Modeling the Universe Chapter 11 Hawley/Holcomb. Adapted from Dr. Dennis Papadopoulos UMCP

Transcription

1 Modeling the Universe Chapter 11 Hawley/Holcomb Adapted from Dr. Dennis Papadopoulos UMCP

2 Spectral Lines - Doppler

3 λ λ em 1+ z = obs z = λ obs λ λ em em

4 Doppler Examples

5 Doppler Examples

6 Expansion Redshifts = R now /R then z=2 three times, z=10, eleven times

7 Expansion Redshifts

8 Expansion - Example

9 Current Record Redshift

10 Hubbleology Hubble length D H =c/h, Hubble sphere: Volume enclosed in Hubble sphere estimates the volume of the Universe that can be in our light-cone; it is the limit of the observable Universe. Everything that could have affected us Every point has its own Hubble sphere Look-back time: Time required for light to travel from emission to observation

11 Gravitational Redshift

12 Interpretation of Hubble law in terms of relativity New way to look at redshifts observed by Hubble Redshift is not due to velocity of galaxies Galaxies are (approximately) stationary in space Galaxies get further apart because the space between them is physically expanding! The expansion of space, as R(t) in the metric equation, also affects the wavelength of light as space expands, the wavelength expands and so there is a redshift. So, cosmological redshift is due to cosmological expansion of wavelength of light, not the regular Doppler shift from local motions.

13 Relation between z and R(t) Using our relativistic interpretation of cosmic redshifts, we write Redshift of a galaxy is defined by z = So, we have λ obs = R present λ em R emitted λobs λ λ em em z = R present R emitted 1= R present R emitted R emitted ΔR R

14 Hubble Law for nearby (z<0.1) objects Thus cz cδr R (ΔR /Δt) = cδt R = d light travel H where Hubble s constant is defined by H = 1 R ΔR Δt = 1 R But also, for comoving coordinates of two galaxies differing by space-time interval d=r(t) D comoving, have v= D comoving ΔR/Δt=(d/R) (ΔR/Δt) Hence v= d H for two galaxies with fixed comoving separation dr dt

15 Peculiar velocities Of course, galaxies are not precisely at fixed comoving locations in space They have local random motions, called peculiar velocities e.g. motions of galaxies in local group This is the reason that observational Hubble law is not exact straight line but has scatter Since random velocities do not overall increase with comoving separation, but cosmological redshift does, it is necessary to measure fairly distant galaxies to determine the Hubble constant accurately

16 Distance determinations further away In modern times, Cepheids in the Virgo galaxy cluster have been measured with Hubble Space Telescope (16 Mpc away ) Virgo cluster

17 Tully-Fisher relation Tully-Fisher relationship (spiral galaxies) Correlation between width of particular emission line of hydrogen, Intrinsic luminosity of galaxy So, you can measure distance by Measuring width of line in spectrum Using TF relationship to work out intrinsic luminosity of galaxy Compare with observed brightness to determine distance Works out to about 200Mpc (then hydrogen line becomes too hard to measure)

18 t Hubble time Once the Hubble parameter has been determined accurately, it gives very useful information about age and size of the expanding Universe Recall Hubble parameter is ratio of rate of change of size of Universe to size of Universe: H = 1 R ΔR Δt = 1 R If Universe were expanding at a constant rate, we would have ΔR/Δt=constant and R(t) =t (ΔR/Δt) ; then would have H= (ΔR/Δt)/R=1/t dr dt ie t H =1/H would be age of Universe since Big Bang R(t)

19 Modeling the Universe

20 BASIC COSMOLOGICAL ASSUMPTIONS Germany 1915: Einstein just completed theory of GR Explains anomalous orbit of Mercury perfectly Schwarzschild is working on black holes etc. Einstein turns his attention to modeling the universe as a whole How to proceed it s a horribly complex problem

21 How to make progress Proceed by ignoring details Imagine that all matter in universe is smoothed out i.e., ignore details like stars and galaxies, but deal with a smooth distribution of matter Then make the following assumptions Universe is homogeneous every place in the universe has the same conditions as every other place, on average. Universe is isotropic there is no preferred direction in the universe, on average.

