Axion Miniclusters in Modified Cosmological Histories
Abstract
If the symmetry breaking leading to the origin of the axion dark matter field occurs after the end of inflation and never restores, then overdensities in the axion field collapse to form dense objects known in the literature as axion miniclusters. The mass of the minicluster is set by the energy density of the axion field inside the horizon at the onset of coherent oscillations, and the radius is fixed by the details of the collapse at matterradiation equality. In the standard cosmological scenario, axion miniclusters are expected to be as massive as the heaviest asteroids in the Solar System and with a size comparable to an astronomical unit; yet, their density would be millions of times larger than the local dark matter density. These estimates strongly depend on the details of the cosmology at which the onset of axion oscillations begin. A modification to these initial conditions would alter the expected mass and radius of an axion minicluster by various orders of magnitudes. Assessing the properties of miniclusters is crucial for experimental interest, since the direct detection of axions at current and future facilities would be sensibly altered if the axion field clumps into substructures instead of existing as a smooth component. In turn, this could potentially lead to an alternative method for exploring the content of the Universe before Big Bang Nucleosynthesis (BBN) occurred, the earliest time from which we have direct data. We have considered some modified exotic cosmologies during which the Universe could have been dominated by some form of energy other than radiation, down to a reheating temperature which is bounded by BBN considerations. If the axion field starts to evolve during this exotic cosmological period then various properties would be altered compared to the standard expectations, including the number of miniclusters formed, their size and mass, and the chance of an encounter with the Earth.
I Introduction
The nature of the cold dark matter (CDM) remains unknown to date despite the growth of evidence in support of its existence coming, on top of the original motivations Rubin et al. (1976a, b), from gravitational lensing Trimble (1987), the cosmic microwave background radiation (CMBR) Ade et al. (2016); Aghanim et al. (2018), also in combination with Lyman and weak lensing Lesgourgues et al. (2007), the hierarchical structure formation of the observable universe Springel et al. (2005), the formation and evolution of galaxies Davis et al. (1985); Efstathiou et al. (1985); Springel et al. (2006), galactic collisions Clowe et al. (2004); Markevitch et al. (2004), and a plethora of other observational techniques.
Among many proposed hypothetical particles which candidate as composing the dark matter constituent is the quantum chromodynamics (QCD) axion Weinberg (1978); Wilczek (1978), the quantum of the axion field arising from the spontaneous breaking of a U(1) symmetry first introduced by Peccei and Quinn (PQ Peccei and Quinn (1977a, b)) to address the strongCP problem Belavin et al. (1975); ’t Hooft (1976); Jackiw and Rebbi (1976); Callan et al. (1976). The fact that the existence of the axion could solve up to two distinct problems in physics makes its search particularly appealing. If the axion field exists, it would have been originated in the early universe as the angular variable of the complex PQ field, after the PQ symmetry breaking occurring at a yet unknown energy scale , where is the number of vacua in the theory which we set to unity. Realizable minimal extensions of the Standard Model that embed the socalled “invisible” axion include the KSVZ or hadronic model Kim (1979); Shifman et al. (1980) in which the axion does not couple to the standard model quarks and leptons at treelevel but it couples with a new family of heavy quarks, and the DFSZ model Zhitnitsky (1980); Dine et al. (1981), where the axion couples to a Higgs doublet.
The axion field interacts with QCD instantons Di Vecchia and Veneziano (1980); Gross et al. (1981), which alter the axion mass and introduce a temperature dependence. Semianalytical estimates of this effect Gross et al. (1981); Turner (1986); Bae et al. (2008); Wantz and Shellard (2010a) have been recently superseded by full lattice computation Bonati et al. (2016); Borsanyi et al. (2016); Petreczky et al. (2016), for which a consensus is yet to be achieved. An accurate evaluation of QCD effects is crucial in assessing the present axion energy density Yamaguchi et al. (1999); Hiramatsu et al. (2011a); Klaer and Moore (2017a, b).
In recent years, considerable effort has been devoted in search for this elusive particle in laboratories Vogel et al. (2013); Budker et al. (2014); Stadnik and Flambaum (2014); Rybka et al. (2015); Arvanitaki and Geraci (2014); Kahn et al. (2016); Stern (2016); Brubaker et al. (2017); Chung (2016); Caldwell et al. (2017); Barbieri et al. (2017); Abel et al. (2017); Stadnik et al. (2018). In most of the proposed experiments, the search for the axion is motivated by the effective axionphoton coupling arising from a fermion loop Kaplan (1985); Srednicki (1985), which also leads to a modification of Maxwell’s equations Sikivie (1983); Wilczek (1987); Raffelt and Stodolsky (1988); Das et al. (2005, 2008); Ganguly et al. (2009); Visinelli (2013); Terças et al. (2018); Visinelli and Terças (2018). Treelevel couplings are proportional to the axion mass at zero temperature , for which an upper bound can be placed using the measured duration of the neutrino signal of the supernova SN1987A Raffelt (2008), the nonobservation of deviations from the modeling of the brightness tip of red giant stars Viaux et al. (2013), or unobserved deviations from the standard theory of the cooling time of white dwarves Raffelt (1986); Giannotti et al. (2016). Other constraints come from considerations on the energy loss of spinning black holes of solar mass Arvanitaki et al. (2010); Arvanitaki and Dubovsky (2011); Arvanitaki et al. (2015).
