hudson m. cosmology lecture notes

I will focus on the current paradigm, the Big Bang model and structureformation in a Universe dominated by dark matter and dark energy.. He measured Cepheid stars in nearby ies such as M

PHYS 787:COSMOLOGY

Winter 2005Mon & Fri 11:30-12:50 PMMain Link Rooms (W: EIT 2053, G: MacN 101)WWW: http://astro.uwaterloo.ca/~mjhudson/teaching/phys787

Instructor: Mike Hudson mjhudson@uwaterloo.ca

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 geous but is not required

(c) Distance; Ages; Volumes

3 Hot Big Bang

(c) Nonlinear models

5 Galaxies and Galaxy Formation

6 Cosmic Microwave Background Fluctuations

7 Gravitational Lensing

8 Inflation

Grading:

Term Paper & Seminar 50%

The course WWW page:

http://astro.uwaterloo.ca/~mjhudson/teaching/phys787will always have the most up-to-date information

2

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 expandingUniverse and the formation of structure from the horizon down to the scale ofgalaxies I will focus on the current paradigm, the Big Bang model and structureformation in a Universe dominated by dark matter and dark energy

A deep knowledge of General Relativity is not necessary, although a familiaritywith GR will make the course more palatable Likewise a basic understanding ofastrophysical processes and some knowledge of basic particle physics are helpful

In an effort to be broad some depth has necessarily been sacrificed, but I hope thatenough background and reference pointers have been provided for the interestedstudent to delve deeper on their own

1 INTRODUCTION TO COSMOLOGY 1

1 Introduction to Cosmology

1.1 A Very Brief History

Early Cosmological Models

I will skip the full treatment of early cosmological models — which would cover the Ptolemaicmodel and the Copernican revolution via Tycho and Kepler — except to note that the “CopernicanPrinciple”, i.e that we do not live in a special place in the Universe, has proved to be influential

Newton’s cosmology was infinite Time and space were absolute and independent of the matter

in the Universe Newton’s 1692 Letter to Richard Bentley:

It seems to me, that if the matter of our sun and planets, and all the matter of the universe,were evenly scattered through all the heavens, and every particle had an innate gravity towardsall the rest, and the whole space throughout which this matter was scattered, was finite, thematter 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 onegreat spherical mass But, if the matter were evenly disposed throughout an infinite space, itcould never convene into one mass, but some of it would convene into one mass and some intoanother, so as to make an infinite number of great masses, scattered great distances from one

to another throughout all that infinite space And thus might the sun and fixed stars be formed,supposing the matter were of a lucid nature

Problems with Newton’s Universe:

• Stability

• Olber’s paradox - an infinite universe would produce an infinite amount of light at our tion, so ”why is the night sky dark?”

posi-Einstein’s Static Model In 1917, before discovery of cosmological redshifts, Einstein proposed

a closed universe with a spherical geometry which was finite in extent, centreless and edgeless Inorder to make this model static, Einstein introduced into GR a small repulsive force known as thecosmological constant

Einstein believed in a static Universe – to the extent that he was willing to add an extra parameter

to his theory Why? (Later he referred to the cosmological constant as his “greatest blunder”).

1 INTRODUCTION TO COSMOLOGY 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, whichLemaitre (1927) independently rediscovered

-Why was the Universe assumed to be homogeneous?

Early Extra-galactic Cosmography

At the beginning of the twentieth century, it was generally accepted that our galaxy was shaped and isolated But what were the spiral “nebulae” like M31 (Andromeda) - were they inside

disk-or outside the Milky Way? Immanuel Kant had speculated that they were other “island” universes

In 1912, Slipher measured spectra from the nebulae, showing that many were Doppler-shifted By

1924, 41 nebulae had been measured, and 36 of these were found to be receding

In 1929, Hubble measured the distances to “nebulae” He measured Cepheid stars in nearby ies such as M31 and then measured the relative distances between M31 and more distant galaxies

galax-by assuming that brightest stars were standard candles

Combining these with the known velocities (corrected to the velocity frame of the Milky Way), heobtained the plot shown in Fig 1.1

Figure 1.1:Hubble’s plot of velocity versus distance

Fitting a straight line,

1 INTRODUCTION TO COSMOLOGY 3

Hubble found H0 = 500 km/s/Mpc, a value about 7 times too large1

The outstanding feature, however, is the possibility that the velocity-distance lation may represent the de Sitter effect, and hence that numerical data may be intro-duced into discussions of the general curvature of space

Units In this section we will use “astronomer” units

1 Megaparsec (Mpc) = 3.26 × 106light years = 3.1 × 1022m

1 year = 3.16 × 107 s

1 Solar Mass (M ) = 1.99 × 1030kg

1.2.2 Expansion of the Universe

Fig 1.2 shows a modern Hubble diagram using Type Ia supernovae as distance indicators Notethe deviations from linearity at large z, we will return to this later Supernovae in all directions in

1 Hubble made two errors First, Hubble assumed that the variable stars he observed in nearby galaxies (Cepheids) were the same as a different class of variable stars (W Virginis) in our galaxy Second, what Hubble thought were bright stars in other galaxies were actually collections of bright stars These errors were not discovered until the 1950s.

