Introduction: Big-Bang Cosmology

Basic Assumptions Principle of Relativity: The laws of nature are the same everywhere and at all times The Cosmological principle: The universe is homogeneous and isotropic Space time is simply connected, can be filled with comoving observers (CO). Each CO performs local measurements of distance and time in it’s own frame of reference, locally flat. No global inertial frame. Cosmic time: synchronized clocks of COs in space at every given time.

North Galactic Hemisphere Lick Survey 1M galaxies isotropy→homogeneity North Galactic Hemisphere

Microwave Anisotropy Probe February 2003, 2004 WMAP Science breakthrough of the year δT/T~10-5 isotropy→homogeneity

Homogeneous Universe: Flat (Euclidean), therefore open, infinite Curved and closed, finite

Three-dimensional Two dimensional One-dimensional Open Closed ??? 123

Olbers Paradox L R f  L/R2 R N  nR2dR dR The simplest assumptions: Homogeneity and isotropy Euclidean geometry Infinite space A static universe

The Universe is evolving in time

Baby galaxies in early universe z=2

the universe is expanding

Discovery of the Expansion 1929 Discovery of the Expansion

Doppler Shift receding source approaching source

Red-shift distance wavelength

Hubble Expansion: V =H R Distance R Velocity V Acceleration

Hubble Expansion V = H R velocity Hubble constant distance

V = H R A special center?

V = H R A special center? Other Origin

Prediction from homogeneity: The Hubble Law V=HR time t2 ax t1 x x distance

distance galaxy 2 galaxy 1 here now time t0 13.7 Gyr The Big Bang

היכן היה המפץ הגדול? בנקודה אחת? בהרבה נקודות? היכן היה המפץ הגדול? בנקודה אחת? בהרבה נקודות? ? ? היכן המפץ

היכן היה המפץ הגדול? בנקודה אחת? בהרבה נקודות? היכן היה המפץ הגדול? בנקודה אחת? בהרבה נקודות? כל הנקודות מתלכדות לנקודה אחת מפץ באינסוף נקודות

The Big Bang model The Steady State model The Steady State model

Cosmic Microwave Background Radiation 1965

COBE 1992 Nobel Prize 2006 to Smoot and Mather Plank black-body spectrum I=energy flux per unit area, solid angle, and frequency interval Spectrum of the CMB

Homogeneity and Isotropy: Robertson-Walker Metric

Metric Distance B A Metric distance Hubble expansion in curved space: local Hubble law t=t1 universal expansion factor comoving distance

Metric Coordinate system: x1, x2 (2d example) In a small neighborhood (locally flat) Line element: The metric: Specifies the geometry uniquely. Exact form depends on the choice of coordinates Orthogonal coordinates: gij=0 for i≠j

Coordinates: cartezian spherical Example: E3 Coordinates: cartezian spherical interval angular distance cartezian spherical Example: a 2D sphere embedded in E3 : r=const. Area:

The Metric of a Homogeneous and Isotropic Universe (Robertson-Walker) t=const. In comoving spherical coordinates Isotropy: Three solutions: Space-time interval

k=0 flat space (E3) infinite volume

k=+1 a closed space For visualization: a 3D sphere embedded in E4: (w, x, y, z) a 3D sphere the embedding is defined by the transformation: consistent with w x y To visualize plot subspace 2D sphere in w,x,y,z=0 a u φ u=const. is a sphere of comoving radius u A grows for 0 < u < π/2 and decreases for π/2 < u < π

k=-1 an open space

Homogeneity and Isotropy  Robertson-Walker Metric expansion factor comoving radius angular area

de Broglie wavelength (particles or photons): Redshift Radial ray local Hubble local flat nearby observers along a light path, separated by δr: For Black-Body radiation, Planck’s spectrum: Also for free massive particles: de Broglie wavelength (particles or photons): like photons: useful: conformal time

Horizon limit exists Example: EdS (k=0, Λ=0) Causality problem: Our Horizon is not causally connected: what is the origin of the isotropy?

Friedman’s Equation and its solutions

Newtonian Gravity shell 3 types of solutions, depending on the sign of E H=0 at maximum expansion, possible only if E<0 independent of r because lhs is

The Friedman Equation Einstein’s equations For the isotropic RW metric Newton’s gravity: space fixed, external force determining motions Einstein’s equations left side of E’s eq. is the most general function of g and its 1st and 2nd time derivatives that reduces to Newton’s equation Gravity is an intrinsic property of space-time. geometry <-> energy density. Particles move on geodesics (local straight lines) determined by the local curvature. For the isotropic RW metric Stress-energy tensor Einstein’s tensor Add λ energy conservation eq. of motion conservation of number of photons mass conservation A differential equation for a(t)

Solutions of Friedman eq. (matter era) radiation era acceleration expansion forever conformal time collapse critical density big bang here & now

Light travel in a closed universe A photon is emitted at the origin (ue=0) right after the big-bang (te=0) photon: conformal time 0 π/2 π north pole u south pole 2π 3π/2 π π/2 η big bang big crunch t

H and t General: Carrol, Press, Turner 1992, Ann Rev A&A 30, 499

Solutions with a Cosmological Constant k=0 t a k=-1 same asymptotic behavior as k=0 3rd-order polynomial – one real root exists t a

Solutions with a Cosmological Constant (cont.) a solution for every a Lematre k=+1 (closed) a critical value: the rhs at as to be >0 a double root at static expanding to inflation Eddington-Lematre t a Einstein’s static universe (unstable) static ac big-bang expanding to static

Solutions with a Cosmological Constant (cont.) k=+1 (closed) a critical value: cos. const. wins t a a2 for a small and large a1 mass attraction wins for a1<a<a2 – no solution t a only attraction

kinetic potential curvature vacuum Friedman Equation kinetic potential curvature vacuum Carrol, Press, Turner 1992, Ann Rev A&A 30, 499 two free parameters closed/open decelerate/accelerate Flat: k=0 a 1

Dark Matter and Dark Energy vacuum – repulsion? 1 decelerate accelerate m /2 - =0 open closed flat m + =1  bound unbound =0 1 2 m mass - attraction

Cosmological Constant: Newtonian Analog vacuum energy density force per mass on shell vs. potential vs. in Einstein’s eqs.

Acceleration, pressure, energy density FRW:  p dV Energy change by work (2) Construct a static model: de Sitter: Quintessence for inflation need to exceed

Future SN Cosmology Project

Dark Energy acceleration WMAP_1  WMAP_3