Steven Weinberg - Cosmology

612 Pages·2008·2.63 MB·English

Cosmology
StevenWeinberg
UniversityofTexasatAustin
1
3
GreatClarendonStreet,OxfordOX26DP
OxfordUniversityPressisadepartmentoftheUniversityofOxford.
ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship,
andeducationbypublishingworldwidein
Oxford NewYork
Auckland CapeTown DaresSalaam HongKong Karachi
KualaLumpur Madrid Melbourne MexicoCity Nairobi
NewDelhi Shanghai Taipei Toronto
Withofficesin
Argentina Austria Brazil Chile CzechRepublic France Greece
Guatemala Hungary Italy Japan Poland Portugal Singapore
SouthKorea Switzerland Thailand Turkey Ukraine Vietnam
OxfordisaregisteredtrademarkofOxfordUniversityPress
intheUKandincertainothercountries
PublishedintheUnitedStates
byOxfordUniversityPressInc.,NewYork
©StevenWeinberg2008
Themoralrightsoftheauthorhavebeenasserted
DatabaserightOxfordUniversityPress(maker)
©Firstpublished2008
Allrightsreserved.Nopartofthispublicationmaybereproduced,
storedinaretrievalsystem,ortransmitted,inanyformorbyanymeans,
withoutthepriorpermissioninwritingofOxfordUniversityPress,
orasexpresslypermittedbylaw,orundertermsagreedwiththeappropriate
reprographicsrightsorganization.Enquiriesconcerningreproduction
outsidethescopeoftheaboveshouldbesenttotheRightsDepartment,
OxfordUniversityPress,attheaddressabove
Youmustnotcirculatethisbookinanyotherbindingorcover
andyoumustimposethesameconditiononanyacquirer
BritishLibraryCataloguinginPublicationData
Dataavailable
LibraryofCongressCataloginginPublicationData
Dataavailable
TypesetbyNewgenImagingSystems(P)Ltd.,Chennai,India
PrintedinGreatBritain
onacid-freepaperby
BiddlesLtd.,King’sLynn,Norfolk
ISBN 978–0–19–852682–7
10 9 8 7 6 5 4 3 2 1
To Louise, Elizabeth, and Gabrielle
This page intentionally left blank
Preface
Researchincosmologyhasbecomeextraordinarilylivelyinthepastquarter
century. Intheearly1980stheproposalofthetheoryofinflationoffereda
solution to some outstanding cosmological puzzles and provided a
mechanismfortheoriginoflarge-scalestructure,whichcouldbetestedby
observations of anisotropies in the cosmic microwave background.
November 1989 saw the launch of the Cosmic Background Explorer
Satellite. Measurements with its spectrophotometer soon established the
thermal nature of the cosmic microwave background and determined its
temperaturetothreedecimalplaces,aprecisionunprecedentedincosmol-
ogy. Alittlelaterthelong-soughtmicrowavebackgroundanisotropieswere
found in data taken by the satellite’s radiometer. Subsequent observa-
tions by ground-based and balloon-borne instruments and eventually by
theWilkinsonMicrowaveAnisotropyProbeshowedthattheseanisotropies
areprettymuchwhatwouldbeexpectedonthebasisofinflationarytheory.
In the late 1990s the use of Type Ia supernovae as standard candles led to
the discovery that the expansion of the universe is accelerating, implying
that most of the energy of the universe is some sort of dark energy, with a
ratioofpressuretodensitylessthan−1/3. Thiswasconfirmedbyprecise
observations of the microwave background anisotropies, and by massive
surveys of galaxies, which together provided increasingly accurate values
forcosmologicalparameters.
Meanwhile, the classic methods of astronomy have provided steadily
improving independent constraints on the same cosmological parameters.
The spectroscopic discovery of thorium and then uranium in the atmo-
spheresofoldstars,togetherwithcontinuedstudyoftheturn-offfromthe
mainsequenceinglobularclusters,hasnarrowedestimatesoftheageofthe
universe. Themeasurementofthedeuteriumtohydrogenratioininterstel-
larabsorptioncombinedwithcalculationsofcosmologicalnucleosynthesis
has given a good value for the cosmic density of ordinary baryonic mat-
ter, and shown that it is only about a fifth of the density of some sort of
mysterious non-baryonic cold dark matter. Observations with the Hubble
SpaceTelescopeaswellasground-basedtelescopeshavegivenincreasingly
precisevaluesfortheHubbleconstant. Itisgreatlyreassuringthatsomeof
theparametersmeasuredbytheseothermeanshavevaluesconsistentwith
thosefoundinstudiesofthecosmicmicrowavebackgroundandlargescale
structure.
Progress continues. In the years to come, we can expect definite
information about whether the dark energy density is constant or evolv-
ing, and we hope for signs of gravitational radiation that would open the
v
Preface
era of inflation to observation. We may discover the nature of dark mat-
ter, either by artificially producing dark matter particles at new large
accelerators,orbydirectobservationofnaturaldarkmatterparticlesimping-
ingontheearth. Itremainstobeseenifinourtimesfundamentalphysical
theory can provide a specific theory of inflation or explain dark matter or
darkenergy.