22 There is clearly large-scale structure Filaments, clumps Voids and bubbles But, homogeneous on very large-scales. So, we have the The Generalized Copernican Principle there are no special points in space within the Universe. The Universe has no center! These ideas are collectively called the Cosmological Principles.

23 Key Assumptions

24 Riddles of Conventional Thinking

25 Stability

26 GR vs. Newtonian

27 Newtonian Universe

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29 Expanding Sphere

30 Fates of Expanding Universe

31 Spherical Universe

32 Friedman Universes

33 Einstein s Greatest Blunder

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35 THE DYNAMICS OF THE UNIVERSE EINSTEIN S MODEL Einstein s equations of GR 8πG G = c 4 G describes the spacetime curvature (including its dependence with time) of Universe here s where we plug in the RW geometries. T T describes the matter content of the Universe. Here s where we tell the equations that the Universe is homogeneous and isotropic.

36 Einstein plugged the three homogeneous/isotropic cases of the FRW metric formula into his equations of GR to see what would happen Einstein found That, for a static universe (R(t)=constant), only the spherical case worked as a solution to his equations If the sphere started off static, it would rapidly start collapsing (since gravity attracts) The only way to prevent collapse was for the universe to start off expanding there would then be a phase of expansion followed by a phase of collapse

37 So Einstein could have used this to predict that the universe must be either expanding or contracting! but this was before Hubble discovered expanding universe (more soon!) everybody thought that universe was static (neither expanding nor contracting). So instead, Einstein modified his GR equations! Essentially added a repulsive component of gravity New term called Cosmological Constant Could make his spherical universe remain static BUT, it was unstable a fine balance of opposing forces. Slightest push could make it expand violently or collapse horribly.

38 Soon after, Hubble discovered that the universe was expanding! Einstein called the Cosmological Constant Greatest Blunder of My Life!.but very recent work suggests that he may have been right (more later!)

39 Sum up Newtonian Universe

40 Newtonian Universe Send r->, k is twice the kinetic energy per unit mass remaining when the sphere expanded to infinite size

41 Fates of Expanding Universe FINITE SPHERE 2 V = 2 GMs / R+ k k = 2E Explore R-> 2 2GM R& = s + R 2E 1. E <0, negative energy per unit mass; expansion stops and re-collapses 2. E =0, zero net energy; exactly the velocity required to expand forever but velocity tends to zero as t and R go to infinity 3. E > 0, positive energy per unit mass; keeps expanding forever; reaches infinity with some velocity to spare BIG LEAP -> CONSIDER SPHERE THE UNIVERSE 4 3 Ms = πr ρ 3 ΔR V = R& Δt ΔV GM R& = g = Δt R s 2 What happens when R->? 4 R& = πgρr R& = πgρr + 2E 3

42 Standard Model From Newtonian to GR 4 R& = πgρr R& = πgρr + 2E 3 The Friedmann Equation R & = πgρr kc 3 Robertson Walker (RW) metric : k=0, +1, -1 In Friedmann s equation R is the scale factor rather than the radius of an arbitrary sphere Gravity of mass and energy of the Universe acts on space time scale factor much as the gravity of mass inside a uniform sphere acts on its radius and E replaced by curvature constant. Term retains significance as an energy at infinity but it is tied to the overall geometry of space

43 Standard Model Simplifications 8 R & = πgρr kc To solve we need to know how mass-energy density changes with time. If only mass ρr 3 =constant. 4 R& = πgρr 3 Now need relativistic equation of mass-energy conservation and equation of state i.e ρ E =f(ρ m ). Notice that here ρ m R 3 =constant but ρ E R 4 =constant. Why the extra R? Mainly photons left out of Big-Bang. Red-shifting due to expansion reduces energy density per unit volume faster than 1/R 3 Photons dominant early in Universe are negligible source of space time curvature compared with mass to day. All models decelerate R & < 0 Also now dr/dt>0 expansion. For all models R=0 at some time. R=0 at t=0. Density -> infinity, and kc 2 term negligible at early times. Great simplification.

