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General Relativity Physics Honours 2008 A/Prof. Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 10 Slide 2 http://www.physics.usyd.edu.au/~gfl/Lecture Lecture Notes 10 Cosmology Chapter 18 As we have seen, it is difficult to determine general analytic solutions to the field equations. However, applying symmetries reduces the complexity of the problem. One place this has proved successful is in cosmology where the symmetries are Isotropy: The spacetime geometry is spherically symmetric about any point. Homogeneity: The spacetime geometry is the same at all points (although this can change with time). These symmetries imply that the energy distributions must also be isotropic and homogeneous (clearly, this breaks down on small scales in the universe, but appear to apply on average on large enough scales. Slide 3 http://www.physics.usyd.edu.au/~gfl/Lecture Lecture Notes 10 Robertson-Walker Metric These symmetries imply that the metric can only take on a specific form, the Robertson-Walker metric; Here, k describes the geometry of space and can take on values of +1 (closed: spherical), 0 (flat) and -1 (open: hyperbolic). The general function a must be determined, but it can only depend upon time. Notice that unlike the previous metrics we have seen, this one is dynamic and changes with time. Hence, in this metric, there is no Killing vector associated with time, but there are with spatial symmetries. Slide 4 http://www.physics.usyd.edu.au/~gfl/Lecture Lecture Notes 10 Cosmological Equations Suppose the universe is filled with a fluid with a particular density and pressure. For an observer at rest with respect to the fluid, the stress-energy tensor is simply We now need to take the metric, calculate the Ricci tensor and scalar to form the Einstein tensor. This is then projected into the orthonormal frame of our observer so that Slide 5 http://www.physics.usyd.edu.au/~gfl/Lecture Lecture Notes 10 Cosmological Equations How to the pressure and density change with time? We can use the conservation equation If we assume our fluid is at rest in our coordinates, then u =(1,0,0,0) and So we can write Slide 6 http://www.physics.usyd.edu.au/~gfl/Lecture Lecture Notes 10 Cosmological Equations The only non-zero Christoffel symbols that matter are Which gives; This can be written as Slide 7 http://www.physics.usyd.edu.au/~gfl/Lecture Lecture Notes 10 Flat Universe Lets start with a spatially flat (k=0) universe. The metric is As a(t) changes, the coordinate distance between any two points remains fixed, but the physical distance changes as In our Universe, a(t) appears to be increasing with time, and so the distance between comoving observers is increasing i.e. the Universe is expanding. This also means that the volume between comoving observers is increasing and the Universe is becoming more dilute. Slide 8 http://www.physics.usyd.edu.au/~gfl/Lecture Lecture Notes 10 Redshift What about photons exchanged between two comoving observers? Writing the metric in polar form and considering photons traveling on radial paths, then Hence, the coordinate distance traveled by the photon is Remembering that the frequency is = 2/t then Where z is the redshift of photon. Slide 9 http://www.physics.usyd.edu.au/~gfl/Lecture Lecture Notes 10 Hubble Law Suppose we have a nearby object at coordinate separation R, so the physical separation at the present time is The time taken by the photon to travel this distance is So we know t e = t o d. Assuming d is small, then This is the Hubble Law which says we should see a linear correlation between distance and redshift. Slide 10 http://www.physics.usyd.edu.au/~gfl/Lecture Lecture Notes 10 Hubble Law Hubbles constant has been measured to be 72km/s/Mpc. Its natural units are inverse time, and so expressing this value in years gives us a Hubble time; We can turn the redshift into a velocity via z~v/c and plot distance verses velocity. In the local universe, this is a linear trend (but not in the larger universe). The slope of the line is Hubbles constant. This is the natural universal time scale. Slide 11 http://www.physics.usyd.edu.au/~gfl/Lecture Lecture Notes 10 How do fluids evolve? We can calculate how the density and pressure of the fluids in the universe evolve with time using Matter: can be represented as a pressureless fluid and so its density simply scales as As a increases, the universe expands and the density of matter drops in proportion to the volume of the expansion. This is what we expect. Slide 12 http://www.physics.usyd.edu.au/~gfl/Lecture Lecture Notes 10 How do fluids evolve? Photons: For a blackbody gas of photons, the energy density and pressure are related via Plugging this into our equation and integrating gives The photon gas thins out and loses additional energy! Where does it go? Additionally, the