Models of Our Universe: The Friedman Equation By Ashleigh Miller November 4, 2014 Southern Methodist...

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Models of Our Universe:The Friedman Equation

By Ashleigh MillerNovember 4, 2014

Southern Methodist UniversityDr. Cooley’s 3305 Modern Physics Class

Modern Physics 3305 Final Presentation November 4,2014 Ashleigh Miller

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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

Modern Physics 3305 Final Presentation November 4, 2014 Ashleigh Miller

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Alexander Friedman (1888-1925)

• 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.


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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


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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


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Components of the Friedman Equation


•H(t) : Hubble parameter•ε(t) : Energy density•κ : Curvature•R0 : Radius of curvature (at present time)

•a(t) : Scale of distance (changes with time)

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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 < 1 -> κ = -1– “empty” universe– Open, infinite, saddle-like shape

• Ω = 1 -> κ = 0– Critical density– Open, infinite, and flat


κ/R0²=(H0²/c²)(Ω0-1)

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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 < 0 -> Ωm,0 > 1: will slow

down and collapse– Λ > 0 -> Ωm,0 < 1: will

expand forever

Λ =ΩΛ,0=1-Ωm,0

H²/H0²=Ωm,0/a³+(1-Ωm,0)


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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


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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.


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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


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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.


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References• Friedman, Aleksandr. "On the Curvature of Space." Zeitschrift Fur

Physik 10 (1992): 377-86. Wwwphy.princeton.edu. Princeton University. Web. 29 Nov. 2014. <http://wwwphy.princeton.edu/~steinh/ph563/friedmann.pdf>.

• NASA/WMAP Science Team. "Our Universe." WMAP- Universe Overview. National Aeronautics and Space Administration, 10 Dec. 2010. Web. 29 Nov. 2014.

• 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. 2014.

• Ryden, Barbara Sue. Introduction to Cosmology. San Francisco: Addison-Wesley, 2003. Print.


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Images• Wright, Edward L. 3geomtrs. Digital image. Astro. University of California at Los Angeles, 03 July 2009. Web. 30 Nov.

2014. <http://www.astro.ucla.edu/~wright/cosmo_03.htm>.• Tropp, Eduard A., Viktor Ya Frenkel, and Artur D. Chernin. Alexander A. Friedmann. Digital image.Cambridge.org.

Cambridge University Press, 2006. Web. 1 Dec. 2014. <http://assets.cambridge.org/97805210/25881/frontmatter/9780521025881_frontmatter.pdf>.

• Greason, Michael R. COBE. Digital image. Nasa.gov. Goddard Space Flight Center, 26 June 2008. Web. 1 Dec. 2014. <lambda.gsfc.nasa.gov/product/cobe/>.

• Hinshaw, Gary. End_of_universe.jpg. Digital image. Nasa.gov. Goddard Space Flight Center, 6 Jan. 2006. Web. 1 Dec. 2014. <http://www.google.com.br/imgres?q=cubo&hl=pt-BR&gbv=2&tbm=isch&tbnid=kRVXDeBtM7wUyM:&imgrefurl=http://computacaografica.ic.uff.br/erratas.html&docid=JGBhW5pzDGjSIM&w=2163&h=1713&ei=bwplTrncOMWCgAfGouCMCg&zoom=1&iact=hc&vpx=1337&vpy=410&dur=156&hovh=200&hovw=252&tx=173&ty=72&page=1&tbnh=133&tbnw=168&start=0&ndsp=55&ved=1t:429,r:30,s:0&biw=1920&bih=899>.

• Eracleous, Mike, Dr. Figure 16.2. Digital image. The Astronomical Universe. Pennsylvania State University, n.d. Web. 1 Dec. 2014. <http://www2.astro.psu.edu/~mce/A001/lect23.html>.

• 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. 2014. Web. 1 Dec. 2014. <https://www.flickr.com/photos/98498293@N05/13740217295/>.

• Greason, Michael R. Timeline. Digital image. Nasa.gov. Goddard Space Flight Center, 26 June 2008. Web. 1 Dec. 2014. <http://map.gsfc.nasa.gov/media/060915/>.

• 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 2008. Web. 1 Dec. 2014. <http://map.gsfc.nasa.gov/media/101082/101082_q_7yr_WMAP_2048.png>.


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Values for the Benchmark Model


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Models of Our Universe: The Friedman Equation By Ashleigh Miller November 4, 2014 Southern Methodist...

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