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INTRODUCTION TO THE THEORY OF THE EARLY UNIVERSE Hot Big Bang Theory
Second Edition INTRODUCTION TO THE THEORY OF THE EARLY UNIVERSE Hot Big Bang Theory Second Edition VALERY A RUBAKOV Russian Academy of Sciences, Russia & Moscow State University, Russia
DMITRY S GORBUNOV Russian Academy of Sciences, Russia
Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Names: Gorbunov, D. S. (Dmitriĭ Sergeevich) | Rubakov, V. A.
Title: Introduction to the theory of the early universe : hot big bang theory / Valery A. Rubakov (Russian Academy of Sciences, Russia & Moscow State University, Russia), Dmitry S. Gorbunov (Russian Academy of Sciences, Russia).
Other titles: Vvedenie v teoriiu ranneĭ Vselennoĭ. English | Hot big bang theory
Description: 2nd edition. | New Jersey : World Scientific, 2017. | Includes bibliographical references and index.
Identifiers: LCCN 2017014302| ISBN 9789813209879 (hardcover : alk. paper) | ISBN 9813209879 (hardcover : alk. paper) | ISBN 9789813209886 (pbk : alk. paper) | ISBN 9813209887 (pbk : alk. paper)
Subjects: LCSH: Big bang theory. | Expanding universe.
Classification: LCC QB991.B54 G6713 2017 | DDC 523.1/8--dc23
LC record available at https://lccn.loc.gov/2017014302
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2018 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.
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ToOlesyaandElvira Prefacetothe2ndEdition Particlephysicsandcosmologyenjoyedrapiddevelopmentbetweenthefirstand thesecondeditionsofthisbook.Experimentalresearchresultedinawealthof newdata,whichforcedustocorrectandinsomeplacesrewritechaptersondark matter,phasetransitionsandbaryonasymmetryoftheUniverse.Wehavealsomade substantialrevisionsinotherchapters.Thenumericalvaluesofparticlephysicsand cosmologicalparametershavebeenupdatedinaccordancetocontemporarydata.
Besidesthat,wehavecorrectednumerousmisprintsanddrawbacksthatexisted inthefirstedition.Weareindebtedtoournumerouscolleaguesfortheirinput,and alsostudentsattheDepartmentofParticlePhysicsandCosmology,whichhasbeen recentlycreatedatthePhysicsFacultyoftheLomonosovMoscowStateUniversity (http://ppc.inr.ac.ru).
Prefacetothe1stEdition Itisclearbynowthatthereisadeepinterconnectionbetween cosmologyandparticlephysics,andbetweenmacro-andmicro-worlds.Thisbookiswrittenprecisely fromthisperspective.Wepresentheretheresultsonthehomogeneousandisotropic Universeatthehotstageofitsevolutionandatsubsequentstages.ThispartofcosmologyisoftendubbedastheHotBigBangtheory.Intheaccompanyingbookwe studythetheoryofcosmologicalperturbations(inhomogeneitiesintheUniverse), inflationarytheoryandthetheoryofpost-inflationaryreheating.
Thisbookgrewfromthelecturecoursewhichhadbeentaughtforanumber ofyearsattheDepartmentofQuantumStatisticsandFieldTheoryofthePhysics FacultyoftheLomonosovMoscowStateUniversity.Thiscourseisaimedatundergraduatestudentsspecializingintheoreticalphysics.Wedecided,however,toadda numberofmoreadvancedChaptersandSectionswhichwemarkbyasterisks.The reasonisthatthereareproblemsincosmology(natureofdarkmatteranddark energy,mechanismofthematter-antimatterasymmetrygeneration,etc.)which havenotfoundtheircompellingsolutionsyet.Mostoftheadditionalmaterialdeals withhypothesesontheseproblemsthatatthemomentcompetewitheachother.
Knowledgeofmaterialtaughtingeneralphysicscoursesisinprinciplesufficient forreadingthemainChaptersofthisbook.So,themainChaptersmustbeunderstandablebyundergraduatestudents.ThenecessarymaterialonGeneralRelativity andparticlephysicsiscollectedintheApp endiceswhich,ofcourse,donotpretend togivecomprehensiveaccountoftheseareasofphysics.Ontheotherhand,some partslabeledbyasterisksmakeuseofthemethodsofclassicalandquantumfield theoryaswellasnonequilibriumstatisticalmechanics,sobasicknowledgeofthese methodsisrequiredforreadingtheseparts.
Literatureoncosmologyishuge,andpresentingsystematicandcomprehensive bibliographywouldbewayoutofthescopeofthisbook.Toorientthereader,at theendofthisbookwegivealistofmonographsandreviewswheretheissueswe touchuponareconsideredindetail.Certainly,thislistisbynomeanscomplete.We occasionallyrefertooriginalliterature, especiallyinthoseplaceswherewepresent concreteresultswithoutdetailedderivation.
x Preface
Bothobservationalcosmologyandexperimentalparticlephysicsdevelopvery fast.Observationalandexperimentaldatawequote,theresultsoftheircompilations andfits(valuesofthecosmologicalparameters,limitsonmassesandcouplingsof hypotheticalparticles,etc.)willmostprobablygetmorepreciseevenbeforethis bookispublished.Thisdrawbackcanbecorrected,e.g.,byusingtheregularly updatedmaterialofParticleDataGroupathttp://pdg.lbl.gov/.
WewouldliketothankourcolleaguesfromtheInstituteforNuclearResearchof theRussianAcademyofSciences—F.L.Bezrukov,S.V.Demidov,V.A.Kuzmin, D.G.Levkov,M.V.Libanov,E.Y.Nugaev,G.I.Rubtsov,D.V.Semikoz, P.G.Tinyakov,I.I.TkachevandS.V.Troitskyforparticipationinthepreparationofthelecturecourseandnumeroushelpfuldiscussionsandcomments.Our specialthanksaretoS.L.Dubovskywhoparticipatedinwritingthisbookatan earlystage.WearedeeplyindebtedtoV.S. Berezinsky,A.Boyarsky,A.D.Dolgov, D.I.Kazakov,S.Y.Khlebnikov,V.F.Mukhanov,I.D.Novikov,K.A.Postnov, M.V.Sazhin,M.E.Shaposhnikov,A.Y.Smirnov,A.A.Starobinsky,R.A.Sunyaev,A.N.Tavkhelidze,O.V.Verkhodanov,A.Vilenkin,M.B.Voloshinand M.I.Vysotskyformanyusefulcommentsandcriticism.
