Constraining axion-like particles using the white dwarf initial-final mass relation
CConstraining axion-like particles using the white dwarfinitial-final mass relation
Matthew J. Dolan , Frederick J. Hiskens , and Raymond R. Volkas ARC Centre of Excellence for Dark Matter Particle Physics, School of Physics, The University of Melbourne, Victoria3010, Australia Corresponding author: [email protected]
Abstract.
Axion-like particles (ALPs), a class of pseudoscalars common to many extensions of the StandardModel, have the capacity to drain energy from the interiors of stars. Consequently, stellar evolution can beused to derive many constraints on ALPs. We study the influence that keV-MeV scale ALPs which interactexclusively with photons can exert on the helium-burning shells of asymptotic giant branch stars, the late-life evolutionary phase of stars with initial masses less than 8 M (cid:12) . We establish the sensitivity of the finalstellar mass to such energy-loss for ALPs with masses currently permitted by stellar evolution bounds. A semi-empirical constraint on the white dwarf initial-final mass relation (IFMR) derived from observation of doublewhite dwarf binaries is then used to exclude part of a currently unconstrained region of ALP parameter space,the cosmological triangle. The derived constraint relaxes when the ALP decay length becomes shorter than thewidth of the helium-burning shell. Other potential sources for stellar constraints on ALPs are also discussed. Axion-like particles (ALPs) are light, weakly interacting pseudoscalars which feature in manyextensions of the Standard Model (SM) of particle physics. They arise as pseudo-Nambu Goldstonebosons (pNGBs) of spontaneously broken symmetries in, for example, the Peccei-Quinn solutionof the strong CP problem [1–4], compactification scenarios in string theory [5–7] and in models ofelectroweak relaxation [8].The properties of specific ALPs, such as their masses and coupling strengths to SM particles,are model-dependent, which has sparked investigations of their influence in a wide phenomenolog-ical range. Light ALPs with masses below the MeV scale impact astrophysical and cosmologicalphenomena [9], such as Big Bang Nucleosynthesis (BBN) [10], the Cosmic Microwave Background(CMB) and stellar evolution [11–19]. As pNGBs, they can be naturally light and weakly interacting,which makes them ideal candidates for cold dark matter (DM) [20].ALPs in the MeV to GeV range, however, are generally too massive to significantly influencecosmology and astrophysics, yet are relevant in aspects of particle physics. It has been suggestedthat ALPs may contribute to the anomalous muon magnetic moment [21–23], or act as a portalbetween the dark sector and SM particles [24]. Theoretical and experimental interest in ALPshas risen significantly with the suggestion that DM ALPs can explain the recent 3 . σ excess inelectron-recoil measured at the XENON1T experiment [25–27].In this work our attention is limited to ALPs which couple exclusively to photons via theinteraction L a = − g aγγ F µν ˜ F µν a, (1) m a [keV]10 − − − − − − − − g a γγ [ G e V − ] HB Stars ElectronBeamDumpsSN1987SN 1987A( a → γγ )WD-IFMRFig. 1: Constraints on ALP mass m a and coupling strength to photons g aγγ in the keV-MeV massrange. Individual bounds are referenced in the text. These are shown at 95% confidence level. Theconstraint derived in this work is labelled ’WD-IFMR’.1 a r X i v : . [ h e p - ph ] J a n here g aγγ is the ALP-photon coupling strength, a is the ALP-field, F µν is the electromagneticfield-strength tensor and ˜ F µν its dual. Specifically, we consider such ALPs with masses m a in thekeV-MeV range, the m a - g aγγ parameter space of which is shown in Figure 1. Constraints for thisregion arise from stellar evolution [16], results at electron beam dumps, in particular the SLACE137 experiment [28, 29] and the neutrino signal associated with the cooling of SN1987A [30] aswell as the visible signal associated with a subsequent decay of ALPs into photons [31].These bounds fail to exclude a small unconstrained triangular region at g aγγ ∼ − GeV − and m a ∼ cosmological triangle . While constraints derived from BBN excludethe cosmological triangle [9, 32], these relax significantly in certain scenarios of non-standard cos-mology [29, 33]. Furthermore, as several approaching experiments will have the capacity to directlyprobe the cosmological triangle [29, 34], it is timely to investigate phenomenological implicationsof ALPs within this region. In this paper we revisit the effects ALPs can exert on stellar evolution.Over nearly four decades, stellar evolution has been frequently deployed to constrain ALPs viathe so-called energy-loss argument . The general mechanism by which this operates is as follows,though a more detailed account of its effects on stellar structure can be found in [13] and specificproduction process will be discussed in Section 2. If sufficiently light and weakly interacting, ALPsproduced in stellar interiors can freely escape the star and act as a local energy-sink in that region.This merely results in cooling of the stellar plasma if the zone in which production occurred hosts nonuclear activity. If, on the other hand, this region is undergoing nuclear burning, the star accountsfor the energy deficit by contracting and heating, which drives the intensity of fusion upwards. Asa result both the rate of consumption of nuclear fuel and the energy-loss rate associated with ALP-production increase and a positive feedback mechanism is defined, which ultimately accelerates theprogression of the entire evolutionary phase. For sufficiently strongly interacting ALPs, this effectintroduces a contradiction between theory and observation, which leads to a constraint.The most stringent stellar energy-loss constraints on ALPs interacting with photons alone havebeen derived from observation of horizontal branch (HB) stars [9, 11, 13, 14, 16]. Specifically,population studies in globular clusters (large gravitationally bound collections of old, metal-poorstars) place limits on the helium-burning lifetime τ He of 0 . M (cid:12) HB stars. If, for a given choice of m a and g aγγ , ALP energy-loss causes τ He to fall below the observed lower bound, it can be excluded.Indeed, this is exactly the basis of the HB star constraint in Figure 1. Notably this relaxes as m a increases beyond the HB star core temperature of 10 keV, owing to the Boltzmann suppression ofALP-production.However, ALPs are known to influence many more aspects of stellar evolution than just HBstars. ALPs as massive as 100 MeV can drain energy from the cores of supernovae (SN). Themagnitude of novel energy-loss, however, is constrained by the SN1987A neutrino signal, the stan-dard astrophysical source of supernova cooling. The SN1987A ALP constraint shown in Figure 1was recently computed using a state-of-the-art SN model in [30]. Note that it does not extend toarbitrarily high values of g aγγ . Instead, ALPs become trapped within the supernova core and con-tribute towards energy transfer. The constraint relaxes when this ALP energy transfer falls belowthat of neutrinos. This defines the lower g aγγ boundary of the cosmological triangle and motivatesthe search for complementary constraints.Beyond these, it is also known that ALP production can alter chemical abundances during nu-cleosynthesis in core collapse supernova progenitors [15], prevent the blue loop evolutionary phasefrom occurring in intermediate mass helium-burning stars [17], and cause significant structuralchanges in late-life intermediate mass stars on the asymptotic giant branch (AGB) [18, 19]. Theseinvestigations, however, have been restricted to low-mass ALPs ( m a (cid:46)
10 keV) and their potentialfor constraining MeV-scale ALPs has never been assessed.We rectify this by exploring the influence of keV-MeV mass ALPs on stars on the asymp-totic giant branch, the late-life evolutionary phase of stars with masses (cid:46) M (cid:12) . It has long beenrecognised that the production of low-mass ALPs within the helium-burning (He-B) shell of suchstars can greatly affect their final mass M f [18]. Though this tendency is likely to have observableconsequences for white dwarfs (WDs) and core collapse supernova (CCSN) progenitors, it has notled to the construction of a robust ALP constraint. The He-B shells of intermediate mass AGBstars, however, are typically hotter than the cores of HB stars in globular clusters, making theman enticing prospect for further constraining the cosmological triangle.We simulate stellar evolution with and without energy-loss to massive ALPs to establish thesensitivity of M f in a region of the ALP-plane unrestricted by the HB star bound, using anedited version of the open source, 1-D stellar evolution code Modules for Experiments in StellarAstrophysics ( MESA ) [35–39].
