JHEP06(2020)132 Springer June 7, 2020 April 7, 2020 June 22, 2020 eV), empha- : : : 12 − c 10 . Received Accepted forest, and show that 0 Published , A α m . eV 15 − -type spectral distortions, and show y Published for SISSA by https://doi.org/10.1007/JHEP06(2020)132 Samuel D. McDermott b [email protected] , [email protected] . 3 , Salvador Rosauro-Alcaraz, d a The Authors. c

Dark photon dark matter will resonantly convert into visible photons when the , [email protected] eV, potentially by several orders of magnitude. Finally, we discuss implications 14 − 10 [email protected] Universidad Aut´onomade Madrid, Cantoblanco, 28049,Theoretical Madrid, Astrophysics Spain Group, Fermi NationalBatavia, Accelerator Laboratory, IL, U.S.A. LUPM, CNRS & Universit´ede Montpellier, F-34095 Montpellier, France E-mail: Instituto de Fisica CorpuscularValencia, (IFIC), Spain CSIC-Universitat de Valencia, Departamento and de T´eorica Instituto F´ısica IFT-UAM/CSIC, de T´eorica, F´ısica b c d a Open Access Article funded by SCOAP strength. We then projectto sensitivity measure for the near-future 21cm cosmological transition surveysthat in that neutral these are hydrogen experiments hoping prior to will∼ reionization, be and extremely demonstrate usefulfor in reionization, early improving star sensitivity formation, to and masses late-time near sizing both the importance ofcosmological inhomogeneous observations energy to injection, the as inhomogeneitiesmodified well themselves. as constraints the More on sensitivity specifically, of darkthe we presence photon derive of dark inhomogeneities matterrange allows that one from would to be the extend obtainable Ly- in constraints the to homogeneous masses limit, outside while of only slightly the relaxing their surrounding gas. Existing work in thising field the has implications been of predominantly focused thesethe on resonant Universe understand- can transitions be in treated the asin limit being the that perfectly the electron homogeneous, i.e. plasma number neglecting frequency density.dark inhomogeneities of photon In dark this matter work in werelevant the focus for presence of on dark inhomogeneous the photons structure implications with (which of masses is heating particularly in from the range 10 Abstract: dark photon mass is equal tocontexts, the this plasma frequency transition of leads the ambient to medium. an In extremely cosmological efficient, albeit short-lived, heating of the Samuel J. Witte, and Vivian Poulin Dark photon dark matterinhomogeneous in structure the presence of JHEP06(2020)132 2003.13698 Cosmology of Theories beyond the SM, Thermal Field Theory ArXiv ePrint: resolve signatures unique tofantastic the potential light for dark a photon positive dark detection. matter scenario, andKeywords: thus offer a that probes which are inherently sensitive to the inhomogeneous state of the Universe could JHEP06(2020)132 3 10 ] for reviews on axions). An alterna- 2 8 , 1 – 1 – refs. [ 23 5 27 e.g. 3 23 ]. From the perspective of particle physics, a particularly simple 21 22 13 – 18 3 1 25 18 observations of the epoch of HeII reionization α 5.4 Late-time spectral distortions 5.1 Reionization 5.2 Star formation 5.3 Dark Ages energy injection 2.1 The homogeneous2.2 Universe The inhomogeneous2.3 Universe Impact of inhomogeneities on resonance gravitationally bound objects.conditions For many has years, been thetive leading the candidate possibility axion that which (see satisfies haslight these gained vector bosons increasing [ interestdark is matter that candidate dark that matter has is a comprised non-trivial of coupling to the Standard Model is a dark for the existence ofyears, physics there has beyond been the anretical Standard increasing prejudice interest Model in in (SM), how the and is darkstone where matter unturned’ still we community has lacking. to search carried remove for with theo- Infrontier. dark it recent matter; a Dark the renewed interested mentality matter in ofbe with the ‘leaving sufficiently a low-energy no cold, / sub-eV high-intensity and mass bosonic must in be order non-thermally to produced fit in the order abundance to observed inside low-mass There is overwhelming evidencetrophysical for scales the based existence solely of onand dark its experimental matter gravitational efforts on influence. over a theof Despite wide dark past extensive matter, four variety theoretical of should decades they as- fication however, exist, non-gravitational of have signatures the yet true to nature be of robustly dark identified. matter, which Therefore is the among identi- the strongest pieces of evidence 1 Introduction 6 Conclusion A Comparison to previous work 4 21cm cosmology 5 Discussion 3 Lyman- Contents 1 Introduction 2 Dark photon conversion in the presence of inhomogeneities JHEP06(2020)132 ] ]. & 5 21 z , 8 ]. Field 26 – 24 ] when the dark 30 ]. These proposals – 17 28 ]. This can be avoided 18 ], and the second relies on , of resonance, is a function 14 9 , – 8 11 time 15 orders of magnitude between big bang ], as shown in refs. [ ∼ 28 , – 2 – 27 , and thus the ]. More recently, a number of novel production p 9 ] because the norm of the vector field, and thus the ω 7 eV. These proposals can be broken into two categories: eV [ µ 21 − & 10 0 A ∼ m the inflation during reheating) [ ]. This also has interesting implications for the recent claim by 22 . While this is likely to be approximately valid at redshifts e.g. , a quantity which scans p ω ]. redshift 23 ], and cosmic voids offer a cosmological laboratory by which these bounds can 2 [ , which kinetically mixes with the SM photon via the renormalizable operator 21 / 0 µ , 8 A 0 µν F In this work, we investigate the extent to which the presence of inhomogeneities mod- The vast majority of previous work on the cosmological implications of this resonance Phenomenologically, light dark photons are unique because the efficiency with which Historically, one of the concerns that has limited the appeal of this candidate was its µν the EDGES collaboration ofline the at observation high of redshift an byphotons anomalous neutral are absorption hydrogen not dip [ cold. of the 21cm ifies the energy injection arising from both resonant and non-resonant conversion of dark at any instant in(larger) time, than the the plasma mean frequencyof plasma in masses frequency voids to of (halos) the undergo canvoids, Universe, resonant be thus as conversion. significantly allowing bounds This smaller for derived is amasses in of [ wider particular the range importance homogeneous in limitbe the typically smoothly case have extended of an [ abrupt edge at low 100, when almost alllower redshift density strongly perturbations violates remainfor this linear, experiments assumption the with and sensitivity formation canimplication to of have of ultra-light structure wide-reaching properly dark implications at photonresonant accounting dark constraints for will matter. the extend One presence over immediate of a wider inhomogeneous range structure of is masses that — this is simply because, visible photons, interactions are resonantly enhanced when their mass ordering changeshave [ effectively assumed that the— plasma the frequency of simplifying the assumptionexclusively Universe is is of homogeneous that [ in the late-universe behavior of light vector boson darkthey matter. interact withrounding SM medium, particles dependsnucleosynthesis strongly (BBN) and on today. the As plasma a result frequency of of the kinetic the mixing sur- between the dark and the fuzzy dark matter scale the first of these exploitsto a a tachyonic decaying instability scalar that ( the arises fact when that the cosmic dark stringsexhibit photon may a couples preferentially wide radiate range dark of possible photons early-universe [ phenomenology, leading to renewed interest by introducing a non-minimalof coupling to introducing the instabilities Ricciexcitations in scalar, induced the but during this longitudinal fix inflationabundance; mode comes were however, of at shown this the the to mechanismmass cost dark be over-predicts of photon primordial capable the [ gravitational of darkmechanisms waves were producing photon if shown the to the be correct efficient in producing dark photons of sub-eV mass down to  F production mechanism. Unlike theefficiently axion, produce the light misalignment vector mechanism bosons cannot [ energy be density, is used to efficiently diluted in the early Universe [ photon JHEP06(2020)132 (2.1) (2.2) ], and forest) α 21 is , 8  . ) At leading order, the 1 and mixing . homo res 0 p t A ω − , m ) t ' ( ], which in the limit of small z 0 ( δ A γ 31 1 − m → 0

is the time at which the resonant homo 2 A  P res t d dz homo p ) dt z ω (  ln d – 3 – homo CDM

ρ 0 eV. Finally, we discuss implications of inhomoge- 2 A ) = (assuming that the probability of conversion is much m 14 ω ] z 2 − ( 6 γ π 10 → time 0 ' dz ∼ homo A ) 0 z ]). The limit obtained here is valid only for a non-relativistic and non- dρ ( A γ 33 m , → 0 homo 32 A is the probability of conversion and we have assumed (as we will P is the plasma frequency and d dt refs. [ /dz ) z homo p ( e.g. ω γ → 0 ), A 2.2 dP More generally, the resonant conversion involves equating the mass with the real part of photon 1 In eq. ( condition is met, which is determined by the condition self-energy (see degenerate plasma. less than unity) isconversion probability given is by given the by [ Landau-Zener formula [ where throughout this work) that thenon-adiabatic conversion dark of photon a accounts dark for photon,conversion the probability which is per entirety the of unit scenario dark of matter. interest, the For differential 2.1 The homogeneousThe Universe differential energy densityhomogeneous per (‘homo’) unit Universe by redshift a introduced dark to photon the with SM mass photon bath in a We first begin bysimplifying reviewing assumption energy thatthen injection the generalize from Universe this dark can formalismstructure. photon be in dark the treated matter following as subsection under homogeneous to the account [ for inhomogeneous unique cosmological signatures thatdark can photon be dark used matter. to constrain or confirm the2 existence of Dark photon conversion in the presence of inhomogeneities Dawn may be able tofor improve upon dark cosmological photon constraints by masses neous a heating few during orders the of epochformation magnitude of of the reionization, first the stars,homogeneous impact the of limit extent dark should to photon be whichfrom heating constraints consider an on derived robust, excess the using heating and the of late-time CMB the spectral in IGM. the distortions This work induced provides novel insights into interesting and constraints derived from excessduring heating the of epoch the ofto intergalactic helium account medium reionization for (IGM) (obtained inhomogeneities, priorconstraints using showing over to observations that a wider and of the range the presence ofscopes Ly- dark of aiming photon structure to masses. serves measure We to the then show 21cm broaden that differential future brightness radio temperature tele- during the Cosmic photon dark matter. We illustrate the potential importance of structure by generalizing JHEP06(2020)132 ) ], ] to 2.4 35 (2.9) (2.7) (2.8) (2.3) (2.4) (2.5) (2.6) , 21 34 ), with forest. ) can be α ~x,z 2.7 ( p ω . ) through ( 3 ) 2.1 , ) = ] 0 z A . ( homo res z m ! − , ) 3 z homo p ) ) ( ω (1 + . 3 0 homo e ) 2755 K the CMB temperature homo p T homo homo n res . k ω z 2 T , ( 2 (3) homo , res 3 EM − ' π ) ζ π z sign[ 0 α z 0 2 4  2 A ( − in a homogeneous Universe inherits T η 2 δ ~x,z ) m s z ( πα e p 2) z ( e e given by 0 4