Prefacetothe2ndEdition vii
Prefacetothe1stEdition ix
1.Cosmology:APreview1
1.1.Units...................................1
1.2.TheUniverseToday..........................3
1.2.1.Homogeneityandisotropy..................4
1.2.2.Expansion...........................4
1.2.3.AgeoftheUniverseandsizeofitsobservablepart.....9
1.2.4.Spatialflatness.........................10
1.2.5.“Warm”Universe.......................10
1.3.EnergyBalanceinthePresentUniverse...............14
1.4.FutureoftheUniverse.........................19
1.5.UniverseinthePast..........................21
1.5.1.Recombination.........................21
1.5.2.BigBangNucleosynthesis(BBN)..............22
1.5.3.Neutrinodecoupling......................23
1.5.4.Cosmologicalphasetransitions................24
1.5.5.Generationofbaryonasymmetry..............25
1.5.6.Generationofdarkmatter..................26
1.6.StructureFormationintheUniverse.................26
1.7.BeforetheHotEpoch.........................28
1.7.1.Argumentfromobservations.................28
1.7.2.DrawbacksoftheHotBigBangtheory...........30
1.8.InflationaryTheory...........................31
1.8.1.Alternativestoinflation....................32
2.HomogeneousIsotropicUniverse35
2.1.HomogeneousIsotropicSpaces....................35
2.2.Friedmann–Lemaˆıtre–Robertson–WalkerMetric...........38
2.3.Redshift.HubbleLaw.........................40
2.4.SlowingDownofRelativeMotion...................44
2.5.GasesofFreeParticlesinExpandingUniverse............47
3.DynamicsofCosmologicalExpansion51
3.1.FriedmannEquation..........................51
3.2.SampleCosmologicalSolutions.AgeoftheUniverse.Cosmological Horizon..................................55
3.2.1.Non-relativisticmatter(“dust”)...............56
3.2.2.Relativisticmatter(“radiation”)...............58
3.2.3.Vacuum.............................60
3.2.4.Generalbarotropicequationofstate p = wρ ........63
3.3.SolutionswithRecollapse.......................64
4.ΛCDM:CosmologicalModelwithDarkMatterandDarkEnergy67
4.1.CompositionofthePresentUniverse.................67
4.2.GeneralPropertiesofCosmologicalEvolution............70
4.3.TransitionfromDecelerationtoAcceleration............71
4.4.TransitionfromRadiationDominationtoMatterDomination...72
4.5.PresentAgeoftheUniverseandHorizonSize............76
4.6.Brightness–RedshiftRelationforDistantStandardCandles....79
4.7.AngularSizesofDistantObjects...................88
4.8. ∗ Quintessence..............................92
4.8.1.EvolutionofscalarfieldinexpandingUniverse.......92
4.8.2.Acceleratedcosmologicalexpansionduetoscalarfield...96
4.8.3.Trackerfield..........................98
5.ThermodynamicsinExpandingUniverse101
5.1.DistributionFunctionsforBosonsandFermions...........101
5.2.EntropyinExpandingUniverse.Baryon-to-PhotonRatio.....109
5.3. ∗ ModelswithIntermediateMatterDominatedStage:Entropy Generation...............................113
5.4. ∗ InequilibriumProcesses........................118
6.Recombination123
6.1.RecombinationTemperatureinEquilibriumApproximation....123
6.2.PhotonLastScatteringinRealUniverse...............128
6.3. ∗ KineticEquilibrium..........................138
6.3.1.Energytransferfromphotonstoelectrons.........139
6.3.2.Coulombscattering:thermalequilibriumofelectron-proton component...........................142
6.3.3.Thermal(in)equilibriumofphotons.............144
6.4.HorizonatRecombinationanditsPresentAngularSize. SpatialFlatnessoftheUniverse....................152
7.RelicNeutrinos159
7.1.NeutrinoFreeze-OutTemperature..................159
7.2.EffectiveNeutrinoTemperature.CosmologicalBound onNeutrinoMass............................161
7.3.SterileNeutrinosasDarkMatterCandidates............165
8.BigBangNucleosynthesis179
8.1.NeutronFreeze-Out.Neutron-ProtonRatio.............179
8.2.BeginningofNucleosynthesis.DirectionofNuclearReactions. Primordial 4 He.............................184
8.3.KineticsofNucleosynthesis......................190
8.3.1.Neutronburning, p + n → D + γ ...............191
8.3.2.Deuteriumburning......................192
8.3.3. ∗ Primordial 3 Heand 3 H...................199
8.3.4. ∗ Productionandburningoftheheaviestelements inprimordialplasma.....................202
8.4.ComparisonofTheorywithObservations..............205
9.DarkMatter209
9.1.Cold,HotandWarmDarkMatter..................210
9.2.Freeze-OutofHeavyRelic.......................213
9.3.WeaklyInteractingMassiveParticles,WIMPs............223
9.4. ∗ OtherApplicationsoftheResultsofSection9.2..........230
9.4.1.Residualbaryondensityinbaryon-symmetric Universe............................230
9.4.2.Heavyneutrino........................231
9.5.DarkMatterCandidatesinParticlePhysics.............232
9.6. ∗ StableParticlesinSupersymmetricModels.............233
9.6.1.Neutralino...........................235
9.6.2.Sneutrino............................244
9.6.3.Gravitino............................244
9.7.AxionsandOtherLong-livedParticles................255
9.8.OtherCandidates............................270
9.8.1. ∗ Superheavyrelicparticles..................271
9.8.2.Exotica.............................272
10.PhaseTransitionsintheEarlyUniverse273
10.1.OrderofPhaseTransition.......................275
10.2.EffectivePotentialinOne-LoopApproximation...........287
10.3.InfraredProblem............................300
10.4. ∗ ElectroweakVaccumatZeroTemperature.............305
11.GenerationofBaryonAsymmetry313
11.1.NecessaryConditionsforBaryogenesis................314
11.2.BaryonandLeptonNumberViolationinParticleInteractions...317
11.2.1.Electroweakmechanism....................318
11.2.2.BaryonnumberviolationinGrandUnifiedTheories....324
11.2.3.ViolationofleptonnumbersandMajorana massesofneutrino.......................333
11.3.AsymmetryGenerationinParticleDecays..............335
11.4.BaryonAsymmetryandNeutrinoMasses:Leptogenesis......345
11.5.ElectroweakBaryogenesis.......................352
11.5.1.Departurefromthermalequilibrium.............352
11.5.2. ∗ Thickwallbaryogenesis...................354
11.5.3. ∗ Thinwallcase........................360
11.5.4.Electroweakbaryogenesis, CP-violationand neutronEDM.........................366
