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Cosmological Constant”, Or “Antigravity” Term, Which He Called “Lambda” ( Λ) to Basically Keep the Universe from Collapsing Into Itself!

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Cosmological Constant”, Or “Antigravity” Term, Which He Called “Lambda” ( Λ) to Basically Keep the Universe from Collapsing Into Itself! JatilaJatila vanvan derder VeenVeen andand PhilipPhilip LubinLubin UniversityUniversity ofof California,California, SantaSanta BarbaraBarbara TheThe StandardStandard CosmologicalCosmological ModelModel ofof thethe 2020thth centurycentury waswas foundedfounded onon threethree observationalobservational pillars:pillars: thethe redred shiftsshifts ofof galagalaxies,xies, thethe abundancesabundances ofof lightlight elements,elements, andand thethe cosmiccosmic microwavemicrowave backgroundbackground radiation.radiation. FromFrom anan initialinitial intenselyintensely hot,hot, densedense phaphase,se, inin whichwhich mattermatter waswas spontaneouslyspontaneously createdcreated outout ofof radiationradiation accordingaccording toto MasterMaster Einstein’sEinstein’s famousfamous E=mcE=mc22,, ourour UniverseUniverse hashas beenbeen coolingcooling andand expaexpanding.nding. Hydrogen,Hydrogen, helium,helium, andand lithium,lithium, thethe lightestlightest elements,elements, werewere producedproduced withinwithin thethe firstfirst 300300 secondsseconds ofof thethe infantinfant universe.universe. AllAll otherother elementselements upup throughthrough Iron-56Iron-56 areare producedproduced inin stars,stars, andand thosethose heavierheavier thanthan IronIron inin greagreatt stellarstellar explosionsexplosions calledcalled supernovae.supernovae. InIn thethe beginning,beginning, radiationradiation dominateddominated overover matter,matter, butbut soonsoon mattermatter tooktook over.over. IfIf thethe initialinitial cosmiccosmic explosionexplosion hadhad susufficientfficient kinetickinetic energyenergy totobalancebalance thethe gravitationalgravitational potentialpotential ofof allall thethe mmaatttter,er, thethe universeuniverse wouldwould expandexpand forever.forever. TheThe cucurrentrrent expansionexpansion raterate ofof 5050 kmkm/sec/Mpc/sec/Mpcwwouldould asymptoticallyasymptotically approachapproach zero.zero. EventuallyEventually thethe UniverseUniverse wouldwould endend itsits gloriousglorious existenceexistence inin emptinessemptiness andand darkness,darkness, afteafterr thethe lastlast starstar fadedfaded away...away... SuchSuch waswas thethe StandardStandard CosmologicaCosmologicall ModelModel ofof thethe lastlast cencenttury..ury.... The red shifts of galaxies discovered by Hubble in 1929 led to the conclusion that the universe is expanding . Edwin Hubble ’ λ0 λ distant galaxy λ' − λ 0 = z Red shift λ 0 defined nearby 500−400 = .25 = z 400 galaxy This example ∆λ v Assuming a simple = z ≈ Doppler shift … which led Hubble to his relationship: λ c v(r) = H0r The recession velocity of a galaxy is proportional to its distance from us. H0 = Hubble’s constant But the red shift really tells us about the stretching of the wavelengths of light due to the expansion of the Universe: ∆λ R = z = 0 −1 λ R So H0 gives us the ratio of the present expansion rate to the present size of the Universe: H “now” R& rate of change of “size” of universe H 0 = “size” of universe “now” R0 Because looking out in space = looking back in time R0 = size R of Universe “now” “then” R = size of Universe when “now” light left source R0 The value of H0 puts an upper limit on the age of the Universe: In the Standard Model of the 20th Century, H0 = 50 km/sec/Mpc , putting the maximum age of the Universe at about 19-20 billion years, which is at least consistent with the ages of the oldest stars. 50km H = = 50×3.24×10−20 sec−1 0 sec⋅ Mpc 1 H =1.68×10−18 sec−1 ≈ 0 t ∴t ≈1.96×1010 yrs Of course, the expansion rate could not have been constant. It was expected that the expansion would have been fastest in the very beginning, slowed down as the universe cooled off, then sped up a bit as matter “took over” from radiation, and has been slowing down ever since...at least, so people thought in the last century... H0 is the expansion rate as we measure it today, from our perspective. H(t) is what we refer to as the Hubble Parameter – denoted as a function of time. (The original value that Hubble himself estimated was about 10x too fast...) Imagine a spherical region of the universe, of radius r and mass M, homogeneous and v (r) isotropic, expanding with velocity v(r) . If this region is expanding with the Hubble “flow”, m M then we can say that v(r) = Hr M = ρV r = 4 πr 3 ρ Let m be a little chunk of 3 matter at radius r, moving with v(r) – the same rate as the general expansion. If the Universe is matter dominated, and has the “critical density” necessary so that the kinetic energy of its expansion just balances the gravitational potential energy of all the mass it contains... mv2 GMm starting with conservation of energy... E = − = 0 total 2 R rearrange 2GM v2 = = ()Hr 2 substitute r M = ρV = 4 πr 3 ρ 3 1 2 get expression ⎡8πGρc ⎤ for H in terms of H = ⎢ ⎥ critical density ⎣ 3 ⎦ From the previous expression, one could – in principle – calculate the critical density of the universe by knowing the Hubble parameter... if all we had to worry about was the matter content . 