energy density is And the blackbody spectrum cools as the universe expands. Slide 13 http://www.physics.usyd.edu.au/~gfl/Lecture Lecture Notes 10 How do fluids evolve? In general, we can define an equation of state for a fluid; For matter, w=0, and photons, w=1/3. For a general fluid, this implies that The quantum vacuum has w=-1 and its energy density is constant with time. While the energy density in the vacuum is unknown, it appears that the dominant energy in our universe is in this form (known as the cosmological constant). General fluids with w Slide 14 http://www.physics.usyd.edu.au/~gfl/Lecture Lecture Notes 10 Friedmann Equation Lets consider a spatially flat universe (k=0), the we can write the Friedmann equation as All we need to do is put how the fluids evolve with time into the equation and solve for a. It is convenient to scale to present day quantities as we know that We can define a useful critical density to be Slide 15 http://www.physics.usyd.edu.au/~gfl/Lecture Lecture Notes 10 Friedmann Equation We can normalize our energy densities by this critical value so And for our spatially flat universe, the sum of the normalized energy densities must equal unity. In the flat model, we have the freedom to define a(t 0 )=1 and and Slide 16 http://www.physics.usyd.edu.au/~gfl/Lecture Lecture Notes 10 Friedmann Equation In this example, at the present time energy is equally distributed between vacuum, matter and radiation. But the mix was different at other epochs, with one essentially dominant. In this picture (as in most models) the universe has a beginning at a finite time in the past when a=0; this is the Big Bang of cosmological models. Slide 17 http://www.physics.usyd.edu.au/~gfl/Lecture Lecture Notes 10 Friedmann Equation Matter dominated: m =1 Photon dominated: r =1 Vacuum dominated: v =1 We can of course generalize these expressions for arbitrary equations of state. Generally, one component will dominate. Slide 18 http://www.physics.usyd.edu.au/~gfl/Lecture Lecture Notes 10 Spatial Curvature When k is not 0 then the universe can possess spatial curvature. If k=+1, then space is closed and has spherical geometry (and it has finite volume which changes with time). With k=-1 the space is open, the geometry is hyperbolic and possesses infinite volume. Read through section 18.6 to understand this spatially curvature. What does it mean for geometry? Slide 19 http://www.physics.usyd.edu.au/~gfl/Lecture Lecture Notes 10 Friedmann Equation In its generalized form, this is Considering the present epoch, we can write So the spatial curvature is determined by the total energy density in the universe. Exceeding the critical value results in a closed universe, while densities less than critical are open. Generally, the cosmological equations do not possess analytic solutions for arbitrary equations of state and mixes of fluids. These must be tackled numerically (especially for interacting models). Slide 20 http://www.physics.usyd.edu.au/~gfl/Lecture Lecture Notes 10 Matter Dominated Universes If we consider universes with only matter, it is possible to derive a parametric form for the expansion. For positively curved universes So the Universe expands to a maximum size (and volume) before recollapsing back to a point (a big crunch). We can derive a similar expression for negatively curved universes, but containing hyperbolic functions. These universes expand for ever, and so the critical density differentiate between eventually collapsing and expanding universes (for those containing matter). Slide 21 http://www.physics.usyd.edu.au/~gfl/Lecture Lecture Notes 10 Generalized Cosmology We can derive a generalized form of the cosmological equation where and Again, this can be generalized further to arbitrary equations of state for more complex fluids. Slide 22 http://www.physics.usyd.edu.au/~gfl/Lecture Lecture Notes 10 Our Universe Our universe appears to be flat, with m ~0.3 and v ~0.7 and negligible radiation. The curve on the left shows the effective potential for our Universe, showing that we are entering a phase of eternal acceleration. On the right is the effective potential for m =0.2 and v =0.02. This universe has two possible evolutions, expanding and then collapsing, or collapsing and then bouncing. With general fluids, we can get similar behaviour. Slide 23 http://www.physics.usyd.edu.au/~gfl/Lecture Lecture Notes 10 What happens? If we assume we only have vacuum energy and matter (which is a good model for our universe as the photon energy density currently very small). Even with these two free parameters at the present time, there are a number of differing evolutions of the universe. Observational evidence (Chapter 19) suggests we are in a flat universe, sitting on the Tot = 1 line. Slide 24 http://www.physics.usyd.edu.au/~gfl/Lecture Lecture Notes 10 The End