2 This convention is quite recent (and still by no means universal) In many sources Ω implicitly refers to matter The contribution from the vacuum is often denoted Ω , Λ, or λ depending on how it is normalized.

In flat universe: ΩM = 0.28 [± 0.085 statistical] [± 0.05 systematic]

Prob of fit to Λ = 0 universe: 1%

Figure 1.2: Hubble diagram for Type Ia Supernovae (Perlmutter et al.)

the sky fit the curve: the expansion is indeed isotropic

The Hubble Space Telescope Key Project measured the flux of Cepheid stars in nearby galaxies toallow a calibration of the distance scale and hence the Hubble constant3

3 In fact, the Hubble constant is neither constant in space n– because of peculiar velocities – nor in time, so it would

be better called the Hubble parameter.

1 INTRODUCTION TO COSMOLOGY 5

from (Freedman et al 2001)

1.2.3 Isotropy and Homogeneity of the Universe

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 thedistribution looks smooth

Figure 1.3: This picture covers a region of sky about 100 degrees by 50 degrees around the South GalacticPole The intensities of each pixel are scaled to the number of galaxies in each pixel, with blue, green andred for bright, medium and faint galaxies (1-mag slices centred on B magnitude 18, 19 and 20) The manysmall dark ‘holes’ are excluded areas around bright stars, globular clusters etc (From the APM survey.)

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 smallscales (1 − 10 Mpc), mass is clumped in galaxies and clusters of galaxies On intermediate scales(10 − 100 Mpc), clusters are grouped into superclusters and are connected by walls and filaments

1.2.4 Cosmic Microwave Background (CMB)

Gamow predicted relic radiation from a primeval fireball in 1948 Penzias & Wilson (Bell LabsEngineers) discovered the CMB in the radio in the 1960s

The spectrum of the CMB is a perfect black body with a temperature of 2.728 ± 0.004K

1 INTRODUCTION TO COSMOLOGY 6

Figure 1.4:The 2-Degree-Field Galaxy Redshift Survey

1 INTRODUCTION TO COSMOLOGY 7

Figure 1.5: Three false color images of the sky as seen at microwave frequencies The orientation of themaps are such that the plane of the Milky Way runs horizontally across the center of each image The topfigure shows the temperature of the microwave sky in a scale in which blue is 0 K and red is 4 Note thatthe temperature appears completely uniform on this scale The middle image is the same map displayed in

a scale such that blue corresponds to 2.721 Kelvin and red is 2.729 Kelvin The ”yin-yang” pattern is thedipole anisotropy that results from the motion of the Sun relative to the rest frame of the cosmic microwavebackground The bottom figure shows the microwave sky after the dipole anisotropy has been subtractedfrom the map This removal eliminates most of the fluctuations in the map: the ones that remain are thirtytimes smaller On this map, the hot regions, shown in red, are 0.0002 Kelvin hotter than the cold regions,shown in blue The band across the centre is emission from our Galaxy

1.2.5 Baryonic Universe

Abundance of Light Elements Big-bang nucleosynthesis allows us a prediction of the dances of helium, deuterium and lithium with only one free parameter (the baryon-to-photon ratio,

abun-or equivalently the density of baryons)

The current best observational measurements of the primordial abundances of these elements ticularly deuterium) suggests

(Burles, Nollett & Turner 01)

Baryon Budget Only a small fraction of the baryons in the Universe are in stars The bulk of thebaryons are likely to be in between galaxies

Note that the density in stars is quite negligible compared to the total baryonic density This implies

that there is baryonic dark matter, primarily in groups/intergalactic medium.