11.6. ∗ Affleck–DineMechanism.......................367
11.6.1.Scalarfieldscarryingbaryonnumber............367
11.6.2.Asymmetrygeneration....................369
11.7.ConcludingRemarks..........................375
12.TopologicalDefectsandSolitonsintheUniverse377
12.1.ProductionofTopologicalDefectsintheEarlyUniverse......378
12.2.’tHooft–PolyakovMonopoles.....................379
12.2.1.Magneticmonopolesingaugetheories............379
12.2.2.Kibblemechanism.......................383
12.2.3.Residualabundance:themonopoleproblem........384
12.3.CosmicStrings.............................386
12.3.1.Stringsolutions........................386
12.3.2.Gasofcosmicstrings.....................394
12.3.3.Deficitangle..........................396
12.3.4.StringsintheUniverse....................401
12.4.DomainWalls..............................405
12.5. ∗ Textures................................408
12.6. ∗ HybridTopologicalDefects......................411
12.7. ∗ Non-topologicalSolitons: Q-balls..................412
12.7.1.Two-fieldmodel........................412
12.7.2.Modelswithflatdirections..................418
AppendixAElementsofGeneralRelativity443
A.1.TensorsinCurvedSpace–Time....................443
A.2.CovariantDerivative..........................447
A.3.RiemannTensor............................452
A.4.GravitationalFieldEquations.....................456
A.5.ConformallyRelatedMetrics.....................461
A.6.InteractionofMatterwithGravitationalField. Energy–MomentumTensor......................464
A.7.ParticleMotioninGravitationalField................470
A.8.NewtonianLimitinGeneralRelativity................473
A.9.LinearizedEinsteinEquationsaboutMinkowski Background...............................476
A.10.MacroscopicEnergy–MomentumTensor...............477
A.11.NotationsandConventions......................478
AppendixBStandardModelofParticlePhysics481
B.1.FieldContentandLagrangian.....................481
B.2.GlobalSymmetries...........................492
B.3.C-,P-,T-Transformations.......................494
B.4.QuarkMixing..............................495
B.5.EffectiveFermiTheory.........................501
B.6.PeculiaritiesofStrongInteractions..................502
B.7.TheEffectiveNumberofDegreesofFreedom intheStandardModel.........................503
AppendixCNeutrinoOscillations505
C.1.OscillationsandMixing........................505
C.1.1.Vacuumoscillations......................505
C.1.2.Three-neutrinooscillationsinspecialcases.........509
C.1.3.Mikheev–Smirnov–Wolfensteineffect............512
C.2.ExperimentalDiscoveries.......................514
C.2.1.SolarneutrinosandKamLAND...............514
C.2.2.Atmosphericneutrinos,K2KandMINOS..........522
C.2.3.Acceleratorandreactorneutrinos: |Ue3 | ...........524
C.3.OscillationParameters.........................525
C.4.DiracandMajoranaMasses.SterileNeutrinos............528
C.5.SearchforNeutrinoMasses......................535
AppendixDQuantumFieldTheoryatFiniteTemperature537
D.1.BosonicFields:EuclideanTimeandPeriodicBoundary Conditions................................537
D.2.FermionicFields:AntiperiodicBoundaryConditions........541
D.3.PerturbationTheory..........................545
D.4.One-LoopEffectivePotential.....................548
D.5.DebyeScreening............................553
Chapter1 Cosmology:APreview ThepurposeofthisChapteristogiveapreviewofthefieldwhichweconsiderin thisandtheaccompanyingbook.Thepresentationhereisatthequalitativelevel, andisbynomeanscomplete.Ourpurposeistoshowtheplaceofoneoranother topicwithintheentireareaofcosmology.
Beforeproceeding,letusintroduceunitsandconventionsthatweusethroughout thisbook.
1.1.Units Wemostlyusethe“natural”systemofunitsinwhichthePlanckconstant,speed oflightandtheBoltzmannconstantareequalto1,
= c = kB =1.
Thenthemass M ,energy E andtemperature T havethesamedimension(since [E ]=[mc2 ],[E ]=[kB T ]).Aconvenientunitofmassandenergyis1eVor1GeV= 109 eV;theprotonmassisthenequalto mp =0 938GeV,and1Kisapproximately 10 13 GeV.Time t andlength l inthenaturalsystemhavedimension M 1 (since [E ]=[ ω ],[ω ]=[t 1 ]and[l ]=[ct]),with1GeV 1 ∼ 10 14 cmand1GeV 1 ∼ 10 24 s.WegivethecoefficientsrelatingvariousunitsinTables1.1and1.2.
Problem1.1. ChecktherelationsgiveninTables 1.1 and 1.2. Whatare 1 Volt (V), 1 Gauss (G), 1 Hertz (Hz) and 1 Angstr¨om ( ˚ A) innaturalsystemofunits?
Innaturalsystemofunits,theNewtongravityconstant G hasdimension M 2 . Thisfollowsfromtheformulaforthegravitationalpotentialenergy V = G m1 m2 r , since[V ]= M ,[r 1 ]= M .ItisconvenienttointroducethePlanckmass MPl in thefollowingway,
Numerically
Table1.1.ConversionofnaturalunitsintoCGSunits.
Energy1GeV=1.6 10 3 erg
Mass1GeV=1.8 10 24 g
Temperature1GeV=1.2 1013 K
Length1GeV 1 =2.0 10 14 cm Time1GeV 1 =6 6 · 10 25 s Particlenumberdensity1GeV 3 =1 3 ·
Table1.2.ConversionofCGSunitsintonaturalunits.
Energy1erg=6.2 102 GeV
Mass1g=5 6 · 1023 GeV
Temperature1K=8 6 · 10 14 GeV
Length1cm=5 1 · 1013 GeV 1
Time1s=1 5 · 1024 GeV 1
Particlenumberdensity1cm 3 =7 7 10 42 GeV3 Energydensity1erg cm 3
andthePlancklength,timeandmassare
Thegravitationalinteractionsareweakpreciselybecause MPl islarge.
Problem1.2. Checktherelations (1.1) and (1.2).
Problem1.3. WhatistheratioofgravitationalinteractionenergytoCoulomb energyfortwoprotons?
Thetraditionalunitoflengthincosmologyismegaparsec, 1Mpc=3.1 · 1024 cm.
Letusalsointroduceaconventionwhichweuseinthisbook.Thesubscript0 denotespresentvaluesofquantitieswhichcandependontime.Asanexample, ρ(t) denotestheenergydensityintheUniverseasafunctionoftime,while ρ0 ≡ ρ(t0 ) isalwaysitspresentvalue.
Thereareseveralunitsoflengththatareusedinastronomy,dependingonsizesofobjects andlengthscalesconsidered.Besidesthemetricsystem,inuseareastronomicalunit(a.u.),
whichistheaveragedistancefromtheEarthtotheSun,
1a.u.=1 5 1013 cm; lightyear,thedistancethataphotontravelsinoneyear,
1year=3.16 · 107 s, 1lightyear=3 · 1010 cm s · 3.16 · 107 s=0.95 · 1018 cm; andparsec(pc)—distancefromwhichanobjectofsize1a.u.isseenatangle1arcsecond,
1pc=2.1 105 a.u.=3.3lightyear=3.1 1018 cm.
ToillustratethehierarchyofspatialscalesintheUniverse,letusgivethedistancesto variousobjectsexpressedintheaboveunits.
