Axions arise generically in string theory [871]. They are similar to the well known QCD axion
[715
, 873, 872, 315, 742
, 866, 911, 2
, 316
, 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
16
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 are entirely determined by its potential
, whose specific form
depends on details in string theory that will not concern us, and two parameters in the four-dimensional
Lagrangian
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 symmetry is spontaneously broken at the scale
the
ALP acquires a vacuum expectation value, the misalignment angle
, uncorrelated across
different causal horizons. However, provided that inflation occurs after symmetry breaking,
and with a reheat temperature
, then the field is homogenized over our entire causal
volume. This is the scenario we will consider. The field
is a PGB and evolves according
to the potential
acquired at the scale
. However, a light field will be frozen at
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
invariance in the Peccei-Quinn mechanism for the QCD axion. Coherent
oscillations about the minimum of
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
[579
, 431, 826].
The underlying shift symmetry restricts to be a periodic function of
for true axions,
but since in the expansion all couplings will be suppressed by the high scale
, and the
specific form of
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 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,
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 [63, 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
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
.
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.
On large scales the pressure becomes negligible, the sound speed goes to zero and the field behaves as
ordinary dust CDM and will collapse under gravity to form structure. However on small
scales, set by , the sound speed becomes relativistic, suppressing the formation of
structure;
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 CDM model is expected.
During matter and radiation eras the scale factor grows as
and
can be altered
away from the
CDM expectation by
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 might account for up to 10% of the dark matter
abundance.
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, , and the fraction of
dark matter,
, through the dependence of
on the initial field value
. Therefore
an accurate measurement of the matter power spectrum in the low-
region where massive
neutrinos corresponding to the WMAP limits on
are expected to suppress structure will
determine whether the expected relationship between
and
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
expected from standard
CDM would place severe limits on ultralight fields in this mass
regime.
Recently, [633] 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
come from weak lensing; when
combined with a galaxy redshift survey, constraining
to 0.1% should be possible, see Figure 45
. 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 [633
].
Finally, the expected suppression of structure caused by ultralight fields should be properly taken into
account in -body simulations. The nonlinear regime of
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
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
for
-body input, and shifting the other variables such as
accordingly.
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Living Rev. Relativity 16, (2013), 6
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