( / ω δ m 2 A p m 0 homo b homo CDM ) Y A ], and m ρ n log z πα n ( 0 homo p homo m − ) 4 36 ) = 2 A 3 ν ω H z ) 245 is the primordial helium abundance [ (  3 – 4 – m s . π )(1 homo e 2 0 ) H homo n res z 3 ' homo ( π z k ' ) = ( ) homo T res p , the redshift at which a dark photon of a given mass H 2 EM z ( ' z ( Y ) ( e ~x,z ) homo γ z ( e homo z π α ν m p → ( homo res 2 2 n 0 ], even if the dark photon mass is small compared to the ) by writing ω z  γ homo homo e A q √ x → P 3 21 0 2.2 4 2(1 + , homo b A homo d dz n ρ = 10 ' ) = d dz γ homo res homo homo , z → ν 0 ( IB A P is the baryon-to-photon ratio [ homo e n is the baryon number density in a homogeneous Universe. Eq. ( 10 − . The probability of absorption from this process in a homogeneous medium can ) we have introduced 2 ]. 10 is the free electron fraction, ) b homo p × 2.4 e 37 ) representing the spatial average. Since n 1 x . /ω 6 As described in refs. [ Anticipating the scenario of interest below, it is trivial to extend eqs. ( 0 ~x,z A ( ' p m with the frequency of electron-ion collisions local plasma frequency, darkbremsstrahlung photons process, can with deposit the energy( rate into of the energy medium depositionbe via being written an suppressed as inverse by the ratio where integrated over a redshift range ofenergy interest injected to obtain per the baryon homogeneousconstrain result at exotic for a the energy specific given injection by cosmological dark epoch, photons which using was observations used of the in Ly- ref. [ to consider the energy deposited per baryon as a function of redshift: Here, η today [ undergoes resonance. This is given by solving where the homogeneous electron number density is given by time-dependence solely via the effectwe of may the further expansion simplify rate eq. on the ( electron number density, In eq. ( which in theω context of a homogeneous Universe reduces to plasma frequency is given by JHEP06(2020)132 ) ≡ b ) in 2.1 (2.12) (2.10) (2.11) 1000. 2.2 . Thus, b ). z < , we find ∆ 1 × − ) homo res ) z , z , ( )  b b z < z ∆ homo e ) is small. Accounting b n 1000)km/s at z, e dz ln ∆ ( ∆ x z/ γ d ) z, → ) = b 0 ( + γ A ∆ z . P → ~x,z 30(1 + 0 ( × 2 3 e A ) d ∼ dz 2 b n z and therefore also the conversion P 1 + ] term has additional contributions v δ ) b . In particular, resonance will be b  d b − dz dz ) ) to the case of energy injection and ∆ ∆ . The differential probability of dark ) /dt z homo b (1 + p below the minimum plasma frequency √ z 2 p z, ω ( 2.7 ) ( ω × z ) ∆ i.e. = ∆ ln z ln([ P )(1 + ( CDM d b d z ρ ) to ( (1 + ) ∆ 1 / dt b dz CDM 1) grows proportionally to (1 + H b the b homo p 2.2 – 5 – δ ∆ ω ∆ ) and in halos at later times ( − ) and eq. ( ' ' d b z, (ii) ( 1 + 2.1 dt ) = dx Z ∆ (∆ homo res 2 p ) ). This allows us to write P = b z ≡ ω b the typical distance travelled by the dark matter per ~x,z ( b ( ∆ b ), and dx ∆ p ln b δ 2 z > z d z, ω d dz ( ln ∆ homo CDM (∆ ∆ ' Z ρ + d 0 P res t = A dt dz ) = m 2 p → z ( ω γ dz ln → will depend on the local value of ∆ 0 homo res d the time at which dark photons undergo resonance now depends on the such that dark photons over a broad range of masses will be capable of t dz A γ p = → dρ ω 0 (i) 2 p A ω P dt ln d , we may decompose the electron number density as b More quantitatively, we generalize the energy injection per unit redshift from eq. ( For instance, this allows us to ignore the bulk relative velocity ¯ ρ 2 / b ; alternatively, one can understand that the effect of inhomogeneities is to induce a spatial where in the last step wefor have assumed the that fact the that time-dependence of an overdensity generalize the time derivative term in eq. ( local overdensity, from the time-dependent evolutiondark of photon over-densities and as baryon well fluids.conversion as Since after the we recombination, focus relativeHubble on motion time the of is case the extremely of small. non-relativistic dark Furthermore, in photon the presence of inhomogeneities, one can where in the second linephotons we have converting assumed ∆ per unitseveral redshift ways: in the inhomogeneous case differs from eq. ( by introducing a probabilitybations density at function a characterizing given the redshift baryonic density pertur- undergoing resonant conversion atρ any given redshift.the Introducing resonance the condition overdensity ∆ probability occur in voids at earlier times ( We now extend this formalisminhomogeneities to broadens the case the of resonance anable such inhomogeneous to that Universe. undergo a resonant The dark conversion presencez over photon of a of redshift a interval rather givendependence than mass of at will a be fixed value of specific energy injection for the inverseThis bremsstrahlung process process is in much a homogeneous lessonly Universe. for efficient dark than photons that with extremely ofof low resonant masses the conversion, ( Universe). and is thus of interest 2.2 The inhomogeneous Universe It is straightforward to generalize eq. ( JHEP06(2020)132 . ) , ) res there (2.18) (2.17) (2.14) (2.15) (2.16) (2.13) z b res per unit z − ∆ 0 − A . high m z monotonically )) high 0 z A )Θ( , m ) Θ( low b z for a given . low . (∆ − z )) 0 2) res − A res z / z p for a particular overdensity res − Y , m z b ] gives the energy injected per z )Θ( . − res ) to emphasize that, for a fixed ( b 0 z δ (∆ A ∆ ) Θ( high b , /dz , b res ) (1 0 2 , z z ] , m ∆ ~x 2 A res ) ∆ b ( , z b b − m low ( z res ∆ (∆ z ∆ ∆ homo p z = ( ) ( P z, ω res δ ) z ( ∆ ) 0 z ( ) p ∆ ln[ A P , ~x z res P d ( ) m z m ) res which is a function of redshift. The differential b ( 2 – 6 – homo b ) H z res e ρ ( 3 ∆ res z π e res res ( z d ) m ) that allows the resonant conversion to take place ( z 0 homo b ) is never more than one-third as large as the part ( ' A n Z z, ~x ) H ) b ( π α n homo CDM 0 homo CDM , m e 2.12 ) ρ 4 A res b ∆ term, and is typically much smaller. In what follows, we ρ x z z b ( m ( z, (∆ 2 ∆ ( H H b γ /dz ) = 3 res π 2 b ] → ∆ z ) 0 d ∆ z z, ~x A ( ) over a redshift interval [ d ( P e homo Z p n d Z ω dz 0 2.16 homo CDM for simplicity. Analogously, one can write the specific energy injected 0 A ρ ln[ 0 A m 3 A d ' 2 m 3 2 ) m ); we discuss the implications of inhomogeneous reionization later in sec- π z z ( ( π e γ = unique redshift ), we have introduced the notation ¯ x and = → 0 b ); explicitly, ' dz A high low . Notice that, for a fixed dark photon mass, one can equivalently conceptualize z z 2.13 | ) high low dρ ρ z z | 2.10 Integrating eq. ( The differential probability of conversion for a dark photon with mass on ∆ 5.1 ∆ , defined as the energy injected per unit volume per unit baryon, as z, ~x inj ( ε e inj res z ε unit volume, where Θ is the Heaviside step function, and we have dropped the explicit dependence of eq. ( This provides the generalization of the homogeneous energy injection rate. We will assumex throughout this worktion that thethis free as having electron a fraction resonantenergy overdensity injection is ∆ per homogeneous unit volume per unit redshift can then be directly determined with where the electron number density can be expressed as (modulo the effect ofmagnitude reionization, of which the we inhomogeneity, discuss the local below),with physical since, redshift baryon regardless density (until decreases of becoming theand non-linear). dark initial photon The mass is value given of by solving In eq. ( dark photon mass, thebaryonic overdensity. resonant transition An occurs essentialwill exist at observation a of a our redshift work determined is by that the local coming from the choose to neglect these small corrections to redshift is therefore Thus, the contribution from eq. ( JHEP06(2020)132 ]. 40 – (2.22) (2.23) (2.19) (2.20) (2.21) 38 , ] ] 0 0 A A on the over- m m , − k − . ] b b 0 T ∆ A ∆ ! √ m √ b ) ) − z 3 z b ( ( ) )∆ , ∆ ; when only adiabatic z 2 , z ( √ β b ) b homo p ) homo p ) 2 z ∆ ω 2 σ ω 1). ( σ (∆ homo e 1+ k ∝ − n sign[ √ sign[ b homo p k b   ω π T ∆ T b b ] to be minimal over the range ( 2 log(1+ 4 3 (∆ EM ∆ δ ∆ α log 22 sign[ 2 2 ) 0  − ) 0 s → z b e 2 A z 2 A ( ) (

