Cosmological perturbations in coherent oscillating scalar field models

The fact that fast oscillating homogeneous scalar fields behave as perfect fluids in average and their intrinsic isotropy have made these models very fruitful in cosmology. In this work we will analyse the perturbations dynamics in these theories assuming general power law potentials $V(\phi)=\lambda \vert\phi\vert^{n}/n$. At leading order in the wavenumber expansion, a simple expression for the effective sound speed of perturbations is obtained $c_{\text{eff}}^2 = \omega=(n-2)/(n+2)$ with $\omega$ the effective equation of state. We also obtain the first order correction in $k^2/\omega_{\text{eff}}^2$, when the wavenumber $k$ of the perturbations is much smaller than the background oscillation frequency, $\omega_{\text{eff}}$. For the standard massive case we have also analysed general anharmonic contributions to the effective sound speed. These results are reached through a perturbed version of the generalized virial theorem and also studying the exact system both in the super-Hubble limit, deriving the natural ansatz for $\delta\phi$; and for sub-Hubble modes, exploiting Floquet's theorem.


I. INTRODUCTION
Rapidly evolving coherent scalar fields have been widely studied in cosmology. Their dynamics is not only important during the reheating epoch after inflation, but they can also support periods of accelerated expansion in both the early universe [1,2] or at late times [3,4]. Concerning the dark matter problem, non-thermal candidates like the axion [5] or other massive scalar [6] or pseudoscalar fields [7] also fall in this class. These models can be interpreted as Bose-Einstein condensates, where the scalar particles occupy the lowest quantum state of the potential [8]. Finally, the possibility of ultra-light scalar fields as dark matter candidates has been explored in different works [9,10] by tuning appropriately the potential and initial conditions [10].
The general analysis of a homogeneous oscillating scalar field in an expanding universe was performed by Turner in [11]. For a power-law potential V (φ) = λ|φ| n /n, the rapid scalar oscillations around the minimum of such a potential behave as a perfect fluid with an effective equation of state ω = (n − 2)/(n + 2). His results can be recovered by means of a generalization of the virial theorem [12]. Recently, it has been shown that a fast oscillating abelian vector [13], non-abelian vector [14] or arbitrary spin field [15] will behave in a very similar way.
The purpose of this work is to analyse the growth of perturbations in these coherent oscillating scalar theories for arbitrary power law potential. This subject has been mainly studied for harmonic potential models that mimic the standard dark matter case [16], as it happens for the axion field [17]. It has been proved by using the linear perturbation theory that the axion was equivalent to CDM for high enough masses [18,19]. However, gravitational instabilities of oscillations in a harmonic potential are suppressed on small scales [17,20]. This analysis determines the cut-off in the matter power spectrum and its deviations with respect to the CDM phenomenology. On the other hand, the dynamical stability (ignoring metric perturbations) of general coherent oscillating scalar dark energy models has been analysed in different works [1,4,12,21], even by considering nonlinear evolutions [22]. They conclude that potentials supporting accelerated expansion are generically unstable with respect to the growth of inhomogeneities.
This work is organized as follows: we will briefly review the standard average approach for the background evolution of a scalar field under a power law potential (Section II), as well as set the equations that rule its perturbations (Section III). After the preliminary discussion, we will analyse the well-known case of a massive scalar by means of an adiabatic expansion approach (Section IV). The perturbations evolution of power-law potential models will be studied following the average approach (Section V). Firstly, we will compute the effective sound speed, which is in general the quantity that rules the evolution, using the perturbed version of the generalized virial theorem. This method allows to extend previous results to an arbitrary power-law potential. Also, exploiting this equation, we will be able to derive a general expression for a possible anharmonic correction in a massive scalar theory. After that (section VI), we will check the validity of the result for the effective sound speed by studying the exact system of equations (non-averaged) in both super-Hubble and sub-Hubble limits. For small wavenumbers, we will show what is the natural ansatz for δφ (subsection A), whereas in the sub-Hubble limit we will study the exact solution of the non expanding equations thanks to Floquet's theorem (subsection B). Finally, we will see that at high k, a cut-off is always expected (subsection C).

