## P. J. E. Peebles

Print publication date: 2020

Print ISBN-13: 9780691196022

Published to Princeton Scholarship Online: January 2021

DOI: 10.23943/princeton/9780691196022.001.0001

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# The Age of Abundance of Cosmological Models

Chapter:
(p.300) Chapter Eight The Age of Abundance of Cosmological Models
Source:
Cosmology's Century
Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691196022.003.0008

# Abstract and Keywords

This chapter examines why in the early 1980s cosmologists co-opted the astronomers' subluminal mass and the particle physicists' nonbaryonic matter in what became known as the standard cold dark matter, or sCDM, cosmological model. The letter “s” might be taken to mean that the model was designed to be simple (as it was) but it instead signified “standard,” not because it was established but because it came first. A large part of the cosmology community soon adopted variants of the sCDM model as bases for exploration of how galaxies might have formed in the observed patterns of their space distribution and motions, and for analyses of the effect of galaxy formation on the angular distribution of the sea of thermal radiation. This widespread adoption was arguably overenthusiastic, because it was easy to devise other models, less simple to be sure, that fit what we knew at the time. And it was complicated by the nonempirical feeling that space sections surely are flat.

THINKING IN THE cosmology community in the early 1980s and later was informed by two old ideas, that our universe is homogeneous on average and is usefully described by general relativity, and by two new ones. The first of these is the cosmological inflation picture discussed in Section 3.5.2. Some took the picture to be too elegant to be wrong, and in this sense, it might be compared to the community faith in general relativity, but with the difference that inflation is a framework in which one may place a considerable variety of theories. The second is the new idea of nonbaryonic CDM discussed in Chapter 7, which opened the possibility of more interesting cosmologies beginning with the CDM model. The motivation for this model, and the list of what it assumes, are discussed in Sections 8.1 and 8.2.

The CDM cosmological model owed its initial popularity to simplicity, which allowed the analytic and numerical explorations of cosmic structure formation reviewed in Sections 8.2 and 8.3. But the model is flawed by its awkwardly large mass density. That could be adjusted, of course: maybe the nonbaryonic matter is warm, or decaying, or self-interacting, or something completely different; maybe the initial conditions suggested by inflation are to be adjusted; or maybe Einstein’s cosmological constant should be reconsidered. This is the subject of Section 8.4.

My thinking about cosmological models in the early 1980s was influenced by results from the search for departures from an exactly homogeneous sea of cosmic microwave radiation. Two groups—Fabbri et al. (1980) and Boughn, Cheng, and Wilkinson (1981)—announced evidence of detection of anisotropy at δ‎T/T ∼ 1 × 10−4 on large angular scales. I took this seriously—the authors of the second paper were colleagues—and crafted a model that fit the measurements by application of the Sachs-Wolfe relation (to be discussed below) to an ansatz about initial conditions (Peebles 1981b). I think this model is reasonably simple, even elegant, but the details are irrelevant, (p.301) because Boughn et al. withdrew the detection, and Fixsen, Cheng, and Wilkinson (1983) placed a new, tighter bound on the anisotropy. I knew about the tighter bound when I introduced the CDM model before publication of the Fixsen, Cheng, and Wilkinson paper. Again, they were colleagues. My paper, Peebles (1982b), on the CDM cosmological model, was meant to save the gravitational instability picture from the tighter anisotropy bound.

# 8.1 Why Is the CMB So Smooth?

In the 1970s, improving upper bounds on the CMB anisotropy were revealing the striking difference between the distinctly clumpy distribution of matter in galaxies and the much smoother distribution of the sea of microwave radiation. I recall informal discussions of the possibility that this situation may falsify the gravitational instability picture for cosmic structure formation. There were declarations of this thought in print. The analysis of the effect of the gravitational growth of the clustering of mass on the CMB by Silk and Wilson (1981) led them to conclude that “already we believe it possible to assert that adiabatic fluctuations in the standard model are untenable for any combination of n and Ω‎0.” This standard model is a universe of radiation and baryons with mass density parameter Ω‎0 (Ω‎m in equation (3.13)) and the initial condition of primeval adiabatic mass fluctuations with a power-law power spectrum with spectral index n (in equations (5.49) and (8.7)).

But a new thought was discussed on page 232 in Section 5.3.6. In the HDM picture, the neutrinos would freely slip through the radiation, minimizing the disturbance to the radiation needed for the gravitational assembly of galaxies and their clustering. Doroshkevich et al. (1981, 37) put it that

The fluctuations of neutrinos that have a mass larger than Mν‎ have an uninterrupted growth. Due to their coupling to photons, the fluctuations of baryons smaller than the horizon can start growing only after recombination. Starting from the same initial amplitude at the moment of recombination, the neutrinos have much larger fluctuation amplitudes than the baryons and photons of the background radiation. At the recombination, the baryon Jeans mass drops very quickly to the value 105M. The growth of baryon fluctuations is accelerated by the large inhomogeneities formed in the neutrino density until the same amplitude is reached. … This process thus provides large fluctuations in baryons and neutrinos and small δ‎T/T in the photon background.

Here Mν‎ is the mass in equation (7.9) of the first generation of mass fluctuations to break away from the expansion of the universe, set by the free streaming of neutrinos in the HDM picture. The resulting top-down structure formation is seriously problematic, as discussed in Sections 5.2.7 and 7.1.1. (p.302) But the broader point is that nonbaryonic dark matter would slip through the radiation.

I introduced the CDM cosmological model in Peebles (1982b) in response to the Fixsen, Cheng, and Wilkinson (1983) improvement in the bound on the CMB anisotropy. This model replaces HDM with the nonbaryonic CDM reviewed in Section 7.2. It was meant to be a counterexample to the apparent challenge to the gravitational instability picture from the tight upper bound on the CMB anisotropy. I was not aware of it at the time, but as just noted, the starting idea had already been introduced: The CMB would be minimally disturbed by the growing clustering of nonbaryonic dark matter, whether HDM or CDM, leaving the more modest disturbances to the CMB caused by gravity and the smaller mass density in baryons.

