Many dark energy models are characterized by attractor or tracker behavior. The goal also is to design the model so that the field energy density is subdominant at high redshift, where it is not wanted, and dominant at low redshift, where that is what seems to be observed.
In the simplest such scalar field models the action has a conventionally normalized scalar field kinetic and spatial gradient term, and the real scalar field is coupled only to itself and gravity. Then the scalar field part of the model is fully characterized by the scalar field potential (along with some broad constraints on the initial conditions for the field, if the attractor behavior is realized). The inverse power law potential model is described in Secs. II.C and III.E, and used in some of the cosmological tests in Sec. IV. Here we list other scalar field potentials now under discussion for these minimal dark energy models, modifications of the kinetic part of the action, possible guidance from high energy particle physics ideas, and constraints on these ideas from cosmological observations in the context of dark energy models.
To those of us not active in this field the models may seem baroque in their complexity, but that may be the way Nature is. And as we accumulate more and better data it will be possible to test and constrain these models.
This is an active area of research, with frequent introduction of new models, so our discussion must be somewhat sketchy. ^{108} We limit discussion to a cosmological model with space sections that are flat thanks to the presence of the dark energy density.
As mentioned in Sec III.E, the simplest exponential potential scalar field model is unacceptable because it cannot produce the wanted transition from sub-dominant to dominant energy density. Ratra and Peebles (1988) consider more complex potentials, such as powers of linear combinations of exponentials of the field . Related models are under active investigation. These include potentials that are powers of cosh() and/or sinh(). ^{109} Sahni and Wang (2000) present a detailed discussion of a specific example, V() (cosh - 1)^{p}, where and p are constants, emphasizing that this potential interpolates from V e^{-p } to V ( )^{2p} as | | increases, thus preserving some of the desirable properties of the simplest exponential potential case. De la Macorra and Piccinelli (2000) consider potentials that are exponentials of more complicated functions of , such as ^{2} and e^{}. Skordis and Albrecht (2002) discuss a model with V() [1 + ( - A)^{2}] exp[- (q / 2)^{1/2}], while Dodelson, Kaplinghat, and Stewart (2000) study a potential V() [1 + A sin(B)] exp[- (q / 2)^{1/2}], and Ng, Nunes, and Rosati (2001) consider a model with V() ^{A} exp[B^{C}], a simple example of a class of supergravity-inspired potentials studied by Copeland, Nunes, and Rosati (2000). Here A, B, and C are parameters. Steinhardt et al. (1999) consider more complicated functions of inverse powers of , such as V() exp(1/) and linear combinations of inverse powers of . Brax and Martin (2000) consider a supergravity-inspired generalization with V() = ^{-} e^{2/2}. ^{110}
It will be quite a challenge to select from this wide range of functional forms for the potential those that have particular theoretical merit and some chance of being observationally acceptable.
Following Dolgov (1983), there have been discussions of non-minimally coupled dark energy scalar field models. ^{111} These Jordan-Brans-Dicke type models have an explicit coupling between the Ricci scalar and a function of the scalar field. This causes the effective gravitational "constant" G (in units where masses are constant) to vary with time, which may be an observationally interesting effect and a useful constraint. ^{112}
In yet another approach, people have considered modifying the form of the dark energy scalar field kinetic and spatial gradient term in the action. ^{113}
We discussed in Sec. III.E the idea that at the end of inflation the dark energy scalar field potential might patch on to the part of the scalar field potential responsible for inflation. In an inverse power law model for the dark energy potential function this requires an abrupt drop in V() at the end of inflation. More sophisticated models, now dubbed quintessential inflation, attempt to smooth out this drop by constructing scalar field potentials that interpolate smoothly between the part responsible for inflation and the low redshift, dark energy, part. These models assume either minimally or non-minimally (Jordan-Brans-Dicke) coupled scalar fields. ^{114}
The dark energy scalar field models we have reviewed here are meant to be classical, effective, descriptions of what might come out of a more fundamental quantum mechanical theory. The effective dark energy scalar field is coupled to itself and gravity, and is supposed to be coupled to the other fields in the universe only by gravity. This might be what Nature chooses, but we lack an understanding of why the coupling of dark and ordinary fields that are allowed by the symmetries are not present, or have coupling strengths that are well below what might be expected by naive dimensional analysis (e.g., Kolda and Lyth, 1999). A satisfactory solution remains elusive. Perhaps this is not unexpected, because it likely requires a proper understanding of how to reconcile general relativity and quantum mechanics.
We turn now to scalar field dark energy models that arguably are inspired by particle physics. Inverse power law scalar field potentials are generated non-perturbatively in models of dynamical supersymmetry breaking. In supersymmetric non-Abelian gauge theories, the resulting scalar field potential may be viewed as being generated by instantons, the potential being proportional to a power of e^{-1/g2()}, where g() is the gauge coupling constant which evolves logarithmically with the scalar field through the renormalization group equation. Depending on the parameters of the model, an inverse power-law scalar field potential can result. This mechanism may be embedded in supergravity and superstring/M theory models. ^{115} This has not yet led to a model that might be compared to the observations. ^{116}
In the model considered by Weiss (1987), Frieman et al. (1995), and others the dark energy field potential is of the form V() = M^{4}[cos( / f )+ 1], where M and f are mass scales and the mass of the inhomogeneous scalar field fluctuation ~ M^{2} / f is on the order of the present value of the Hubble parameter. For discussions of how this model might be more firmly placed on a particle physics foundation see Kim (2000), Choi (1999), Nomura, Watari, and Yanagida (2000), and Barr and Seckel (2001).
