Cosmology with General Relativity Foundations Special relativity Shortest paths between two points (geodesics) can have various shapes Mass alters the shapes of geodesics. The Metric: How to specify the geometry of space Let dl specify an incremental distance within some coordinate system. The metric is the mathematical description of dl. 2-D Euclidian space dl 2 dr 2 r d . 2 3-D Euclidian space-time ds 2 c 2 dt 2 dl 2 c 2 dt 2 dr 2 r d . 2 Light travels along null distances in space-time, since for it, and it alone, dl c dt ds 0 . For any particle, on the other hand, dl < c dt ds > 0. 2-D Curved space (sphere) dl 2 d 2 RC Sin 2 d 2 , 2 where RC . The coordinate measures the distance from an observer at location O on the surface. It is the analog of the comoving coordinate – fixed forever by the location of each point on the surface. Define a new coordinate x such that x RC Sin R . C Eliminate coordinate increment d by noting that d dx RC Cos R R Cos R d , C C C so that 2 2 dx 2 Cos 2 R d 1 Sin 2 R d , C C and therefore dx 2 d 2 . 1 Sin 2 R C Now define the curvature parameter k: k 1 RC . 2 Clearly, for spherical surfaces k > 0. Then x2 1 Sin 2 R 1 2 1 kx2 , C RC and the incremental distance (the metric) can thus be written as dx 2 dl 2 x 2 d 2 . 1 kx 2 The proper distance dlP is the length of a line segment that subtends angle d at O. Evidently that distance corresponds to dx = 0, since in this case dlP dl xd RC Sin d , which is the length of a line segment perpendicular to . 4-D Curved space-time dr02 2 ds c dt R(t ) r0 d r0 sin d 2 2 2 2 2 1 kr0 which is the Robertson-Walker metric for spherical cosmological models. Newtonian models: k specifies the total energy of an arbitrary mass shell in an expanding space. GR models: k determines the geometry of space-time: k < 0: negatively curved space (analog of a saddle) k = 0: ordinary, flat Euclidian space k > 0: positively curved space (analog of a spherical surface) The Scale Factor in GR It all starts with the Einstein’s Field Equation: 1 8G R g R 2 T , where 2 c R = the Ricci Tensor, contains info about velocities, accelerations, g = the Metric Tensor, specifies the local curvature of space-time, T = the Stress-Energy Tensor, contains info about momentum, energy. Einstein noticed that a more general form of the Field Equation could be written as 1 8G R g R 2 T g . 2 c Here is an arbitrary constant that introduces additional curvature. In Newtonian models it represents a kind of “potential energy”. Solutions: Assume that space is homogeneous and isotropic. The field equation then reduces to two equations for the scale factor, one of which is 2 8G 2 1 R R R 2 kc2 . 3 3 or 2 R 8G 1 R 2 kc2 , R 3 3 which, with = 0, is identical to the Newtonian result. We recover equation 7) exactly! Set = 0 for now. The Size of the Observable Universe (the horizon distance, dh) A DISTANCE DICTIONARY Comoving coordinate (r0): A fixed coordinate assigned to each particle (galaxy) in the Universe. Neglecting proper motions, the expansion has no effect on comoving coordinate values, and the coordinate distances between particles are forever fixed. In other words, the coordinate system expands with space itself. Proper distance: The instantaneous (dt = 0) distance between any two points. The distance between any two events which occur simultaneously. As the Universe ages, observers have time to receive light from increasingly large distances. In other words, the size of the causally connected part of the Universe grows. The general expression for an increment of proper distance is given by the Robertson-Walker metric with dt = 0: ds 2 c 2 02 R 2 dl 2 . For a radial path, the angular parts of dl2 vanish, and so dr02 dl R 2 2 1 kr0 . We therefore evaluate the integral d r0 dr0 ' 20) d (ds) R 2 0 0 1 kr02 . Recall that r0 tracks the same particle as the Universe expands. NOTES: The proper distance itself is a function of time, through R(t). As the Universe expands, galaxies really do become farther apart. At any time, the distance between galaxies depends on both the “size” and “shape” of the Universe. The shape is given by the value of k. k = 0: d R(t ) r0 Now return to the task of finding the horizon