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Understanding the UniverseUnderstanding the UniverseSummary LectureSummary Lecture
Volker Beckmann
Joint Center for Astrophysics, University of Maryland, Baltimore County
& NASA Goddard Space Flight Center, Astrophysics Science Division
UMBC, December 12, 2006
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OverviewOverview● What did we learn?● Dark night sky● distances in the Universe● Hubble relation● Curved space● Newton vs. Einstein : General Relativity● metrics solving the EFE: Minkowski and Robertson-Walker metric● Solving the Einstein Field Equation for an expanding/contracting Universe: the Friedmann equation
Graphic: ESA / V. Beckmann
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OverviewOverview● Friedmann equation● Fluid equation● Acceleration equation● Equation of state● Evolution in a single/multiple component Universe● Cosmic microwave background● Nucleosynthesis● Inflation
Graphic: ESA / V. Beckmann
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Graphic by Michael C. Wang (UCSD)
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The velocity-distance relation for galaxies found by Edwin Hubble.
Graphic: Edwin Hubble (1929)
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Expansion in a steady state Universe
Expansion in a non-steady-state Universe
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The effect of curvature
The equivalent principle: You cannot distinguish whether you are in an accelerated system or in a gravitational field
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Newton vs. EinsteinNewton vs. Einstein
Newton:- mass tells gravity how to exert a force, force tells mass how to accelerate (F = m a)Einstein:- mass-energy (E=mc²) tells space time how to curve, curved space-time tells mass-energy how to move (John Wheeler)
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The effect of curvature
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The effect of curvature
Left side (describes the action of gravity through the curvature of space time):Rab = Ricci curvature tensor (4x4, 10 independent)R = Ricci scalar (each point on a Riemannian manifold assigned a real number describing intrinsic curvature)gab = metric tensor (~ gravitational field)Solutions to the EFE are called “metrics of spacetime” (“metrics”) In flat space time for example we get the Minkowski metric:
… and the metric tensor becomes:
A glimpse at Einstein’s field equation
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The Minkowski metric(a solution to the Einstein field equation for flat space time)
ds2 = 0 represents the path of light (null geodesic)
assume movement along r only :c2 dt2 = dr2
or
dr / dt = +- c
ds2 = -c2 dt2 + dL2 = -c2 dt2 + dr2 +r2 dΩ2
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The effect of curvatureRobertson-Walker metricHoward Percy Robertson (1903-1961) &Arthur Geoffrey Walker (1909-2001)
Which metric solves the Einstein equation if there is curvature?Robertson-Walker metric is an exact solution for General Relativity
r : comoving distance from observerrr r : proper motion distanceRC : absolute value of the radius of curvaturehomogeneous, isotropic expanding universe
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Fundamental Principles of GRFundamental Principles of GR
- general principle of relativity- principle of general covariance- inertial motion is geodesic motion- local Lorentz invariance- spacetime is curved- spacetime curvature is created by stress-energy within the spacetime
(1 and 3 result in the equivalent principle)
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The effect of curvatureFriedmann EquationAleksander Friedmann (1888-1925)
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The effect of curvature
Friedmann Equation
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The effect of curvature
Friedmann equation with cosmological constant
Fluid equation (stays the same) :
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The effect of curvature
Key question: what is the average energy density of virtual particles in the universe?
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Look for the lecture about dark energy on the course webpage (under “Literature”). Fast connection required (170 MByte; 45 minutes).
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Density parameter and curvature
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The effect of curvature
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The first part of the lectureThe first part of the lecture- Universe is expanding (Hubble relation)- Newton’s not enough: Einstein’s idea about space-time- General relativity for curved space-time- Four equations to describe the expanding/contracting universe
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The second part of the lectureThe second part of the lecture
- How to model the Universe- the Friedmann equation as a function of density parameters- Matter-, radiation-, lambda-, curvature- only universe- mixed-component universes- the important times in history:ar,m and a m,
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The second part of the lectureThe second part of the lecture
- How to measure the Universe- the Friedmann equation expressed in a Taylor series: H0 and q0 (deceleration parameter)
- luminosity distance, angular size distance- distance ladder: parallax, Cepheids, SuperNova Type Ia- results from the SuperNova measurements
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The second part of the lectureThe second part of the lecture
- What is the matter contents of the Universe?- matter in stars- matter between stars- matter in galaxy clusters- dark matter
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The effect of curvature
Scale factor a(t) in a flat, single component universe
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The effect of curvature
Universe with matter and curvature only
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Curved, matter dominated UniverseCurved, matter dominated Universe
0 < 1 = -1 Big Chill (a t)0 = 1 = 0 Big Chill (a t2/3)0 > 1 = -1 Big Crunch
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The effect of curvature
Universe with matter, , and curvature(matter + cosmological constant + curvature)
0 = m,0 + ,0
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The effect of curvature
Flat Universe with matter and radiation(e.g. at a ~ arm)
0 = m,0 + r,0
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The effect of curvatureDescribing the real Universe - the “benchmark” model
0 = L + m,0 + r,0
= 1.02 ± 0.02 (Spergel et al. 2003, ApJS, 148, 175)
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The effect of curvatureThe “benchmark” model
0 = L + m,0 + r,0
= 1.02 ± 0.02 (Spergel et al. 2003, ApJS, 148, 175)
“The Universe is flat and full of stuff we cannot see”
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The effect of curvature
The “benchmark” model
“The Universe is flat and full of stuff we cannot see - and we are even dominated by dark energy right now”
Important Epochs in our Universe:
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The effect of curvature
The “benchmark” model
Some key questions:
- Why, out of all possible combinations, we have 0 = + m,0 + r,0 = 1.0 ?
