2.12 Ultra-light scalar fields

Ultra-light scalar fields arise generically in high energy physics, most commonly as axions or other axion-like particles (ALPs). They are the Pseudo-Goldstone bosons (PGBs) of spontaneously broken symmetries. Their mass remains protected to all loop orders by a shift symmetry, which is only weakly broken to give the fields a mass and potential, through non perturbative effects. Commonly these effects are presumed to be caused by instantons, as in the case of the QCD axion, but the potential can also be generated in other ways that give potentials that are useful, for example, in the study of quintessence [705Jump To The Next Citation Point]. Here we will be considering a general scenario, motivated by the suggestions of [63Jump To The Next Citation Point] and [452Jump To The Next Citation Point], where an ultralight scalar field constitutes some fraction of the dark matter, and we make no detailed assumptions about its origin.

Axions arise generically in string theory [871Jump To The Next Citation Point]. They are similar to the well known QCD axion [715Jump To The Next Citation Point, 873, 872, 315, 742Jump To The Next Citation Point, 866, 911, 2Jump To The Next Citation Point, 316Jump To The Next Citation Point, 908, 932], and their cosmology has been extensively studied [see, for example, 84]. String axions are the Kaluza–Klein zero modes of anti-symmetric tensor fields, the number of which is given by the number of closed cycles in the compact space: for example a two-form such as BMN16 has a number of zero modes coming from the number of closed two-cycles. In any realistic compactification giving rise to the Standard Model of particle physics the number of closed cycles will typically be in the region of hundreds. Since such large numbers of these particles are predicted by String Theory, we are motivated to look for their general properties and resulting cosmological phenomenology.

The properties of the axion 𝜃 are entirely determined by its potential U, whose specific form depends on details in string theory that will not concern us, and two parameters in the four-dimensional Lagrangian

fa2 2 4 ℒ = -2-(∂𝜃) − Λ U (a), (2.12.1 )
where fa is the scale at which the Peccei–Quinn-like symmetry – an additional global U (1) symmetry – is broken, also referred to as the axion decay constant, and Λ is the overall scale of the potential. In terms of the canonically normalized field ϕ = fa𝜃, we find that the mass is given by
Λ2- m = f . (2.12.2 ) a
The values of these parameters are determined by the action S of the non-perturbative physics that generates the potential for a given axion, and it was argued in [63Jump To The Next Citation Point] that this scales with the volume/area of the closed cycle giving rise to that axion, S ∼ A. fa and S are related by
fa ∼ Mpl-. (2.12.3 ) S
fa is typically of order 1016 GeV and can be considered constant for all string axions [871Jump To The Next Citation Point]. However, the mass of each axion depends exponentially on S from
Λ4 = μ4e −S, (2.12.4 )
where μ sets the scale of the non-perturbative physics (essentially, the Planck Scale in the string case), and so, as S varies from axion to axion depending on the cycle areas in the compact space, we expect axion masses to evenly distribute on a logarithmic mass scale all the way down to the Hubble scale today, − 33 H0 ∼ 10 eV [63Jump To The Next Citation Point].

There will be a small thermal population of ALPs, but the majority of the cosmological population will be cold and non-thermally produced. Production of cosmological ALPs proceeds by the vacuum realignment mechanism. When the Peccei–Quinn-like U (1 ) symmetry is spontaneously broken at the scale fa the ALP acquires a vacuum expectation value, the misalignment angle 𝜃i, uncorrelated across different causal horizons. However, provided that inflation occurs after symmetry breaking, and with a reheat temperature T ≲ fa, then the field is homogenized over our entire causal volume. This is the scenario we will consider. The field 𝜃 is a PGB and evolves according to the potential U acquired at the scale μ. However, a light field will be frozen at 𝜃i until the much later time when the mass overcomes the Hubble drag and the field begins to roll towards the minimum of the potential, in exact analogy to the minimum of the instanton potential restoring 𝒞𝒫 invariance in the Peccei-Quinn mechanism for the QCD axion. Coherent oscillations about the minimum of U lead to the production of the weakly coupled ALPs, and it is the value of the misalignment angle that determines the cosmological density in ALPs [579Jump To The Next Citation Point, 431, 826].

The underlying shift symmetry restricts U to be a periodic function of 𝜃 for true axions, but since in the expansion all couplings will be suppressed by the high scale f a, and the specific form of U is model-dependent, we will make the simplification to consider only the quadratic mass term as relevant in the cosmological setting, though some discussion of the effects of anharmonicites will be made. In addition, [705] have constructed non-periodic potentials in string theory.

Scalar fields with masses in the range −33 −22 10 eV < m < 10 eV are also well-motivated dark matter candidates independently of their predicted existence in string theory, and constitute what Hu has dubbed “fuzzy cold dark matter”, or FCDM [452]. The Compton wavelength of the particles associated with ultra-light scalar fields, λ = 1∕m c in natural units, is of the size of galaxies or clusters of galaxies, and so the uncertainty principle prevents localization of the particles on any smaller scale. This naturally suppresses formation of structure and serves as a simple solution to the problem of “cuspy halos” and the large number of dwarf galaxies, which are not observed and are otherwise expected in the standard picture of CDM. Sikivie has argued [827] that axion dark matter fits the observed caustics in dark matter profiles of galaxies, which cannot be explained by ordinary dust CDM.