MESA is a suite of modules containing up-to-date astrophysics such asopacity tables, nuclear reaction rates and equations-of-state. Its stellar evolution module
MESAstar has shown remarkable versatility at modelling stars over a wide range of initial masses. It has beenwidely tested and compared to astrophysical observation and other stellar evolution codes.2 constraint on ALPs based on reducing M f is then established by using the white dwarfinitial-final mass relation (IFMR). The IFMR maps the initial mass with which a star forms M init to the final mass of the white dwarf into which it ultimately evolves. It is used in age and distancedetermination in globular clusters and informs our understanding of supernovae rates [40], galacticchemical evolution and the field white dwarf population. Numerous constraints on the IFMR exist[41–47] and have previously been used to restrict stellar physics on the AGB [48]. We redeploy oneof these constraints, derived from wide double white dwarf binaries, to produce a robust bound onALPs derived from AGB stars.This paper has the following structure. In Section 2 we describe the mechanisms of ALP photo-production in a stellar plasma. We then use the results of our simulations in MESA to explore theimpact of MeV scale ALPs on the AGB in Section 3. In Section 4 we construct our constraint,account for the possibility of ALP-decay and detail some of the systematic uncertainties affect-ing our analysis. Other possible observational constraints are mentioned in Section 5 before wesummarise and conclude in Section 6. We describe our treatment of ALP energy-loss in MESA,as well as our adopted input physics in Appendices A.1 and A.2 respectively. Further discussionof the systematic uncertainties relevant to this work is included in Appendix B while Appendix Cincludes an alternative probabilistic approach to deriving a constraint.
The impact of ALPs on stellar structure varies with their lifetime τ a = Γ − a and production cross-section. The interaction in Equation 1 leads to the decay width Γ a = g aγγ m a π . (2)For sufficiently small values of m a and g aγγ , this lifetime is large and ALPs freely escape the stellarinterior and contribute to energy-loss within the star. When the ALP mass and photon-couplinggrow large, however, ALPs decay within the star and contribute towards radiative energy-transfer. The production of freely-escaping ALPs affects stellar structure by reducing the local energy-gainrate per unit mass (cid:15) . The magnitude of the ALP energy-loss rate (cid:15) a is given as a sum over therelevant ALP-production processes. For ALPs in the keV-MeV range which couple only to photonstwo such mechanisms of significance exist, Primakoff production and photon coalescence (or fusion).ALP-Primakoff production refers to the conversion of a photon into an ALP in the presence ofan external electromagnetic field [13, 49]. In a stellar interior this is facilitated by the Coulomb fieldof the constituent charged particles in the plasma. The transition rate of a photon with momentum (cid:126)k and energy ω into an ALP of momentum (cid:126)p is [50] Γ Pγ → a = g aγγ T κ π kω (cid:32) (( k + p ) + κ )(( k − p ) + κ )4 kpκ ln (cid:18) ( k + p ) + κ ( k − p ) + κ (cid:19) − ( k − p ) kpκ ln (cid:18) ( k + p ) ( k − p ) (cid:19) − (cid:33) , (3)where k = | (cid:126)k | , p = | (cid:126)p | and T is the temperature of the stellar plasma. Importantly, the Primakofftransition rate is subject to plasma screening effects, the scale of which is set by the Debye-H¨uckelwave number κ = 4 παT ρm u (cid:18) Y e + (cid:88) j Z j Y j (cid:19) , (4)where ρ is the local mass density, m u is the atomic mass unit, Y e is the number of electrons perbaryon and Y j is the number per baryon of nuclear species with charge Z j .In the stellar plasma, the effective photon mass is given by the plasma frequency ω pl ≈ παn e /m e . In all scenarios we consider, however, this is small compared with the average pho-ton energy and is therefore neglected. Furthermore, we assume that the mass of the produced ALPis small compared with that of the charged particle. Consequently, its recoil can be ignored andEquation 3 simplifies to Γ Pγ → a = g aγγ T κ π (cid:32) ( m a − κ ) + 4 ω κ ωpκ ln (cid:18) ( ω + p ) + κ ( ω − p ) + κ (cid:19) − m a ωpκ ln (cid:18) ( ω + p ) ( ω − p ) (cid:19) − (cid:33) (5)3
100 200 300 400 500 m a [keV] − − − − − l og (cid:15) a / g PrimakoffCoalescence (a) . . . . . . . T [K] − l og (cid:15) a / g m a = 10 keV m a = 316 keV (b) Fig. 2: The magnitude of energy-loss rate per unit mass associated with ALP-production for thePrimakoff and Coalescence production mechanisms as a function of ALP mass m a given conditionswithin the He-B layer of a 4 M (cid:12) AGB star (a). A comparison between the total energy-loss ratesper unit mass is also shown as a function of temperature given two different ALP masses, 10 keVand 316 keV (b).where p = (cid:112) ω − m a .This contribution of this process to (cid:15) a can then be computed as [11, 13] (cid:15) Pa = 2 ρ (cid:90) dp p π Γ γ → a ωf ( ω ) , (6)where the factor of 2 accounts for the photon polarisations and f ( ω ) is the thermal photon energydistribution, here given by the Bose-Einstein distribution [50] f ( ω ) = 1 e ω/T − . (7)Substituting Equation 5 into the expression for (cid:15) Pa then gives (cid:15) Pa = g aγγ T πρ F ( ξ , µ ) (8)where dimensionless parameters µ = m a /T and ξ = κ/ (2 T ) have been defined. The function F involves an integral over photon phase-space and encompasses the entire m a - and κ -dependence of (cid:15) a . So long as the kinetic threshold m a ≥ ω pl is met, photon coalescence γγ → a can contributeto ALP production in stellar interiors. The production rate due to this process is [50] dN a dω = g aγγ m a π (cid:112) ω − m a e − ω/T , (9)which corresponds to an energy-loss rate (cid:15) Ca = 1 ρ (cid:90) ω dN a dω dω = g aγγ T πρ G ( µ ) . (10)Here G ( µ ), like F ( ξ , µ ) above, contains the entire m a -dependence of (cid:15) Ca . The total energy-lossrate per unit mass to ALP production is then given by (cid:15) a = g aγγ T πρ ( F ( ξ , µ ) + G ( µ )) . (11)Both functions F ( ξ , µ ) and G ( µ ) contain integrals over photon phase-space which must beevaluated numerically.The relative importance of these two production mechanisms was comprehensively discussed in[16]. Primakoff production was found to dominate energy-loss in HB star cores for low ALP masses( ≈
30 keV). When ALPs with masses of 80 keV were considered, however, photon coalescencecontributed most significantly towards (cid:15) a . As can be seen in Figure 2a, the same is true when theconditions of the He-B shell of a 4 M (cid:12) AGB star are adopted ( T ≈
16 keV, ρ ≈ . × g cm − , κ ≈
35 keV) when normalised by g ≡ ( g aγγ / [10 − GeV − ]) . Note that these values have beentaken from our models. 4oth production mechanisms are Boltzmann suppressed for heavy ALPs. Consequently thetemperature sensitivity of (cid:15) a is enhanced significantly as m a increases. This is depicted for 10 keVand 316 keV ALPs in Figure 2b given the conditions of the HeB shell. Production of the latter,which is dominated by photon fusion, rapidly increases between log T = 8 . . ξ = k s / (2 T ) decreases, which somewhat inhibits Primakoff ALP pro-duction, resulting in total energy-loss for the 10 keV ALP being underestimated. Note that thevalue of 316 keV has been chosen because 316 = 10 . and we shall be investigating the impact ofALPs logarithmically spaced in mass. Strongly interacting ALPs, which decay before departing the local stellar region, modify stellarstructure by contributing an additional term to the radiative opacity of the medium. The magnitudeof this term is given by summing over the Rooseland mean opacities of the inverse Primakoff processand direct decay to photons. For the former this is given by [9] κ Pa = (cid:82) ∞ m a dω ω β ∂∂T ω/T ) − ρ (cid:82) ∞ m a dω ω λ a β ∂∂T ω/T ) − (12)where ω now refers to the ALP energy and β is the ALP velocity. The ALP mean-free path λ a isgiven by [9] λ − ω = (cid:88) Z n Z σ bcZ ( ω ) . (13)Here σ bcZ ( ω ) is the cross-section for the inverse Primakoff process, or back-conversion (bc), for atarget of charge Ze and n Z refers to its number density. This cross-section is given by [29] σ bcZ = 2 β σ PZ ( ω ) , (14)where σ PZ is the production cross-section. The contribution towards ALP Rooseland mean opacitydue to decay can be calculated in a similar fashion, although only the high-mass limit ( m a /T (cid:29) κ Pa = (2 π ) / ρ (cid:18) Tm a (cid:19) / exp( m a /T ) Γ a . (15)The total ALP opacity is then κ a = κ Pa + κ Da [29]. For MeV scale ALPs direct decay is the dominantcontribution towards their energy transfer. The total radiative opacity κ Rad is then given as κ − = κ − γ + κ − a , (16)where κ γ is the photon opacity. Note that the ALP contribution to energy transport would onlyimpact the structural evolution of a star if the affected stellar region is radiative rather thanconvective. Asymptotic giants are a class of cool, luminous star which have evolved beyond the phase ofcentral helium burning. The asymptotic giant branch (AGB) is an evolutionary stage experiencedonly by stars in the approximate mass range of 0 . − M (cid:12) [52], which are massive enough forhelium burning to occur, but insufficiently massive to support non-degenerate carbon fusion. Acomprehensive review can be found at [53].An AGB star has at its centre a core composed of carbon and oxygen, the products of heliumfusion. Surrounding the core is a helium-rich layer at the base of which is a shell supportinghelium-burning. This shell is a remnant of the previous phase of central He-B. The outer stellarenvelope is composed primarily of hydrogen and hosts a convective layer which penetrates from5ig. 3: Kippenhahn diagram showing the evolution of the He-B