The history and the properties of the present axion field strongly depend on the moment at which the breaking of the PQ symmetry occurs with respect to inflation Linde (1988, 1991); Turner and Wilczek (1991); Wilczek (2004); Tegmark et al. (2006); Hertzberg et al. (2008); Freivogel (2010); Mack (2011); Visinelli and Gondolo (2009); Acharya et al. (2010). This is parametrized by the relative strength of the axion energy scale to the Hubble rate at the end of inflation . In this paper, we only consider an axion field that has originated after the end of inflation, . Under this assumption, the axion field acquires different initial values within a Hubble patch, owing to the small correlation length between different regions that are nonetheless in causal contact. Such mechanism creates topological defects like axionic strings and walls which connect and separate different patches Vilenkin and Everett (1982); Kibble et al. (1982), which have to decay prior to recombination since there is no measurable effect of such defects onto the observed CMBR spectrum. In this scenario i) the axion field contains both zero and nonzero modes, ii) the decay of topological defects eventually increases the cold axion population, iii) spatial variations in the axion field form large overdense regions.
If the PQ symmetry breaking occurs after inflation, a fraction of the total axions component is expected to organize into gravitationally bound structures known as axion “miniclusters” Hogan and Rees (1988); Kolb and Tkachev (1993, 1994a, 1994b), prompted by the inhomogeneities in the axion field in this scenario. Axion miniclusters are compact objects with a density of various orders of magnitude higher than the present local CDM density. It is inside axion minicluster that another type of exotic structure, an axion star Kaup (1968); Ruffini and Bonazzola (1969); Das (1963); Feinblum and McKinley (1968); Teixeira et al. (1975); Colpi et al. (1986); Seidel and Suen (1991); Tkachev (1991); Braaten et al. (2016a); Eby et al. (2016a, b); Levkov et al. (2017); Helfer et al. (2016); Braaten et al. (2016b, c); Bai et al. (2016); Eby et al. (2017); Desjacques et al. (2017); Visinelli et al. (2018); Krippendorf et al. (2018) could possibly form. Axion miniclusters erase the power spectrum at scales of the order of solar masses, corresponding to the mass enclosed within a Hubble horizon at the QCD phase transition. This scale is much smaller than the smallest clump that weakly interacting massive particles form due to the freestreaming cutoff, and thus provides an unique detection signature. As structure formation evolves, axion miniclusters are expected to hierarchically assemble into dark matter halos of galactic size, forming minicluster halos. This claim has yet to be addressed numerically, as well as the possibility that miniclusters might have not survived tidal disruption. Some studies suggest that miniclusters survive hierarchical structure formation to date Zurek et al. (2007); Tinyakov et al. (2016); Hardy (2017); Dokuchaev et al. (2017), claiming that it is possible to bound the fraction of dark matter in halos using microlensing data Fairbairn et al. (2017, 2018). Semianalytic results on the mass function of axion miniclusters are available today Enander et al. (2017), with refined numerical work in progress.
Today, miniclusters are gravitationally bound clumps of axions with masses of the order of the largest asteroids like Vesta or Pallas, and of size comparable to an astronomical unit. It is usually expected that a sizable fraction of axions is bound into minicluster structures, the remaining part forming a homogeneous halo. Despite the large number of clumps expected, the Earth would rarely encounter one such object within a galactic year. If is close to unity, the direct detection of axions in microwave cavity searches would then be severely affected. A negative search by a cavity experiment could then be an indication that axions are mostly organized in miniclusters. The existence of axion minicluster could then possibly alter the expectations for a direct detection. Since axion detection is sensitive to the local CDM energy density, a clumpy axion distribution would lead to spikes in the axion detection spectrum and be relevant for a direct detection technique. The interest in all of the upcoming axion detectors then lies in the present phasespace distribution of the axion CDM, which is not expected to be homogeneous even at the interstellar scale. A reliable detection must take into account the possibility of a inhomogeneous CDM distribution either in space (axion miniclusters and stars) or in momentum (lowdispersion filaments from tidal stripping). Numerical simulations show that the most massive filaments could indeed be detected by ADMX Vogelsberger and White (2011). If the axion is discovered, the spatial granularity of the distribution can be probed by a devoted network of detectors, while the momentum distribution would be revealed by enabling axion directional detection Irastorza and Garcia (2012); Irastorza and Redondo (2018); Knirck et al. (2018). Miniclusters could suffer tidal disruption by stars, with the value of diminishing since the initial value fixed at the QCD phase transition and with the appearance of axionic streams. The determination of and of the axion phasespace is then crucial in correctly interpreting the outcome of the various experiments that will start looking for axion CDM in the near future.
Owing to the importance of miniclusters in axion astrophysics and laboratory searches, a determination of their properties like the mass and the size is crucial. These quantities strongly depend on the details of the cosmological model during which the axion field acquires a nonzero mass. It has been already noticed that the value of the axion mass for which the axion explains the totality of the observed CDM might depend not only on the mechanisms of production (vacuum realignment and the decay of topological defects), but also on the cosmological model that describes the expansion of the Universe at the time when coherent oscillations of the axion field begin Visinelli and Gondolo (2010). In this paper, we show how a choice of a different cosmological model also alters the properties of the axion clumps that later form and might affect their direct detection Tinyakov et al. (2016) and the microlensing from miniclusters Fairbairn et al. (2018). In this work we assume that all of the cold axion population falls into miniclusters and that axions make up the totality of the cold dark matter budget observed. We have summarized the relevant results in Table 1 below, where we discuss the modifications to the cosmological scenario (see text for details), the alterations to the mass and size of miniclusters in modified scenarios and the frequency and duration of their encounter with the Earth.