II. BACKGROUND EVOLUTION
We are interested in studying the cosmological evolution of a homogeneous scalar field which is rapidly oscillating around the potential minimum. Let us then consider a scalar field theory in cuved space-time with The equation of motion can be written as where V represents the derivative with respect to its argument. By considering a Friedmann-Lemaître-Robertson-Walker metric in conformal time η, the equation of motion takes the form where H =ȧ/a and˙≡ ∂/∂η . The corresponding energy-momentum tensor reads Following the same approach as in previous works [11], we are interested in the cosmological evolution generated by the effective energy-momentum tensor obtained after averaging over the fast scalar oscillations. Thus, we will concentrate on the average Einstein equations given by In the particular case of a homogeneous scalar field in flat RW background, they reduce to In order to obtain the average equation of state of the oscillating scalar, we apply a generalization of the virial theorem [12]. Let us consider that the typical frequency of the φ oscillations is ω eff H and let us calculate the average in a certain time interval T such that it is large compared to the oscillation period but small compared to the Hubble time [13], i.e., H −1 T ω −1 eff . We start by calculating the average of the total derivative given by ∂ 0 φ φ =φ 2 +φφ. Thus, if the φ oscillations are bounded Thus, comparing with φ2 ∼ O(ω 2 eff φ 2 ), we see that (8) is suppressed by a factor 1/(ω eff T ). Using (3) and neglecting H terms, we can write φ2 +φφ The error introduced by neglecting the total derivative can be reduced by taking large T , so that the minimum limit is set by ≡ H/ω eff By using these equations, we can reach a useful expression for the the average equation of state: Considering a power-law potential V (φ) = λ|φ| n /n, the last expression results (see [11] for an alternative discussion). Therefore, from the conservation equatioṅ we can show that the evolution of the average energy density is so that from the Friedmann equation (6) we get:

III. FIRST ORDER PERTURBATIONS
Let us consider now a perturbation on the homogeneous evolution studied above for φ: with δφ a small perturbation. The only sourced metric perturbations are scalars. Therefore we can write where we have chosen the longitudinal gauge for the computation. By taking the Fourier transformation in the spatial coordinates, the equation of motion for the perturbation results and, following the same approach as for the background, the average Einstein equations (5) read to first order in perturbations, where the perturbed energy-momentum tensor components read and the effective sound speed is: In the following section, we will analyse this system for the simple case of a massive scalar. By making an adiabatic expansion, we will obtain the background solution of φ and compute the averages of Eqs. (6)-(7) explicitly, together with the corresponding effective sound speed.

