Dark photon dark matter from an oscillating dilaton

We present a mechanism for generating ultralight dark photon dark matter in the early Universe via a dilatonlike scalar field coupled to the dark photon's kinetic term. Energy is initially stored in the condensate of the dilaton, which resonantly produces dark photons when it begins oscillating in the early Universe. While similar scenarios with axion--dark-photon couplings require large coupling coefficients to fully populate the dark photon, the dilatonic coupling features a unique regime: When the dark photon's mass is half that of the dilaton, dark photons are copiously produced even when the dilaton undergoes small-amplitude oscillations. Scenarios consistent with the cosmic microwave background allow for ultralight vector dark matter with mass as light as $10^{-20}$ eV.


I. INTRODUCTION
Ultralight, massive dark photons (i.e., spin-1 vector bosons) are a curious candidate for the dark matter (DM) in our Universe.Like scalar fuzzy DM [1][2][3][4], ultralight dark photons exhibit wavelike properties on macroscopic scales, λ ≲ 10 pc, for masses m γ ′ ≳ 10 −21 eV [5,6].Halos supported by dark photons can therefore feature vector solitonic cores [5] reminiscent of those in scalar fuzzy DM models [1].However, these vector solitons are distinguished due to their intrinsic spins [7,8].The vectorlike nature of the condensate also allows for distinctive higher energy solitonic configurations similar to Proca stars with radially directed vector fields [5].Furthermore, stable starlike solutions are possible when self-interactions of the dark photons are included [9,10].Vector dark matter's predictions for small-scale structure also differ considerably from its scalar cousin due to the nature of its production mechanisms, which typically result in a highly peaked power spectrum on small scales [11][12][13][14][15].Such a peaked spectrum leads to rich small-scale structure, with significant energy density stored in boson stars [8,16].
Recent studies on dynamical heating of ultrafaint dwarf galaxies via coherent fluctuations in fuzzy dark matter put pressure on the very low mass end of the mass spectrum, requiring m fdm > 10 −19 eV [17].Slightly weaker limits are expected to apply to the vector scenario due to the reduced interference of the multiple polarization states [8].Reference [18] argues that a generic bound m fdm > 10 −18 eV applies when dark matter is produced by a causal process after inflation.While scalar [19][20][21][22] and vector [23,24] masses in the range 10 −13 − 10 −11  eV are constrained by solar-mass black hole superradiance bounds (and lighter masses could be probed by supermassive black holes [22]), these can be evaded by self-interactions of the vector field [15], leaving a wide range of masses viable.
Production of dark photon DM in the ultralight particle mass range is a longstanding problem.Early models that produced dark photons from a misalignment mechanism analogous to that of scalar dark matter production [25] were later shown to require nonminimal couplings to the Ricci scalar [26][27][28] that can lead to violations of unitarity at relatively low energy scales in longitudinal gravitonphoton scattering [15].Massive dark photons minimally coupled to Einstein gravity (but otherwise decoupled from other matter fields) are produced during slow-roll inflation, but their abundance matches the one of DM only if m γ ′ ≳ 10 −5 eV [11,[29][30][31].Finally, an oscillating Higgs [32] or an oscillating, misaligned axion [15,33] allow for the resonant production of ultralight dark photons with the correct DM abundance.While these resonant vector DM production models are based on well-motivated theories, they pose questions about naturalness and choices of couplings.In particular, the dimensionless coupling constants should respect a hierarchy in the case of the oscillating Higgs [32], and in the axion models, large couplings between the axion and the Chern-Simons term F F of the vector field are exacerbated by the small gauge couplings that are required [15,33].
In this work, we identify a new variant of the resonant mechanism for generating dark photon DM that does not require unnaturally large couplings.In addition to the dark photon, we consider a scalar (or "dilaton") ϕ kinetically coupled to the dark photon via an interaction L ⊃ W (ϕ)F µν F µν /4.This form of interaction was studied in Refs.[34,35], where ϕ played the role of the inflaton.During slow-roll inflation, ultralight dark photons can be produced with the right DM abundance provided that the effective dependence of the coupling on the Friedmann-Lemaître-Robertson-Walker (FLRW) scale factor is W (ϕ(a)) ∝ a −n for n close to 4. Reference [36] considered a spectator ϕ whose evolution during inflation produces large-wavelength dark photons.In this work, we instead consider ϕ to be a light spectator field during inflation whose postinflationary oscillations (rather than its inflationary dynamics) give rise to the resonant production of vector DM.