1 2 ⎡8πGρc ⎤ H = ⎢ ⎥ rearrange ⎣ 3 ⎦ 3H 2 ρ = c 8πG But, if... mv2 GMm That is, if the kinetic energy of E = − ≠ 0 expansion and the gravitational total 2 R potential energy are not balanced... Then, after a bit of rearranging, and using our assumption of a spherical chunk of expanding universe with density ρ... M = ρV 4 3 = 3 πr ρ We get this relationship: 2E 8πGρ total = H 2 − mR2 0 3 continuing with this 2Etotal 2 8πGρ line of reasoning... = H − mR2 0 3 Substitute in the definition of H R& 0 H = 0 R And use the first order “Einstein Equation” from General Relativity theory ... 2 ⎛ R& ⎞ 8πG ρ k Where k is a ⎜ ⎟ = − curvature parameter... ⎜ ⎟ 2 ⎝ R ⎠ 3 R 2 Next substitute this expression for H0 into the equation at the top of this page... we get... 2E 8πG k 8πG total = ρ − − ρ mR2 3 R2 3 2E total = −k m which is another way of saying that the total matter/energy content of the Universe determines the curvature of spacetime, denoted by “-k”. This leads to three possibilities for the geometry of space : k < 0 V(t) |KE|>|PE| Et > 0 “open”” t k = 0 |KE|=|PE| “flat”” Et = 0 k > 0 |KE|<|PE| “closed” Et < 0 Einstein said this in his General Relativity theory, in a more complicated way. He predicted that the geometry of space could be completely specified if we knew the total mass and energy content of the Universe. This is summarized in his famous equation: 1 Rµν − 2 Rgµν = 8πGTµν Left hand side = Right hand side = the matter geometry of spacetime and energy contained in the where “R” here is called = Universe, where “T” is the the “Ricci tensor” and “stress-energy tensor” which describes the shape of describes the distribution of spacetime in x,y,z, and matter and energy in spacetime, time in x,y,z, and time Hmmm… but something was missing... Any new theory that is developed in Physics has to agree with Newton’s Laws in the low-energy limit. According to Newton, gravity is always attractive, and has infinite range. GMm F = − g r 2 So... Einstein reasoned... if mass always attracts mass, and if (as was believed before Hubble) the Universe is static and unchanging, why had not everything attracted everything else by this time? Why is there a Universe? To answer this, Einstein postulated a “cosmological constant”, or “antigravity” term, which he called “Lambda” ( Λ) to basically keep the Universe from collapsing into itself! After Hubble announced his discovery in 1929, Einstein retracted his cosmological constant, calling it “the biggest blunder of his career”. So the idea of a lambda term in the equations faded into the background for the time being. It was not part of the Standard Cosmological Model of the 20th Century, but neither was it totally forgotten…. Meanwhile...the Big Bang nucleosynthesis and abundances of light elements were predicted by Ralph Alpher in his 1948 Ph.D. thesis. And in 1964 Arno Penzias and Robert Wilson, then at Bell Labs, noticed a small discrepancy in their microwave instruments that indicated an excess of radiation coming in from space. Not content to ignore it, they soon made one of the profound discoveries of the 20th century:.. TheyThey hadhad foundfound thethe afterafter glowglow ofof thethe enormousenormous heatheat leftleft overover fromfrom thethe “BIG“BIG BANG”BANG” inin thethe formform ofof thethe CosmicCosmic MicrowaveMicrowave Background.Background. Since the Cosmic Microwave Background radiation (CMB), fit a black body spectrum with a precision of 1 part in 104, the question became: If the universe started in some uniform hot, dense state, how did the structures that we observe today form? So people started looking for small variations in the CMB. To make a long story short, the landmark measurement of the 20th century was the COBE satellite, which measured variations in the microwave background of approximately 10 to 30 microKelvin, and an angular resolution of about 20 degrees. Red = warmer cyan = cooler above and below a background temperature of 2.726 K So, the Standard Cosmological Model of the 20th Century favored the following values for the cosmological parameters: H0 = 50 km/sec/Mpc; CMB black body temperature = 2.726K,, no cosmological constant, and “flat” geometry. We define the dimensionless density parameter of the universe, “Omega” ρ Ω = total =1 ρc Where ρtotal = total average density of the all the components of the universe, considered at sufficiently large scales to look homogeneous. The only problem was that the critical density needed to make the Standard Cosmological Model work was MORE than the density of matter that has been measured between galaxies, of approximately 1 proton per cubic meter... Using this expression for critical density 3H 2 for a flat, matter-dominated Universe, ρc = with no cosmological constant, 8πG and the using the value we derived for -18 -1 H0 , 1.68 x 10 sec , for the Standard Cosmological Model of the 20th century, and using G = 6.67 x 10-11 N-m2-kg -2 , this value turns out to be about 5 x 10 –27 kg/m3 you can verify this yourself..
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