1 INTRODUCTION TO COSMOLOGY 9

Figure 1.6: CMB temperature fluctuations as a function of multipole number l, showing data from manyrecent experiments The curve shows a cosmological model which fits the data From Wayne Hu’sCMBexperiment page

1.2.6 Evidence for Dark Matter

As noted above, the total density of baryonic matter is small: Ωb ∼ 0.04 if h ∼ 0.7 Various lines

of evidence suggest that there is significantly more dark matter in the Universe Some of these are:

Dynamics of Galaxies in Clusters As early as the thirties, Zwicky (1937) applied the virialtheorem to the orbits of galaxies in clusters and argued that there was evidence for the presence ofdark matter The temperature of X-ray emitting plasma in clusters leads to the same conclusion

1 INTRODUCTION TO COSMOLOGY 10

Figure 1.7: Predicted abundances (by mass) of4He, D,3He and Li

Rotation Curves of Spiral Galaxies How large are the DM haloes of galaxies?

Fig 1.9 shows the rotational velocity of stars in the Milky Way It is approximately constant overthe range of radii which have been measured A similar behaviour is seen in other large spiralgalaxies Beyond the edge of the light distribution, one expects the rotation velocity to fall as

r−1/2 However at large radii, well beyond the edge of the disk, the rotation curve is observed to

be approximately constant This implies M (< r) ∝ r This believed to be due to a dark matter

“halo”

1 INTRODUCTION TO COSMOLOGY 11

Figure 1.8:The Baryon “Budget” (Fukugita, Hogan & Peebles, 1998 ApJ, 503,518)

Gravitational Lensing Strong gravitational lensing leads to the formation of multiple images

and giant arcs

Weak gravitational lensing uses the small distortions of background galaxies to map the dark

mat-ter distribution Because galaxies are not intrinsically round, there is considerable noise in thismethod

Microlensing is strong gravitational lensing where the multiple images are not resolved This is

typically the case when the stars are the lenses Has been used to place constraints on the abundance

of dark compact objects in the halo of our Galaxy

Cosmic Flows: Deviations from uniform Hubble expansion Deviations from homogeneitylead to deviations from uniform expansion The latter can be used to measure the former, providedthe scales are sufficiently large

Observations of peculiar velocities of individual galaxies are noisy, so large numbers need to beobserved to obtain reliable results

A comparison between mass and galaxies yields

1 INTRODUCTION TO COSMOLOGY 12

Figure 1.9: The rotation curve of the Milky Way Galaxy from a compilation of data (From Fich &Tremaine 1991)

Figure 1.10:Light and mass in the cluster Cl 0024+17

if fluctuations in the mass follow fluctuations in the light

1 INTRODUCTION TO COSMOLOGY 13

Figure 1.11: Density fluctuation fields of POTENT mass (left-hand column) vs IRAS galaxies (middlecolumn), both smoothed G12 Contour spacing is 0.2 in δ; the heavier contour is δ = 0; solid contours mark

δ > 0 and dashed contours δ < 0 The density is also indicated by shading Also drawn is the difference

field in units of the error, where the contour spacing is unity (right-hand column) The maps are drawnout to a radius of 80h−1 Mpc, and the very thick contour marks the boundaries of the Re = 40h−1 Mpccomparison volume From Sigad et al., 1998, ApJ, 495, 516

1.2.7 Ages of the oldest stars

Globular clusters are compact balls of stars orbiting in the halo of the Milky Way and other ies The stars appear to be co-eval, and are thus easier to date by fitting to models of stellarevolution Allowing for 0.8 Gyr from the formation of the Universe to the formation of these stars,Krauss and Chaboyer(2002) find that the age of the Universe > 11.2 Gyr at 95% CL, with a bestfit age of 13.4 Gyr

galax-5.1 shows the Hertzsprung-Russell diagram for the globular cluster M5 The three solid lines arethe model predictions for 10, 12 & 14 Gyr

1.2.8 Galaxies

Luminosity function and mass functions The galaxy luminosity function (i.e the probabilitydistribution function for galaxy luminosities), φ, is a power law at low luminosities and has anexponential break at high luminosities

The predicted mass function of virialized (dark-matter dominated) objects has a similar behaviour,

but breaks at larger masses corresponding to rich clusters

1 INTRODUCTION TO COSMOLOGY 14

Figure 1.12: Isochrone fit to M92.The best fit age is 15±1 Gyr From Harris et al 1997

What physics causes the break in the galaxy spectrum?

Morphologies There are two different types of galaxies: spirals are dominated by a supported, gas and dust rich disk of young stars; ellipticals are gas-poor and are dominated byold stars on random orbits (“pressure supported”) However, many spirals have a small elliptical-like bulge and many ellipticals have a weak disk Ellipticals are found mainly in rich clusters ofgalaxies, whereas spirals are found in low density regions

rotationally-How is galaxies related to mass on different scales?

