FERMILAB-PUB-19-565-T

The Cosmological Evolution of Light Dark Photon Dark Matter

Samuel D. McDermott1 and Samuel J. Witte2

1Theoretical Astrophysics Group, Fermi National Accelerator Laboratory, Batavia, IL, USA2Instituto de Fısica Corpuscular (IFIC), CSIC-Universitat de Valencia, Spain

Light dark photons are subject to various plasma effects, such as Debye screening and reso-nant oscillations, which can lead to a more complex cosmological evolution than is experienced byconventional cold dark matter candidates. Maintaining a consistent history of dark photon darkmatter requires ensuring that the super-thermal abundance present in the early Universe (i) doesnot deviate significantly after the formation of the CMB, and (ii) does not excessively leak into theStandard Model plasma after BBN. We point out that the role of non-resonant absorption, whichhas previously been neglected in cosmological studies of this dark matter candidate, produces strongconstraints on dark photon dark matter with mass as low as 10−22 eV. Furthermore, we show thatresonant conversion of dark photons after recombination can produce excessive heating of the IGMwhich is capable of prematurely reionizing hydrogen and helium, leaving a distinct imprint on boththe Ly−α forest and the integrated optical depth of the CMB. Our constraints surpass existingcosmological bounds by more than five orders of magnitude across a wide range of dark photonmasses.

I. INTRODUCTION

As many once-favored models of particle dark matterbecome increasingly constrained (see e.g. [1–5]), candi-dates other than those resulting from weak-scale thermalfreeze-out have been the subject of growing focus and de-velopment. One candidate of recent interest is the darkphoton, A′ [6–19], which arises from an abelian groupoutside of the Standard Model (SM) gauge group. Thisparticle may “kinetically mix” with the SM photon viathe renormalizable operator ε Fµν F ′µν / 2 [20], with ‘nat-

ural’ values of ε typically ranging from 10−16 to 10−2 [21–23].

Historically, one of the more problematic features oflight vector dark matter has been the identification ofa simple, well-motivated production mechanism. Earlywork on the subject suggested that such a candidatecould be produced via the misalignment mechanism [10],similar to that of axion dark matter (see e.g. [24, 25]),but it was later pointed out that this mechanism is in-efficient at generating the desired relic abundance un-less one also introduces a large non-minimal couplingto the curvature R [11, 12, 18]. Such a coupling, how-ever, can introduce ghost instabilities in the longitudinalmodes [26–28]; while it may be possible to avoid this fea-ture, proposed solutions come at the cost of additionalmodel complexity [19]. The work of [12] provided a com-pelling alternative production mechanism due to fluctu-ations of the metric during a period of early-universe in-flation, but the non-observation of primordial gravita-tional waves constrain this mechanism from producing aviable dark matter population if mA′ . µeV. More re-cently, [13–17] showed that a dark photon coupled to ahidden sector (pseudo)scalar field can generate the en-tire dark matter with masses as light as mA′ ∼ 10−20 eV.This super-thermal population of dark photons is gener-ated by temperature-dependent instabilities or defects inthe (pseudo)scalar field. Given that various works havenow provided more compelling mechanisms to generate

what had perhaps previously been a more speculativedark matter candidate, we find it timely to revisit old,and develop novel, cosmological constraints on (and po-tential signatures of) light dark photon dark matter.

The observational signatures of dark photon dark mat-ter are quite distinct from canonical weak-scale particles.Various cosmological effects of light dark photon darkmatter have been investigated over the years, typicallyfocusing exclusively on the observational consequencesarising from the resonant transition between dark andvisible photons that occurs when the plasma frequencyωp is approximately equal to the mass of the dark pho-ton mA′ [11]. These constraints, however, are typicallyonly applicable for mA′ ≥ ω0

p ∼ 10−14 eV, ω0p being the

background plasma frequency today. More recently, lim-its on very light dark photons were obtained using theobservation that the kinetic mixing allows for an off-shell(non-resonant) absorption of dark photons, subsequentlyheating baryonic matter; if this heating is sufficientlylarge, it may destroy the thermal equilibrium of the MilkyWay’s interstellar medium [29] or that of ultra-faint dwarfgalaxies such as Leo T [30]. This idea has also been usedto project the sensitivity that could be obtained from fu-ture 21 cm experiments which observe absorption spectraduring the cosmic dark ages [31].

In this work, we put forth a simple cosmological pic-ture of dark photon dark matter, requiring only that (i)dark matter is not overly depleted after recombinationand (ii) the energy deposited into the SM plasma doesnot produce unwanted signatures in BBN, the CMB, orthe Ly-α forest. We identify (and describe in a unifiedmanner) the resonant and non-resonant contributions toboth of these classes of observables. We find that thesesimple and robust requirements lead to extremely strin-gent constraints for light photon dark matter, coveringdark photon masses all the way down to ∼ 10−22 eV.Our constraints are stronger than existing bounds acrossa wide range of masses (in some cases by more than fiveorders of magnitude), and are robust against astrophys-

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This manuscript has been authored by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office of High Energy Physics.

