Abstract
Guth (Phys. Rev. D 23:347–56, 1981) provided a persuasive rationale for inflationary cosmology based on its ability to solve fine-tuning problems of big bang cosmology. Yet one of the most important consequences of inflation was only widely recognized a few years later: inflation provides a mechanism for generating small departures from uniformity, needed to seed formation of subsequent structures, by “freezing out” vacuum fluctuations to form classical density perturbations. This paper recounts the historical development of this aspect of inflation and puts it in context of the development of ideas on structure formation in relativistic cosmology, before turning to the comparison between inflation and a competing account of structure formation based on topological defects. One aim is to assess in what sense inflation is empirically tested through its account of the formation of structure, in light of persistent debates among cosmologists regarding whether inflation is “falsifiable.”
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Notes
- 1.
The meaning of Mach’s principle and its status has been a focal point for foundational discussions since Einstein’s day; see Barbour and Pfister (1995) for an entry point to the recent literature.
- 2.
The steady-state theory was no longer a serious rival to the standard “big bang” model by this time, although a small group of proponents (including Hoyle, Narlikar, and others) continued to explore the idea and to challenge the empirical underpinnings of the big bang model (see Kragh 1996).
- 3.
Recombination refers to the process by which nuclei capture free electrons and form neutral hydrogen and helium (although “re-” is misleading, as there was no earlier time, in the standard model, at which stable nuclei existed). During recombination, photons decouple from matter as the cross section for Thomson scattering drops to zero.
- 4.
- 5.
- 6.
Lifshitz (1946) analyzed the behavior of small perturbations for two different equations of state, corresponding to radiation-dominated expansion, i.e., p = ρ∕3, where p is the pressure and ρ the energy density, and matter-dominated expansion with p = 0. See, e.g., Longair (2007) for a modern treatment.
- 7.
- 8.
Gamow and Teller (1939) advocated an account of structure formation based on gravitational instability that is undermined by Lifshitz’s results (as Lifschitz explicitly noted). Gamow (1952) and Gamow (1954) are the original papers on the turbulence theory; see Peebles (1971) for a critical review of Gamow’s proposal and other similar ideas. Two other problems with the gravitational instability account were also important in motivating the search for alternatives. First, there is no preferred length or mass scale in general relativity (with the cosmological constant set to zero), so it is unclear how to introduce scales such as the mass of a typical galaxy (see Harrison (1967a) and Harrison (1967b) for a detailed discussion of this point). Second, alternative accounts often claimed to give natural explanations of features of galaxies, such as their rotation and spiral structure.
- 9.
This terminology is due to Zel’dovich and his collaborators. The factor of \(\frac {4}{3}\) arises since the energy density of radiation is ∝ T 4, compared to T 3 for matter (where T is the temperature). These are called “adiabatic” perturbations since the local energy density of the matter relative to the entropy density is fixed. A third mode—tensor perturbations, representing primordial gravitational waves—was not usually included in discussions of structure formation, since they do not couple to energy-density perturbations.
- 10.
In general, for a scale-invariant power spectrum, the Fourier components of the perturbations obey a power law, |δ k |2 ∝ k n; the Harrison-Peebles-Zel’dovich spectrum corresponds to a choice of n = 1 (given that the volume element in the inverse Fourier transform is \(\frac {dk}{k}\); for the other conventional choice, k 2dk, we then have n = −3). The Hubble radius has the appropriate dimension, length: restoring c, it is given by \(\frac {c}{H}\), and the Hubble constant H has units of km per second per megaparsec.
- 11.
Since the perturbations grow with time, at a “constant time” the shorter wavelength perturbations have greater amplitudes for this spectrum. The difficulty with defining the spectrum of density perturbations in terms of “amplitude at a given time” is that it depends on how one chooses the constant time hypersurfaces.
- 12.
It is defined as R H = H −1, where H is the Hubble “constant” (\(H= \frac {\dot {R}}{R}\)); it is also called the speed of light sphere, given that objects moving with the expansion, at a distance R H , appear to move at speed c.
- 13.
A horizon is the surface in a time slice t 0 separating particles moving along geodesics that could have been observed from a world line γ by t 0 from those which could not (Rindler 1956). The distance to this surface, for signals emitted at a time t e , is given by
$$\displaystyle \begin{aligned} d = R(t_0)\int_{t_e}^{t_0}\frac{dt}{R(t)} \end{aligned} $$(9.1)Different “horizons” correspond to different choices of limits of integration. The integral converges for R(t) ∝ t n with n < 1, which holds for matter or radiation-dominated expansion. Thus the integral for the particle horizon (\(\lim _{t_e\rightarrow 0}\)) converges for the FLRW models (e.g., Ellis and Rothman 1993).