# 8.2 The Counterexample: CDM

A counterexample to save the phenomenon is best kept simple. Thus I used the Einstein–de Sitter parameters Ω‎ = 0 and Ω‎m = 1, even though the measured galaxy relative velocity dispersion in the CfA redshift survey had already convinced me that the mean mass density likely is lower than this. (This is entered in row 13 in Table 3.2. It and other considerations that in 1982 convinced me the mass density likely is less than Einstein–de Sitter were published the following year, in Davis and Peebles 1983a,b.) For simplicity, I took the nonbaryonic DM to be initially cold. The massive neutrino—the WIMP—introduced in 1977, would do. The warm DM that Pagels and Primack (1982) and Blumenthal, Pagels, and Primack (1982) had introduced would have done equally well, but cold is simpler. The model assumes primeval adiabatic Gaussian scale-invariant initial conditions. This follows from a simple interpretation of the recently introduced concept of cosmological inflation. But again, I was more influenced by the argument from simplicity for these initial conditions independently suggested by Harrison (1970) and Peebles and Yu (1970), and developed by Zel’dovich (1972). My computation of the mass-fluctuation power spectrum took account of radiation and DM, but it ignored the modest mass in baryons. That made the computation easier and again simplified the model by eliminating a parameter: Ω‎baryon need only be much less than unity.1

(p.303) The computation of the predicted anisotropy of the microwave background radiation and the spatial distribution of the matter in this CDM model must follow evolution from conditions at high redshift, when baryons, CDM, neutrinos, and radiation are assumed to have the same space distribution, through the early stage of evolution in which baryons and radiation behave as an approximation to a viscous fluid, then through decoupling as radiation starts to diffuse through the baryons and then breaks free as the plasma combines. Subsequently the radiation propagates through almost purely neutral baryons and the slightly dented spacetime, while the baryonic matter joins the growing clustering of the cold dark matter.

These are a lot details to consider. But the important point is that there is not the complexity of seriously nonlinear processes, such as turbulence or star formation, which are impossible to analyze from first principles. In the CDM model and its variants, the departures from homogeneity on large scales are small and computable in perturbation theory, and the predicted evolution on the smaller scales of clusters of galaxies can be modeled to reasonable accuracy by numerical simulations. This, along with serious advances in the observations, made possible demanding cosmological tests.

The form for the power spectrum of the mass distribution well after decoupling, introduced in Peebles (1982b), is

(8.1)
$Display mathematics$

It has the primeval scale-invariant shape Pk on large length scales, small k. It swings to Pk−3 on smaller scales, larger k. The origin of this small-scale behavior follows by the argument in footnote 13 on page 231. The computation, adapted from Peebles (1982a), takes account of the tight coupling of radiation and plasma as the mode wavelengths start to oscillate, but it does not take into account the effect of the mode oscillations, an approximation that was good enough for the immediate need. Successive improvements to equation (8.1), taking account of the mass in the baryons and the effect of massless neutrinos, are in Bond and Efstathiou (1984); Davis et al. (1985); Bardeen et al. (1986); and Efstathiou, Bond, and White (1992).

Computing the predicted large-scale CMB anisotropy in this CDM model requires the normalization A in equation (8.1). I used results from the program of measurements of the galaxy two-point correlation function summarized in Peebles (1980). This statistic is usefully summarized as the standard deviation (root-mean-square value) of the count of galaxies in randomly placed spheres of a given size. The convenient choice of the sphere radius was 8h−1 Mpc, about the smallest at which the growth of the fractional fluctuations (p.304) in the mass density δ‎M/M might be reasonably well approximated by linear perturbation theory, and about the largest for which the measurements of the two-point function we had then could be trusted. The normalization in Peebles (1982b) assumes galaxies trace mass on this scale:

(8.2)
$Display mathematics$

The numerical value for the radius is rounded from equation (2.18). Later discussions that take account of the idea that σ8(mass) may be significantly less than σ8(galaxies) are discussed in Section 8.4.

Sachs and Wolfe (1967) derived the wanted relation between the largescale mass distribution and the CMB anisotropy in an Einstein–de Sitter universe with adiabatic initial conditions. The form of the Sachs-Wolfe relation I used, from Peebles (1980, equation (93.25)), relates the angular distribution of the CMB to the Fourier transform of the mass distribution.2 In the linear perturbation theory used in this computation, the second moments of the mass determine the second moments of the CMB. The latter are usefully expressed in terms of the spherical harmonic expansion of the CMB temperature as a function of angular position across the sky.3 The result in the 1982 CDM model is4

(8.3)
$Display mathematics$

The CMB temperature anisotropy spectrum $〈|alm|2〉$ in this equation is the Sachs-Wolfe gravitational effect on the angular distribution of the CMB by primeval adiabatic departures from homogeneity in the CDM model with primeval power spectrum Pkk normalized to the present universe by (p.305)

Figure 8.1. The Bond and Efstathiou (1987) computation of the CMB temperature and polarization angular power spectrum in the CDM model.

By permission of Oxford University Press on behalf of the Royal Astronomical Society.

the assumption that galaxies trace mass. It is independent of the Hubble parameter h. And since the CMB anisotropy in this model is well below the upper bounds on anisotropy we had at the time, it was sufficient for a counterexample.

The quadrupole l = 2 anisotropy in this first computation can be compared to the Bennett et al. (2003) first-year quadrupole measurement,

(8.4)
$Display mathematics$

which is reasonably close.

Computation of the CMB anisotropy on smaller angular angular scales must take account of the effect of the acoustic oscillations of the plasma-radiation fluid prior to decoupling. That was discussed in more primitive notation (and neglecting the nonbaryonic dark matter that had not yet been invented) in Peebles and Yu (1970). Bond and Efstathiou (1987) computed the CDM model prediction of the CMB angular power spectrum $〈|alm|2〉1/2$ as a function of the spherical harmonic degree to larger l, beyond the quadrupole. Their result is shown in Figure 8.1. The dashed curves in the figure are from a convenient approximation to the computation of the solid curves.

The measure of the CMB anisotropy on the vertical axis of Figure 8.1, $l2Cl=l2〈|alm|2〉$, is motivated by the following consideration. Since $〈|alm|2〉$ is independent of m, the mean square value of the departure of the CDM temperature from isotropy is, in the spherical harmonic expansion in equation (8.3), with the convention that the integral of $|Ylm|2$ over the sphere is unity,

(8.5)
$Display mathematics$

(p.306) The variance of the temperature per logarithmic interval in l is well approximated by $l(2l+1)〈|alm|2〉/(4π)$. It is a historical accident that 2l + 1 was replaced by 2(l + 1), bringing a standard measure of the variance per logarithmic interval of l to

(8.6)
$Display mathematics$

The CMB prediction of the CMB anisotropy spectrum in Figure 8.1 peaks at l ∼ 200, an angular scale of roughly 1 degree. The predicted value of l at the peak depends on Hubble’s constant, the mass density, and the curvature of space sections. The detection of the peak near the turn of the century was an important part of the evidence that drove the promotion of the CDM model, with the addition of Einstein’s cosmological constant, to the standard cosmology.

