Chapter 16. The Hubble Expansion

Chapter 16 The Hubble Expansion The observational characteristics of the Universe coupled with theoretical interpretation to be discussed further in subsequent chapters, allow us to formulate a standard picture of the nature of our Universe. 16.1 The Standard Picture The standard picture rests on but a few ideas, but they have profound significance for the nature of the Universe. 435

436 CHAPTER 16. THE HUBBLE EXPANSION 16.1.1 Mass Distribution on Large Scales Observations indicate that the Universe is homogeneous (no preferred place) and isotropic (no preferred direction), when considered on sufficiently large scales. When averaged over distances of order 50 Mpc, the fluctuation in mass distribution is of order unity, δm/m 1 When averaged over a distance of 4000 Mpc (comparable to the present horizon), δm/m 10 4 Thus, averaged over a large enough volume, no part of the Universe looks any different from any other part. The idea that the Universe is homogeneous and isotropic on large scales is called the cosmological principle. The cosmological principle, as implemented in general relativity, is the fundamental theoretical underpinning of modern cosmology.

16.1. THE STANDARD PICTURE 437 The cosmological principle should not be confused with the perfect cosmological principle, which was the underlying idea of the steady state theory of the Universe. In the perfect cosmological principle, the Universe is not only homogeneous in space but also in time. Thus it looks the same not only from any place, but from any time. This idea once had a large influence on cosmology but is no longer considered viable because it is inconsistent with modern observations that show a Universe evolving in time.

438 CHAPTER 16. THE HUBBLE EXPANSION 16.1.2 The Universe Is Expanding Observations indicate that the Universe is expanding; the interpretation of general relativity is that this is because space itself is expanding The distance l between conserved particles is changing according to the Hubble law v dl dt = H 0l, deduced from redshift of light from distant galaxies. H 0 is the Hubble parameter (or Hubble constant, but it changes with time; the subscript zero indicates that this is the value at the present time). The Hubble parameter can be determined by fitting the above equation to the radial velocities of galaxies at known distances. The uncertainty in H 0 is sometimes absorbed into a dimensionless parameter h by quoting H 0 = 100h km s 1 Mpc 1 = 3.24 10 18 h s 1, where h is a dimensionless parameter of order 1. The currently accepted value of the Hubble constant is H 0 = 72 ± 8 km s 1 Mpc 1, corresponding to h = 0.72. Note: units of H 0 are actually (time) 1.

16.1. THE STANDARD PICTURE 439 We may define a Hubble length L H through L H = c H 0 4000 Mpc. Thus, for a galaxy lying a Hubble length away from us, v = dl dt = H c 0 = c, H 0 This implies that the recessional velocity of a galaxy further away than L H exceeds the speed of light, if the observed redshifts are interpreted as Doppler shifts. We shall find that the redshift of the receding galaxies is not a Doppler shift caused by velocities in spacetime, but is a consequence of the expansion of space itself, which stretches the wavelengths of all light. The light speed limit of special relativity applies to velocities in spacetime; it does not apply to spacetime itself.

440 CHAPTER 16. THE HUBBLE EXPANSION It is important to understand that local objects are not partaking of the general Hubble expansion. The Hubble expansion is is not caused by a force. It only occurs when forces between objects are negligible. Smaller objects, such as our bodies, are held together by chemical (electrical) forces. They do not expand. Larger objects like planets, solar systems, and galaxies are also held together by forces, in this case gravitational in origin. They generally do not expand with the Universe either. It is only on much larger scales (beyond superclusters of galaxies) that gravitational forces among local objects are sufficiently weak to cause negligible perturbation on the overall expansion.

16.1. THE STANDARD PICTURE 441 16.1.3 The Expansion is Governed by General Relativity It is possible to understand much of the expanding Universe using only Newtonian physics and insights borrowed from relativity. However, in the final analysis there are serious technical and philosophical difficulties that eventually arise and that require replacement of Newtonian gravitation with the Einstein s general theory of relativity for their resolution. Central to these issues is the understanding of space and time in relativity compared with that in classical Newtonian gravitation. In relativity, space and time are not separate but enter as a unified spacetime continuum. Even more fundamentally, space and time in relativity are not a passive background upon which events happen. Relativistic space and time are not things but rather are abstractions expressing a relationship between events. Thus, in this view, space and time do not have a separate existence apart from events involving matter and energy. On fundamental grounds, the gravitational curvature radius of the Universe could be comparable to the radius of the visible Universe general relativity.

