Wolfram Cloud Document

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(*ThisMathematicanotebook usesGUTCPtocalculatehowlongtheuniversehasbeenexpanding(inthecurrentcycle)andwhatthecosmicmicrowavebackgroundtemperaturewasatthestartoftheexpansion.Theanswersare90billionyearsagoatatemperatureof4.4Kelvin.Yourcommentsandcriticismsaresolicited.Caveat:RandellMillshasspecificallydeniedme"permission to use my copyrighted equations and theory in an algebraic manipulation to invert them to fundamental constants which are the input parameters. The same could be done with all observables and my exact equations."Charitableinterpretationsofhisdenial:1)Anerroronmypartcouldresultinmaterialdamagetohisbusiness.Havinggrantedhimthischaritableinterpretation,I'llassertthatthefollowingmayverywellbeinerrorinsomepotentiallydamagingwaywhichiswhyI'dlikenowtoaskforhelpinidentifyingtheerrorinthefollowingderivation.2)Evenifthefollowingderivationisbasicallycorrect,itmaybemisconstruedasattributingtome,ratherthanRandellMillsandcolleagues,amajorAdvanceincosmology.Thederivation,ifcorrect,isatrivialresultofthenon-trivialGUTCP.ThefactthatImayhavepublishedtheresultfirstshouldn'tbetakentoassigncredittomeforthisadvance.Afterall,Millsandcolleagueshavetheirhandsfullfightingbattlessothereareboundtobecorrectionsinthedetailsofthepreviouslypublishedbooksand/orpapersinvolvingGUTCP.Considerthiswork,evenifcorrect,littlemorethanproof-reading. Withthatcaveatsaid,letmeexplainwhatIdid:TheJWSTobservationsmotivatedmetolookintousingGUTCP'sformulastoderiveatheoreticvalueofH0(whichwecancallH0GUTCP)fromphysicalconstants(c,G,sigmaande(missivityofblackbody))andtheobservedcosmicmicrowavebackgroundradiationtemperature.ItturnsoutthereisasolutionfortheHubbleconstantwithinGUTCPusingonlyNewton'sgravitationalconstantandthemicrowavebackgroundtemperatureasparameterswithanyuncertainty.Otherphysicalconstantshavebeenrelegatedtozerouncertaintybythe2019CODATA.Thisavoidsrelyingontheestimatedmassoftheuniverse(mU)providedbyobservationalextrapolationsindependentlyoftheCMBRTemperature.TheformulaforH0GUTCPcanbederivedbyfirstsolvingformU*intermsofCMBRTemperatureandthenusingthattosolveforHthensubstituting10^10yearstogetH0.Therelevantformulasare:1)Thealeph(radiusoftheuniverse)equationunder"The Expanding Universe and the Microwave Background"section.2)Thealeph_minequation(32.146)3)Settingt=0foralephsothatexpressioncanbeset==aleph_mintosolveformU4)SubstitutethesolutionformUinto(32.156)toobtainHasafunctionoftime.5)InHsubstitute10^10yearsforttoobtainH0.*mU,whenitappearsincalculationsasaconstant(asopposedtoafunctionoftime)designatesthemaximummassoftheuniverseasitoccursattheminimumdiameteroftheuniversewhichoccursatt=0.HOWEVER,whenIranthecalculation,theH0GUTCPthatIcameupwithwaslessthanafactorof2thatofthelowestcurrentlyredshift-measuredestimateofH0.ThismademesuspectthateithertheCMBRTemperatureATTHESTARTOFTHECURRENTEXPANSIONorthetimesincethestartoftheexpansionorBOTHparametersneededadjustment.ThatiswhatmotivatedmetowritethefollowingMathematicascriptwhichpurportstosolveforBOTHparameters(tNow,CMBRTempAtStart)intermsofphysicalconstantsanddirectmeasurements.Thecalculationsayswe're>80billionyearsintothepresentexpansionandtheCMBRTemperatureatthestartoftheexpansionwas>4Kelvins.*)

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aleph[t_]=(2GmUAtStart)/c^2+(cmUAtStart/(c^3/(4PiG)))-(cmUAtStart/(c^3/(4PiG)))*Cos[2Pit/(2PiGmUAtStart/c^3)]PU[t_]=c^5*(1+Cos[2Pit/(2GmUAtStart/c^3)])/(8GPi)AU[t_]=4Pialeph[t]^2RU[t_]=PU[t]/AU[t]TU[t_]=(RU[t]/(esigma))^(1/4)alephMin=Sqrt[c^5/((4Pi)^2*Gesigma*CMBRTempAtStart^4)]alephAtStart=aleph[0]mUAtStart=First[mUAtStart/.Solve[alephMin==alephAtStart,mUAtStart]]

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2GmUAtStart

2

c

+

4GmUAtStartπ

2

c

-

4GmUAtStartπCos

3

c

t

GmUAtStart



2

c

Out[]=

5

c

1+Cos

3

c

πt

GmUAtStart



8Gπ

Out[]=

4π

2

2GmUAtStart

2

c

+

4GmUAtStartπ

2

c

-

4GmUAtStartπCos

3

c

t

GmUAtStart



2

c

Out[]=

5

c

1+Cos

3

c

πt

GmUAtStart



32G

2

π

2

2GmUAtStart

2

c

+

4GmUAtStartπ

2

c

-

4GmUAtStartπCos

3

c

t

GmUAtStart



2

c

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1/4

5

c

1+Cos

3

c

πt

GmUAtStart



eGsigma

2

2GmUAtStart

2

c

+

4GmUAtStartπ

2

c

-

4GmUAtStartπCos

3

c

t

GmUAtStart



2

c

2

1/4

2

π

Out[]=

5

c

4

CMBRTempAtStart

eGsigma

4π

Out[]=

2GmUAtStart

2

c

Out[]=

2

c

5

c

4

CMBRTempAtStart

eGsigma

8Gπ

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(*uncG=UnitConvert[Around@@Entity["PhysicalConstant","GravitationalConstant"][{"Value","StandardUncertainty"}]]*)uncG=QuantityMagnitude[UnitConvert[Entity["PhysicalConstant","GravitationalConstant"][{"Value"}]]] (*uncCMBRTempNow=Around@@Entity["PhysicalConstant","CosmicMicrowaveBackgroundTemperature"][{"Value","StandardUncertainty"}]+Quantity[1.8,"KelvinsDifference"]*) uncCMBRTempNow=QuantityMagnitude[UnitConvert[Entity["PhysicalConstant","CosmicMicrowaveBackgroundTemperature"][{"Value"}]]]

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{6.674×

-11

10

}

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{2.73}

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alephRate[t_]=4PicSin[2Pit/(2PiGmUAtStart/c^3)]

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4cπSin

8cπt

5

c

4

CMBRTempAtStart

eGsigma



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(*H=alephRate[UnitConvert[Quantity[10^10,"Year"]]]*)H[t_]=alephRate[t]/(ct)LightRadius[t_]=c/H[t]

Out[]=

4πSin

8cπt

5

c

4

CMBRTempAtStart

eGsigma



t

Out[]=

ctCsc

8cπt

5

c

4

CMBRTempAtStart

eGsigma



4π