- 14.
Energy conditions are constraints on what is taken to be a reasonable source for the gravitational field equations. Roughly speaking, the strong energy condition requires that the stresses in matter will not be so large as to produce negative energy densities. Formally, \(T_{ab}\xi ^{a}\xi ^{b}\geq \frac {1}{2} \mbox{Tr}(T_{ab})\) for every unit timelike ξ a; for a perfect fluid, this implies that ρ + 3p ≥ 0, where ρ is the energy density and p is the pressure. As Bardeen notes, if the strong energy condition fails then there are solutions such that the integral in Equation (9.1) diverges.
- 15.
For example, Blau and Guth (1987) compare the density contrast imposed at t i = 10−35 seconds to the fluctuations obtained by evolving backward from the time of recombination implies Δ ≈ 10−49 at t i , nine orders of magnitude smaller than thermal fluctuations.
- 16.
- 17.
“Hot” vs. “cold” refers to the thermal velocities of relic particles for different types of dark matter. Hot dark matter decouples while still “relativistic,” in the sense that the momentum is much greater than the rest mass, and relics at late times would still have large quasi-thermal velocities. Cold dark matter is “nonrelativistic” when it decouples, meaning that the momentum is negligible compared to the rest mass and relics have effectively zero thermal velocities.
- 18.
De Sitter spacetime is a solution to Einstein’s field equations with a stress-energy tensor given by T ab = −ρ v g ab , where ρ v is the vacuum energy density. The scale factor then expands exponentially, with \(\chi ^2 = \frac {8\pi }{3}\rho _v\). During inflation the stress-energy tensor has approximately this form. Given that the vacuum energy density remains constant during the expansion while “ordinary” matter and energy is rapidly diluted, the vacuum energy dominates the expansion and the solution, roughly speaking, approaches de Sitter spacetime.
- 19.
The stress-energy tensor stated in the previous footnote does not satisfy the strong energy condition formulated in footnote 14; the fact that the vacuum energy density does not dilute with expansion reflects this. A stress-energy tensor that violates this condition is a necessary condition for exponential expansion within classical GR.
- 20.
Guth discovered inflation while focusing on a third problem, the monopole problem. GUTs from the late 1970s predicted the existence of magnetic monopoles, and the relic abundance of the monopoles would be many orders of magnitude greater than the observed energy density of the universe. See Guth (1997a) for his account of how he discovered inflation. Unlike the monopole problem, which arises for the combination of cosmology and these GUTs, the flatness and horizon problems are problems for the cosmological standard model.
- 21.
See Smeenk (2005) for a discussion of how these two features of the FLRW models were treated prior to Guth’s identification of them as problems to be solved by inflation.
- 22.
- 23.
\(\varOmega =: \frac {\rho }{\rho _{c}}\), where the critical density ρ c is the value required for the attraction of gravity due to positive matter-energy density to precisely balance the initial expansion and cosmological constant: \(\rho _{c} = \frac {3}{8\pi }\left (H^2 - \frac {\varLambda }{3}\right )\).
- 24.
More precisely,
$$\displaystyle \begin{aligned} \frac{|\varOmega - 1|}{\varOmega} \propto R(t)^{3\gamma - 2}, \end{aligned} $$(9.2)where γ is used to classify different types of perfect fluids. The equation of state of a perfect fluid is p = (γ − 1)ρ, where p is the pressure and ρ the density. For radiation, γ = 4∕3 and for “dust” γ = 1 (corresponding to zero pressure). For “normal” matter, satisfying the energy conditions defined in footnote 14, γ > 2∕3.
- 25.
During inflation, the strong energy condition is violated, and γ = 0; it is clear from Equation (9.2) that Ω is then driven toward 1.
- 26.
This point was first made in response to Misner’s “chaotic cosmology,” which like inflation proposed new dynamics (in Misner’s case, damping of anisotropies due to neutrino viscosity) in order to insure that an isotropic universe emerges from a large range of anisotropic initial conditions. In response to Misner, Collins and Stewart (1971) showed that one can always pick an arbitrarily large anisotropy at a given time t 0 and find a solution of the relevant system of equations as long as there are no processes which could prevent arbitrarily large anisotropies at some t i < t 0. A similar criticism applies to inflation, as Madsen and Ellis (1988) have emphasized. Guth (1997b) has acknowledged this point: “…I emphasize that NO theory of evolution is ever intended to work for arbitrary initial conditions. …In all cases, the most we can hope for is a theory of how the present situation could have evolved from reasonable initial conditions” (pp. 240–241, emphasis in the original).