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ical uncertainties1.This work is organized as follows. We begin by out-

lining the relevant on- and off-shell conversion processesthat alter the energy and number densities of the darksector and SM plasma. We then discuss various cosmo-logical implications for the existence of light dark photondark matter, including modifications to the evolution ofthe energy density after neutrino decoupling, spectral dis-tortions produced in the CMB, dark matter evaporation,and modifications to the Ly-α forest from the heatingof the IGM. We conclude by discussing more speculativeways in which sensitivity can be extended to the low massregime.

II. PLASMA MASS AND (DARK) PHOTONCONVERSION

Dark photons and SM photons can interconvertthrough cosmic time. Accurately treating this conver-sion requires accounting for plasma effects: the SMphoton has a modified dispersion relation in a chargedplasma, given by ω2 = ReΠ(ω, k, ne) + k2. The di-mensionful scale that governs the SM photon dispersionrelation is the plasma mass ReΠ(ω, k, ne) ∝ ω2

p(z) =4παEM

∑ni(z)/EF,i; here, ni is the number density of

species i and EF,i =√m2i + (2π2ni)2/3 is the charged

particle Fermi energy. We will focus on cosmologicalepochs for which the only relevant species is the electron,with number density given by

ne = Xe(z)

(1− Yp

2

)η

2ζ(3)

π2T 3

0 (1 + z)3 . (1)

In Eq. (1), Xe(z) is the free electron fraction, Yp is theprimordial helium abundance, η is the baryon to photonratio, and T0 is the temperature of the CMB today. Thefunction Xe(z) can be obtained using the open sourcecode class [33], and we fix Yp = 0.245 [34, 35] and T0 =2.7255 K [36].

In general, dark photons and SM photons will convertwith equal probability. An asymmetry in energy flow istherefore possible only due to initial conditions: at thetime of the formation of the CMB the SM photons aredescribed to good precision by a blackbody at a temper-ature T0(1 + zCMB), while dark photons that constitutethe cold dark matter must be a collection of non-thermalparticles with a number density far larger than nγ andan energy spectrum peaked very close to mA′ (for thesake of completeness, we will also address the possibleexistence of dark photons with a very small initial num-ber density). The total energy taken from the reservoir

1 We choose here to neglect bounds from superradiance whichin principle could constrain dark photons with masses below∼ 10−11 eV [32], as the existence of such bounds require self-interactions of the new gauge boson to be small [13].

of cold dark photons and introduced to the SM photonbath is

∆ρA′→γ =

∫dz PA′→γ(z)× ρA′(z) , (2)

where PA′→γ(z) is the redshift-dependent probability ofconversion from an A′ to a SM photon and ρA′(z) isthe redshift dependent energy density of dark photons.Later, we will consider the energy injected normalized tothe number density of baryons, which is given by Eq. (2)with the simplifying substitution ρA′(z)→ ρA′(z)/nb(z).If the conversion probability is small, one can approx-imate ρA′(z) ∼ (1 + z)3ρ0

A′ , with ρ0A′ being the mean

dark matter density today; however, in some cases, theprobability is sufficiently large that dark matter densityprior to conversion is significantly greater than the darkmatter density after, in which case the aforementionedapproximation is not valid.

Similarly to Eq. (2), we may write the energy extractedfrom the SM photon bath as [6, 9]

∆ργ→A′(E) =T 4

0

π2

∫dzdx

x3(1 + z)4

ex − 1Pγ→A′(x, z), (3)

where x ≡ E/T , and we have explicitly included the en-ergy dependence in the conversion probability since theCMB spectrum is far broader than that of cold dark mat-ter, and is well-measured near the peak.

We will use Eqs. (2) and (3) to constrain the existenceof dark photons. As we show below, the most sensitiveprobes are from limits on the heating of the SM bathafter recombination. Before deriving these bounds, wefirst discuss the different routes by which a dark photoncan convert to a SM photon.

III. ON-SHELL AND OFF-SHELLCONVERSION

A dark photon can convert either to an on-shell SMphoton (via oscillation or 2-to-2 processes) or to a vir-tual SM photon (through a 3-to-2 process). Examplesare shown in Fig. 1. While the 3-to-2 process is naıvelynegligible due to the extra phase space and the factorof αEM, it can dominate in some regimes of parameterspace, depending on kinematic matching considerations.

