Decay of ALP condensates via gravitation-induced resonance

Oscillating scalar field condensates induce small amplitude oscillations of the Hubble parameter which can induce a decay of the condensate due to a parametric resonance instability [1]. We show that this instability can lead to the decay of the coherence of the condensate of axion-like particle (ALP) fields during the radiation phase of standard cosmology for rather generic ALP parameter values, with possible implications for certain experiments aiming to search for ALP candidates. As an example, we study the application of this instability to the QCD axion. We also study the magnitude of the induced entropy fluctuations.


I. INTRODUCTION
Axions and ALPs are amongst the well-motivated candidates for dark matter (see, e.g., Refs.[2,3] for recent reviews of axions as dark matter, and Ref. [4] for reviews of ALPs).It is often assumed that the associated fields are misaligned relative to the minium of their zero temperature potential in the early universe and then at some time begin to coherently oscillate about their ground state values 1 .For certain parameter ranges, these coherently oscillating ALP of axion fields can provide an important contribution to Dark Matter (DM).As was pointed out recently in Ref. [1] (see also Ref. [5] for some earlier related work), a coherently oscillating scalar field condensate φ in the early universe will induce oscillations in the Hubble expansion parameter superimposed on the overall decreasing expansion rate.In a similar way that a coherently oscillating inflaton condensate will lead to a parametric resonance instability of the condensate to the production of quanta of any field coupled to the inflaton [6,7] 2 , in particular to quanta of the inflaton field itself, the oscillations in the Hubble parameter induced by the oscillations of the homogeneous condensate of φ will induce resonant production of φ quanta.This instability can destroy the coherence of the φ condensate.
In this paper we study the application of this general effect to proposed dark matter condensate fields, e.g., axions or ALPs, or more generally "wave dark matter".We find that in a wide range of parameter space of interest to DM model building, the instability is effective and the coherence of the condensate is destroyed.While this effect does not change the overall energy density of matter (the energy density of the condensate and of an assembly of individual low momentum particle quanta both redshift as matter), coherence effects are washed out.This may have implications for certain searches for wave DM candidates.
We are interested in condensates which are present in the radiation phase of standard cosmology.We work in the context of a homogeneous, isotropic and spatially flat metric with scale factor a(t), where t is the physical time (with associated temperature T ).We will be focusing on the radiation phase of Standard Big Bang cosmology when a(t) ∼ t 1/2 .This phase ends at the time t eq of equal matter and radiation (with associated temperature T eq ∼ 1eV.The expansion rate H(t) of space is related to the temperature via the Friedmann equation where m pl denotes the Planck mass.One notes that the mechanism described here is model-independent and applies to any oscillating condensate field.The mechanism is also operative in the absence of nonlinear interactions of φ quanta.The instability of axion condensates in the presence of nonlinearities has been studied in Ref. [13], and the analysis generalizes to ALPs, as mentioned in that same reference.In the presence of couplings of φ to other fields, parametric resonance instabilities have been analyzed in various works.In particular, in Ref. [14], the standard coupling of the QCD axion to photons was supplemented with a coupling to a dark photon, and the tachyonic instability (closely related to the parametric resonance instability studied here) of the dark photon mode equation allowed an opening up of the axion window (see also [15]).In [16] the decay of an ALP condensate into photons has been analyzed.However, the resonance is model-dependent and was found to be of "narrow resonance" type, and its effects hence are much reduced when the expansion of space is taken into account.The instability which we find is, one the other hand, of "broad resonance" type and hence robust towards taking the expansion of space into account.In spirit, our work is related to that of Ref. [17], where the transfer of energy from low momentum modes to momenta of the order of the ALP mass was considered, in that case triggered by the cosmological fluctuations set up in the primordial universe.There is also related work on kinetic fragmentation of an ALP condensate [18].This work is organized as follows.In section II, we first review the basic effect, following the discussion in [1].We next (section III) apply the analysis to ALPs and show that, at least in the class of models which we study, the coherence of the ALP condensate is destroyed.The process is robust and can happen generically in the radiation dominated phase and prior to the time of matterradiation equality.In section IV, we then study in more detail the special case of the QCD axion.We study the contribution of the produced axion fluctuations to the entropy in section V, and in section VI, we demonstrate that, although entropy fluctuations on infrared scales are induced by our effect, they are too small to have an interesting effect on the amplitude of the curvature fluctuations.Finally, in section VII, we conclude with a discussion of the implications for experiments designed to search for ALPs and axions.
Throughout this paper, we work with the natural units, in which the speed of light, Planck's constant and Boltzmann's constant are all set to 1, c = = k B = 1.