- 27.
Brawer’s case is based on published discussions of these problems, as well as the extensive interviews with cosmologists published in Lightman and Brawer (1990).
- 28.
Penrose’s original terse statement of this criticism appeared in a book review of the conference proceedings of the Nuffield workshop (discussed below), and he has discussed it further in Penrose (1989, 2004). Although I do not have the space to discuss Penrose’s objection, several more recent papers pursue the issues raised by Penrose, including Unruh (1997) and Hollands and Wald (2002).
- 29.
The stress-energy tensor for a scalar field is given by
$$\displaystyle \begin{aligned} T_{ab} = \nabla_a\phi \nabla_b\phi - \frac{1}{2} g_{ab}\left(g^{cd}\nabla_c\nabla_d\phi - V(\phi)\right); \end{aligned} $$(9.3)inflation requires that the field is “potential-dominated” in the sense that the field is sufficiently uniform that the derivative terms are negligible, V (ϕ) >> g cd∇ c ∇ d ϕ. If this condition holds, T ab ≈−V (ϕ)g ab as required to produce exponential expansion.
- 30.
- 31.
- 32.
At roughly the same time, Stephen Hawking and Ian Moss proposed an alternative solution to the transition problem. Although Hawking and Moss (1982) is sometimes cited as a third independent discovery of new inflation, it differs substantially from the other proposals. The aim of the paper is to show that including the effects of curvature and finite horizon size leads to a different description of the phase transition. This phase transition proceeds from a local minimum at ϕ = 0 to the global mimimum ϕ 0 via an intermediate state ϕ 1; rather cryptic arguments lead to the conclusion that “the universe will continue in the essentially stationary de Sitter state until it makes a quantum transition everywhere to the ϕ = ϕ 1 solution” (p. 36). They further argue that following this transition to a coherent Hubble scale patch, ϕ will “roll down the hill” (for an appropriate values of parameters in the effective potential), producing an inflationary stage long enough to match Guth’s success.
- 33.
- 34.
- 35.
The description is taken from the invitation letter to the conference (Guth 1997a, 223). The Nuffield Foundation had previously sponsored conferences in quantum gravity but shifted the focus to early universe cosmology in response to interest in the inflationary scenario. A 1981 conference in Moscow on quantum gravity also included numerous discussions of early universe cosmology (Markov and West 1984), but Nuffield was the first conference explicitly devoted to the early universe.
- 36.
Synchronous gauge is also known as “time-orthogonal” gauge: the coordinates are adapted to constant time hypersurfaces orthogonal to the geodesics of comoving observers. All perturbations are confined to spatial components of the metric, i.e., the metric has the form ds 2 = R 2(t)(dt 2 − h ij dx idx j), with i, j = 1, 2, 3. The coordinates break down if the geodesics of co-moving observers cross.
- 37.
- 38.
- 39.
- 40.
Rocky Kolb used such a slide in a talk at the Pritzker Symposium (Chicago, 1998); for an example of such a list, see Shellard (2003, figure 41.3).
- 41.
I have left aside one important aspect of the comparison between inflation and topological defect theories, namely, the role of different types of dark matter in each scenario. The mechanisms for structure formation are part of package deal, including assumptions about the overall matter budget and other factors more significant for later stages of structure formation.
- 42.
This means that, roughly speaking, for all g ∈ G, the Hamiltonian of the system is invariant under the action of g, but the vacuum or ground state of the system is not. (This is only a rough gloss; in quantum mechanics the action of a symmetry g is usually represented by a unitary operator on the Hilbert space, but in the case of broken symmetry, there is not a well-defined operator mapping between degenerate vacua, as these each define different Hilbert spaces.) The degenerate vacuum states are labeled by different values of the “order parameter” of the transition. The order parameter is the thermodynamic quantity that changes discontinuously through the transition and characterizes different phases, corresponding to degenerate vacua in this case; it is the vacuum expectation value of the relevant field(s).
- 43.
The relevant structure is given by the homotopy groups of the space. For further discussion, see, e.g., Vilenkin and Shellard (2000).
- 44.
Additional types of defects arise due to the distinction between gauge and global symmetries and the possibility of “hybrid” defects. Defects formed in a transition breaking a global symmetry tend to have energy density distributed throughout a region, whereas those formed by gauge symmetry breaking are more localized. Hybrid defects are produced by a series of phase transitions, leaving an interacting network of defects of different kinds. See, e.g., Vilenkin and Shellard (2000), for further discussion.