10a.u.istheaveragedistancetoSaturn,30a.u.isthesameforPluto,100a.u.isthe estimateofmaximumdistancewhichcanbereachedbysolarwind(particlesemittedby theSun).100a.u.isalsotheestimateofthemaximumdistancetocosmicprobes(Pioneer 10,Voyager1,Voyager2).FurtheroutistheOortcloud,thesourceofthemostdistant comets,whichisatthedistanceof104 –105 a.u. ∼0.1–1pc.
Theneareststars—ProximaandAlphaCentauri—areat1.3pcfromtheSun. ThedistancetoArcturusandCapellaismorethan10pc,thedistancestoCanopusand Betelgeuseareabout100pcand200pc,respectively;CrabNebula—theremnantof supernovaseenbynakedeye—is2kpcawayfromus.
Thenextpointonthescaleofdistancesis8kpc.ThisisthedistancefromtheSunto thecenterofourGalaxy.OurGalaxyisofspiraltype,thediameterofitsdiscisabout 30kpcandthethicknessofthediscisabout250pc.Thedistancetothenearestdwarf galaxies,satellitesofourGalaxy,isabout30kpc.Fourteenofthesesatellitesareknown; thelargestofthem—LargeandSmallMagellanicClouds—are50kpcaway.Searchfor new,dimmersatellitedwarfsisunderway;wenoteinthisregardthatonlyeightofMilky Waysatelliteswereknownby1994.
Themassdensityoftheusualmatterinusual(notdwarf)galaxiesisabout105 higher thantheaverageovertheUniverse.
Thenearestusualgalaxy–thespiralgalaxyM31inAndromedaconstellation—is 800kpcawayfromtheMilkyWay.Despitethelargedistance,itoccupiesasizeablearea onthecelestialsphere:itsangularsizeislargerthanthatoftheMoon!Anothernearby galaxyisintheTriangulumconstellation.OurGalaxytogetherwiththeAndromedaand Triangulumgalaxies,theirsatellitesandother35smallergalaxiesconstitutetheLocal Group,thegravitationallyboundobjectconsistingofmorethan50galaxies.
Thenextscaleinthisladderisthesizeofclustersofgalaxies,whichis1–3Mpc. Richclusterscontainthousandsofgalaxies.Themassdensityinclustersexceedsthe averagedensityovertheUniversebyafactorofahundredandevensometimesathousand. Thedistancetothecenterofthenearestcluster,whichisintheVirgoconstellation,is about15Mpc.Itsangularsizeisabout5degrees.Clustersofgalaxiesarethelargest gravitationallyboundsystemsintheUniverse.
1.2.TheUniverseToday
Webeginourpreviewwiththebriefdiscussionofthepropertiesofthepresent Universe(moreprecisely,ofitsobservablepart).
1.2.1. Homogeneityandisotropy TheUniverseishomogeneousandisotropicatlargespatialscales.Thesizesofthe largeststructuresintheUniverse—superclustersofgalaxiesandgiganticvoids— reach1 tensofmegaparsec.Atscalesexceeding200Mpc,allpartsoftheUniverse lookthesame(homogeneity).Likewise,therearenospecialdirectionsintheUniverse(isotropy).Thesefactsarewellestablishedbydeepgalaxysurveyswhich collecteddataonmorethanamilliongalaxies.
About20superclustersareknownbynow.TheLocalGroupbelongstoasuperclusterwith thecenterinthedirectionofVirgoconstellation.Thesizeofthissuperclusterisabout 30Mpc,andbesidestheVirgoclusterandLocalGroupitcontainsaboutahundredgroups andclustersofgalaxies.Superclustersareratherloose:thedensityofgalaxiesinthemis onlytwicehigherthantheaverageintheUniverse.ThenearesttoVirgoisthesupercluster intheHydraandCentaurusconstellations;itsdistancetotheVirgosuperclusterisabout halfahundredmegaparsec.
Thelargestcatalogofgalaxiesandquasarsuptodateisthefreelyavailablecatalog ofSDSS[2](SloanDigitalSkySurvey).Thiscatalogistheresultoftheanalysisofthe datacollectedduringalmost8yearsofoperationofadedicatedtelescope,2.5metersin diameter,whichiscapableofmeasuringsimultaneouslyspectraof640astrophysicalobjects in5opticalbandpasses(photonwavelengths λ =3800 9200 ˚ A).Thecatalogincludes millionscelestialobjects.Mostofthedatahasbeenprocessedbynow;measurementsof spectraofmorethan1.8milliongalaxiesandmorethan300thousandquasarsresulted inthecreationofa3-dimensionalmapcoveringalargepartofthevisibleUniverse.Its areaexceedsaquarterofthesky.Thereareothercatalogswhichcoversmallerpartsof theUniverse(see,e.g.,Ref.[3]forthenext-to-largestcatalogbasedonthe2dFGalaxy RedshiftSurvey).
TheearlySDSSresultsareillustratedinFig.13.1incolorpages,wherepositions of40thousandgalaxiesand4thousandquasarsarepresented.Thecoveredpartofthe celestialspherehastheareaof500squareddegrees.Recognizableareclustersofgalaxies andvoids.IsotropyandhomogeneityoftheUniversearerestoredatspatialscalesoforder 100Mpcandlarger.Colorofeachdotreferstothetypeoftheastrophysicalobject. Thedominationofonetypeoverothersis,generallyspeaking,causedbypeculiaritiesof structureformationandevolution.Thus,whatoneobservesispartiallypicturedin time ratherthaninspace.
Indeed,fromthedistanceof1.5Gpc,wherethedistributionofbrightredellipticalgalaxies (reddotsinFig.13.1)isatmaximum,lighttravelstotheEarthforabout5billionyears.At thatepoch,theUniversewasdifferent(forinstance,therewasnoSolarsystemyet).One morereasonforchoosingobjectsofacertaintypeisthefinitesensitivityofatelescope.In particular,onlyhighlyluminousobjectscanbedetectedatthelargestdistances,whilethe highest-luminosity,continuouslyshiningobjectsintheUniversearequasars.
1.2.2. Expansion TheUniverseexpands:thedistancesbetweengalaxiesincrease.2 Looselyspeaking, thespace,beingalwayshomogeneousandisotropic,stretchesout.
1 Thisisasomewhatloosestatement:mostaccurateestimatesareobtainedfromthegalaxycorrelationfunction,whichfallsoffaspower-lawatlargeseparations.