-Why is ~ 1 ? - What is the dark matter?-What is the dark energy?- What is the evidence from observations for the benchmark model?
“The Universe is flat and full of stuff we cannot see”
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How do we verify our models with observations?
Hubble Space Telescope Credit: NASA/STS-82
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The effect of curvature
Scale factor as Taylor Series
q0 = deceleration parameter
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The effect of curvature
Deceleration parameter q0
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The effect of curvature
How to measure distance
M101 (Credits: George Jacoby, Bruce Bohannan, Mark Hanna, NOAO)
Measure flux to derive the luminosity distanceMeasure angular size to derive angular size distance
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The effect of curvature
Luminosity DistanceIn a nearly flat universe:
How to determine a(t) :
- determine the flux of objects with known luminosity to get luminosity distance
- for nearly flat: dL = dp(t0) (1+z)
- measure the redshift
- determine H0 in the local Universe
q0
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The effect of curvature
Angular Diameter Distance
dA = length / = dL / (1+z)2
For nearly flat universe:dA = dp(t0) / (1+z)
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Distance measurementsDistance measurements
- Distance ladder:- measure directly distances in solar system- nearby stars with parallax (up to 1kpc) - nearby galaxies with variable stars (Cepheids)- distant galaxies with Supernovae Type Ia (“standard candles”) up to redhsift z~2- see also “literature” on the course webpage
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The effect of curvature
For nearlyflat Universe:
Credit: Adam Block
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Chandrasekhar limit
Electron degeneracy pressure can support an electron star (White Dwarf) up to a size of ~1.4 solar masses
Subrahmanyan Chandrasekhar (1910-1995)
Nobel prize winner
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Super Nova Type Ia lightcurves
Corrected lightcurves
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Perlmutter et al. 1999
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Perlmutter et al. 1999
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The effect of curvature
Dark matter can “seen” in rotation curves of galaxiesand in velocity dispersion of galaxies in clusters
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The third part of the lectureThe third part of the lecture- Cosmic microwave background- Nucleosynthesis; the first 3 minutes- inflation: solving the flatness, horizon, and magnetic monopol problem by inflating the universe exponentially for a very short time starting at ti = 10-35 s
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Penzias & Wilson (1965): 3˚ K background radiation
Bell Telephone Lab in Murray Hill (New Jersey)
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The 2.75 ˚K background as seen by WMAP
Credit: NASA/WMAP Science Team
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Acoustic Oscillations
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Where do the elements come from???(beats every chemist or biologist!!!)
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Neutron decay involves the weak force: small cross section whichdecreases with time (proportional to 1/t). Thus the neutron fraction freezes out after the universe is 1 second old.The neutron fraction ends up to be ~ 0.15
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4He is a “dead end” for primordial nucleosynthesis. Helium amountsto ~24% of all baryonic matter
Illustration: Christine M. Griffin
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Primordial nucleosynthesis is finished after 1000 seconds
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Why is the baryon-to-photon fraction = 5.5 10-10 ?
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Our Universe model works fine so far, but…
- flatness problem (nearly flat but not totally flat)- horizon problem (isotropic CMB)- monopole problem (there are no magnetic monopoles)
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Inflation of the Universe
- phase of the Universe with accelerated expansion
Alan Guth, MIT (*1947)
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Final ExamFinal Exam
- Thursday, 8:00 a.m. - 10 a.m. (here)- show up 5 minutes early - bring calculator, pen, paper- no books or other material- formulas provided on page 4 of exam - questions are ordered in increasing difficulty
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Final exam preparationFinal exam preparation
- Ryden Chapter 2 - 11 (incl.)- look at homework (solutions)- look at midterm #1 and #2- check out the resources on the course web page (e.g. paper about benchmark model, videos of lectures)