IV. PERTURBATIONS OF A MASSIVE SCALAR
Homogeneous scalars are widely considered in cosmology, particularly fast oscillating massive scalars are specially relevant since its perturbations mimic those of dust perfect fluids. This fact makes them a good candidate for solving the dark matter problem.
Assuming a quadratic potential (n = 2) and redefining the fieldφ = a(η)φ in (3), the equation of motion turns intoφ Making an adiabatic expansion [23] with = H/ma 1, the solution to the leading adiabatic order results with φ s and φ c integration constants, and Setting the origin of time adequately, the background field reads The average energy-momentum tensor is equivalent to a dust perfect fluid (see Eq. (11) or [11]) The problem of the perturbations of the massive oscillating scalar has been previously studied in [18,19]. In our case, in order to solve the system, we will combine Equations (19) and (20), obtaining an equation which together with (18), form an algebraic system for Φ k and Ψ k as a function of φ and δφ.
notice that the origin of the error comes from the use of the virial theorem (9) and (6) to pass the termφ 2 Ψ k to the left-hand side on (19). This expression gives Ψ k as a function of the field and its perturbation, thus, using (27) we can obtain the solution of (17) for δφ k in an adiabatic expansion.
As we have just seen the background equations of motion have been solved using a WKB approximation (27) which can be written as an expansion in the so called adiabatic parameter ≡ H/ma. In order to solve the perturbation equations we can similarly use this method. In this case, there is a new scale, k, independent of the other two: H and ma. Therefore, we have two different expansion parameters: the same = H/ma as in the background case and the new parameter k/H. In order to simplify the calculations and work with a single parameter, we will assume that the new parameter k/H is related to by k/H = O( α ). Thus we will work in different regimes by assigning different values to the exponent α. Thus for example, as we are interested in cosmological perturbations, we typically expect k ∼ H, i.e. both H and k of the same adiabatic order, or in other words, k/H ∼ 0 . However, as we will see below, this is not the only interesting range in k. For example, we will show that when k 2 ∼ Hma, the behaviour of perturbations changes. In such a case, k/H ∼ −1/2 . Finally we will also solve the system for scales k ∼ ma, i.e., k/H ∼ −1 .
FIG. 1: Sub-Hubble mode of a massive scalar field with k ma. Those graphics are the numerical solution to the non-averaged Equations (62−64), with V (φ) = m 2 φ 2 /2. We work in 8πG/3 = 1 and η ini = 1 units. In order to calculate a representative mode, we have set m = 3000, k = 100, δφ(η = 1) k = 10 −6 ,δ φ k (η = 1) = −10 −6 , a(η = 1) = 1. The two plots on the top show that the scalar perturbation of the metric tends to a constant. The energy density contrast (gray) grows as η 2 ∼ a, as expected for the perturbations of a dust-like perfect fluid. Finally, the ratio δp k /δρ k (purple) oscillates around zero. The parameter α on the bottom left corner has been introduced to show explicitly the asymptotic behaviour of δ.
A. α ≥ 0 We will proceed as in the background by assuming an adiabatic ansatz for the field perturbation, The amplitudes can be expanded in the adiabatic parameter, where δφ (0) s,c and δφ (1) s,c are of adiabatic order 0 and respectively. In the standard case α = 0 (k ∼ H), we have access to super-Hubble and sub-Hubble modes provided k 2 maH. From the leading order of the equation of motion (17), we obtain and once we take this constraint into account, the equation implies By solving this system, we can easily compute the perturbations to the leading order, As it can be seen the solution mimics the perturbations of dust perfect fluids, i.e. it has vanishing effective sound speed, constant Φ k and δρ k / ρ ∝ a at late times.
The case α > 0 corresponds to super-Hubble modes and has the same solutions (37)-(40)  Let us first consider the α = −1/2 (k 2 ∼ Hma) case. Expanding the equations of motion in we obtain the system where C 1 and C 2 are integration constants. The cosmological perturbations read In this case Φ k oscillates slowly compared to ω eff with a decaying amplitude at early times and is constant at late times. On the other hand δρ k / ρ also oscillates with decaying amplitude at early times and grows as a at late times thus matching the dusty behaviour.
Finally, for modes with α ≤ −1, δφ oscillates with a higher frequency than φ and all perturbations go to zero in average. Consequently, a cut-off is expected as we consider larger-k modes.

V. PERTURBATION OF POWER LAW POTENTIAL THEORIES
In general for a power-law potential, the background field equation is non linear and we cannot obtain the explicit adiabatic expansion of the solutions as in the quadratic case. We can still define a typical frequency of the background oscillations ω eff and compare it with the other two scales in the problem H and k. Moreover, in this case, the pressure is not negligible in comparison with the energy density. This fact makes easy to compute the effective sound speed c eff , which is the quantity that controls the evolution of sub-Hubble perturbations.