II. THE BASIC EFFECT
We consider a scalar field φ with a potential V (φ) which is quadratic about its minimum The equation of motion of a homogeneous condensate of φ in an expanding universe is It is convenient to consider the rescaled field ψ defined via whose equation of motion is The condensate φ will be frozen by Hubble friction until the time t o when H ∼ m.Thereafter, it will oscillate with an amplitude A which scales like The rescaling of φ was done such that the amplitude A of ψ remains constant.We will normalize the scale factor a(t) such that a(t o ) = 1.This implies that A is the amplitude of φ at the time t o .Let us consider some time t d after the condensate has started to oscillate.For times after t o , the gravitational terms in Eq. ( 5) can be neglected if we are interested in processes that take place on a time scale shorter than the Hubble expansion time.Hence, we have that The oscillations of the condensate will (via the Friedmann equations) induce an oscillating contribution to the scale factor, which is superimposed on the usual radiation phase scaling.Following [1], we make the ansatz where a 0 (t) is the usual radiation phase evolution of the scale factor, and b(t) is a perturbation whose amplitude is obviously suppressed by the ratio of the energy density of the condensate divided by the total energy density.As shown in [1], to leading order the solution for b(t) is where η (η d ) is the conformal time associated with t (t d ).
As pointed out in [1], the oscillations in a(t) can induce a gravitational parametric resonance instability in all fields.In particular, it can excite inhomogeneous modes of φ.If the resonance is effective, this process will destroy the coherence of φ.The equation of motion for the Fourier modes Inserting Eqs. ( 8) and ( 9) in Eq. ( 10) and neglecting terms with time derivates of a 0 (i.e., focusing on the oscillatory term), the mode equation (10) for infrared modes (modes with k/a < m) becomes ) This is the equation of motion of a harmonic oscillator with an oscillating contribution to the mass.Except for the fact that this contribution to the mass has an amplitude which grows in time, this is the usual Mathieu equation [19], which has a parametric resonance instability.In [1] it was shown that this instability persists also when taking the time dependence of the amplitude into account 4 .An intuitive way to understand this result is the following: the time scale of the instability is (for the parameter range in which the instability is effective) short compared to the Hubble time scale, the time scale on which the coefficient varies.Hence, within the instability time scale the variation of the coefficient is negligible.
In terms of a new time variable z = mt, the mode equation (11) takes the form (where in this equation the overdot stands for the derivative with respect to z) with The condition for the parametric resonance instability to be effective is q > 1, i.e., Since A is constant, the parametric resonance condition can be satisfied at later times t d even if the condition is not satisfied when the condensate oscillations start.
If the decay condition ( 14) is satisfied, then all modes with k p < m (where k p is the physical momentum) will undergo exponential amplification with with µ = √ qm.Note that the modes acquire a high occupation number.
The second efficiency condition is the requirement that the instability is rapid on the Hubble time scale, i.e.,