b ∆ b m m 1 2 ∆ ∆ ) 0 log ) , homo p z homo p 2 A 2 ( b ω ω res σ given by ). Explicitly, the energy deposition rate is m  z  ( ν 3 ) ) ∆ ) 2.8 ) homo p ) ∆ z z – 7 – ) b ( ( , z ω P , z 1 b ) 3. As in the case of resonant conversion, the rates , z ∆ b  H z / b ) log(1 + ( ) (∆ z, homo e z (∆ ( π ν , z (∆ k n = 2 b 2 ∆ ) ν T b ) homo b P 2 β e p z (∆  n b 2 EM ∆ ( ) ν m ) 2(1 + z ∆ ) z, ) z ( b ( π α z ( p ) = × ( 2 b from the baryon number density. The homogeneous limit can , z homo CDM H ∆ 3 ∆ H b ρ ) b P d √ ∆ ) 2 z b  z 4 (∆ z, homo CDM Z ∆ ( ρ P ) b 2 ∆ b  z ) = P ∆ ( b 2(1 + ∆ d 2(1 + d ∆ dz Z dz z, homo CDM Z ( ρ × ν Z × Z ' = ) is the variance of the density field. In general, deviations from this simple = z γ ( 2 inj → γ σ 0 ε → A 0 ρ The probability distribution function (PDF) characterizing the baryon overdensity is It is also possible to generalize the expressions for non-resonant absorption of dark dz (IB) A ∆ dρ where parameterization are expected, but were found in ref. [ the final ingredient necessary to describematter the density energy field injection from is darkWe will photons. known assume to The here dark that approximately onthe follow scales underlying sufficiently a larger dark log-normal than matter the distributionassume density Jeans the scale [ distribution. the baryon Thus, baryons overdensity track for PDF the is given sake by of being concrete, we respectively. and cooling is relevant, the solutionof is energy injection and specific energy injection are Here, we have included andensity. explicit Generically dependence one of expects the the matter temperature temperature to obey with the frequency of electron-ion collisions lation with the factor ofbe ∆ straightforwardly recovered by taking photons via inverse bremsstrahlung in eq. ( The explicit dependence on the overdensity parameter has dropped out due to the cancel- JHEP06(2020)132 the (2.24) (2.25) ) that the log- Jeans kR ( λ 1 W ], and an analytic 46 – 50 are small on the 41 , ), obtained from the ) 7 the Universe is fully . b , with adiabatic index that differences in the p z ∆ k, z ( , z < k /m P . k ) T class z, T = 1; that is, the relative width ). B ( s kR c ( γk res 2 8 3 2.24 ) and at p 4 ], including: a log-normal distribution, a W 12 due to the complicated nature of r − ) as a function of redshift for three dark = 22 ) . z 10 ], where the gas temperature contains s ] ) c 2 res z z k, z × ], and likely could not have started before 52 ( 52 ( π 12 the free electron fraction of the Universe 2 2 ) in eq. ( . P 53 H (1 + k 3 ∼ π k z > ,T – 8 – b = ]. = 6 [ k dk ∆ J , π ]), while very large smoothing scales reproduce the 49 z – k 2 z, ] prescription, with a window function R Z ( 22 47 51 [ ) = ) = k Jeans ref. [ λ ,T b R, z = ( e.g. ∆ 2 for further discussion). We overlay on figure figure R σ z, Halofit ( 5.1 Jeans evaluated at the redshift for which ∆ Thus, we restrict our attention to this range of over-densities in this λ b the mean molecular weight [ 3 . µ 2 ] with the we plot the resonant overdensity ∆ 10 ]). We omit redshifts between 7 50 1 [ ≤ 56 – , and b is the speed of sound which in general depends on redshift, baryon tempera- µ 54 ∆ s 3 ) by convolving the non-linear matter power spectrum c / ≤ class 12 [ In figure 2 Various types of overdensity PDFs were considered in [ R, z 3 = 5 ( − & 2 reionization (see section normal PDFs for ∆ Gaussian distribution, a distributionapproximation extracted based from on a spherical hydrodynamic collapse simulations [ [ photon masses. We assume that atis redshifts consistent with the pre-reionizationionized value (these ( boundaries are notthe so Universe well-known, must though be current fullyz observations reionized suggest by that In this section weenergy briefly injection. illustrate Our the primary impactover goal that which is energy inhomogeneities to injection have takes provide on place,capable the dark as of reader photon well undergoing with as resonance an an as idea idea structure of of which begins the dark to timescales photon form. masses are homogeneous limit. Welinear have baryon also spectrum verified from explicitlyscales the of dark using interest, matter and powerFor at spectrum these larger at reasons, redshifts we the take result tends toward2.3 the homogeneous result. Impact of inhomogeneities on resonance an implicit dependence onbaryon distribution redshift is and expected over-density.to to the differ On presence significantly scales of fromaccurate. smaller pressure that forces, than of Taking and the slightly thus dark largerenergy the matter injection smoothing history adopted owing scales (see power typically spectrum has will a become minimal in- impact on the where ture, and over-density. Inγ the following, we take Since we assume inmatter, we our adopt formalism the smallest thatvalid: smoothing the this scale is for baryon which the distribution this Jeans assumption traces scale, is which that expected is to of given be by the [ dark work, which in turnσ translates into a conservativecode result. We compute thesmooths mass the variance distribution on the scale 10 JHEP06(2020)132 12] have , [7 . The lines 0 ∈ A z for further dis- m eV. We plot the 13 5.1 − , one can see that the b = 10 ∆ 0 evaulated at the redshift(s) A √ ∆ m ∝ P and 58, and we illustrate using p , ω 54 , 50 , 46 , = 42 z – 9 – 30 spans roughly a factor of two in ∼ ) for various dark photon masses. Redshifts z z line with the edges of a particular band, one can approximate ( res res ) (black line) for a dark photon with z ( res , while the one at 0 A m . Evolution of ∆ . Resonant over-density ∆ = 1) are shown for comparison. b (∆ res width of these PDFsPDFs valid in at redshift a hasby single the no redshift). vertical physical size The of meaning broadeningPDF the — PDFs. of at they the small Using the PDF redshifts simply scalingmagnitude at accommodates represent relation in low resonant 1-D transitions redshifts spanning is roughly reflected one order of log-normal overdensity PDF (bluered regions) bands the at intervals containingIdentifying the 50% intersection and of the 90% ∆ the of fraction the of PDF energy (assuming injected equal in weight a in particular each redshift tail). interval. of these distributions displays the relative value of the PDF (we stress that the absolute Figure 2 Figure 1 been removed to avoidcussion). complications Normalized associated log-normal with probability reionizationz distribution (see functions section JHEP06(2020)132 ); ∆ P line 2.17 res cosmology. α 5 eV / baryon . 3, while in the 0 ∼ ≤ z ∆ 3. For masses larger  − weighting in eq. ( b eV. We overlay 5 PDFs = 1. 13 ; this bound, however, assumed res − 1, where in both cases a majority 15 − ∼ ) energy is injected inhomogeneously i z 10 = 10 ( ∆ 0 = 1). For dark photons that experience × and 56. We highlight in red the regions A  b ] was able to constrain the kinetic mixing 2 , m 21 (∆ 52 . , res for z 50 – 10 – ] for a review). Exotic heating of the IGM during , res 57 46 [ , required to achieve resonance with the dark photon b , which shows that most of the energy is injected for e.g. = 42 2 50% of their energy in an interval z ∼ observations are not uniformly sensitive to all phases of the α ) Ly- 100, with the baryon power spectrum having a significantly smaller ), and it does not account for the extra ∆ ii ]. Using this bound, ref. [ the evolution of ∆ z ( 58. This result is typical of models which have resonance in the range ( ∼ 62 2 – H . observations of the epoch of HeII reionization z z 58 α ]. The former of these effects can be treated using the formalism derived in . 6 [ eV the redshift range over which energy is deposited decreases, albeit quite ; we address the proper treatment of the latter effect below. ) and 50. We have explicitly verified that dark photons undergoing resonance prior to . 13 observations requires two primary modifications to account for the presence of z 64 , ( − z . 2.2 α 63 z . In order to illustrate the rate at which energy is deposited for a particular model, CDM . ρ on Ly- inhomogeneities: one must account forinto the the fact that IGM, and IGM [ section of the temperature of theto IGM astrophysical during uncertainties (see and afterthe the epoch post-reionization of epoch reionization, can arefor therefore quite 2 be robust constrained toof the dark level of photons undergoinga resonance homogeneous to Universe, be which is not exact. A more precise derivation of a bound based 3 Lyman- Recent years have shown significantVarious progress analyses have in shown the that field these of observations, high-redshift which indirectly Ly- probe the evolution than 10 slowly; this occurs becausesignificantly the by dark matter andvariance baryon than power that spectrum of the beginprecisely, dark as to matter. it diverge does However, we not do have not any attempt significant to impact quantify for this this effect study. of the energy injectionresonance is at centered very around late times,and part thus of the the total energy energydark injected injected photon “should” is masses take suppressed place capable relative in tothis of the the resonantly suppression future, homogeneous converting is case, in but typically the (for never homogeneous much limit more by today) than a factor of 2 using the 90% intervalredshifts in 43 figure 10 reionization typically deposit post reionization epoch this interval decreases to of the PDFs whichcontains contain the 50 remaining 25 and and 90% 5%,over of respectively. which 50% From the or these, 90% weight, one of can defined theoverlaps estimate energy such a the is that particular timescale injected by each red finding of regionof the range — the over this tails which the is ∆ notthis exact method as serves only it as neglects a the reasonable redshift proxy dependence for more exact solutions. This is illustrated of the mass indicated byline that provides color. a visual The solution location to where the these lines redshift cross forwe the which show dashed ∆ in black figure characterizing overdensities at in this diagram show the value of ∆ JHEP06(2020)132 α (3.2) (3.3) (3.1) spec- α . ) produce large res b z − )) = 1. We illustrate z ( 2)Θ(6 S , − 95 . ) b 0 observations are sensitive to res ∆ 100%, one loses sensitivity to z ≤ α 5 eV/baryon. z, , . ( ) τ 0 ∼ b 100% absorption, no line will be 95 05 )Θ( − . . ∆ b e ≤ 0 0 z, b ∼ ( ∆ τ , > < ∆ ) based on the absorption probability ) − ) ) b ) res e b b b observations will be extremely sensitive z res ) such that Max( ∆ ∆ ∆ ( z b ) ≤ ∆ z z, z, ( α ( ∆ z, ( ( ∆ ( τ τ z, res P A H 05 spectrum observed at a given redshift is only ∂ ( z . S − − ) ) ( b e e frequency as it travels toward Earth, one can α ∂τ – 11 – res ∆ z α ) , ( z homo CDM ( res ρ optical depth. First, we assume that the Ly- 0 10 0 observation on the thermal state of a particular over- z A ( homo b α        α n ) = b × S b ) = ∆ b ∆ d ∆ comes from the fact that we take the derivative with respect to ) that is constrained to be z, ( b Z z, ( ). Specifically, we adopt the following S 3.1 0  S ) is the Ly- observations to the temperature of the IGM is characterized by a A b − observations directly depends on the transmitted flux, it stands to m α 3 ∆ 2 lines whose height and width characterize the properties of the IGM (1 α z, π α ( ≤ τ ), and thus the energy injection which Ly- ]. For an alternative approach, we adopt a sensitivity function that varies ) = b b forest is an absorption phenomenon that occurs when light produced from α 61 ∆ ∆ α z, z, ( Ly- ( τ , and we choose the normalization ε b S − , where e ) b ≤ ∆ density, temperature, etc.). If the photons traverse large over-densities they will be We make two choices for the function The Ly- z,  ( τ − e.g. proportional to the derivative of the absorption probability. In this case, we take where the extra factor of ∆ the log ∆ reason that the sensitivity ofdensity a depends Ly- directly onthe the extent density to field which isto the a varied absorption particular (that probability inhomogeneity changes ischanges if when to in small absorption perturbations say, probability). Ly- about Concretely, that this density means ∆ adopting a sensitivity function where the sensitivity thresholdsobservations are [ (roughly) basedsmoothly on with the overdensity. signal-to-noise Since ratioextracted the of from extent Ly- Ly- to which the properties of the IGM can be trum is only sensitivei.e. to scales for which absorption is neither too large nor too small, It is the quantity in eq. ( e sensitivity of the Ly- function is given by observed here at Earth;over-densities responsible consequently for it near-total is absorption.densities not Conversely, will photons possible easily traversing to under- passthe characterize through; properties the if of properties the theextracted. of transmission IGM Thus, is as one well, expectssensitive since that to the there a Ly- is finite no range line of from inhomogeneities. which information To can account be for this, we assume that the distant QSOs (quasi-stellar objects,the or spectrum quasars) redshifts passes throughobserve through many the neutral Ly- Ly- hydrogen.( Since preferentially absorbed, and in the extreme case of JHEP06(2020)132 x ]. = n ν 65 , the (3.8) (3.6) (3.7) (3.4) (3.5) φ , , and 4 is the 0 − ν 0 61 = 2 and i.e. , 10 b ν z 58 ∼ ] optical depth b = 5 [ 2] typically lie . 64 α , 1 for D ν [6 3 , ∼ ∈ 2 γ  z 0 D ν = 6 will have dropped ν − is the neutral hydrogen ∆ z ν  H − , . e n ) b  transition, ∆ D 1. ) . ν ∆ z 0 α H ∆ z, and central line frequency ¯ n z ( ∼ π , via τ H ν √ (1 + ). Since the width of the line profile ) 9) frequency at b b b φ . ) ) to ensure observations match the re- eff  z /m τ σ , α z D ∆ of photons to a given process is defined as k ( x ν 1 = 1 T τ z, τ ∆ 2 ( Erf z 1 ( ) ∆ d` n lu p ) (1 + z f P eff ( z H 2 b Z τ e – 12 – ( = ¯ n H observations in the interval ∆ m b H πe = 2 d 9993. Consequently, we can approximate the entire α . τ e.g. . Converting the line of sight integration to a redshift 0 = lu dz Z ν f D σ ν e 2 = 5 ∆ emit m ) = z πe z / z 0. This issue can be resolved using the method described 0 ) ( ] to fix 0 Z ' ν ∼ eff i.e. 63 0 τ τ lu − H f ν b ν n 2 e is the temperature at the mean density and ], which is in good agreement with the approximation adopted in 0 m πe = ( ], we assume that the temperature of an over-density scales like T 59 x = 65 6 [ , τ 3 by a redshift 416 is the oscillator strength of the Ly- . The cross section for resonant line scattering is given by [ . . ], which relies on renormalizing ∼ 61 , with b , d` , where π ∆ 1 64 58 = 0 , √ − is photon frequency at a particular redshift, and ¯ / b γ . is the cross section for a photon to scatter from a species with number density 63 ) lu 2 ). ∆ ). Generally speaking, the optical depth 5 ν f σ 0 . b x T 3.3 − ∆ In order to compute the sensitivity functions, we must estimate the Ly- is not a known quantity; to leading order the Universe is fully ionized at these red- is the line profile which we model here as the Gaussian core of a Voigt profile, = z, = 6. Optimal over-densities for Ly- H ν ( ¯ sults of hydrodynamical simulations.determined In by other defining words, an this effective optical unknown depth normalization can be and using the result of ref. [ T This problem is complicatedn by the fact thatshifts, the and average thus local ¯ na¨ıvely neutralin hydrogen refs. fraction [ contribution as being local, Following [ along the line ofis sight extremely (which narrow, is thesource. proportional For absorption to example, ∆ is given that dominatedexponential typical term by Doppler parameters for a are a narrow onby photon the a redshift which order factor region has of the near Ly- the exp( integration, we find that the optical depth can be expressed as where where doppler width with Dopplerφ parameter where along a path z near 0 eq. ( τ the behavior of these sensitivity functions in the right panel of figure JHEP06(2020)132 4 α , ) is 4 ; the 4 3.1 10 ≤ = 6. 1, and thus ] to account b we plot the eV or much z ] and used the ∆ & 4 21 ] recently per- 13 ]. Here, we have b 61 − ≤ 22 64 4 10 − , where we plot the × = 2 and 3 4 z , from [ 1 to 10 5 when ∆ − ' 2 γ b 0 & ∆ 10 ). We see that the effect of A 0 z in figure T m ], which in principle can be used to ≤ 3.3 τ b 66 ≡ ) (red) for − ) (assumed to be the default sen- e ∆ ) b 3.3 3.3 tail. Note that ref. [ ≤ (∆ 0 2 A T − m . In the right panel of figure 2 for redshifts A 6]. These constraints are shown in figure . , 0 forest flux as computed in ref. [ [2 . α