The on-shell processes of interest are oscillation andsemi-Compton absorption. These can operate efficientlyif mA′ & ωp, but A′ → γ is strongly suppressed for acold dark photon bath if mA′ < ωp. On-shell phenom-ena are most pronounced at a level crossing, occurringat mA′ ' ωp(z) for traverse modes and ω ' ωp(z) forlongitudinal modes. In practice, these occur at the sameredshift for on-shell conversion of dark photon dark mat-ter, since ω ' mA′ ; note that this need not be true foroff-shell conversion or for conversion to non-cold darkphotons. The probability of a transition at the timeof level crossing is governed by the non-adiabaticity of

3

Sam McDermott

August 1, 2019

A′ γ

×

(A)

e−

A′ e−

γ

(B)

e−

e−

γ∗e−, I+

A′

(C)

1

FIG. 1. Processes by which photons and dark photons interconvert. Since the dark matter is inherently cold, the processeslabelled (A) and (B) require mA′ ≥ ωp. In the case of inverse bremsstrahlung, shown in panel (C), the fact that the photoncan be off-shell allows dark matter to be absorbed even when mA′ ωp.

the change in ωp(z), and is approximately given by theLandau-Zener expression [9, 37, 38]

P(res)A′→γ '

π ε2m2A′

ω (1 + z)H(z)

∣∣∣∣∣d logω2p(z)

dz

∣∣∣∣∣−1

δ(z − zres). (4)

Eq. (4) is valid only when PA′→γ 1. When this con-dition is violated we adopt the general expression, whichcan be found e.g. in [11]. The delta function in Eq. (4)makes the redshift integral in Eq. (2) trivial. A similarexpression holds for resonant γ → A′ conversion.

In contrast to resonant conversion, an off-shell pro-cess like inverse bremsstrahlung will operate even formA′ ωp, and can dominate the heating rate despiteentering at a lower order in αEM. This process can occuroff resonance and is not forbidden by energy conservationbecause the outgoing photon is not on-shell. This pro-cess leads to a heating of the plasma proportional to thenumber of dark matter particles absorbed. As describedin [29], this process is subject to Debye screening whenmA′ 6= ωp, and thus the rate of loss of energy from thecold dark photon reservoir is given by

P(nonres)A′→γ ' ε2ν

2(1 + z)H(z)

[m2A′

ωp(z)2

]sign[ωp(z)−mA′ ]

, (5)

with the frequency of electron-ion collisions ν given by

ν =4√

2π α2EM ne

3√me T 3

e

log

(√4π T 3

e

α3EM ne

). (6)

The fact that Eq. (5) is proportional to ν is related to thefact that this is an inherently off-shell process. This ratedecouples like (εmA′/ωp)

2 for mA′ < ωp (and, conversely,like (εωp/mA′)2 for mA′ > ωp), but even an arbitrarilylight dark photon may participate, and the rate does notabruptly drop to zero.

In the following, we derive constraints on the kineticmixing parameter for light to ultra-light dark photons,assuming either that dark photons do or do not comprisethe entirety of dark matter. We analyze both resonantand non-resonant processes that lead to either a depo-sition of energy into or removal of energy from the SM

plasma. By including off-shell dark photon absorption,we find that there exist stringent cosmological bounds onthe kinetic mixing of the dark photon dark matter at allrelevant masses.

IV. PRE-CMB CONSIDERATIONS

Resonant conversions between photons and dark pho-tons at temperatures T . O(MeV) and prior to recom-bination can leave discernible signatures in the energydensity inferred from BBN and the CMB. In the absenceof a dark photon population, CMB photons will reso-nantly convert and populate a relativistic dark sector,producing a positive shift in the effective number of lightdegrees of freedom Neff . Such a bound was first derivedin [6], and is reproduced in Fig. 2.

Alternatively, should dark photons contribute signifi-cantly to the cold dark matter energy density, conversionsfrom the dark sector into the SM photon bath will be themore efficient process (owing to the large dark photonnumber density, and the fact that low-energy photonswith ω T can be produced). In fact, resonant pro-duction of photons can be so efficient that nearly all ofthe dark matter can be converted into radiation. Naıvelythis appears problematic for the existence of dark mattertoday; however, the earliest measurement of cold darkmatter energy density comes from the CMB, and thematter energy density before this time is basically un-constrained. For this scenario to remain consistent withobservations, one may postulate the existence of an ini-tial population of cold dark photons much larger thanwhat would be expected given a (1 + z)3 extrapolationof Ω0