III. PARAMETRIC INSTABILITY OF ALP CONDENSATES DUE TO HUBBLE CONSTANT VARIATIONS
Let us now establish that the mechanism proposed in the previous section is generic during the radiation dominated regime 5 .Here we consider a coherently oscillating scalar field condensate for a field φ of mass m and amplitude of oscillation A(t).The oscillations begin when the Hubble damping in the equation of motion for φ becomes negligible.This happens when H ∼ m.Recalling that we are considering the radiation phase of standard cosmology, the Friedmann equation immediately yields for the temperature T o when the oscillations start.Equivalently, Next we want to determine at what temperature T d the parametric resonance instability sets in.The condition was given in the previous section, by Eq. ( 14).Note that the amplitude A in that condition is the amplitude A of the oscillation of the condensate φ when the oscillations set in.
Assuming that the amplitude of the oscillation of the condensate is chosen such that the condensate can provide all of the DM, we have (from the Friedmann equation evaluated at the time t eq ), and, hence (using the scaling of A(t) discussed in the previous section), Inserting this result into Eq.( 14), solving for t d and expressing the result in terms of the temperature T d (again making use of the Friedmann equation) yields Note that in the above equation we have expressed the ratios in terms of T eq since we want to determine for which values of m the onset of the instability will be before the time of equal matter and radiation.
Writing the condensate mass as m = m 1 eV (i.e., in units of eV) and inserting the values of T eq and m pl we obtain and The observational lower bound on the condensate mass is m 1 > 10 −20 .Inserting the value of the lower bound into the two above equations, we see that for this value of m we have T eq < T d < T o , and the scaling with m 1 immediately shows that for all allowed values of m 1 we have We have thus seen that for the allowed parameter range of models considered here, ALP condensates suffer the parametric resonance instability well during the radiation period of standard cosmology.It is easy to verify that the efficiency condition ( 16) is also satisfied.