– 13 –

��� lxAbsorbed Flux ∈ ) , defined as τ z ) (blue) and eq. ( γ − 3.2 and 0 �� ), we extend the constraints obtained in ref. [ photons as a function of overdensity for various redshifts. Forest Ly-α T α 2.22 ) from eq. ( ) = 1 (dashed) or is given by eq. ( , z � � b eV will be heavily suppressed. b ] to normalize the optical depth at each redshift. Right: adopted Ly- Δ ∆ shows that exp( 14 63 (∆ ) (dubbed ‘Flat window’). For the default sensitivity function, we also − 3 z, S ( ) and eq. ( 3.2 10 S × 3.1 ���� ]. observations are effectively insensitive to energy injection in halos. We conclude 11

. Left: absorption probability of Ly-

α 3 = z 2 = z 6 = z 5 = z 4 = z lxTransmitted Flux Using eq. ( We illustrate the behavior of the suppression factor

����

We do not show bounds derived from black hole superradiance [ ��� ��� ��� ��� ��� ���

� 4 τ - The field of 21cmhydrogen cosmology in aims the to understand Universe the by spatiotemporal studying evolution the ofconstrain hyperfine neutral ultralight transition dark between photonsdependent with the [ masses ground near and this range, as the existence of such bounds is model assuming either structure and the sensitivitynon-resonant function absorption have bounds. a minimal effect on the4 net sensitivity of the 21cm cosmology show the effect of increasing thedifference integration is from negligible 10 everywhere exceptformed the a low- similar analysis, howevercomparison reached of a the rather two approaches differentbounds in result from appendix — non-resonant we absorption make in a both detailed the homogeneous and inhomogeneous limit, for the presence oftion. inhomogeneous structure, This both is forequal done resonant to and by 0.5 eV non-resonant requiring / absorp- baryonassuming that in the the the sensitivity observable interval functions energy aresitivity) given injected and as eq. given in ( in eq. ( eq. ( Interestingly, figure that Ly- that constraints on darksmaller photons than 5 with masses larger than taken the redshift dependencemeasurements of in ref. [ sensitivity functions absorption probability of Ly- Figure 3 JHEP06(2020)132 is -�� b �� that (4.2) (4.1) i y pumping. CMB α -�� �� ]. , and using two differ- [��] 4 -�� 21 �� 6; obtained by requiring �� 10 � . ≤ z b ∆ ) (green). These constraints are . -�� ≤ �� 3.3 forest (purple) assuming a homoge- 4 , as . k − α k T Inhomogeneous Homogeneous T , . Focusing our attention on neutral ) α s α B or 10 y y -�� /T 2 -� -� -� -� �� /k

+

�� �� �� �� + 21 ϵ 10 21 T k k y − y ≤ hν e b -�� ) (blue) and eq. ( – 14 – + ( 1 + �� ∆ ≡ ). Constraints are derived assuming the PDF of ∆ = 3 R 3.2 ≤ , defined via T 1 0 21 observations between 2 3.1 s 2 n n T T − ' α s ]). Right: non-resonant constraints derived with and without CMB -�� T �� 21 [��] -�� Ly-α �� �� � represent the excited and ground state, the factor of 3 is the degen- 0 -�� n 2 4 �� 10 ≤ 10 ≤ b b , and the (kinetic) temperature of matter ] for more extensive reviews. Δ ≤ Δ ≤ r -2 -4 and T 10 10 window Flat 5 eV/baryon, as defined in eq. ( . Left: constraints from Ly- 69 . 1 – 0 n ). Grey region denotes parameter space excluded using the CMB [ -�� 67 �� < -�� -�� -�� -�� -�� The amount of absorption or emission of the gas is determined by the relative occupa-

3.3

α �� �� �� �� �� ϵ Ly- The spin temperature cancharacterize be the expressed efficiency in ofradiation terms each of of effective these coupling processes, coefficients the temperature of the background eracy factor thehydrogen excited in state, the and IGMble during of the changing dark thespin ages, temperature. ratio only of These a are: groundlisional small excitation/de-excitation, spontaneous and number and emission, of excited indirect stimulated processes states, absorption/emission, excitation/de-excitation are col- or via capa- Ly- equivalently of changing the tion number of the groundwith and the excited so-called states, spin a temperature quantity which is typically parameterized where took place, and the intensityhydrogen, provides such various pieces as of itsfraction. information temperature, on We the review number the state density, basics ofrefs. line-of-sight of neutral [ velocity, 21cm cosmology and below, ionization but refer the interested reader to eq. ( first excited state (thisto is measure also the known evolution asobserved of the line the “21cm” provides intensity direct transition). of information on The the the goal, redshifted epoch simply 21cm at put, line; which the is the absorption frequency or of emission an ε log-normal and is valident in sensitivity the functions range as 10 compared divided with in those eq. derived ( neous from Universe the (as CMB derived (grey) ininhomogeneities, and ref. and Ly- [ including (solid) and excluding (dashed) the default sensitivity function in Figure 4 JHEP06(2020)132 (4.3) (4.4) the differential and absorption i.e. cmb ) that the differential > T s 4.4 , T and constrain the maximum mK , 0) if )  cmb 0 ν > s τ ]. b cmb T ). We devote the remainder of this − ) one can see that the spin temper- < T 70 e T δT k 4.4 4.2 − − T 1 ref. [ i.e. (1 , respectively) are positive semi-definite ] will measure the 21cm power spectrum  α b 25. Radio interferometers have a clear ad- y 73 [ cmb z e.g. in eq. ( ∆ T . 6 k – 15 – z − and T 1 + + 1 10 s k z T → y s r T ) = HI ν x ( ] and SKA b 27 ] claimed the first detection of the global 21cm differential 72 δT rather than intensity itself (the two are directly related by , ' 27 couplings ( b b 71 . Furthermore, from eq. ( [ T α δT 5 cmb < T s T 15), one can immediately infer that & is the neutral hydrogen fraction. It is clear in eq. ( 0) if is the optical depth of the 21cm line. Since the optical depth is small, the z < 0 HI ν b τ x δT The simplest and least expensive 21cm experiments are comprised of single antennas Radio experiments searching for the redshifted 21cm line are only sensitive to the rel- https://reionization.org/ https://www.skatelescope.org/ 5 6 i.e. low multipoles, which aremore easy information to than remove. thepowerful In differential tool addition, brightness in the temperature, constraining power and exotic spectrum thus physics. contains can far Incorporating be the a effect more of inhomogeneous with the ΛCDM prediction. Consequently,differential in brightness this temperature work we isometers will as assume such yet that as unknown. the HERA truefrom In 21cm reionization the to near redshifts future,vantage as radio over high interfer- single as dish antennas in that smooth radio backgrounds contribute only to heating induced by dark photons should they observe thethat 21cm attempt signal to in measure absorption. the the EDGES sky-averaged collaboration differential [ brightnessbrightness temperature. temperature. At Recently, theof moment, the hot validity debate of this due measurement the is complicated still nature a matter of foreground removal and its incompatibility ature is bounded to beAn between immediate the consequence temperature is ofpected the that, for CMB if and the the 21cm mattertemperature signal temperature. of is the observed IGM insection by absorption taking to (as investigating is the ex- extent to which future 21cm experiments can constrain the where brightness temperature will be seen( in emission ( where exponential term can beline expanded, of yielding sight a equal differential to brightness temperature along a the CMB intensity.brightness Following the temperatures convention inthe radio Rayleigh-Jeans astronomy limit to ofbrightness work the temperature) with blackbody can effective relation), be the expressed as 21cm signal ( quantities whose definitions can be found in ative difference between the intensitywhich produced by is these expected transitions to andbody. radio be background, dominated As by such, photons the in signal the is Rayleigh-Jeans typically tail expressed of as CMB a black- differential measurement relative to The collisional and Ly- JHEP06(2020)132 , ] ]), ) = /dt 77 b 74 , b (4.5) (4.6) ∆ ]). 76 (∆ [ d k 16 T , ]), while at low mK , ) x-ray heating and 79 Recfast++ ,  He ) f ) ) we neglect the term 78 z , z e.g. , + ( b ] and the other being a e inj 4.6 21 x (∆ cmb 73 Q s 2 T T b (1 + b ) ∆ n b 3 ∆ = ]: z, e ( ), which in most cases will be an overly dt x 74 ∆ , z e P b K, the cooling rate of the medium changes x b k 4 T (∆ ∆ 10 k d 1 + T – 16 – ∼ Z + 0 mK across a range of redshifts. The model is k →  b (similar sensitivity estimates have been made using ] to understand the expected contribution of x-ray T ) ≤ ∆ i − dt 75 b d , z 1 b b k  δT T h ∆ = 1. For high redshift resonances (left panel), it is difficult (∆ ref. [ s ]. We assume that these experiments measure absorption 2 3 b + 1 T 10 82 − z – e.g. k 50 mK or r 80 ), we can account for the presence of inhomogeneities in the glob- ) for ∆ HT HI , since we are focusing on the epoch prior to reionization where the x 4.4 4.6 ), as well as the exotic energy injection from resonant conversion. + 2 i ≤ − /dt b e k 4.6 = 27 dt δT dx i dT h b we show the evolution of matter temperature relative to that of ΛCDM and 5 δT is the heating rate per unit volume (which includes h is the mean adiabatic temperature. In solving eq. ( inj (which reproduces the solution that when only adiabatic cooling is relevant [ ad 3 Q T / 2 b In order to project potential sensitivity of 21cm experiments measuring the global In figure Generalizing eq. ( ∆ ad brightness temperature to dark photonconfigurations, dark one matter, we consistent adoptlunar-based with two radio an potential array experimental SKA-like [ at experiment the [ level of to significantly elevate thefor matter which temperature 21cm observations above will that soonthe of exist. threshold the for This CMB is collisional because during ionization and if the the one epoch net heats the heating mediumdifferent. saturates. above One The can right easily panel heatwith shows the the that medium ionization at above threshold. late the CMB times without the encountering story any is issue quite to the CMB temperature for(right) dark redshift. photons which For undergo the(including resonance at a high high contribution redshift (left) from resonance, andredshift collisional low we we solve ionization solve eq. as ( this in using refs. [ proportional to free electron fraction is slowlyas changing. this We term also for neglect typicaladiabatic the overdensities cooling. term is proportional expected We to to dohand be explicitly side small include relative of the to eq. Compton the ( cooling contribution contribution of in the right where Compton cooling as wellT as dark photon heating).where We adopt initial conditions conservative estimate (see heating by stellar sources). Weat can each then possible solve overdensity for using the evolution of the matter temperature where we have includedand the over-density. explicit Since dependence we arewe of can interested the make in spin the determining substitution temperature the on maximal the level of redshift absorption, postpone this to future work.ferential brightness In temperature this study, wethe instead average focus differential brightness on temperature the globally in averaged the dif- homogeneous limitally in averaged ref. differential [ brightness temperature [ energy injection from dark photons in this case is, however, rather involved, and thus we JHEP06(2020)132 . This shows 6 .  assuming homogeneous energy injection at k eV and various values of T = 5). Right: evolution of matter temperature at 14 ] with an amplitude less than or equal to some − z 10 max – 17 – , z × min z = 3 [ 0 A ∈ m z 50 mK). − = 1, for b . Left: evolution of matter temperature . Parameter space that could be excluded should an experiment observe the sky-averaged (modeled using Gaussian with a width ∆ z Finally, it is worth mentioning that near-future radio interferometers will hope to to probe the existence of dark photon dark matter. move well beyond thethe globally 21cm averaged power spectrum. differential brightness Farspectrum more temperature than information in and is the contained measure global ininhomogeneous signal, the way. particularly evolution when Consequently, of energy radio the injection interferometers power proceeds will in provide a a largely great opportunity in this redshift range.such a In statement reality, would constraints rely maylittle on is be complicated currently astrophysical significantly known. modeling stronger Potential atthat than future high 21cm sensitivity this, redshift are observations but where illustrated could indark be figure photons. extremely useful in extending sensitivity to lower-mass Figure 6 21cm temperature inthreshold an (being interval here 0 mK or assumed to be falsifiable if the dark photon heats the medium above this level at all points high low redshift, assuming ∆ Figure 5 JHEP06(2020)132 ) & as- iii z ( ) late- i.e. iv ( ) complicates the spa- ~x,z ( e x ], until the bubbles grow and merge, at ) the expectations for resonant conversion i ( 87 – 84 [ – 18 – 08), while the IGM outside the bubbles remained . 4 1 − ∼ 10 ) potential implications for star formation rates, e ii x ∼ ( e x ] does not allow for significant changes to the free electron fraction at ]. On the other hand, the integrated optical depth of the CMB measured 83 53 ]. Between these epochs, UV radiation (likely) sourced from the first collapsed 6 [ 56 -type spectral distortions due to excess heating of the IGM. – ∼ y Here, we briefly sketch two possibilities for approximating the effect of energy injec- The dramatically inhomogeneous nature of reionization has implications for resonant 54 z and non-ionized regions? tion from resonant darktheoretical photon uncertainties conversion associated during tothis this modeling section epoch. should reionization, not the be However, interpreted constraints given as obtained the robust. in large Instead, this discussion is intended merely suming the co-existence ofis a consistent fully with ionized the mediumneglect standard with the background a boundaries evolution medium (without betweena whose reionization ionized reasonable ionization sources), and approximation level and given non-ionized theneutral regions. medium. short mean The We question free believe then path becomes: that of this how the should is one ionizing map photons over-densities in to ionized the dark photon conversion, since the inhomogeneoustial structure and of temporal understandingthe of globally averaged the value resonance. ofmasses the allow Naively, free for one electron resonant may fractionwrong conversion be to during since understand tempted the reionization which reionization to is darktreating epoch. use photon an the Unfortunately intrinsically Universe this inhomogeneous during is this process. epoch A is better to work attempt in at a two-phase approximation, the Universe is fully ionizednearly (with unaffected. The free electronby fraction the outside pre-reionization of the value which ionized point bubbles reionization would is be complete. given Understanding how this process beganStill, and there evolved are is some currently featuresionizing an which photons active appear were area consistent likely of among producedpaths leading research. in theories: in over-densities the namely, and the neutral hadimagine media relatively that short that reionization mean they proceeded free were via ionizing. the formation Consequently, of to ionized first bubbles, order inside one of can which in the spectrum ofby quasars provide convincingby evidence Planck that [ reionization had15 completed [ objects is expected to permeate the Universe and rapidly change the free electron fraction. time 5.1 Reionization The epoch ofbeing reionization predominately refers neutral to to fully the ionized. period Measurements during of which the Gunn-Peterson the trough Universe evolved from In this section weappear comment in on this a model. numberduring In of the particular we additional epoch discuss signatures ofexpected reionization, and modifications features due that to may bounds derived from the CMB optical depth, and 5 Discussion JHEP06(2020)132 (5.1) (5.2) . ion of the Universe ) is determined − ion pre e, 2.14 V x during the epoch of = e e x x

