Axion constraints in non-standard thermal histories
It is usually assumed that dark matter is produced during the radiation dominated era. There is, however, no direct evidence for radiation domination prior to big-bang nucleosynthesis. Two non-standard thermal histories are considered. In one, the low-temperature-reheating scenario, radiation domination begins as late as MeV, and is preceded by significant entropy generation. Thermal axion relic abundances are then suppressed, and cosmological limits to axions are loosened. For reheating temperatures , the large-scale structure limit to the axion mass is lifted. The remaining constraint from the total density of matter is significantly relaxed. Constraints are also relaxed for higher reheating temperatures. In a kination scenario, a more modest change to cosmological axion constraints is obtained. Future possible constraints to axions and low-temperature reheating from the helium abundance and next-generation large-scale-structure surveys are discussed.
The Peccei-Quinn (PQ) solution to the strong-CP problem yields the axion, a dark-matter candidate Crewther et al. (1979); Baluni (1979); Peccei and Quinn (1977). If the axion mass , axions will be produced thermally, with cosmological abundance
where is the effective number of relativistic degrees of freedom when axions freeze out Chang and Choi (1993); Turner (1987); Kolb and Turner (1990); Moroi and Murayama (1998). Axions with masses in the range would contribute to the total density in roughly equal proportion to baryons.
Axions in the mass range are relativistic when they decouple at Chang and Choi (1993). Free streaming then erases density perturbations, suppressing the matter power spectrum on scales smaller than the axion free-streaming length Hu et al. (1998); Eisenstein and Hu (1997); Kolb and Turner (1990); Hannestad (2006). Light axions would also contribute to the early integrated Sachs-Wolfe (ISW) effect Hu and Sugiyama (1995). Data from large-scale structure (LSS) surveys and cosmic microwave-background (CMB) observations have been used to impose the constraint to light hadronic axions Hannestad et al. (2005, 2007); Melchiorri et al. (2007). These arguments apply to any particle relativistic at matter-radiation equality or cosmic microwave background (CMB) decoupling, thus imposing the similar constraint to the sum of neutrino masses Crotty et al. (2004); Tegmark et al. (2004a); Barger et al. (2004); Seljak et al. (2005); Hannestad (2006); Fogli et al. (2007); Kristiansen et al. (2006); Fukugita et al. (2006); Spergel et al. (2007); Elgaroy and Lahav (2006); Pierpaoli (2003).
These constraints rely on abundances computed assuming that radiation domination began earlier than the chemical freeze-out of light relics. There is, however, no direct evidence for radiation domination prior to big-bang nucleosynthesis (BBN) Kamionkowski and Turner (1990). The transition to radiation domination may be more gradual than typically assumed. In such a modified thermal history, two effects may cause relic abundances to change. First, the Hubble expansion rate scales differently with temperature until radiation domination begins, leading to a different freeze-out temperature. Second, entropy may be generated, suppressing relic abundances.
The universe could have reheated to a temperature as low as , with standard radiation domination beginning thereafter Kawasaki et al. (2000, 1999); Ichikawa et al. (2005); Giudice et al. (2001a); Hannestad (2004); Kolb et al. (2003). This low-temperature reheating (LTR) scenario may be modeled simply through the entropy-generating decay of a massive particle into radiation, with fixed rate and initial value of the Hubble parameter. The scalar may be the inflaton, oscillating as inflation ends and decaying into standard-model particles, or it might be a secondary scalar, produced during preheating Dolgov and Linde (1982); Abbott et al. (1982); Kofman et al. (1994, 1997); Shtanov et al. (1995). This decay softens the scaling of temperature with cosmological scale factor , increasing the Hubble parameter and leading to earlier freeze-out for certain relics. Entropy generation then highly suppresses these relic abundances.
Kination models offer another alternative to the standard thermal history, invoking a period of scalar-field kinetic-energy dominance Salati (2003), but no entropy production. Without entropy generation, abundances change more modestly.
Past work has shown that cosmological constraints to neutrinos, weakly interacting massive particles, and non-thermally produced axions are relaxed in LTR Giudice et al. (2001b, a); Yaguna (2007). Non-thermally produced axions () would be produced through coherent oscillations of the PQ pseudoscalar Turner (1987); Kolb and Turner (1990); Turner (1986). In this paper, we obtain constraints to thermally-produced hadronic axions in the kination and LTR scenarios. While kination modestly loosens limits, LTR dramatically changes the cosmologically allowed range of axion masses.