and H-B shells (red), CO core(blue), convective zone (grey) of a 4 M (cid:12) star throughout the E-AGB. The extent of these regionsat time 0 . M r , i.e. the mass enclosed in a spherical shell of radius r . Also shown is the H/Hediscontinuity (green dotted line).the surface deep within the star and efficiently mixes its contents. A hydrogen-burning shell existsat the bottom of this, which has persisted from the end of the main sequence throughout centralHe-B.The evolution of the inner 2 M (cid:12) of a 4 M (cid:12) star from the point of exhaustion of central heliumis shown in the Kippenhahn diagram Figure 3. A Kippenhahn diagram depicts changes in stellarstructure across evolutionary periods. At a given moment in time, the extent of stellar regions (e.g.the core, convective and burning regions) can be read along the vertical axis and are defined interms of the radial mass coordinate M r . For example, when the star has been on the AGB for 0.5Myr, the innermost 0 . M (cid:12) , shown in blue, is occupied by the CO core. This is surrounded by thehelium-rich zone which extends from 0 . M (cid:12) to ≈ . M (cid:12) , with the He-B shell (red) occupyingthe bottom 0 . M (cid:12) of this. The entire region external to this is occupied by the outer hydrogenenvelope. The convective layer extends as far down as M r ≈ . M (cid:12) , and a thin hydrogen-burningshell remains at M r ≈ . M (cid:12) . The dotted green line indicates the mass coordinate of the boundarybetween hydrogen- and helium-rich zones. The early-AGB:
The onset of the AGB coincides with considerable structural change within thestar. Once the He-B shell has been established surrounding the CO core, its substantial energyoutput prompts the expansion and cooling of the entire He-rich layer. Consequently, nuclear activitywithin the superior H-B shell is suppressed and, if M init (cid:38) M (cid:12) , extinguished. What follows is aperiod of stable helium shell burning and CO core growth, known as the early-AGB (E-AGB).For helium fusion to be sustained throughout the E-AGB, the He-B shell must graduallyprogress outward through the He-rich region. Throughout this process the He-B shell begins tothin, causing its temperature to increase and nuclear activity to intensify. Much of the associatedenergy-flux drives further expansion and cooling of the outer layers, enabling the convective zoneto penetrate more deeply into the stellar envelope.A critical point is reached when the convective zone reaches the H/He discontinuity ( ∼ . M init (cid:46) M (cid:12) , the still functional H-B shell prevents any deeperincursion of the convective zone and the H/He discontinuity is left unaltered. However in moremassive stars, where the H-B shell is dormant, the convective zone breaches the H/He discontinuityand delves into the helium-rich region below, dispersing its contents (namely helium and nitrogen)throughout the outer-envelope. This event is termed the second dredge-up . Notably the seconddredge-up disperses a substantial amount of fuel for the He-B shell, which restricts CO core growththroughout the E-AGB. The thermal pulsating-AGB:
The E-AGB is brought to a close when the He-B layer approachesthe H/He discontinuity and its supply of nuclear fuel dwindles. The associated decline in helium Stellar structure equations are typically defined in terms of the radial mass coordinate, i.e. the mass interior to a sphericalshell of radius r , rather than the radius itself. Spherical symmetry has been assumed. The first dredge-up occurs at the end of the main-sequence when a star approaches the red-giant branch. . . . . . . l og L i / L (cid:12) L H L He Fig. 4: The evolution of the luminosities L H (blue) and L He (yellow) throughout a typical pulsecycle for a 4 M (cid:12) star. Time has been set to zero when quiescent hydrogen burning begins.burning activity allows the outer-envelope to contract, reigniting the dormant hydrogen shell.Interestingly, the geometrically thin He-B is thermally unstable, which facilitates the developmentof pulsations within the star’s outer layers. These thermal pulsations characterise the second phaseof the AGB, the thermal pulsating-AGB (TP-AGB).A typical pulse cycle, illustrated in terms of the hydrogen (blue) and helium (yellow) lumi-nosities, is shown in Figure 4 for the same 4 M (cid:12) star whose evolution is illustrated in Figure 3.Sparse fuel supply in the He-B shell causes nuclear activity therein to dwindle, giving way to a longperiod of quiescent hydrogen shell burning (the inter-pulse period). The helium produced duringthis time settles onto the helium-rich region below, increasing its mass and causing the pressureand temperature at its base to rise.Once the mass of this inter-shell region reaches a certain threshold, helium is re-ignited inan unstable event known as the helium shell flash . Such flashes are brief, occurring on scales of O (1 yr), and suppress hydrogen-shell burning. They are followed by a period of stable He-B whichis sustained for a few hundred years. Once its fuel has been exhausted, nuclear activity within theHe-B shell again diminishes, giving way to quiescent H-burning. The duration of the inter-pulseperiod depends on the core mass, with more massive cores supporting more rapid pulsations [54,55].Many thermal pulsations occur during the TP-AGB, each of which produces a non-trivialamount of helium, carbon and oxygen which increase the masses of the CO and hydrogen-depletedcores (with boundary defined by the H/He discontinuity) outwards. These can be accompanied byfurther dredge-up events, in which the convective layer again penetrates into a region containing theashes of helium-burning (the third dredge-up), which reduces the growth of the cores [56]. The TP-AGB, and indeed the AGB itself, is ultimately halted by strong stellar wind, which progressivelystrips the outer envelope and leaves only the remnant white dwarf. In low-mass stars ( (cid:46) M (cid:12) ),the TP-AGB is sufficiently long for thermal pulses to contribute to the final stellar mass M f byas much as 30% [48]. More massive stars, however, shed their envelopes much more rapidly andexperience only marginal core growth during the TP-AGB [57]. For the 4 M (cid:12) star shown in Figure3, the hydrogen-depleted core grows only from 0 . M (cid:12) to 0 . M (cid:12) , an increase of approximately2%. To probe the impact of ALPs on the AGB, evolutionary simulations of 4 M (cid:12) stars were computedfor two choices of m a and g aγγ values. We distinguish between the cases of low and high massALPs below. The first case corroborates the results presented in [18], though we include only theALP-photon interaction. We then show that this behaviour persists to heavier ALPs. Light ALPS:
The E-AGB phase from the simulation with g aγγ = 0 . × − GeV − and m a = 10 keV is shown in the lower panel of Figure 5. The upper panel contains the same phasegiven standard astrophysics alone. The inclusion of these ALPs within the simulation expeditesthe E-AGB.The primary culprit for this is ALP-production and escape in the He-B shell. Energy loss withinthis region prompts it to contract and heat, which accelerates nuclear fuel consumption and causesthe entire evolutionary phase to occur more rapidly. More intense nuclear burning also sparksa premature and deeper second dredge-up event, which displaces a greater mass of helium-richmaterial throughout the convective region. This naturally increases the surface abundances of the7 . . . . . . M r / M (cid:12) .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 .
25 1 .
50 1 . . . . . . M r / M (cid:12) Fig. 5: Comparison between the E-AGB evolution of stellar structure for 4 M (cid:12) stars with (bottom)and without (top) the inclusion of 10 keV ALPs with g aγγ = 0 . × − GeV − . All elements ofthe plot are defined in Figure 3.remnants of nuclear burning (He and N). A deeper dredge-up event also results in a smallerH-depleted core mass M c , which reduces possible CO core growth during the E-AGB. This isprecissely what is seen in our models, where the terminal E-AGB value of M c and M CO decreasesfrom 0 . M (cid:12) and 0 . M (cid:12) to 0 . M (cid:12) and 0 . M (cid:12) respectively.The addition of ALPs also disrupts the evolution of the TP-AGB. As shown in Figure 6, whenALPs are included in the model, the length of the inter-pulse period increases substantially. Thisis primarily caused by the decreased core mass at the onset of pulsation [18]. Furthermore, energy-loss within the He-B shell during these pulses produces more extreme third dredge-up events.This produces a potentially observable signature, as the surface abundances of the products ofhelium-burning increase [18]. Potential ramifications of lengthening the inter-pulse period as wellas deeper third dredge-up events are discussed in Section 5. Heavy ALPs:
In order to investigate the impact of heavier ALPs we recompute the evolution ofthe 4 M (cid:12) star with m a = 316 keV and g aγγ = 10 − GeV − , well outside the region constrained byHB stars. The E-AGB evolution of this star is shown in Figure 7.The production of heavier ALPs in the He-B shell is Boltzmann suppressed and consequentlythere is minimal reduction in the duration of the early asymptotic giant branch phase. Like thelight ALP case, however, the second dredge-up penetrates more deeply into the helium-rich layerand, consequently, an observed reduction in M c and M CO is retained (0 . M (cid:12) and 0 . M (cid:12) to0 . M (cid:12) . M (cid:12) ), though this effect is of moderate strength only. − l og L i / L (cid:12) g = 00 10 20 30 40 50 60Time since TP-AGB termination [kyr] − l og L i / L (cid:12) g = 0 . Fig. 6: Comparison between the thermal pulses of the TP-AGB for 4 M (cid:12) stars with g = g aγγ / (10 − GeV − ) = 0 . . . . . . . M r / M (cid:12) .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 .