442 CHAPTER 16. THE HUBBLE EXPANSION 16.1.4 There Is a Big Bang in Our Past Evidence suggests that the Universe expanded from an initial condition of very high density and temperature. This emergence of the Universe from a hot, dense initial state is called the big bang. The popular (mis)conception that the big bang was a gigantic explosion is in error because it conveys the idea that it happened in space and time, and that the resulting expansion of the Universe is a consequence of forces generated by this explosion. The general relativistic interpretation of the big bang is that it did not happen in spacetime but that space and time themselves are created in the big bang. What happened before the big bang? or what is the Universe expanding into? are meaningless because these questions presuppose the existence of a spacetime background upon which events happen. The big bang should be viewed not as an explosion but as an initial condition for the Universe. Loosely we may view the big bang as an explosion because of the hot, dense initial state. But then we should view the explosion as happening at all points in space: there is no center for the big bang.

16.1. THE STANDARD PICTURE 443 General relativity implies that the initial state was a spacetime singularity. Whether general relativity is correct on this issue will have to await a full theory of quantum gravitation, since general relativity cannot be applied too close to the initial singularity (on scales below the Planck scale) without incorporating the principles of quantum mechanics. However, for most (but not all!) issues in cosmology the question of whether there was an initial spacetime singularity is not relevant. For those issues, all that is important is that once the Universe expanded beyond the Planck scale it was very hot and very dense. This hot and dense initial state is what we shall mean in simplest form when we refer to the big bang.

444 CHAPTER 16. THE HUBBLE EXPANSION 16.1.5 Particle Content Influences Evolution of the Universe The Universe contains a variety of particles and their associated fields that influence strongly its evolution. The ordinary matter composed of things that we find around us is generally termed baryonic matter. Baryons are the strongly interacting particles of half-integer spin such as protons and neutrons. Although baryonic matter is the most obvious matter to us, data indicate that only a small fraction of the total mass in the Universe is baryonic. The bulk of the mass in the Universe appears to be in the form of dark matter, which is easily detected only through its gravitational influence. We don t know what dark matter is, but there is good reason to believe that it is primarily composed of as-yet undiscovered elementary particles that interact only weakly with other matter and radiation. There is also growing evidence that the evolution of the Universe is strongly influenced by dark energy, which permeates even empty space and causes gravity to effectively become repulsive. We do not know the origin of dark energy.

16.1. THE STANDARD PICTURE 445 A fundamental distinction for particles and associated fields is whether they are massless or massive. Lorentz-invariant quantum field theories require that massless particles must move at light speed and that particles with finite mass must move at speeds less than that of light. Therefore, massless particles like photons, gravitons, and gluons, and nearly massless particles like neutrinos, are highly relativistic. Roughly, particles with rest mass m are non-relativistic at those temperatures T where kt << mc 2 Electrons have a rest mass of 511 kev and they are nonrelativistic at temperatures below about 6 10 9 K. Protons have a rest mass of 931 MeV and they remain nonrelativistic up to temperatures of about 10 13 K. Conversely (massless) photons, gluons, gravitons, and (nearly massless) neutrinos are always relativistic. In cosmology, it is common to refer to massless or nearly massless particles as radiation. Conversely, massive particles have v << c (unless temperatures are extremely high) and are non-relativistic. In cosmology nonrelativistic particles are termed matter (or dust). Non-relativistic matter has low velocity and exerts little pressure compared with relativistic matter. The energy density of the present Universe is dominated by non-relativistic matter and dark energy. But this was not always true: the early Universe was dominated by radiation.