In the 1980s, it was not at all clear that the CDM model is a useful approximation,5 or if so, that the angular power spectrum is the best way to compare the CMB anisotropy theory and measurements. Thus Bond and Efstathiou (1987) and Efstathiou and Bond (1987) considered isocurvature as well as adiabatic primeval conditions, and they computed the CMB two-point correlation functions as well as power spectra.6

When I introduced the CDM model in Peebles (1982b), I considered it an example of what might have happened, likely to be one of many to be explored. I did not imagine that it might so readily grow into a convincing picture of the early universe. But I might have expected that the very simplicity of the CDM model would draw interest. That was aided by a common interpretation of the inflation picture (Section 3.5.2), which would have the flat space sections and initially adiabatic, Gaussian, and scale-invariant initial conditions assumed in the CDM model. But I was surprised at how seriously the model was taken, and was uneasy about it, because I saw no reason to be confident that nature (p.307) shared our ideas of simplicity. It had not taken me long to think up this model and compute the CMB anisotropy in equation (8.4), and I could see how to set up other models, maybe not quite as simple, that could equally well fit the observational constraints (which were not all that tight then). I continued to invent such models until the late 1990s, when the CMB anisotropy measurements started to reveal the anisotropy peak predicted in the CDM model and illustrated in Figure 8.1. That went a long way toward persuading me that nature may have taken our simplest way, apart from the curious presence of Einstein’s cosmological constant, Λ‎, and the hypothetical nonbaryonic dark matter. The many tests since then continue to agree with the CDM model with the addition of Λ‎. It is a remarkable advance, although at the time of writing, the natures of Λ‎ and CDM remain unknown.

# 8.3 CDM and Structure Formation

A viable cosmology must offer a platform for an acceptable analysis of how the galaxies formed in all their rich phenomenology. This is a cosmological test, but one that is difficult to assess, because the galaxies formed by complex nonlinear processes, at the heart of which are the still very poorly characterized properties of star formation. It means that studies of galaxy formation must rely on prescriptions to be explored by analytic and numerical methods and adjusted to fit the observations.

Before the revolution in cosmology, some studies of cosmic structure formation tested cosmological models; the prime example is the analyses of clusters of galaxies that helped lead us to a low-density universe (summarized in categories B1 and B2 in Table 3.3). That test was persuasive, because the gravitational assembly and evolution of clusters seems to be simple enough to be modeled in numerical simulations sufficiently accurately to serve as a guide to the nature of the cosmology.

The more common tradition has been to explore how a picture for structure formation may be framed to fit a given cosmological model. An early example is the study of the gravitational origin of the rotation of galaxies (Section 5.2.3). Still earlier is the Eggen, Lynden-Bell and Sandage (1962) picture of the formation of our Milky Way galaxy (Section 5.2.3) that Partridge and Peebles (1967) placed in the context of the gravitational instability of the expanding big bang cosmological model, along the lines reviewed in Section 5.1.7 Elliptical galaxies contain little gas and few young stars, which might suggest they formed (p.308) by the gravitational gathering of matter that had already been largely converted to stars. Larson (1969) and Gott (1975) placed exploration of this mode of formation of an elliptical galaxy in the context of an expanding universe.

Galaxies are much more complicated than clusters of galaxies. Gas and plasma in a newly forming galaxy can be expected to gain internal energy from shocks and turbulence driven by gravitational collapse along with radiation and winds from massive young stars, lose energy by thermal bremsstrahlung emission by plasma and by radiative decay of collisionally excited atoms, and be rearranged by winds and explosions. This would happen in the messy early concentrations of mass that gravity may have gathered. In a series of papers, Richard Larson pioneered exploration of the budget for loss and gain of energy in galaxy formation and evolution (e.g., Larson 1969, 1976 and 1983). Spitzer (1956) offered the thought that the collapse of a protogalaxy might have left thermally supported coronae of plasma in the outer parts at densities low enough that the energy dissipation times exceed the Hubble expansion time. But plasma at the mean baryon mass density at our position in the Milky Way, n ∼ 1protoncm−3, with the plasma temperature T ∼ 107 K that can be gravitationally confined by the mass of the galaxy, has a cooling time ∼ 107 years, shorter than the collapse time ∼ 108 years. Thus the radiative loss of energy by plasma in a young galaxy can be quite significant. Binney (1977), Rees and Ostriker (1977), and Silk (1977) introduced considerations of how the relative rates of cooling and free gravitational collapse in a protogalaxy can determine the nature of its evolution.

Gunn et al. (1978) and White and Rees (1978) introduced considerations of the role that a subluminal massive halo might play in the formation of the luminous parts of a galaxy (as discussed on page 218). The CDM cosmological model introduced in Peebles (1982b) offered a more definite basis for exploration of how gravity might grow concentrations of nonbaryonic matter, in which the baryons might dissipatively settle to form a galaxy within a massive dark matter halo. Early explorations are in Peebles (1984a), and in greater detail, Blumenthal et al. (1984) and Bardeen et al. (1986). The Blumenthal et al. considerations of how the CDM model can accommodate the rich phenomenology of cosmic structure on the broad range of scales from galaxies to superclusters of galaxies continue to be widely cited.

In the 1970s, applications of statistical measures of the space distribution and motions of the galaxies offered the prospect of a cosmological test: compare the observed measures of the galaxy spatial distribution to what would be expected from the gravitational build-up of mass clustering in an expanding universe. Ed Groth and I spent a lot of time exploring numerical N-body simulations of the evolution of the mass distribution in an expanding model universe. We meant to compare the mass correlation functions in the (p.309) simulations to the measurements we were making of the galaxy low-order position and velocity correlation functions, as discussed in Davis, Groth, and Peebles (1977). With scale-invariant initial conditions in the scale-invariant Einstein–de Sitter model, the mass distribution in pure gravity N-body simulations will relax to a scaling solution for the mass correlation functions, after enough time has elapsed to allow transients to die down. We could not find this scaling behavior, and what we had did not seem to be close to convergence to scaling. Our few publications include a conference proceeding early on (Peebles 1973b) and a conference abstract (Groth and Peebles 1975) presented at about the time we were giving up.

Others took up the challenge of simulating the evolution of the distributions of mass and galaxies in an expanding universe; examples include Press and Schechter (1974); Doroshkevich and Shandarin (1976); Aarseth, Gott, and Turner (1979); Efstathiou, Fall, and Hogan (1979); Miller and Smith (1981); Centrella and Melott (1983); Melott et al. (1983); Miller (1983); Kauffmann and White (1992); and Governato et al. (2010). The results from a lot of work show that the parameters meant to describe what the baryons are doing can be adjusted to fit the observations, including the properties of the galaxies and their space distribution and motions, in impressive detail. The Λ‎CDM cosmology passes this test. But the complexity makes it difficult to assess the weight of this case from the study of cosmic structure formation.

Davis et al. (1985), in a celebrated paper known as DEFW, presented early numerical simulations of the evolution of structure in the CDM model (before it became the Λ‎CDM cosmology). This is along the lines of Groth and Peebles (1975) but with the far more capable numerical computers of a decade later. It was attractive to turn to the CDM model and away from simulations of structure formation in a universe of baryons, because baryons behave in the complex ways Richard Larson had been exploring. In the CDM model, the dominant mass interacts only with gravity, so one might hope to arrive at a useful first approximation to cosmic evolution by ignoring the complications of the baryons. It is easy to set up numerical simulations of the evolution of the distribution of a gas of particles that move under the influence of gravity alone. Accurate computation of the gravitational accelerations given the particle positions is time-consuming, but one can work on that. And simulations of the CDM model are convenient, because initial conditions that are close to scale invariant grow into a first generation of nonlinear structures that form across a broad range of mass scales at about the same time, reducing the span of cosmic time called for in simulations.8 Adding to all this was the motivation (p.310) from the community interest in the CDM model on the basis of simplicity, inflation, and maybe promise.

