Chapter 29 The Hubble Expansion -...

Chapter 29

The Hubble Expansion

The observational characteristics of the Universe coupledwith theoretical interpretation to be discussed further insubsequent chapters, allow us to formulate astandard pic-ture of the nature of our Universe.

29.1 The Standard Picture

The standard picture rests on but a few ideas, but they have profoundsignificance for the nature of the Universe.

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922 CHAPTER 29. THE HUBBLE EXPANSION

29.1.1 Mass Distribution on Large Scales

Observations indicate that the Universe ishomogeneous(no preferred place) andisotropic (no preferred direction),when considered on sufficiently large scales.

• When averaged over distances of order50 Mpc, thefluctuation in mass distribution is of order unity,δM/M ≃ 1

• When averaged over a distance of4000 Mpc(com-parable to the present horizon),δM/M ≤ 10−4

• Thus, averaged over a large enough volume,no partof the Universe looks any different from any otherpart.

• The idea that the Universe is homogeneous andisotropic on large scales is called thecosmologicalprinciple.

The cosmological principle as implementedin general relativity is the fundamental theo-retical underpinning of modern cosmology.

29.1. THE STANDARD PICTURE 923

The cosmological principle should not be confused withthe perfect cosmological principle, which was the under-lying idea of thesteady state theory of the Universe.

• In the perfect cosmological principle, the Universe isnot only homogeneous in space but also in time.

• Thus it looks the same not only from any place, butfrom any time.

• This idea once had a large influence on cosmologybut is no longer considered viable because it is incon-sistent with modern observations that show a Uni-verse evolving in time.


29.1.2 The Universe Is Expanding

Observations indicate that theUniverse is expanding; theinterpretation of general relativity is that this is becausespace itself is expanding

• The distanceℓ between conserved particles is chang-ing according to theHubble law

v ≡dℓdt

= H0ℓ,

deduced from redshift of light from distant galaxies.

• H0 is theHubble parameter (or Hubble “constant”,but it changes with time; the subscript zero indicatesthat this is the valueat the present time).

• The Hubble parameter can be determined by fittingthe above equation to the radial velocities of galaxiesat known distances.

• The uncertainty inH0 is sometimes absorbed into adimensionless parameterh by quoting

H0 = 100h km s−1 Mpc−1 = 3.24×10−18h s−1,

whereh is a scaling parameter of order one.

• The currently accepted value of the Hubble constantis H0 = 72±8 km s−1 Mpc−1, corresponding toh =

0.72.

• Note: units ofH0 are actually(time)−1.


• We may define aHubble length LH through

LH =c

H0≃ 4000 Mpc.

• Thus, for a galaxy lying a Hubble length away from us,

v =dℓdt

= H0c

H0= c,

• This implies that the recessional velocity of a galaxy furtheraway thanLH exceeds the speed of light, if the observed redshiftsare interpreted as Doppler shifts.

• We shall find that the redshift of the receding galaxies isnot aDoppler shift caused by velocities in spacetime, but is a con-sequence of theexpansion of space itself, which stretches thewavelengths of all light.

• The light speed limit of special relativity applies to velocities inspacetime; it does not apply to spacetime itself.


It is important to understand that local objects are not par-taking of the general Hubble expansion.

• The Hubble expansion isis not caused by a force. Itonly occurs when forces between objects are negligi-ble.

• Smaller objects, such as our bodies, are held togetherby chemical (electrical) forces. They do not expand.

• Larger objects like planets, solar systems, and galax-ies are also held together by forces, in this case gravi-tational in origin. They generally do not expand withthe Universe either.

• It is only on much larger scales (beyond superclus-ters of galaxies) that gravitational forces among lo-cal objects are sufficiently weak to cause negligibleperturbation on the overall expansion.


29.1.3 The Expansion is Governed by General Relativity

• It is possible to understand much of the expanding Universeus-ing only Newtonian physics and insights borrowed from relativ-ity.

• However, in the final analysis there are serious technical andphilosophical difficulties that eventually arise and that requirereplacement of Newtonian gravitation with the Einstein’s gen-eral theory of relativity for their resolution.