2 Ofcourse,thisdoesnotapplytogalaxiesthataregravitationallyboundtoeachotherinclusters.
Todescribethisexpansion,oneintroducesthescalefactor a(t)whichgrowsin time.ThedistancebetweentwofarawayobjectsintheUniverseisproportionalto a(t)andthenumberdensityofparticlesdecreasesas a 3 (t).Therateofthecosmologicalexpansion,i.e.,therelativegrowthofdistancesinunittime,ischaracterized bytheHubbleparameter,
Hereafter,thedotdenotesthederivativewithrespecttothecosmictime t.The Hubbleparameterdependsontime;itspresentvalue,accordingtoourconvention, isdenotedby H0
TheexpansionoftheUniversegivesrisealsotothegrowthofthewavelength ofaphotonemittedindistantpast.Likeotherdistances,thephotonwavelength increasesproportionallyto a(t);thephotonexperiencesredshift.Thisredshift z is determinedbytheratioofphotonwavelengthsatabsorptionandemission,
Clearly,thisratiodependsonthemomentoftheemission(assumingthatthephoton isdetectedtodayontheEarth),i.e.,onthedistancetothesource.Redshiftis adirectlymeasurablequantity:thewavelengthatemissionisdeterminedbythe physicsoftheemissionprocess(e.g.,byenergiesofanexcitedandthegroundstate ofanatom),while λab isthemeasuredwavelength.Thus,oneidentifiesthesystem ofemission(orabsorption)linesanddetermineshowmuchtheyareshiftedtothe redspectralregion,andinthiswayonemeasurestheredshift.
Inreality,theidentificationoflinesmakesuseofpatternswhicharecharacteristic ofparticularobjects,seeFig.1.1,Ref.[5].Ifthespectrumcontainsabsorptiondips, asinFig.1.1,thentheobjectwhoseredshiftisbeingmeasuredisbetweenthe emitterandobserver.3 Thepeaksinthespectrum—emissionlines—meanthat theobjectisanemitteritself.
For z 1,thedistancetothesource r andtheredshiftarerelatedbythe Hubblelaw
Atlarger z theredshift-distancerelationismorecomplicated,whichwewilldiscuss indetailinthisbook.
Thedeterminationofabsolutedistancestofarawaysourcesisacomplicated problem.Oneofthemethodsistomeasurethephotonfluxfromasourcewhose absoluteluminosityisassumedtobeknown.Thesesourcesaresometimescalled standardcandles.
3 Photonsofdefinitewavelengthsexperienceresonantabsorptionbyatomsandions,withsubsequentisotropicemission.Thisleadstothelossofphotonsreachingtheobserver.
Fig.1.1.Absorptionlinesofdistantgalaxies[5].Theupperpanelshowsthemeasurementofthe differentialenergyfluxfromafarawaygalaxy(z =2.0841).Theverticallinesshowthepositionof atomiclineswhoseidentificationhasbeenusedtomeasureredshift.Thespectraofnearergalaxies havemorepronounceddips.Theplotwiththespectraofthesegalaxies,shiftedtocomovingframe, isshowninthelowerpanel.
Systematicuncertaintiesinthedeterminationof H0 werenotparticularlywell knownuntilrecentlyandtheyarestillfairlylarge.Wenoteinthisregardthatthe valueoftheHubbleconstantasdeterminedbyHubblein1929was550km/(s · Mpc). Thecontemporarydeterminationsgive[1]
Problem1.4. RelatethedimensionlessredshifttodistanceexpressedinMpc.
LetuscommentonthetraditionalunitfortheHubbleparameterusedin(1.5). AnaiveinterpretationoftheHubblelaw(1.4)isthattheredshiftiscausedby
theradialmotionofgalaxiesfromtheEarthwithvelocitiesproportionaltothe distances,
(1.6)
Thentheredshift(1.4)isinterpretedasthelongitudinalDopplereffect(at v c, i.e., v 1innaturalunits,theDopplershiftequalsto z = v ).Accordingtothis interpretation,thedimensionoftheHubbleparameter H0 is[velocity/distance].We stress,however,thattheinterpretationofthecosmologicalredshiftintermsofthe Dopplereffectisunnecessary,andofteninadequate.Therightwayistousethe relation(1.4)asitis.
Problem1.5. ConsiderasystemofmanyparticlesinNewtonianmechanics.Show thatitisspatiallyhomogeneousandisotropicifandonlyifthedensityoftheparticlesisconstantoverspace,andtherelativevelocityofeachpairofparticles i and j isrelatedtothedistancebetweenthembythe “Hubblelaw”
ij = H0 rij ,
where H0 isindependentofspatialcoordinates.Hereafter,boldfacelettersdenote three-vectors, v =(v1 ,v2 ,v3 )
Thequantity H0 isusuallyparameterizedinthefollowingway,
where h isadimensionlessparameteroforderone(see(1.5)), h =0.673 ± 0.012.
Weusethevalue h =0.7innumericalestimatesthroughoutthisbook.
OnetypeofobjectsusedformeasuringtheHubbleparameterareCepheids,starsof variablebrightnesswhosevariabilityisrelatedtoabsoluteluminosityinaknownway. ThisrelationshipismeasuredbyobservingCepheidsincompactsystemslikeMagellanic Clouds.SinceCepheidsinoneandthesamesystemare,togoodapproximation,atthe samedistancefromus,theratiooftheirvisiblebrightnesstoabsoluteluminosityisthe sameforeverystar.TheperiodsofCepheidpulsationsrangefromadaytotensofdays, andduringthisperiodthebrightnessvarieswithinanorderofmagnitude.Theresults ofobservationsshowthatthereisindeedawell-definedrelationbetweentheperiodand luminosity:thelongertheperiod,thebrighterthestar.Hence,Cepheidsserveasstandard candles.
Cepheidsaregiantsandsuper-giants,sotheyarevisibleatlargedistancesfromour Galaxy.Bymeasuringtheirspectra,onefindsredshiftofeachofthem,andbymeasuringtheperiodofpulsationsoneobtainstheabsoluteluminosityandhencethedistance. Usingthesedata,onemeasurestheHubbleconstantin(1.4).Figure1.2showstheHubble diagram—redshift-distancerelation—obtainedinthisway[10].
BesidesCepheids,thereareotherobjectsusedasstandardcandles.Theseinclude,in particular,supernovaeoftypeIa.ThedeterminationoftheHubbleparameterfromthe observationsofremotestandardcandlesisshowninFig.1.3.
H0 = h 100 km s · Mpc , (1.7)
Fig.1.2.HubblediagramforCepheids[10].ThesolidlineshowstheHubblelawwiththeHubble constant H0 =75km/(s · Mpc),asdeterminedfromtheseobservations.Thedashedlinesshowthe uncertaintyinthedeterminationoftheHubbleparameter.
Fig.1.3.HubblediagramforremotestandardcandlesincludingsupernovaeoftypeIa[10].The solidlineshowstheHubblelawwiththevalueoftheHubbleparameter H0 =72km/(s Mpc)as determinedfromthesedata.DashedlinescorrespondtoexperimentaluncertaintyintheHubble parameter.