DEFW introduced a change of thinking: identify galaxies with the mass concentrations in the simulation, and compare the galaxy space distribution to the distribution of these mass density peaks, not to the measures of the mass distribution. Carlos Frenk (in a personal communication, 2018) recalls that

We started working on what became DEFW towards the end of 1982 when the department in Berkeley had just bought a then amazingly powerful new computer, a VAX 780. … In the first part of the DEFW project we assumed that the distribution of galaxies traced the distribution of mass and we were upset to find that, with this assumption, an open model with Ω‎m ∼ 0.2 gave an acceptable match to the CfA survey while the much more appealing flat Ω‎m = 1 model did not. The problem with the latter, as you know, was that matching the observed galaxy clustering amplitude required a very large amplitude of mass fluctuations and thus rms pairwise peculiar velocities much larger than those that you and Marc [Davis] had measured in the CfA redshift survey. As consolation we ran a flat model with a cosmological constant which looked as good as the open model from the point of view of clustering and peculiar velocities and at least had the virtue of having a flat geometry, even if at the expense of the then highly unattractive inclusion of Λ‎. In 1984, with the simulations in hand and seeing what Nick [Kaiser] had come up with to explain the large clustering amplitude of galaxy clusters, we figured out that if we extended his ideas to galaxies and deployed the same “high-peak” trick to them, then the required mass fluctuations would be reduced by b2 and we could then reconcile the Ω‎m = 1 CDM model with the CfA galaxy two-point correlation function and the measured rms velocities. This is how the idea of “biased galaxy formation” came about but we soon convinced ourselves that, regardless of the desire to have Ω‎m = 1, the idea that galaxies formed in high peaks made physical sense and would be relevant for any value of Ω‎m.

The early DEFW numerical simulations were pure dark matter, and the peaks were identified in the initial conditions. A satisfactory study of how the galaxies formed must take account of the complex behavior of the baryons and identify the model galaxies as they formed. But DEFW set the direction of thinking.

The long history of ideas about how the galaxies formed includes the early debates reviewed in Chapter 5 on whether gravity can assemble the mass concentration of a galaxy in an expanding universe, and if so, whether gravity can cause the galaxy to rotate. The behavior of a concentration of baryons with the dimensions of a galaxy is complex. The idea that most of the mass of a galaxy is CDM allowed an easy entry to the problem: Ignore the baryons at (p.311) first, and then introduce analyses of the behavior of the baryons in increasing detail guided by what is found to work. It is difficult to argue that the results added much weight to the cosmological tests, but they did help keep the community interested in the CDM model and its variants in the 1990s.

# 8.4 Variations on the Theme

The CDM cosmology in Peebles (1982b) assumes the Einstein–de Sitter model with Ω‎ = 0and Ω‎m = 1. That was for simplicity, and despite two reservations. First, the evidence seemed to me to be that the mass density is less than Einstein–de Sitter. Second, I was not at all confident that simplicity is a reliable guide to a better cosmology. I was right about the first, wrong about the second. But arriving at this second conclusion required sifting through the ideas to be reviewed this section. For this purpose, let us follow the standard practice of naming the 1982 model sCDM, for standard, though it never was that.

Warm dark matter, WDM, was introduced in the same year as sCDM, by Pagels and Primack (1982) and Blumenthal, Pagels, and Primack (1982). They suggested that a characteristic mass set by WDM streaming might account for the masses of large galaxies. The idea has not had much effect so far, but the idea, with a smaller characteristic mass, continues to be discussed. Variants of what might be termed sWDM could have been considered along with the variants of sCDM to be discussed here, but I have found little evidence that this has happened.

Recall the growing list of measures summarized in Table 3.2 and Figure 3.5 that indicate the cosmic mean mass density is about a third of the Einstein–de Sitter value. The evidence includes dynamical measures of the mass clustered with galaxies on scales from ∼ 0.3 Mpc to ∼ 10 Mpc; the cluster mass function, evolution, spatial correlation function, and baryon mass fraction; and the positive spatial correlation of galaxy positions extending to ∼ 50 Mpc (this last assuming initial conditions are adiabatic and scale-invariant, as in the sCDM model). But this low mass density was counter to the common opinion in the years around 1990 that the mass density surely is Einstein–de Sitter: Ω‎m = 1. That would have to mean that these mass density measurements are all biased low (Section 3.5.3). Was it reasonable to have assumed that biasing had such a systematically similar effect on estimates of Ω‎m on scales ranging from ∼ 0.3 Mpc to ∼ 30 Mpc? Or was it more reasonable to take it that the mass density likely is only a third of the sCDM model?

The issue grew more interesting with the Smoot et al. (1992) announcement of detection of the CMB anisotropy by the differential microwave radiometers (DMR) experiment on the NASA Cosmic Background Explorer satellite, COBE. The background radiation temperature anisotropy in the first-year data is δ‎T/T = 1.1 × 10−6 measured on an angular scale θ‎ ∼ 10°. The (p.312) detection of such a small departure from isotropy is impressive, though later measurements have done even better. The detection was exciting, because the degree of anisotopy is about what one would expect in the sCDM cosmology and its variants. It was important, because it is a measure of the departure from a homogeneous mass distribution (depending on the initial conditions, of course, usually assumed to be adiabatic). And it was a new constraint on cosmological models.

Bunn and White (1997) reported that the sCDM model normalized to the COBE anisotropy in the 4-year CMB measurements (Bennett et al. 1996) requires the mass fluctuation amplitude (defined in equation (8.2)) to be the equivalent of σ8 = 1.22. But they pointed out that the Viana and Liddle (1996, Fig. 3) condition to fit the abundance of rich clusters of galaxies requires σ8 ≃(0.6 ± 0.1)Ω‎m−0.4. Since Ω‎m = 1 in sCDM, this is a considerable discrepancy that might be added to the other evidence that Ω‎m is well below unity. But there were other ideas to consider.

## 8.4.1 TCDM

Bunn and White (1997, 20) concluded that the

COBE-normalized “standard” CDM [that is, sCDM] … predicts significantly too much small-scale power and is therefore ruled out. However, any of several slight changes to the model can easily resolve this inconsistency. Perhaps the simplest solution is a slight tilt to the power spectrum. Inflationary models typically predict spectral indices slightly less than unity, and a value of n of 0.8 or even less is quite natural in such models.