This paper is organized as follows. Sec. II is devoted to reviewing the production of cosmological axions. In Sec. III we fix our notation for the description of a generic cosmological model and we provide results for the equation of motion and the axion population from both the misalignment mechanism and string decay in a modified cosmology. In Sec. IV we obtain analytic results for the size and the mass of the miniclusters, and we estimate the size of the freestreaming length at the matterradiation equality showing that miniclusters are not erased by the freestreaming mechanism.
Ii Axion cosmology
ii.1 Production mechanisms
In this section, we review the axion physics and cosmological production. For an excellent introduction to the subject we refer to Ref. Kolb and Turner (1994), while thorough reviews are found in Refs. Raffelt (2007); Peccei (2008); Sikivie (2008); Kim and Carosi (2010); Duffy and van Bibber (2009); Ringwald (2012); Kawasaki and Nakayama (2013); Marsh (2016) and in the appendix to Ref. Dias et al. (2014).
Populations of cosmological axions are produced through five main mechanisms: thermalization Turner (1987), the decay of a parent particle Higaki and Takahashi (2012); Cicoli et al. (2012, 2013); Hasenkamp and Kersten (2013); Conlon and Marsh (2013), vacuum realignment Preskill et al. (1983); Abbott and Sikivie (1983); Dine and Fischler (1983), the decay of topological string defects Davis (1985); Battye and Shellard (1994); Harari and Sikivie (1987); Hagmann and Sikivie (1991); Chang et al. (1999); Yamaguchi et al. (1999); Hiramatsu et al. (2011a), and wall decay Nagasawa and Kawasaki (1994); Chang et al. (1999); Hiramatsu et al. (2011b, 2012); Kawasaki et al. (2015). Of these mechanisms, only the latter three contribute to a sizable cold dark matter population. We briefly revise these production methods.

Thermal axions
Thermal axions are produced in the early Universe mainly through the process Chang and Choi (1993). Similarly to neutrinos, thermal axions would contribute the hot dark matter component. For this reason, an upper bound can be placed from the requirement that thermal axions do not overclose the Universe Hannestad et al. (2010); Archidiacono et al. (2013); Di Valentino et al. (2016).

Decay of a parent particle
A decaying massive particle or a modulus coupled to the axion field would lead to an increment of the hot dark matter or dark radiation components, through the decay of the modulus into two axions. An effective model for the massive modulus would be a lowenergy manifestation of a larger theory involving both supersymmetry and extra dimensions, thus dark radiation from a string model Cicoli et al. (2013, 2012) is able to constrain string and Mtheory compactification scenarios through the change in the effective number of relativistic degrees of freedom Mangano et al. (2002), with constraints coming from both the CMB polarization and big bang nucleosynthesis. In some models, the parent particle is a modulus field which, if it dominates the Universe, must decay prior Big Bang Nucleosynthesis (BBN) at a reheat temperature Kawasaki et al. (1999, 2000); Hannestad (2004); Ichikawa et al. (2005); Bernardis et al. (2008)
(1) in order to avoid the socalled “moduli decay problem” Coughlan et al. (1983); Acharya et al. (2008). The limit on results from general considerations on the successes of BBN, and it is then a general lower bound below which the Universe has to be dominantly filled with radiation. Here we do not treat further the possibility that axions are produced from the decay of parent particles, since axions as dark radiation do not pile up to the present CDM budget.

Vacuum realignment
Vacuum realignment, one of the main mechanisms to produce a cold axion population, occurs after the breaking of the PQ symmetry that sets the axion field at the bottom of a “Mexican hat” potential Preskill et al. (1983); Abbott and Sikivie (1983); Dine and Fischler (1983). Axions are massless from the breaking of the PQ symmetry down to temperatures of the order of the QCD phase transition, when instanton effects generate an effective axion potential Di Vecchia and Veneziano (1980); Grilli di Cortona et al. (2016),
(2) where is an angular variable, is the topological susceptibility, and with the ratio of the up and down quark masses . The square of the axion mass at zero temperature is then Weinberg (1978); Wilczek (1978)
(3) We discuss the vacuum realignment mechanism in a generic cosmological scenario in Sec. A.

Decay of topological strings
Topological strings are produced because the angular variable takes different values at each spatial point after the breaking of the PQ symmetry, through the Kibble mechanism Kibble (1976). After production, the energy density in strings scales with the string unit length, and the string continuously emit lowfrequency modes axions which eventually contribute to the present cold dark matter energy density. The actual emission spectrum is crucial in determining the present abundance of cold axions, which is computed in Refs. Davis (1985); Battye and Shellard (1994) by using an energy spectrum with a sharp peak at the horizon scale, and in Refs. Harari and Sikivie (1987); Hagmann and Sikivie (1991); Chang et al. (1999) by using a spectrum proportional to the inverse of the axion momentum . Results are often expressed in terms of the ratio of the present energy density of cold axions from axionic strings and that from axions produced via the misalignment mechanism . Refs. Davis (1985); Battye and Shellard (1994) report , while Refs. Harari and Sikivie (1987); Hagmann and Sikivie (1991); Chang et al. (1999) report , thus the estimation of the CDM axion mass differs by order of magnitudes in the two models. The controversy between these different models is solved with lattice QCD numerical simulations Yamaguchi et al. (1999); Hiramatsu et al. (2011a), which show that the energy spectrum peaks at the horizon scale and is exponentially suppressed at higher momenta. This method yields an intermediate value . However the recent numerical simulations in Refs. Klaer and Moore (2017a, b) find an order of magnitude discrepancy with the results in Refs. Yamaguchi et al. (1999); Hiramatsu et al. (2011a), showing that a consensus on the detail on the axion string radiation into a spectrum of axions has not been reached yet. All of the results discussed are valid in a radiationdominated cosmology, however the value of also depends on the properties of the cosmological model before BBN Visinelli and Gondolo (2010).