CDM. Since the energy density of radiation redshiftsmore quickly than that of cold dark matter, one mustalso be concerned about the possibility of having a pe-riod of early matter domination during BBN. In orderto ensure a successful nucleosynthesis, we require the ini-tial matter density at T ∼ MeV to be no larger thanthe energy density stored in new effective light degreesof freedom, which are constrained during this epoch to

be ∆N(BBN)eff . 0.5 [39]. This constraint was first de-

rived in [9], and since it is logarithmically sensitive to

4

FIG. 2. Bounds that apply for low (or zero) initial abundanceof dark photons arising from constraints on µ- and y-type dis-tortions using the Green’s function formalism of [42, 43]. Alsoshown are existing constraints from spectral distortions [9],5th force experiments [44, 45], modifications to ∆Neff [6],stellar cooling constraints [46–48], and the CROWS experi-ment [49]. Finally, we project the sensitivity of experimentslike PIXIE and PRISM to µ- and y-type distortions. Theredshift for which a dark photon with mass mA′ undergoesresonant conversion zres is shown on the top x-axis for com-parison (neglecting reionization).

the constrained value of ∆Neff , the bounds derived hereare effectively identical to those obtained nearly a decadeago.

Remaining consistent with the thermal history as in-ferred from measurements of BBN and the CMB pro-duces the strongest bounds on the kinetic mixing for val-ues of the dark photon mass mA′ ∼ 10−4 eV. We derivethe bounds shown in Fig. 2 and Fig. 3 using the latestconstraints on ∆Neff from Planck [40] and BBN [35, 41].

V. CMB SPECTRAL DISTORTIONS

Light dark photons depositing energy in the SMplasma at z . 2 × 106 (i.e. temperatures T . 500 eV)will produce distortions in the CMB blackbody spectrum.For redshifts z & 2 × 106, double Compton (DC) scat-tering and bremsstrahlung are efficient at producing lowenergy photons which are subsequently up-scattered viaComptonization (see e.g. [42, 50–52] for an overview).This process of thermalization erases any spectral distor-tions that could arise as a result of the energy injectionfrom dark sectors, and, because thermal equilibrium dic-tates the number density of photons as well as their spec-trum, spectral distortions are possible only after photon-number-changing processes become inefficient. We pro-vide a review of the signatures imprinted on the CMBfrom energy transfers between the dark and visible sec-

tors in the Appendix, and focus below only the formalismadopted for computing the current limits and projectedsensitivity.

Spectral distortions are constrained by various experi-ments, most notably COBE/FIRAS [53], to the level of|y| ≤ 1.5× 10−5 and |µ| ≤ 6× 10−5 [52]. Future experi-ments such as PIXIE [54] and PRISM [55, 56] could en-hance the sensitivity of these spectral distortions to thelevel of |y|, |µ| . 10−8. Should dark photons not con-tribute to the dark matter, blackbody photons can reso-nantly convert and lead to a depression of the spectrumat the measured frequencies [9]. The analysis performedin [9], however, focuses only on resonant conversions oc-curring in the frequency band observable by FIRAS. Thebound derived using this method is clearly conservative,as conversions at frequencies below what is observable byFIRAS still occur, and for z & 103 can still induce spec-tral distortions since Compton and bremsstrahlung pro-cesses are still partially active and lead to a modificationof the blackbody spectrum. Similarly, should dark pho-tons account for the entirety of dark matter, the energydeposited in the SM plasma will create µ- and/or y-typedistortions, depending on when this process takes place(see Appendix to understand for which redshifts energydeposition results in µ and y-type distortions, and theeffects they induce on the black body spectrum). Exist-ing constraints were derived on this energy deposition ina heuristic way in [11]; here, we attempt provide a moredetailed a rigorous analysis of this effect.

We compute constraints on dark photons from bothresonant and non-resonant energy deposition and extrac-tion using the Green’s function formalism [42, 43, 57, 58];the results of these analyses are summarized in Fig. 2 andFig. 3 for the case in which the initial dark photon den-sity is ∼ 0 or equal to that of dark matter, respectively.Existing constraints on µ- and y-type distortions comefrom COBE/FIRAS, and we also project future boundsfor a PIXIE/PRISM-like experiment. Specifically, thelevel of spectral distortions can be accurately approxi-mated by convolving the energy deposition rate with aseries of visibility functions accounting for the fraction ofinjected energy that produces a particular type of distor-tion. These expressions are given by:

y ' 1

4

∫ Jy(t)

ργ(t)

dρ(t)

dtdt (7)

µ ' 1.401

∫ Jbb(t)Jµ(t)

ργ(t)

dρ(t)

dtdt , (8)

with dρ/dt the energy density injected to or extractedfrom the plasma per unit time, and the visibility func-