446 CHAPTER 16. THE HUBBLE EXPANSION 16.1.6 The Universe is Permeated by a Microwave Background As noted in the preceding section, radiation plays a role in the evolution of the Universe. The most important feature beyond the Hubble expansion is that the Universe is filled with a microwave photon background that is extremely smooth and isotropic. Any theoretical attempt to understand the standard picture must as a minimal starting point account for the expansion of the Universe and for this cosmic microwave background (CMB) that fills all of space. Conversely, precision measurements of tiny fluctuations in the otherwise smooth CMB are presently turning cosmology into a highly quantitative science. Although the CMB currently peaks in the microwave region of the spectrum, its wavelength has been steadily redshifting since the big bang and it was originally much higher energy radiation. For example, near the time when the temperature dropped low enough for electrons to combine with protons the spectrum of what is now the microwave background peaked in the near-infrared region. The CMB accounts for more than 90% of the photon energy density (less than 10% in starlight).

16.2. THE HUBBLE LAW 447 Time (Billion Years) Scale Factor Relative to Today Now 80 60 72 0 1 60 72 80 3 5 10 16.2 12.2 13.5 Age (10 9 years) Figure 16.1: 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 Universe are indicated below the lower axis. Redshift is indicated on the right axis. 16.2 The Hubble Law Hubble Law : v dl dt = H 0l H 0 72 km s 1 Mpc 1, The Hubble expansion is most consistently interpreted in terms of an expansion of space itself. Convenient to introduce a scale factor a(t) that describes how distances scale because of the expansion of the Universe. Hubble s law for evolution of the scale factor is illustrated in Fig. 16.1. The slopes of the straight lines plotted there define the Hubble constant H 0.

448 CHAPTER 16. THE HUBBLE EXPANSION We shall interpret the Hubble constant as being characteristic of a space (thus constant for the Universe at a given time) but having possible time dependence as the Universe evolves. The subscript zero on H 0 is used to denote that this is the value of the Hubble constant today, in anticipation that the coefficient governing the rate of expansion changes with time. One often refers to the Hubble parameter H = H(t), meaning an H that varies with time, and to the Hubble constant H 0 to mean the value of H(t) today. It is also common to use the term Hubble constant loosely to mean a parameter that is constant in space but that may change with time. Hubble s original value was H 0 = 550 km s 1 Mpc 1. This is approximately an order of magnitude larger than the presently accepted value of about 72 km s 1 Mpc 1. The large revision (which implies a corresponding shift in the perceived distance scale of the Universe) was because Hubble s original sample was a poorly-determined one based on relatively nearby galaxies. There was confusion over the extra-galactic distance scale at the time because of issues like misinterpreting types of variable stars and failing to account for the effect of dust on light propagation.

16.2. THE HUBBLE LAW 449 Table 16.1: Some peculiar velocities in the Virgo Cluster Galaxy Redshift (z) v r (km s 1 ) IC 3258 0.001454 436 M86 (NGC 4406) 0.000901 270 NGC 4419 0.000854 256 M90 (NGC 4569) 0.000720 216 M98 (NGC 4192) 0.000467 140 NGC 4318 +0.004086 +1226 NGC 4388 +0.008426 +2528 IC 3453 +0.008526 +2558 NGC 4607 +0.007412 +2224 NGC 4168 +0.007689 +2307 M99 (NGC 4254) +0.008036 +2411 NGC 4354 +0.007700 +2310 Source: SIMBAD 16.2.1 Redshifts If a galaxy has a spectral line normally at wavelength λ emit that is shifted to a wavelength λ obs when we observe it, the redshift z is z λ obs λ emit λ emit. A negative value of z corresponds to a blueshift. A positive value of z corresponds to a redshift. Since the Universe is observed to be expanding, the Hubble law gives rise only to redshifts. Thus, any blueshifts correspond to peculiar motion of objects with respect to the general Hubble flow (Table 16.1). Peculiar = a property specific to an object ( = strange ).

450 CHAPTER 16. THE HUBBLE EXPANSION The few galaxies observed to have blueshifts are nearby, in the Local Group or the Virgo Cluster, where peculiar motion is large enough to partially counteract the overall Hubble expansion. The Andromeda Galaxy (M31), which is part of our Local Group of galaxies, is moving toward us with a velocity of about 300 km s 1 and will probably collide with the Milky Way in several billion years. The most extreme blueshifts (negative radial velocities) found in the Virgo Cluster are the largest blueshifts known with respect to our galaxy.