25 1 .
50 1 . . . . . . M r / M (cid:12) Fig. 7: Comparison between the E-AGB evolution of stellar structure for 4 M (cid:12) stars with (bottom)and without (top) the inclusion of 316keV ALPs with g aγγ = 10 − GeV − . This choice of ALPparameters is well outside the HB star bound in Figure 1.The persistence of ALP effects on the second dredge-up can be understood by examining theevolution of temperature in the He-B layer during the E-AGB, indicated by the pink line in Figure8. For the majority of the E-AGB the temperature of the shell is below T = 10 . K ≈ . × Kand little energy-loss to ALP-production occurs (cf. Figure 2b). As the layer thins, its temperaturesteadily increases, alleviating the Boltzmann suppression of ALP production. This triggers thepositive-feedback loop, which facilitates a more intense spike in helium luminosity and the rapidestablishment of a deep dredge-up event.Figure 9 depicts the evolution of the H/He discontinuity (solid line) and M CO (dashed) for316 keV ALPs with couplings strengths of g aγγ = 3 . × − GeV − (purple), g aγγ = 3 . × − GeV − (yellow) and with ALP-production switched off (navy). As expected, increasing the valueof g aγγ causes M CO and M c to decrease. Notably, approximately the same reduction in CO coremass is obtained when both 10 keV ALPs with g aγγ = 0 . × − GeV − g aγγ = 3 . × − GeV − M (cid:12) star in Figure 8), reach higher temperatures still during the E-AGB. Such objects should thereforeshow even greater sensitivity to heavy ALPs. This is precisely what we see in our simulations,which predict a 16% reduction M CO and a 46% reduction in E-AGB duration for a 6 M (cid:12) starexperiencing energy-loss to 316 keV ALPs with g aγγ = 10 − GeV − , compared with 5% and 1%respectively for the 4 M (cid:12) stars in Figure 7.Compared with the He-B shell of AGB stars, the cores of the 0 . M (cid:12) HB stars simulated in [16]are cooler across all stages of their respective evolutionary phases (see the yellow line in Figure8). It is precisely this which gives AGB stars great potential to further constrain the cosmologicaltriangle. Although the HB star bound has been consistently refined over time, any constraint wederive based on the established effects on M f will simply be sensitive to heavier ALPs.It should be noted that ALPs retain the capacity to elongate the inter-pulse period during theTP-AGB, as this is principally a function of M c . However, this does not significantly influencethe constraint derived in Section 4 and as such discussion of its potential constraining power isdeferred to Section 5. The IFMR relates the initial mass with which a star forms to the mass of the white dwarf intowhich it ultimately evolves. IFMRs calibrated to observation therefore provide a constraint on totalmass loss throughout a stellar lifetime, as well as free parameters of stellar modelling [48, 58]. The9 . . . . . . . . . . l og T [ K ] . M (cid:12) H B S t a r M (cid:12) A G B S t a r M (cid:12) A G B S t a r Fig. 8: The evolution of temperature at thelocation of maximum helium burning inmodels of 4 M (cid:12) and 6 M (cid:12) AGB stars as wellas a 0 . M (cid:12) horizontal branch star in the ab-sence of ALPs. Evolution is depicted as afraction of the total duration of the phase. . . . . . . . . . . . . M r / M (cid:12) g aγγ = 0 . − g aγγ = 3 . × − GeV − g aγγ = 3 . × − GeV − Fig. 9: The mass coordinate of the H/Hediscontinuity (solid) and M CO (dashed) in4 M (cid:12) stars for 316 keV ALPs with g aγγ =3 . × − GeV − (pink), g aγγ = 3 . × − GeV − (yellow) and with the ALP-photoninteraction switched off (dark blue).IFMR is instrumental in age and distance determination in globular clusters, our understandingof supernovae rates [40], galactic chemical evolution and the field white dwarf population [59]. Numerous constraints on the IFMR exist, the majority of which are derived using WDs in starclusters (e.g. [42, 47, 60, 61]). We provide a general description of the construction of these starcluster IFMRs, though a more detailed account can be found in [47].1. Spectroscopic analysis of the WDs enables their effective temperatures T eff and surface gravitylog g to be derived. These can be converted to white dwarf masses M f and cooling age τ C viaapplication of theoretical white dwarf cooling models.2. If the age of the cluster τ SC which hosts the WD is known, the progenitor lifetime can bedetermined as τ P = τ SC − τ C . Cluster age determination is typically achieved through use ofisochrone fitting.3. Finally stellar evolution models of appropriate metallicity are used to determine the initialstellar mass M init associated with the progenitor lifetime τ P . Repeating this process over anentire sample of WDs yields an IFMR calibrated to observation.The central values of two star-cluster IFMRs derived in [47] are shown in Figure 10. In thisanalysis two different sets of isochrones and stellar evolution models were used, which producesthe observed differences at large values of M init . These are the PARSEC isochrones [62], computedfrom the Padova stellar evolution models, as well as those of MESA
Isochrones and Stellar Tracks(MIST) [63, 64] which are based on
MESA simulations.A large source of uncertainty in these semi-empirical IFMRs arises from the determinationof cluster ages, which can vary significantly between stellar models with different treatments ofrotation and core overshoot (see Appendix B) [44]. As our principal aim is to constrain physicsbeyond the Standard Model, the use of an IFMR constraint which removes much astrophysicaluncertainty is desirable.Recently there has been interest in finding complementary constraints on the IFMR. In [46]an empirical measurement of the IFMR was sought via analysis of a sample of 1100 WDs fromthe
Gaia
Data Release 2 [65, 66]. This is shown in blue in Figure 10. However it was argued in[47] that, by restricting their data to white dwarfs which have previously been spectroscopicallyidentified, non-trivial selection biases are introduced. Furthermore, their derived IFMR is sensitiveto the choice of initial-mass function (IMF) for large initial masses, precisely where ALP effectsare most significant.An appropriate IFMR constraint for our purposes was derived in [44] from a sample of 14 widedouble white dwarf binary systems of solar metallicity. Like their counterparts in open clusters,the masses and cooling times of these WDs can be determined by application of theoretical WDcooling models. Instead of relying on the computation of absolute progenitor lifetime, however, Isochrones are a complementary tool to the evolutionary models discussed thus far. While stellar tracks report informationpertaining to the evolution of a single star with given initial mass M init and metallicity Z , isochrones detail the propertiesof a cluster of items at a fixed age, as a function of their mass. Implicit in their use is the assumption that all objectsdescribed by a single isochrone have formed out of the same homogeneous gas cloud, and consequently have an identicalcomposition. M init /M (cid:12) . . . . . . M f / M (cid:12) MIST IFMRPARSEC IFMRDWD BinariesGaia DA WDs
Fig. 10: A sample of existing constraints on the IFMR, with the full range of the doublewhite dwarf binary constraint with flexible breakpoints [44] shown in light green. Alsoincluded are the 95% confidence interval of the constraint from 1100 Gaia hydrogen-richwhite dwarfs within 100pc [46] in blue and the star cluster semi-empirical IFMRs derivedusing stellar models and isochrones from MIST (yellow) and PARSEC (purple) [47]only the relative lifetime ∆τ P = τ P, − τ P, is necessary. Consequently, the constraint derived fromthis analysis is independent of cluster ages. As these binaries are wide, we can assume that theyhave evolved independently of one another as single stars. This detail is crucial, as binary stellarevolution modelling is beyond the scope of this work.The constraint [44] was determined in the following manner.1. Spectroscopic analysis of the WD atmospheres when combined with synthetic WD coolingtracks enabled the determination of their final masses M f , , M f , and cooling times τ C, , τ C, .2. The relative cooling time ∆τ C = τ C, − τ C, can then be determined. As binary companions,each pair of stars can be assumed to be the same age. Consequently the difference in progenitorlifetime is given by ∆τ P = − ∆τ C .3. An initial parametric model for the IFMR is then assumed, modelled as a three-piece linearrelation, which allows estimates for initial masses M init , , M init , to be determined for eachbinary WD.4. Stellar evolution models can then be used to convert each initial mass to a theoretical progenitorlifetime τ (cid:48) P, , τ (cid:48) P, . How well the difference between these ∆τ (cid:48) P matches the observed ∆τ P is usedto define the likelihood that the parametric model of the IFMR matches the observational data.5. When this is iterated upon, the best-fit parametric model can be determined.In [44] this was repeated many times to derive a posterior sample of semi-empirical IFMRs.For the majority of [44] breakpoints at 2 M (cid:12) and 4 M (cid:12) are assumed in the three-piece fit. Themotivation for this choice is physical. Stars with with M init (cid:46) M (cid:12) experience a degenerate helium-flash, while in the range 2 M (cid:12) (cid:46) M init (cid:46) M (cid:12) helium burning proceeds in a stable, non-degenerateconvective core [67]. If M init (cid:38) M (cid:12) , a second dredge-up event can occur, which flattens the IFMR.The posterior sample for this three-piece fit was made available by the authors of [44].When the values of the breakpoints are allowed to vary, however, a wider spread of IFMRs isobtained, particularly for high initial stellar masses. Given that the value of these breakpoints varyin the literature (e.g. in the MIST IFMR of [47], upper and lower breakpoints of 2 . M (cid:12) and 3 . M (cid:12) respectively produce the best fit), we conservatively favour this less restrictive constraint. Theposterior sample in this case was not made available, which