5

FIG. 3. Limits on dark photon dark matter from: Neff (purple); µ- and y-type distortions (resonant and non-resonant correspondto teal and yellow, respectively); the depletion of dark matter at the level of 10% (resonant and non-resonant correspond toblue and green, respectively), as in Eq. (12); energy deposition during the cosmic dark ages (pink solid) and enhancements inthe integrated optical depth produced by resonant conversions (pink dotted), as in Eq. (14); and heating of the IGM around theepoch of helium reionization (resonant and non-resonant correspond to brown and red, respectively), as in Eq. (15). Existingcosmological constraints on modifications to ∆Neff during BBN and recombination [11], spectral distortions [11], the depletionof dark matter [11], stellar cooling [46–48], and the Ly-α forest [59], are shown in grey for comparison. Dashed black lines denoteastrophysical bounds derived from thermodynamic equilibrium of gravitationally collapsed objects: the Milky Way [29] (labeled‘Dubovsky et al’) and the ultra-faint dwarf galaxy Leo T [30] (labeled ‘Wadekar et al’). The mean plasma frequency today isshown for reference with a vertical line, along with the redshift dependence of the plasma frequency, neglecting reionization, onthe upper axis. We include alongside this publication an ancillary file outlining the strongest constraint for each dark photonmass in order to ease reproduction of our bounds.

tions Ji are given by

Jbb(t) = Exp

[−(z

zµ

)5/2]

(9)

Jy(t) =

[1 +

(1 + z

6× 104

)2.58]−1

(10)

Jµ(t) = 1− Jy . (11)

Here, zµ = 1.98×106 (Ωbh2/0.022)−2/5 [(1− Yp/2)/0.88]

−2/5

is the redshift at which DC begins to become ineffi-cient. These equations are only valid for z & 103,explaining the somewhat unphysical truncation ofbounds derived from resonant transitions shown inFig. 2 and Fig. 3 at mA′ ' 10−9 eV. We confirm theexisting bounds from the FIRAS instrument in the

range 10−14 eV . mA′ . 10−9 eV [11], and we scalethese to future sensitivity expected by PIXIE/PRISM.In the scenario that dark photons constitute the entiretyof dark matter, we show for completeness in Fig. 3constraints derived from non-resonant dark photonabsorption, obtained by combining Eq. (5) with Eqs. (7)and (8).

VI. DARK MATTER SURVIVAL

After recombination, dark photon dark matter can bedepleted via the processes shown in Fig. 1. The totalchange in the dark matter energy density is given by in-tegrating Eq. (2) using Eqs. (4) and (5) from redshift 0to z ∼ 103. Should this change in density be sufficiently

6

high, the relative abundance of dark matter observed to-day would differ from the value inferred by observationsof the CMB. Maintaining consistency with current ob-servations requires, at a minimum, that the density ofdecaying dark matter particles changes by no more than' 2 − 3% after matter-radiation equality [60, 61]. Webegin here by deriving a conservative bound, imposingthat off-shell processes change the dark matter densityby no more than 10%, i.e.

∆ρ0≤z≤1000A′ ≤ 0.1× ρ0

A′ . (12)

A similar bound has been derived with on-shell (reso-nant) conversion for dark photon masses mA′ & 10−12.5

eV in [11]. At lower masses, the resonant bound canno longer be applied and the off-shell process becomesdominant, albeit with a increasing suppression due toDebye screening, exhibiting the expected decoupling be-havior with respect to mA′ . In the case of the reso-nant conversion, we derive a more rigorous bound usingthe latest CMB observations by Planck [62]. Specifi-cally, we modify class to include an abrupt change inthe dark matter energy density, modeled using a tanhfunction of width ∆z = 1, and perform an MCMC us-ing montepython [63]. Our combined likelihood includesthe Planck-2018 TTTEEE+low`TT+lowE+lensing likeli-hood [62] and observations of baryonic acoustic oscilla-tions (BAOs) from the 6DF galaxy survey [64], the MGSgalaxy sample of SDSS [65], and the CMASS and LOWZgalaxy samples of BOSS DR12 [66]. We adopt flat priorson log10mA′ and log10 ε in the range of [−9,−14] and[−12,−7], respectively. The resultant 2σ bound is sig-nificantly stronger than the off-shell constraint across allmasses for which resonant conversions can occur.

Of particular interest in cosmology today is the so-called Hubble tension, which is a 4 − 6σ disagreementbetween the value of H0 inferred using local measure-ments [67–72] and that inferred from early Universe cos-mology [40] (see also e.g. [73]). It has been pointed outthat resolving this tension seems to require early Universephysics [74], and in particular favors a modification tothe energy density near the time of recombination. Giventhat this model is capable of generating an abrupt changein the matter density (and thus the expansion rate) at thetime of recombination, it is natural to wonder whetherthe effect could address any outstanding discrepanciesbetween early and late Universe cosmology. As we willshow in the following section, the impact of the energyinjection from this resonant conversion process actuallyproduces constraints sufficiently strong so as to eliminatethe possibility of an O(1)% change in ΩCDM, as would benecessary to noticeably impact the inferred value of H0.