1 INTRODUCTION TO COSMOLOGY 15

1.3 The “Standard Model”

in which the light elements were synthesized 1960s 99%

Structure was seeded by Gaussian irregularities 1980-90s 75%

which are the relics of quantum fluctuations, 1980s 80%

The dominant matter is “cold dark matter” 1980s 80%

but Λ (or “quintessence”) is dynamically important late 1990s 67%

(Adapted from Peebles Sci Am Jan 01)

The fundamental paradigm is the expansion of Universe governed by gravity and by the equation

of state of its constituents and the growth of structure within the Universe driven by gravity butcounteracted by pressure

There are a number of parameters which characterize the standard model At present few of these

are predicted a priori, and so must be fixed by observation These include:

• The present-day expansion rate, h or H0[1 parameter]

• The present-day densities (Ω) of matter, baryonic matter, radiation and of the vacuum [4parameters]

• The characteristics of the baryonic matter are described by the standard model of particlephysics One could add the mass of the dark matter particle(s) [1 parameter?] and its inter-actions In the simplest standard model, the DM particle interacts only weakly

• The equation of state of the dark energy [1 parameter?]

• The amplitude of primordial fluctuations, characterized by A and n, where the latter eter describes how these fluctuations behave as a function of scale [2 parameters]4 To these

param-we might also add a gravitational wave background [2 more parameters] Certain models ofinflation make specific predictions for combinations of the latter parameters

4 Of course it is possible that the primordial fluctuation spectrum is not described by a simple power law as a function of scale

1 INTRODUCTION TO COSMOLOGY 16

The simplest viable model contains only a few parameters that need to be measured (h, A, Ωm,

Ωb) A more general model might have as many as a dozen parameters Occam says that we shouldprefer the former unless we have evidence to think otherwise

Some observations which might have disproved the Standard Model

• A non-blackbody spectrum for the CMB

• CMB fluctuations ∆T /T  10−5

• CMB fluctuations without the “acoustic peaks”

• A star with helium abundance Y  23%

• Non-Gaussian galaxy or CMB power spectra

• Direct evidence of violations of GR

Status of the cosmological parameters (2002)

CMB data alone gives: ΩΛ+ Ωm = 1.04 ± 0.04 The Universe appears to be very close to flat!The current best-fitting parameters from CMB anisotropy experiments plus the 2dF galaxy surveygive:

(From Wang et al 2002)

For the best fitting Hubble constant, the above yields: Ωm = 0.28, Ωb = 0.04

Also note that Ωb/Ωm = 0.17 ± 0.027 A universe consisting only of baryons is ruled out at > 6σ.Dark matter must be mostly non-baryonic

• Supernovae give: Ωm = 0.28 in a flat Universe Also in agreement with assessments fromweak lensing and large-scale flows Errors remain large, though

• Big-bang nucleosynthesis gives: Ωbh2 = 0.02 in agreement with above

1 INTRODUCTION TO COSMOLOGY 17

• Hubble constant is: 0.72 ± 0.08 in agreement with the above

• These numbers yield an age of the Universe of 12.2 Gyr, in agreement with the globularcluster ages

Some outstanding issues

• The Very Early Universe

– What physics drove inflation?

∗ scalar field(s)?

∗ collisions in the brane world?

– How did the “Bang” begin?

∗ quantum gravity

• The Dark Sector

– What is the dark energy?

∗ how much is there?

∗ what is its equation of state?

∗ is it related to inflation? to dark matter?

– What is the dark matter?

∗ how much is there?

∗ what are the mass and interactions of the particle?

• Formation of Baryonic Objects

– When did the Universe re-ionize (when was “First Light”)?

– Were the first luminous objects stars or galaxies?

– When did galaxies assemble?

– Why are there two types of galaxies: spirals and ellipticals?

– Is there a connection between galaxy formation and active galactic nuclei (AGN)

[mas-sive black holes]

2 FRIEDMANN-ROBERTSON-WALKER COSMOLOGICAL MODEL 18

2.1 Cosmological Principle

Cosmological models are based on

• the observation that the Universe appears isotropic and;

• the assumption that observers on other galaxies also see an isotropic Universe

This cosmological principle is a “Copernican” assumption that the Earth (and the Milky Way

galaxy ) are not at a “special” location (For example that we are not at the exact center of an

explosion)

Isotropy demands that on the surface of sphere whose radius is at a given “distance”, the localmatter and radiation densities, the local expansion rate, as well as the redshift of light and theticking rate of clocks must be independent of direction

If we require all observers also find that the Universe is isotropic, this places strong restrictions onthe metric as we will see below In particular once we have isotropy around all points, the Universe

must be homogeneous in the above properties as well.