We begin by reviewing these modified thermal histories and calculating axion relic abundances. We then generalize cosmological constraints to axions, allowing for low-temperature reheating and kination. For reheating temperatures , LSS/CMB limits to the axion mass are lifted; constraints are also relaxed for higher . Constraints from the total matter density are also relaxed, but not completely lifted. For , the new constraint is , while for , we find that . If , standard results are recovered. After estimating the ability of future large-scale-structure surveys to further constrain axion masses for a variety of reheating temperatures, we derive modestly relaxed constraints to axions in the kination scenario. We conclude by considering future possible constraints to the relativistic energy density of axions in a low-temperature reheating model.
Ii Two non-standard thermal histories: Low-temperature reheating and kination
We now review the low-temperature reheating (LTR) scenario. We consider the coupled evolution of unstable massive particles , which drive reheating, and radiation , both in kinetic equilibrium. The relevant distribution functions obey a Boltzmann equation with a decay term, and may be integrated to yield Chung et al. (1999); Giudice et al. (2001b, a):
where and denote the energy densities in the scalar field and radiation, respectively, is the decay rate of the scalar to radiation, and is the cosmological scale factor. The evolution of the scale factor is given by the Friedmann equation, which is well before matter or vacuum-energy domination. The reheating temperature is defined by Giudice et al. (2001b); Chung et al. (1999); Kolb and Turner (1990)
where is the Planck mass and is the effective number of relativistic degrees of freedom when . In our calculation of the expansion history in LTR, we use calculated using the methods of Refs. Kolb and Turner (1990); Gondolo and Gelmini (1991), as tabulated for use in the DarkSUSY package Gondolo et al. (2004). We neglect the axionic contribution to for simplicity and assume massless neutrinos. The resulting error in leads to a comparable fractional error in the resulting axion relic abundance, and is thus negligible at our desired level of accuracy.
where is the temperature today. At the beginning of reheating, dominates the energy density and radiation is negligible. Thus, as initial conditions, we use and , where is the initial value of the Hubble parameter. The two physical free parameters in this model are and . The temperature is related to the radiation energy density by Kolb and Turner (1990)
We numerically integrate Eqs. (2) to obtain the dependence of on , and the results are shown in Fig. 1. As the scalar begins to decay, the temperature rises sharply to a maximum at , where is the initial value of the scale factor, and then falls as . This shallow scaling of temperature with scale factor results from the continual dumping of scalar-field energy into radiation, and yields an unusually steep dependence of scale factor on temperature. As shown in Figs. 1 and 2, when the comoving radiation energy density overtakes near , the epoch of radiation domination begins, with the usual scaling.
where and is the maximum temperature obtained (see Fig. 1). Here and are the reheating temperature and initial value of the Hubble parameter, in units of and , respectively.
At a given temperature, the universe thus expands faster during reheating than it would during radiation domination, and the equilibrium condition is harder to establish and maintain. Relics with freeze-out temperature will thus have highly suppressed abundances because they never come into chemical equilibrium. Relics with come into chemical equilibrium, but then freeze out before reheating completes. Their abundances are then reduced by entropy production during reheating. In either case, species with have highly suppressed relic abundances.
Less radical changes to abundances follow in kination scenarios. During epochs dominated by the kinetic energy of a scalar field, the energy density scales according to Salati (2003). Thus or , where is the standard radiation-dominated and denotes the transition temperature from kination to radiation domination. Kination yields relic freeze-out temperatures somewhere between the standard and LTR values. There is, however, no entropy generation during kination, leading to a less dramatic change in relic abundances. Note that these conclusions are rather general, as we have not relied on any detailed properties of the kination model, but only on the scaling Kamionkowski and Turner (1990).
Iii Axion production in non-standard thermal histories
Axions with are thermally produced in the standard radiation-dominated cosmology. We now show that in LTR models, these axions have suppressed abundances. We consider standard hadronic axions, which do not couple to standard-model quarks and leptons at tree level but do have higher-order couplings to pions and photons Kim (1979); Vainshtein et al. (1980). We do not consider flaton models, or other scenarios in which PQ symmetry breaking is related to supersymmetry breaking Chun et al. (2000, 2000, 2004); Murayama et al. (1992); Moxhay and Yamamoto (1985). For temperatures , the dominant channels for axion production are , , and . Axion scattering rates are suppressed relative to particle-number-changing interactions by factors of and thus decouple very early. Thus, axions stay in kinetic equilibrium because of , and kinetically decouple when they chemically freeze out.