This simple solution became known as the tilted cold dark matter or TCDM model. The only difference from sCDM is the primeval mass fluctuation power spectrum, tilted from scale-invariance to

(8.7)
$Display mathematics$

To see how this remedies some problems with sCDM, note that the angular resolution of the COBE measurement, θ‎ ∼ 10°, subtends a comoving length scale of roughly 500 Mpc at high redshift, when expanded to the present epoch. Under adiabatic initial conditions, the CMB anisotropy translates to the departures from a homogeneous mass distribution on this scale. The departures from homogeneity on much smaller scales, about 10 Mpc, are relevant for the gravitational formation of clusters. The sCDM overprediction of fluctuation power on the scales of clusters thus is remedied by tilting the primeval power spectrum. As White, Efstathiou, and Frenk (1993) had anticipated, this resolves the problem with clusters. It does leave the evidence that the mean mass density is less than Einstein–de Sitter, from the array of (p.313) measures summarized in Figure 3.5, but a common feeling at the time was that this might be dealt with separately.

The attention to inflation is worth noting. It is natural to expect a slowing of the expansion rate during inflation, which would tilt initial conditions from scale invariance in the direction wanted for the TCDM model. But the value of n is not predicted by inflation; its implementation can be chosen to make n larger or smaller than unity. This flexibility is illustrated by the titles of two papers: “Designing Density Fluctuation Spectra in Inflation” (Salopek, Bond, and Bardeen 1989), and “Arbitrariness of Inflationary Fluctuation Spectra” (Hodges and Blumenthal 1990). A broad variety of initial conditions is afforded by the choice of the potential V (φ‎) as a function of the scalar field φ‎ in a single-field inflation model. Two scalar fields offer still greater variety. Thus one response to the failure of sCDM was to adjust the model for inflation, as in the Bunn and White tilt.

One can postulate the tilt in equation (8.7) or offer a model. If the expansion parameter during inflation is a power law in physical time, atp, it produces the power law initial condition in equation (8.7) with n =(p − 3)/(p − 1) (Lucchin and Matarrese 1985; Liddle, Lyth, and Sutherland 1992). In another approach, Freese, Frieman, and Olinto (1990) introduced a single-field model they termed “natural inflation,” because their field potential is an arguably natural form in a grand unified model for particle physics. The primeval mass fluctuations in natural inflation are close to equation (8.7), but for some, the pedigree is better. Cen et al. (1992); and Adams et al. (1993) concluded that TCDM and the almost-equivalent natural inflation model look promising but require work, including significant galaxy position biasing to get around the dynamical evidence for Ω‎m ∼ 0.3: perhaps a trace of HDM, perhaps a cosmological constant.

## 8.4.2 DDM and MDM

In the decaying dark matter (DDM) model, the dark matter has been decaying, perhaps into other kinds of nonbaryonic matter, perhaps into something that interacts with baryonic matter in an observable way. Early examples are the Dicus, Kolb, and Teplitz (1977 and 1978) and Sato and Kobayashi (1977) discussions of massive neutrinos with possibly interesting radiative decay lifetimes.

Davis et al. (1981) proposed that the early universe may have contained both HDM, the thermally produced sea of neutrinos with rest mass of a few tens of electron volts discussed in Section 7.1, and WIMPS, another neutrino family with the much larger rest mass discussed in Section 7.2. In this model, the more massive neutrinos are assumed to have decayed after their gravity had served to amplify the growth of structure on the scale of galaxies, leaving the HDM to provide most of the present Einstein–de Sitter mass density. Free streaming may have left the HDM close enough to smooth on a scale of about (p.314) 10 Mpc (as in the HDM cosmology in Section 7.1.1) that this mass component would not have been detected in dynamical probes on smaller scales. That is, the mass in and around galaxies could be small enough to account for their small relative velocity dispersion. The theme was pursued in early papers by Doroshkevich and Khlopov (1984); Fukugita and Yanagida (1984); Gelmini, Schramm, and Valle (1984); Hut and White (1984); and Olive, Seckel, and Vishniac (1985). Turner, Steigman, and Krauss (1984) proposed a variant: After the massive dark matter particles had gravitationally boosted structure formation, their decay might have produced a sea of relativistic nonbaryonic matter. The dominant mass density in this relativistic dark matter would be close to smooth across the Hubble length, leaving subdominant mass clustered in and around the galaxies. The galaxies would have the observed small relative velocities driven by their small contribution to the mean mass density. The effect would be roughly similar to that of a cosmological constant, except that the cosmic expansion time would be considerably shorter, a serious but at the time perhaps not fatal problem.

The MDM model (also known as mixed dark matter; cold plus hot dark matter, as in CHDM; and two-component dark matter) places the stable CDM largely in massive halos around the galaxies, while the greater mass density in the more smoothly distributed HDM brings the total to the Einstein–de Sitter value. This is the Davis et al. (1981) two-component scheme but without decay of any of the nonbaryonic matter. The model received considerable attention; early examples are Fang, Li, and Xiang (1984); Shafi and Stecker (1984); Umemura and Ikeuchi (1985); and Valdarnini and Bonometto (1985).

## 8.4.3 Λ‎CDM and τ‎CDM

Models with Ω‎m = 1 are challenged by the considerable variety of evidence summarized in Figure 3.5 that the mass density is well below unity, Ω‎m ∼ 0.3. The Λ‎CDM model accepts this and postulates that Einstein’s cosmological constant Λ‎ serves to keep space sections flat. The move in this direction was foreshadowed by Gunn and Tinsley (1975), who marshaled empirical indications of “An Accelerating Universe.” It was first proposed in the context of CDM by Peebles (1984b); Rees (1984); Turner, Steigman, and Krauss (1984); and Kofman and Starobinsky (1985). It is a simpler, more direct way than biasing to interpret the considerable variety of probes of the mass density discussed in Section 3.6.5while preserving the flat space sections indicated by inflation (and, arguably, the elegance of this spacetime geometry).

The τ‎CDM model for the power spectrum of the mass distribution is designed to allow examination of effects of adjustments of cosmological parameters in a variety of models, including Λ‎CDM. Efstathiou, Bond, and White (1992) present a generalization of equation (8.1) for the mass fluctuation power spectrum:

(p.315)

(8.8)
$Display mathematics$

The unit of the wavenumber k is 1h Mpc−1. The amplitude B and the number Γ‎ are free parameters, and in Λ‎CDM, we have Γ‎ = Ω‎mh. Efstathiou et al. give other expressions for Γ‎ for other models. I take the wonderfully detailed equation (8.8) to be a good illustration of the ingenuity devoted to the search for a believable picture of the behavior of the universe, from galaxies to the largest observable scales.

The Λ‎CDM model is counter to the long tradition of nonempirical arguments against Λ‎ reviewed in Section 3.5.1, but it proves to pass the revolutionary advances in the cosmological tests at the turn of the century. Before discussing how this came about, we should consider still more ideas about how to reach a well-established cosmology.

## 8.4.4 Other Thoughts

Gott et al. (1974) discussed the case for a low-density cosmological model with open space sections. In the 1970s, this model was not considered unreasonable. The later change of thinking was largely inspired by the interpretation of cosmological inflation in the early 1980s, which held that the universe is close to homogeneous because the great expansion during inflation swept away space curvature gradients and with them the curvature of space sections. Even though Abbott and Schaefer (1986) shared the common opinion that a cosmology with open space sections is contrary to inflation, they felt it is worth considering anyway. And Ellis, Lyth, and Mijić (1991) felt they could craft an acceptable model for inflation in an open universe.

