Cosmology lecture notes

127 Pages·2005·3.093 MB·English

by  Hudson M.

PHYS 787:COSMOLOGY
Winter 2005
Mon & Fri 11:30-12:50 PM
Main Link Rooms (W: EIT 2053, G: MacN 101)
WWW: http://astro.uwaterloo.ca/~mjhudson/teaching/phys787
Instructor: Mike Hudson [email protected]
Office: Physics 252 (ext 2212)
Textbook: The primary textbook is Structure Formation in the Universe,
T. Padmanabhan, 1993, Camb. Univ. Press.
Other useful references are listed on the P787 WWW references page
Prerequisites: None. Some knowledge of General Relativity is advanta-
geous but is not required.
Syllabus:
1. Observational Overview
2. Homogeneous Universe
(a) Metric; Redshift
(b) Dynamics
(c) Distance; Ages; Volumes
3. Hot Big Bang
(a) Thermodynamics
(b) Recombination
(c) Nucleosynthesis
4. Structure Formation
(a) Linear Perturbation Theory
(b) Statistics of LSS
1
(c) Nonlinear models
5. Galaxies and Galaxy Formation
6. Cosmic Microwave Background Fluctuations
7. Gravitational Lensing
8. Inflation
Grading:
Assignments 50%
Term Paper & Seminar 50%
The course WWW page:
http://astro.uwaterloo.ca/~mjhudson/teaching/phys787
will always have the most up-to-date information.
2
Preamble
About this course
This course aims to give a broad review of modern cosmology. The emphasis is
on physical cosmology, i.e. its content, the physical processes in the expanding
Universe and the formation of structure from the horizon down to the scale of
galaxies. I will focus on the current paradigm, the Big Bang model and structure
formationinaUniversedominatedbydarkmatteranddarkenergy.
A deep knowledge of General Relativity is not necessary, although a familiarity
with GR will make the course more palatable. Likewise a basic understanding of
astrophysicalprocessesandsomeknowledgeofbasicparticlephysicsarehelpful.
Inanefforttobebroadsomedepthhasnecessarilybeensacrificed,butIhopethat
enough background and reference pointers have been provided for the interested
studenttodelvedeeperontheirown.
1 INTRODUCTIONTOCOSMOLOGY 1
1 Introduction to Cosmology
1.1 A Very Brief History
EarlyCosmologicalModels
I will skip the full treatment of early cosmological models — which would cover the Ptolemaic
modelandtheCopernicanrevolutionviaTychoandKepler—excepttonotethatthe“Copernican
Principle”,i.e. thatwedonotliveinaspecialplaceintheUniverse,hasprovedtobeinfluential.
Newton’s cosmology was infinite. Time and space were absolute and independent of the matter
intheUniverse. Newton’s1692LettertoRichardBentley:
Itseemstome,thatifthematterofoursunandplanets,andallthematteroftheuniverse,
wereevenlyscatteredthroughalltheheavens,andeveryparticlehadaninnategravitytowards
all the rest, and the whole space throughout which this matter was scattered, was finite, the
matter on the outside of this would by its gravity tend towards all the matter on the inside,
and by consequence fall down into the middle of the whole space, and there compose one
great spherical mass. But, if the matter were evenly disposed throughout an infinite space, it
couldneverconveneintoonemass,butsomeofitwouldconveneintoonemassandsomeinto
another, so as to make an infinite number of great masses, scattered great distances from one
toanotherthroughoutallthatinfinitespace. Andthusmightthesunandfixedstarsbeformed,
supposingthematterwereofalucidnature.
ProblemswithNewton’sUniverse:
• Stability
• Olber’s paradox - an infinite universe would produce an infinite amount of light at our posi-
tion,so”whyisthenightskydark?”
Einstein’sStaticModel In1917,beforediscoveryofcosmologicalredshifts,Einsteinproposed
a closed universe with a spherical geometry which was finite in extent, centreless and edgeless. In
order to make this model static, Einstein introduced into GR a small repulsive force known as the
cosmologicalconstant.