44 Fate of Universe-Standard Model While early time independent of curvature factor ultimate fate critically dependant on value of k, since mass-energy term decreases as 1/R. Fate of Universe in Newtonian form depended on value of E. In Friedmann Universe it depends on value of curvature k. All models begin with a BANG but only the spherical ends with BANG while the other two end with a whimper.

45 Theoretical Observables Friedmann equation describes evolution of scale factor R(t) in the Robertson-Walker metric. i.e. universe isotropic and homogeneous. Solution for a choice of ρ and k is a model of the Universe and gives R(t) We cannot observe R(t) directly. What else can we observe to check whether model predictions fit observations? Need to find observable quantities derived from R(t). Enter Hubble H = v/ l = R & / R Since R and its rate are functions of time H function of time. NOT CONSTANT. Constant only at a particular time. Now given symbol H 0 8 R& = πgρr 3 kc R& R kc R = H = πgρ 2 2 Time evolution equation for H(t) Replaces scale factor R by measurable quantities H, ρ and spatial geometry

46 Observing Standard Model Average mass density critical parameter why? kc R 8πGρ = ( 1) Measurement of H 0 and ρ 0 give H H0 Explore equation: curvature constant k 1. Empty universe ρ=0, k negative hyperbolic universe, expand forever 2. Flat or require matter or energy. 3. k=0 -> critical density 8 π G ρ 3 H H 0 ρ c = 8 π G ρ 0 Ω M = ρ Ω = c M 1

47 Critical Density 8π Gρ 3H H0 ρc = 8π G ρ0 Ω M = ρ Ω = c M 1 If H km s -1 Mpc -1 critical density is 2x10-26 kg/m 3 or 10 Hydrogen atoms per cubic meter of space Scales as H 2 50 km/sec Mpc gives ¼ density Current value of 72 km/sec Mpc gives critical density kg/m 3 Ω M =1 gives boundary between open hyperbolic universes and closed, finite, spherical universe In a flat universe Ω is constant otherwise it changes with cosmic time

48 Deceleration Parameter q R R& = H0 Ω ( ) kc R M q R& Deceleration Parameter. Now q 0. All 2 RH cosmological constant to change it standard models decelerate q.0. Need 1 q = Ω For standard models specification of q determines M 2 geometry of space and therefore specific model

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50 Summary - Definitions

51 Review How does R(t) (and H) change in time? And what is the value of the curvature k? Need to solve Einstein s equation! 8πG G = c 4 T

52 STANDARD COSMOLOGICAL MODELS In general Einstein s equation relates geometry to dynamics That means curvature must relate to evolution Turns out that there are three possibilities k=-1 k=0 k=+1

53 Important features of standard models All models begin with R=0 at a finite time in the past This time is known as the BIG BANG Space and time come into existence at this moment there is no time before the big bang! The big bang happens everywhere in space not at a point!

54 There is a connection between the geometry and the dynamics Closed (k=+1) solutions for universe expand to maximum size then re-collapse Open (k=-1) solutions for universe expand forever Flat (k=0) solution for universe expands forever (but only just barely almost grinds to a halt).