distance. Let a photon be emitted at time t = 0 from a source at comoving coordinate r0 = r0,e. At some later time t the photon is received by an observer at comoving coordinate r0 = 0 (just for convenience). Since photons travel along null world lines, i.e. paths for which ds = 0, the Robertson- Walker metric which describes the path is dr02 0 c dt R 2 2 2 1 kr0 . The integral of this is dt' t r0 ,e dr0 ' 21) c 0 R (t ' ) 0 1 kr02 . Note that the limits on the RHS are reversed because the photon is moving towards smaller r0. The proper distance of the source follows from using 20) to eliminate the integral on the RHS of 21). The result is dt' t 22) d d h R (t ) c 0 R (t ' ) . Choose a value for k, and hence for R(t), and evaluate the RHS. k = 0: Use equation 11), which is 2/3 t 2/3 3 R t . 2 H Plug in for R(t ) in order to evaluate the integral in 22). Then plug in again for R(t). The result is 23) d h 3ct . Express the RHS in terms of the corresponding redshift using 17), still for k = 0: 3 / 2 t t H 1 z . 2 3 Plugging into 23) gives the result 2c d h ( z) H 0 1 z . 3/ 2 For us, z = 0. Taking H0 = 71 gives 2c 2 3 105 kms1 d h (0) 8450Mpc . H0 71kms1 Mpc1 FOOTNOTE: For k = 0 we just found that dh t, whereas R(t) t2/3. Thus, the horizon distance expands faster than the Universe, and more of the latter becomes visible. Observational Cosmology THE COSMOLOGICAL REDSHIFT What happens to light emitted “long ago in a galaxy far, far away…”? Let a wave packet of duration t1 be emitted from a source at r0 at time t1. An observer at r0 = 0 at the present epoch, time t0, receives the packet over a duration t0. Remembering that light travels along null world paths (ds = 0), the Robertson-Walker metric is: dr02 0 c dt R 2 2 2 1 kr0 . The path taken by the leading edge of the wave packet is described by t0 dt' r0 dr0 ' c t1 R(t ' ) 0 1 kr02 As before, choosing the coordinate origin at the observer introduces an extra minus sign, reversing the order of integration on the RHS. The trailing edge of the wave packet, which follows the same path, moves according to t0 t0 dt' r0 dr0 ' c t1 t1 R(t ' ) 0 1 kr02 . Because the duration of the packet at each end of its trip is exceedingly small, the LHS can be well approximated by t0 dt' ct0 ct1 r0 dr0 ' c t1 R(t ' ) R(t0 ) R(t1 ) 0 1 kr02 . Equating the two expressions to eliminate the RHS, and remembering that R(t0) = 1, we find that ct0 ct1 1 R(t1 ) , or just t1 t 0 R (t1 ) . RESULT: The expansion of space during the packet’s trip has increased its duration ( because R(t1) < 1). The same argument applies to all time periods, including those defined by photon frequencies. In particular, photon frequencies are lowered, and wavelengths lengthened. THE LUMINOSITY DISTANCE How can we find values for 0, H0 ?? L dL 2 4 f , where L is the luminosity of some remote source whose measured flux is f. Prof. Einstein tells us that the observed flux is written L 1 f 4 r02 1 z 2 . The factors of (1+z) are courtesy of the Cosmological Redshift: a) As we have seen, time dilation expands the intervals between successive photon oscillations, i.e. lowers their frequencies. That lowers the energy per photon (E = h). Furthermore, b) the same time dilation increases the time interval between the arrival of successive photons. This further reduces the received flux. From the above we see that 4r02 1 z , 1 L 1 dL 2 2 4 f 4 or in other words 24) d L r0 1 z . Now we need to find out how r0 depends on the choice of world model. Start with equation 21): dt' t r0 ,e dr0 ' c 21) 0 R (t ' ) 0 1 kr02 . We’d rather integrate the LHS directly over R, so use the fact that dt dR , R which allows us to write R0 dR r0 ,e dr0 ' c Re RR 0 1 kr02 . Equations 6) and 7), and the one for the critical density, can be used to show that 2 R H 0 0 0 1 . 