• Central to these issues is the understanding of space and time inrelativity compared with that in classical Newtonian gravitation.

– In relativity, space and time are not separate but enter as aunified spacetime continuum.

– Even more fundamentally, space and time in relativity arenot a passive background upon which events happen.

– Relativistic space and time are not “things” but rather areabstractions expressing a relationship between events.

Thus, in this view, space and time do not have a separate exis-tence from events involving matter and energy.

• On fundamental grounds, the gravitational curvature radius ofthe Universecould be (it actually isn’t) comparable to the radiusof the visible Universe→ general relativity.


29.1.4 There Is a Big Bang in Our Past

Evidence suggests that the Universe expanded from an ini-tial condition of very high density and temperature.

• This emergence of the Universe from a hot, denseinitial state is called thebig bang.

• The popular (mis)conception that the big bang was agigantic explosion is in error because it conveys theidea that it happened in space and time, and that theresulting expansion of the Universe is a consequenceof forces generated by this explosion.

• The general relativistic interpretation of the big bangis that it did not happenin spacetime but thatspaceand time themselves are created in the big bang.

• Thus, questions of “what happened before the bigbang?” and “what is the Universe expanding into?”become meaningless because these questions presup-pose the existence of a space and time backgroundupon which events happen.

• The big bang should be viewed not as an explosionbut as an initial condition for the Universe.

• (Very) loosely we may view the big bang as an “ex-plosion” because of the hot, dense nature of the ini-tial state. But then we should view the explosion ashappening at all points in space, so that it makes nosense to talk about a “center” for the big bang.


General relativity implies that the initial state was aspace-time singularity.

• Whether general relativity is correct on this issue willhave to await a full theory of quantum gravitation,since general relativity cannot be applied too closeto the initial singularity (on scales below the Planckscale) without incorporating the principles of quan-tum mechanics.

• However, for most (but not all!) issues in cosmologythe question of whether there was an initial spacetimesingularity is not relevant.

• For those issues, all that is important is that once theUniverse expanded beyond the Planck scale it wasvery hot and very dense.

• This hot and dense initial state is what we shall meanin simplest form when we refer to the big bang.


29.1.5 Particle Content Influences Evolution of the Universe

The Universe contains a variety of particles and their as-sociated fields that influence strongly its evolution.

• The “ordinary” matter composed of things that wefind around us is generally termedbaryonic matter.

• Baryons are the strongly interacting particles of half-integer spin such as protons and neutrons.

• Although baryonic matter is the most obvious matterto us, data indicate thatonly a small fraction of thetotal mass in the Universe is baryonic.

• The bulk of the mass in the Universe appears to bein the form ofdark matter, which is easily detectedonly through its gravitational influence.

• We don’t know what dark matter is, but there is goodreason to believe that it is primarily composed ofas-yet undiscovered elementary particles that inter-act only weakly with other matter and radiation.

• There is also growing evidence that the evolution ofthe Universe is strongly influenced bydark energy,which permeates even empty space and causes grav-ity to effectively become repulsive.

• We do not know the origin of dark energy but a rea-sonable guess is that it is associated with quantum-mechanical fluctuations of the vacuum, or possiblywith an as-yet undiscovered elementary particle field.


A fundamental distinction for particles and associated fields is whetherthey aremassless or massive.

• Lorentz-invariant quantum field theories require that masslessparticles must move at light speed and that particles with finitemass must move at speeds less than that of light.

• Therefore, massless particles like photons, gravitons, and glu-ons, and nearly massless particles like neutrinos, are highly rel-ativistic. Roughly, particles with rest massm are non-relativisticat those temperaturesT wherekT << mc2

– Electrons have a rest mass of511 keV and they are non-relativistic at temperatures below about6×109 K.

– Protons have a rest mass of931 MeVand they remain non-relativistic up to temperatures of about1013 K.

– Conversely (massless) photons, gluons, gravitons, and (nearlymassless) neutrinos are always relativistic

• In cosmology, it is common to refer to such massless or nearlymassless particles asradiation.

• Conversely, particles with significant mass have velocities wellbelow that of light (unless temperatures are extremely high) andare non-relativistic. In cosmology non-relativistic particles aretermedmatter (or dust).