IV. DECAY OF AN AXION CONDENSATE
In this section we consider a special case, namely the QCD axion.A key difference compared to the previous analysis is that the potential for the condensate is not present at all times, but sets in during a particular phase transition, namely the Peccei-Quinn symmetry [20] breaking.Another difference compared to the analysis of the previous section is that we will not assume that the axion makes up all of the DM, but we will study the potential instability of the axion condensate more generally as a function of the axion parameters.
We will assume that the axion dark matter is produced through the misalignment mechanism, where there is a coherent initial displacement of the axion field.In this case, at early times when the Hubble friction is large compared to the axion mass, H > m φ , the axion field is overdamped and is frozen at some initial value.Later, when H < m φ , the axion field becomes underdamped and oscillations can begin.For small oscillations, we can approximate the axion potential as a quadratic potential.Thus, the equation of state oscillates around w φ = 0, and the energy density scales as ρ φ ∝ 1/a 3 .This behavior is similar to that of ordinary matter.This is why misalignment axions can then be considered as valid DM candidates.The misalignment scenario for axion DM production is expected to happen in the radiation dominated universe, between the TeV and the QCD scales, depending on the value of the axion decay constant f a .
The axion equation of motion is (for amplitudes φ f a ), where, for a radiation dominated universe, where g * (T ) is the number of relativistic degrees of freedom (DoF) at the temperature T .The axion mass is a function of the temperature [21] where χ 0 ≈ (75.6MeV) 4 , n ≈ 4.08 and T QCD ≈ 153MeV.The axion mass is approximately constant below the QCD phase transition, and its value will be denoted by m a .
Taking m φ (T o ) = 3H(T o ), we can determine the temperature when the axion condensate starts to oscillate coherently.Using Eqs. ( 26) and ( 27) (taking n = 4 in Eq. ( 27) for simplicity), we find where we have used g * ∼ 75 at around the GeV scale [22].The number of relativistic DoF g * is well known from temperatures about the electroweak scale down to today.In Fig. 1 we show the variation of g * as a function of the temperature.1.The number of relativistic DoF g * as a function of the temperature.An interpolation of the data given in Ref. [22] has been used.
For all T < T o the axion zero mode will satisfy m a > 3H and will be oscillating around the minimum of its potential.Let us parametrize the initial amplitude of oscillation as where c a < 1 is a positive constant describing the fraction of the maximal amplitude.Using the scaling of the oscillation amplitude as a function of temperature, we can then estimate the contribution of the oscillating axion condensate to the dark matter density.A careful analysis [2] yields where Ω a is the fractional contribution of axions to the total energy density, and h is the value of the current Hubble constant in units of 100kms −1 Mpc −1 .The allowed parameter space for the axion is then determined by the range of values of c a and f a for which Ω a ≤ Ω m , where Ω m is the total fractional contribution of matter to the energy density budget of the current universe.
From the results of section II of this paper, we know that the oscillations of the scale factor that the oscillating condensate induces lead to parametric resonance growth of long wavelength axion fluctuations.The resonance is efficient if Recall that A is independent of time.Hence, the instability condition is easier to satisfy the later we look.
Inserting the expression for the axion mass (27), making use of the Friedmann equation to express t in terms of the temperature, and normalizing by T eq yields the efficiency condition on the temperature when the resonance can effectively set in, Thus, we see that for the range 10 8 GeV < f a < 10 12 GeV of the axion decay constant which is usually considered, the axion condensate will be unstable towards the resonance effect considered here.The axion fluctuations grow exponentially in time t with the rate where A is the initial amplitude of oscillation of the condensate.We must also check that the instability is efficient on the time scale of the expansion of space, i.e., µ H 1 . ( Expressing H and t in terms of the temperature T , using the Friedmann equation and normalizing quantities by T eq , we obtain From the above we conclude that the axion condensate is rather generically unstable to the parametric instability which we are discussing here. We can now estimate the production of axion quanta due to the coherent oscillation of the condensate around the minimum of the potential.Since it is infrared modes with k ≤ m a which are excited, the energy density in produced quanta can be estimated to be (see also [1]) where in the first line we have assumed that the modes begin in their quantum vacuum state (this gives the k −1 factor, while two powers of k come from the phase space volume, the other two powers come from the k 2 in the energy of an individual mode).
The resonant particle production is expected to stop when its backreaction becomes significant, i.e., when the produced energy density Eq. (36) approaches that of the oscillating scalar field, which gives for the time interval τ over which this particle production mechanism is effective Although the argument inside the logarithm in the above equation is large, its logarithm is of the order 10 2 .Hence, as long as µ > 10 2 H, the time scale of backreaction is smaller than the Hubble scale (which is a condition for our analysis to be self-consistent).
Let us now take a slightly closer look at the efficiency condition for the resonance, namely that the resonance time scale must be smaller than the Hubble time.Since the ratio Γ of particle production is determined by the coefficient µ in Eq. ( 36), or more precisely, Γ = 2µ, the process will be efficient if Γ > H.
Recalling again that the window bounds on the axion decay constant is typically 10 8 GeV f a 10 12 GeV.For these values of f a , from Eq. ( 28), then the value for T o where the axion starts oscillating is typically T o ∼ 1 GeV.Note that for T > T QCD , the axion mass is determined by the temperature dependent expression in Eq. ( 27).Then, in the interval T QCD T 1GeV, we find that while for temperatures below the QCD scale, the mass of the axion is determined by its zero temperature term in Eq. ( 27).The ratio Γ/H then now becomes, for T T QCD , Γ/H 3.6 10 12 GeV f a (41) Note again that axion production gets more and more efficient as the temperature decreases.
From the data given in Fig. 1 and using Eqs.( 40) and (41), we show in Fig. 2 the region of parameters in terms of the temperature and axion decay constant where Γ/H > 1 is satisfied.Taking for instance the temperature T = 100 MeV, for which g * 17.7, and for f a = 10 10 GeV, we obtain that Γ/H 8.2 × 10 3 .

V. CONTRIBUTION OF AXION PRODUCTION TO THE ENTROPY
In this section we estimate the abundance and entropy of axion DM particles produced via our instability mechanism and compare the result to the entropy of the thermal bath.The abundance is defined by the ratio of the number density of created particles by the entropy density, The entropy density of the thermal bath is where g s, * (T ) is the number of relativistic DoF for the entropy at the temperature T .The increase of entropy density for the axion dark matter particles can be estimated as where s k (t) is the entropy per mode, which can be defined by the von Neumann entropy, with f (k, t) being the occupation number.
For the resonant particle production, we can estimate the occupation number as For f (k, t) 1, (47) From Eqs. (38) and (43), we can estimate the ratio For typical values of parameters (for a QCD axion), we have s prod s thermal .
It is useful to estimate the ratio (48) at the time when the axion starts oscillating, m φ = 3H.This gives a minimum value for the produced entropy (recalling that particle production will occur for some T < T osc , see Fig. 2), where we have considered g s ∼ g * .