prevents a probabilistic interpretationof the results. Consequently, we show the entire range of IFMRs allowed in Figure 10.A degree of tension exists between constraints for low initial masses. When M init (cid:46) . M (cid:12) ,the central fit for the MIST and PARSEC IFMRs is approximately 0 . M (cid:12) higher than the upperboundary of the Gaia constraint. This discrepancy is worse for DWD binaries, the upper limit ofwhich falls approximately 0 . M (cid:12) below the central values of the cluster IFMRs when M init (cid:46) . M (cid:12) .Multiple sources of this tension have been suggested, including errors in star cluster ages or thepresence of unresolved binaries in the WD samples [44]. This first possibility in particular motivatesour choice of the DWD constraint, which is independent of star cluster ages and consequently moregeneral. Clearly this constraint is far less restrictive than both the cluster IFMRs and the Gaiabound, particularly for the large initial masses which are sensitive to the influence of ALPs. As we11 M init /M (cid:12) . . . . . . . . . . M f / M (cid:12) Fig. 11: The M f values from our simulations without ALPs for initial masses in 0 . M (cid:12) intervalsbetween 2 − M (cid:12) (black points). A three-piece linear fit with flexible breakpoints has been applied(solid black line). The double white dwarf binary constraint [44] is provided for comparison (green).shall see, however, even when a conservative approach is adopted, a substantial region of the ALPcosmological triangle can be ruled out. The constraint [44] has been selected to ease the comparison of our simulations with observation.However, some dependence on theoretical calculations is retained in [44] and we now discuss theirimpact.Theoretical stellar evolution tracks are used to determine progenitor lifetimes in [44]. However,constraints on the IFMR are primarily used to place limits on the AGB evolutionary phase, whichaccounts for less than 1% of the progenitor lifetime according to our models. Consequently, thesemodels can be used freely as long as they constrain model parameters which do not significantlyaffect the Main Sequence lifetime of the star.This fact also makes such IFMRs appropriate for constraining axion-like particles. Although theenergy-loss associated with ALP-production reduces evolutionary timescales in stars, this affectsonly the AGB and the latter stages of central helium-burning for the ALPs we consider. Thesehave a collective duration less than 3% of the progenitor lifetime according to our models, and assuch do not prevent semi-empirical IFMRs being used to constraint ALPs.The synthetic white dwarf evolutionary models of [68] are also used in the derivation of [44]to generate the necessary cooling times. White dwarf evolution is a fairly well-understood coolingprocess dominated in phases by neutrino emission, gravothermal settling and crystallisation (for adetailed explanation see [68, 69]). As such, cooling models are often used to determine the ages ofstellar populations (see e.g. [70]) and place limits on neutrinos [71, 72].However, anomalies in white dwarf cooling may exist. For example, tension exists between theobserved and theoretical white dwarf luminosity functions (WDLFs) in certain stellar populations,which probe WD cooling [73–76]. Interestingly, this tension can be explained by additional energy-loss to a DFSZ [77, 78] type axion with m a ∼ . M (cid:12) increments over the 2 − M (cid:12) range without energy-loss to ALPs. These were allowed to run fromthe pre-Main Sequence until the termination of the AGB, where mass loss has reduced the outerenvelope to 1% of the total stellar mass. The adopted input physics for these simulations mimicsthat used to calculate the MIST isochrones [63, 64].The initial and final stellar masses for these simulations are shown in as the black pointsin Figure 11, alongside a theoretical IFMR derived by applying a three-piece linear fit, where the12 M init /M (cid:12) . . . . . . M f / M (cid:12) g aγγ = 0 . × − GeV − g aγγ = 0 . × − GeV − g aγγ = 10 − GeV − Fig. 12: The three-piece IFMRs generatedfrom our
MESA simulations given the in-clusions of energy-loss to 10 keV ALPswith g aγγ = 0 . × − GeV − (pur-ple), 0 . × − GeV − (pink) and 10 − GeV − (yellow). The full range of the con-straint [44] is included (green), along withour three-piece IFMR given standard astro-physics alone (black). M init /M (cid:12) . . . . . . M f / M (cid:12) g aγγ = 3 . × − GeV − g aγγ = 10 − GeV − g aγγ = 3 . × − GeV − Fig. 13: The three-piece IFMRs generatedfrom our
MESA simulations given the inclu-sions of energy-loss to 316 keV ALPs with g aγγ = 3 . × − GeV − (purple), 10 − GeV − (pink) and 3 . × − GeV − (yel-low). The full range of the constraint [44] isincluded (green), along with our three-pieceIFMR given standard astrophysics alone(black).breakpoints remained unfixed. While this fits well within the DWD binary bound, it is considerablylower than the Gaia, MIST and PARSEC IFMR constraints. This discrepancy is well documented(see e.g. [48]) and can be somewhat mitigated by the inclusion of rotation in the stellar models(Section 4.5) and core overshoot (Appendix B). To quantify the effects of axion-like particles on the IFMR, the series of simulations detailed inSection 4.2 were repeated including energy-loss to ALP-production. Initially 10 keV ALPs with g aγγ = 0 . × − GeV − were considered and a new set of M f values generated. The samethree-piece fit specified in Section 4.2 was applied, the results of which are shown in Figure 12.The departure from standard astrophysics due to deeper dredge-up events becomes evident for M init (cid:38) M (cid:12) . This effect becomes more stark when larger values of g aγγ are adopted. Ultimately,for g aγγ = 0 . × − GeV − , the theoretical IFMR falls outside the DWD constraint and thischoice of ALP parameters can be excluded.Figure 13 depicts the results of repeating this analysis for 316 keV ALPs. The addition of ALPenergy-loss again flattens the IFMR, however, this effect is considerably stronger in the 4-8 M (cid:12) range, owing to the increased temperatures of their He-B shells (see Figure 8). Note that eventhe choice of parameters g aγγ = 3 . × − GeV − associated with lowest IFMR in Figure 13 isunconstrained by the HB star bound.In order to construct a constraint, theoretical IFMRs were generated at logarithmically spacedintervals in m a and g aγγ . For each ALP mass, the smallest value of g aγγ associated with an excludedtheoretical IFMR was recorded. Our simulations do not take into account ALP decay. For someof these apparently excluded points the ALP will decay inside the nuclear burning region, sothat energy loss does not result. We take this into account below to derive a bound on the ALPparameter space. Any constraint derived from ALP influence on the He-B shell of AGB stars would not extend toarbitrarily high values of g aγγ , as ALP-decay becomes an important factor in that region. In orderto quantify this, we follow the example of [16], who recently addressed ALP-decay in their HBstar bound. In their treatment it was argued that the energy-loss constraint should relax when thedecay-length λ a = 5 . × − g − aγγ m − ωm a (cid:115) − (cid:18) m a ω (cid:19) R (cid:12) (17)falls below the HB star core radius R c ≈ . R (cid:12) , where m keV = ( m a / m a [keV]10 − − − − − − − − g a γγ [ G e V − ] HB Stars ElectronBeamDumpsSN1987SN 1987A( a → γγ )WD-IFMRFig. 14: A suite of constraints in the keV-MeV region of the ALP plane. Our preliminary constraintincluding the impact of ALP-decay, is given by the solid dark blue line. The upper boundary ofthe nominal region in which ALPs are relevant for energy transfer in AGB stars is indicated bythe dashed dark blue line. See text below for discussion.stars when λ a falls below the shell thickness R He ≈ . R (cid:12) . Taking this into account yields thesolid dark blue line in Fig. 14.Unlike HB cores, however, the He-B shell of AGB stars is radiative rather than convective.Consequently, ALPs can be expected to influence AGB stellar structure beyond the boundary ofthis constraint, though a thorough treatment of this requires dedicated simulations which includeALP contributions to energy transfer, which is beyond the scope of this work.However, we are able to identify a region of parameter space in which ALPs still influence AGBstructure. This is achieved by following the example of [51] and insisting that photons contributedominantly towards radiative energy transfer in the He-B shell ( κ a > κ γ ). The upper-boundaryof this region is indicated by the blue line in Figure 14, using values for T , ρ , k s and κ γ takenfrom our models. Though this does not include any part of the cosmological triangle, it would beinteresting to examine how this could be improved upon through detailed stellar simulations. We have carefully selected the constraint [44] to minimise dependence on astrophysical uncer-tainties such as star cluster age determination and the initial mass function. Nevertheless, as ourinvestigative tool is stellar modelling, there are systematic uncertainties which we now discuss.The AGB phase is notoriously difficult to model accurately. The TP-AGB particularly presentssignificant challenges, owing to its dependence on a combination of intricate processes such as massloss and enhanced mixing from core overshoot and rotation. An extensive discussion of these canbe found in [80].Typically the uncertainty surrounding free parameters in models of the AGB phase is mitigatedby comparison with observation or a solar calibration. Examples of model input physics which fallinto this category include the efficiency of mass loss rates and core overshoot. Although theirinfluence on the IFMR can be considerable, observation limits these free parameters to a thinrange. Consequentially a discussion of their relevance is deferred to Appendix B.Unlike these, however, rotation - and the enhanced mixing it elicits - is a reality of stellarphysics. Rather than taking one simple value, the angular velocity of stars will vary according toa probability distribution (e.g. that of [81]). We therefore present a discussion of the importanceof rotational mixing to the IFMR.For simplicity, it has been assumed that all stars simulated in this work are 1-D and do notrotate. The impact of rotation can affect both theoretical predictions for the IFMR and the deriva-tion of its constraints [48]. Rotational mixing increases the supply of hydrogen available when theWD progenitor is evolving on the Main-Sequence, which produces larger stellar cores. This, alongwith an associated increase in the duration of the E-AGB, means that progenitor stars experienc-ing rapid rotation will have a higher WD masses than their more slowly rotating counterparts [82].Consequently, theoretical predictions of the IFMR including rotation lie above those where it isneglected.The effects of rotation on theoretical IFMRs was recently investigated in [48]. 40,000 syntheticstars were generated spaced uniformly in initial mass, with rotation drawn from the distribution14 M init /M (cid:12) . . . . . . M f / M (cid:12) MIST non-rotatingMISTSYCLISTMIST/ATON