Effective sound speed
Combining (19) and (21), we get a single equation for the metric perturbation: If φ dominates the background energy density, (48) can be rewritten as here, H and ω correspond to the average background evolution.
In order to compute the effective sound speed, let us consider the following average in a period T , which veri- If the field evolution is periodic or bounded, averaging during long enough periods, the left-hand term of the last equation will be negligible in comparison with the average of φ +δ φ k 2 .
In principle, except for a quadratic potential, δφ will have some growing modes. Those modes make ∂ 0 φ δφ + φδ φ not to vanish, but as long as it oscillates around zero, the following discussion holds at leading order.
Focusing on the first order of perturbations and introducing the equation of motion (17): where we have neglected the scale factor derivatives as we are considering time intervals much smaller than the inverse of the expansion rate.
By using (51), we obtain the following expression for the effective speed of sound In the second equation the terms proportional to Ψ k are averaged out as their fast oscillating part can be expressed as a total derivative or coincide with the background generalization of the virial theorem (9). If a power-law potential, V (φ) = λ|φ| n /n, is considered: • When ω eff k, i.e. the frequency of φ is almost equal to the perturbation frequency, the behaviour is similar to a perfect fluid with constant equation of state (see Fig. 1 for the case n = 2 and Fig. 2 for n = {4, 6, 8}), i.e. up to O( ): Notice that for n < 2, there is an instability in agreement with [12]. In this case V (φ) is not well defined when φ = 0.
• For massive scalar fields n = 2, Reproducing the result obtained in [17][18][19]: Although this expression seems to be valid for all k, when k is comparable with ma fails since the averaging time interval has the same order of the δρ oscillation period. See curve (c) in Fig. 3 at early times. The high k limit will be discussed in section VIII.
The corresponding comoving Jeans length λ J = 2π/k J , which satisfies c 2 eff (k J )k 2 J = 4πG ρ a 2 reads: • Because of the vanishing of the potential terms in the numerator of (52) in the harmonic case, we can also consider anharmonic corrections, V (φ) = m 2 φ 2 /2+λφ l /l in a simple way. Due to the averaging process the sine mode of δφ k gives a negligible contribution: The same situation occurs with the term k 2 φδφ k . Therefore, only the cosine mode contributes and we will be able to extract the common factor φ 0 δφ c both in the numerator and the denominator and simplify the expression. We will only consider even powers l = 2p since odd exponents make the sinusoidal resulting function to oscillate around zero and are suppressed by the average. Finally, For example, for the case p = 2, we recover the sound speed obtained in [12].

VI. COMPARING WITH THE NON-AVERAGED SOLUTIONS
In order to check the validity of the results obtained above from the averaged equations, in this section we will compare them with exact results in certain particular limits.
Let us then consider the Einstein equations with the exact (non-averaged) energy-momentum tensor. At the background level we have: whereas to first order in perturbations they read, where the expression for the perturbed energymomentum tensor components are given in (22), (23) and (24). Notice that in this section all the geometric quantities are not averaged so that in order to compare with the result of the previous section we will explicitly take the corresponding average of the obtained results.

A. Super-Hubble analytic approach
In the super-Hubble limit k H, we can neglect the terms proportional to k in (63), and introducing (64) It is interesting to note that any time the oscillatingφ is zero, the perturbation δφ k must be zero too. Therefore, it is natural to assume the ansatz δφ k = f k (η)φ. When it is substituted in the previous equation and by using Eq. (3), we can write the metric perturbation as: If we take into account (64), we can obtain the following equation for f k (η): By combining equations (60) and (61), we can integrate the last equation obtaining: wherec 0 k is an integration constant. Solving this equation we obtain where c 0 k and c 1 k are integration constants. Substituting in (67) As it can be seen in Fig. 4 for the case of a quadratic potential, Ψ k does not oscillate in an important way. On the other hand, computing the energy density contrast: Notice that sinceφ 2 oscillates around a non-vanishing value, the same behaviour is expected for δ k . This agrees with the numerical result in Fig. 4. Finally, it can be seen that averaging the previous expression, we find a perfect agreement with the results in previous section. Thus, substituting (14) in (71) and (70), and averaging in (72) we find that at late times: where the O( ) error comes from the average inφ 2 . This result agrees with the standard expression for super-Hubble modes in perfect fluid cosmologies with constant equation of state. Notice that (73) is in good concordance with the numerical solution shown in Fig. 4.