Nucleonic channels are negligible at these temperatures. If pions are in chemical equilibrium and Bose enhancement can be neglected, the axion production rate is Hannestad et al. (2005); Chang and Choi (1993); Kolb and Turner (1990); Berezhiani et al. (1992)
where is the dimensionless pion momenta, is the dimensionless pion energy, is the pion distribution function, is the dimensionless axion-pion coupling constant, and is the Riemann -function. The energy scale is the PQ scale, normalized by the PQ color anomaly , and is the mass of a neutral pion Yao et al. (2006a). The PQ scale can be expressed in terms of the axion mass Weinberg (1978):
Evaluating Eq. (8) for and numerically solving Eq. (2) for , we estimate the axion freeze-out temperature using the condition . As the reheating temperature is lowered, axions freeze out at higher temperatures due to the higher value of , as shown in Fig. 3. As the reheating temperature is increased, the epoch becomes increasingly irrelevant, and the freeze-out temperature of the axion asymptotes to its standard radiation-dominated value. Examining Eq. (8), we see that , so higher-mass axions keep up with the Hubble expansion for longer and generally decouple at lower temperatures. Thus, for higher , a more radical change to the thermal history (even lower ), is needed to drive to a fixed higher value.
As axions are spin- relativistic bosons, their number density at freeze-out is . If we assume that axion production ceases at freeze-out, the density of axions at any subsequent time is just , where is the value of the cosmological scale factor at axion freeze-out. The reheating time scale, , is much shorter than the Hubble time for , and so it is a good approximation to treat the break between the and epochs as instantaneous at . Doing so, we apply Eq. (6) prior to the completion of reheating and afterwards, to obtain
where is the axion mass in units of .
Low reheating temperatures drive up the freeze-out temperature. When , the present mass density in axions is severely suppressed, because of the sharper dependence of the scale factor on during reheating. This is a result of entropy generation. Using the numerical solution for from Sec. II, we obtain , accounting for the smooth transition between the and regimes. In Fig. 4, we show normalized by its standard value, , as a function of . At reheat temperatures just a factor of a few below the usual axion freeze-out temperature for a given axion mass, the axion abundance is suppressed by a factor of . For , the axion abundance asymptotes towards its standard value.
In the case of kination, axion freeze-out temperatures are still raised, but there is no additional entropy production. Axion abundances are given by Eq. (1), but with the higher values appropriate at higher values of .
For the LTR case, our results do not depend on the initial value of the Hubble parameter. As seen in Fig. 1, changes to determine the moment of the fast rise to , but have little influence on the expansion history for . For convenience, we choose for our calculations, corresponding to .
Our calculation is valid only if axions are produced in equilibrium by thermal pions. This requirement imposes the restriction . Outside this range, our assumptions break down. For sufficiently low values of and , pionic cross sections lead to , earlier than the quark-hadron phase transition. The absence of hadrons then necessitates the use of quark-axion production cross sections.
Furthermore, for , pions will decay before they can come into equilibrium. In both cases, axion abundances are suppressed relative to our calculation. For axion masses saturating our upper limits and , we have checked that we are well within the equilibrium regime. We restrict ourselves to this range, noting that for , more suppressed abundances will lead to an even more dramatic relaxation to cosmological axion limits. Finally, coherent oscillations of the axion field produce a condensate that behaves as cold dark matter Giudice et al. (2001b), but the resulting additional abundance is negligible for at all values of under consideration here Giudice et al. (2001b).