• Non-relativistic matter has low velocity and exerts little pressurecompared with relativistic matter.

The present energy density of the Universe is dominated by non-relativistic matter and by vacuum energy density (dark energy). How-ever, this was not always the case: the early Universe was dominatedby radiation.


29.1.6 The Universe is Permeated by a Microwave Background

As noted in the preceding section, radiation plays a role in the evolu-tion of the Universe.

The most important feature of the standard picture, otherthan the Hubble expansion, is that the Universe is filledwith a photon background lying in the microwave regionof the spectrum that is extremely smooth and isotropic.

• Any theoretical attempt to understand the standardpicture must as a minimal starting point account forthe expansion of the Universe and for thiscosmic mi-crowave background (CMB) that fills all of space.

• Conversely, precision measurements of tiny fluctu-ations in the otherwise smooth CMB are presentlyturning cosmology into a highly quantitative science.

• Although the CMB currently peaks in the microwaveregion of the spectrum, its wavelength has beensteadily redshifting since the big bang and it wasoriginally much higher energy radiation.

• For example, near the time when the temperaturedropped low enough for electrons to combine withprotons the spectrum of what is now the microwavebackground peaked in the near-infrared region.

• The CMB accounts for more than 90% of the photonenergy density (less than 10% in starlight).


80

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Age (109 years)

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Time (Billion Years)

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Figure 29.1:Expansion of the Universe for three values of the Hubble constant( km s−1 Mpc−1). The corresponding Hubble times estimating the age of the Uni-verse are indicated below the lower axis. Redshift is indicated on the right axis.

Hubble Law : v ≡dℓdt

= H0ℓ H0 ≃ 3.24×10−18h s−1,

The Hubble expansion is most consistently interpreted in terms of anexpansion of space itself.

• Convenient to introduce ascale factora(t) that describes howdistances scale because of the expansion of the Universe. Hub-ble’s law for evolution of the scale factor is illustrated inFig. 29.1.

• The slopes of the straight lines plotted there define the HubbleconstantH0.


We shall interpret the Hubble constant as being characteristic of aspace (thus constant for the Universe at a given time) but having pos-sible time dependence as the Universe evolves.

• The subscript zero onH0 is used to denote that this is the valueof the Hubble constanttoday, in anticipation that the coefficientgoverning the rate of expansion changes with time.

• One often refers to theHubble parameter H = H(t), meaning anH that varies with time, and to theHubble constant H0 to meanthe value ofH(t) today.

• It is also common to use the term Hubble constant loosely tomean a parameter that is constant in space but that may changewith time.

Hubble’s original value wasH0 = 550 km s−1 Mpc−1.This is approximately an order of magnitude larger thanthe presently accepted value of about72 km s−1 Mpc−1.The large revision (which implies a corresponding shift inthe perceived distance scale of the Universe) was because

• Hubble’s original sample was a poorly-determinedone based on relatively nearby galaxies.

• There was confusion over the extra-galactic distancescale at the time because of issues like misinterpret-ing types of variable stars and failing to account forthe effect of dust on light propagation.


Table 29.1: Some peculiar velocities in the Virgo ClusterGalaxy Redshift (z) vr (km s−1)IC 3258 −0.001454 −436

M86 (NGC 4406) −0.000901 −270NGC 4419 −0.000854 −256

M90 (NGC 4569) −0.000720 −216M98 (NGC 4192) −0.000467 −140

NGC 4318 +0.004086 +1226NGC 4388 +0.008426 +2528

IC 3453 +0.008526 +2558NGC 4607 +0.007412 +2224NGC 4168 +0.007689 +2307

M99 (NGC 4254) +0.008036 +2411NGC 4354 +0.007700 +2310

Source: SIMBAD

29.1.7 Redshifts

If a galaxy has a spectral line normally at wavelengthλemit that isshifted to a wavelengthλobs when we observe it, theredshift z is

z ≡λobs−λemit

λemit

.

• A negative value ofz corresponds to ablueshift.

• A positive value ofz corresponds to aredshift.