VII. ENERGY DEPOSITION DURING COSMICDARK AGES

The energy per baryon stored in the dark sectoris, on average, greater than 109 eV (i.e. ρCDM/nb ∼

ΩCDM/Ωb×mp ∼ 5×109 eV). For most dark matter can-didates, the relevant processes allowing energy flow intothe SM sector decouple well before the formation of theCMB. In the case of the dark photon, however, resonanttransitions can concentrate this energy in a narrow win-dow, leading to enhanced observable effects. Specifically,if the energy is deposited in the SM plasma after recom-bination, the induced heating can raise the temperatureof the gas above the threshold for the collisional ioniza-tion of hydrogen, and induce an early, albeit short-lived,period of reionization. This will affect the integrated op-tical depth of the CMB, currently measured by Planckto be τ = 0.054±0.007 [40]. In order to assess the extentto which dark photon resonant transitions enhance theoptical depth, we modify the equations tracking the tem-perature of the medium to include a near-instantaneousenergy injection from resonant dark photon conversion.More specifically, the evolution of the gas temperature Tis given by

dT

dz=

2

3

T

nb

dnbdz− T

1 + xe

dxedz

+

2

3kB(1 + xe)

1

(1 + z)H(z)

∑j

εheat,j , (13)

where nb is the baryon number density, xe the free elec-tron fraction, and εheat,j the heating rate per baryon(arising from Compton cooling, X-ray heating, exotic en-ergy injection, etc.). Dark photon conversion can be in-cluded as a term in the final summation, with εj given by

P(res)A′→γ×ρCDM/nb×f(z)×dz/dt, where f(z) encodes the

time dependence of the energy injection. Here, we modelf(z) as a narrow gaussian centered on the resonance andwith a width of ∆z = 0.5. Formally, we include thiscontribution in the latest version of Recfast++ [75, 76],and use this open-source program to determine the evo-lution of T . As mentioned before, the energy depositedgoes directly into heating the medium; however, oncethe temperature of the gas is sufficiently high, the gascan become collisionally ionized. The evolution of thefree-electron fraction must be solved simultaneously withEq. (13), since these equations are coupled. We illustratethe evolution of the free electron fraction as a functionof redshift for a dark photon with mass mA′ = 10−12

eV and various mixings in Fig. 4. It is clear that theeffect of the resonance can be substantial, and is able tosignificantly increase the integrated optical depth.

We perform a first estimate of this effect by jointlysolving for xe(z) and T (z), as described above, and com-puting the optical depth by [77–79]

τ =

∫dz

dt

dzσT n

0H(1 + z)3

(xe(z)− x0

e(z)), (14)

where σT is the Thompson scattering cross section, n0H

is the number density of hydrogen today, xe is the freeelectron fraction as computed here, and x0

e is the free elec-tron fraction left over after recombination. We include

7

FIG. 4. Evolution of the free electron fraction for scenariosthat account for the resonant conversion of dark photons ofmass mA′ = 10−12 eV and various kinetic mixings.

in xe the effect of late time reionization by astrophysicalsources using the tanh reionization model (see e.g. [79])with a width of 0.5 and a central value of z = 7, near theminimum allowed given late time observations of reion-ization (chosen so as to be maximally conservative). Anadditional tanh function is included at z = 3.5 to accountfor the second ionization of helium.

In order to assess the robustness of this estimate, wemodify class to include the effect of heating (in addi-tion to that of dark matter depletion, since these effectsmust occur simultaneously to be self-consistent). Onceagain using montepython, we perform an MCMC withthe Planck-2018 TTTEEE+low`TT+lowE+lensing likeli-hood [62]. For models with sufficiently large or latetime energy injection, computing the background ther-modynamics requires increasing the redshift sampling inclass. In order to avoid issues with computation speed,we limit our priors on log10mA′ and log10 ε to be between[−9,−13] and [−12.5,−15], respectively. We show the2σ bound (labeled ‘Dark Ages’) derived from this anal-ysis (solid) when applicable, and extend to lower massesusing the 2σ bound obtained using only the Planck pos-terior on τ (dotted), computed using Eq. (14), in Fig. 3(pink). These are among the most stringent constraintsfor dark photons with masses 10−14 . mA′ . 10−10 eV,losing sensitivity at lower masses as the effect is maskedby astrophysical reionization, and at higher masses byrecombination. Notice that while the extension of thecontour below the ωp(today) is perhaps counterintuitive,it is nevertheless correct – the process of reionizationincreases the plasma frequency such that ωp(today) isslightly above the pre-reionization value.

Similar to our analyses above, we predict that the heat-ing induced via non-resonant inverse bremsstrahlung mayalso yield a strong constraint. However, computing thiscontribution is more complicated than in the case of res-onant conversion due to the fact that the frequency of

electron-ion collisions will induce a feedback effect: i.e.,increasing temperature decreases the rate of energy in-jection due to the Te dependence in ν. We estimate thatthis bound may be a factor of a few stronger than thenon-resonant bound derived from helium reionization atmasses mA′ . 10−14 eV, discussed in the next section,but we leave a rigorous treatment of the implications ofnon-resonant energy injection in the cosmic dark ages tofuture work.