There has been much recent interest in the idea of inflation in the brane scenario. Dvali and Tye (1999) note that the potential of the scalar field which describes the relative separation between branes can be of a form that leads to inflation, and will include some inverse power-law scalar field terms. ^{117} It will be interesting to learn whether these considerations can lead to a viable dark energy scalar field model. Brane models allow for a number of other possibilities for dark energy scalar fields, ^{118} but it is too early to decide whether any of these options give rise to observationally acceptable dark energy scalar field models.
Building on earlier work ^{119}, Hellerman, Kaloper, and Susskind (2001), and Fischler et al. (2001) note that dark energy scalar field cosmological models have future event horizons characteristic of the de Sitter model. This means some events have causal futures that do not share any common events. In these dark energy scalar field models, some correlations are therefore unmeasurable, which destroys the observational meaning of the S-matrix. This indicates that it is not straightforward to bring superstring/M theory into consistency with dark energy models in which the expansion of the universe is accelerating ^{120}.
At the time of writing, while there has been much work, it appears that the dark energy scalar field scenario still lacks a firm, high energy physics based, theoretical foundation. While this is a significant drawback, the recent flurry of activity prompted by developments in superstring/M and brane theories appears to hold significant promise for shedding light on dark energy. Whether this happens before the observations rule out or "confirm" dark energy is an intriguing question.
^{108} In particular, we do not discuss variable mass term models, complex or multiple scalar field models, or repulsive matter. We also omit non-scalar field aspects of brane models, Kaluza-Klein models, bimetric theories of gravitation, quantum mechanical running of the cosmological constant, the Chaplygin gas, and the superstring tachyon. Back.
^{109} For examples see Chimento and Jakubi (1996), Starobinsky (1998), Kruger and Norbury (2000), Di Pietro and Demaret (2001), Ureña-López and Matos (2000), González-Díaz (2000), and Johri (2001). Back.
^{110} For still more examples see Green and Lidsey (2000), Barreiro, Copeland, and Nunes (2000), Rubano and Scudellaro (2002), Sen and Sethi (2002), and references therein. Back.
^{111} See Uzan (1999), Chiba (1999), Amendola (1999, 2000), Perrotta, Baccigalupi, and Matarrese (2000), Bartolo and Pietroni (2000), Bertolami and Martins (2000), Fujii (2000), Faraoni (2000), Baccigalupi et al. (2000), Chen, Scherrer, and Steigman (2001), and references therein. Back.
^{112} Nucleosynthesis constraints on these and related models are discussed by Arai, Hashimoto, and Fukui (1987), Etoh et al. (1997), Perrotta et al. (2000), Chen et al. (2001), and Yahiro et al. (2002). Back.
^{113} See Fujii and Nishioka (1990), Chiba, Okabe, and Yamaguchi (2000), Armendariz-Picon, Mukhanov, and Steinhardt (2001), Hebecker and Wetterich (2001), and references therein. Back.
^{114} Early work includes Frewin and Lidsey (1993), Spokoiny (1993), Joyce and Prokopec (1998), and Peebles and Vilenkin (1999); more recent discussions are given by Kaganovich (2001), Huey and Lidsey (2001), Majumdar (2001), Sahni, Sami, and Souradeep (2002), and Dimopoulos and Valle (2001). Back.
^{115} In the superstring/M theory case, since the coupling constant is an exponential function of the dilaton scalar field, the resulting potential is usually not of the inverse power-law form. However, it is perhaps not unreasonable to think that after the dilaton has been stabilized, it or one of the other scalar fields in superstring/M theory might be able to play the role of dark energy. Gasperini, Piazza, and Veneziano (2002) and Townsend (2001) consider other ways of using the dilaton as a dark energy scalar field candidate. Back.
^{116} See, Davis, Dine, and Seiberg (1983), and Rossi and Veneziano (1984) for early discussions of supersymmetry breaking, and Quevedo (1996), Dine (1996), Peskin (1997), and Giudice and Rattazzi (1999) for reviews. Applications of dynamical non-perturbative supersymmetry breaking directly relevant to the dark energy scalar field model are discussed in Binétruy (1999), Masiero, Pietroni, and Rosati (2000), Copeland et al. (2000), Brax, Martin, and Riazuelo (2001), de la Macorra and Stephan-Otto (2001), and references therein. Back.
^{117} Recent discussions of this setup include Halyo (2001b), Shiu and Tye (2001), Burgess et al. (2002), Kyae and Shafi (2002), García-Bellido, Rabadán, and Zamora (2002), Blumenhagen et al. (2002), and Dasgupta et al. (2002). Back.
^{118} See Uzawa and Soda (2001), Huey and Lidsey (2001), Majumdar (2001), Chen and Lin (2002), Mizuno and Maeda (2001), Myung (2001), Steinhardt and Turok (2002) and references therein. Back.
^{119} See Maldacena and Nuñez (2001), Bousso (2000), Banks and Fischler (2001), and references therein. Back.
^{120} Other early discussions of this issue may be found in He (2001), Moffat (2001), Deffayet, Dvali, and Gabadadze (2002), Halyo (2001a), and Kolda and Lahneman (2001). The more recent literature may be accessed from Larsen, van der Schaar, and Leigh (2002), and Medved (2002). Back.