This new excitement in cosmology came as if on cue for elementary
particlephysicists. Bythe1980stheStandardModelofelementaryparticles
and fields had become well established. Although significant theoretical
and experimental work continued, there was now little contact between
experiment and new theoretical ideas, and without this contact, particle
physics lost much of its liveliness. Cosmology now offered the excitement
thatparticlephysicistshadexperiencedinthe1960sand1970s.
In1999Ifinishedmythree-volumebookonthequantumtheoryoffields
(cited here as “QTF”), and with unaccustomed time on my hands, I set
myselfthetaskoflearningindetailthetheoryunderlyingthegreatprogress
incosmologymadeintheprevioustwodecades. AlthoughIhaddonesome
researchoncosmologyinthepast,gettinguptodatenowturnedouttotake
afairamountofwork. Reviewarticlesoncosmologygavegoodsummaries
of the data, but they often quoted formulas without giving the derivation,
and sometimes even without giving a reference to the original derivation.
Occasionally the formulas were wrong, and therefore extremely difficult
for me to rederive. Where I could find the original references, the articles
sometimeshadgapsintheirarguments,orreliedonhiddenassumptions,or
used unexplained notation. Often massive computer programs had taken
the place of analytic studies. In many cases I found that it was easiest to
workouttherelevanttheoryformyself.
This book is the result. Its aim is to give self-contained explanations
oftheideasandformulasthatareusedandtestedinmoderncosmological
observations. The book divides into two parts, each of which in my exp-
erience teaching the subject provides enough material for a one-semester
graduatecourse. Thefirstpart,Chapters1through4,dealschieflywiththe
isotropic and homogeneous average universe, with only a brief introduc-
tiontotheanisotropiesinthemicrowavebackgroundinSection2.6. These
chaptersaremore-or-lessinreversechronologicalorder;Chapter1concen-
tratesontheuniversesincetheformationofgalaxies,correspondingroughly
toredshiftsz < 10;Chapter2dealswiththemicrowavebackground,emit-
ted at a redshift z (cid:1) 1,000; Chapter 3 describes the early universe, from
the beginning of the radiation-dominated expansion to a redshift z ≈ 104
whenthedensityofradiationfellbelowthatofmatter;andChapter4takes
up the period of inflation that is believed to have preceded the radiation-
dominated era. The second part, Chapters 5 through 10, concentrates on
the departures from the average universe. After some general formalism
vi
Preface
in Chapter 5 and its application to the evolution of inhomogeneities in
Chapter6,IreturninChapter7tothemicrowavebackgroundanisotropies,
andtakeupthelargescalestructureofmatterinChapter8. Gravitational
lensing is discussed late, in Chapter 9, because its most important cosmo-
logical application may be in the use of weak lensing to study large scale
structure. ThetreatmentofinflationinChapter4dealsonlywiththeaver-
agepropertiesoftheuniverseintheinflationaryera;Ireturntoinflationin
Chapter10,whichdiscussesthegrowthofinhomogeneitiesfromquantum
fluctuationsduringinflation.
To the greatest extent possible, I have tried throughout this book to
presentanalyticcalculationsofcosmologicalphenomena,andnotjustreport
resultsobtainedelsewherebynumericalcomputation. Thecalculationsthat
are used in the literature to compare observation with theory necessarily
take many details into account, which either make an analytic treatment
impossible,orobscurethemainphysicalfeaturesofthecalculation. Where
this is the case, I have not hesitated to sacrifice some degree of accuracy
for greater transparency. This is especially the case in the hydrodynam-
ical treatment of cosmic fluctuations in Sections 6.2 through 6.5, and in
the treatment of large scale structure in Chapter 8. But in Section 6.1
and Appendix H I also give an account of the more accurate kinetic the-
ory on which the modern cosmological computer codes are based. Both
approachesareappliedtothecosmicmicrowavebackgroundanisotropiesin
Chapter7.
So much has happened in cosmology since the 1960s that this book
necessarily bears little resemblance to my 1972 treatise, Gravitation and
Cosmology. On occasion I refer back to that book (cited here as “G&C”)
for material that does not seem worth repeating here. Classical general
relativityhasnotchangedmuchsince1972(apartfromagreatstrengthen-
ing of its experimental verification) so it did not seem necessary to cover
gravitationaswellascosmologyinthepresentbook. However,asaconve-
niencetoreaderswhowanttorefreshtheirknowledgeofgeneralrelativity,
and to establish my notation, I provide a brief introduction to general rel-
ativityinAppendixB.Otherappendicesdealwithtechnicalmaterialthatis
neededhereandthereinthebook. Ihavealsosuppliedatthebackofthis
bookaglossaryofsymbolsthatareusedinmorethanonesectionandan
assortmentofproblems.
