Ast 207 F2009 Dec-09Ast 207 F2009Weighing the Universe with Supernovae.Discovery of Dark Energy/ Cosmological Constant / Review — 11 Dec• Final exam– Final from 2008 (with answers) is on angel.– Covers entire course with emphasis on 20thcentury cosmology (Oct 28 to end of term, Hwk 7−10)– One 8.5×11” cheat sheet– Mon, 14th, 3:00-5:00, 1410 BPS (large classroom next door)• Office hours– Normal time on Mon.• Please fill out on-line SOCT http://rateyourclass.msu.edu– Will close when grades are submitted.Ast 207 F2009Observations• Lower plot compares data to a model with density parameterΩ = PE/KE = 0.2• Distant SN are 20% fainter than model with Ω = 0.2.• Distant SN are 15% fainter than model with Ω =0!–f=L/D2–Fainter ⇒ distance is greater – Longer time to expand than for a universe having no mass at all!– Longer time means expansion sped up.a=0.99a=0.9a=0.5compared to model with Ω =0.2Ast 207 F2009 Dec-09Ast 207 F2009Observations• Distant SN are 15% fainter than model with Ω =0!–f=L/D2–Fainter ⇒ distance is greater – Longer time to expand than for a universe having no mass at all!– Longer time means expansion sped up.• Einstein (about 1920): I thought of a kind of stuff where gravity repels. E called it “cosmological constant.”a=0.99a=0.9a=0.5-1 -0.8 -0.6 -0.4 -0.2 0H × time00.20.40.60.81ecnatsiD/ woNecnatsiDNowLots of massLittle massa = ½Time for lightto reach usActual U is a line to the left of this lineAst 207 F2009Observations• Distant SN are 15% fainter than model with Ω =0!–f=L/D2–Fainter ⇒ distance is greater – Longer time to expand than for a universe having no mass at all!– Longer time means expansion sped up.• Einstein (about 1920): I thought of a kind of stuff where gravity repels. E called it “cosmological constant.”a=0.99a=0.9a=0.5-1.0-0.8-0.6-0.4-0.2 0.00.00.20.40.60.81.0Hμtim ea=DistanceêDistancenowAst 207 F2009 Dec-09Ast 207 F2009How did Einstein think of “cosmological constant,” stuff that repels?• Example of how physicists create new ideas.• Figure out the appropriate mathematics to describe gravity.– Newton: Vectors describe gravity.– Einstein’s happiest thought: “A man falling from a roof does not feel gravity” (until he hits the ground). Years of thought. Tensors describe gravity.• Figure out the consequences of the theory.Ast 207 F2009Einstein’s General Relativity• What causes gravity?• Newton’s answer: mass.– Force of gravity between what’s in the sphere and mass mF = G M m/R2.• Einstein’s answer: mass and pressure– Force of gravity between what’s in the sphere and mass mF = G (M + 3PV/c2) m/R2vMRmAst 207 F2009 Dec-09Ast 207 F2009Source of Gravity• Einstein’s answer: mass and pressureF = G (M + 3PV/c2) m/R2.• Newton’s Law of gravityF = G M m/R2.• Einstein’s Law of gravityCurvature of space = 8πG (Mass-Pressure tensor)G = 8πG T– Object feels curvature of space and changes its momentum– G and T are tensors having 16 components– If on average the material is at rest, then T has 4 components. The source of gravity, (M + 3PV/c2), is the sum of the 4 terms on the diagonal of T V.• Ordinary matter has little pressure because speed is much smaller than c.3PV/c2= M (v/c)2is negligible compared with M.• Radiation has positive pressure3PV/c2= M.F = G 2M m/R2.⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛=222matter)(0000)(0000)(00001/cvcvcvzyxVMT⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛=313131rad0000000000001/VMT⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛=222/0000/0000/0000/cPcPcPVMTzyxAst 207 F2009Cosmological Constant• Einstein’s answer: mass and pressureForce: F = G (M + 3PV/c2) m/R2.– If on average the material is at rest, then Thas 4 components. source of gravity, (M + 3PV/c2), is the sum of the 4 terms on the diagonal of T V.• Einstein in 1920s: My equations of gravity allow a special tensor. E called it a “cosmological constant.”–Tcchas same mathematical properties as Tmatterand Trad.– The CC has mass and negative pressure. Normal pressure pushes. CC sucks.– The CC may exist in Nature. • “What is not forbidden is mandatory”—W Pauli⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛=313131rad0000000000001/VMT⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛−−−=1000010000100001/CCVMT⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛=222matter)(0000)(0000)(00001/cvcvcvzyxVMTAst 207 F2009 Dec-09Ast 207 F2009Cosmological Constant• Einstein’s answer: mass and pressureForce: F = G (M + 3PV/c2) m/R2.– If on average the material is at rest, then T has 4 components. The source of gravity, (M + 3PV/c2), is the sum of the 4 terms on the diagonal of T V.1. Write the force of gravity for the case of the cosmological constant. (Watch the signs.)A. F=G M m/R2B. F=G (2M) m/R2C. F=G (– 2M ) m/R2⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛=313131rad0000000000001/VMT⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛−−−=1000010000100001/CCVMT⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛=222matter)(0000)(0000)(00001/cvcvcvzyxVMTAst 207 F2009Cosmological Constant• Einstein’s answer: mass and pressureForce: F = G (M + 3PV/c2) m/R2.• Einstein’s Law of gravityCurvature of space = 8πG Mass-PressureG = 8πG T– Object feels curvature of space and changes its momentum• Einstein in 1920s: My equations of gravity allow “cosmological constant”–Tcchas same mathematical properties as Tmatterand Trad.– Pauli: “What is not forbidden is mandatory.”–F = G (M –3M ) m/R2F = – G 2M m/R2.– Repulsive gravity• Einstein tried to make his theory of gravity prevent expansion or contraction of the universe. The cosmological constant balances gravity of matter.– In 1929, Hubble discovered the expansion of the U. Einstein said the cosmological constant was “his greatest blunder.”– Had he lived to 1998, he would have called it his greatest discovery.Ast 207 F2009 Dec-09Ast 207 F2009Summarizing questions• What is the evidence for dark energy? What was measured. If the result of the measurements were ___, there would be no evidence for dark energy.• Ideas needed to answer the question:– SN are fainter than if U had no dark energy.– Flux of SN is related to distance.– With no DE, distance to SN is shorter.– Redshift of SN determines the amount U expands.– SN have the same luminosity: They are standard candles.– Astronomers can model flux vs redshift for different density parameters.– What plot did we
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