. Solving the Friedmann Equation, and the Equation of State 10

11 Solving The Friedmann Equation In order to solve it, we also need to define the behavior of mass/energy density ρ(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 ρ w could be a constant, or a function of something 11

12 The EOS Parameter w Defined by the equation p = w ρ 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 1

13 Equation of State (EOS) Some simple EOS we can consider: Matter, dust, galaxies : p = 0 Radiation: p = ρ c / 3 Cosmological Constant : p = ρ c 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

14 Evolution of the Density Generally, ρ ~ R -3(w+1) Matter dominated (w = 0): ρ ~ R -3 Radiation dominated (w = 1/3): ρ ~ R -4 Cosmological constant (w = 1): ρ = constant Dark energy with w < 1 e.g., w = : ρ ~ 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

15 What is Dominant When? Matter dominated (w = 0): ρ ~ R -3 Radiation dominated (w = 1/3): ρ ~ R -4 Dark energy (w ~ 1): ρ ~ 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 domination Matter domination Dark energy domination 15

16 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

17 3.3 Examples of Cosmological Models 17

18 Specific Models Consider several simple models: k = 0, matter dominated, Einstein de Sitter k = 0, radiation dominated k < 0, ρ > 0 k > 0 Λ dominated 18

19 k = 0, matter dominated Einstein de Sitter πgρ a a 3 8 = # $ % & ' ( = " # $ % & ' a G a a ρ π = a G a ρ π = ± a G t a ρ π ± = dt G da a ρ π a 3 = ±(3 / ) 8 3 πgρ 0 t a t /3 Friedman Equation with k = 0 19

20 k = 0, radiation dominated Friedman Equation with k = 0 ' % & a a $ " # = ( a & ' a % # $ 8 4 πgρ 0a 3 a = 8 πgρ 3 a 1 ada t a t a t 1/ 0

21 Matter Dominated Model 1

22 Positive Curvature Model: k = +1 Friedman Equation: ' % & a a $ " # = 8 πgρ 3 kc a a a 8 = πgρa 3 c if 3 ρ a or 4 ρ a at some point a = 0 The acceleration equation is: then a a ρa a n 4 & P = πg$ ρ % c And since all other quantities are positive, ( n =1,)!! a < 0 #! " decreases Therefore, a collapse is inevitable

23 Cosmological Constant (Λ) Dominated Friedman Equation: ' % & a a $ " # = 8 πg 3 ρ Λ kc a Which can be written as: a = C ρa 0 kc Assuming that a is allowed to grow, then eventually C0a dominates over kc If > 0!a > ρ Λ (Note that if then 0 ρ Λ < 0 a no matter what value of k Universe expands for ever things get more complicated) 3

24 Models With Both Matter & Radiation Harder to solve for ρ (t) However, to a good approximation, we can assume that k = 0 and either radiation or matter dominate a( t) γ-dom 1/ t 3 3/ ρm a t 4 ργ a t m-dom t t t /3 8/3 log ρ Radiation Domination Matter Domination logt now Generally, 8πGρ = H 0 Λ,0 m,0 γ,0a 3 ( 3 Ω + Ω a + Ω 4 ) 4

25 Dynamics of the Universe R(t) ~ t /[3(w+1)] Matter dominated (w = 0): R ~ t /3 Decelerating Radiation dominated (w = 1/3): R ~ t 1/ Decelerating Cosmological constant (w = 1): R ~ e λ t (special) Accelerating Where is the transition? w > 1/3 decelerating w < 1/3 accelerating 5

26 Pure Λ High density 6

27 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 7

28 3.4 Distances in Cosmology 8

29 Distances in Cosmology A convenient unit is the Hubble distance, D H = c / H 0 = 4.83 h Gpc = 1.3Î10 8 h cm and the corresponding Hubble time, t H = 1 / H 0 = h Gyr = 4.409Î10 17 h s 9

30 Distances in Cosmology A convenient unit is the Hubble distance, D H = c / H 0 = 4.83 h 70-1 Gpc = 1.3Î10 8 h 70-1 cm and the corresponding Hubble time, t H = 1 / H 0 = h 70-1 Gyr = 4.409Î10 17 h 70-1 s At low z s, distance D z D H. But more generally, the comoving distance to a redshift z is: where 30

31 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 Ω k is defined by: 31

32 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 dθ and dφ = 0, so: Taking the square root of both sides and integrating: 3

33 Distances in Cosmology In general this is non-analytic. In a special case of a Λ = 0 Universe, we have q 0 = Ω 0 /, and: For a non-zero Λ universe: Assuming Ω k <0, if Ω k >0 then the sinh becomes a sin and if Ω k =0 then the sinh and the Ω k drop out and all that s left is the integral, which has to be evaluated numerically. 33

34 Comoving Distance Ω m = 0.05, Ω Λ = 0 Ω m = 0., Ω Λ = 0.8 Ω m = 1, Ω Λ = 0 34

35 Luminosity Distance In relativistic cosmologies, observed flux (bolometric, or in a finite bandpass) is: f = L / [ (4π D ) (1+z) ] 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 D L = D (1+z), so that f = L / (4π D L ). For a specific flux, however, (since Angstroms are also stretched by 1+z) 35

36 Luminosity Distance Ω m = 0.05, Ω Λ = 0 Ω m = 0., Ω Λ = 0.8 Ω m = 1, Ω Λ = 0 36

37 Angular Diameter Distance Angular diameter of an object with a fixed comoving size X is by definition θ = 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 θ = (1+z) X / D Thus, we define the angular diameter distance D A = D / (1+z), so that the angular diameter of an object whose size is fixed in proper coordinates is θ = X / D A 37

38 Angular Diameter Distance Ω m = 0.05, Ω Λ = 0 Ω m = 0., Ω Λ = 0.8 Ω m = 1, Ω Λ = 0 38

39 Volume Element This is useful, e.g., when computing the source counts. Generally, it has to be evaluated numerically. The total volume out to some z, over the whole sky, is: 39

40 Volume Element Ω m = 0.05, Ω Λ = 0 Ω m = 0., Ω Λ = 0.8 Ω m = 1, Ω Λ = 0 40

41 Age and Lookback Time The time elapsed since some redshift z is: Generally it has to be integrated numerically, except in some special cases, such as Λ = 0. Integrating to infinity gives the age of the universe, and the difference of the two is the age at a given redshift. 41

42 Age and Lookback Time Ω m = 0., Ω Λ = 0.8 Ω m = 0.05, Ω Λ = 0 Ω m = 1, Ω Λ = 0 4

43 The Basis of Cosmological Tests R(t)/R 0 = 1/(1+z) 1 D(z) ~ c [t 0 -t(z)] Big bang at z = 0 Big bang t 0 now t 0 now z All cosmological tests essentially consist of comparing some measure of (relative) distance (or look-back time) to redshift. Absolute distance scaling is given by the H 0. 43

44 Cosmological Tests: Expected Generic Behavior of Various Models R(t) R(t)/R 0 0 t 0 t 0 t - t 0 Models with a lower density and/or positive Λ expand faster, are thus larger, older today, have more volume and thus higher source counts, at a given z sources are further away and thus appear fainter and smaller Models with a higher density and lower Λ behave exactly the opposite 44

45 The luminosity distance is: A. Equal to the Hubble length B. Greater than the angular diameter distance C. Smaller than the angular diameter distance D. Measured in the comoving units E. Measured in the proper units 45