2.2 The Metric

In General Relativity (GR), space-time is described by a metric ds2 = c2dt2− dl2 This is pactly written ds2 = gαβdxαdxβ where Greek indices run from 0 to 3 Index 0 denotes time (so

com-dx0 could also be written dt) and the indices 1 to 3 indicate the spatial dimensions

Light travels along paths with ds = 0 Other particles follow geodesics, which can be thought of

as shortest paths in space time

To gain some intuition about the nature of the metric, first consider 2-d surfaces at a fixed time.Clearly one possible metric which satisfies the assumptions of isotropy and homogeneity is the

“x-y” plane, i.e dl2 = dx2+ dy2

Another is the 2-d surface of a 3-d sphere (a “2-sphere”)

In the 3-d space in which the 2-sphere is embedded, dl2 = dx2+ dy2 + dz2 If we consider only

2 FRIEDMANN-ROBERTSON-WALKER COSMOLOGICAL MODEL 19

distances along the surface of the sphere, we can substitute for dz using equation (2.1) to get

If instead we choose the usual polar coords θ, φ to be our 2-d coordinate system, ie

x = a sin(θ) cos(φ); y = a sin(θ) sin(φ); z = cos(θ) (2.6)then in terms of these coordinates

dl2 = a2(dθ2+ sin(θ)2d(φ)2) (2.7)

Note that in addition to the flat and 2-sphere metrics, there is also a homogeneous hyperbolicsurface (with negative curvature) This has metric a plus sign in the denominator of equation (2.5)and has a sinh(θ) in place of the sin(θ) in the above equation

2.3 Robertson-Walker metric

The 2-d arguments can be extended to a 3-sphere embedded in a fictitious 4th dimension

Instead let us re-derive the form of the RW metric a different way, following Gunn (1978) Isotropyand the cosmological principle require that the spatial part of the RW metric has the following form

a2(t)b2

2 FRIEDMANN-ROBERTSON-WALKER COSMOLOGICAL MODEL 20

where dψ2 = dθ2 + sin2θdφ2 This last step is necessary so that distances in the plane of the skyare the same in both the θ and φ directions

Because of isotropy, we cannot have a different expansion rate in the r and ψ directions, so thea(t) term has to be the same for both

Similarly b, f and a cannot be functions of θ and φ

Now let us choose a new radial coordinate χ so that

y

h x

l

θ

Figure 2.1:Gunn’s “isosceles” triangle

We will now work out S(χ) using an argument from Gunn (1978) S(χ) is related to the “angulardiameter distance”5; it is a function that allows us to map a given angle dψ into a proper length.For Euclidean space, S(χ) = χ, but as we saw above it need not be so

Consider the three comoving observers O, P and Q in Fig 2.1 Suppose P is slightly displacedfrom the ray OQ so that γ and θ are small angles Note that by isotropy the two γ’s must be equal.Then

5The angular diameter distance in an expanding Universe is a little more complicated than simply S(χ) Here,

however, we are consider angles along the spatial surface at fixed cosmic time t.

2 FRIEDMANN-ROBERTSON-WALKER COSMOLOGICAL MODEL 21

Figure 2.2: In order to estimate the proper size of an object which subtends a given angle, we need thegeometry of space-time

differentiating with respect to x and setting x = 0 yields

(2.16)

where the constant K is called the curvature constant Note also that if A is a constant, ASK(χ/A)

is also a solution to equation (2.15) However, usually it is easier to absorb A into the definition ofa(t)

For example, K = +1 is positive curvature and the metric is a three-dimensional generalization ofthe two-dimensional surface of sphere This metric has a finite volume, in the same way that thesurface of a sphere has a finite area This model is often called “closed”

2 FRIEDMANN-ROBERTSON-WALKER COSMOLOGICAL MODEL 22

Similarly K = −1 is a generalization of a hyperbolic or saddle-shaped geometry It is infinite and

There is a special class of observers — those who a fixed value of r, θ and φ

• It can be shown that such observers follow geodesics and are therefore “freely falling”

• Note also that the proper time for all comoving observers is the same and is identical to thecosmic variable time, t

• These observers have no velocity with respect to the local matter

Such observers are labeled fundamental or comoving observers and the coordinates r are comoving

coordinates

2.4 Kinematics of the Expansion

Hubble’s Law

At small distances, Hubble’s law should hold Consider two nearby comoving galaxies, one at

r = 0 and another at a small comoving distance ∆r They are separated by a proper radial distance

2 FRIEDMANN-ROBERTSON-WALKER COSMOLOGICAL MODEL 23

∆l = (a ∆r) The rate of change of the proper distance is

How is redshift related to the FRW metric?