) , z in eq. ( ) ( γ ion res → V z 0 dz A − dρ (1 ) ion ion − V pre − e, defining the boundary between ionized and x , while the right region shows the evolution of the 5. + 4 . + (1 − – 19 – thresh 08 ion . 10 = 0 V ) z ∼ =1 e e He x

x f 08. The mean free electron fraction is assumed to be a tanh ) . 1 z ( is the value of the free electron fraction assuming ionization ∼ γ (1 + e → 0 ion x ∼ dz A − e means that the resonant redshift dρ x pre e, ··· x = ion e = 10 with width ∆ V x | z ) = z 08 and ( . γ 0 → 0 ∼ dz A . Probability distribution function characterizing distribution of plasma frequencies in the He dρ f Adopting the two phase approximation, and assuming we have a measurement of the to be produced in collapsed objects,outwards. meaning At they the originate from converse over-densitiesdensest extreme, and objects, expand we and might thus proceedscenario expect from as ionization over-densities the to to ‘strongly first under-densities.redshift occur inhomogeneous’ the We threshold in scenario. will of the refer over-densities In to ∆ this this case, we can identify at each Here, the notation with a particular value‘homogeneous’ of reionization. the As free previously electron mentioned, however, fraction. we expect We ionizing photons will refer to this scenario as the where has not modified the evolutionunder-densities of and the IGM. over-densities Thedifferential are simplest rate assumption equally of one energy likely can injection make to is of be that ionized. This amounts to a global free electron fraction, oneis can estimate ionized what via volumetric fraction predominantly ionized regions function centered at as an exercisereionization to might illustrate affect the how bounds considering derived inhomogeneities elsewhere. in Figure 7 IGM during reionization, assuming theassumed ‘strongly to inhomogeneous’ progress scenario from (in over-densitiesof to which the under-densities). reionization predominantly is neutral The medium left region captures the evolution JHEP06(2020)132 − 5. . 13 (5.3) . − = 0  10 z 08 ion . ∼ − 0 =1 . e A pre ) x b e,

m x ) ∆ b = e ∆ z, x (

) z, ∆ b ( P γ ∆ b → 1 eV/ baryon. We adopt 0 z, ionized at same rate), or in ∆ ( ], which is consistent with A eV the difference in these d b 10] is less than 1 eV / baryon, < γ P , 88 14 → 0 [6 d − inj dz = 10 and of width ∆ A ε all ∆ ∈ with time. b thresh 10 P z z ∆ d . i.e. dz ) ∆ Z b 0 b thresh A ∆ ' m ) ) ∆ z, b ( between . The bounds obtained for ion ∆ ∆ V 8 inj P z, ε b ( − ∆ ∆ d – 20 – P b ∆ d thresh ∆ Z ) or (1 b thresh + ∆ ∆ ]). We take the requirement , where we plot the evolution of the PDF characterizing z, Z ( 61 7  ∆ ) P z b ( ∆ d homo CDM ρ thresh ∆ ) = Z z ( eV are similar, but for smaller masses ' γ 12 → . Bound that could be derived assuming 0 quasar measurements. We plot the bounds obtained for both the homogeneous − ion 10] (which would also assume that the temperature evolution of the IGM is well dz A V , z 10 dρ [6 − For both reionization scenarios discussed, we derive an illustrative ‘bound’ on dark × during reionization in the strongly inhomogeneous scenario, assuming the mean free ∈ p a recent modeling of thehigh mean free electron fraction fromand ref. strongly [ inhomogeneous scenarios infew figure treatments can be significant. We have highlighted the twocut phases in via the the distributions illustrates labels the ‘non-ionized’ evolution and of ‘ionized’. ∆ photon The dark sharp matter byz requiring that theunderstood, IGM which is it is not not over-heated [ in the redshift range An important consequence ofa the gap strongly in inhomogeneous theThis scenario evolution is is of illustrated the theω PDF appearance in characterizing of figure the plasmaelectron frequency fraction of is given the by Universe. a tanh function centered at non-ionized regions by solving In this case, one can write the differential rate of energy injection Figure 8 and assuming reionization proceeded eithera homogeneously strongly ( inhomogeneous manner (ionizing over-densities first, and under-densities last). JHEP06(2020)132 4 is 10 µ is as (5.5) (5.6) (5.7) (5.4) & J . λ  z 10 1 +  3 / 1 ,  2 2 ) z π 1) , ( c 3 18 − ∆  , one can relate this threshold ) m m,z b m,z . vir Ω Ω ∆ T Λ 2  z, ( 39(Ω ' 3 3 / J ) + Ω 2 k − λ z T 3  )  1) 1 z ) − b − (1 + h m h