- 45.
However, the sense in which the two theories are scale invariant is different; see, e.g., § 5.1.1 of Martin and Brandenberger (2001). Many defect models are scale invariant only over a limited dynamical range; for example, in models of defect formation via strings scale invariance is broken at the matter-radiation transition.
- 46.
See Vilenkin and Shellard (2000, Chapter 11) for an overview; the closing section (p. 342) emphasizes the changes in the account due to numerical simulations of the evolution of string networks.
- 47.
Several groups published calculations at around this time supporting the general picture I summarize here; see, e.g., Magueijo et al. (1996) and Pen et al. (1997). See Durrer et al. (2002) for a comprehensive review of this area with further references to the original papers and Brandenberger (1994) for an earlier review.
- 48.
Mukhanov et al. (1992) is the canonical review article regarding structure formation.
- 49.
The equation can be derived from the action for the scalar field minimally coupled to gravity (with various simplications, such as neglecting metric perturbations):
$$\displaystyle \begin{aligned} \frac{d^2 \phi_k}{dt^2} + 3H\frac{d\phi_k}{dt} + \frac{k^2}{R^2}\phi_k=0. \end{aligned} $$(9.8) - 50.
Although I do not have space to discuss the issue further here, this step involves a quantum to classical transition.
- 51.
The angular power spectrum characterizes the variations in temperature of the CMBR, i.e., the amount of temperature variation across different points of the sky versus the angular frequency ℓ. Small values of ℓ correspond to temperature variations with a large angular scale. See, e.g., Liddle and Lyth (2000, § 5.2), for further discussion of the angular power spectrum.
- 52.
Peter Scheuer made this remark in the course of warning a student, Malcolm Longair, about the current status of cosmology in 1963; the list included (1) that the sky is dark at night, (2) that the galaxies recede, and (2 1/2) that the universe is evolving (qualified as a half fact due to its uncertainty).
- 53.
Observations seem to rule out topological defects as the primary mechanism for generating large-scale structure. However, defects might still play a role as part of the full account of the formation of structure or in other aspects of early universe cosmology, such as baryogenesis.
- 54.
The falsifiability of inflation, focusing in part on flatness, is addressed quite directly in a number of papers in Turok (1997), in particular the contributions by Linde, Steinhardt, Guth, and Albrecht. This has been a perennial subject of debate since the early days of inflation.
- 55.
The question was particularly pressing throughout the 1990s, when the evidence seemed to favor open cosmological models with Ω 0 ≈ 0.2–0.3, although there was not a general consensus. See, e.g., Coles and Ellis (1997) for a detailed argument in favor of an open universe. However, the consensus had begun to shift in favor of a flat universe by 1998. Peebles and David Schramm were invited to convene a “great debate” on the issue in April of 1998. Due to Schramm’s death, the debate was rescheduled for October of 1998, with Michael Turner taking Schramm’s place. But given that Peebles and Turner both agreed that the evidence decisively favored a flat universe, they changed the subject of the debate to “Is Cosmology Solved?” (Peebles 1999a; Turner 1999).
- 56.
The distinction is perhaps too quick, given that there are some predictions related to initial conditions. For example, inflation predicts that the observed universe is topologically simply connected; inflation is incompatible with compact topology at sub-horizon scales. Evidence that the universe is multiply connected would rule out inflation.
- 57.
The modes will be “born” at different times, continually “emerging out of the spacetime foam” (or whatever description the full theory of quantum gravity provides), with the modes relevant to large-scale structure born at times much earlier than the Planck time. By way of contrast, in the usual approach the modes at all length scales are specified to be in a ground state at a particular time, such as the Planck time. But the precise time at which one stipulates the field modes to be in a vacuum state does not matter given that the sub-horizon modes evolve adiabatically.
- 58.
The apsidal angle θ is the angle through which the radius vector rotates between two consecutive apsides, which are points on the orbit of maximum (aphelion) or minimum (perihelion) distance from the force center. Newton establishes (Book I, Proposition 45) that for approximately circular orbits under a centripetal force varying as f ∝ r n−3, the apsidal angle is given by \(n = \left (\frac {\theta }{\pi }\right )^{2}\). For stable orbits, the radius vector rotates through π between the aphelion and perihelion, such that n = 1 and f ∝ r −2; and for nearly stable orbits, the force is approximately f ∝ r −2.
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Smeenk, C. (2018). Inflation and the Origins of Structure. In: Rowe, D., Sauer, T., Walter, S. (eds) Beyond Einstein. Einstein Studies, vol 14. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-7708-6_9
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