Iv Constraints to axions
Most constraints to the axion mass are obtained from its two-photon interaction. This interaction is parameterized by a coupling constant , given by Raffelt (1996); Kolb and Turner (1990); Kaplan (1985); Srednicki (1985); Kephart and Weiler (1987); Bershady et al. (1991); Turner (1987); Ressell (1991); Grin et al. (2007); Gnedin et al. (1999); Andriamonje et al. (2007); Asztalos et al. (2002)
where is a model-dependent parameter and is the fine-structure constant. The tightest constraint to ,
comes from the helium burning lifetime of stars in star clusters Raffelt (1996); Raffelt and Dearborn (1987). The parameter is given by Raffelt (1996); Buckley and Murayama (2007); Srednicki (1985); Kaplan (1985)
Here and are weighted sums of the electric and PQ charges of fermions that couple to axions. In existing axion models, Moroi and Murayama (1998); Buckley and Murayama (2007), while is poorly constrained and lies in the range Buckley and Murayama (2007); Yao et al. (2006b); Maltman et al. (1990); Kaplan and Manohar (1986):
As a result, for any axion model in the range , there are experimentally allowed values for which vanishes (see Eq. 13), and so constraints to axions from star clusters, helioscope, RF cavity, and telescope searches may all be lifted Buckley and Murayama (2007); Moroi and Murayama (1998). Such a cancellation is fine tuned, but even for other values of , constraints to the axion mass are loosened.
In contrast, the hadronic couplings do not vanish for any experimentally allowed values Moroi and Murayama (1998). Axion searches based on these couplings are underway, and have already placed new upper limits to the axion mass in the range Ljubicic et al. (2004) (and references therein). These couplings also determine the cosmological abundance of axions, and so useful constraints may be obtained from cosmology.
Although the hadronic coupling determines the relic abundance of axions, will determine the lifetime of the axion, which may have implications for cosmological constraints. For the high axion masses allowed by our new constraints and certain values of and , the decay may no longer be negligible on cosmological time scales. Such an axion would be completely unconstrained by limits to from the total matter density or the hot dark matter mass fraction. In the following calculation, we neglect axion decay. Consistent with a vanishing two-photon coupling for , we use the value . We have verified that our results for vary by only for variations in of about , as the dependence of the axion production rate and on is weak (see Eq. 8) compared with the dependence on the Boltzmann factor.
iv.1 Constraints to the axion mass from
In the standard cosmology, a conservative constraint is obtained by requiring that axions not exceed the total matter density of Spergel et al. (2007), yielding the limit , using a concordance value for the dimensionless Hubble parameter . In LTR scenarios, axion abundances are highly suppressed, as shown in Eq. (10) and Fig. 4. Mass constraints to thermal axions from cosmology are thus considerably relaxed.
To obtain abundances in the LTR scenario, we apply the numerical freeze-out and abundance calculation of Section III. Axion mass limits resulting from the requirement that axions not exceed the total dark matter density are shown by the dot-dashed hashed region in Fig. 5. If , constraints are considerably relaxed. For example, if , the axion mass constraint is . When , we obtain , equal to the standard radiation-dominated result. As already discussed, abundances are only slightly changed in the case of kination, so imposes the constraint if .
iv.2 Constraints to the axion mass from CMB/LSS data
Axions will free stream at early times, decreasing the matter power spectrum on length scales smaller than the comoving free-streaming scale, evaluated at matter-radiation equality. For sufficiently low masses, axions will also contribute to an enhanced early ISW effect Hu and Sugiyama (1995) in the CMB. This suppression is given by if Hannestad et al. (2005, 2007); Hannestad and Raffelt (2004) (and references therein). The matter power spectrum thus imposes a constraint to .
First we review the constraints imposed by these effects in a standard thermal history. In this case, both and the free-streaming scale, , depend only on in a hadronic axion model. Using Sloan Digital Sky Survey (SDSS) measurements of the galaxy power spectrum Tegmark et al. (2004b) and Wilkinson Microwave Anisotropy Probe (WMAP) Spergel et al. (2003) 1-year measurements of the CMB angular power spectrum, Refs. Hannestad et al. (2005, 2007); Hannestad and Raffelt (2004) derived limits of . Axions in the mass range of interest () are non-relativistic at photon-baryon decoupling, and so this constraint essentially comes from measurements of the galaxy power spectrum Crotty et al. (2004). As a result, we do not expect that an analysis including more recent WMAP results would make a substantial difference in the allowed axion parameter space. In order to lift this constraint, the relationship between and or must be changed.
