Fate of the Universe Relativity and Astrophysics Lecture 20 Terry Herter Outline Skateboard paradox Fate of the Universe Critical Density Dark Matter Three Omegas (really two) Dark Energy Problem L-13, page 117 (due Today/Monday) Will hand back on Monday if you hand it in today A2290-20 Fate of the Universe 2 A2290-20 1
Paradox of Skateboard & Grid: L-12 We have a 1-m skateboard (in rest frame of rider) and a 1-m hole (in rest frame of lab). Proper lengths are the same A person on a skateboard moves very fast (causing ativistic contraction). The skateboard passes over the hole. Will it fall in? Paradox Lab person sees skateboard contracted => should fall in Ride sees hole contracted => should pass over hole skateboard Lab Frame Skateboard Frame skateboard Which person is correct? Idealize problem to a one-meter rod sliding lengthwise over a flat table. Let the Lorentz contraction factor be 10 so that the rod is 10 cm in the lab frame, thus L = L o /, where L = length of rod in lab frame, L o is rest length of rod, and is the Lorentz factor. Assume that the skateboard does not tilt as it falls in the lab frame, that is, once it is completely over the hole it starts to drop. A2290-20 Fate of the Universe 3 Skateboard & Grid (cont d) Define t = 0 and t = 0 when back of skateboard is even with back of hole. In the lab frame the motion of the front and back are given by: x v t L & x v t We now transform to the skateboard frame. For distances xf xf vt vt L vt L x x x v t v t v t 0 Then times t F t B B skateboard hole B F L L o / Thus the front starts to fall before the back. The paradox is removed since the skateboard drops over the edge and goes through the hole. B Lab frame x B F L o As expected 2 2 t v xf t vt vl tf t1 v vlo 2 2 t v x t v t t t v B B 1 A2290-20 Fate of the Universe 4 A2290-20 2
Skateboard paradox (graphically) From Sartori (Understanding Relativity). Here strings, which keep the board from tilting, are cut simultaneously in lab frame (left) but not in skateboard frame (right). Note: ativity does not allow the skateboard to be infinitely rigid. A2290-20 Fate of the Universe 5 Skateboard paradox (graphically) The previous graphs picture the skateboard from the viewpoint of the lab where it is always horizontal. If we start in the skateboard frame this will not be the case. At left is an (incorrect) attempt to picture what is happening from the skateboard viewpoint When the first thread is cut the board begins to droop The board at second thread doesn t know it has been cut The simultaneous cutting in the lab frame means spacelike so the points can t communicate. From Sartori (Understanding Relativity). Here strings, which keep the board from tilting, are cut simultaneously in lab frame (left) but not in skateboard frame (right). Note: ativity does not allow the skateboard to be infinitely rigid. A2290-20 Fate of the Universe 6 A2290-20 3
Fate of the Universe The universe is currently expanding. What will ultimately happen to it? A2290-20 Fate of the Universe 7 What will happen to the universe? Consider a planet; shoot a spacecraft from the surface (single shot) High Velocity Distance Spacecraft Escapes A2290-20 Fate of the Universe 8 A2290-20 4
What happens (cont d) Low Velocity Distance Spacecraft falls back At the escape velocity the spacecraft just escapes. A2290-20 Fate of the Universe 9 The Evolution of the Universe Looking at the evolution of the universe similar to escaping from a planet. Distance Present un marginally Big Bang A2290-20 Fate of the Universe 10 A2290-20 5
The Critical Density Will the universe keep expanding forever or stop expanding and collapse? If the mass density is: large eventual collapse () small expansion forever (un) The critical density is the dividing line between the and un cases. For H o = 75 km/sec/mpc crit ~ 10-29 g/cm 3, or about 6 H-atoms/m 3 This corresponds to 1 MW galaxy/mpc 3 Define M as the ratio of the actual density to the critical density M actual density critical density M M 1 un 1 A2290-20 Fate of the Universe 11 M and H o M determines the fate of the universe. Expanding forever vs. eventual collapse H o determines the age of the universe. M = 0 age = 1/H o : Empty Univ. M = 1 age = (2/3) (1/H o ) : Critical Univ. We ve already discussed H o. What is M? numerically A2290-20 Fate of the Universe 12 A2290-20 6
The Age of the Universe (again) Distance Present Empty un marginally (critical) 1/H o 2/3(1/H o ) A2290-20 Fate of the Universe 13 Measuring M Two ways to determine M are: Galaxy Counts - Directly measure amount of mass in the universe. Deceleration of the Universe - Measure how much faster galaxies were moving in the past. A2290-20 Fate of the Universe 14 A2290-20 7
Galaxy Counts & Dark Matter Counting galaxies and adding up their mass yields M ~ 0.045. But... This misses most of the mass. Masses can be found from (gravity) the motions of stars in galaxies, the motions of galaxies in clusters. Gravity implies much more mass than we see in stars/galaxies dark matter. We can t see > 90% of the mass in the universe, except by gravity! MACHOs (Massive Compact Halo Objects)? WIMPs (Weakly Interacting Massive Particles)? Correcting for the missing mass due to dark matter gives M ~ 0.3. A2290-20 Fate of the Universe 15 Deceleration of the Universe We need a standard candle to measure distances and recessional velocities independently. It is difficult to find a standard candle. Galaxies are different in the past. Type Ia Supernovae now appear to work very well. (Caused by accretion onto a white dwarf.) A funny thing happened... Type Ia SN results indicate that the universe is accelerating! Acceleration => the expansion rate is increasing Contrary to our initial bias that it should be decelerating Deceleration => the expansion rate is slowing (due to gravity) A2290-20 Fate of the Universe 16 A2290-20 8
The Cosmological Constant () Introduced by Einstein into GR Thought universe was static Can be repulsive or attractive Called it his biggest mistake But now we may need it! As a repulsive force A2290-20 Fate of the Universe 17 The Three Omegas Besides H o we need three more parameters to determine the evolution of the universe M,, and k M is defined as before defines Cosmological constant part k defines geometry of the universe However k can be written in terms of the other two so we really only need 2 Omegas k = 1 M A2290-20 Fate of the Universe 18 A2290-20 9
The Two Omegas So now we only need to know M and. With zero (no) cosmological constant Have M and k with k = 1 M With a non-zero cosmological constant Have M,, and k with k = 1 M Determine Dynamics and Geometry M and set acceleration/deceleration k sets geometry ( k = 1 M ) A2290-20 Fate of the Universe 19 0 Accelerating empty Distance Present un marginally Look back in time Expansion velocity in the past depends upon M and. Galaxies would be moving faster in the past for a non-empty universe. A2290-20 Fate of the Universe 20 A2290-20 10
Type Ia SN results M = 0, = 1 M = 0.3, = 0.7 M = 0, = 0 M ~ 0.5, = 0 Figure from: Riess and Turner, SciAm, Feb. 2004 A2290-20 Fate of the Universe 21 Dark Energy The acceleration of the Universe driven by an enormous amount of energy. What causes the acceleration? Or what is the cosmological constant? Unknown! This unknown energy is call dark energy. Attempts to estimate it from particle physics are off by at least a factor of 10 55! Even more of the Universe is unknown to us than we thought! Illustration from WMAP website A2290-20 Fate of the Universe 22 A2290-20 11
The Fate of the Universe The best estimates now yield H o = 71 km/sec/mpc M = 0.27, = 0.73 k = 0 Age of universe is 13.67 Gyr This implies The Universe will go on expanding forever Because > 0, it will expand ever faster! A2290-20 Fate of the Universe 23 A2290-20 12