Decay of domain walls
When the primordial plasma undergoes the QCD phase transition, the effective axion potential in Eq. (2) takes place, showing minima separated by domain walls attached to strings. Similarly to what discussed for axions from strings, there has been some controversy regarding the spectrum of axion radiated from domain walls. Ref. Nagasawa and Kawasaki (1994) claims that the energy spectrum peaks around the axion mass, while in Refs. Chang et al. (1999) a larger axion population is obtained by using an emission spectrum proportional to the axion wave number. The evolution of the stringwall network with has been explored in Refs. Hiramatsu et al. (2011b, 2012), where numerical simulations have been performed to settle the controversy and a spectrum peaking at a wave number of the order of the axion mass is obtained. The contribution of cold axions from wall decay is found as Hiramatsu et al. (2012); Kawasaki et al. (2015).
Iii Axion population in modified cosmological scenarios
iii.1 Parametrizing a cosmological model
In the standard cosmological picture, temperature and scale factor are related by the conservation of the entropy density in a comoving volume,
(4) 
where is the number of effective entropy degrees of freedom at temperature . Eq. (4) applies when a single species dominates at temperature . Such conservation law is not guaranteed when the cosmology modifies. For example, in the early preBBN Universe the expansion rate could have been controlled by some exotic form of energy rather than radiation. A popular example is the early domination by a massive modulus, which would lead to an early matterdominated epoch Kolb et al. (1999); Giudice et al. (2001). The decay of the massive modulus field would inject entropy into the system, thus altering the conservation law in Eq. (4). The effect of a nonstandard cosmological history might vary the present value of the axion energy density by orders of magnitude Visinelli and Gondolo (2010), depending on the equation of state for the species that dominates the expansion rate before the standard scenario and to the amount of entropy injection. These modifications lead to an axion that begins to oscillate at a temperature that is different from what obtained in the standard picture, because of a different relation between temperature and time in the modified cosmology Visinelli and Gondolo (2010); Visinelli (2017a); Nelson and Xiao (2018).
We modify the relation for entropy conservation in a modified cosmology as
(5) 
where is a new parameter that enters the relation between the scale factor and the temperature of the plasma . Entropy conservation is assured for . In the following, we neglect any change coming from a variation in for and we write equalities as when this approximation is applied.
We introduce a generic relation between time and scale factor,
(6) 
where we assume , corresponding to the range in which the expansion is subluminal. Eq. (6) does not include the important case in which , which we do not treat. The Hubble rate during the modified cosmological epoch immediately follows from Eq. (6) as . In order to express the Hubble rate as a function of temperature, we assume that when the temperature of the plasma cools down to the reheating temperature , the Hubble rate is
(7) 
where is the Planck mass and is the number of relativistic degrees of freedom at temperature . In writing Eq. (7), we have assumed that the expression for the Hubble rate in a radiationdominated cosmology is valid up to the reheat temperature. To find the expression for at early times, we combine Eqs. (5) and (6) to obtain , so that the expression for valid for reads
(8) 
iii.2 Examples of modified cosmologies
We consider two modified scenarios which are justified in some extensions of the Standard Model.

Early matter domination
In a matterdominated cosmology, the energy density of the component dominating the expansion rate of the Universe scales as matter, that is . One example of such scenario is the domination by massive moduli field, or lowreheat temperature (LRT) cosmology, which consists in a matterdominated () period prior the standard scenario Dine and Fischler (1983); Steinhardt and Turner (1983); Turner (1983); Scherrer and Turner (1985); Lazarides et al. (1987, 1990), where the Universe is dominated by heavy decaying moduli. The decay leads to a nonconservation of the entropy density , in particular the model predicts Giudice et al. (2001). One such tractable scenario embed in string theory is the large volume scenario Balasubramanian et al. (2005); Conlon et al. (2005), where a unique modulus field appears.

Kination domination
In the kination cosmology scenario Barrow (1982); Ford (1987); Spokoiny (1993); Joyce (1997); Salati (2003); Profumo and Ullio (2003), the expansion of the Universe before the standard radiationdominated cosmology begins is driven by the kinetic energy of a scalar field. The field is a “fastrolling” field with an equation of state relating the pressure and energy density of the fluid as . The energy density in the field scales faster than the radiation energy density contribution , so that the contribution from the energy density redshifts away and radiation becomes important at temperature . The thermal production of a weakly interacting massive particle during kination has been discussed in Refs. Salati (2003); Profumo and Ullio (2003); Pallis (2005, 2006); Gomez et al. (2009); Lola et al. (2009); Lewicki et al. (2016); Artymowski et al. (2017); Redmond and Erickcek (2017); D’Eramo et al. (2017); Visinelli (2017b); Redmond et al. (2018). Since the kination field does not decay but it redshifts away, we assure entropy conservation by setting , while . We also consider a decaying field as in Ref. Visinelli (2017b), for which and .