z, δ (1 + ( m,z Ω M ] – 21 – ¯ M ρ m ] 8 π 95 Ω 3 4 10 94 , =  + 82(Ω 70 ) =  2 b 22 for a neutral medium). The virial overdensity based m,z . π 22 ∆ Ω 1 µ . 1 z, ∼ (  J ) = 18 ) is given by [ K z M z 4 ( ( c c 10 ∆ . As before, we will assume that the baryon density approximately × 8 . 2.2 4 ' ] in the near future. ) , z 90 h is the average density in the radius of interest, and the Jeans length , ], as this is the threshold for which molecular hydrogen cooling becomes efficient. M ρ 89 99 ( – If dark photon heating is active near the epoch of star formation, however, it may be Star formation is typically expected to be efficient when the kinetic temperature of We conclude this section by emphasizing that a detailed treatment of reionization will vir 96 T where ¯ defined in section possible that thermal pressuresay, if prevents the the medium formation iswhere of heated star star to forming a forming halos sufficientmay halos. would degree, occur, never the we That have Jeans compute is collapsed. mass the to may Jeans To increase mass assess to as the a a extent level function to of which redshift this and overdensity via Efficient star formation is onlyK[ expected to proceed in halosWhile with there virial temperatures is uncertainty inis both clear how that to a treat sufficientto this number efficiently of threshold produce collapsed and star objects where forming exactly with regions. it these lies, masses must it exist in order where We have assumed the neutral hydrogenis fraction approximately is given given by bythe the that IGM mean of value, molecular the a halo weight truncatedon profile ( isothermal spherical collapse sphere, ∆ and we recall that subsequent collapse, of gas becomes efficient.is Assuming approximately the given kinetic by temperaturedirectly the in to the virial halo the temperature host halo mass via [ servations of high-redshift star formationstar do formation not rate yet exist, mightwhich we be might simply be suppressed, outline testable and here usingscope how the observations the [ from interesting 21cm regions telescopes of or parameter the space James Webb tele- the gas in a gravitationally bound object exceeds some threshold at which the cooling, and be needed to move beyondsimulations these to two understand extremal the examples, correlations and between will ionized likely regions5.2 require and numerical over-densities. Star formation Dark-photon-induced heating may have strong implications for star formation. Since ob- JHEP06(2020)132 , ], vir 67 the M 9 eV and 14 − 25], assum- , ; thus, given 10 over the entire 4 cooling becomes [15 × 2 vir ∈ H z ). The temperature > M b over this interval, star Ω J / M vir This final assumption will M 7 CDM . b (1 + Ω K). Should b 4 ∆ b ρ = ¯ eV. The minimum virial mass required for star the evolution of the Jeans mass in an over- ρ 14 9 − – 22 – 10 over the entirety of the interval × ]. vir min 90 , 89 > M J , one can determine the redshift-dependent evolution of the b K. Typically, star formation is expected to begin near redshifts 1 at these redshifts should be extremely rare, and thus we expect the curves M . 4 b  b -weighted average over these curves. However, if high-redshift 21cm ex- b 25, we expect star formation can be notably modified. Right: estimated sensitivity . when the virial temperature of a halo is 10 z = 1 as a function of redshift, assuming a dark photon mass 3 25, and thus if the Jeans mass sufficiently exceeds . b 1 to be the most relevant to future surveys. . Left: comparison of Jeans mass as a function of redshift assuming energy injected at i.e. . & b z We compare in the left panel of figure Stars in regions with ∆ = 1 from a dark photon with mass 3 . 7 b perhaps the environmental dependence canbe be a isolated. detectable We signature alsocome emphasize in online that the in this James the could Webb near Space future Telescope [ (JWST) which is likelynear to ∆ as those models for which ing star formation islikely observed not in be an met in isolated mostsomething over-density experiments, of ∆ and a in ∆ reality oneperiments should in expect the the distant sensitivity future to find be themselves capable of achieving 21cm tomography [ with a temperature of 10 15 formation rates can bedark dramatically photons altered. on thedark In star photon order parameter formation space to rate, capable assess we of the highlight suppressing in potential star formation, the impact which right of we panel define of here figure Jeans mass for any dark photon candidate. density ∆ various mixings. We compare thiswhich mass is scale obtained with by the assuming redshift-dependent the virial matter mass profile is given by a truncated isothermal profile follows that of theevolution of dark a matter, particular overdensity andsolving in write equations the presence for ¯ of the darka evolution photon of large-scale heating the is overdensity matter obtained ∆ by temperature as in section ∆ formation is shown in blackefficient, dashed line (assuming starinterval 15 formation onsets when to the modification of theover Jeans and mass assuming under-densities modification ∆ to star formation is observed in various Figure 9 JHEP06(2020)132 30. ∼ z 10 at ∼ z it occurs over a short i.e. 5 (solid), 5 (dashed), or . = 10 is large relative to the z = 0 ) there is nearly no difference 0 z A it is clear that the presence of m 2 sweeps through the overdensity PDF. = 10 energy injection actually produces large z res we plot the evolution of the free electron i.e. and figure 10 1 – 23 – , we expect the differences here to produce tiny changes  ], but we argue here that such concerns are unwarranted. 22 ] that strong bounds can be placed on dark photon dark matter 5. From figure . 21 ] that strong constraints can be derived on dark photons that res- ]. The reason is simply that the resonant timescale is short relative = 0 21 21 z ] was treated assuming the redshift dependence followed a Gaussian 21 The second concern is related to potential back-reaction. In the case of a homogeneous There are two potential causes for concern. The first arises from the fact that the energy It was also shown in ref.resonantly [ converting into photons prior to recombination from the non-observation of rate, and therefore weprocess expect is beyond back-reaction the to scopeto be of future this negligible. studies. work, and A we proper leave a treatment detailed of exploration5.4 of this such effects Late-time spectral distortions resonance, but ratherHowever, by in the order ratemust for at reionization which back-reaction in ∆ to someproduced over-density occur in take in this place process on the mustthe short quickly inhomogeneous electrons time and are scenario, scales, efficiently always diffuse but not non-relativistic, to free we only larger electrons expect over-densities. this Since to be slow relative to the Hubble Universe it was shown that thereto can prematurely be end no [ back-reactionto which would that cause of the collisional resonance complicated, ionization. because The the case resonance of timescale an is inhomogeneous not Universe is dictated slightly solely more by the width of the resonance redshift itself, andon partially the because maximal level the of efficiency ionizationthe of obtained peak during cooling ionization the fraction processes energy is injection depends largerextremely process for sensitive (notice the to that narrow the Gaussian). value of in Since the the energy derived injection limit. is fraction for two differentmodeled dark with photon a parameters, Gaussian10 assuming (dotted). distribution the in energy For the injection redshiftbetween high all can with redshift of be ∆ the resonance curves. ( larger At asymptotic low redshifts, values the of ∆ sets the of curves free partially electron arises fraction. from the The fact difference that a between value these of two ∆ injection in ref. [ distribution with ∆ inhomogeneities broadens theThis energy concern injection is such easy that to it address. spans ∆ In figure period of time, afterHowever, the which asymptotic atoms free are electronof allowed fraction ΛCDM. to will This be cool), can significantlytremely and be larger sensitive strongly some to than the constrained atoms in free using electron the willbound the fraction case recombine. was between CMB recombination recently since and criticized reionization. the [ optical This depth is ex- It was shown in ref.onantly [ convert during theefficiently deposit dark their ages. energy intion baryons, Dark heating threshold the and photons medium subsequently above converting causingof the during the a collisional medium homogeneous ioniza- this to Universe, the epoch prematurely energy re-ionize. will injection In is not the sustained case ( 5.3 Dark Ages energy injection JHEP06(2020)132 ], 91 with [ (5.8) z 7 − 10 . ∼ 6 in order to ) . cmb z T . − ) ) , one would naturally ) to be much stronger b z (∆ 5.3 k T ( ) (1 + z b , assuming reionization begins , although this number should ( 7 4 − H ) ∆ recombination constraints will likely be stronger z z 10 ( α . × 5 z homo e n . e i ≥ T c σ m y h ) b reion z – 24 – ]. In order to heuristically estimate what type of ∆ 1 eV / baryon for redshifts 3 93 z, ∼ ( ∆ -type spectral distortions at the level of P y b ∆ d ]. 21 Z ] or PRISIM [ dz assuming energy is injected following a Gaussian distribution in 92 e x max z min z Z = ], which when averaged over ∆ yields i c 91 y 10 (note this is probably a rather optimistic assumption, but is sufficient for h . Change in z < 5, 5, or 10, and for two masses and choices of mixing. Computation includes collisional . For heating that occurs at redshifts = 0 type distortions can still be induced after recombination; consequently, it is in general parameter [ z − We define this signal tobe be understood observable when purelymay as alter a the rough ΛCDM expectation estimate, by as a factor uncertainties of associated a with few (exploring reionization the detailed degeneracies sensitivity these experiments might have toevolve exotic the heating temperature from dark of photon theonly conversion, IGM we at as described inillustrative section purposes). We theny compute the contribution of this heating to the Compton be injected in orderwhich to to produce look dramatic for changes.already these Perhaps expected effects the to is most at contributewhich promising redshifts to regime is near in slightly reionization. aboveexperiment In like the ΛCDM, PIXIE reionization expected [ is threshold for a futuristic CMB spectral distortion than those arising from late-timethe spectral IGM distortions. at a Similarly, oneproduce level must much observable greater inject spectral heat than distortions, into in implying this Ly- regime.cosmological uncertainty. At In addition, lower the redshifts, IGM is one much hotter, suffers meaning more from energy must a vast array of astrophysical and spectral distortions iny the CMB. Shouldalso the possible gas to be constrain dark heated photons to from late-time a heatingexpect sufficiently of constraints high the from IGM. level, early reionization (discussed in section Figure 10 ∆ ionization as described in ref. [ JHEP06(2020)132 on 3 ], revisiting the 21 ] which had neglected 21 50. Accounting for inhomogeneities . z eV. 14 − forest, we derived constraints in section 10 α – 25 – ∼ . Constraint is derived assuming reionization has not yet 7 − 10 ∼ y . Depending on the details of reionization, this method may provide 11 10, and dark photons supply the only source of heating. 5 may produce resonant conditions for dark photons with masses spanning ∼ . Projected sensitivity for an experiment like PIXIE or PRISIM capable of measuring ∼ z z Using current observations of the Ly- e.g. energy into) cosmic voids. the excess heating ofthe the presence IGM, of extending inhomogeneities.extend these on constraints Accounting to the masses for work approximately the an of order presence ref. of magnitude of [ smaller structure than were allows us to tion assumes (up tothe effects redshift associated of the withat resonance reionization) and a the one-to-onemultiple dark mapping orders photon of between magnitude. mass, The whilelower implications inhomogeneous masses, are structure as most prominent suchthe for particles homogeneous dark approximation, photons may but with appear in to reality can have resonantly no convert in chance (and to thus undergo inject resonance in robustness of existing bounds andsensitive investigating to novel signatures the which inhomogeneous may features behas arising particularly important at implications forphotons, dark since the photons resonance capablefrequency occurs of of when resonantly the the converting ambient dark medium. to photon visible Consequently, mass the equals homogeneous the Universe local approxima- plasma 6 Conclusion In this work wethe have cosmology investigated the of extent an to ultra-lightmation dark which of photon inhomogeneous the dark structure CMB. matter can This that alter generalizes injects and energy expands after upon the existing for- work [ between reionization histories andthe scope exotic of energy the injectionexperiment current in from work). figure dark We show photonsa the probe is sensitivity of beyond for dark a photons future at masses PIXIE/PRISIM-like below Figure 11 spectral distortions to abegun level of by JHEP06(2020)132 50 is . z forest, respectively. α − eV. We expect that radio in- 14 − 10 ∼ 0 eV. A m 15 − for 10 – 26 – 12, we show that experiments might be able to ∼ 15 − & 10 z ∼ we investigated the extent to which future radio observations of the 21cm 4 In summary, properly accounting for the existence of baryonic structure at Lastly, we have commented on other signatures that may arise at late times. In par- In section Acknowledgments We thank Nick Gnedin,Mishra-Sharma for Hongwan discussions. Liu, SJW Joshfor their thanks Ruderman, valuable Luisa discussions Andrea on Lucie-Smith structureSJW Caputo, and formation and and Andreu and SR the Font-Ribera Lyman acknowledge Siddarth support from the European Union’s Horizon 2020 research and phasize that experiments whichstate themselves are of inherently the sensitive Universe toton could the dark provide inhomogeneous matter, striking and signaturesthey thus that exist. offer are a unique tantalizing to opportunity light for dark a pho- positive detection, should crucial for understanding the extentlight dark to photon which dark cosmology matter. canis probe The that predominant it the effect makes existence cosmological of ofprobe observations including ultra- sensitive lower these to masses, inhomogeneities a while wider relaxing range some of constraints masses, at allowing larger to mass. Furthermore, we em- these regions. Such effectsor may using potentially 21cm be tomography. observable Finally,PRISIM with we (which have the hope shown James to that Webb measureheating future telescope spectral of experiments the distortions like IGM in PIXIE at the redshifts or sensitivity CMB) just to prior may to dark be reionization, photons sensitive potentially with to gaining unprecedented masses the tion, but robust analysisof will ionized require and a non-ionized detailedcalized regions. understanding heating of produced We from spatial havehave dark inhomogeneities also shown photon that commented absorption efficient on can heating the maythreshold, increase actually extent particularly the raise in Jeans to the under-densities, mass which Jeans — significantly mass lo- suppressing above we the star star formation forming rates in spectrum, we defer a complete treatment of observables fromticular, this we epoch have to discussed future theunderstanding work. epoch the of distribution reionization and andshown evolution the that of complications future plasma associated data frequencies. with should In allow this one case, to we study have exotic energy injection during reioniza- sion. By conservatively assuming thatwhich future is experiments observe a the signal genericconstrain in expectation mixings absorption, at for theterferometers level looking of to measureprojected the sensitivity 21cm of power spectrum 21cmof will observations modeling significantly to the improve light the contribution dark of photons. inhomogeneous Due dark to photon the heating complexity to the 21cm power the constraining power obtainedconstrain in larger the masses, homogeneous however, limit.preferentially was absorbed We severely in found limited overdensities. that by the the ability fact to thatabsorption quasar signal fluxes at are high redshift might be able to constrain resonant dark photon conver- obtained in the homogeneous approximation, with only minor (but real) corrections to JHEP06(2020)132 ] b 22 in the ). Our photons, 4 3 α ) is nearly two ] writes down a 22 ], meaning they are 2.25 ) with the adiabatic 21 2.25 ). The formalism of ref. [ 0 ~x ( b . Specifically, ref. [ 3 we show our result from figure ] appeared on the arXiv, presenting ideas 12 22 8 ] were correct, there would be two immediate – 27 – 22 one should neglect the observational sensitivity to ∆ 6, which has large implications for the constraints that ∼ (ii) forest. In figure z α ] for discussions on these topics. ). We believe, however, that this cannot be the case, as the 22 ), and ] uses the baryonic power spectrum of hydrodynamic simulations 3 observations of the IGM temperature (as done in section inj 22 ε α when computing the energy injection per unit baryon, one should neglect 10 kpc at low redshifts. The value found using eq. ( ∼ (i) dependence in the baryon number density, since energy is ultimately shared between b one does not need to be concerned with the optical depth of the Ly- ) does not have any significant impact on a different ∆ In addition to the apparent disagreement about the potential thermalization of the One of the key points lying at the heart of this disagreement is the understanding of ~x We thank the authors of [ ( 8 b e.g. orders of magnitude largercan at be derived usingthick the Ly- black line; wetemperature contrast of this the with IGM the (neglecting impact heating of from using reionization, eq. purple ( dashed). We also at odds with the fact that electrons are non-relativistic. injected energy, ref. [ to determine the Jeans scaleto and be baryon around PDF. The value of the Jeans scale is determined as discussed inphotons section produced in resonant transitions haveinstantaneously short absorbed mean free by paths the [ remain local non-relativistic. medium, and theoccur, In electrons the diffusion order that timescale must absorb for be the small efficient relative energy to thermalization Hubble time, across which seems inhomogeneities inherently to everywhere. If theimplications: assumption of ref.the [ ∆ under- and over-densities (this directly modifiesin the importance the of calculation under- and of over-densities ( the final fate ofthat the injected energy energy. injection In is∆ this a paper, local we phenomenon. haveon worked the That under other the is hand assumption to implicitly assumes say, that energy energy injection injected at anywhere is a quickly given thermalized that overlap with some offormalism those to discussed account in for section theconverting to injection photons of in energy the from presenceconstraints dark of from photon inhomogeneities, Ly- dark and uses matterresults, this resonantly formalism however, are to derive not in exact agreement. 2016-0597. Fermilab is operated byAC02-07CH11359 Fermi Research with Alliance, the LLC United under States Contract No. Department DE- of Energy. A Comparison to previousWhile work this work was being completed, ref. [ 674896. SJW acknowledges support under Spanish85985-P grants of FPA2014-57816-P the and MINECO FPA2017- and PROMETEOalso II/2014/050 acknowledges support of the from Generalitat the Valenciana.the “Spanish EU SR Agencia “Fondo Estatal Europeo de de Investigaci´on”(AEI) and 78645-P; Desarrollo and Regional” through (FEDER) the through the Centro de project Excelencia FPA2016- Severo Ochoa Program under grant SEV- innovation program under the Marielodowska-Curie Sk grant agreements No. 690575 and JHEP06(2020)132 , ]. Phys. 02 , SPIRE IN JCAP ) = 1) (blue, sensitivity is ]. ][ , ]. , z α b K (black solid), or (∆ 4 SPIRE ]. S SPIRE IN -11 IN 10 ][ = 10 ][ k SPIRE T taking IN ][ ]. i.e. arXiv:1510.07633 ]. We find strong differences in CMB [ -12 10 ) using 22 ]. SPIRE 3.3 IN ][ (2016) 1 arXiv:1801.08127 arXiv:0804.4157 SPIRE [ [ IN [��] -13 643 10 Ly-α ][ �� arXiv:0901.0014 � [ Bosonic super-WIMPs as keV-scale dark matter Signatures of a hidden cosmic microwave – 28 – ]. (2018) 89 Microwave background constraints on mixing of photons -14 102 (2008) 131801 SPIRE obtained with eq. ( 10 arXiv:0807.3279 Phys. Rept. IN (2009) 026 4 [ New experimental approaches in the search for axion-like , ) = 1 (blue dot-dashed). A comparison is made to the constraint ][ ), which permits any use, distribution and reproduction in 101 Massive hidden photons as lukewarm dark matter 03 Dark light, dark matter and the misalignment mechanism , z b arXiv:1105.2812 [ (∆ -15 S 10 JCAP -12 -13 -14 -15 -11 ,