If is allowed to vary freely, the constraints may be relaxed. In particular, using Eq. (1) we can see that increasing (and thus ) will decrease . Furthermore, if the free-streaming length of the axion is less than the smallest length scale on which the linear power-spectrum may be reliably measured (), its effect on the matter power spectrum is not observable Hannestad et al. (2005, 2007); Hannestad and Raffelt (2004). The comoving free-streaming length scale at matter-radiation equality111This differs from the expression in Refs. Hannestad and Raffelt (2004); Hannestad (2006) because we assume, as is the case in our parameter region of interest, that , the temperature at matter-radiation equality. In Ref. Hannestad (2006), is assumed. Kolb and Turner (1990),
is set by the ratio between the axion and neutrino temperatures,
so that if (i.e., if axions freeze out considerably before neutrinos), the constraint to axion masses is lifted Hannestad et al. (2005).
In the case of a modified thermal history, the relationship between and acquires dependence on an additional parameter (, in the case of LTR, or , in the case of kination) thus allowing us to loosen the constraints. At a series of values of , Refs. Hannestad et al. (2005, 2007); Hannestad and Raffelt (2004) determine the maximum values of consistent with WMAP measurements of CMB power spectra and SDSS measurements of the galaxy power spectrum. We begin by mapping these contours, from Fig. 5b in Ref. Hannestad et al. (2005) (which do not include constraints from the Lyman- forest), into the plane. For a fixed or , scales monotonically with , and thus serves as a proxy for .
In the domain and , we calculate for hadronic axions in LTR, using the full numerical solution described in Sec. III. We also calculate . Since axions freeze out while relativistic, their energy will redshift as . They will have temperature . Meanwhile, the temperature of the coupled radiation redshifts as until radiation domination begins. Thus entropy generation modifies the relationship between the axion and neutrino temperatures to
if . To obtain all of our constraints we use the more precise scaling accounting for the smooth transition between the and regimes. The dominant change to the free-streaming length comes from the modified axion temperature, while the modified expansion rate itself induces negligible fractional changes of order , where is the cosmic temperature at which the axion goes non-relativistic.
For each pair , we calculate and to trace out the region forbidden with confidence. When , outside the domain of Ref. Hannestad and Raffelt (2004), we extrapolate, assuming that the contour asymptotes to a line of constant axion free-streaming wavelength . Such a trend is noted in Ref. Hannestad and Raffelt (2004), and at the maximum value of of the contour obtained from Ref. Hannestad et al. (2005), the maximum allowed free-streaming length is consistent with our assumed asymptote.
We obtain the upper limit to the axion mass as a function of , shown in Fig. 5. Existing LSS/CMB constraints are severely relaxed in the LTR scenario, and lifted completely for . For , for , and so the axion mass is unconstrained. It will still be subject to phase-space constraints if it saturates the bound , and is to compose all the dark matter in galactic halos Tremaine and Gunn (1979); Madsen (1991, 2001). At high reheating temperatures, the constraint from LSS/CMB data () supercedes the constraint .
The narrow allowed region between the LSS/CMB and total matter density constraints in Fig. 5 () may be simply understood. Axions in this narrow window are cold and massive enough to evade large-scale structure constraints (i.e., ), and dilute enough to evade constraints from the total matter density. We note that the CMB/LSS limits asymptote to their standard value of for .
Future instruments, such as the Large Synoptic Survey Telescope (LSST), will measure the matter power-spectrum with unprecedented precision ( Zhan et al. (2006); Tyson (2006). This order of magnitude improvement over past work Tegmark et al. (2004, 2006) will improve the constraint to by an order of magnitude, resulting in the improved sensitivity to axion masses and reheating temperatures shown by the dotted line in Fig. 6. To estimate possible constraints to axions from LSST measurements of the power spectrum, we recalculated our limits using the approximate scaling , assuming for .
We also estimate the possible improvement offered by including information on smaller scales (), as may be obtained from measurements of the Lyman- flux power spectrum Viel et al. (2004), also shown in Fig. 6. We include this effect by replacing with this lower minimum length scale. This is indicated by the dashed line in Fig. 6. We can see that more massive axions are probed because of information on smaller length scales, as are lower reheating temperatures.
In the case of kination, a much less severe relaxation of limits to axions is obtained. As there is no entropy generation in the kination case, the abundance and temperature of the axion are still given by Eqs. (1) and (16), with the value of appropriate at the new freeze-out temperature. In the range of parameter space explored, , and so the variation in as a result of kination is . For , the new allowed regions are and . These conclusions apply to any non-entropy-generating scenario in which at some early epoch, and not only to kination Kamionkowski and Turner (1990). If , standard results are recovered.