• Since the Universe is observed to be expanding, the Hubble lawgives rise only to redshifts.

• Thus, any blueshifts correspond topeculiar motion of objectswith respect to the general Hubble flow (Table 29.1).

• “Peculiar” = “a property specific to an object” ( 6= “strange”).


The few galaxies observed to have blueshifts are nearby,in the Local Group or the Virgo Cluster, where peculiarmotion is large enough to partially counteract the overallHubble expansion.

• The Andromeda Galaxy (M31), which is part of ourLocal Group of galaxies, is moving toward us witha velocity of about300 km s−1 and will probablycollide with the Milky Way in several billion years.

• The most extreme blueshifts (negative radial ve-locities) found in the Virgo Cluster are the largestblueshifts known with respect to our galaxy.


29.1.8 Expansion Interpretation of Redshifts

The redshifts associated with the Hubble law may beap-proximately viewed as Doppler shifts forsmall redshifts.

• This interpretation is problematic for large redshifts.

• The Hubble redshifts (large and small) are most con-sistently interpreted in terms of the expansion ofspace, which may be parameterized by thecosmicscale factor a(t).

• If all peculiar motion is ignored the time dependenceof the expansion is lodged entirely in the time depen-dence ofa(t), andall distances simply scale with thisfactor.

• A simple analogy on a 2-dimensional surface will beexploited in later discussion:distances between dotsplaced on the surface of a balloon all scale with theradius of the balloon as it expands.

• In the general casea(t) may be interpreted as settinga scale for all cosmological distances.

• In thespecial case of a closed universe, we may thinkof a(t) loosely as a radius for the universe.


As we shall show, light traveling between two galaxies separated bycosmological distances follows the null curve

c2dt2 = a2(t)dr2

where the scale factora(t) sets the overall scale for distances in theUniverse at timet andr is the coordinate distance. Therefore,

cdta(t)

= dr.

Consider a wavecrest of monochromatic light with wavelength λ ′ thatis emitted at timet ′ from one galaxy and detected with wavelengthλ0

at time t0 in the other galaxy. Integrating both sides of the aboveequation gives

c∫ t0

t ′

dta(t)

=∫ r

0dr = r.

The next wavecrest is emitted from the first galaxy att = t ′+λ ′/c andis detected in the second galaxy at timet = t0+λ0/c. For the secondwave crest, integrating as above assuming that the intervalbetweenwavecrests is negligible compared with the timescale for expansionof the Universe gives

c∫ t0+λ0/c

t ′+λ ′/c

dta(t)

=∫ r

0dr = r.

Since from above

c∫ t0

t ′

dta(t)

= r c∫ t0+λ0/c

t ′+λ ′/c

dta(t)

= r

we may equate the left sides to obtain∫ t0

t ′

dta(t)

=

∫ t0+λ0/c

t ′+λ ′/c

dta(t)

,


which may be rewritten as

∫ t ′+λ ′/c

t ′

dta(t)

+∫ t0

t ′+λ ′/c

dta(t)

︸︷︷︸Cancel

=∫ t0

t ′+λ ′/c

dta(t)

︸︷︷︸Cancel

+∫ t0+λ0/c

t0

dta(t)

∫ t ′+λ ′/c

t ′

dta(t)

=

∫ t0+λ0/c

t0

dta(t)

.

Because the interval between wave crests is negligible compared withthe characteristic timescale for expansion, we may bring the factor1/a(t) outside the integral to obtain

1a(t ′)

∫ t ′+λ ′/c

t ′dt =

1a(t0)

∫ t0+λ0/c

t0dt −→

λ ′

λ0=

a(t ′)a(t0)

,

This demonstrates explicitly

• the stretching of wavelengths caused by the expansion of theUniverse

• that the cosmological redshift isnot a Doppler shift (no veloci-ties appear in the formula).

From the definition for the redshiftz,

1+ z = 1+λ0−λ ′

λ ′=

λ0

λ ′=

a(t0)a(t ′)

−→ z =a(t0)a(t ′)

−1.

It is conventional to normalize the scale parameter so that its value inthe present Universe is unity,a(t0) ≡ 1, in which case

z =1

a(t ′)−1.