When the dilaton oscillates with large amplitude [ϕ/M ≫ 1, where M is the mass scale associated with the coupling function W (ϕ)], dark photons are efficiently produced via broad resonance, closely resembling models that feature a coupling to an axion [15,33].Typically, such parametric resonances become inefficient for small oscillation amplitudes ϕ/M < 1, below which As such, large couplings are required to completely deplete the dilaton (or axion) condensate into the vector.However, the dilatonic coupling exhibits a unique regime.When the vector mass is half the dilaton mass, m γ ′ = m ϕ /2, a low-momentum instability becomes efficient at late times, i.e., under small amplitude oscillations ϕ < M .We show this explicitly for the benchmark model W (ϕ) = e ϕ/M , but our main results depend only on the small-amplitude behavior being of the form Eq. ( 1).With M ≈ 10 17 GeV, the model successfully realizes ultralight vector DM scenarios (m γ ′ ∼ 10 −21 eV) consistent with cosmic microwave background (CMB) bounds on isocurvature perturbations and the scale of inflation.Unlike existing resonance mechanisms [15,33], this mechanism therefore does not require a large coupling between the dark photon and the scalar, instead requiring a specific tuning between the masses of the scalar and dark photon.
In the remainder of this paper, we detail the governing equations for a massive vector coupled to a dilaton (Section II), study the resonant production of dark photons due to an oscillating dilaton (Section III), and present the resulting relic abundance of dark photon dark matter and viable parameter space given various cosmological constraints (Section IV).We conclude in Section V. Throughout, we work with an FLRW metric of the form with a(t) the scale factor.We use natural units in which ℏ = c = 1 and the reduced Planck mass m Pl = 1/ √ 8πG.Greek spacetime indices are contracted via the Einstein summation convention, while repeated Latin spatial indices are contracted with the Kronecker delta function (regardless of their placement).Dots denote derivatives with respect to cosmic time t, and the Hubble rate is H ≡ ȧ/a.

II. MODEL AND DYNAMICS
We consider a dilatonlike scalar ϕ coupled to a vector A µ via the action where Equation (4b) implies that which coincides with the Lorenz gauge choice (for gauge theories) but is instead here a constraint imposed by consistency of the equations of motion.In FLRW spacetime, Eqs. ( 2), (4a) and (4b) reduce to We expand the vector in Fourier modes as where the polarization vectors satisfy and we likewise expand the dilaton into To linear order in spatial fluctuations, Eq. (6c) then decomposes into using the shorthand W = W ( φ).The Lorenz constraint Eq. ( 5) sets which combines with Eq. (6b) to give Note that the third-order differential equation yielded by substituting Eq. ( 12) for A ∥ in Eq. ( 11b) is proportional to the time derivative of Eq. ( 13) plus 3H + Ẇ/ W times Eq. ( 13); i.e., the system is self-consistent.
The dynamics of the transverse modes and A 0 are most conveniently studied in terms of the rescaled fields for which Eqs. (11a) and ( 13) respectively reduce to In general, W (ϕ) → 1 at late times (as ϕ decays with expansion), and A ± and √ aA ± then coincide, but an instability analysis is more straightforward in terms of the former.In addition, Eq. ( 12) sets i.e., one can study the (relatively simpler) dynamics of Eq. (15b) in place of the longitudinal mode.

III. STABILITY ANALYSIS
To characterize the (in)stability of vector modes in an oscillating dilaton background, we first consider the limiting case of Minkowski spacetime.We then apply these results to FLRW spacetime, relying on the fact that (for the parameter space of interest) the relevant dynamics occur on timescales much shorter than the instantaneous Hubble rate.

