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thefriedmannequations
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The Fate of the Universe How one set of equations changed an entire field of science Brian Kay PHY 495
Objective • Examine the Friedmann equation and its impact on our understanding of the evolution of the universe • Produce numerical and analytical solutions to the Friemann equation • The results will provide us with the geometry, current age, and ultimate fate of the universe
Outline • Part 1
- Early 20th century cosmology
• Part 2 – Understanding the Friedmann Equation • Part 3 – Solving the Friedmann equation
The Friedmann Equation
Part I The origins of cosmology http://hubblesite.org/newscenter/archive/releases/2004/07
Alexander A. Friedmann (1888-1925) The Man Who Made the Universe Expand
http://en.wikipedia.org/wiki/File:Aleksa ndr_Fridman.png
Soviet mathematician and meteorologist Most famous for contributions to cosmology First person to mathematically predict an expanding universe (1922) Derived from Einstein's general relativity Einstein initially dismissed Friedmann’s equations as nothing more that a mathematical “curiosity” Developed his equation based on an isotropic and homogenous universe
Expanding Universe First general relativistic models by Einstein predicted a contracting universe (1917) Introduction of Cosmological Constant by Einstein 1922 the Friedmann equation predicted an expanding universe
1928 Georges Lemaitre independently concluded the same Lemaitre was the first to propose a ‘primeval atom’ theory, later known as the Big Bang
http://en.wikipedia.org/wiki/File:CMB_Timeline300_no_WMAP.jpg
Edwin Hubble •
Confirmed the existence of other galaxies besides the Milky way (1924)
•
Vesto Slipher first discovered the Doppler shift in the spectra of galaxies, Hubble used this to prove the universe is expanding (1929)
•
Velocities of receding galaxies from Earth are proportional to their distance from Earth
•
Caused Einstein to abandon his cosmological constant
http://en.wikipedia.org/wiki/File:Hubble.jpg
Hubble’s Constant • Gives the speed of a galaxy in km/s that is 1 Mpc away • The inverse H0-1 is known as Hubble time referred to as the expansion time scale • Hubble first calculated H0 to be 500 km s-1 Mpc-1 • H0 = 2.26 x 10-18 s-1 • H0-1 ≈14 Gyr
Cosmic Microwave Background Radiation (CMB) • • • • • • • •
Radiation from the “Big Bang” first theorized in 1948 by Gamow, Alpher, and Herman Steady State (SS) was an alternative theory develpoed in 1948 by Hoyle, Gold, and Bondi SS claimed that new matter was continuously being created as the universe expanded to keep the average density of matter equal over time Discovered in 1964 by radio astronomers Arno Penzias and Robert Wilson CMB caused most cosmologists to accept “Big Bang” and reject SS COBE (1989-1996), WMAP (2003-2010), Plank Spacecraft -No data yet Uniform distribution suggesting an isotropic and homogeneous universe The mean temperature is 2.725 ± 0002 K
http://www.daviddarling.info/encyclopedia/L/large-scale_structure.html
http://map.gsfc.nasa.gov/media/060913/index.html
Rate of Expansion http://scienceblogs.com/startswithabang/2010/01
•
Two teams in the 1990’s set out to show the rate deceleration of the universe
•
They both independently concluded the expansion is accelerating
•
Appears to be a repulsive force countering gravity now dubbed dark energy
•
Now cosmologists are reevaluating the importance of the cosmological constant
Part II Understanding the Friedmann Equation
Robertson-Walker Metric •
Derived from Einstein’s general relativity
•
A space-time metric that satisfies the principle of a homogeneous, isotropic universe
•
The geometric structure should remain constant with time in comoving coordinates (r, θ, φ) and physical distances scale with a(t)
•
Represents the κ value in the Friedmann equation with values of either (1, -1, 0)
Cosmic Scale Factor a(t) • Represents relative expansion of the universe • Dimensionless function of time • Also called the Robertson-Walker scale factor • Relates proper distance between a pair of objects undergoing isotropic and homogeneous expansion
Curvature http://map.gsfc.nasa.gov/media/990006/index.html
•
Open, closed, flat
•
Data suggests we live in a flat universe
•
Derived from Robertson-Walker metric
•
Κ is the curvature constant
•
R0 is the radius of curvature
•
Determines how the universe will end
Density is Destiny! •
Consists of matter, radiation, and most likely a cosmological constant (Λ)
•
Einstein’s biggest blunder might have been correct all along. (Dark energy)
•
More convenient to use the ratio of the density to the critical density
•
κ/ R0 = 0 gives the critical density
•
The density parameter Ω ultimately governs whether the curvature is: – – –
negative (Ω < 1) positive (Ω > 1) flat (Ω = 1)
Rewriting the Friedmann equation in integral form
Part III Analytic and numerical solutions to the Friedmann equation
Matter Only ( κ = 0) Einstein-de Sitter Einstein de Sitter 2
0=1
1.8 1.6 1.4
a(t)
1.2 1 0.8 0.6 0.4 0.2 0 -0.5
0
0.5
1 H0t
1.5
2
Matter Only ( κ = 1) Exact Solution •
These are closed universes with a finite time At some time in the future expansion will cease and all matter will collapse in on itself called the “Big Crunch”
Matter Only Positve Curvature 12
0 1.1 0 1.3
10
0 1.7 0 1.9
8
a(t)
•
0 2.0
6
4
2
0
0
20
40
60 H0t
80
100
120
Matter Only ( κ = -1) Exact Solution • Expansion will continue for all time. Universe ends in the “Big Chill” Matter Only Negative Curvature 2
0 0.1
1.8
0 0.3
1.6
0 0.5
1.4
0 0.7 0 0.9
a(t)
1.2
0 0.999
1 0.8 0.6 0.4 0.2 0
0
0.2
0.4
0.6
0.8
1 H0t
1.2
1.4
1.6
1.8
2
Matter Only Numerical Solutions Matter Only Numerical 1.8
0 0.1 1.6
0 0.3 0 0.5
1.4
0 0.7
1.2
0 0.9 0 1.0
a(t)
1
0 1.1
0.8
0 1.5 0 1.8
0.6
0 2.0
0.4 0.2 0 -0.5
0
0.5
1 H0t
1.5
2
Matter + Λ (κ = 0)
Matter + Λ (κ = 0)
Matter + Universe
Matter + Universe
2.5
30
25
2
20
a
a
1.5 15
1 10
0.5
5
0
0
0
1
2
3
4 H0t
5
6
7
0
1
2
3
4
5 H0t
6
7
8
9
10
Matter + Λ Curved
• Loitering is when during expansion a remains almost constant for a long period of time • Could be a “big bounce” universe, however, we cannot calculate a present time with no beginning time • Could lead to unphysical, forbidden range of the scale factor
Benchmark model • Considered the best fit to the currently available observational data with t0 = 14Gyr
Benchmark Plot 2 benchmark benchmark no radiation
1.8
a(t)
• Is spatially flat and contains 1.6 radiation, matter, and a dominant 1.4 cosmological constant 1.2 1 0.8 0.6 0.4 0.2 0
0
0.2
0.4
0.6
0.8
1 H0t
1.2
1.4
1.6
1.8
2
Conclusion
www.sciencenews.org/view/download/id/72459/name/Cosmic_Armageddon_
Bibliogrpahy • Ryden, Barbara Sue. Introduction to Cosmology. San Francisco: Addison-Wesley, 2003. Print • Tropp, Eduard A. Alexander a Friedmann The Man Who Made the Universe Expand.Cambridge: Cambridge Univ Pr, 2006. Print.