The large phase space density of ultralight scalar fields causes them to form Bose–Einstein condensates [see 828, and references therein] and allows them to be treated as classical fields in a cosmological setting. This could lead to many interesting, and potentially observable phenomena, such as formation of vortices in the condensate, which may effect halo mass profiles [829, 484], and black hole super radiance [63Jump To The Next Citation Point, 64, 772], which could provide direct tests of the “string axiverse” scenario of [63]. In this summary we will be concerned with the large-scale indirect effects of ultra-light scalar fields on structure formation via the matter power spectrum in a cosmology where a fraction f = Ωa∕Ωm of the dark matter is made up of such a field, with the remaining dark matter a mixture of any other components but for simplicity we will here assume it to be CDM so that (1 − f)Ωm = Ωc.

If ALPs exist in the high energy completion of the standard model of particle physics, and are stable on cosmological time scales, then regardless of the specifics of the model [882] have argued that on general statistical grounds we indeed expect a scenario where they make up an order one fraction of the CDM, alongside the standard WIMP candidate of the lightest supersymmetric particle. However, it must be noted that there are objections when we consider a population of light fields in the context of inflation [605, 606]. The problem with these objections is that they make some assumptions about what we mean by “fine tuning” of fundamental physical theories, which is also related to the problem of finding a measure on the landscape of string theory and inflation models [see, e.g., 583], the so-called “Goldilocks problem.” Addressing these arguments in any detail is beyond the scope of this summary.

We conclude with a summary of the most important equations and properties of ultra-light scalar fields.

Numerical solutions to the perturbation equations indeed show that the effect of ultralight fields on the growth of structure is approximately as expected, with steps in the matter power spectrum appearing. However, the fits become less reliable in some of the most interesting regimes where the field begins oscillations around the epoch of equality, and suppression of structure occurs near the turnover of the power spectrum, and also for the lightest fields that are still undergoing the transition from cosmological constant to matter-like behavior today [632]. These uncertainties are caused by the uncertainty in the background expansion during such an epoch. In both cases a change in the expansion rate away from the expectation of the simplest ΛCDM model is expected. During matter and radiation eras the scale factor grows as a ∼ τp and p can be altered away from the ΛCDM expectation by 𝒪 (10)% by oscillations caused during the scalar field transition, which can last over an order of magnitude in scale factor growth, before returning to the expected behavior when the scalar field is oscillating sufficiently rapidly and behaves as CDM.

The combined CMB-large scale structure likelihood analysis of [37] has shown that ultralight fields with mass around 10 −30–10 −24 eV might account for up to 10% of the dark matter abundance.

2.12.1 Requirements

Ultralight fields are similar in many ways to massive neutrinos [37], the major difference being that their non-thermal production breaks the link between the scale of suppression, km, and the fraction of dark matter, fax, through the dependence of fax on the initial field value ϕi. Therefore an accurate measurement of the matter power spectrum in the low-k region where massive neutrinos corresponding to the WMAP limits on Ω ν are expected to suppress structure will determine whether the expected relationship between Ω ν and km holds. These measurements will limit the abundance of ultralight fields that begin oscillations in the matter-dominated era.

Another powerful test of the possible abundance of ultralight fields beginning oscillations in the matter era will be an accurate measure of the position of the turn over in the matter power spectrum, since this gives a handle on the species present at equality. Ultralight fields with masses in the regime such that they begin oscillations in the radiation-dominated era may suppress structure at scales where the BAO are relevant, and thus distort them. An accurate measurement of the BAO that fits the profile in P (k) expected from standard ΛCDM would place severe limits on ultralight fields in this mass regime.

Recently, [633Jump To The Next Citation Point] showed that with current and next generation galaxy surveys alone it should be possible to unambiguously detect a fraction of dark matter in axions of the order of 1% of the total. Furthermore, they demonstrated that the tightest constraints on the axion fraction fax come from weak lensing; when combined with a galaxy redshift survey, constraining fax to 0.1% should be possible, see Figure 45View Image. The strength of the weak lensing constraint depends on the photometric redshift measurement, i.e., on tomography. Therefore, lensing tomography will allow Euclid – through the measurement of the growth rate – to resolve the redshift evolution of the axion suppression of small scale convergence power. Further details can be found in [633Jump To The Next Citation Point].

View Image

Figure 45: Marginalized uncertainty in fax for CMB (green), a galaxy redshift survey (red), weak lensing (blue) and the total (black) evaluated for four different fiducial axion masses, for the cosmology ΛCDM+fax+ν. Image reproduced by permission from [633Jump To The Next Citation Point], copyright by APS.

Finally, the expected suppression of structure caused by ultralight fields should be properly taken into account in N-body simulations. The nonlinear regime of P (k) needs to be explored further both analytically and numerically for cosmologies containing exotic components such as ultralight fields, especially to constrain those fields which are heavy enough such that km occurs around the scale where nonlinearities become significant, i.e., those that begin oscillation deep inside the radiation-dominated regime. For lighter fields the effects in the nonlinear regime should be well-modelled by using the linear P (k) for N-body input, and shifting the other variables such as Ωc accordingly.

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