1.2.3. AgeoftheUniverseandsizeofitsobservablepart
TheHubbleparameterinfacthasdimension[t 1 ],sothepresentUniverseischaracterizedbythetimescale
andthescaleofdistances
Crudelyspeaking,alldistancesintheUniversewillbecometwicelargerinabout10 billionyears;galaxiesatdistanceoforder3Gpcfromusmoveawaywithvelocities comparabletothespeedoflight.Wewillseethatthetimescale H 1 0 givesthe orderofmagnitudeestimatefortheageoftheUniverse,andthedistancescale H 1 0 isroughlythesizeofitsobservablepart.Wewilldiscussthenotionsoftheage andsizeoftheobservablepartinthecourseofpresentation,andherewepointout thatboldextrapolationofthecosmologicalevolutionbackintime(madeaccording totheequationsofclassicalGeneralRelativity)leadstothenotionoftheBig Bang,themomentatwhichtheclassicalevolutionbegins.Thentheageofthe UniverseisthetimepassedfromtheBigBang,andthesizeoftheobservablepart (horizonsize)isthedistancetravelledbysignalsemittedattheBigBangand movingatthespeedoflight(moreaccurateestimateofthehorizonsizeis15Mpc). WenoteinpassingthattheactualsizeofourUniverseislarger,andmostprobably muchlargerthanthehorizonsize;thespatialsizeoftheUniversemaybeinfinite inGeneralRelativity.
Irrespectiveofthecosmologicaldata,thereexistobservationallowerboundsontheageof theUniverse t0 .Variousindependentmethodsgivesimilarboundsatthelevel t0 13billionyears=1 3 1010 yrs (1.10)
Oneofthesemethodsmakesuseofthedistributionofluminositiesofwhitedwarfs.White dwarfsarecompactstarsofhighdensity,whosemassesaresimilartothesolarmass.They slowlycooldownandgetdimmer.Therearewhitedwarfsofvariousluminositiesinthe Galaxy,butthenumberofthemsharplydropsoffbelowacertainluminosity.Thismeans thatthereisamaximumageofwhitedwarfs,which,ofcourse,shouldbesmallerthanthe ageoftheUniverse.Thismaximumageisfoundfromtheenergybalanceofawhitedwarf (see,e.g.,Ref.[12]).Inthiswaytheboundlike(1.10)isobtained.
OthermethodsincludethestudiesoftheradioactiveelementabundancesintheEarth core,inmeteorites(see,e.g.,Ref.[13]),andinthemetal-poor 4 stars(e.g.,Ref.[14]), thecomparison(e.g.,Ref.[15])ofthestellarevolutioncurveformain-sequencestars
4 Theterm“metals”inastrophysicsisusedforallelementsheavierthanhelium.
ontheHerzsprung–Russeldiagram(luminosity-colororbrightness-temperature)withthe abundanceoftheoldeststarsinmetal-poorglobularclusters,5 theanalysisofrelaxation processesinstellarclusters,measurementoftheabundanceofhotgasinclustersof galaxies,etc.
1.2.4. Spatialflatness HomogeneityandisotropyoftheUniversedonotimply,generallyspeaking,thatat eachmomentoftimethe3-dimensionalspaceisEuclidean,i.e.,thattheUniversehas zerospatialcurvature.Besidesthe3-plane(3-dimensionalEuclideanspace),there aretwohomogeneousandisotropicspaces,3-sphere(positivespatialcurvature)and 3-hyperboloid(negativecurvature).Afundamentalobservationalresultofrecent yearsisthefactthatthespatialcurvatureofourUniverseisverysmall,ifnotexactly zero.Our3-dimensionalspaceisthusEuclideantoaverygoodapproximation.We willrepeatedlygetbacktothisstatement,bothforquantifyingitandforexplaining whichobservationaldatasetboundsonthespatialcurvature.Weonlynotehere thatthemainsourceoftheseboundsisthestudyofthetemperatureanisotropyof theCosmicMicrowaveBackground(CMB),andthatatthequalitativelevel,these boundsmeanthattheradiusofspatialcurvatureismuchgreaterthanthesizeof theobservablepartoftheUniverse,i.e.,muchgreaterthan H 1 0
WenoteherethatCMBdataarealsoconsistentwiththetrivialspatialtopology.Ifour Universehadcompacttopology(e.g.,topologyof3-torus)anditssizewereoftheorder oftheHubblelength,CMBtemperatureanisotropywouldshowacertainregularpattern. Suchapatternisabsentinmeasuredanisotropy,seeRef.[11].
1.2.5. “Warm”Universe ThepresentUniverseisfilledwithCosmicMicrowaveBackground(CMB),gasof non-interactingphotons,whichwaspredictedbytheHotBigBangtheoryand discoveredin1964.ThenumberdensityofCMBphotonsisabout400percubic centimeter.Theenergydistributionofthesephotonshasthermal,Planckianspectrum.ThisisshowninFig.1.4[16].TheCMBtemperatureis[1]
Thetemperatureofphotonscomingfromdifferentdirectionsoncelestialsphereis thesameatthelevelofbetterthan10 4 (modulodipolecomponent,seebelow); thisisyetanotherevidenceforhomogeneityandisotropyoftheUniverse.
Still,thetemperaturedoesdependonthedirectioninthesky.Theangular anisotropyoftheCMBtemperaturehasbeenmeasured,asshowninFig.1.5[38] (seeFig.13.2oncolorpages).Itisoforder δT/T0 ∼ 10 4 10 5 .
5 Globularclustersarestructuresofsizesoforder30pcinsidegalaxies;theycancontainhundreds ofthousandandevenmillionsofstars.
Wavelength (cm)
(GHz)
Fig.1.4.CMBspectrum.ThecompilationofthedataismadeinRef.[16].Thesolidlineshows thePlanckian(blackbody)spectrum.
Fig.1.5.WMAPdata[38]:angularanisotropyofCMBtemperature,i.e.,variationofthetemperatureofphotonscomingfromdifferentdirectionsinthesky;seeFig.13.2forcolorversion.The averagetemperatureanddipolecomponentaresubtracted.Theobservedvariationoftemperature isatthelevelof
WewillrepeatedlycomebacktoCMBanisotropyandpolarization,since,on theonehand,theyencodealotofinformationaboutthepresentandearlyUniverse and,ontheotherhand,theycanbemeasuredwithhighprecision.
LetusnotethattheexistenceofCMBmeansthatthereisaspecialreference frameinourUniverse:thisistheframeinwhichthegasofphotonsisatrest. ThesolarsystemmoveswithrespecttothisframetowardsHydraconstellation. ThevelocityofthismotiondeterminesthedipolecomponentofthemeasuredCMB anisotropy[18],
Problem1.6. Makinguseofthevalueofthedipolecomponent, estimatethevelocityofmotionoftheSolarsystemwithrespecttoCMB.
Problem1.7. EstimatetheseasonalmodulationoftheCMBanisotropycausedby themotionoftheEartharoundtheSun.