iii.3 Finite temperature effects on the axion mass
At high temperature, QCD becomes less nonperturbative and the effects of instantons become severely suppressed. This reflects onto the value of the axion mass, which decreases strongly with an increasing value of the temperature of the plasma Gross et al. (1981). Below a confinement crossover temperature the axion mass levels off to the value. The exact masstemperature relation has been recently assessed through latticeQCD computations Bonati et al. (2016); Borsanyi et al. (2016); Petreczky et al. (2016), with some discrepancies among different groups. Here, we parametrize this dependence as Gross et al. (1981)
(9) 
where is the mass of the axion at zero temperature. In the following, we use or to take into account the temperature dependence of the axion mass. We model the temperaturedependent exponent as
(10) 
where the exponent has been obtained with various techniques in the literature, either in lattice computations Bonati et al. (2016); Borsanyi et al. (2016); Petreczky et al. (2016), in the framework of the dilute instanton gas approximation Gross et al. (1981); Turner (1986); Bae et al. (2008), or in the interacting instanton liquid model Wantz and Shellard (2010a). In the QCD axion theory, the mass depends on the axion energy scale , which represents the energy at which the PecceiQuinn U(1) symmetry breaks, through Eq. (3), . Here, we choose to present results in terms of the axion mass , in place of the energy scale . Here, we choose the parameters and , in line with the latest and most accurate results Borsanyi et al. (2016).
The axion potential becomes a relevant term in the equation of motion at the temperature when the Universe has sufficiently cooled so that the axion mass is of the same order as the Hubble rate, . From Eqs. (8) and (9), we obtain the temperature at which oscillations take place,
(11) 
We introduced the Hubble rate that the Universe would experience if its expansion were dominated by radiation at temperature . Using and , we obtain . The value of is obtained by reinserting Eq. (11) into Eq. (8),
(12) 
iii.4 Present axion energy density
In Appendix A we show that the energy density stored in the collective oscillations of the zeromode of the axion field, the socalled vacuum realignment mechanism, redshifts as matter in any cosmological model, . The main result, quoted in Eq. (64), is the axion energy density at the onset of coherent oscillations,
(13) 
where we used Eq. (9) to express in terms of . In Eq. (13), is the average of the initial misalignment angle squared, including the contribution from the nonharmonic terms in the axion potential in Eq. (2) Bae et al. (2008); Visinelli and Gondolo (2009); Wantz and Shellard (2010b); Gondolo and Visinelli (2014).
We now turn to the contribution from the decay of topological defects. Axions radiated from axionic strings may contribute a sizable amount of the present energy density of these particles. We parametrize the string contribution as
(14) 
where is the present energy density computed from the misalignment mechanism, and is the present energy density from cosmic strings decay. The proportionality between the two energy densities is justified by the fact that both and scale with to the same exponent Sikivie (2008); Hiramatsu et al. (2011b); Klaer and Moore (2017b). Following Refs. Sikivie (2008); Hiramatsu et al. (2011b), we also parametrize the contribution from walls through the ratio , so that we indicate the total axion energy density as
(15) 
where . This parametrization allows for the case in which the axion CDM yield is even smaller than the misalignmentonly contribution, a scenario supported by recent simulations Fleury and Moore (2016); Klaer and Moore (2017a); Gorghetto et al. (2018).
iii.5 Cosmological bounds on the axion mass
We require that the present abundance of axions explains the totality of the CDM observed, that is the energy density in the cold axion population is equal to the CDM energy density Ade et al. (2016). The present axion energy density is obtained by imposing the conservation of the number of axions inside a comoving volume and results in
(16) 
The ratio of the two scale factors is expressed in terms of temperatures as
(17) 
which can be interpreted as a product of the total number of axions in a comoving volume times the ratio of the different volumes in the cosmologies Visinelli and Gondolo (2010). In Eq. (17), the ratio is evaluated using Eq. (5), and the ratio is evaluated using Eq. (4). Inserting the expressions for the axion energy density in Eq. (13) and the temperature in Eq. (11) into Eq. (16) gives the present axion energy density
(18) 
where we have defined the quantity
(19) 
corresponding to the axion energy density from the misalignment mechanism at temperature , if the coherent oscillations of the axion field begin at temperature . For reference, setting and gives
(20) 
The constrain relates the three parameters , and ; when this constrained is solved for the axion mass which explains the observed CDM budget, we obtain
(21) 
If the axion mass is higher than , the axion is a subdominant CDM component, while values are excluded by cosmology. In the reminder of the paper we assume that axions account for all of the CDM, or . Note that the value of depends on the relative contribution from topological defects which is uncertain, as well as the reheat temperature . The bound in Eq. (21) applies whenever the PQ energy scale is lower that the Hubble rate at the end of inflation, which is also the requirement for miniclusters to be produced around the QCD phase transition. In the top panel of Fig. 1 we show the value of the axion mass in Eq. (21) as a function of the reheat temperature, setting . The region marked “Astrophysical consideration” labels the portion excluded by the energy transport in different astrophysical environments Raffelt (1986, 2008); Viaux et al. (2013). The kink for the LRT scenario is due to the change in the temperature dependence of the axion mass, see Eq. (9), which switches from being independent on to the left of the kink to to the right of the kink. All values of become equal at GeV independently of the cosmology, because for higher values the coherent oscillations of the axion field begin in the standard scenario. To further investigate this fact, we first compute from Eq. (11), obtaining . The mass at temperature is then
(22) 
independently of the parameters and describing the cosmology. The bottom panel in Fig. 1 shows the value of as a function of for the same cosmologies and for the same color coding as in the top panel. All values become equal to when . In the kination cosmology, we always have and the two lines overlap. We have cut the plot for in the decaying kination scenario at the value of corresponding to the bound on the axion mass coming from the astrophysical considerations. In the modified cosmologies considered, the axion begins to oscillate at a lower temperature than in the standard cosmology.