Physical argument Consider the light emitted from a nearby comoving galaxy (receding at avelocity v) to the origin The change in the frequency is

2 FRIEDMANN-ROBERTSON-WALKER COSMOLOGICAL MODEL 24

From the above, the redshift can be seen as a kinematic effect: an integral of small Doppler shifts6.See also 3.3

Metric argument We can obtain the same result straight from the metric The comoving radialcoordinate distance χ from which the light arrived can be obtained using the fact that photons movealong null geodesics, so that c dt = a(t)dχ Integrating this gives

Consider for example a pulsar located in a distant galaxy pulsing at a regular rate It emits a pulse

at tewhich is received at toand then another pulse at te+ dtewhich is received at to+ dto Sincethe comoving distance r is a constant, we require dte/dto = ae/ao We have found a formula forcosmological time dilation

Of course time dilation also applies to the periods (and hence frequencies) of light

Choices of the Time Coordinate

It is sometimes convenient to change to time coordinate to conformal time η defined via dη =

c a−1(t) dt Then the RW metric becomes

ds2 = a2(t)dη2− dχ2− S2(χ)dψ2

In this coordinate system, light travels along 45◦lines: ds = 0, so dη = dχ and hence η = χ.While the time variable t is the proper time measured by a comoving observer, t is difficult toobserve for distant galaxies

If a(t) is a monotonically increasing function of time, we could also use a as a time variable.However, a is also not observable, but the redshift z(t) = a0/a(t) − 1 is an observable

6 It is incorrect to use the special-relativistic Doppler formula to obtain v from z because, in general, v is not a constant: H(t) is a function of time In the limit of low z, H is approximately constant, so the special-relativistic formula works See also CP Section 3.3

2 FRIEDMANN-ROBERTSON-WALKER COSMOLOGICAL MODEL 25

Figure 2.3: (a) Observed light-curve width factor w vs 1 + z The blue squares correspond to low redshift SNe, and the red circles are for high-redshift SNe used in this paper The band delineated by the black dash-dotted lines corresponds to stretch values 0.7 - 1.3, which encompass the bulk of the data, except for two outliers The green line shows the best linear fit to the data The band delineated by the two green dashed curves corresponds to the ±1σ values (b) Stretch s vs 1 + z Stretch is defined as the observed light-curve width w divided by 1 + z for each SN The points and lines are defined as in (a) From Goldhaber et al 1997

2.5 Dynamics of the Expansion

Conservation of Energy

Consider a homogeneous universe filled with material with density ρ and pressure P

2 FRIEDMANN-ROBERTSON-WALKER COSMOLOGICAL MODEL 26

The first law of thermodynamics (conservation of energy) requires that as a

The Friedmann Equation

Now we will obtain a(t) using a quasi-Newtonian argument

Consider a sphere of radius a(t) A comoving observer on the surface of the sphere feels the gravityfrom matter inside the sphere7

In the limit of small distances and velocities, the total energy of this observer is a constant givenby

2 FRIEDMANN-ROBERTSON-WALKER COSMOLOGICAL MODEL 27

Using equation (2.31) and dividing by ˙a gives

The acceleration is similar to the Newtonian case, but with an active gravitational mass density

ρ + 3P/c2 This says that pressure gravitates in GR At first, it might seem that the pressure Pshould cause an acceleration as opposed to the deceleration given above This would be true ifone was sitting on the surface of a balloon with high pressure inside and low pressure outside In

this case, however the force arise due to the pressure gradient – in the FRW context, however, the

pressure is homogeneous so there are no gradients

Note that both sides of equation (2.34) are linear in a So if a(t) is a solutions, this means that

a0 = αa is also a solution for arbitrary α The radii of all shells in the dust cloud evolve in thesame way so if initially of uniform density, the sphere will remain uniform density

General Relativity is needed to obtain the relationship between the integration constant E and thecurvature parameter K This gives Friedmann’s equation:

 (Ω − 1)K

− 1

(2.41)

As Ω → 1, the scale factor becomes larger than the Hubble length c/H and goes to infinity

2 FRIEDMANN-ROBERTSON-WALKER COSMOLOGICAL MODEL 28

2.6 Solutions to Friedmann equation

Solutions for matter-dominated K = 0

Here we assume Ωv= Ωr= 0 and Ωm = 1 This is also known as “Einstein-de Sitter”

Let us introduce the parameter

Note that a∗ is a constant if the universe is matter-dominated.