VIII. HELIUM II REIONIZATION

Finally, we address the possibility that dark photonconversion takes place at relatively late times, after bary-onic structures have collapsed and UV and X-ray emis-sion from stars and supernovae play an important role inthe life of baryons. In particular, we focus on the epochin which helium is reionized. Dark photon conversionat this time could lead to an abnormal heating of theIGM. Measurements of the Ly-α forest have been usedto infer the temperature evolution of the IGM across therange of redshifts 2 . z . 6. Convincing evidence of anon-monotonic heating of the plasma of the IGM aroundz ∼ 3.5 [80–82] has been interpreted as evidence of thereionization of HeII. Although the magnitude of this fea-ture varies at the ∼ O(50%) level in recent analyses [82–84], a consensus seems strong that the IGM was heatedby no more than ∆T . 104 K ' 0.8 eV. Since the major-ity of this heating is surmised to come from the partialionization of helium atoms, bounds on anomalous heat-ing of the IGM of size ∼ 0.5 eV per baryon in the range2 ≤ z ≤ 5 were presented in [82]. Anomalous heatingof the IGM on a comparable level can be constrained forredshifts extending to the end of hydrogen reionization,occurring near z ∼ 6 [85, 86].

In this work, we will impose a conservative limit

∆ρ2≤z≤6A′ ≤ 1 eV×nb, (15)

where nb is the total number density of baryons. Onlya small fraction of baryons at these redshifts are con-tained in collapsed objects, so we approximate nb by thecosmic average [87]. We consider both resonant and non-resonant absorption of dark photons, as in Eqs. (4) and(5), corresponding to conversion to an on-shell photonor off-shell inverse-bremsstrahlung, respectively. For alldark photon masses mA′ . 10−14 eV, this turns out tobe the strongest constraint on the dark photon parameterspace. Thus, dark photon dark matter that could poten-tially be heating collapsed structures such as the MilkyWay (as suggested by [29]) or its satellites (as suggestedby [30]) would in fact also have unacceptably heated theIGM at redshift 2 ≤ z ≤ 6.

For the range of dark photon masses coinciding withthe SM photon plasma mass in this redshift range, mA′ ∼10−13 eV, this bound is stronger than previous cosmo-logical limits [11] by 5 orders of magnitude and stronger

8

than bounds on local collapsed objects [29, 30] by 4 or-ders of magnitude. We note that these bounds will scalequadratically in ε, so the bound for ∆ρA′ ≤ 0.5 eV×nbis trivially obtained by rescaling our HeII limit by

√2.

IX. CONCLUSIONS

In this work we have revisited cosmological constraintson light to ultralight dark photon dark matter. Since thedark photon mixes with the SM photon, this dark mat-ter candidate is subject to plasma effects such as reso-nant photon-dark photon conversion and Debye screen-ing, making its phenomenology more diverse than con-ventional cold dark matter candidates. We have de-rived novel constraints that cover a far broader massand mixing range than previously appreciated. We showthat very simple and robust cosmological bounds arisingfrom the non-resonant evaporation of dark photons con-strain masses as low as ∼ 10−20 eV. Very strong boundscan be attained by requiring dark bremsstrahlung pro-cesses not significantly heat the IGM at redshifts forwhich Ly-α forest measurements probe the epoch of he-lium reionization (i.e. 2 . z . 6). We also demon-strate that resonant bounds derived from helium andpost-recombination reionization significantly strengthenexisting bounds in the range 10−14 . mA′ . 10−9 eV.Collectively, the bounds derived here robustly excludelarge regions of previously unexplored parameter spacefor light dark photon dark matter.

One point not directly addressed here, but perhapsworth serious consideration, is the role of plasma inho-

mogeneities in resonant dark photon conversion. Cos-mological studies to date have assumed the plasma fre-quency is well-characterized by a mean electron num-ber density. This naive assumption likely works quitewell when mA′ ∼ ωp, where ωp indicates the cosmologi-cally averaged value at a given redshift; however, electronunder-densities that inevitably exist within the plasmashould allow for dark photons with mA′ < ωp to reso-nantly convert, a process which is strongly suppressed.The necessary existence of such under-densities impliesresonance constraints, typically much stronger than theirnon-resonant counterparts, extend to a much broadermass range. Depending on the abundance and distribu-tion of these under-densities, it may be possible to derivefar more stringent constraints in the low mass regime.We leave the prospect of understanding the role of con-versions in inhomogeneities to future work.