V Axions as relativistic degrees of freedom at early times
Future limits to axions in the standard radiation-dominated and LTR thermal histories may follow from constraints to their contribution to the energy density in relativistic particles at . Axions are relativistic spin- bosons, and so Kolb and Turner (1990). We can express the total relativistic energy density in terms of an effective neutrino number
Treating the transition between the and epoch as instantaneous, we solve for the photon and axion temperatures, and then obtain
For sufficiently high masses, the axionic contribution saturates to at high reheating temperatures Chang and Choi (1993). In Fig. 7, we show , the effective neutrino number evaluated at the axion mass which saturates the LSS/CMB bounds, for , or saturates the constraint for lower . The behavior of the curve may be readily understood. As can be seen from Fig. 5, as we increase , the maximum allowed decreases. For , even though the maximum allowed is large (which corresponds to a lower , since ), the amount of entropy production between and leads to a small axionic contribution to . As increases, the interval between freeze-out and reheating decreases. This lessens the impact of entropy generation, and leads to the rise in . Finally, for , the impact of entropy generation is nearly negligible, and falls as the maximum allowed value of decreases, as in the standard case (due to earlier freeze-out).
A comparison between the abundance of He and the predicted abundance from BBN places constraints to the radiative content of the Universe at MeV Cardall and Fuller (1996); this can be stated as a constraint to . At early times, axions will contribute to the total relativistic energy density (through ), and thus constraints to He abundances can be turned into constraints on and , as shown in Fig. 7.
In terms of the baryon-number density , we write the primordial He abundance as . In order to translate measurements of to constraints on and we use the scaling relation Steigman (2007)
Constraints to from direct measurements of , including a determination of from CMB observations, lead to the 68% confidence level upper limit of Cyburt et al. (2005); Ichikawa and Takahashi (2006); Ichikawa et al. (2007). From Fig. 7 and Eq. (19), we see that this bound cannot constrain or . If future measurements reduce systematic errors, constraints to will be obtained for the lighter-mass axions.
Constraints to and may also follow from indirect CMB measurements of . The presence of He affects CMB anisotropies by changing the ionization history of the universe Trotta and Hansen (2004). The Planck satellite is expected to reach , yielding a sensitivity of , while CMBPol (a proposed future CMB polarization experiment) is expected to approach , leading to the sensitivity limit Trotta and Hansen (2004); Kaplinghat et al. (2003); Ichikawa et al. (2007); Eisenstein et al. (1999). As shown in Fig. 7, for , such measurements of may impose more stringent limits on the axion mass. Also, if axions with mass in the range are directly detected, might impose a surprising upper limit to Andriamonje et al. (2007); Rosenberg (2004).
The lack of direct evidence for radiation domination at temperatures hotter than has motivated the introduction of kination, low-temperature reheating, and other scenarios for an altered pre-BBN expansion history. In the case of kination, the change in axion abundances and thus cosmological constraints is modest. Low-temperature reheating will suppress the abundance of thermally-produced hadronic axions, once the reheating temperature . This is rather intuitive once we recall that the axion freeze-out temperature in a radiation dominated cosmology is . If the reheating temperature crosses this threshold, axion densities are severely reduced by dramatic entropy production during reheating.
Total density, large-scale structure, and microwave background constraints to axions are all severely loosened as a result, possibly pushing the the axion mass window to very high values; for , the new constraint is . For , standard radiation dominated results are recovered. The inclusion of information on smaller scales will probe higher axion masses and lower reheat temperatures. More precise measurements of the matter power spectrum on all scales will probe lower axion masses. Kination also relaxes constraints to axions, though much less markedly. Future probes of primordial helium abundance will either lead to further constraints on axion properties, or, if axions are directly detected, provide a new view into the thermal history of the universe during the epoch .
Acknowledgements.D.G. was supported by the Gordon and Betty Moore Foundation and acknowledges helpful discussions with Mark Wise, Stefano Profumo, and Sean Tulin. T.L.S. and M.K. were supported by DoE DE-FG03-92-ER40701, NASA NNG05GF69G, and the Gordon and Betty Moore Foundation.
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