16.2. THE HUBBLE LAW 451 16.2.2 Expansion Interpretation of Redshifts The redshifts associated with the Hubble law may be approximately viewed as Doppler shifts for small redshifts. This interpretation is problematic for large redshifts. The Hubble redshifts (large and small) are most consistently interpreted in terms of the expansion of space, which may be parameterized by the cosmic scale factor a(t). If all peculiar motion is ignored the time dependence of the expansion is lodged entirely in the time dependence of a(t), and all distances simply scale with this factor. A simple analogy on a 2-dimensional surface will be exploited in later discussion: distances between dots placed on the surface of a balloon all scale with the radius of the balloon as it expands. In the general case a(t) may be interpreted as setting a scale for all cosmological distances. In the special case of a closed universe, we may think of a(t) loosely as a radius for the universe.

452 CHAPTER 16. THE HUBBLE EXPANSION As we shall show later, light traveling between two galaxies separated by cosmological distances follows the null curve defined by c 2 dt 2 = a 2 (t)dr 2 where the scale factor a(t) sets the overall scale for distances in the Universe at time t and r is the coordinate distance. Therefore, cdt a(t) = dr. Consider a wavecrest of monochromatic light with wavelength λ that is emitted at time t from one galaxy and detected with wavelength λ 0 at time t 0 in the other galaxy. Integrating both sides of the above equation gives t0 dt r c t a(t) = dr = r. The next wavecrest is emitted from the first galaxy at t = t +λ /c and is detected in the second galaxy at time t = t 0 + λ 0 /c. For the second wave crest, integrating as above assuming that the interval between wavecrests is negligible compared with the timescale for expansion of the Universe gives t0 +λ 0 /c dt r c t +λ /c a(t) = dr = r. 0 Since from above two equations, t0 c t dt a(t) = r c t0+λ 0/c t +λ /c we may equate the left sides to obtain t0 t dt t0+λ 0/c a(t) = dt t +λ /c a(t), 0 dt a(t) = r

16.2. THE HUBBLE LAW 453 The result may be rewritten as t0 t dt t0+λ 0/c a(t) = dt t +λ /c a(t), t +λ /c t dt a(t) + t0 t +λ /c t +λ /c t dt a(t) = t0 t +λ /c dt t0+λ 0/c a(t) = dt t 0 a(t). dt a(t) + t0+λ 0/c t 0 dt a(t) Because the interval between wave crests is negligible compared with the characteristic timescale for expansion, we may bring the factor 1/a(t) outside the integral to obtain 1 t +λ /c a(t dt = 1 t0 +λ 0 /c dt λ /c ) t a(t 0 ) t 0 a(t ) = λ 0/c a(t 0 ), and thus that λ λ 0 = a(t ) a(t 0 ).

454 CHAPTER 16. THE HUBBLE EXPANSION The result that we have just obtained, demonstrates explicitly that λ λ 0 = a(t ) a(t 0 ), The stretching of wavelengths (redshift) is caused by the expansion of the Universe (the change in a(t) while the photon is propagating). The cosmological redshift is not a Doppler shift (no velocities appear in this formula). From the definition for the redshift z and the preceding result, 1+z = 1+ λ 0 λ λ = λ 0 λ = a(t 0) a(t ) z = a(t 0) a(t ) 1. It is conventional to normalize the scale parameter so that its value in the present Universe is unity, a(t 0 ) 1, in which case z = 1 a(t ) 1, where a(t ) is the scale factor of the Universe when the light was emitted.

16.2. THE HUBBLE LAW 455 Thus the redshift that enters the Hubble law Depends only on the ratio of the scale parameters at the time of emission and detection for the light, 1+z = a(t 0) a(t ). It is independent of the details of how the scale parameter changed between the two times. The ratio of the scale parameters at two different times is determined by the cosmological model in use. Measuring the redshift of a distant object is then equivalent to specifying the scale parameter of the expanding Universe at the time when the light was emitted from the distant object, relative to the scale parameter today. Thus, measuring redshifts tests cosmological models. EXAMPLE: If from the spectrum for a distant quasar one determines that z = λ/λ = 5, the scale factor of the Universe at the time that light was emitted from the quasar was equal to 1 6 of the scale factor for the current Universe: a(t ) a(t 0 ) = 1 z+1 = 1 6.