Iv Axion miniclusters
The density seeds of miniclusters form at the onset of axion oscillations Hogan and Rees (1988), because of the large axion isocurvature perturbations Turner (1986), as confirmed by numerical computations Kolb and Tkachev (1993, 1994a, 1994b). Axions miniclusters are peculiar since they would greatly affect direct detection Sikivie (1983), and are potentially detectable through gravitational lensing Kolb and Tkachev (1996); Zurek et al. (2007); Fairbairn et al. (2018, 2017).
iv.1 Density of the minicluster
Axion overdensities form at temperature , however these fluctuations separate out as gravitationally bound miniclusters around matterradiation equality Kolb and Tkachev (1993), when the temperature of the plasma is . We follow the notation in Ref. Kolb and Tkachev (1994b), where an overdensity in the axion energy density is indicated as
(23) 
where is the mean axion energy density discussed in Sec. III.5 and is the density of the minicluster. corresponds to the mean axion density. A minicluster seed of overdensity enters into matterdomination at a redshift given by ; the overdensity grows linearly with the scale factor until it becomes of order unity and collapses. Since is typically of order unity, the collapse is almost immediate. Note that values much larger than unity are also possible, although these regions tend to be smaller. The computation of the density of the minicluster in the standard cosmology proceeds as follows. We parametrize the temperature prior matterradiation equality at which the minicluster collapses as , so that the energy density of the minicluster at the moment of collapse is
(24)  
where is the axion energy density at . A detailed calculation that follows from the dynamics of the collapse obtains an extra factor of 140 Kolb and Tkachev (1993, 1994b), so the expression we use in place of Eq. (24) for the energy density of the minicluster is
(25) 
In a modified cosmology, the result in Eq. (25) is valid only for , or for fluctuations smaller than . We can thus safely assume that the gravitational collapse of fluctuations occur during the radiationdominated epoch.
iv.2 Mass of the miniclusters
The mass of the axion minicluster is of the order of the dark matter mass enclosed within the Hubble length at temperature Hogan and Rees (1988); Kolb and Tkachev (1993)
(26) 
where we have used Eqs. (13) to specify and we expressed the CDM energy density using Eq. (16). The quantity is the size of the Universe that is in causal contact at temperature . The value of in Eq. (26) depends on through and , see Eqs. (11) and (12), which differ from their value in the standard scenario in the case where the onset of axion oscillations take place in a modified cosmological model. Depending whether we impose the constraint that the axion explains the totality of the observed CDM, we also have a dependence on coming from the axion mass in Eq. (21). On the other hand, if we insist on a particular value of the axion mass, for instance in the case of an experimental discovery, we could assume that the contribution from topological defects has the required value to attain 100% CDM. In general, we then expect the mass and size of a minicluster to be altered according to the cosmological model studied, depending on the values of , , and on the reheat temperature. In formulas, for and for axion CDM we have
(27) 
where is the mass of an axion minicluster forming in the standard scenario, described in Eq. (33) below. In a modified scenario, the mass of the minicluster increases with the reheating temperature if
(28) 
iv.3 Radius of the minicluster
Given the results for the mass and the density of the minicluster, we can estimate the typical size of a minicluster as
(29) 
where we set . The temperature at matterradiation equality has been estimated as eV.
V Axion miniclusters in the standard cosmology
In the standard radiationdominated picture, the expansion rate of the Universe is governed by the relativistic degrees of freedom at time , with the parameters and . Using Eq. (18), the present energy density of CDM axions is
(30) 
which, as expected, is completely independent on . Notice that, setting (), Eq. (30) gives (), valid for (), as reported in Ref. Visinelli and Gondolo (2009). Given a value of , the axion mass that corresponds to 100% of the CDM in axions, Eq. (21), reads
(31) 
Notice that such value of the CDM axion mass is mildly dependent on the choice of . On the other hand, the contribution from the decay of topological defects might modify the result by up to two orders of magnitude, depending on their specific decay mechanism. Setting for example , the mass of the CDM axion is .
Given the bound obtained in Eq. (31), in this section we safely focus on the relevant case . In addition, the temperature at which coherent oscillations begin for is , thus the coherent oscillations begin when the axion mass is still varying with temperature as . Assuming that all of the axions population collapses into miniclusters, the size of the miniclusters in this model is obtained from the last line of Eq. (29),
(32) 
where is the mean distance between the Sun and the Earth. In the standard scenario, the size of the minicluster at the matterradiation equality is thus approximately equal to the EarthSun distance, consistently with previous estimates Kolb and Tkachev (1993). The mass enclosed at radius is given by Eq. (26),
(33) 
where is the mass of the Sun.
The clustering of axion has observational consequences, since detectors would be triggered by a larger energy density when Earth passes through one such substructures. In fact, the local energy density of CDM is , while the density obtained in Eq. (25) for the minicluster is given in Eq. (25). The density enhancement during the encounter with a minicluster is then
(34) 
Assuming the mass of the Milky Way as , the number of miniclusters in the halo is
(35) 
Since we assume that all of the CDM is in the form of miniclusters, the local number density of axion miniclusters is then
(36) 
Given the size of the minicluster in Eq. (32) and a virial velocity of the Solar System around the Galactic centre km/s, the encounter lasts up to
(37) 
where we have assumed that the relative velocity between the Solar System and the minicluster is of the order of . During a complete revolution around the galactic halo, the Solar System transverse a length , where is the distance of the Solar System from the galactic center. On this path, the Earth encounters a number of miniclusters equal to
(38) 
Given a galactic year , the time between two encounters can be estimated as
(39) 
During the encounter, the energy density in the minicluster is enhanced by the factor given in Eq. (34).