Fig. 15: Two-piece fits from the statistical IFMRs of [48] generated using the MIST (yellow),SYCLIST (pink) and MIST/ATON (dark blue) rotating stellar evolution models. The full range ofthe double white dwarf IFMR constraint [44] and our standard theoretical IFMR (generated fromMIST input physics) are also included in green and black respectively. Individual stellar modelsare referenced in the text.of [81]. Values of M f were then determined through application of three different rotating stellarmodels - MIST, SYCLIST [83, 84] and a combination of ATON/MIST. The two-piece fits of theresulting statistical IFMRs are shown in Figure 15.The MIST tracks have relatively inefficient rotational mixing, selected to reproduce surfacenitrogen abundances in Main Sequence stars [64]. Consequently, the statistical IFMR determinedin [48] is weighted strongly towards lower final masses and the two-piece fit in Figure 15 is offsetfrom the non-rotating MIST model by an average of 0 . M (cid:12) for initial masses between 3 . M (cid:12) and6 M (cid:12) .The SYCLIST rotating models, however, employ more efficient rotational mixing than thoseof MIST. Consequently the resulting statistical IFMR is weighted heavily towards larger values of M f . This corresponds to a mean upward shift of 0 . M (cid:12) in the same initial mass range. The mostsignificant shift of 0 . M (cid:12) is achieved when MIST rotation is applied to ATON models [85], whichhave enhanced convective overshoot (see Appendix B for a discussion).In an analysis which accounts for the effects of rotation, the standard astrophysical IFMRis shifted upward from its non-rotating counterpart. Consequently, for theoretical IFMRs to falloutside the region of the DWD constraint, larger values of g aγγ are needed than those derived inSection 4.3. This could be achieved by repeating our simulations for different stellar rotations andperforming an analysis in the spirit of [48]. This approach would be computationally cumbersomeand would risk underestimating the influence of rotation owing to the inefficiency of rotationalmixing in the MIST input physics. Consequently, we adopt a much more simplistic approach andsimply apply an upward shift to our theoretical IFMRs.We select the magnitude of this upward shift to be 0 . M (cid:12) , equal to that of the ATON/MISTmodels and record the new value of g aγγ for specific ALP masses which cause the theoretical IFMRto fall outside the DWD constraint. The corresponding constraint is shown as a light blue linein Figure 16. While less restrictive than its non-rotating counterpart (indicated by the dark blueline), especially for low ALP masses, we have nevertheless ruled out a significant region of thecosmological triangle.We stress that a conservative approach has been adopted throughout this analysis. We choseto use a version of the DWD binary constraint of [44] over others as it is insensitive to star clusterages and has unfixed breakpoints which leads to a less restrictive range of IFMRs. We have alsoallowed for the possibility of efficient rotational mixing affecting our predictions. Naturally, there isscope for this constraint to improve dramatically when theoretical uncertainties surrounding AGBphysics and rotational mixing are better understood. Furthermore, millions of binary systems havebeen resolved in the Gaia data release 3 [86]. Given this analysis was performed with only 14 binarysystems, it would be of great interest for the method of [44] to be applied to the wide subset ofthe 1400 double white dwarf binaries identified. In Section 4 we elected to use the white dwarf initial-final mass relation as the basis of ourconstraint. However, the behaviour discussed in Section 3 has other potential observable effects.15 m a [keV]10 − − − − − − − − g a γγ [ G e V − ] HB Stars ElectronBeamDumpsSN1987SN 1987A( a → γγ )WD-IFMR(Rotation)WD-IFMR(No Rotation)Fig. 16: A comparison between our derived constraints including (light blue) and excluding (darkblue) a conservative estimate of the effects of efficient rotational mixing on the IFMR in thekeV-MeV region of the ALP-plane.Two of these, which were proposed in [19], reference the impact ALPs can have on the ultimatefates of stars.The final state into which a star evolves depends sensitively on the nature of carbon fusion,which is principally governed by the CO core mass. Stars with M CO < . M (cid:12) never reach therequisite conditions for carbon ignition and end their lives as CO white dwarfs. Alternatively, if1 . M (cid:12) < M CO < . M (cid:12) , the CO core becomes partially degenerate and carbon ignition occursin an off-centre flash. The inner boundary of this burning region then advances to the centre ofthe star, paving the way for stable carbon burning and the development of an oxygen-neon (O-Ne)core. Such stars are termed Super-AGB stars and are the progenitors for O-Ne white dwarfs. Ifthe CO core is still more massive after the exhaustion of central helium ( M CO > . M (cid:12) ), carbonis ignited in a stable, convective core. Such stars experience all further episodes of nuclear burningand are the progenitors of core collapse supernovae (CCSN) .A great deal of work has been conducted in astrophysics to identify the masses, M up and M (cid:48) up ,which correspond to the minimum mass of O-Ne white dwarf and CCSN progenitors respectively(for a review see [88]). These values naturally vary between models, though it is believed that theylie between 7-8 M (cid:12) and 10-12 M (cid:12) respectively [52].As the predicted value of M CO varies significantly when ALPs are included in the stellar model(see Section 3), so too do the values of M up and M (cid:48) up . Specifically, given that ALP-productionreduces the mass of M CO , much larger initial masses are required for the formation of O-Ne WDand CCSN progenitors. This impact, in the context of ALPs below the keV-MeV scale, is thesubject of [19], wherein it is suggested that observational upper limits on M (cid:48) up , or the rates ofType Ia SN could be used to constrain ALPs. Here we shall briefly comment on the prospectsand potential pitfalls of constructing a constraint on ALPs based on these and two other pieces ofobservational evidence. Core Collapse Supernova Progenitors:
Theoretical values of M (cid:48) up , the minimum mass ofCCSN progenitors, vary between 10-12 M (cid:12) . As such stars reach temperatures substantially higherthan in the He-B shells of AGB stars, constructing a constraint based on heavy ALP production intheir interiors is an encouraging prospect. However, whether or not such observational constraintscan be used depends on the method of their construction. There appear to be two sources of suchconstraints used in the literature.The first of these is via the investigation of pre-explosion images. If a CCSN is detected ina region which has previously been photographed, the progenitor candidate can be analysed andan initial mass estimated through a theoretical initial mass-final luminosity relation. In [89] thisled to a constraint of 8 +1 . − . M (cid:12) . The initial mass-final luminosity relation, however, relies on thepredictions of stellar evolution models during the AGB and have already been shown to vary whenALP-production is included in the simulations [90]. To prevent self-consistency issues, the analysisin [89] would have to be re-derived using an initial mass-final luminosity relation which takes theALP-dependence into account. Not all stars in this mass range do experience a supernova (see e.g. [87]), however this is recognised as the minimum COcore mass for such an event. m a [keV]10 − − − g a γγ [ G e V − ] HB StarsElectron BeamDumps SN 1987AWD-IFMR
DUNE GAr 1yrDUNE LAr 1yrBelle II (50 ab − )SN 1987AModified Luminosity Fig. 17: The state of the ALP plane in the keV-MeV region including our constraint (Section 4), themodified luminosity SN1987A constraint (shaded purple) [30]. The projected sensitivity of Belle II[29], and DUNE liquid (LAr) and gaseous argon (GAr) detectors [34] are indicated by the regionabove the dashed black, blue and red lines respectively.Another source of constraints on M (cid:48) up comes from the analysis of supernova remnants (SNRs),e.g. [91] which finds M (cid:48) up = 7 . +0 . − . M (cid:12) . Key components used in the derivation of this limit arestar formation histories (SFHs), which require the use of stellar isochrones. As these isochrones arethemselves derived from stellar modelling, it would be necessary to ascertain the degree to whichthese depend on ALP properties before this constraint could be used. As a minimal requirement,consistency demands that the SFHs be determined using isochrones derived from the same stellarevolution code used to analyse the impact of ALPs (in our case the MIST isochrones). This isfurther complicated by the dependence of M up and M (cid:48) up on the C+ C reaction rate, which ispresently a source of great uncertainty in stellar simulations [92].