10 10 10 10 10

i.e. ϵ CC-BY 4.0 (2008) 115012 Axion cosmology This article is distributed under the terms of the Creative Commons 78 Phys. Rev. Lett. arXiv:0811.0326 (2011) 103501 , [ ] (light brown, dashed). Prog. Part. Nucl. Phys. , 22 84 . Our result from figure (2009) 005 with hidden photons Rev. D background Phys. Rev. D particles A. Mirizzi, J. Redondo and G. Sigl, A.E. Nelson and J. Scholtz, J. Jaeckel, J. Redondo and A. Ringwald, M. Pospelov, A. Ritz and M.B. Voloshin, J. Redondo and M. Postma, D.J.E. Marsh, I.G. Irastorza and J. Redondo, 1 Mpc, and strong differences at high masses when the effect of the Ly- [6] [7] [3] [4] [5] [1] [2] References not included. Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. show the implications of neglecting thedot sensitivity dashed). function ( We comparethe to constraints the derived bound at derived∼ low in masses ref. [ when varying the Jeans scale between 10 kpc and Figure 12 using the adiabatic temperaturedashed of purple). the The IGM resultthe (neglecting is sensitivity function, the also compared heating with inducedobtained the by during [ bound reionization, that would be obtained if we neglect JHEP06(2020)132 ]. ]. ]. ]. , ]. Phys. , SPIRE SPIRE ]. JCAP SPIRE , IN IN SPIRE SPIRE (2020) IN ][ IN ][ IN [ ][ ][ SPIRE 801 coupling to IN [ ]. ]. 6 ]. ]. / ) 2 (2019) 123511 (2009) 123530 (1986) 196 SPIRE SPIRE SPIRE RA SPIRE 99 ( IN IN IN IN 80 (2019) 019 ][ ][ ][ ][ 166 ]. Phys. Lett. B arXiv:1201.5902 arXiv:0809.2779 ]. 10 , [ [ Vector dark matter production arXiv:1905.09836 arXiv:1812.10919 Dark photon oscillations in our [ ]. [ SPIRE SPIRE arXiv:1903.12190 ]. IN JCAP , IN ]. Phys. Rev. D , ][ , ][ Phys. Rev. D SPIRE (2012) 013 , IN Galactic Center gas clouds and novel Phys. Lett. B SPIRE , (2020) 014 ][ 06 IN (2009) 111301 SPIRE Ghost instabilities of cosmological models with Instability of anisotropic cosmological solutions arXiv:1810.07195 IN arXiv:1504.02102 ][ arXiv:1810.07208 arXiv:1810.07196 Vector dark matter from inflationary [ 02 ]. Misalignment & Co.: (pseudo-)scalar and vector [ [ [ (2019) 023001 [ Dark photon dark matter produced by axion 102 Bounds on ultralight hidden-photon dark matter Parametric resonance production of ultralight vector Heating up the galaxy with hidden photons JCAP – 29 – , 100 SPIRE JCAP ]. , IN ][ arXiv:1911.05086 (2019) 015 Cosmological evolution of light dark photon dark matter arXiv:1810.05912 [ On the health of a vector field with [ SPIRE (2019) 035036 (2016) 103520 arXiv:1007.1426 04 First direct astrophysical constraints on dark matter (2019) 075002 ]. ]. IN [ ]. Dark photon dark matter from a network of cosmic strings ][ 99 93 cm signal at cosmic dawn 99 arXiv:1901.03312 Phys. Rev. Lett. Phys. Rev. D arXiv:2002.05165 [ , , 21 , SPIRE SPIRE JCAP (2018) 67 SPIRE ’s and epsilon charge shifts , IN IN An absorption profile centred at 78 MegaHertz in the sky-averaged IN ][ ][ (2020) 063030 ][ (2010) 023 Relic abundance of dark photon dark matter 555 U(1) arXiv:1509.00039 WISPy cold dark matter 11 Vector coherent oscillation dark matter 101 [ Phys. Rev. D Phys. Rev. D Phys. Rev. D Two , , (2019) 063529 , ´ Alvarez, T. Hugle and J. Jaeckel, Nature , arXiv:1810.07188 JCAP 99 [ , (2015) 054 arXiv:0909.3524 arXiv:1907.06243 arXiv:1809.01139 fluctuations [ gravity spectrum supported by vector fields vector fields nonminimally coupled to the curvature interactions with ordinary matter at very low velocities Phys. Rev. D inhomogeneous universe dark matter with curvature couplings [ from observation of the [ Rev. D at the end of inflation bounds on ultralight darksuper-heavy photon, dark vector matter portal, strongly interacting, composite and 135136 dark matter oscillations 12 P.W. Graham, J. Mardon and S. Rajendran, P. Arias et al., M. Karciauskas and D.H. Lyth, J.D. Bowman et al., B. Holdom, B. Himmetoglu, C.R. Contaldi and M. Peloso, B. Himmetoglu, C.R. Contaldi and M. Peloso, S.D. McDermott and S.J. Witte, A. Caputo, H. Liu, S. Mishra-Sharma and J.T. Ruderman, G. Alonso- K. Nakayama, D. Wadekar and G.R. Farrar, E.D. Kovetz, I. Cholis and D.E. Kaplan, A.J. Long and L.-T. Wang, M. Bastero-Gil, J. Santiago, L. Ubaldi and R. Vega-Morales, A. Bhoonah, J. Bramante, F. Elahi and S. Schon, J.A. Dror, K. Harigaya and V. Narayan, R.T. Co, A. Pierce, Z. Zhang and Y. Zhao, S. Dubovsky and G. Hern´andez-Chifflet, P. Agrawal et al., [9] [8] [26] [27] [23] [24] [25] [21] [22] [18] [19] [20] [16] [17] [14] [15] [12] [13] [10] [11] JHEP06(2020)132 ] , . , ]. ]. 707 Mon. Not. SPIRE SPIRE , IN IN (1932) 696 ]. (2018) 031103 ][ Phys. Lett. B ][ , 121 SPIRE astro-ph/0105218 A 137 IN [ ][ Astrophys. J. arXiv:1812.05609 ]. , (2020) 135420 , Baryonic effects on the Mon. Not. Roy. Astron. Soc. 805 , SPIRE (2001) 22 10830 on helium abundance ]. λ ]. IN arXiv:1405.3749 arXiv:1801.08023 [ ][ Room for new physics in the Phys. Rev. Lett. [ 561 The BAHAMAS project: calibrated , ]. SPIRE SPIRE IN ]. IN Big-Bang nucleosynthesis after Planck ][ arXiv:1510.01320 ][ Proc. Roy. Soc. Lond. (2018) 1 [ Phys. Lett. B SPIRE ]. Precision Big Bang nucleosynthesis with , (2014) 175 ]. , IN Constraining dark photons and their [ SPIRE ]. Revisiting supernova 1987A constraints on dark 754 IN Astrophys. J. 445 SPIRE ][ , ]. SPIRE – 30 – The effects of He I ]. IN [ (2016) 72 IN arXiv:1503.08146 SPIRE [ ][ IN New stellar constraints on dark photons 15 ][ SPIRE SPIRE Probability distribution function of cosmological density IN IN Properties of the cosmological density distribution function Phys. Rept. arXiv:1611.03864 astro-ph/9403028 , ]. ]. [ ][ [ (2015) 011 arXiv:2003.10465 , SPIRE SPIRE 07 A lognormal model for the cosmological mass distribution (1991) 1 [ arXiv:1912.01132 IN IN [ ][ ][ The EAGLE simulations of galaxy formation: public release of halo and (2017) 107 arXiv:1910.03597 Effective photon mass and (dark) photon conversion in the (1995) 479 248 Astron. Comput. , arXiv:1603.02702 The IllustrisTNG Simulations: public data release , arXiv:1302.3884 [ JCAP 01 Introducing the Illustris project: the evolution of galaxy populations across [ , 443 Mon. Not. Roy. Astron. Soc. The temperature of the Cosmic Microwave Background arXiv:0911.1955 , [ (2020) 010 Nonadiabatic crossing of energy levels JHEP ]. ]. , 03 (2017) 2936 (2013) 190 SPIRE SPIRE IN IN arXiv:1803.07048 arXiv:2002.08942 465 cosmic time matter bispectrum [ galaxy catalogues hydrodynamical simulations for large-scale structure cosmology fluctuations from Gaussian initialmodel condition: predictions comparison with of n-body one- simulations [ and two-point log-normal (2009) 916 Roy. Astron. Soc. Astrophys. J. improved Helium-4 predictions JCAP photons determinations inhomogeneous Universe 725 Rayleigh-Jeans tail of the[ cosmic microwave background connection to 21 cm[ cosmology with CMB data S. Genel et al., S. Foreman, W. Coulton, F. Villaescusa-Navarro and A. Barreira, S. McAlpine et al., I.G. McCarthy, J. Schaye, S. Bird and A.M.C. Le Brun, I. Kayo, A. Taruya and Y. Suto, D. Nelson et al., P. Coles and B. Jones, F. Bernardeau and L. Kofman, B.D. Fields, K.A. Olive, T.-H. Yeh and C. Young, D.J. Fixsen, J.H. Chang, R. Essig and S.D. McDermott, E. Aver, K.A. Olive and E.D. Skillman, C. Pitrou, A. Coc, J.-P. Uzan and E. Vangioni, A.A. Garcia et al., C. Zener, H. An, M. Pospelov and J. Pradler, K. Bondarenko, J. Pradler and A. Sokolenko, M. Pospelov, J. Pradler, J.T. Ruderman and A. Urbano, [44] [45] [42] [43] [40] [41] [38] [39] [36] [37] [33] [34] [35] [30] [31] [32] [29] [28] JHEP06(2020)132 ] ]. ]. 54 494 ]. ]. SPIRE ] ] SPIRE with the IN IN 5 ][ SPIRE SPIRE ][ IN IN (2006) 117 z > ][ ]. ][ (2016) 1885 ]. , Cambridge arXiv:1208.2701 132 [ 460 SPIRE IN SPIRE (2019) 13 Was there an early IN ][ Revising the halofit model arXiv:1712.02807 Mon. Not. Roy. Astron. Soc. arXiv:1811.07913 ][ [ , [ 872 (2017) 106. (2012) 152 arXiv:1708.04913 arXiv:1104.2933 Astron. J. [ [ , Mon. Not. Roy. Astron. Soc. Ann. Rev. Astron. Astrophys. astro-ph/0107126 , , arXiv:1802.08177 837 761 [ [ Models of the thermal evolution of the New constraints on IGM thermal (2018) 024 (2019) 009 opacity with a sample of 62 quasarsat α 04 Exploring the effects of galaxy formation on (2017) 028 (2011) 034 03 ]. Astrophys. J. Non-perturbative probability distribution arXiv:1609.04788 , 11 07 [ (2002) 412 astro-ph/0109408 forest spectra (2018) 1055 [ ]. Astrophys. J. JCAP Astrophys. J. ]. JCAP Galaxy formation and evolution α SPIRE – 31 – Self-consistent modeling of reionization in , , , Mon. Not. Roy. Astron. Soc. Complete reionization constraints from Planck 2015 , 382 IN , The Cosmic Linear Anisotropy Solving System 479 JCAP JCAP ][ , , SPIRE SPIRE IN IN ][ (2002) 477 ][ (2017) 023513 ]. ]. ]. Witnessing the reionization history using Cosmic Microwave 382 95 forest power spectrum quasars. 