VI. ENTROPY FLUCTUATIONS INDUCED BY ALP CONDENSATE DECAY
It is well known that axion fluctuations can induce primordial entropy fluctuations [23].A similar effect also arises for ALP fluctuations.While it is known that there is no parametric instability of adiabatic fluctuations on super-Hubble scales (e.g., induced by the oscillation of the inflaton field at the onset of reheating (see e.g.[24]), it is possible to have amplification of super-Hubble entropy modes (see, e.g., [25]).The oscillations of the ALP or axion condensates which we have studied in this paper induce fluctuations in a sub-dominant matter component, i.e., entropy fluctuations.In this section, we estimate the magnitude of the entropy fluctuations induced by the ALP condensate decay process which we have studied, and compute the induced corrections to the amplitude of the curvature fluctuations.Since our process produces high occupation states for infrared modes, one could worry that the induced curvature fluctuations are too large.
It is well known that entropy fluctuations induce growing curvature fluctuations (see, e.g., [26,27] for reviews of the theory of cosmological fluctuations).Specifically, a non-adiabatic pressure fluctuation δP nab will induce a growth of the usual curvature fluctuation variable ζ via the equation (see [28]) where the subscript k indicates the comoving momentum mode which we are considering, and p and ρ are background pressure and energy density, respectively.In our case, the background is the radiation fluid, and the perturbation is given by the ALP field φ.In this case 6 , the non-adiabatic pressure fluctuation is [28] where the fluctuation terms are the φ pressure and energy density fluctuations, and the background terms are from the radiation background.Since φ approximately pressureless, we obtain We see from the above equation that the result is suppressed by two important factors: first the ratio of H to µ, and secondly the ratio of the density in ALPs to the total radiation density.The power spectrum P ζ (k) of the induced curvature fluctuations on a scale k is, hence, given by the power spectrum of the ALP density fluctuations on that scale, which is obtained by integrating ρ φ from k = 0 to k (see Eq. 49).The result is On the far infrared scales relevant to cosmological observations, the induced power spectrum of curvature fluctuations is, thus, suppressed by an additional factor of (k/m) 4 .Hence, we conclude that our mechanism does not lead to a dangerous amplitude of entropy fluctuations.

VII. DISCUSSION
We have suggested that axion and ALP condensates in the radiation phase of Standard Big Bang cosmology are unstable to a model-independent parametric resonance instability, which is triggered by the contribution of a periodic variation of the effective mass of the axion or ALP mode functions.This effect comes as a result of the periodic variation of the Hubble constant due to the oscillation of the condensate.We have indicated that the instability is of the "broad resonance" type and, hence, robust taking the expansion of space into account.Our analysis applies to situations when the condensate is oscillating coherently on the Hubble scale.
Our effect will destroy the temporal oscillations of axion and ALP fields coherent over all space.However, the modes which are excited all have momenta smaller than the axion or ALP mass m, are highly excited and hence can be viewed as classical states (see e.g.[30]) which are oscillating on a time scale of m −1 or larger.Many existing and planned experiments for ALP detection search for signals involving periodic variations on a time scale of m −1 (see, e.g., Ref. [31] for a recent overview).These should not be effected by the instability of the global condensate.Experiments which, on the other hand, search for oscillatory signals which are coherent over large spatial scales, may need to be reconsidered 7 .
In the case of the QCD axion, the effect does not appear to have any implications for standard axion detection experiments that look for interactions of individual axion quanta with photons.The loss of coherence will, however, impact the suggested signatures [32] of axion DM on galactic scales.
FIG.1.The number of relativistic DoF g * as a function of the temperature.An interpolation of the data given in Ref.[22] has been used.

FIG. 2 .
FIG.2.The region of parameters for which Γ/H > 1 (blue) and for when the axion starts oscillating (green).
To estimate the magnitude of the induced curvature fluctuations we can integrate this equation from the time t d , when the instability sets in (see section III) until the time t d + τ when back-reaction shuts off the instability.Since the exponential increase in δρ is rapid and terminates on a Hubble time scale, we can approximate ζ k as