Type IA Supernova Rates:
Type Ia SN are believed to occur when the mass of a CO whitedwarf in a binary system exceeds the Chandrasekhar limit due to accretion from a main sequenceor giant partner (single degenerate pathway) or a merger with a second white dwarf (doubledegenerate pathway). For a detailed review of these pathways see [93].Given the inclusion of ALPs in stellar models leads to larger M up values, their existence wouldfavour greater populations of CO white dwarfs and, consequently, a higher incidence of Type IaSN. It is therefore conceivable that bounds on the latter could enable the properties of ALPs tobe constrained or, as suggested in [19], might even hint to their existence.Constraints on Type Ia SN are typically presented in terms of their delay time distribution,i.e. their rate of incidence as a function of time following a normalised burst of star formation(see [94] for a more detailed description). It would be interesting to explore the manner in whichALP-production affects the predicted shape of the delay time distribution. Such an analysis would,naturally, require ALPs to be included in binary stellar modelling. A complete analysis on the topicwould naturally have to investigate both production pathways. Mira Variable Drifting:
Mira variable stars are a sub-class of asymptotic giants which areunstable to radial pulsations with periods that are O (100 days). Certain Mira variables havedescriptive data which go back over a century. As a result, it has been possible to detect period-drifting in these stars, the magnitude of which can be stark. For instance, the period of R Hya haschanged from approximately 500 days, as measured in 1700, to 387 at the year 2000 [53]. Similarly,R Aql’s period has decreased from approximately 320 to 267 days in recordings between 1915 and2000.It was shown in [95] that this drifting is consistent with helium shell flashes occurring duringthe larger-scale thermal pulsations of the TP-AGB, though this is still an open area of debatein the literature [96]. It has previously been shown [18], and we confirmed in Section 3, that thethermal pulses of stars with a given initial mass vary when ALPs are included in the model. Itwould be interesting to investigate whether these changes introduce tension between the agreementof period-drifting and long-term evolution. This likely requires the inclusion of ALP-productionwithin dedicated TP-AGB models. 17 bundance Ratios: A further consequence of the inclusion of ALP-production in stellar mod-elling is the impact it has on elemental abundance ratios within the star. By accelerating periodsof helium burning (both central and shell), the relative abundance of carbon and oxygen in thecore is likely to change. A deeper dredge-up event is also likely to increase the presence of fusionproducts in the surface composition. In fact, the presence of light ALPs in the late-evolutionaryphases of 16 M (cid:12) has already been found to dramatically increase the abundance of oxygen, magne-sium and neon for values of g aγγ as low as 10 − GeV − [15] in the case of the latter. This is likelyexacerbated by ALP-enhanced second and third dredge-up events which increase the presence offusion products at the stellar photosphere.Observational measurements of these ratios could therefore constitute a potent source of con-straints on ALP parameter space. Unfortunately information about AGB stars themselves is some-what scarce, with examples limited to individual post-AGB stars, e.g. [97]. This could be mitigatedthrough comparison of theory with abundance constraints from spectroscopic analysis of whitedwarfs such as [98] or via the investigation of the composition of planetary nebula (see for example[99, 100]).It is likely, owing to the sensitivity of dredge-up events to the adopted prescription of coreovershoot (see Appendix B) that this would introduce significant systematic uncertainty. Despitethis, the use of spectroscopic analysis is a particularly compelling prospect, as it does not rely onany of the results of stellar evolution theory, circumventing any issues of self-consistency. Stellar evolution has a well-established pedigree in constraining physics beyond the StandardModel, most notably for axions and axion-like particles. In this work we provide a detailed in-vestigation of the effects of keV-MeV scale ALPs on stellar evolution simulations.The photo-production of such axion-like particles in the keV-MeV mass range significantlyimpacts the evolution of asymptotic giant branch stars, the late evolutionary phase of stars withinitial masses (cid:46) M (cid:12) . Specifically, the free streaming of ALPs produced in the helium-burningshells of these stars facilitates more rapid and deeper dredge-up events, which significantly reducetheir final masses.This behaviour has been constrained by appealing to semi-empirical measurements of the whitedwarf initial-final mass relation. In particular, analysis of 14 wide double white dwarf binarysystems conducted in [44] enabled us to construct a new bound on the ALP-plane which provesmore restrictive for large m a than that derived from horizontal branch stars, most notably in theunconstrained cosmological triangle, even when the effects of stellar rotation have been considered.We expect these results to improve in the near future if the method of [44] were applied to a subsetof the 1400 double white dwarf binaries identified in the Gaia early Data Release 3 [86].For sufficiently large values of g aγγ and m a , the axion-like particle decay-lengths fall belowthe width of the helium-burning layer and the foundational criterion of the energy-loss argumentis no longer met. This reduces our initial constraint to the green shaded region in Figure 1. Asenergy transfer within the helium-burning layer is radiative, more strongly interacting axion-likeparticles still influence the structural evolution of asymptotic giant branch stars. We can estimatethe region of the ALP-plane in which this is relevant by insisting that the Rooseland mean opacityof axion-like particles be larger than that of photons. A conclusive statement about these effectswould require the addition of axion-like particle energy transfer to stellar models.Within the last year there has been a resurgent interest in the cosmological triangle. In additionto the recent HB star bound [16], which our work complements, the constraints which define itsother boundaries have been revisited. For instance, the constraint derived from the neutrino signaland observed cooling of SN1987A was recomputed recently with a state-of-the-art supernova model[30], which we show in Figures 1, 14 and 17. The work in question included a second calculation,based on the condition that only the part of the axion-like particle luminosity that can be readilyconverted to neutrino energy is relevant for the SN1987A bound [101, 102]. When this so-called modified luminosity criterion is applied a new region above the pre-existing constraint is excluded,and the cosmological triangle shrinks further (see Figure 17).The definition of new boundaries for the cosmological triangle is timely, as future experimentswill be able to directly probe this region. The Belle II experiment, for example, has estimatedsensitivity within the relevant mass range to couplings as low as g aγγ ∼ − GeV − at a luminosityof 50 ab − [29]. Future neutrino experiments, such as DUNE, will also be able to probe thecosmological triangle with both liquid argon (LAr) and gaseous argon (GAr) detectors [34].This work, like many before it, employs robust observational astrophysics and ever-more acces-sible stellar modelling to investigate the impact of axion-like particles on stellar evolution. Thoughwe have directed these tools towards this class of particle, the impact of other weakly interacting18articles can be probed in this manner. Stars, as ubiquitous objects in the universe, have a vitalrole to play in constraining physics beyond the Standard Model. Acknowledgements
We would like to thank Katie Auchettl for her invaluable insight and assistance, as well as JeffAndrews, Pierluca Carenza and Felix Kahlhoefer for helpful comments and discussion. This workwas supported in part by the Australian Research Council and the Australian Government ResearchTraining Program Scholarship initiative.