In order to keep the book to manageable proportions, I decided to
exclude material that was highly speculative. Thus this book does not go
intocosmologicaltheoryinhigherdimensions, oranthropicreasoning, or
holographic cosmology, or conjectures about the details of inflation, or
many other new ideas. I may perhaps include some of them in a follow-
up volume. The present book is largely concerned with what has become
mainstream cosmology: a scenario according to which inflation driven by
vii
Preface
oneormorescalarfieldsisfollowedbyabigbangdominatedbyradiation,
colddarkmatter,baryonicmatter,andvacuumenergy.
Ibelievethatthediscussionoftopicsthataretreatedinthisbookisup
to date as of 200n, where n is an integer that varies from 1 to 7 through
differentpartsofthebook. Ihavetriedtogivefullreferencestotherelevant
astrophysicalliteratureuptothesedates,butIhavedoubtlessmissedsome
articles. Themereabsenceofaliteraturereferenceshouldnotbeinterpreted
as a claim that the work presented is original, though perhaps some of it
is. Where I knew them, I included references to postings in the Cornell
archive, http://arxiv.org, as well as to the published literature. In
somecasesIhadtolistonlytheCornellarchivenumber,wherethearticlein
questionhadnotyetappearedinprint,orwhereithadneverbeensubmit-
tedtopublication. Ihavequotedthelatestmeasurementsofcosmological
parametersknowntome,inpartbecauseIwanttogivethereaderasense
of what is now observationally possible. But I have not tried to combine
measurementsfromobservationsofdifferenttypes,becauseIdidnotthink
thatitwouldaddanyadditionalphysicalinsight,andanysuchcosmological
concordancewouldverysoonbeoutofdate.
I owe a great debt to my colleagues at the University of Texas, includ-
ingThomasBarnes,FritzBenedict,WillyFischler,KarlGebhardt,Patrick
Greene, Richard Matzner, Paul Shapiro, Craig Wheeler, and especially
Duane Dicus, who did some of the numerical calculations and supplied
manycorrections. IamgratefulaboveallamongthesecolleaguestoEiichiro
Komatsu, who read through a draft of the manuscript and was a never-
failing source of insight and information about cosmological research.
I received much help with figures and calculations from my research stu-
dentRaphaelFlauger,andIwaswarnedofnumerouserrorsbyFlaugerand
other students: Yingyue Li Boretz, Kannokkuan Chaicherdsakul, Bo Li,
IanRoederer,andYukiWatanabe. MatthewAndersonhelpedwithnumeri-
calcalculationsofcosmologicalnucleosynthesis. Ihavealsobenefitedmuch
from correspondence on special topics with Ed Bertschinger, Dick Bond,
Latham Boyle, Robert Cahn, Alan Guth, Robert Kirshner, Andrei Linde,
Eric Linder, Viatcheslav Mukhanov, Saul Perlmutter, Jonathan Pritchard,
Adam Riess, Uros Seljak, Paul Steinhardt, Edwin Turner, and Matias
Zaldarriaga. Thanks are also due to Jan Duffy and Terry Riley for many
helps. Ofcourse,Ialoneamresponsibleforanyerrorsthatmayremaininthe
book. Ihopethatreaderswillletmeknowofanymistakestheymaynotice;
I will post them on a web page, http://zippy.ph.utexas.edu/
˜weinberg/corrections.html.
Austin,Texas
June2007
viii
Notation
Latinindicesi,j,k,andsoongenerallyrunoverthethreespatialcoordinate
labels,usuallytakenas1,2,3.
Greek indices µ,ν, etc. generally run over the four spacetime coordinate
labels1,2,3,0,withx0 thetimecoordinate.
Repeatedindicesaregenerallysummed,unlessotherwiseindicated.
Theflatspacetimemetricηµν isdiagonal,withelementsη11 = η22 = η33 =
1, η = −1.
00
Spatialthree-vectorsareindicatedbylettersinboldface.
A hat over any vector indicates the corresponding unit vector: Thus, vˆ ≡
v/|v|.
Adotoveranyquantitydenotesthetime-derivativeofthatquantity.
∂2 ∂2 ∂2
∇2 istheLaplacian, + + .
∂(x1)2 ∂(x2)2 ∂(x3)2
Exceptonvectorsandtensors,asubscript0denotesthepresenttime.
On densities, pressures, and velocities, the subscripts B, D, γ, and ν refer
respectivelytothebaryonicplasma(nucleipluselectrons),colddarkmatter,
photons,andneutrinos,whilethesubscriptsM andR referrespectivelyto
non-relativisticmatter(baryonicplasmapluscolddarkmatter)andradia-
tion(photonsplusneutrinos).
The complex conjugate, transpose, and Hermitian adjoint of a matrix or
vector A are denoted A∗, AT, and A† = A∗T, respectively. +H.c. or +c.c.
attheendofanequationindicatestheadditionoftheHermitianadjointor
complexconjugateoftheforegoingterms.
Beginning in Chapter 5, a bar over any symbol denotes its unperturbed
value.
In referring to wave numbers, q is used for co-moving wave numbers, with
anarbitrarynormalizationoftheRobertson–Walkerscalefactora(t),while
k is the present value q/a of the corresponding physical wave number
0
q/a(t). (N.B. This differs from the common practice of using k for the
ix