456 CHAPTER 16. THE HUBBLE EXPANSION Time (Billion Years) 80 72 Scale Factor Relative to Today Now 60 0 1 60 72 80 3 5 10 16.2 12.2 13.5 Age (10 9 years) The preceding discussion indicates that we may use the scale factor a(t) or the redshift z interchangeably as time variables for a universe in which the scale parameter changes monotonically (compare the right and left axes of the above figure).

16.2. THE HUBBLE LAW 457 Time (Billion Years) 80 72 Scale Factor Relative to Today Now 60 0 1 60 72 80 3 5 10 16.2 12.2 13.5 Age (10 9 years) 16.2.3 The Hubble Time The Hubble parameter has the dimensions of inverse time: [H 0 ] = [km s 1 Mpc 1 ] = time 1. Thus, 1/H 0 defines a time called the Hubble time τ H, τ H 1 = 9.8h 1 10 9 y. H 0 If the Hubble law is obeyed with a constant value of H 0, the intercept of the curve with the time axis gives the time when the scale factor was zero. Hence, the value of τ H = 1/H 0 is sometimes quoted as an estimate of the age of the Universe.

458 CHAPTER 16. THE HUBBLE EXPANSION This is simply a statement that if the expansion rate today is the same as the expansion rate since the big bang, the time for the Universe to evolve from the big bang to today is the inverse of the Hubble constant. In general the Hubble time is not a correct age for the Universe because the Hubble parameter can remain constant only in a Universe devoid of matter, fields, and energy. The realistic Universe contains all of these and the expansion of the Universe is accelerated (positively or negatively, depending on the details) because of gravitational interactions. In later cosmological models we shall see that the age of the Universe may be substantially longer or shorter than τ H, depending on the details of the matter, fields, and energy contained in the Universe.

16.2. THE HUBBLE LAW 459 16.2.4 A Two-Dimensional Hubble Expansion Model Figure 16.2 (next page) illustrates a two-dimensional Hubble expansion as viewed from two different vantage points (Show interactive animation of this). See: http://csep10.phys.utk.edu/guidry/cosmonew/

460 CHAPTER 16. THE HUBBLE EXPANSION Figure 16.2: The same two-dimensional Hubble expansion as viewed from two different vantage points.

16.2. THE HUBBLE LAW 461 Figure 16.3: Hubble parameter extracted from observations. 16.2.5 Measuring the Hubble Constant The Hubble constant may be determined observationally by measuring the redshift for spectral lines and comparing that with the distance to objects at a range of distances sufficiently large that peculiar motion caused by local gravitational attraction is small compared with the motion associated with the Hubble expansion. Figure 16.3 illustrates the determination of the Hubble constant from a variety of observations. The adopted value is corresponding to h = 0.72. H 0 = 72 ± 8 km s 1 Mpc 1,

462 CHAPTER 16. THE HUBBLE EXPANSION 16.3 Limitations of the Standard World Picture We shall demonstrate that the standard picture has been remarkably successful in describing many features of our Universe. However, there are two aspects of this picture suggesting that it is (at best) incomplete: 1. In order to get the big bang to produce the present universe, certain assumptions about initial conditions must be taken as given. While not necessarily wrong, some of these assumptions seem unnatural by various standards. 2. As the expansion is extrapolated backwards, eventually one would reach a state of sufficient temperature and density that a fully quantum mechanical theory of gravitation would be required. This is the Planck era, and the corresponding scales of distance, energy, and time are called the Planck scale. Since we do not yet have a consistent theory of quantum gravity, the presently understood laws of physics may be expected to break down on the Planck scale. Thus the standard picture says nothing about the Universe at those very early times. In later chapters we shall address these issues, to consider whether modifications of the standard picture can alleviate some of these problems.