ThepresentUniverseistransparenttotheCMBphotons:6 theirmeanfreepath wellexceedsthehorizonsize H 1 0 .ThiswasnotthecaseintheearlyUniverse,when photonsactivelyinteractedwithmatter.
Problem1.8.Greisen–Zatsepin–Kuzmineffect [20, 21]. Interactionofphotonwithprotonatsufficientlyhighenergiesmayleadtotheabsorptionofphotonand creationof π -meson.Letthecross-sectionofthelatterprocessinthecenter-of-mass framebe (infact,thisisaprettyreasonableapproximationforthisproblem)
where √s isthetotalenergyofphotonandproton, mΔ =1200MeV (Δ isthe resonancemass), 1mb=10 27 cm2 .
FindthemeanfreepathofaprotoninthepresentUniversewithrespecttothis processasafunctionofprotonenergy.Atwhatdistancefromthesourcedoesproton lose 2/3 ofitsenergy?Ignoreallphotons (e.g., emittedbystars), exceptforCMB.
SincetheCMBtemperature T dependsonthedirection n ofcelestialsphere,itis convenienttoperformitsdecompositionoversphericalharmonics Ylm (n).Thelatter formthebasisoffunctionsonasphere,andthedecompositionistheclosestanalog oftheFourierdecomposition.Thetemperaturefluctuation δT inthedirection n is convenientlydefinedas
(n)= T (n) T0
anditsdecompositionis
δT (n)= l,m al,m Ylm (n), wherethecoefficients al,m obey a∗ l,m =( 1)m al, m ,sothattemperatureisreal.The multipoles l correspondtofluctuationsofcharacteristicangularsize π/l .Thecurrent measurementsarecapableofstudyingangularscalesrangingfromthelargestones tolessthan0 1◦ (l ∼ 1000,seeFig.1.6[39]).
6 ThisisnotcompletelytrueinsomeregionsoftheUniverse.Asanexample,photonsscatter offhotgas(T ∼ 10keV)inclustersofgalaxiesandgainsomeenergy.Thus,CMBiswarmerin thedirectionstowardsclusters.ThisiscalledtheSunyaev–Zeldovicheffect[19].Itissmallbutis measuredinobservations.
δT
δTdipole
Fig.1.6.CMBangularanisotropyasmeasuredbyPlanckexperiment[39].Thetheoreticalcurve isobtainedwithintheΛCDMmodel(seeChapter4);finitewidthofthiscurve(shadow)illustrates cosmicvariance,whichisduetothefactthatonlyone(our)Universeisobserved.
Theobservationaldataareconsistentwiththepropertythattemperaturefluctuations δT (n)areGaussianrandomfield,i.e.,thatthecoefficients al,m arestatisticallyindependentfordifferent l and m,
wherebracketsmeanaveragingoveranensembleofUniverseslikeours.Thecoefficients Clm donotdependon m inisotropicUniverse, Clm = Cl .Theydetermine thecorrelationoftemperaturefluctuationsindifferentdirections,
(n1 )δT (n2 )
where Pl aretheLegendrepolynomials,functionsoftheangle θ betweenthevectors n1 and n2 .Inparticular,thetemperaturefluctuationis
Thus,thequantity Dl ≡ l(l+1)Cl 2π determinesthecontributiontothefluctuationof adecimalintervalofmultipoles.ItisthisquantitythatisshowninFig.1.6. ItisimportantthatthemeasurementoftheCMBanisotropygivesnotjusta number,butalargesetofdata,thevaluesof Cl fordifferent l .Thissetisdetermined bynumerousparametersofthepresentandearlyUniverse,henceitsmeasurement providesalotofcosmologicalinformation.Additionalinformationcomesfromthe measurementofCMBpolarization.
1.3.EnergyBalanceinthePresentUniverse AdimensionalestimateoftheenergydensityintheUniversemaybeobtained inthefollowingway.Giventheenergydensity ρ0 ,thedensityof“gravitational charge”isoforder Gρ0 .SincethedynamicsoftheUniverseisgovernedbygravity, the“charge” Gρ0 mustsomehowberelatedtothepresentexpansionrate.The “charge”hasdimensionof M 2 ;thesamedimensionas H 2 0 .Thissuggeststhat ρ0 ∼ H 2 0 G 1 = M 2 Pl H 2 0 .Indeed,wewillseethatthepresentenergydensityina spatially flat Universeisgivenby
Withprecisionbetterthan1%,thisistheenergydensityinourUniversetoday.7 Numerically
Accordingtothedataofcosmologicalobservationswhichwewilldiscussindue course,thecontributionofbaryons(protons,nuclei)intothetotalpresentenergy densityis8 about4.6%,
Thecontributionofrelicneutrinosofalltypesisevensmaller;thecosmological boundis
wherethesumrunsoverthethreespeciesofneutrinos νe , νμ , ντ andanti-neutrinos νe ,¯ νμ ,¯ ντ .Weemphasizethatthereisstillno cosmologicalevidencefortheneutrinomass;itisratherlikelythattheneutrinocontributionisquiteabitsmaller thantheright-handsideofEq.(1.15).Otherknownstableparticlesgivenegligible contributiontothepresenttotalenergydensity.Thus,thedominatingmaterialin thepresentUniverseissomethingunknown.
This“somethingunknown”infactconsistsoftwofractions,oneofwhichis capableofclustering,andanotherisnot.Theformercomponentiscalled“dark matter”.Itscontributiontotheenergydensityisabout25%.
Wewilldiscusstheresults(BigBangNucleosynthesis,CMBanisotropy, structureformation)whichshowthatdarkmattercannotconsistofknownparticles. Mostprobablyitismadeofnewstableparticleswhichwerenon-relativisticinvery distantpastandremainnon-relativistictoday(cold,orpossiblywarmdarkmatter). ThisisoneofafewexperimentalevidencesforNewPhysicsbeyondtheStandard
7 This1%hastodowithobservationallyallowedeffectofspatialcurvature.
8 Notethatonly10%ofbaryonsareinstars.Mostofbaryonsareinhotgas.
Modelofparticlephysics.Directdetectionofdarkmatterparticlesisanextremely important,andyetunsolvedproblemofparticlephysics.
Accordingtocurrentviewpoint,therestofenergyinthepresentUniverse, about70%,ishomogeneouslyspreadoverspace.Thisisnotmatterconsistingof someunknownparticles,butratheraunconventionalformofenergyofvacuum type.Itiscalledbydifferentnames:darkenergy,vacuum-likematter,quintessence, cosmologicalconstant,Λ-term.Wewillusetheterm“darkenergy”andwilluse theterms“quintessence”and“cosmologicalconstant”fordarkenergywithspecific properties:inthecaseofcosmologicalconstanttheenergydensitydoesnotdepend ontime,whileforquintessenceweakdependence,instead,exists.