A. Small-amplitude broad resonance in Minkowski spacetime
In Minkowski (i.e., nonexpanding) spacetime, the homogeneous component of the dilaton evolves according to solved by φ(t) = ϕ 0 cos(m ϕ t) under the initial condition φ(t) → ϕ 0 and φ(t) → 0 for t ≪ 1/m ϕ .For concreteness, we set for which Eqs. (15a) and (15b) (after substituting a = 1 and H = 0) are Because φ(t) is periodic, Eqs.(19a) and (19b) describe harmonic oscillators with periodic frequencies.By the Floquet theorem [37,38], their solutions are of the form P + (t)e µt + P − (t)e −µt , where µ is the Floquet exponent and P ± (t) = P ± (t + T ), with T being the period of the background.Modes for which ℜ(µ) ̸ = 0 are unstable and grow exponentially with time.We next study the structure of parametric instabilities in these equations.
For massless dark photons, unstable solutions are present only for sizeable oscillation amplitudes, ϕ 0 ≳ M .While massive dark photons also experience these largeamplitude instabilities (provided the mass is not too large, i.e., m γ ′ ≲ Ẇ/ W ∼ m ϕ ), the presence of the mass term gives rise to a novel low-momentum (k ≪ m ϕ ) instability in the small-amplitude (ϕ 0 ≪ M ) regime when m γ ′ /m ϕ = 1/2.To see how this instability arises, we first note that in this limit and to leading order in ϕ 0 /M , Eqs. (19a) and (19b) tend to the Mathieu equation where The solution to the Mathieu equation [37], P+ (t)e μz + P− (z)e −μz , is unstable for small q provided that p = n 2 for integer n.For instance, the case n = 1, corresponding to m γ ′ = m ϕ /2 for modes k ≪ m ϕ , gives rise to an unstable solution in the limit of small q (and therefore of small amplitude).The Floquet exponent of the Mathieu equation in this regime is known analytically to be for the solutions to Eqs. (19a) and (19b) when ϕ 0 ≪ M and k ≪ m ϕ .
Figure 1 depicts the Floquet exponents for the transverse [Eq.(19a)] and longitudinal [Eq.(19b)] components as a function of the vector's wave number and the dilaton's oscillation amplitude.In addition to the broad instability bands for large momenta and amplitudes, Fig. 1 exhibits the above-described band at small k/m ϕ extending down to ϕ 0 /M → 0. We note that the unstable band does not extend to arbitrarily small amplitudes if m γ ′ is not precisely equal to m ϕ /2.In particular, the width of the instability band in k in the small-q limit is |p n=1 − 1| = |q| [37].Quantifying the mass tuning as the width of the instability band is The low-momentum instability band vanishes for ϕ 0 /M < 8δ (though if δ < 0, a narrow instability band persists to arbitrarily small ϕ 0 /M over wave numbers satisfying Below we estimate the level of mass tuning this implies for viable dark photon dark matter scenarios.

B. Parametric resonance in FLRW
The above results for the resonant instabilities in Minkowski spacetime can be extended to the expanding Universe by accounting for the redshifting of physical momenta k/a and of the dilaton's oscillation amplitude.Specifically, in an FLRW spacetime, the dilaton begins to oscillate when H(t i ) ≈ m ϕ with an initial amplitude ϕ 0,i that subsequently redshifts as 2 ϕ 0 (t) ≈ 1.5ϕ 0,i a(t) a i −3/2 (25) in the radiation-dominated era.The corresponding time dependence of the coefficients in the equations of motion Eqs.(19a) and (19b), which nominally brings them away from the form of the Mathieu equation Eq. ( 20), is negligible when the dilaton's oscillation rate is much faster than the expansion rate (which, in the radiation-dominated era, evolves as H ∝ 1/a 2 ).[Likewise, the terms proportional to the Hubble rate in Eqs.(15a) and (15b) are also negligible in this limit.]That is, at sufficiently late times, the dilaton's oscillation amplitude and each mode's physical wave number are effectively constant over each oscillation.The Minkowski-space Floquet exponent Eq. ( 22) therefore captures the instantaneous growth rate in FLRW spacetime to good approximation upon replacing the constant ϕ 0 with the (slowly) decaying amplitude Eq. ( 25).