2 R We also know that R0 = 1, and that Re 1 z , 1 so that, for any GR cosmology we have 1 dR r0 ,e dr0 ' c 1 25) 1 z H 0 0 R 0 1R 2 0 1 kr02 . k = 0: This is equivalent to 0 = 1. Therefore 25) looks simpler! c 1 dR r0 ,e H0 1 1 z R dr0' 0 . This is easy to integrate and solve for r0,e r0. The result is 26) r0 2c H0 1 1 z 1 / 2 . k 0: The starting point is the same, but the derivation more complex. C&O present the process on pages 1267,68. The bottom line is 27) r0 2c 1 H 0 1 z 0 2 0 z 2 0 1 0 z 1 . which turns out to hold for both open and closed universes. CHECK: Do we recover equation 26) if 0 = 1? Yes! The luminosity distance of a source at redshift z can now be determined by using 26) or 27) to replace r0 in equation 24). Collect the pieces for the general case: 24) d L r0 1 z , 27) r0 2c 1 H 0 1 z 0 2 . 0 z 2 0 1 0 z 1 Doing the substitution gives 28) dL H 0 0 2 0 z 2 0 1 0 z 1 . 2c 1 Check to see if this makes sense in the familiar limits. 0 = 1: Then 28) becomes dL 2c H0 z 1 1 z 1 2c H0 z 1 1 z . z << 1: Use the binomial series expansion for the square root, keeping just first-order terms in z: dL 2c z 1 1 1 z 2c 1 z cz , 2 H0 H0 2 H0 which looks familiar! THE REDSHIFT-MAGNITUDE RELATION If we can find nice standard candles at large distances, equation 28), together with the distance modulus relation, will predict the run of apparent magnitude with redshift for various world models. 29) m M 5 log d L 5 . So one has m M 5 5 log d L , with 28) dL 2 0 z 2 0 1 0 z 1 . 2c 1 H 0 0 The entire derivation can (and has!) been repeated including a cosmological constant, . Supernova Ia hunters at Berkeley and CTIO are trying to push beyond z = 1 in order to measure both 0 and . Recent results, in conjunction with studies of the CMB fluctuations, are pointing towards TOTAL 1 , with 0.7 , and B 0.3 The Cosmological Constant (An idea whose time has come…and gone…and come…and ?…) The most general solution of the Field Equation is 2 8G 2 1 30) R R R 2 kc2 . 3 3 Newtonian interpretation: The third term on the LHS represents a new “potential energy” added to the original equation for energy conservation in the expanding shell. The original equation would look like 1 2 GM r m 1 1 mv mc 2 r 2 kmc2 r02 . 2 r 6 2 The new potential energy term corresponds to a force given by ~ F U r , ˆ 1 2 2 mc r r , ˆ r 6 1 mc 2 rr ˆ 3 Result: If > 0, then an outward force exists which counteracts gravity, as Einstein originally intended. Furthermore, the force grows as the Universe expands! A candidate for - vacuum energy Classical Physics: Vacuum energy is irrelevant! E K U , and the force on a particle can be written as F U U const. , ~ so the motions of particles are not affected by choosing another zero point for potential energy. E(minimum) need not be zero! Modern Physics: Vacuum energy is required! Particle Physics View: “Vacuum” is just the lowest energy state of any physical theory. If the lowest energy state looks the same for all observers, then it must be invariant under the Lorentz transformation (“lowest energy” must represent the same thing in all inertial reference frames). Consequence: the “vacuum” is a fluid whose equation of state (pressure-density relation) must be PVAC VAC c 2 . Quantum Mechanics View: Uncertainty Principle allows spontaneous appearance and disappearance of virtual particles. Particle Physics views these as just the “vacuum “ states of real particles. These particles, then, become the source of mass-energy. Uncertainty Principle also forbids any particle from being isolated in a state with zero total energy. The vacuum must correspond to a non-zero energy state. Incorporating in Cosmological Theory Equation 30) can be rewritten as 2 8G 2 8G 2 R R R kc2 , 3 3 where is the mass density associated with . In the same way as the customary density parameter is defined, one can define 8G , 3H 0 which means that 3H 0 . 2 Following the spirit of the Newtonian formalism, the generalized form of equation 10) is H 0 0 1 kc2 . 2 So the condition for a flat universe, k = 0, is that 0 1 . As we have seen, there are powerful reasons for wanting k = 0 (i.e. for believing in Inflation). The Age of the Universe Making > 0 increases the age of the Universe for k = 0 cosmologies. The current age is given by 2 1 1 1 1 / 2 t0 1 / 2 ln 1 1 / 2 . 3 H 0 With H0 = 71, the age is 1 1 1 / 2 1 1 / 2 Gyr. t0 9.2 1 / 2 ln Studies of CMB fluctuations and remote supernova currently point toward 0.7 , which means that the age is 1 0.71 / 2 t0 9.2 0.7 1 / 2 0.31/ 2 Gyr, ln 9.2 1.45 13.3 Gyr. This greatly relieves, and perhaps solves, the “age problem”.