B. Sub-Hubble analytic approach
In this section, we will consider a different approach to check the results of previous section. This approach is valid for sub-Hubble modes of power law potential theories with n = 2. Fortunately, as we shown, the n = 2 case is well understood.
Using (3) and (17), we can write the perturbation of the energy density and pressure as, Multiplying (63) by the expected effective sound speed c 2 eff = (n − 2)/(n + 2) and subtracting it from (65): By using (74) and (75), with V (φ) = m 2 φ 2 /2. We work in the same units of Fig. 1. In order to calculate a representative mode we have set m = 10, k = 0, δφ k (η = 1) = 10 −5 ,δ φ k (η = 1) = 0, a(η = 1) = 1. It can be seen that the scalar perturbation of the metric (blue) tends to a constant. The perturbation of the energy density (gray) decays as η −6 ∼ a −3 , consequently the average contrast of density is constant as expected for the perturbations of a dust-like perfect fluid during the super-Hubble limit. Finally, the pressure perturbation (purple) oscillates around zero.
Notice that the order of the φ amplitude can be estimated through (6), thus, 16πG 3 (n + 2) So Eq. (77) can be approximated bÿ Moreover, in time intervals ∆η H −1 , we can write approximatelÿ The solution of this equation is decomposed in a solution for the homogeneous equation, and a particular solution for the inhomogeneous one. In order to obtain it let us take the Fourier transform of (81),Ψ k,part (ω) = −8πG where * denotes the convolution operation.
In the limit ω eff k, φδφ k oscillates with ω eff and, thus,Ψ By taking the inverse Fourier transform, we reach the general solution, where z = ω eff η/2. As it can be seen, Ψ k,part ∼ O(8πGφδφ k ), however for sub-Hubble modes, and, consequently, the particular solution can be neglected. In addition, the source on (80) can also be neglected, thus obtaining the same approximated equation (48) of the effective model. A more rigorous analysis can be done thanks to Floquet's theorem. Within time intervals ∆η H, (17) takes the form, with Q(z) ≡ 4(n − 1)λ|φ| n−2 /ω 2 eff and k ≡ 2k/ω eff . Notice that as φ is periodic, so is Q(z) and the Floquet's theorem gives us the general form of the solution of (87).
As an example, let us consider the quartic potential case V (φ) = λφ 4 /4, with φ 0 and ∆ integration constants and sn[z, i] the corresponding Jacobi Elliptic function. If k = 0, Notice that, as Jacobi Ellliptic functions are periodic with period 4K, making the change z = 4Kx/π, we can see that this solution corresponds to the τ 1k = τ 2k = 1 case of the Floquet's theorem: by comparing with (91). However, in general, when k = 0, τ 1k = τ 2k and it can be seen that for modes with ω ef f k, δφ is an oscillating function modulated by a periodic function of frequency O k/ λφ 2 0 . In Fig Fig. 1. The initial conditions are δφ k (0) = 0 andδ φ k (0) = −10 −6 , for λ = 10 6 . The solution with k = 0 (black) shows the growing mode of (94), which corresponds to the second case of Floquet's theorem. As it can be seen for the cases with k = 1 (gray) and k = 10 (lighter gray), in general the characteristic equation roots will be different and the first case of Floquet's theorem applies. of nodes is multiplied by a factor of 10 when k is increased from k = 1 to k = 10.
For a general power law potential, if k = k = 0, the analogous solution to (94) can be written as which also corresponds to the second case of Floquet's theorem. For k = 0, δφ k behaves analogously to the n = 4 case and, thus, we will assume that δφ k has the form: As δφ k ∈ R, solutions can be divided in [25]: 1. Stable solutions with Re(α k ) = 0: 2. Unstable solutions with Re(α k ) = µ and Im(α k ) = l ∈ Z: δφ k (z) = e µ k z e ilz ξ 1k (z) + e −µ k z e −ilz ξ 2k (z) . (100) Notice that e ilz ξ 1k (z) and e −ilz ξ 2k (z) are real πperiodic functions.
Due to the periodic properties of φ and δφ k , their product can be expanded in Fourier series as, From this series, we can obtain a simple expression for φ * δφ k , We can obtain the general solution of (81), Notice that in the limit in which we are interested (ω eff k), c mk 1. Therefore the particular solution is at leading order equal to φδφ k and is negligible in comparison with the homogeneous one, in agreement with our preliminary analysis (86).
From the general solution (104), we can obtain an expression for the effective sound speed up to O( ): where we have considered only the m = 0 mode in (104) since the m = 0 modes vanish when taking the average. This expression depends not only on α k but also on the initial conditions given by b 0k and Ψ 0k . However, from (86) we know that |b 0k /Ψ 0k | O(k 2 /ω 2 ef f ). On the other hand we have to determine the size of the parameter α k . With this purpose, we will follow the discussion made in [25] writing Q(z) and a possible solution δφ (1) k as Fourier expansions: We introduce the tentative solution δφ (1) k in (107), result-ing the following system: We eliminate c (2m)k , obtaining the determinant ∆ k (α k ) and an equation for α k , ∆ k (α k ) = 0. This equation can be rewritten as We know the solution when k = 0, where one of the modes is periodic. Consequently k → 0 ⇒ α k → 0. Notice that we have also checked this behaviour numerically in Fig. 5. Expanding (110) in α k , Even if we can not compute ∆ k (0) analytically for k = 0, we know that expanding in k the first correction of the matrix elements is O 2 k . Thus from equation (111), we expect that α k ∼ O( k ). Therefore, the correction to the effective sound speed given by (106) is: which is a generalization of (56) for n = 2.