55 t Hubble time We can relate this to observations Once the Hubble parameter has been determined accurately from observations, it gives very useful information about age and size of the expanding Universe Recall Hubble parameter is ratio of rate of change of size of Universe to size of Universe: H = 1 R ΔR Δt = 1 R If Universe were expanding at a constant rate, we would have ΔR/Δt=constant and R(t) =t (ΔR/Δt) ; then would have H= (ΔR/Δt)/R=1/t i.e. t H =1/H would be age of Universe since Big Bang dr dt R(t)

56 Hubble time for nonuniform Slope of R(t) curve is dr/dt expansion Big Bang NOW Hubble time is t H =1/H=R/(dR/dt) Since rate of expansion varies, t H =1/H gives an estimate of the age of the Universe This tends to overestimate the actual age of the Universe since the Big Bang

57 Terminology Hubble distance, D=ct H (distance that light travels in a Hubble time). This gives an approximate idea of the size of the observable Universe. Age of the Universe, t age (the amount of cosmic time since the big bang). In standard models, this is always less than the Hubble time. Look-back time, t lb (amount of cosmic time that passes between the emission of light by a certain galaxy and the observation of that light by us) Particle horizon (a sphere centered on the Earth with radius ct age ; i.e., the sphere defined by the distance that light can travel since the big bang). This gives the edge of the actual observable Universe.

58 Friedmann Equation Where do the three types of evolutionary solutions come from? 8πG Back to Einstein s eq. G = 4 When we put the RW metric in Einstein s equation and go though the GR, we get the Friedmann Equation this is what determines the dynamics of the Universe 8 G 2GM R& π = ρr kc = kc 3 R What are the terms involved? G is Newton s universal constant of gravitation R & is the rate of change of the cosmic scale factor same as ΔR/Δt for small changes. in time ρ is the total matter and energy density k is the geometric curvature constant c T

59 If we divide Friedmann equation by R 2, we get: R& 2 2 8π G kc ( ) H = ρ R 3 R Let s examine this equation H 2 must be positive so the RHS of this equation must also be positive. Suppose density is zero (ρ=0) Then, we must have negative k (i.e., k=-1) So, empty universes are open and expand forever Flat and spherical Universes can only occur in presence of (enough) matter. 2 2

60 Critical density What are the observables for flat solution (k=0) Friedmann equation then gives 2 8πG H = ρ 3 So, this case occurs if the density is exactly equal to the critical density 3H 2 ρ = ρcrit = 8πG Critical density means flat solution for a given value of H, which is the most easily observed parameter

61 H 2 8π G = ρ 3 kc R 2 2 In general, we can define the density parameter Ω ρ 8πGρ ρ crit 3H 2 Can now rewrite Friedmann s equation yet again using this we get Ω = 1+ 2 kc 2 H R 2

62 Omega in standard models Ω = 1+ 2 kc 2 H R 2 Thus, within context of the standard model: Ω<1 if k=-1; then universe is hyperbolic and will expand forever Ω=1 if k=0; then universe is flat and will (just manage to) expand forever Ω>1 if k=+1; then universe is spherical and will recollapse Physical interpretation: If there is more than a certain amount of matter in the universe (ρ>ρ critical ), the attractive nature of gravity will ensure that the Universe recollapses!

63 Value of critical density For present best-observed value of the Hubble constant, H 0 =72 km/s/mpc critical density is equal to ρ critical =10-26 kg/m 3 ; i.e. 6 H atoms/m 3 Compare to: ρ water = 1000 kg/m 3 ρ air =1.25 kg/m 3 (at sea level) ρ interstellar gas = kg/m 3

64 The deceleration parameter, q The deceleration parameter measures how quickly the universe is decelerating (or accelerating) In standard models, deceleration occurs because the gravity of matter slows the rate of expansion For those comfortable with calculus, actual definition of q is: R& q = 2 RH

65 Matter-only standard model In standard model where density is from rest mass energy of matter only, it turns out that the value of the deceleration parameter is given by q = Ω 2 This gives a consistency check for the standard, matter-dominated models we can attempt to measure Ω in two ways: Direct measurement of how much mass is in the Universe -- i.e. measure mass density and compare to critical value Use measurement of deceleration parameter Measurement of q is analogous to measurement of Hubble parameter, by observing change in expansion rate as a function of time: need to look at how H changes with redshift for distant galaxies