Depleting the dilaton's energy (so that the majority of the dark matter comprises dark photons) requires parametric resonance to be efficient [38], i.e., that the exponential growth rate is significantly greater than the Hubble rate (ℜ(µ)/H ≫ 1) for a sufficiently long duration.Crucially, the expansion rate decays faster with expansion than the dilaton's oscillation amplitude [Eq.(25)] in the radiation-dominated era-namely, the exponential growth rate Eq. ( 22) relative to the expansion rate is Hence, the efficiency (i.e., per e-fold of expansion) of resonance grows with the expansion of the Universe.Accordingly, ℜ[µ(t)]/H is approximately equal to the quantity plotted in Fig. 1 times a(t)/a i • 3ϕ 0,i /2M .Furthermore, the comoving width of the instability band also grows with time, bounded above by (k/m ϕ ) 2 ≤ ϕ 0,i /8M • (a/a i ) 1/2 [via Eq. ( 24)] and below by the comoving horizon scale m ϕ /(a/a i ) (corresponding to the gray lines in Fig. 1).Equation (26) shows that, even if the initial oscillation amplitude is small, after the Universe expands by a factor 2 More precisely, the solution to Eq. (6a) for the homogeneous mode φ(t) in a radiation background is Γ(5/4)ϕ 0,i J 1/4 (m ϕ t)/ 4 m ϕ t/2, where Jν is the order-ν Bessel function.The asymptotic oscillation amplitude is 2 3/2 Γ(5/4)ϕ 0,i /(a/a i ) 3/2 √ π.
∝ (ϕ 0,i /M ) −2 , the rate of particle production becomes efficient.More precisely, production of the vector completes soon after its energy density becomes comparable to the dilaton's.Via Eq. ( 25), ρ ϕ ≈ m 2 ϕ ϕ 2 0,i /(a/a i ) 3 .Since the vector is produced well after a i with a comoving wave number of order m ϕ , it is nonrelativistic at production; its energy density is then approximately where Before production, A λ is in the vacuum state [39,40], Setting the growth rate to that from Eq. ( 26) in Eq. ( 27), taking the integral over wave number to be dominated by modes with k ∼ m ϕ , and noting that W ≈ 1 at late times, The dilaton and vector have comparable energy density at a scale factor For ϕ 0,i /M = 1 and m γ ′ = 10 −18 eV (choosing ϕ 0,i to match the relic abundance of dark matter-see Section IV), a ⋆ /a i ≈ 7.1 × 10 4 , decreasing only by a factor of 2.3 for m γ ′ = 10 −6 eV.The above results apply only while modes remain in the small-amplitude resonance band.While the instability extends to arbitrarily small amplitudes when the vector's mass is precisely half the dilaton's, resonance will terminate early if the mass ratio is not so precisely tuned.Concretely, the n = 1 band must have nonzero width in k at sufficiently small |q| ∝ ϕ 0 /M .Requiring that low-momentum modes do not leave the instability region by the time resonance becomes efficient-namely, that Eq. ( 24) remains satisfied for some range of k until the Universe expands by a ⋆ /a-we arrive at for masses m γ ′ ∼ 10 −18 eV.While other narrow-band resonances exist (e.g., if m γ ′ is slightly smaller than 2m ϕ or for other mass ratios m γ ′ = nm ϕ /2), they are centered at nonzero k (in contrast to the low-momentum n = 1 band) and shrink in width as the dilaton's amplitude decays.The redshifting of momenta in FLRW spacetime therefore prevents any mode from growing large enough to deplete the dilaton background of its energy density in such cases.