Itisnotexcludedthatobservationaldatawhicharequotedasshowingthe presenceofdarkenergycanbeexplainedinanalternativeway.Onepossibilityis thatgravitydeviatesfromGeneralRelativityatcosmologicaldistanceandtime scales.Thereistheoreticalactivityinthelatterdirectionindeed,butitisoutof thescopeofthisbooktodiscussitinanydetail.Wewillassumethroughoutthat gravitationalinteractionsaredescribedbyGeneralRelativity.
Wewillfurtherdiscussdarkenergyandobservationsleadingtothisnotionindue course.Herewementionthepropertythatunliketheenergy(mass)densityofnonrelativisticparticleswhichdecaysas a 3 (t)astheUniverseexpands,darkenergy densityeitherdoesnotdependontimeatall,ordependsontimeveryweakly.Hence, atsomestageofthecosmologicalevolutiondarkenergystartstodominate.The transitionfrommatterdominatedtodarkenergydominatedexpansionoccurredin ourUniverseat z 0 5.
Densityofbaryonsanddarkmatterinclustersofgalaxiesisdeterminedbyvariousmethods ofmeasurementofthegravitationalpotential,i.e.,totalmassdistribution.Asanexample, theleftpanelofFig.1.7(seeFig.13.4oncolorpages)showsmassdistributionina
Fig.1.7.ClusterofgalaxiesCL0024+1654[22];seeFig.13.4forcolorversion.
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The Project Gutenberg eBook of Woman free This ebook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this ebook or online at www.gutenberg.org. If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook.
Title: Woman free
Author: Ellis Ethelmer
Release date: August 8, 2022 [eBook #68715]
Language: English
Original publication: United Kingdom: Women's Emancipation Union, 1893
Credits: Richard Tonsing and the Online Distributed Proofreading Team at https://www.pgdp.net (This file was produced from images generously made available by The Internet Archive) *** START OF THE PROJECT GUTENBERG EBOOK WOMAN FREE ***
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WOMAN FREE BY ELLIS ETHELMER
1893 PUBLISHED BY THE WOMEN’S EMANCIPATION UNION Hon. Sec.:—M . WOLSTENHOLME ELMY B H , C
[PRICE FIVE SHILLINGS, POST FREE]
WYMAN AND SONS, LIMITED, LONDON AND REDHILL.
“Le philosophe, en étudiant les lois de la Nature, acquiert chaque jour la conviction que de leur violation seule naissent tous les maux dont gémit l’humanité.”
“The philosopher, in studying the laws of Nature, acquires more deeply every day the conviction that from their abuse alone spring all the evils from which humanity is groaning.”
D . M P
(Histoire de la Femme; Vol. III., p. 3).
WOMAN FREE. I. Source of the Light that cheers this later day, Science calm moves to spread her sovereign sway; Research and Reason, ranged on either hand, Proclaim her message to each waiting land; In truths whose import stands but part revealed, Till man befit himself those truths to wield; Since to high Knowledge duties high belong, As to the poet’s power the task of worthy song.
II. And man, from every stage of slow degree, Amendment for his previous rule may see; His keener conscience in our fuller time Perceives the whilom careless act a crime, Or finds some fancied fault to progress tend, By wiser vision traced to truer end; Till, growing shrewder in the growing light, We know no lack of good but our own lack of sight.
III. Thus, sad at first, we mark each evil deed, Of ignorance or will, bear fatal seed Of suffering to others in its train, The guileless share its penalty of pain, And man ’ s worst misery ofttimes is brought By trespass he himself nor did nor thought; Austere the fiat, yet therefrom we learn A purer life to frame, lest myriads mourn in turn.
IV. Deep though the teaching that this truth reveals Of fellowship of man with all that feels, Remains the riddle that, though inmost ken Of humblest creatures and of rudest men Has sense of freedom as an instinct strong, Resenting injury as act of wrong, Man listed not this monitor’s still voice, But gave his wanton wish the guilty force of choice.
V. Dark looms the record of his earlier years, A troubled tale of infamy and tears; For, of the ill by man primeval wrought, Shows forth predominant with anguish fraught, And long disaster to the ensuant race, The direful course of degradation base, Where freedom, justice, right, at one fell blow, In woman ’ s life of slave were outraged and laid low.
VI. The inklings gleaned of prehistoric hour Speak woman thrall to man ’ s unbridled power; Than brute more gifted, he, with heinous skill, Subdued her being to his sensual will; Binding her fast with ties of cunning weight, By mother’s burden forced to slavish fate; Thus woman was, and such her man-made doom, Ere yet the dawn of love illumed the soulless gloom.
VII. Ere Evolution, in unhasting speed, Trained man ’ s regard to larger life and need; By Art his feelings waked to functions higher, Disclosed within his clay the veins of fire, Taught him his pleasures of the flesh to find But presage of the mightier joys of mind; Evoked the soul from fume of mortal dust, The vestal flame of love from lower flush of lust.
VIII. The eye that once could note but food or foe Grew wise to watch the landscape’s varied glow; To gaze beyond our earthly temporal bars, And track the orbit of the wandering stars: The voice erst roused by hunger or by rage Now tells the nobler passions of the age, Till with love’s language is uplifted love To high and selfless thought all sensuous aim above.
IX. But not at once such life and love to know, For progress strives through many an ebb and flow; Man’s kindling sense, though stirred by call of Art, Still missed the motive of her deepest heart; ’Twas in her gracious embassy to give A fairer faith and fate to all that live, Neglecting none, yet man, ’twixt lust and pride, Due portion in the boon to woman still denied.
X.Æons of wrong ere history was born, With added ages passed in slight and scorn, Maintained the chains of primal womanhood, And clogged in turn man ’ s power of greater good; Egypt or Greece in vain sought heavenly light While woman ’ s soul was held from equal flight, Her path confined by man to sordid end, As subjugated wife, or hireling transient friend.
XI. Marriage which might have been a mateship sweet, Where equal souls in hallowed converse meet, Each aiding each the higher truths to find, And raising body to the plane of mind, Man’s baser will restrained to lower grade, And woman ’ s share a brainless bondage made; Her only hope of thought or learning wide, Some freer lot to seek than yoke forlorn of bride.
XII. Yet, as hetaira, comrade, chambermate, (The ambiguous word bespoke her dubious state), She, craving mental food, might but be guest By paying with her body for the quest; Conceding that, might lead a learned life, A licence vetoed to the legal wife, Might win great wealth, or build a lasting fame, Not due to her the guilt that left the tinge of shame.
XIII. What guilt was there, apportion it aright
To him who fixed the gages of the fight; Blame man, who, reckless of the woman ’ s fate, In greed for meaner pleasure lost the great; Blame him, the vaunted sage, who knew her mind Peer to his own in skill and wit refined, Yet left the after-ages to bemoan
The waste of woman worth that dawned and die unknown.
