Models of Our Universe: The Friedman Equation By Ashleigh Miller November 4, 2014 Southern Methodist University Dr. Cooley’s 3305 Modern Physics Class 1Modern Physics 3305 Final Presentation November 4,2014 Ashleigh Miller

What to Look Forward to: What is the Friedman Equation and what does it mean? Components: – What do they mean/do? Predictions of the end The Benchmark Model 2Modern Physics 3305 Final Presentation November 4, 2014 Ashleigh Miller

Alexander Friedman ( ) Russian physicist and mathematician Published “On the Curvature of Space” in 1922 – He was the first to present a mathematical description of the universe that was non- static. 3Modern Physics 3305 Final Presentation November 4, 2014 Ashleigh Miller

The Friedman Equation Derived from Einstein's theory for General Relativity Based on Robertson-Walker Metric – It assumes a universe that is isotropic and homogeneous at very large scales (distances of 100 Mpc and greater) It can be used to derive many different models of universes by the variation of its components 4Modern Physics 3305 Final Presentation November 4, 2014 Ashleigh Miller

The Possibilities The Friedman Equation can produce complex models as well as more simple, single component models. – Curvature only (empty) – Matter only – Radiation only – Constant energy density only Modern Physics 3305 Final Presentation November 4, 2014 Ashleigh Miller5

Components of the Friedman Equation Modern Physics 3305 Final Presentation November 4, 2014 Ashleigh Miller6 H(t) : Hubble parameter ε(t) : Energy density κ : Curvature R 0 : Radius of curvature (at present time) a(t) : Scale of distance (changes with time)

Considering Matter and Curvature Ω is the matter density parameter of the universe. Ω > 1 -> κ = +1 – Matter dominated universe – Closed and sphere-like Ω κ = -1 – “empty” universe – Open, infinite, saddle-like shape Ω = 1 -> κ = 0 – Critical density – Open, infinite, and flat 7Modern Physics 3305 Final Presentation November 4, 2014 Ashleigh Miller κ/R 0 ²=(H 0 ²/c²)(Ω 0 -1)

Considering Matter and Lambda Λ represents the cosmological constant which also defines the expansion of the universe. When considering the expansion of a flat universe: – Λ Ω m,0 > 1: will slow down and collapse – Λ > 0 -> Ω m,0 < 1: will expand forever Λ =Ω Λ,0 =1-Ω m,0 H²/H 0 ²=Ω m,0 /a³+(1-Ω m,0 ) 8Modern Physics 3305 Final Presentation November 4, 2014 Ashleigh Miller

All Good Things Must Come to an End… The Big Chill: – Aka: The Big Freeze – The universe expands forever and cools off making it too cold to sustain life The Big Crunch: – Expansion slows down, reverses, and collapses – This will either cause a massive black hole singularity or another big bang Modern Physics 3305 Final Presentation November 4, 2014 Ashleigh Miller9

The Benchmark Model Best fit to currently available data Flat (κ=0). Contains radiation, matter, and a cosmological constant Energy density is dominated by the cosmological constant and most of the matter in the universe is dark matter. 10Modern Physics 3305 Final Presentation November 4, 2014 Ashleigh Miller

How the Data is Obtained Cosmic Microwave Background – Matter and energy density – Curvature Use redshift of super nova to measure distances and age of universe Standard candles and the expansion of the universe – Hubble Constant 11Modern Physics 3305 Final Presentation November 4, 2014 Ashleigh Miller

Summary The Friedman Equation is a powerful tool for predicting the properties of many different models of the universe. These models provide a basis upon which we are able to build our understanding of the past, present, and end of our universe as we currently understand it. 12Modern Physics 3305 Final Presentation November 4, 2014 Ashleigh Miller

References Friedman, Aleksandr. "On the Curvature of Space." Zeitschrift Fur Physik 10 (1992): Wwwphy.princeton.edu. Princeton University. Web. 29 Nov NASA/WMAP Science Team. "Our Universe." WMAP- Universe Overview. National Aeronautics and Space Administration, 10 Dec Web. 29 Nov O'Connor, J. J., and E. F. Robertson. "Aleksandr Aleksandrovich Friedmann." Friedmann Biography. School of Mathematics and Statistics University of St Andrews, Scotland, n.d. Web. 29 Nov Ryden, Barbara Sue. Introduction to Cosmology. San Francisco: Addison-Wesley, Print. 13Modern Physics 3305 Final Presentation November 4, 2014 Ashleigh Miller

Images Wright, Edward L. 3geomtrs. Digital image. Astro. University of California at Los Angeles, 03 July Web. 30 Nov Tropp, Eduard A., Viktor Ya Frenkel, and Artur D. Chernin. Alexander A. Friedmann. Digital image.Cambridge.org. Cambridge University Press, Web. 1 Dec Greason, Michael R. COBE. Digital image. Nasa.gov. Goddard Space Flight Center, 26 June Web. 1 Dec Hinshaw, Gary. End_of_universe.jpg. Digital image. Nasa.gov. Goddard Space Flight Center, 6 Jan Web. 1 Dec Eracleous, Mike, Dr. Figure Digital image. The Astronomical Universe. Pennsylvania State University, n.d. Web. 1 Dec Mahoney, Ray. Timeline_of_the_far_future. Digital image. Expanding Universe/contracting Universe, the Big Rip/the Big Crunch/the Big Freeze, the Future. Flickr, 9 Apr Web. 1 Dec Greason, Michael R. Timeline. Digital image. Nasa.gov. Goddard Space Flight Center, 26 June Web. 1 Dec All photographs of formulas were taken by Ashleigh Miller and were originally printed in the Introduction to Cosmology text. Greason, Michael R. WMAP_Microwave_Sky_Band. Digital image. Nasa.gov. Goddard Space Flight Center, 26 June Web. 1 Dec Modern Physics 3305 Final Presentation November 4, 2014 Ashleigh Miller14

Values for the Benchmark Model Modern Physics 3305 Final Presentation November 4, 2014 Ashleigh Miller15