Vi Miniclusters in modified scenarios
In a modified cosmological model, the parameter space contains an additional information which corresponds to the temperature at which the government of the expansion transitions to a radiationdominated one.
In Table 1 we have summed up the results for the relevant astrophysical quantities of a typical axion minicluster as obtained in the cosmologies we account for, by repeating the analysis we sketched in Sec. V. In the following, we have set for convenience . In Table 1, we have computed the same quantities presented in Sec. V by assuming that all of the axions clump into miniclusters structures and that axions make up the totality of the CDM budget which, in our notation, is equivalent to assuming that given in Eq. (21). We have set to account for the relative contribution to the axion energy density from the decay of topological defects. For these reasons, the results only depend on the reheat temperature . The mass of the axion for which we have 100% CDM, here , differs by various orders of magnitude among the different cosmologies, as first noted in Ref. Visinelli and Gondolo (2010). Likewise, the mass and size for a minicluster in either the standard and LRT scenarios can have similar ranges.
Scenario  STD  LRT  Kination (no decay)  Kination (w/ decay) 

1  3/8  1  3/4  
1/2  2/3  1/3  1/3  
90  
Enhancement  
The relative quantities describing miniclusters are sketched in more detail in Fig. 2, where we show the mass and the size of the minicluster in the case whether the early cosmological scenario is standard (solid black line), matterdominated or LRT (dashed red line), kination without (dotted blue line) and with the decay of the field (dotdashed green line). We have cut the plots for the quantities obtained in the kination scenaria at the value of for which the axion mass exceeds the bound from the astrophysical considerations. For , the mass and the size are steadily smaller than the standard value for kination cosmologies, while it is higher than what obtained in the standard scenario for the matterdominated model. In more details, miniclusters formed when the axion field starts to oscillate in the LRT cosmology have a mass and a radius which is up to two orders of magnitude larger than what obtained in the standard scenario, while in the kination scenario these quantities can be up to times smaller than in the standard scenario. The miniclusters obtained when considering the kination cosmology are both lighter and more compact, thus making it more frequent for the Earth to come into the vicinity of these objects. As we obtained in Fig. 1, when the axion field starts to oscillate in the standard scenario and we recover the standard results.
In Fig. 3 we report the density plot showing the mass of the axion minicluster, in units of , depending on and . The largest variations in mass are shown for the kination models, for which the mass of the minicluster ranges between to solar masses over the allowed range. The range over which the mass of the minicluster varies is much more contained in the standard cosmology, for which , and in the LRT cosmology for which varies by just two orders of magnitude around the standard value. The white region marks the area where the axion mass is excluded by astrophysical considerations. The dotdashed line marks the region where , where the modified cosmology takes place to the left of the dotdashed line, and the region where the axion field starts to oscillate in the standard radiationdominated cosmology, for which is given by the value in the standard cosmological scenario.
vi.1 Freestreaming length
The velocity dispersion of CDM particles directly influence structure formation, primordial perturbations below a certain length scale are erased. This scale, called the freestreaming length which depends on the mass of the CDM candidate as well as on its formation mechanism, is usually defined as the length traveled by the particle before primordial perturbations begin growing substantially at matterradiation equality . In formulas
(40) 
where the formula applies for any CDM candidate provided is the time of particle production ^{1}^{1}1Results are strengthen if some collapse of the overdensities occurs during an early matterdominated stage, for which the freestreaming length in Eq. (40) is an upper bound.. Here, is the moment at which axions acquire a significant mass fraction when coherent oscillations start, while the velocity dispersion can be obtained from Eq. (5) in the form , where is the momentum dispersion which is approximately equal to at the onset of oscillations, so that
(41) 
Since the dependence of on temperature differs in different cosmological models, we separate the contributions from the modified and RD cosmologies in Eq. (40). We obtain the leading contribution coming from a logarithmic term, regardless of the cosmological model considered, as
(42) 
In oder to check whether axion inhomogeneities get erased, we compare the freestreaming length with the size of the minicluster (the size of inhomogeneities at time ). We find that the ratio is steadily below for the axion mass in any cosmological model and for any reheat temperature, so we expect that the formation of axion miniclusters solidly proceeds in the early Universe in any of these scenarios.
Vii Discussion and conclusions
In this paper, we have discussed the properties of axion miniclusters emerging in different cosmological scenarios before Big Bang Nucleosynthesis (BBN) took place. In particular, we have considered different scenarios in which the cosmology before BBN was governed by either i) a matter component ii) a fastrolling field leading to a kination period, or iii) a decaying kination field . Using assumptions commonly made in the literature, we have obtained the mass and size of the minicluster, as well as the enhancement in axion density over the local CDM background, in different cosmological setups. We have summarized results in Fig. 2 as a function of the temperature at which the modified cosmology transitions to the standard radiationdominated scenario. In general, we expect less dense miniclusters if an early matterdominated stage occurred, and denser miniclusters in preBBN kination models. In turns, the astrophysical quantities of relevance for a detection like the frequency of encounter and the time between two subsequent encounters would alter among different scenarios and different values of the reheating temperature. Results are in Fig. 4, where we show the maximum duration in days of a single encounter of the Earth with the axion minicluster (vertical left axis) or equivalently the expected interval between two consecutive encounters (vertical right axis), as a function of . For , both modified cosmologies shows detection advantages and disadvantages compared to the standard result. If the axion starts oscillating in a kination model, the encounter would only last up to a few minutes owing to the small size of the minicluster itself; on the other hand, the frequency of encounter in the kination cosmology is enhanced by an factor with respect to the standard case, with the encounters possibly being as frequent as one per a few years. On the contrary, for an axion field that begin to oscillate in a matterdominated scenario, the encounter would last up to days, although one such encounter during a Galactic year would be much more rare. Finally, we have showed that the freestreaming length is much smaller than the size of inhomogeneities for any cosmological model, making the prediction of the formation of such axion minicluster rather solid since these structures are not homogenized by the freestreaming of axions before matterradiation equality.