Thus the redshift that enters the Hubble law

• dependsonly on the ratio of the scale parameters at the time ofemission and detection for the light,

1+ z =a(t0)a(t ′)

.

• It is independent of the details of how the scale parameter changedbetween the two times.

• The ratio of the scale parameters at two different times is deter-mined by the cosmological model in use.

• Measuring the redshift of a distant object is then equivalent tospecifying the scale parameter of the expanding Universeat thetime when the light was emitted from the distant object, relativeto the scale parameter today.

• Thus, measuring redshifts tests cosmological models.

EXAMPLE: If from the spectrum for a distant quasar onedetermines thatz = ∆λ/λ = 5, the scale factor of the Uni-verse at the time that light was emitted from the quasarwas equal to16 of the scale factor for the current Universe:

a(t ′)a(t0)

=1

z +1=

16.


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The preceding discussion indicates that we may use thescale factora(t) or the redshiftz interchangeably as timevariables for a universe in which the scale parameterchanges monotonically (compare the right and left axesof the above figure).


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29.1.9 The Hubble Time

The Hubble parameter has the dimensions of inverse time:

[H0] = [km s−1 Mpc−1] = time−1.

Thus,1/H0 defines a time called theHubble time τH,

τH ≡1

H0= 9.8h−1×109 y.

• If the Hubble law is obeyed with a constant value ofH0, theintercept of the curve with the time axis gives the time when thescale factor was zero.

• Hence, the value ofτH = 1/H0 is sometimes quoted as theage ofthe Universe.


• This is simply a statement that if the expansion rate today is thesame as the expansion rate since the big bang, the time for theUniverse to evolve from the big bang to today is the inverse ofthe Hubble constant.

• In general the Hubble time is not a correct age for the Universebecause the Hubble parameter can remain constant only in a Uni-verse devoid of matter, fields, and energy.

• The realistic Universe contains all of these and the expansion ofthe Universe is accelerated (positively or negatively, dependingon the details) because of gravitational interactions.

• In later cosmological models we shall see that the age of theUni-verse may be substantially longer or shorter thanτH, dependingon the details of the matter, fields, and energy contained in theUniverse.


29.1.10 A Two-Dimensional Hubble Expansion Model

Figure 29.2 illustrates a two-dimensional Hubble expansion as viewedfrom two different vantage points (Show interactive animation of this).

See:http://csep10.phys.utk.edu/guidry/darkEnergy/universeExpander.html


Figure 29.2:The same two-dimensional Hubble expansion as viewed from twodifferent vantage points.


Figure 29.3:Hubble parameter extracted from observations.

29.1.11 Measuring the Hubble Constant

The Hubble constant may be determined observationallyby measuring the redshift for spectral lines and comparingthat with the distance to objects at a range of distances suf-ficiently large that peculiar motion caused by local grav-itational attraction is small compared with the motion as-sociated with the Hubble expansion.

Figure 29.3 illustrates the determination of the Hubble constant froma variety of observations. The adopted value is

H0 = 72±8 km s−1 Mpc−1,

corresponding toh = 0.72.

29.2. LIMITATIONS OF THE STANDARD WORLD PICTURE 947

29.2 Limitations of the Standard World Picture

We shall demonstrate that the standard picture has been remarkablysuccessful in describing many features of our Universe. However,there are two aspects of this picture suggesting that it is (at best) in-complete:

1. In order to get the big bang to produce the present universe, cer-tain assumptions about initial conditions must be taken as given.While not necessarily wrong, some of these assumptions seemunnatural by various standards.

2. As the expansion is extrapolated backwards, eventually one wouldreach a state of sufficient temperature and density that a fullyquantum mechanical theory of gravitation would be required.

• This is thePlanck era, and the corresponding scales of dis-tance, energy, and time are called thePlanck scale.

• Since we do not yet have a consistent theory ofquantumgravity, the presently understood laws of physics may beexpected to break down on the Planck scale.

• Thus the standard picture says nothing about the Universeat those very early times.

In later chapters we shall address these issues, to considerwhethermodifications of the standard picture can alleviate some of these prob-lems.

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