The Friedmann equation, equation (2.36) becomes

t = 2

3H

−1

Solutions for matter dominated universe with K = ±1

The solutions to equation (2.36) can be obtained is we transform to “conformal time” dη = c dt/a

a02 = 8πG3c2 ρ a4 − K a2

(2.48)where primes denotes derivatives with respect to conformal time

Using the definition of a∗from above, equation (2.48) is

2 FRIEDMANN-ROBERTSON-WALKER COSMOLOGICAL MODEL 29

For K = +1, solutions to equation (2.48) are

Consider a vacuum-dominated Universe with ρm = ρr = 0 Note that P = −ρc2 This is known

as the “de Sitter” model In this case, equation (2.34) becomes

Provided ρV > 0, then the first two terms will become large compared to the term on the rhs, so

we will be driven towards a behaviour which differs infinitesimally from the K = 0 case

If K = 0 then the solution is

with H =p8πGρv/3 This will be useful later when we consider inflation

8 Note error in Peacock

2 FRIEDMANN-ROBERTSON-WALKER COSMOLOGICAL MODEL 30

Summary

Since ρr ∝ a−4, ρm ∝ a−3 and ρv = const, one can see that radiation will always dominate atearly times and the vacuum (if non-zero) will always dominate at late times The observed matterdensity is such that over much of the Universe’s history matter has dominated the density andhence the expansion

Note that at early times, both terms on the l.h.s of equation (2.36) become large in comparisonwith the r.h.s which is a constant: −K c2 So for a matter and radiation dominated Universe, astearly times the dynamics behaves as if K = 0

2.7 Horizons

Let us consider the comoving distance a photon can travel from time t1to time t0 Since the motion

of a photon along a radial ray is a dχ = c dt we can obtain χ in terms of t

It is interesting to consider the particle horizon, the maximum comoving distance a photon can

travel from t = 0 till a later time t This corresponds to the comoving size of the observable

Universe

For the flat matter-dominated case, note that as z → ∞, a0r → H2c

0 = 3000h−1 Mpc This is thecomving size of the observable Universe9

Aside: note that the particle horizon scales as H−1, which is proportional to the age of the verse So at earlier times, the particle horizon was much smaller For example, at the time ofrecombination the horizon size corresponded to a patch with angular diameter ∼ 1◦ Yet the CMB

Uni-9 This scale has only recently come within the range of cosmological simulations.

2 FRIEDMANN-ROBERTSON-WALKER COSMOLOGICAL MODEL 31

is smooth on much larger scales (360◦) if these patches were not in causal contact, how did they establish homogeneity?

(This is know as the “horizon problem”)

3 CLASSIC OBSERVATIONAL COSMOLOGY 33

3 Classic Observational Cosmology

The classic cosmological tests are based on the global proprties of the FRW modeland its expansion history More modern tests based on the growth of structure will

be discussed later

3.1 Distance Measures in Cosmology

Conceptually, the simplest distance is the proper distance, i.e the ruler-measureddistance between two points This has little practical value since there is no way

We would like to calculate the observed bolometric flux (total energy per unit time

per unit area) As we saw before, the frequency at which the photons are received

is shifted but so is their rate, so that the bolometric flux received is

3 CLASSIC OBSERVATIONAL COSMOLOGY 34

where L is the bolometric luminosity of the source This allows us to define a

L = 1, so this quantity is independent of distance This

is one reason why Olbers’ paradox does not apply to an expanding universe.

3 CLASSIC OBSERVATIONAL COSMOLOGY 35

Comoving radial coordinate distance, χ

Typically, we do not observe χ directly — but we do observe z Therefore, it ismore convenient in practice to obtain expressions for χ and SK(χ) in terms of z.The former can be obtained by noting that for photon traveling from a distancesource to us c dt = −a dχ, so

χ =

Z z 0

c

a0H(z0)dz

0

All that remains is to plug in the expression for H(z) and to integrate from z0 = 0

to z0 = z Note that there is no closed-form solution for the general case in which

Ωv 6= 0 For the case of matter and radiation only it is possible to obtain sions in closed form (see Peacock Section 3.4) Of course it is straightforward tonumerically integrate the appropriate equations as a function of z

expres-For an Ωm= 1, Ωv = 0 cosmology, the angular diameter distance has the propertythat it reaches a maximum at z = 1.25 Objects at distances beyond z = 1.25

therefore begin to increase in angular size! See Peacock, Fig 3.7.

