Lecture 34 The Shape of Space General Relativity Curvature of Space Critical Density Dark Energy Apr 17, 2006 Astro 100 Lecture 34 1 General relativity So far, just needed simple Newton's Gravity. Because the gravity of the universe is so large, and relative speeds of things approach the speed of light, one needs to use Einstein s General Relativity ( GR ) for the full story: Special Relativity. Uniform motion approaching the speed of light: time and space measurements are distorted General Relativity. Accelerated motion approaching the speed of light: Euclid's geometry no longer works ("space is curved")! Gravity is a special force: recall that gravity is proportional to mass, so that all masses are accelerated the same. Einstein postulated: Apr 17, 2006 Astro 100 Lecture 34 2
Mass Causes Curvature of Space Equivalence Principle: you cannot distinguish between being at rest in a gravitational field and being accelerated in a gravity-free environment. In Einstein's view: Mass causes curvature of space-time which looks like acceleration. Best way of visualizing this: "toy" universe with only 2 space dimensions instead of 3. In absence of mass, space is a flat sheet, and objects travel in straight line from one edge to another. With mass present, sheet is depressed around mass. Objects travel along trajectory which is shortest distance between two points (called a "geodesic"). Near the mass, the trajectory is bent towards mass. Apr 17, 2006 Astro 100 Lecture 34 3 Curvature of Universe For Big-Bang with general relativity, the big addition is that the slight curving of space by gravity from the matter is important. The curvature could (in principle) be measured by summing the angles in a very large triangle defined by light beams. There are only three types of geometry, depending on how much stuff is in the Universe Open big-bang: The universe is infinite (has infinite mass and volume). The angles in a triangle sum to < 180 deg ("negatively curved"). 2D analog is saddle surface. Flat big-bang:. The universe is infinite. The geometry is Euclidean (angles in a triangle sum to 180 deg). 2D: plane. Closed big-bang: The universe is finite (but unbounded!), The angles in a triangle sum to > 180 deg ("positively curved"). 2D: sphere Apr 17, 2006 Astro 100 Lecture 34 4
The Beginning In Big-Bang Model, what is most distant (earliest) thing we can see? Predicted by Alpher, Hermann, and Bethe in 1940's: As you look out farther and farther, see universe when things were more dense. Eventually, the universe was so dense that it was opaque (like stellar photosphere). Found this happens for Age of universe 1 million years, Temperature of universe 3000 K. Light emitted at this time has spectrum of a blackbody at T = 3000 K. Apr 17, 2006 Astro 100 Lecture 34 5 Cosmic Microwave Background Since universe has just become transparent, light will travel in all directions and will fill universe. Should be observable today as uniform background, but redshifted by a factor of 1000. Maximum of blackbody curve shifted from λ = 600 nm (T=3000 K, red light) to λ = 0.6 mm (T=3 ºK, microwave light) 3 K Cosmic Microwave Background ( CMB ) was observed just as predicted by Dicke and Peebles in 1960 COBE Satellite: spectrum an accurate 3 K blackbody; very uniform This is the most convincing verification of the Big Bang Model and the existence of a beginning Apr 17, 2006 Astro 100 Lecture 34 6
The Universe is Flat Observing the curvature of the Universe using the CMB The CMB is not perfectly uniform because of lumps in the density of the Universe at the beginning The largest possible physical size of the lumps can be calculated from the age of the Universe when it became visible (it is the distance sound can travel in that time) The observed angular size of these lumps depends on how the Universe is curved Result: the Universe is flat Good article about relativity and the Big Bang: Misconceptions about the Big Bang Scientific American March 2005 Apr 17, 2006 Astro 100 Lecture 34 7 Critical Density and Geometry GR relates the geometry to the amount of mass - energy in the Universe (energy is converted to mass by E = Mc 2 ) Big-Bang Model. At any time, there is a "critical density" of mass energy which depends on the current rate of expansion H at that time: Critical Density = 3 H 2 / (8 π G) For us, now H = 65 km/sec/mpc Critical Density ~ 10-29 gm/cm 3, about 1 H atom/cubic meter < critical: open = critical: flat we must have critical density in mass - energy > critical: closed Apr 17, 2006 Astro 100 Lecture 34 8
Dark Energy So how much stuff is out there? From luminous matter (stars in galaxies) mass density = 0.04 critical But, recall "missing mass" from Milky Way rotation curve and from anomalously high motions of galaxies in clusters. Estimates of amount of "Dark Matter" mass density = 0.29 critical Current evidence: there is only 1/3 critical density in mass With just mass, Universe should be open, slowly decelerating, contradicting evidence 1) To be flat, the remaining must be energy energy density = 0.67 critical Like dark matter, we don't know what this is, call it "Dark Energy" 2) Next time: this energy is also thought to be responsible for the expansion accelerating, rather than decelerating Apr 17, 2006 Astro 100 Lecture 34 9 Curvature of Space Figure 8.13, p254, Arny Apr 17, 2006 Astro 100 Lecture 34 10
Curved 2-D Universes Figure 11.13, p350, Arny Closed Positively curved Flat Open Negatively curved Apr 17, 2006 Astro 100 Lecture 34 11 Spectrum of the Cosmic background Figure 11.7, p344, Arny Apr 17, 2006 Astro 100 Lecture 34 12
Cosmic Microwave background Raw Removing motion of Sun Removing Milky Way Radiation Apr 17, 2006 Astro 100 Lecture 34 13 Lumps in the Beginning Apr 17, 2006 Astro 100 Lecture 34 14