Note added: During the completion of the present work, Ref. Nelson and Xiao (2018) appeared, with the material discussing an early matterdominated epoch overlapping with our work.
Acknowledgements.
The author would like to thank J. Redondo (U. of Zaragoza), P. Sikivie (U. of Florida), S. Baum (Stockholm U.), and S. Vagnozzi (Stockholm U.) for the useful discussions and comments that led to the present work. The author would like to thank the University of Zaragoza, where part of this work was conducted, for hospitality. The author acknowledges support by the Vetenskapsrådet (Swedish Research Council) through contract No. 63820138993 and the Oskar Klein Centre for Cosmoparticle Physics.Appendix A Vacuum realignment mechanism
The axion field originates from the breaking of the PQ symmetry at a temperature of the order of . Axions, which are the quanta of the axion field, are massless from the moment of production down to the temperature of QCD transition, when the mass term in Eq. (9) turns in. In this picture, the equation of motion for the angular variable of the axion field at any time is
(43) 
where is the Laplacian operator with respect to the physical coordinates . We rescale time and scale factor so that these quantities are dimensionless, and , so that Eq. (43) in these rescaled quantities reads
(44) 
where , see Eq. (9). Writing the Laplacian operator with respect to the comoving spatial coordinates , and we use the scale factor as the independent variable related to time by , defining , and setting
(45) 
Eq. (44) is rewritten as
(46) 
where a prime indicates a derivation with respect to . The expression above is the generalization of the equation of motion for the axion field in any cosmological model, and reduces to the usual expression in the radiationdominated limit ,
(47) 
Eq. (47) coincides with the results in Ref. Kolb and Tkachev (1994a), where the conformal time is used as the independent variable in place of the scale factor . We remark that this choice is possible in the radiationdominated cosmology because , whereas in a generic cosmological model this relation reads and the use of as the independent variable leads to a more complicated form of Eq. (46). Thus, in a modified cosmology the choice of the scale factor as the independent variable leads to a simpler form of the equation of motion. Taking the Fourier transform of the axion field as
(48) 
we find
(49) 
Eq. (49) expresses the equation of motion for the axion field in the variable and it is conveniently written to be solved numerically.
a.1 Approximate solutions of the equation of motion
Analytic solutions to Eq. (49) can be obtained in the limiting regime , where Eq. (49) reads
(50) 
with the wave number
(51) 
An approximate solution of Eq. (50), valid in the adiabatic regime in which higher derivatives are neglected, is given by setting
(52) 
where the amplitude is given by
(53) 
Finally, an approximate solution to Eq. (50) is Kolb and Tkachev (1994a); Sikivie (2008)
(54) 
Each of the three terms appearing in Eq. (51) is the leading term in a particular regime of the evolution of the axion field. We analyze these approximate behavior in depths in the following.

Solution at early times, outside the horizon
At early times prior to the onset of axion oscillations, the mass term in Eq. (50) can be neglected since . Defining the physical wavelength , we distinguish two different regimes in this approximation, corresponding to the evolution of the modes outside the horizon () or inside the horizon (). In the first case , Eq. (50) at early times reduces to
(55) with solution ()
(56) One of the two solutions to Eq. (55) is thus a constant value , while the second solution drops to zero for cosmological models with . Regardless of the cosmological model considered, the axion field for modes larger than the horizon is “frozen by causality”. For example, in a radiationdominated model with , Eq. (56) coincides with the result in Ref. Sikivie (2008),
(57) 
Solution at early times, inside the horizon

Solution for the zero mode at the onset of oscillations
An approximate solution of Eq. (50) for the zeromomentum mode , valid after the onset of axion oscillations when , is obtained by setting
(61) so that the adiabatic solution for in Eq. (54) in this slowly oscillating regime gives the axion number density
(62) where is the number density of axions from the misalignment mechanism at temperature ,
(63) Eq. (62) shows that, regardless of the dominating cosmological model, the axion number density of the zero modes after the onset of axion oscillations scales with . The energy density at temperature is obtained as
(64) where we have used Eq. (9) to express in terms of .
Appendix B A note on primordial black hole formation
Primordial black holes formed through various mechanisms, of which one consists in the growing of large inhomogeneities around the QCD phase transition. The question is, should axion inhomogeneities also form black holes instead of condensing into miniclusters? To answer this question, we compute the Schwarzschild radius for the primordial plasma and for the axion energy density at the onset of oscillations.
When overdensities in the primordial plasma grow larger than one, a condition for the formation of primordial black holes is met. At time , the mass enclosed within a Hubble radius is , and the ratio between the Schwarzschild radius and the horizon length is