The open CDM (OCDM) variant of sCDM accepts the evidence for low mass density, Ω‎m ∼ 0.3, and assumes open space sections with Ω‎ = 0. Through the mid-1990s, this model could be adjusted to reasonably acceptable fits to the constraints (e.g., Wilson 1983; Blumenthal, Dekel, and Primack 1988; Kamionkowski et al. 1994; Ratra and Peebles 1995). It was convincingly falsified at the turn of the century.

The primeval isocurvature baryon model, PIB, introduced in Peebles (1987a,b) is even more iconoclastic; it was introduced as a counterexample to the idea that the observations require nonbaryonic matter. The model assumes primeval isocurvature conditions, meaning that a clumpy distribution of the baryons in the early universe is compensated by tiny perturbations to the radiation that serve to eliminate primeval spacetime curvature fluctuations. The matter density, all baryons, is chosen to be consistent with the big bang nucleosynthesis (BBNS) constraint (Section 4.6), and with Ω‎ = 0, this requires open space sections. The power spectrum of the primeval baryon space distribution is a power law, and the power law index and amplitude are free parameters. In the mid-1990s, the model was considered to be seriously constrained but (p.316) perhaps not ruled out (e.g., Efstathiou and Bond 1987; Cen, Ostriker, and Peebles 1993). Hu, Bunn, and Sugiyama (1995, L62) conclude that

At present, none of the simplest models for structure formation fares well in comparison with the combined observations of the CMB and large-scale structure; it is, therefore, perhaps unwise to dismiss this scenario [PIB] as entirely unviable.

I introduced a more elaborate isocurvature model (Peebles 1999) just in time for it to be falsified, along with PIB, in a particularly manifest way by the detection of the acoustic oscillations illustrated in Figure 5.2 and shown in Plates I and II.

Sahni, Feldman, and Stebbins (1992) reconsidered Lemaître’s (1931d) proposal that the expanding universe passed through the near-static hovering phase discussed in Section 3.6. Lemaître made great contributions to cosmology, but this one has not proved to be promising. Messina et al. (1992) discussed the interesting effect on structure formation in the sCDM cosmology by seriously large positive or negative skewness of the primeval mass density fluctuations. The large mass density in sCDM is problematic, of course, but at the time, one could have considered Λ‎CDM with skewness. That line of thought was ruled out at the turn of the century by the close-to-Gaussian CMB anisotropy. Bartlett et al. (1995) pointed out that many of the challenges to the sCDM cosmology would be relieved if the extragalactic distance scale were about twice the astronomers’ measurements. But the low dynamical estimates of Ω‎m in Table 3.2 are not sensitive to the distance scale, and it certainly was difficult to imagine how the astronomers’ distance calibrations could have been so far off. The idea had to be considered and seen to be seriously challenged.

We see in this review the broad range of inventive ideas put forth in the search for clues to a better cosmological model. Motivations surely differed, but I think a common feeling was excitement at the thought that we may be approaching constraints tight enough to make the case that a particular cosmology is a reasonably convincing approximation to what actually happened. Let us consider now examples of the evolution of thoughts about how close we may have been to this end game in the 1990s.

# 8.5 How Might It All Fit Together?

Dekel, Burstein, and White (1997, 176) argue that

the order by which more specific models should be considered against observations, are guided by the principle of Occam’s Razor, i.e., by simplicity and robustness to initial conditions. The caveat is that different researchers might disagree on the evaluation of “simplicity”.

(p.317) It is commonly assumed that the simplest model is the Einstein–de Sitter model, Ω‎m = 1 and Ω‎Λ‎ = 0. One property that makes it robust is the fact that Ω‎m remains constant at all times with no need for fine tuning at the initial conditions (the “coincidence” argument [2]).

The most natural extension according to the generic model of inflation is a flat universe, Ω‎tot = 1, where Ω‎m can be smaller than unity but only at the expense of a nonzero cosmological constant.

These simple models could serve as useful references, and even guide the interpretation of the results, but they should not bias the measurements.

These cautious statements give a sensible picture of the state of the art in the mid-1990s.

The reference [2] in these statements is to Bondi (1960), who pointed out that if space curvature and Λ‎ vanish, then the density parameter for the total mass is Ω‎m = 1, independent of time. As discussed in Section 3.5.1, this would mean we need not have flourished at some particular epoch during the course of evolution of the universe. The community also was quite aware of the problem that a value of Λ‎ acceptable for cosmology appears to be distinctly unsuitable from the point of view of quantum physics, for the reasons considered on page 59. Kofman, Gnedin, and Bahcall (1993, 8) put it that “from both cosmology and fundamental physics, there is a major difference whether the cosmological constant is exactly zero or very small (in Planck units).” Preference for the former is seen in the comment in the paper Davis, Efstathiou et al. (1992) given in Section 3.5.1. Marc Davis, whose many contributions to this subject are discussed at length in this history, recalls (in a personal communication, 2018) that

I remember when we got together in Cambridge, and we were very reluctant to allow Omega less than 1. It just wasn’t done, the title of this paper says it all: “The end of cold dark matter?”

Another caveat is understood, of course: nature must agree. Marc Davis, again in a personal communication (2018), said that

If we had been willing to accept more than one parameter in our models, yes we could have made more progress. But we absolutely had to be forced, with data unambiguously pointing at Lambda.

Taylor and Rowan-Robinson (1992, 396) presented a comparison of the observational constraints on seven variants of sCDM, including Λ‎CDM. Their conclusion is that

We find only one completely satisfactory model, in which the Universe has density Ω‎ = 1, with 69% in the form of cold dark matter, 30% (p.318) provided by hot dark matter in the form of a stable neutrino with mass 7.5eV, and 1% baryonic.

This is MDM, the mix of CDM and HDM discussed in Section 8.4.2, with Λ‎ = 0. Numerical simulations of cosmic structure formation led Davis, Summers, and Schlegel (1992, 359) to a similar conclusion:

The MDM model thus seems to resolve a long-standing problem of large-scale structure, namely the disparate estimates of Ω‎ on small and large scales. Velocity fields are reduced on small scales and increased on large scales, increasing the Mach number of the cosmic velocity field.31 This with its other successes in matching large-scale structure in the Universe8,32 makes the model worthy of serious consideration.

The references are to Ostriker and Suto (1990), who characterized the small dispersion around the mean galaxy flow as a large cosmic Mach number; van Dalen and Schaefer (1992); and Taylor and Rowan-Robinson (1992). The last two papers also present arguments for the MDM model. The paper by Klypin et al. (1993, 1) presents a similar assessment:

C+HDM looks promising as a model of structure formation. The presence of a hot component requires the introduction of a single additional parameter beyond standard CDM—the light neutrino mass or, equivalently, Ω‎ν‎—and allows the model to fit essentially all the available cosmological data remarkably well. The τ‎ neutrino is predicted to have a mass of about 7 eV, compatible with the MSW explanation of the solar neutrino data together with a long-popular particle physics model.

