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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.
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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
WMAP Microwave Anisotropy Probe February 2003, 2004 Science breakthrough of the year δT/T~10-5 isotropy→homogeneity
Homogeneous Universe: Curved and closed, finite Flat (Euclidean), therefore open, infinite
123 Three-dimensional Two dimensional One-dimensional Open Closed ???
L R f L/R2 R N nR2dR dR Olbers Paradox The simplest assumptions: Homogeneity and isotropy Euclidean geometry Infinite space A static universe
the universe isexpanding Expanding
1929 Discovery of the Expansion
Doppler Shift receding source approaching source
Red-shift distance wavelength
Acceleration V =HR Hubble Expansion: Distance R V Velocity
Hubble Expansion distance velocity V=HR Hubble constant
V = H R A special center?
Other Origin V = H R A special center?
ax ax t2 Prediction from homogeneity: The Hubble Law V=HR time t1 x x distance
galaxy 2 galaxy 1 The Big Bang t0 13.7 Gyr distance here now time
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
TheSteady State model The Steady State model The Big Bang model
Spectrum of the CMB COBE 1992 Nobel Prize 2006 to Smoot and Mather Plank black-body spectrum I=energy flux per unit area, solid angle, and frequency interval
Hubble expansion in curved space: local Hubble law comoving distance universal expansion factor B Metric Distance A Metric distance t=t1
In a small neighborhood (locally flat) Line element: The metric: Metric Coordinate system: x1, x2 (2d example) Specifies the geometry uniquely. Exact form depends on the choice of coordinates Orthogonal coordinates: gij=0 for i≠j
Coordinates: cartezian spherical interval angular distance cartezian spherical Example: a 2D sphere embedded in E3 :r=const. Area: Example: E3
In comoving spherical coordinates Isotropy: Three solutions: Space-time interval The Metric of a Homogeneous and Isotropic Universe (Robertson-Walker) t=const.
k=0 flat space (E3) infinite volume
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 To visualize plot subspace 2D sphere in w,x,y,z=0 y u=const. is a sphere of comoving radius u u x a A grows for 0 < u < π/2 and decreases for π/2 < u < π φ k=+1 a closed space
Homogeneity and Isotropy Robertson-Walker Metric expansion factor comoving radius angular area
Radial ray nearby observers along a light path, separated by δr: local flat local Hubble For Black-Body radiation, Planck’s spectrum: like photons: Also for free massive particles: de Broglie wavelength (particles or photons): useful: conformal time Redshift
limit exists Example: EdS (k=0, Λ=0) Causality problem: Horizon Our Horizon is not causally connected: what is the origin of the isotropy?
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 Newtonian Gravity
Newton’s gravity: space fixed, external force determining motions Einstein’s equations Gravity is an intrinsic property of space-time. geometry <-> energy density. Particles move on geodesics (local straight lines) determined by the local curvature. 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 For the isotropic RW metric Einstein’s tensor Stress-energy tensor eq. of motion energy conservation conservation of number of photons mass conservation The Friedman Equation Add λ A differential equation for a(t)
radiation era acceleration expansion forever conformal time critical density collapse Solutions of Friedman eq. (matter era) big bang here & now
photon: conformal time 2π 3π/2 π π/2 0 big crunch η t big bang 0 π/2 π north pole u south pole Light travel in a closed universe A photon is emitted at the origin (ue=0) right after the big-bang (te=0)
General: H and t Carrol, Press, Turner 1992, Ann Rev A&A 30, 499
a a t t k=-1 same asymptotic behavior as k=0 3rd-order polynomial – one real root exists Solutions with a Cosmological Constant k=0
k=+1 (closed) a critical value: a Lematre a solution for every a t a double root at a Einstein’s static universe (unstable) static expanding to inflation Eddington-Lematre ac static big-bang expanding to static t Solutions with a Cosmological Constant (cont.) the rhs at as to be >0
k=+1 (closed) a critical value: a t only attraction a t Solutions with a Cosmological Constant (cont.) cos. const. wins a2 for a small and large a1 mass attraction wins for a1<a<a2– no solution
Friedman Equation two free parameters closed/open decelerate/accelerate 1 0 Flat: k=0 a kinetic potential curvature vacuum Carrol, Press, Turner 1992, Ann Rev A&A 30, 499
Dark Matter and Dark Energy accelerate decelerate m /2 - =0 m + =1 =0 unbound bound closed flat open vacuum – repulsion? 1 0 0 2 1 m mass - attraction
force per mass on shell vs. vs. potential in Einstein’s eqs. Cosmological Constant: Newtonian Analog vacuum energy density
FRW: Energy change by work (2) p p dV Construct a static model: de Sitter: Quintessence for inflation need to exceed Acceleration, pressure, energy density
Dark Energy acceleration WMAP_1 WMAP_3