EinsteinbelievedinastaticUniverse–totheextentthathewaswillingtoaddanextraparameter
tohistheory. Why? (Laterhereferredtothecosmologicalconstantashis“greatestblunder”).
1 INTRODUCTIONTOCOSMOLOGY 2
Shortly afterward de Sitter discovered an expanding but empty solution of Einstein’s equations -
motion without matter. Friedmann (1922) found solutions with both expansion and matter, which
Lemaitre(1927)independentlyrediscovered.
WhywastheUniverseassumedtobehomogeneous?
EarlyExtra-galacticCosmography
At the beginning of the twentieth century, it was generally accepted that our galaxy was disk-
shapedandisolated. Butwhatwerethespiral“nebulae”likeM31(Andromeda)-weretheyinside
oroutsidetheMilkyWay? ImmanuelKanthadspeculatedthattheywereother“island”universes.
In1912,Sliphermeasuredspectrafromthenebulae,showingthatmanywereDoppler-shifted. By
1924,41nebulaehadbeenmeasured,and36ofthesewerefoundtobereceding.
In1929,Hubblemeasuredthedistancesto“nebulae”. HemeasuredCepheidstarsinnearbygalax-
ies such as M31 and then measured the relative distances between M31 and more distant galaxies
byassumingthatbrighteststarswerestandardcandles.
Combiningthesewiththeknownvelocities(correctedtothevelocityframeoftheMilkyWay),he
obtainedtheplotshowninFig.1.1.
Figure1.1: Hubble’splotofvelocityversusdistance
Fittingastraightline,
v = H r, (1.1)
0
1 INTRODUCTIONTOCOSMOLOGY 3
HubblefoundH = 500km/s/Mpc,avalueabout7timestoolarge1
0
The outstanding feature, however, is the possibility that the velocity-distance re-
lation may represent the de Sitter effect, and hence that numerical data may be intro-
ducedintodiscussionsofthegeneralcurvatureofspace.
(Hubble1929)
1.2 Review of Observational Cosmology
1.2.1 PreliminaryDefinitions
ΩdenotesadensitydividedbythecriticaldensityneededtoclosetheUniverse,
3H2
ρ = (1.2)
crit
8πG
Subscripts m,b,r,v denote the densities of matter, baryonic matter, radiation and vacuum. No
subscriptindicatesthetotaldensity2
Subscript 0 denotes the present-day value of a parameter, e.g. H is the present-day value of the
0
Hubbleconstant.
Units Inthissectionwewilluse“astronomer”units.
1Megaparsec(Mpc)=3.26×106 lightyears=3.1×1022 m
1year=3.16×107 s
1SolarMass(M(cid:12))=1.99×1030 kg
1.2.2 ExpansionoftheUniverse
Fig. 1.2 shows a modern Hubble diagram using Type Ia supernovae as distance indicators. Note
the deviations from linearity at large z, we will return to this later. Supernovae in all directions in
1Hubblemadetwoerrors. First,Hubbleassumedthatthevariablestarsheobservedinnearbygalaxies(Cepheids)
were the same as a different class of variable stars (W Virginis) in our galaxy. Second, what Hubble thought were
brightstarsinothergalaxieswereactuallycollectionsofbrightstars.Theseerrorswerenotdiscovereduntilthe1950s.
2Thisconventionisquiterecent(andstillbynomeansuniversal). InmanysourcesΩimplicitlyreferstomatter.
ThecontributionfromthevacuumisoftendenotedΩ ,Λ,orλdependingonhowitisnormalized.
