Models of Our Universe:
The Friedman Equation
By Ashleigh Miller
November 4, 2014
Southern Methodist University
Dr. 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 nonstatic.
Modern Physics 3305 Final Presentation
November 4, 2014
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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
Modern Physics 3305 Final Presentation
November 4, 2014
Ashleigh Miller
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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
Modern Physics 3305 Final Presentation
November 4, 2014
Ashleigh Miller
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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)
Modern Physics 3305 Final Presentation
November 4, 2014
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Considering Matter and Curvature
κ/R0²=(H0²/c²)(Ω0-1)
• 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
Modern Physics 3305 Final Presentation
November 4, 2014
Ashleigh Miller
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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 =ΩΛ,0=1-Ωm,0
H²/H0²=Ωm,0/a³+(1-Ωm,0)
– Λ < 0 -> Ωm,0 > 1: will slow
down and collapse
– Λ > 0 -> Ωm,0 < 1: will
expand forever
Modern Physics 3305 Final Presentation
November 4, 2014
Ashleigh Miller
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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
Modern Physics 3305 Final Presentation
November 4, 2014
Ashleigh Miller
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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.
Modern Physics 3305 Final Presentation
November 4, 2014
Ashleigh Miller
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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
Modern Physics 3305 Final Presentation
November 4, 2014
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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.
Modern Physics 3305 Final Presentation
November 4, 2014
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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
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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=ptBR&gbv=2&tbm=isch&tbnid=kRVXDeBtM7wUyM:&imgrefurl=http://computacaografica.ic.uff.br/erratas.html&doc
id=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>.
Modern Physics 3305 Final Presentation
November 4, 2014
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Values for the Benchmark Model
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November 4, 2014
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