Acknowledgments: We would like to thank PrateekAgrawal, Jeff Dror, Olga Mena, Sergio Palomares-Ruiz,Matt Reece, and Lorenzo Ubaldi for various discus-sions and their comments on the manuscript. SJWacknowledges support under Spanish grants FPA2014-57816-P and FPA2017-85985-P of the MINECO andPROMETEO II/2014/050 of the Generalitat Valenciana,and from the European Union’s Horizon 2020 researchand innovation program under the Marie Sk lodowska-Curie grant agreements No. 690575 and 674896. Thismanuscript has been authored by Fermi Research Al-liance, LLC under Contract No. De-AC02-07CH11359with the United States Department of Energy.

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Appendix A: Photon Optical Depth

Here, we briefly discuss the fate of photons producedfrom the resonant conversion of non-relativistic dark pho-tons. We are interested in studying the resonance thatoccurs in both the transverse and longitudinal modeswhen ω ' ωp ' mA′ . Since both modes are on-resonance, both modes will be produced in appropriateratios, i.e. one-third longitudinal and two-thirds trans-verse. Longitudinal modes, however, don’t propagate

and are thus immediately absorbed by the plasma. Fortransverse modes, one must compute the optical depthalong the direction of propagation in order to determinewhether or not these photons can be treated with the on-the-spot approximation (i.e. they are absorbed instan-taneously). For the energies studied here (Eγ ≤ 10−2

eV), the relevant process dictating the mean free path ofa resonantly produced photon is simply bremsstrahlungabsorption (also known as free-free absorption). The in-tegrated optical depth from production at zi to some finalredshift zf is given by [43]

τBR(Eγ , z) =

∫ zi

zf

dzΛBR(z, Eγ)(1− e−Eγ/Te(z))

(Eγ/Te(z))3

× σT neH(z)(1 + z)

(A1)

where ΛBR = (αλ3c/2π

√6π)np θ

−7/2e gBR(Eγ) is related

to the bremsstrahlung emissivity, and Te is the temper-ature of the plasma. Here, λc is the electron’s Comptonwavelength, θe = Te/me, np is the proton number den-sity, and gBR is the bremsstrahlung Gaunt factor, whichwe take from [88] (see also [89] for a more generalizedtreatment of soft bremsstrahlung processes).

In Fig. 5 we show the optical depth for a photoncreated with energy mA′ at the redshift of resonance(i.e. we take ωp(zi) = mA′) and taking zf = 10. Weadopt zf = 10 rather than e.g. zf = 0 because thepost-reionization epoch requires a detailed description ofreionization and evolution of the IGM, which is stronglymodel dependent. The conclusions drawn here, however,are entirely independent of these details. The coloredcircles in Fig. 5 denote the point of production. Asis clear, the change in τBR at the point of productionover a narrow range of z is always large, regardlessof the dark photon mass, and consequently we alwaysexpect resonantly produced photons to be absorbedinstantaneously. Notice that if dark photons are rel-ativistic at conversion, they do not necessarily suffersuch large optical depths. Such dark photons cannotthemselves constitute dark matter, but, as shown in [90],they may nonetheless have cosmological consequences,such as explaining the anomalously large absorptiondip observed in the 21cm spectrum by the EDGEScollaboration [91].

Appendix B: CMB Spectral Distortions

Here, we provide a brief description of the origin ofspectral distortions of the CMB due to energy injected toor extracted from the SM plasma. The interested readercan find a more extensive discussion in [50].

Around z ∼ 106, photon production from DC andbremsstrahlung become inefficient at producing high en-ergy photons, although at lower frequencies equilibriumcan still be maintained. Compton scattering, however,

11

maintains kinetic equilibrium with the SM plasma; thisimplies a blackbody spectrum cannot be established. Thepartial efficiency of thermalization processes are suchthat the photon distribution can be well-described bya Bose-Einstein distribution with a frequency-dependentchemical potential. For this reason, spectral distor-tions of this sort are known as µ-type. At lower red-shifts, namely 103 . z . 104, Compton scattering losesefficiency, implying kinetic equilibrium can no longerbe maintained. That is, photons injected from inversebremsstrahlung tend to stay, at least approximately, lo-cally distributed near the frequencies at which they areinjected. This results in a lower (higher) temperature

decrement at lower (higher) frequencies, and produceswhat are known as y-type distortions.

In the epoch between 104 . z . 105, there exists acomplex interplay of processes such that the distortionsare not purely µ-type nor y-type, but rather a complexadmixture. For example, i−type distortions, which aredistinct from both µ- and y-type [51, 92], uniquely ap-pear during this epoch. Determining the implications ofenergy injection during this period on the spectrum typ-ically require a complex numerical study; however, sinceour formalism neglects i-type distortions, we caution thereader that the constraints derived result in a somewhatconservative estimation of the sensitivity.