Λ
1 INTRODUCTIONTOCOSMOLOGY 4
e)
m
n ti
k i
c
ER(cid:13)er)(cid:13)er ba
Thh
FAIN(Fart(Furt

Perlmutter, et al. (1998)
(W W ) = (cid:13)
M, L
26 ( 0, 1 )
(0.5,0.5) (0, 0)
( 1, 0 ) (1, 0)
24 (1.5,—0.5) (2, 0)
Supernova(cid:13)
CProosjmecotlogy(cid:13) Flat L = 0
22
B
m
e 20
ectiv C(Haalamnu/Tyo elto alol,(cid:13) (cid:13)
eff 18 A.J. 1996)
16
14
00..0022 00..0055 00..11 00..22 00..55 11..00
redshift z
MORE REDSHIFT(cid:13)
(More total expansion of universe (cid:13)
since the supernova explosion)
In flat universe: W = 0.28 [– 0.085 statistical] [– 0.05 systematic](cid:13)
M
(cid:13)
Prob. of fit to L = 0 universe: 1%
Figure1.2: HubblediagramforTypeIaSupernovae(Perlmutteretal.)
theskyfitthecurve: theexpansionisindeedisotropic.
TheHubbleSpaceTelescopeKeyProjectmeasuredthefluxofCepheidstarsinnearbygalaxiesto
allowacalibrationofthedistancescaleandhencetheHubbleconstant3
h = H /(100km/s) = 0.72±0.08 (1.3)
0
3Infact,theHubbleconstantisneitherconstantinspacen–becauseofpeculiarvelocities–norintime,soitwould
bebettercalledtheHubbleparameter.
1 INTRODUCTIONTOCOSMOLOGY 5
from(Freedmanetal. 2001).
1.2.3 IsotropyandHomogeneityoftheUniverse
The Universe is observed to be isotropic on very large scales. Fig. ?? plots a sample of distant
galaxies on the sky: clustering is evident on small angular scales but on the largest scales the
distributionlookssmooth.
Figure1.3: Thispicturecoversaregionofskyabout100degreesby50degreesaroundtheSouthGalactic
Pole. The intensities of each pixel are scaled to the number of galaxies in each pixel, with blue, green and
redforbright, mediumandfaintgalaxies(1-magslicescentredonBmagnitude18, 19and20). Themany
smalldark‘holes’areexcludedareasaroundbrightstars,globularclustersetc. (FromtheAPMsurvey.)
By obtaining redshifts of galaxies and using Hubble’s law, we can plot the distribution of galaxies
in 3D, as in Fig. 1.4. On the largest scales, the distribution of galaxies is homogeneous. On small
scales (1−10 Mpc), mass is clumped in galaxies and clusters of galaxies. On intermediate scales
(10−100Mpc),clustersaregroupedintosuperclustersandareconnectedbywallsandfilaments.
1.2.4 CosmicMicrowaveBackground(CMB)
Gamow predicted relic radiation from a primeval fireball in 1948. Penzias & Wilson (Bell Labs
Engineers)discoveredtheCMBintheradiointhe1960s.
ThespectrumoftheCMBisaperfectblackbodywithatemperatureof2.728±0.004K.
1 INTRODUCTIONTOCOSMOLOGY 6
Figure1.4: The2-Degree-FieldGalaxyRedshiftSurvey
1 INTRODUCTIONTOCOSMOLOGY 7
Figure 1.5: Three false color images of the sky as seen at microwave frequencies. The orientation of the
maps are such that the plane of the Milky Way runs horizontally across the center of each image. The top
figure shows the temperature of the microwave sky in a scale in which blue is 0 K and red is 4. Note that
thetemperatureappearscompletelyuniformonthisscale. Themiddleimageisthesamemapdisplayedin
a scale such that blue corresponds to 2.721 Kelvin and red is 2.729 Kelvin. The ”yin-yang” pattern is the
dipoleanisotropythatresultsfromthemotionoftheSunrelativetotherestframeofthecosmicmicrowave
background. The bottom figure shows the microwave sky after the dipole anisotropy has been subtracted
from the map. This removal eliminates most of the fluctuations in the map: the ones that remain are thirty
times smaller. On this map, the hot regions, shown in red, are 0.0002 Kelvin hotter than the cold regions,
showninblue. ThebandacrossthecentreisemissionfromourGalaxy.