2. A sample of 19 quasars SPIRE SPIRE SPIRE IN IN IN 6 α New constraints on Lyman- Astron. Astrophys. ][ ][ ][ ∼ , Probing the thermal state of the intergalactic medium at arXiv:1906.00968 z Constraining the evolution of the ionizing background and the epoch of [ The evolution of the intergalactic medium arXiv:2001.10018 Phys. Rev. D Dynamics of gravitational clustering. 2. Steepest-descent method for the Dynamics of gravitational clustering v. subleading corrections in the quasi-linear arXiv:1512.00086 [ , [ ]. ]. ]. Mon. Not. Roy. Astron. Soc. Astron. Astrophys. , 7, . (2020) 2424 5 SPIRE SPIRE SPIRE arXiv:1808.04367 arXiv:1511.05992 IN astro-ph/0512082 IN IN transmission spikes in high-resolution(2020) Ly 5091 z > cosmological hydrodynamical simulations evolution from the Ly [ (2016) 313 intergalactic medium after reionization [ Background observation from Planck reionization component in our universe? [ reionization with [ polarization for the nonlinear matter[ power spectrum University Press, Cambridge U.K. (2010). quasi-linear regime regime (CLASS) II: approximation schemes matter clustering through a491 library of simulation power spectra function for cosmological counts in[ cells P. Gaikwad et al., S.E.I. Bosman et al., J. J. O˜norbe, F. Hennawi and Z. Luki´c, M. Walther, J. J.F. O˜norbe, Hennawi and Z. Luki´c, M. McQuinn, P.R.U. Sanderbeck, A. D’Aloisio and M.J. McQuinn, D.K. Hazra and G.F. Smoot, P. Villanueva-Domingo, S. Gariazzo, N.Y. Gnedin and O. Mena, X.-H. Fan et al., C.H. Heinrich, V. Miranda and W. Hu, R. Takahashi, M. Sato, T. Nishimichi, A. Taruya and M. Oguri, H. Mo, F. Van den Bosch and S. White, P. Valageas, P. Valageas, D. Blas, J. Lesgourgues and T. Tram, M.M. Ivanov, A.A. Kaurov and S. Sibiryakov, M.P. van Daalen, I.G. McCarthy and J. Schaye, [61] [62] [59] [60] [57] [58] [55] [56] [53] [54] [51] [52] [48] [49] [50] [47] [46] JHEP06(2020)132 , ] ]. ]. α ]. , (2015) SPIRE ] SPIRE ]. , IN [ IN 446 ][ SPIRE IN The Lyman- (2019) 043540 ][ arXiv:1912.09488 ]. [ ]. . Springer, Germany Mon. Not. Roy. 100 arXiv:1010.3631 Constraining the Publ. Astron. Soc. Pac. (2012) 086901 , ]. [ , arXiv:1410.5427 [ SPIRE 75 ]. arXiv:1201.1700 SPIRE Detection of extended He II IN arXiv:1106.5194 , IN SPIRE ][ [ IN ][ ]. ][ SPIRE (2011) 748 (2020) 083502 On the maximum amplitude of the IN Phys. Rev. D (2015) 128 astro-ph/0608032 , ][ [ 412 101 SPIRE Mon. Not. Roy. Astron. Soc. (2012) 433 IN 800 , ]. ]. ][ 49 Rept. Prog. Phys. (1971) 416. , 46 (2006) 181 SPIRE arXiv:1804.03888 IN [ arXiv:1704.05081 Black hole superradiance signatures of ultralight – 32 – [ ][ astro-ph/9608010 A new calculation of the recombination epoch Phys. Rev. D 433 21-cm tomography of the intergalactic medium at high [ , Dissipation of primordial turbulence and thermal Astrophys. J. Cosmology at low frequencies: the 21 cm transition and , Astrophysics of the diffuse universe astro-ph/9909275 ]. [ HI epoch of reionization arrays Adv. Space Res. arXiv:1008.2622 arXiv:1210.0197 Towards a complete treatment of the cosmological Inhomogeneous HeII reionization in hydrodynamic simulations ]. , [ [ 21-cm cosmology ]. ]. SPIRE (2018) 103533 (1997) 429 Phys. Rept. IN (2017) 035019 , (1999) L1 ][ 97 Mon. Not. Roy. Astron. Soc. Prog. Theor. Phys. SPIRE 475 SPIRE SPIRE , 96 , Adding context to James Webb Space Telescope Surveys with current IN IN IN Hydrogen Epoch of Reionization Array (HERA) arXiv:1606.07473 [ ][ 523 Reionization and the cosmic dawn with the Square Kilometre Array (2013) 235 [ ][ Probing the first stars and black holes in the early universe with the Dark (2011) 1096 EDGES result versus CMB and low-redshift constraints on ionization 36 410 Phys. Rev. D ]. ]. Astrophys. J. , Phys. Rev. D , , arXiv:1406.6361 (2017) 045001 [ SPIRE SPIRE IN arXiv:1906.07735 IN arXiv:1109.6012 (2013). Ages Radio Explorer (DARE) Astrophys. J. Lett. recombination problem history of the universe high-redshift 21-cm absorption feature [ histories 129 Exper. Astron. primordial black hole abundance[ with 21-cm cosmology and future 21cm radio[ observations the high-redshift universe [ arXiv:2002.05733 vectors redshift reionization in the temperature evolutionAstron. of Soc. the intergalactic medium forest in optically thin3697 hydrodynamical simulations M.A. Dopita and R.S. Sutherland, J.O. Burns et al., L.J. Greenhill and G. Bernardi, J. Chluba and R.M. Thomas, T. Matsuda, H. Sato and H. Takeda, S. Witte et al., S. Seager, D.D. Sasselov and D. Scott, D.R. DeBoer et al., G. Mellema et al., P. Villanueva-Domingo, O. Mena and J. Miralda-Escud´e, O. Mena, S. Palomares-Ruiz, P. Villanueva-Domingo and S.J. Witte, A.P. Beardsley et al., S. Furlanetto, S. Oh and F. Briggs, J.R. Pritchard and A. Loeb, M. Baryakhtar, R. Lasenby and M. Teo, P. Madau, A. Meiksin and M.J. Rees, Z. Luki´c,C.W. Stark, P. Nugent, M. White, A.A. Meiksin and A. Almgren, P.U. Sanderbeck and S. Bird, G.D. Becker, J.S. Bolton, M.G. Haehnelt and W.L.W. Sargent, [79] [80] [81] [77] [78] [75] [76] [72] [73] [74] [70] [71] [68] [69] [66] [67] [64] [65] [63] JHEP06(2020)132 ]. , (1997) 1 , SPIRE ]. IN 474 ][ (1992) 365. ]. ]. ]. SPIRE (2006) 485. Mon. Not. Roy. IN 264 , , ][ Mon. Not. Roy. ]. ]. , SPIRE 123 IN [ (2009). Astrophys. J. SPIRE SPIRE , IN IN ][ ][ 2010 astro-ph/0505065 ]. ]. ]. [ arXiv:1105.2044 ]. arXiv:1302.6553 [ [ Astron. Astrophys. Space Sci. Rev. SPIRE SPIRE , , astro-ph/9710107 SPIRE SPIRE [ IN IN IN IN [ ][ ][ The origin of the galaxy luminosity function ][ arXiv:1907.13130 (2005) 1031 , (2011) 025 ]. arXiv:0704.0946 07 (1998) 80 363 astro-ph/0604177 [ (2013) 1330014 [ – 33 – (1990) 349 22 495 SPIRE JCAP The radiative feedback of the first cosmological objects IN 363 , [ . (2006) 12 (2007) 663 Efficient simulations of early structure formation and Statistical properties of X-ray clusters: Analytic and astro-ph/0610094 ]. Unavoidable CMB spectral features and blackbody photosphere astro-ph/9810164 astro-ph/9903336 [ ]. [ A model for the postcollapse equilibrium of cosmological [ Taxing the rich: recombinations and bubble growth during 654 669 Astrophys. J. SPIRE Planck 2018 results. VI. Cosmological parameters (2014) 006 , SPIRE IN Astrophys. J. Peering into the Dark (Ages) with low-frequency space interferometers: IN The James Webb Space Telescope The morphology of HII regions during Reionization [ , ][ 02 How small were the first cosmological objects? (2000) 11 First light and reionization: open questions in the post-JWST era Rapid reionization by the oligarchs: the case for massive, UV-bright, (1999) 203 (2007) 1043 Int. J. Mod. Phys. D The Primordial Inflation Explorer (PIXIE): a nulling polarimeter for cosmic , Simulations and analytic calculations of bubble growth during hydrogen arXiv:1908.04296 534 , Formation and evolution of X-ray clusters — A hydrodynamic simulation of the Mon. Not. Roy. Astron. Soc. Astrophys. J. Astrophys. J. 307 377 JCAP , , , , PRISM (Polarized Radiation Imaging and Spectroscopy Mission): an extended collaboration, astro-ph/9603007 Astrophys. J. and the thermal evolution of the intergalactic medium [ Astron. Soc. numerical comparisons intracluster medium microwave background observations white paper structure: truncated isothermal spheres from top-hat density perturbations astro2010: The Astronomy and Astrophysics Decadal Survey of our Universe reionization star-forming galaxies with high escape fractions reionization Astron. Soc. using the 21-cm signal(astro)physics of neutral hydrogen from the infantPlanck Universe to probe fundamental arXiv:1807.06209 reionization A. Blanchard, D. Valls-Gabaud and G. Mamon, M. Tegmark et al., Z. Haiman, T. Abel and M.J. Rees, G.L. Bryan and M.L. Norman, A.E. Evrard, P. Andr´e, P.R. Shapiro and I.T. Iliev, M. Stiavelli et al., R.A. Sunyaev and R. Khatri, A. Kogut et al., A. Mesinger and S. Furlanetto, R.P. Naidu et al., J.P. Gardner et al., O. Zahn et al., M. McQuinn et al., S.R. Furlanetto and S. Oh, L. Koopmans et al., [97] [98] [99] [95] [96] [93] [94] [90] [91] [92] [87] [88] [89] [85] [86] [83] [84] [82]