The title of a later review of the situation by Primack (1997) is “The Best Theory of Cosmic Structure Formation is Cold + Hot Dark Matter.” In this cosmological model, the CDM component would be more strongly clustered around individual galaxies, and the HDM component more broadly spread, because the neutrinos were streaming about at high speeds at high redshift. This is in the direction of reconciling the small mass density inferred from observations at relatively small scales and the COBE normalization on large scales with the Einstein–de Sitter mass density Ω‎m = 1 assumed in these models. But of course, there still was the problem with the considerable variety of evidence in Table 3.2 for a lower mass density.

Primack’s (1997) paper is the last carefully organized argument for MDM that I have found; interest in the model had quite abruptly faded. I have not been able to find a direct demonstration at that time that MDM had been falsified, or looked likely to be. And in that year, Perlmutter et al. (1997) presented evidence from measurements of the supernova redshift-magnitude relation that Ω‎m is close to unity. It was withdrawn, but at the time, reconciling the (p.319) apparently large value of Ω‎m with the smaller mass densities indicated by the probes of the mean mass density summarized in Figure 3.5 called for some special arrangement. The MDM model seems to have been worth considering. But instead, the community was growing attached to the simplicity of adding Einstein’s cosmological model Λ‎ to keep space sections flat while lowering the mean mass density from sCDM to agree with the evidence for that, despite the unlikely quantum physics. Later evidence is that we must indeed learn to live with Λ‎, and that while neutrinos do have nonzero rest masses, they are well below those considered in the MDM models.

Arguments for Λ‎CDM include Efstathiou, Sutherland, and Maddox (1990, 705), who conclude from their measurement of the galaxy correlation function (Section 3.6.5) that

the successes of the CDM theory can be retained and the new observations accommodated in a spatially flat cosmology in which as much as 80% of the critical density is provided by a positive cosmological constant, which is dynamically equivalent to endowing the vacuum with a non-zero energy density.

S. White et al. (1993, 432) came to this conclusion from their estimate of the cluster baryon mass fraction (Section 3.6.4). Their thought is that “The flat universe required by the inflation model can be rescued by a non-zero cosmological constant, a possibility which has other attractive features41 but which still conflicts with dynamical evidence for large Ω‎0.” Reference 41 is to the Efstathiou, Sutherland, and Maddox (1990) evidence for mass density less than Ω‎m = 1 from the range of positive correlations of galaxy positions. The mention of evidence of a large mass density is not explained, but at the time, the result from the POTENT method was widely discussed. It argued for consistency with Ω‎m = 1 (Dekel et al. 1993), though we have seen that it is difficult to reconcile this large mass density with the bulk of the evidence illustrated in Figure 3.5. And the improvements in the data and methods of analysis of the mass density later that decade, in particular from method E in Table 3.3, led Willick et al. (1997) to conclude that the mean mass density likely is in the range 0.16 ≲ Ω‎m ≲ 0.34 for reasonable values of Hubble’s constant.

Recall now that at a given value of the mass density parameter less than unity, perhaps Ω‎m ≈ 0.3, the OCDM model with Ω‎ = 0 and open space sections and the Λ‎CDM model with flat space sections equally well fit mass density measurements on scales ∼ 1 Mpc, clusters on scales ∼ 10 Mpc, and the cutoff of correlated galaxy positions at ∼ 100 Mpc. This is the geometrical degeneracy discussed in footnote 9.2 on page 334 in Chapter 9. Kofman, Gnedin, and Bahcall (1993, 2) argued that the degeneracy between OCDM and Λ‎CDM might be broken by the consideration that “a positive cosmological constant helps to overcome the (possible) ‘age problem’ of the universe; (p.320) i.e., the age of the oldest globular clusters … is larger than the age of the universe for Ω‎ = 1 and h ≃ 0.5 (t0 = 13 Gyr).” This issue also led Krauss and Turner (1995) and Chaboyer et al. (1996) to argue for Λ‎CDM and against OCDM. But the Kofman, Gnedin, and Bahcall (1993) qualifier “possible” is to be noted: A reliable value of the Hubble parameter h was difficult to establish.

Gott et al. (1974) assembled the parameter constraints shown in Figure 3.2 on page 76. Ostriker and Steinhardt (1995) updated the figure; they had tighter constraints on Hubble’s constant and the expansion time, and they replaced the baryon mass density by the network of constraints on the total mass density reviewed in Section 3.6.4. The case for mass density parameter in the range 0.2 ≲ Ω‎m ≲ 0.4 had grown reasonably persuasive, as they argue, though not yet generally accepted. Ostriker and Steinhardt proposed that the degeneracy between OCDM and Λ‎CDM is broken by the OCDM prediction of an unacceptably large CMB anisotropy. This is an important addition to the constraints, but at the time, the significance of the CMB anisotropy measurements could be debated.

The bottom figure in Plate I shows the progress of CMB anisotropy measurements compiled by Peebles, Page, and Partridge (2009).9 Kamionkowski, et al. (1994, Fig. 2) concluded that at Ω‎m = 0.3 and Ω‎ = 0, and with scale-invariant initial conditions, the COBE-normalized peak anisotropy is δ‎Tl = 55μ‎K at l = 400 This is allowed by the measurements up to the year 1999 in Plate 1. Ostriker and Steinhardt (1995) argued that since the mass fluctuation spectrum grows more slowly in OCDM than in Λ‎CDM, the primeval spectrum in OCDM should be tilted to n = 1.15, which would increase the power on small scales. They found that this with Ω‎m = 0.375 predicts δ‎Tl ≈ 95μ‎K at l = 400, which is unacceptably large. This consideration can be compared to the Bunn and White (1997) analysis. They used the four-year COBE CMB anisotropy data. In their OCDM model with n = 1, the COBE normalization indicates σ8 = 0.64. This is smaller than their estimate of what is needed to account for the abundance of rich clusters, σ8 = 0.87 ± 0.14 at Ω‎m = 0.4. Bunn and White pointed to tilting the other way, to n = 0.8 with Ω‎m = 1. This fits the cluster abundance but requires biasing on smaller scales.

We must consider two other issues. First, reionization following the dark ages has suppressed the CMB anisotropy by scattering by free electrons. The effect is now known to be modest, but Figure 2 in Kamionkowski, Spergel, and Sugiyama (1994) shows the considerable suppression of the anisotropy that could be contemplated in the mid-1990s. Second, the constraint from the (p.321) CMB anisotropy assumes primeval adiabatic departures from homogeneity. The PIB primeval isocurvature model discussed on page 315 was thought to be viable until the late 1990s.

