Spatially Non-Flat Cosmologies Intro Cosmology Short Course Lecture 4 Paul Stankus, ORNL

Robertson-Walker Metric 1.The  coordinate system is uniform (not necessarily Euclidean) Recall from Lecture 2: General spatial metric Euclidean case What other isotropic, uniform 3-D spaces are possible?

Life in a 2-D plane How do you test if your space is flat/Euclidean? Saddle Bowl Flat Globally Euclidean Slope = 2  Characterize by C(r) Circumference Radius Assume space is locally Euclidean Straight lines are well-defined Circles can be constructed and their radii and circumferences measured 22 11

My childhood astronomy book Part of the “Foundations of Science” series (Greystone, MCMLXVI) Flat Bowl or Sphere Saddle

Metrics in flat 2-D space dy dx dr rd 

R Sphere C(r)C(r) r 0  R Sphere /2  R Sphere Distance Metric in 2-D on Surface of a Sphere Generalization to Saddle Surface Euclid in 2-D r r r

22 11 33 Hyper-sphere “Closed” Euclidean “Flat” Hyper-saddle “Open” Closed Flat Open Robertson-Walker Metric in 3+1D

Common Alternate Notation Our notation (for k  a(t) Dimensionless  Length k ~1/R 0 2 1/Length 2 Older notation (e.g. Weinberg ) R(t) Length r~  R 0 Dimensionless k  -1,0,+1 Dimensionless

Spatial Curvature and Dynamics The Friedmann Equation  0 GR

Three Matter-Filled Universes Assume all  in matter, then  (t)  0 /a(t) 3 a(t)a(t) t a(t)a(t) t a(t)a(t) t k0k0 k0k0 k0k0 Invert directly a ( t )  t 2/3, a ( t )  t -1/3 a ( t ) must hit 0 Recollapse! a ( t )  Constant Flat Closed Open...

Do we live at a special time ( 1 )? t 1 0 (t)(t) (t)(t) k(t)k(t) k(t)k(t) (t)(t) k(t)k(t)  0 t Now a(t)a(t) Flat Closed Open Now?  M Obs

t Now a(t)a(t) Flat Closed Open Then ( ) Now ( ) Working belief  0k0k0 Big question Open, closed or flat?Accelerating or decelerating? Key parameters  M,  k , d  /dt; w, dw/dt Coincidence?  M ~  k  M ~  

Flat (Generic) 00 Open Closed Generic 00 Open

Points to take home Uniform but non-Euclidean spaces: “bowls and saddles” Bowl: C(r) 2  r, A(r)>4  r 2 Generalize Robertson-Walker metric Friedmann equation connects density, expansion, curvature Matter-dominated: open, flat, or closed? expansion or recollapse? Do we live at a special time? Part 1