A Details of simulations
A.1 Treatment of energy-loss in
MESA
Our treatment of energy-loss to ALP-production within
MESA followed the example set by [17],which augmented the module responsible for thermal neutrino rates ( neu ) and added an additionalterm corresponding to (cid:15) a . The rationale behind this was simply that power loss to neutrinos andfreely escaping ALPs yield virtually identical phenomenological effects on stellar evolution, in thatthey both contribute a negative term to the total energy production rate (cid:15) .The functions F ( ξ , µ ) and G ( µ ), defined in Equations 8 and 10 respectively, contain the entirechemical composition and ALP mass dependence of (cid:15) a . Recall that ξ = k s / (2 T ) and µ = m a /T .We took the sum of these functions H ( ξ , µ ) = F ( ξ , µ ) + G ( µ ) (18)and evaluated it at fixed points, logarithmically spread in ξ and µ , to define a grid which is loadedthe first time the neu routine is called. Within each cell of the MESA model, our augmented neu routine determines the appropriate values of ξ and µ (cell average properties) and interpolatesbetween our grid points to find the corresponding value of H ( ξ , µ ) and hence (cid:15) a . This is addedto the energy-loss rate to thermal neutrinos, as determined by the standard neu routine.The aforementioned interpolation is carried out using the PSPLINE bicubic spline algorithmin MESA ’s own interp 2d routine (see [35] for more details). We compared the (cid:15) a values in severalof our models to its numerically evaluated counterpart and found it matched to within 0.2%.As per the MESA manifesto, we shall make our run star extras file, which contains the modified neu routine, as well as our grid of pre-calculated H ( ξ , µ ) values available for download. A.2 Adopted input physics
In addition to making our extension to MESA publicly available, we also include the inlist containing our input physics, which has been taken from the MIST project. The details of thesechoices as well as their motivation can be found in [63, 64]. As all simulations have initial massesbelow 10 M (cid:12) , we choose the prescription intended for low and intermediate mass stars. Aspects ofour run star extras file also originated with MIST, particularly those which assist with conversionof the stellar model and the switching of boundary conditions after 100 steps of the simulation. MESA was compiled using the
MESA
SDK [103]. All data was analysed using the mesa reader Python package . B Systematic uncertainties
In Section 4.5 we accounted for the influence of rotation on our constraint. However, there aremultiple free parameters in our models which can influence the IFMR. Here we discuss two ofthese - mass loss and convective overshoot - and estimate their influence on our constraint.
Mass loss:
The MIST models [63, 64] on which we base our input physics adopt the Reimers[104] and Bl¨ocker [105] prescriptions for mass loss for the RGB and AGB respectively. These aregiven by ˙ M R = 4 × − η R ( L/L (cid:12) )( R/R (cid:12) )( M/M (cid:12) ) M (cid:12) yr − (19) https://w3.pppl.gov/˜pshare/help/pspline.htm https://github.com/wmwolf/py mesa reader M init /M (cid:12) . . . . . . . . . . M f / M (cid:12) Fig. 18: A comparison between double white dwarf constraints from [44]. The full range of thebound where breakpoints are allowed to move is shown in green. The 95% confidence interval forfixed breakpoints at 2 M (cid:12) and 4 M (cid:12) is shown in red. Its average is indicated by the red line. Theblack solid line is a two-piece linear fit derived from our simulations (the black points in Figure 11),constrained with a breakpoint fixed at 4 M (cid:12) .and ˙ M B = 4 . × − η B ( L/L (cid:12) ) . ( M/M (cid:12) ) . ˙ M R η R M (cid:12) yr − (20)where η R and η B are O (1) parameters. In the MIST models, values of η R = 0 . η B = 0 . η B does impact the lower-IFMR ( M init (cid:46) M (cid:12) ), with more efficientmass loss leading to lower WD masses for a given value of M init [48]. For larger initial masses,however, the IFMR remains relatively insensitive to the adopted mass loss rate as the TP-AGB ofsuch stars is too rapid for significant core growth to occur [105]. Convective overshoot:
Fluid parcels in a convective region are accelerated as they approachits boundary and do not begin to decelerate until they enter the radiative zone. Consequently, ifbraking is insufficient, they can penetrate a non-negligible distance beyond the convective boundaryand increase the efficiency of mixing in this region. This phenomenon is known as convectiveovershoot [52].The presence of convective overshoot has two main results on the IFMR. Firstly, the enhancedmixing caused by overshoot facilitates the formation of more massive CO core at the onset of theTP-AGB, which directly shifts the IFMR upwards [106]. On the other hand, increased overshootduring the TP-AGB causes deeper third dredge-up events, which reduces core growth during thisphase [48, 106].The MIST models treat overshoot as a time-dependent, diffusive process, the strength of whichis governed by a parameter f ov . For the core, envelope and shell, values of f ov , core = 0 .
016 and f ov , env = f ov , shell = 0 . f ov , core ) and a solar calibration ( f ov , env ) [64].The ATON models of [85] similarly model overshoot as a diffusive process. However, these have f ov = 0 .
02 for hydrogen and helium burning and 0 .
002 during the AGB. As a result, they predictan IFMR which is higher than that of the MIST models by an average of 0 . M (cid:12) for initialmasses between 3 . . M (cid:12) . This contributes to the mean upward shift of 0 . M (cid:12) correspondingthe combined MIST/ATON models, which was chosen for our constraint in Section 4.5.It should be noted, however, that the MeV scale ALPs we consider do not influence the MainSequence turn off as they are too heavy to be produced until the end of central helium burning.Consequently they have no impact on the selection of overshoot parameters. C A probabilistic approach
In Section 4 we adopted the less restrictive IFMR constraint from [44] in which the breakpoints areallowed to vary. Given the posterior sample pertaining to this constraint was not available, instead20 m a [keV]10 − − − − − − − − g a γγ [ G e V − ] HB Stars ElectronBeamDumpsSN1987SN 1987A( a → γγ )WD-IFMRWD-IFMR(Fixed Breakpoints)Fig. 19: A comparison between our derived constraint from 4.5 (light blue) and that when a proba-bilistic approach is adopted (dark blue) in the keV-MeV region of the ALP-plane. Both constraintsconservatively account for rotational mixing.we conservatively insisted that theoretical IFMRs must fall within its entire range. The posteriorsample with fixed breakpoints at 2 M (cid:12) and 4 M (cid:12) , however, was made available, which enables usto establish what a probabilistic approach would entail.We first define a 95% confidence interval about the mean for uniformly spaced values of M init .This is shown by the red region in Figure 18, with mean indicated by the solid red line. The regioncorresponding to unfixed breakpoints is also included in green. The black solid line is a two-piecefit derived from our simulations (the black points in Figure 11), constrained with a breakpointfixed at 4 M (cid:12) .We can then repeat the analysis detailed in Section 4.3 for a two-piece fit with breakpoint at4 M (cid:12) in order to generate a new result. The corresponding constraint, adjusted for ALP-decay andthe influence of efficient rotation, is shown in Figure 19 in dark blue. Also included is the constraintfrom Section 4.5 in light blue.By adopting this probabilistic approach, our constraint becomes considerably more restrictive.While the HB star bound is still more restrictive at low mass, it is worth noting that we haveaccounted for efficient rotational mixing in this calculation. If the treatment of rotation is relaxedentirely our constraint moves down to g aγγ = 0 . × − GeV − , below that of the HB starbound g aγγ = 0 . × − GeV − . A definitive statement on the matter, however, requires adeeper understanding of rotational mixing in stars.Although we do not adopt this probabilistic approach for our main constraint, it demonstratesclearly the possible constraining power of the IFMR. It should be noted that the constraint [44]could be improved in the near future if it is applied to the larger double white dwarf binary datasetrecently identified in the Gaia early Data Release 3 [86]. Such an analysis would be of great interestto particle physicists hoping to constrain axion-like particles. References [1] R.D. Peccei et al. “CP Conservation in the Presence of Pseudoparticles”. In:
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