3.2 Cosmological tests based on standard candles and rods

From the 30s to the early 60s, observational cosmology was focused primarily onmeasure the constants H0and q0(a deceleration parameter closely related to Ωm).These classical tests used standard candles or standard rods observed at differentredshifts to obtain DLor DAand hence Ωm The standard objects were galaxies.However, there is now good evidence that galaxies evolve with time, certainly viaevolution of their stellar populations and probably by merging The uncertainty

in the evolution was a lest as large as the cosmological effects! These classicalmethods have been revitalized by the use supernovae of Type Ia (SNIa) as standardcandles and the acoustic peaks as standard rods

3 CLASSIC OBSERVATIONAL COSMOLOGY 36

Type Ia Supernovae Type Ia supernovae are due to accreting white dwarf stars

in binary systems which become unstable and explode The Chandrasekhar mass

of 1.4 M for the stability of a white dwarf gives some theoretical justificationfor the standard candle assumption However, the effects of chemical composition

on the explosion are, as yet, not well understood Since the overall abundance ofheavy elements increases with time, one might worry that the SNe at high z todiffer from those at low z

The results of Perlmutter et al.1999 give the degenerate constraint

0.8Ωm,0− 0.6Ωv,0 = −0.2 ± 0.1 (3.11)

A second group ‘(“the high-z supernova team”) finds consistent results

CMB Acoustic Peaks The best standard “rod” is the acoustic sound speed zon at the time of recombination This yields a constraint on the sum of Ωm,0 +

hori-Ωv,0= Ω = 1.04 ± 04 with some weak assumptions about h, Ωb etc

Both distance measures have degeneracies in the Ωm–Ωv plane However, theSNIa and CMB tests are at different redshifts, this degeneracy is broken11

Number Counts Finally, a volume test can also be applied For example, pose that individual galaxies are not good standard candles individually but thepopulation is, i.e the luminosity function is independent of time Then, if theircomoving number density is constant, we can calculate the number of sources as

sup-a function of flux, S by considering shells of with volume dV = 4πsup-a3

0SK(χ)2dχand convert the luminosities to fluxes via DL This method has the advantage thatredshifts are not required

In practice, it is likely that both luminosities and comoving number densitiesevolve Indeed, one of the puzzles of galaxy formation is that these seem to evolve

much more strongly than expected based on the simplest evolutionary models.

It is possible to use other objects as density markers For example, the comovingspace density of massive clusters of galaxies (above some mass threshold) alsoevolves with time Unlike the case of galaxies, however, this evolution is driven

by accretion of dark matter It is well known how this evolution behaves for agiven cosmological model

11 Although if the vacuum energy is actually quintessence, there is a further parameter.

3 CLASSIC OBSERVATIONAL COSMOLOGY 37

Figure 3.1: Experimental determinations of Ωmand ΩΛ The shaded regions showthe range of values allowed by current experiments The smooth “blimp” goingfrom lower left to upper right show the results from the high-redshift supernovae(labeled “SNIa”) The jagged shaded regions running from near the upper leftdown to the lower right show the results from from the CMB experiments (labeled

“Boomerang 98 + MAXIMA-I”) The jagged black contour lines show how much

Ωmand ΩΛare allowed when the results from both types of experiment are bined The straight diagonal line indicates combinations with Ω = Ωm+ ΩΛ = 1.Figure taken from Jaffe et al 2000, submitted to Physical Review Letters

com-3 CLASSIC OBSERVATIONAL COSMOLOGY 38

3.3 Ages and the Hubble Constant

The Age of the Universe in FRW models

To obtain the age of the Universe as a function of z, we can use the propagation

of light along null geodesics to show that

The age t0 is obtained by letting z → ∞ Note that the age of the Universe, t0,

scales with the Hubble time

is just Ωm

Observational Constraints on the Age of the Universe

1 Globular clusters arguably the best clock, as discussed in Chapter 1 denBerg et al., 1996 (Annu Rev Astron Astrophys 34, 461) conclude

Van-t0 > 12 Gyr at ∼ 95% CL Krauss and Chaboyer(2002) find that the age ofthe Universe > 11.2 Gyr at 95% CL, with a best fit age of 13.4 Gyr

2 White dwarf cooling gives and age of 7.3 ± 1.5 Gyr for the age of the

Galac-tic disk and 12.7 ± 0.7 (95% CL) for the age of the globular cluster M4(Hansen et al 2002) The age of the disk corresponds to z ∼ 1.5 in thecurrent best-fit cosmology

3 Decay of radioactive isotopes with long half-lives can be used to age-date

stars Observation of238U in a single old star gives it an age of 12.5 ± 3 Gyr(Cayrel et al 2001)

1 INTRODUCTION TO COSMOLOGY 9

Figure 1.6: CMB... 35

3 CLASSIC OBSERVATIONAL COSMOLOGY 33

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1 INTRODUCTION TO COSMOLOGY 7

Figure 1.5: Three false color images of the sky as seen at microwave frequencies