IV. DARK MATTER ABUNDANCE
We now determine the parameter space for which dark photons make up all of the dark matter.The dark matter must have been produced sufficiently long before scales observable in the CMB become dynamical, so production takes place deep in the radiation-dominated era.The dilaton begins oscillating when H ≈ m ϕ ; at this stage, it contains a negligible fraction of the total energy of the Universe (else the matter-dominated era begins too early).Once the dilaton transfers a substantial fraction of its energy to the dark photons, the subsequent evolution is nonlinear-in particular, the excited vector field modes rapidly backreact on the dilaton background.We assume for simplicity that most of the dilaton energy is transferred into the dark photons 3 so that the dilaton makes a negligible contribution to the dark matter abundance.If the dark photons are nonrelativistic when produced, 4 their abundance today is simply taking g ⋆ = 10.75 effective number of relativistic degrees of freedom in the plasma at the time of production.CMB observations place two further constraints on parameter space.First, ensuring that the dark matter was produced before scales observed in the CMB reenter the horizon imposes a lower limit on the vector mass in the m γ ′ = m ϕ /2, small-amplitude regime because the Universe expands by a substantial amount before resonance becomes efficient.Namely, using conservation 3 Classical lattice simulations demonstrate efficient depletion of an oscillating dilaton into massless dark photons [12,14,41,42] in the strong coupling regime.Furthermore, in similar models with an axial coupling rather than a dilatonic one, most of the energy is transferred to the dark photons, regardless of whether they are massless [13,[43][44][45][46][47][48][49][50] or not [15].Determining whether the same conclusion holds in the small-amplitude regime would require dedicated numerical simulations. 4As discussed previously, in the small-amplitude regime with m γ ′ = m ϕ /2, the dark photons are necessarily nonrelativistic at production, since a⋆/a i ≫ 1 via Eq.(31).In the broad resonance regime (for any m γ ′ ≲ m ϕ and sufficiently large ϕ 0,i /M > 1), the dark photons are instead mildly relativistic at production, leading to a corresponding dilution in their abundance since they redshift more rapidly than matter until they become nonrelativistic.In this case, the predictions should be analogous to dark photons resonantly produced by axions [15].
of entropy and Eq. ( 31), the redshift of production z ⋆ = a 0 /a ⋆ − 1 satisfies (34) (dropping additional logarithmic dependence on m ϕ and ϕ 0,i ).Since the redshift of matter-radiation equality is ≈ 3400 [51], for masses m γ ′ ≲ 10 −17 eV, dark photons are not produced early enough if ϕ 0,i /M < 1.However, lighter masses can be accommodated by a modest increase in amplitude ϕ 0,i /M ∝ m Second, in this scenario, the vacuum expectation value of the dilaton arises from quantum fluctuations during inflation.The value of ϕ 0,i varies across causally disconnected Hubble patches and generates an isocurvature perturbation after the decay into dark photons.The power of the isocurvature perturbation from inflation is , where H I is the Hubble scale during inflation.Assuming Ω γ ′ = Ω DM , CMB constraints on isocurvature perturbations [52] bound the Hubble scale during inflation to be below V. DISCUSSION AND CONCLUSIONS We have demonstrated that a massive vector field kinetically coupled to an oscillating scalar field exhibits a novel nonlinear decay channel.Namely, in addition to a strong-coupling regime reminiscent of axion-darkphoton models [15,33], an oscillating dilaton twice as heavy as its coupled dark photon induces substantial vector production even for small oscillation amplitudes.This effect allows for the efficient production of dark photon dark matter without invoking large couplings, naturally giving rise to dark photon dark matter over a wide range of masses.Efficient parametric resonance in the small-coupling regime requires a finely tuned mass ratio m γ ′ /m ϕ = 1/2-as severe as one part in 10 7 for order-unity couplings-barring a UV model that gives rise to both the dark photon and dilaton's masses and explains the coincidence.
In Section IV we showed that requiring consistency with the CMB jointly constrains the vector mass and coupling strength, but only modestly large couplings are required to achieve ultralight masses ∼ 10 −20 to 10 −18 eV.To go beyond these estimates (e.g., to characterize the relic abundance of dark photons and dilatons as a function of the particle masses and initial field amplitudes and to compute the resulting spectra of dark photons) requires 3 + 1D numerical simulations.Preliminary simulations confirm the numerical and parametric estimates of Sections III B and IV, but we leave a more thorough investigation of these questions to future work.