The nonempirical argument from inflation certainly favors Λ‎CDM over OCDM with its curved space sections, though we have seen that to some, the nonempirical consideration from quantum physics argued against Λ‎CDM. The empirical evidence supported Λ‎CDM over OCDM because it allows a larger expansion time that is an easier fit to stellar evolution ages, and perhaps a better fit to the CMB anisotropy. But in the mid-1990s, the interpretations of these measurements were not secure. Some argued that the addition of Λ‎ could be avoided by turning to MDM or isocurvature models, and others pointed out that since the astronomers were finding estimates of Hubble’s constant in the range 50 to 100 km s−1 Mpc−1, it was not unreasonable to consider H0 ∼ 40 km s−1 Mpc−1, which would help reconcile sCDM with the expansion time and CMB anisotropy. Others remained agnostic, of course.

The state of the empirical evidence changed at the end of the 1990s when the measurements of the CMB anisotropy were in better condition, and results were emerging from the programs to measure the cosmological redshift-magnitude relation. These are the two empirical lines of research that drove the revolutionary consolidation of evidence discussed in Chapter 9. Bahcall et al. (1999) conclude that this assembled evidence is converging to the Λ‎CDM cosmology. In their assessment of the evidence, Bond and Jaffe (1999, 61) took a more cautious line: The count of parameters in the cosmological models they were considering “is thus at least 17, and many more if we do not restrict the shape of 𝒫Φ‎(k) through theoretical considerations of what is ‘likely’ in inflation models.”

Bond and Jaffe took account of the accumulated constraints from the COBE four-year CMB anisotropy data and emerging measurements on smaller angular scales, and they had preprints from the two teams measuring the redshift-magnitude relation. But Bond and Jaffe chose not to make a pronouncement on the most promising cosmological model; they instead concluded with analyses of the prospects for tighter tests from measurements in progress.

We see in this section the great energy devoted to ingenious model-building. The community in the 1980s and 1990s does not seem to have been seriously dismayed by the excess of ideas over observations, though that tended to be my feeling. Morale was maintained by the ongoing input from observational and experimental programs and by the perception that the models under discussion were not unpromising. Thoughts of elegance can inspire, too, and it was exciting to think that the right way forward may bring us to a well-founded cosmology that would be seen to be elegant in a sense to be discovered.

(p.322) A natural tendency to be conservative may help explain why most ideas about a viable cosmology explored in the two decades leading up to the turn of the century are minimal departures from the original sCDM version, along with minimal variations from the early ideas about inflation. That approach could have missed the right direction forward, but the evidence to be summarized in the next chapter is that it did not.

## Notes:

(1.) The baryons are needed for the acoustic oscillations, illustrated in Figure 5.2, that are important in later cosmological tests, but not for the CMB anisotropy on large angular scales that worried us in the early 1980s. It might be noted that Peebles (1981a) computed the matter power spectrum for adiabatic initial conditions in linear perturbation theory, taking account of radiation and baryons (Figure 5.3); Peebles (1982a,b) computed the spectrum for radiation and CDM, ignoring baryons and massless neutrinos; and Blumenthal et al. (1984) and Vittorio and Silk (1984) reported computations for the full case of radiation, CDM, baryons, and massless neutrinos. Bond and Efstathiou (1987) demonstrated the baryon acoustic oscillations caused by the interaction of plasma and radiation in the CDM model.

(2.) I take this opportunity to apologize for not thinking to transfer the reference to Sachs and Wolfe (1967) in Peebles (1980) to Peebles (1982b).

(3.) The use of spherical harmonics to represent the angular distributions across the sky of galaxies and clusters of galaxies was introduced in Yu and Peebles (1969). Peebles (1982b) introduced its application beyond dipole to the variation of the CMB temperature across the sky. Note here that at l > 0, the real and imaginary parts of the spherical harmonic $Ylm$ have zeros spaced at minimum separation π‎/l radians in the azimuthal and polar directions. The problem of converging lines of zeros toward the poles is solved, because $Ylm$ is close to zero at polar angle θ‎ ≲ m/l. Note also that the ensemble average value $〈|alm|2〉$ for a statistically isotropic process is independent of m.

(4.) Turner, Wilczek, and Zee (1983) independently set down elements of the CDM model, but their inadequate estimate of the predicted CMB anisotropy led them to conclude that the model could not accommodate scale-invariant primeval conditions. Abbott and Wise (1984) independently found that the CMB anisotropy spectrum scales as $〈|alm|2〉∝[l(l+1)]−1$ for scale-invariant initial conditions, by a method different from but physically equivalent to Peebles (1982b). Abbott and Wise did not consider the normalization. Bond and Efstathiou (1987) wrote down the scaling of the angular power spectrum $〈|alm|2〉$ with degree l for the generalization of the primeval power spectrum to Pkkn.

(5.) Simpson and Hime (1989) found indications of detection of a new neutrino species with mass mν‎ = 17 keV that mixes in the electron-type interaction. It led Bond and Efstathiou (1991) and others to consider how this neutrino would affect the predicted distributions of matter and radiation. Bond and Efstathiou found tentative advantages of adding to the cosmological model this neutrino with lifetime postulated to be about a year, but interest in this possibility for a slightly more complicated dark sector faded.

(6.) The correlation function and angular power spectrum are mathematically equivalent, but their utility in practice may be quite different. I learned this from the book by Blackman and Tukey (1959), The Measurement of Power Spectra from the Point of View of Communications Engineering. Blackman and Tukey showed that power spectra are more informative than correlation functions for many applications. That led me to use power spectra based on spherical harmonic expansion rather than Fourier expansion in our first analyses of the spatial distributions of extragalactic objects. But N-point correlation functions usually prove to be the more useful statistics in this situation. The angular distribution of the CMB is best characterized by power spectra and higher moments, as Blackman and Tukey recommended, in spherical harmonic expansion.

(7.) The Partridge and Peebles estimate of the spectrum one might expect of a young galaxy included the prominent atomic hydrogen Ly-α‎resonance emission line. We proposed that this line could be a good marker for the search for distant young galaxies, seen as they were in the past because of the light-travel time, and at redshifts large enough to bring the Ly-α‎ line into the optical. I don’t remember worrying that these resonance photons have a large cross section for scattering by hydrogen atoms, so they may be absorbed by dust as the photons diffuse around a newly forming galaxy with a halo of atomic hydrogen. But the Ly-α‎ line proves to be a useful marker for young galaxies.

(8.) Recall the discussion in footnote 13 in Section 5.2.6: Under scale-invariant initial conditions, the mass fluctuation power spectrum on the scale of galaxies approaches P(k)∝ k−3 prior to the onset of formation of nonlinear mass concentrations, meaning the amplitude of the mass fluctuations scales only as the logarithm of the length scale.

(9.) To convert the measure of anisotropy in Ostriker and Steinhardt (1995) and other papers at the time to the value of δ‎Tl in micro Kelvins on the vertical axes in Plate I, take the square root of the anisotropy and multiply by 2.725 × 106.