66 Direct observation of q Deceleration shows up as a deviation from Hubble s law A very subtle effect have to detect deviations from Hubble s law for objects with a large redshift

67 Newtonian interpretation is therefore: k=-1is positive energy universe (which is why it expands forever) k=+1 is negative energy universe (which is why it recollapses at finite time) k=0is zero energy universe (which is why it expands forever but slowly grinds to a halt at infinite time)

68 k=-1 k=0 k=+1

69 Expansion rates For flat (k=0, Ω=1), matter-dominated universe, it turns out there is a simple solution to how R varies with t : R(t) = R(t 0 ) This is known as the Einstein-de Sitter solution In solutions with Ω>1, expansion is slower (followed by re-collapse) In solutions with Ω<1, expansion is faster t t 0 2/3

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71 Modified Einstein s equation But Einstein s equations most generally also can include an extra constant term; i.e. the T term in 8πG c G = 4 has an additional term which just depends on space-time geometry times a constant factor, Λ This constant Λ (Greek letter Lambda ) is known as the cosmological constant ; Λ corresponds to a vacuum energy, i.e. an energy not associated with either matter or radiation Λ could be positive or negative Positive Λ would act as a repulsive force which tends to make Universe expand faster Negative Λ would act as an attractive force which tends to make Universe expand slower Energy terms in cosmology arising from positive Λ are now often referred to as dark energy T

72 Modified Friedmann Equation When Einstein equation is modified to include Λ, the Friedmann equation governing evolution of R(t) changes to become: 8 G R R & π Λ = H R = ρr + kc Dividing by (HR) 2, we can consider the relative contributions of the various terms evaluated at the present time, t 0 The term from matter at t 0 has subscript M ; Two additional Ω density parameter terms at t 0 are defined: Ω M ρ 0 ρ crit ρ 0 (3H 0 2 /8πG) Ω Λ Λ 3H 0 2 Altogether, at the present time, t 0, we have Ω k kc 2 R 0 2 H = Ω M + Ω Λ + Ω k

73 Generalized Friedmann Equation in terms of Ω s The generalized Friedmann equation governing evolution of R(t) is written in terms of the present Ω s (density parameter terms) as: R0 R R& = H0 R0 Ω M +Ω Λ +Ω R R0 The only terms in this equation that vary with time are the scale factor R and its rate of change R & Once the constants H 0, Ω M, Ω Λ, Ω k are measured empirically (using observations), then whole future of the Universe is determined by solving this equation! Solutions, however, are more complicated than when Λ=0 k

74 Special solution Static model (Einstein s) Solution with Λ c =4πGρ, k=λ c R 2 /c 2 No expansion: H=0, R=constant Closed (spherical) Of historical interest only since Hubble s discovery that Universe is expanding! R& / R = (8 πgρ/3) + ( Λ/3) ( kc / R ) R& = 0

75 Effects of Λ Deceleration parameter (observable) now depends on both matter content and Λ (will discuss more later) This changes the relation between evolution and geometry. Depending on value of Λ, closed (k=+1) Universe could expand forever flat (k=0) or hyperbolic (k=-1) Universe could recollapse

76 Consequences of positive Λ Because Λ term appears with positive power of R in Friedmann equation, effects of Λ increase with time if R 2 keeps increasing π G 2 ΛR 2 R & = H R = ρr + kc 3 3 Positive Λ can create accelerating expansion!

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79 de Sitter model: Solution with Ω k =0 (flat space!), Ω M =0 (no matter!), and Λ >0 Hubble parameter is constant Expansion is exponential THE DE SITTER UNIVERSE R = R 0 e Ht / t 0

80 Steady solution: Constant expansion rate Matter constantly created No Big Bang Ruled out by existing observations: Distant galaxies (seen as they were light travel time in the past) differ from modern galaxies Cosmic microwave background implies earlier state with uniform hot conditions (big bang) Observed deceleration parameter differs from what would be required for steady model

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