The rapid growth of vector modes and nonlinear dynamics at the end of the production phase would source a grav-itational wave background, providing a possible probe of the model.Though a quantitative prediction would again require dedicated numerical study, the signals are likely to resemble those from dark photons resonantly produced by a rolling axion [49,50,[53][54][55].However, the signal amplitude is unlikely to be promisingly large.Stochastic backgrounds are parametrically suppressed by both the fraction of Universe's net energy density contained by the source and the gravitational-wave wavelength relative to the horizon size at production [56].In this scenario, the vector and dilaton are massive relics whose abundance is ∝ a ⋆ /a eq , i.e., the ratio of the scale factors at production and matter-radiation equality.One might then expect the long delay from oscillation (a i ) to production (a ⋆ ) to enhance the resulting signals, but gravitational wave emission occurs at fixed comoving scales ∼ m ϕ that are a ⋆ /a i times farther inside the horizon at production.These two effects turn out to cancel each other, and even highly efficient gravitational wave production is unlikely to exceed present-day abundances of Ω GW,0 ∼ 10 −15 (see, e.g., Ref. [57]).
Here we specifically invoke a dilatonic coupling to achieve dark matter predominantly comprising dark photons.In principle, the dilaton could instead be the dark matter and simply happen to have such a coupling with parameter values that fail to achieve efficient conversion to dark photons before the CMB forms.For the same reason that resonance could become efficient at sufficiently late times in the radiation era even for small couplings [as discussed after Eq. ( 26)], in this alternative scenario, substantial dark photon production will never occur: As the Universe transitions to matter domination, the Hubble rate instead decays as a −3/2 just like the dilaton, and the growth rate per Hubble time asymptotes to a constant.In other words, if the dilaton has not efficiently produced dark photons by the time the CMB forms, then it never will.(The marginal regime-resonance becoming efficient as CMB modes enter the horizon-would likely be incompatible with the cold, collisionless dark matter required by CMB observations.) 5hile we have not assumed any particular origin for the mass of the dark photon, in the case where the mass arises from the Higgs mechanism, the production of vortices challenges the viability of dark photon dark matter [58].Moreover, as noted by [15,59,60], Stueckelberg masses are restricted to m γ ′ ≳ meV in string theory, implying that smaller masses must arise from the Higgs mechanism.To our knowledge, all proposed production mechanisms for light dark photon dark matter are afflicted by vortex formation constraints.However, it is possible that the small-amplitude regime we point out could evade such issues: The long delay between dilaton oscillations and dark photon production could ensure the vector's energy density never exceeds the critical value for vortex formation.We defer a full exploration of the parameter dependence of vortex formation constraints, along with the phenomenology of kinetic mixing with the Standard Model photon and subsequent plasma effects, to future work. 6hroughout this work, we assumed that the dark photon mass is ϕ independent.This dependence, like the form of the kinetic coupling W (ϕ), is ultimately determined by the UV physics.Providing a UV completion of the theory is beyond the scope of this work, but it is conceivable that both the kinetic and mass terms attain a ϕ dependence, coming from, e.g., radiative corrections.[In fact, the small-amplitude resonance is present if the coupling function W (ϕ) multiplies the mass term rather than the kinetic term of the vector since the pertinent terms in the equation of motion have the same form.]Such interactions, as well as dark photon self-interactions, could have consequences for the efficiency of dark photon production and the viable parameter space for dark photon dark matter.We leave the investigation of such effects for future work.
FIG. 1. Real part of the Floquet exponent µ k for the solutions of the equations of motion describing the transverse [Eq.(19a)] and longitudinal [Eq.(19b)] sectors.Both panels fix m ϕ = 2m γ ′ , for which a region with ℜ(µ k ) > 0 extends to arbitrarily small ϕ0/M and k/m ϕ (depicted in the inset panels).Results are scaled by ϕ0/M to facilitate interpreting results in FLRW spacetime (see the main text for details).Note that the horizontal axis corresponds to the physical wave number k/a in FLRW spacetime.Gray lines depict the trajectory of the physical horizon H(t) = m ϕ /[a(t)/ai] 2 for three different initial amplitudes ϕ0,i = M , √ 5M , and 5M , roughly indicating the low-wave-number cutoff for resonance in the radiation era as a function of time.