C. High-k modes
We have seen that for ma k an oscillating scalar field with power-law potential behaves as a perfect fluid for which c 2 eff = ω. Let us now consider the opposite limit with k ma. For sub-Hubble modes Ψ k is well approximated by In this case the field perturbation oscillates much faster than the background field. The typical frequency of δφ k oscillations would be k [24], thus We can also reach an equivalent expression using (104).
In this limit, we can neglect V (φ) from Eq. (87) in comparison with 4k 2 /ω 2 eff . It implies α k 2k/ω eff . If we assume that the lowest m coefficients are responsible of the main contribution to the Fourier expansion c mk m + 2 m , for |m| α k .
The gravitational potential, and accordingly the density perturbation, oscillates around zero as shown in Fig.  6. Because of this fact, all the perturbations vanish in average but the effective sound speed is c 2 eff = 1 (see Fig. 6) according to (52), since in both δp k and δρ k , the kinetic term dominates.

VII. CONCLUSIONS
In this work we have shown that a coherent homogeneous scalar field oscillating in a power-law potential behaves as an adiabatic perfect fluid with constant equation of state both at the background and perturbation levels. Thus, scalar perturbations are shown to propagate, to the leading order in k/ω eff , with a sound speed given by c 2 eff = ω = (n − 2)/(n + 2). The first correction to this expression is shown to be O k 2 /ω 2 eff . The robustness of the result has been shown by studying the exact system in the sub-Hubble and super-Hubble limits as well as in the numerical computations.
These results extend previous analysis done in the massive case n = 2 and opens the possibility of using this kind of models as perfect fluid analogues in different cosmological contexts. In the general case, we have shown that there are departures from the perfect fluid behaviour on small scales with a cut-off around k ω eff very much as in the harmonic case. Notice that for n < 2 the negative value of c 2 eff suggests the generation of instabilities as found in previous works in the homogeneous case.
The analysis performed in this paper could be extended to higher-spin oscillating fields. In particular in the massive vector case, which as shown in [13] behaves as nonrelativistic matter at the background level, this study would allow to determine the growth of structures and its viability as dark matter candidate. Work is in progress in this direction [26].