PLB31487S03702693(15)00867910.1016/j.physletb.2015.11.020The AuthorsPhenomenologyFig. 1Demonstration that volume is sufficiently large to have negligible finite volume corrections on Q2. The data for T/Tc=2 is multiplied by 3 for better visibility of the comparison.Fig. 2Demonstration that Wilson flow/lattice renormalization are under control and that the dependence of the results on the choice of the flow time decreases when the continuum limit is approached.Fig. 3Lattice data on the topological susceptibility at Nt=5,6,8 and lattice continuum extrapolation together with fit of simple power law.Fig. 4Lattice data on the anharmonicity coefficient b2 of the axion potential compared to its DIGA prediction. The data points are shifted a bit horizontally for better visibility.Fig. 5Prediction of the topological susceptibility in the DIGA: comparison between oneloop and twoloop RGI results. We used the fourloop expression for the running coupling in the modified minimal subtraction scheme as given in the appendix of Ref. [31] and the central value of Tc/ΛMS‾(nf=0)=1.26(7) as determined from the lattice in Ref. [20].Fig. 6Rescaled oneloop and twoloop RGI DIGA results compared to lattice continuum extrapolation. The DIGA results shown in Fig. 5 were scaled by a factor K of order ten such that they coincide at T/Tc=2 with the central value of the lattice continuum data.Fig. 7Consequence of our findings for axion dark matter from quenched (top) and full QCD (bottom). The dark yellow region is excluded because RA>1 just from the misalignment mechanism, cf. Eq. (7), assuming a flat distribution of initial misalignment angles in the observable universe θ∈[−π,π]. The light yellow region indicates RA>1 when string contributions according to Ref. [33] are included. In the dark green region axions even including strings give a small contribution (RA<0.1) to dark matter. The light green region indicates RA<0.1 just from the misalignment mechanism. In order to compare the quenched results with the axion dark matter scenario we had to transform dimensionless quantities into dimensionful ones. For illustrative purposes we use, in the quenched case, Tc=294 MeV. Our quenched lattice results are shown by the blue points and the twoloop RGI DIGA prediction by the gray band in the top panel. Since the transition in full QCD is a crossover [34], Tc is not unambiguous even in the unquenched case. Using the transition temperature defined by the chiral susceptibility, Tc=147(2)(3) MeV [35–37], we need K=9.22±0.6 in order that the twoloop RGI DIGA matches χ(Tc)/χ(0)=1. The blue and gray bands in the lower panel correspond to the twoloop RGI DIGA predictions with K=1 and K=9.22±0.6, respectively. The dashed red line shows the IILM prediction of Ref. [11]. The dashed black lines correspond to fixed axion masses (in units of μeV). Using K=9.22 and again assuming at least ten percent axion contribution to dark matter we can read off from the lower panel the range 40 μeV≲mA≲930 μeV, while using K=1 we would get 50 μeV≲mA≲1100 μeV, i.e. only a ∼20% correction for an O(10) K factor uncertainty. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)Table 1List of simulation points: temperatures, lattice sizes Nt=1/(aT), Ns=L/a, number of sweeps Nsweeps, and estimated integrated autocorrelation time tint,Q are given.T/TcNtNsNsweepstint,Q
0.912532 K3.3(2)
12648 K7.1(4)
168170 K36.4(32)

1.012548 K4.6(2)
12664 K12.6(9)
168180 K53.9(52)

1.112548 K5.3(3)
166160 K12.7(8)
208330 K65.3(71)

1.316564 K5.7(3)
166220 K15.4(9)
248550 K70.2(87)

1.516596 K5.8(2)
206210 K14.5(10)
248660 K63.8(75)

1.7205420 K6.9(2)
2061300 K18.9(6)
2888200 K88.1(42)

2.0205440 K7.5(2)
2461900 K18.0(6)
3288400 K90.5(51)

2.3245740 K8.0(3)
2862400 K18.7(5)
3688500 K60.4(31)

2.6285960 K6.9(2)
3263500 K17.8(5)
4488000 K51.7(35)

3.03251500 K6.2(2)
3665700 K16.0(5)
48811 000 K54.3(46)

3.53652200 K5.3(1)
4465200 K15.6(6)
56812 000 K<45.0

4.04052600 K<5.0
4865900 K15.6(8)
64812 000 K<45.0
Table 2Temperature slopes of the topological susceptibility predicted in the twoloop RGI DIGA, for a range of renormalization scales according to Eq. (24).T/Tc1.52345
b (κ=0.6)−6.04−6.26−6.43−6.50−6.55
b (κ=1)−6.37−6.46−6.55−6.59−6.62
b (κ=2)−6.55−6.59−6.64−6.67−6.69
Table 3Strong coupling constant at μr=1/ρm for the temperature range covered by the lattice.T/Tc12345
αMS‾(1/ρm)0.360.230.190.170.16
Axion cosmology, lattice QCD and the dilute instanton gasSz.BorsanyiaM.DieriglbZ.FodoracdS.D.Katzde⁎katz@bodri.elte.huS.W.MagesfcD.NogradidegJ.RedondohiA.RingwaldbK.K.SzaboacaDepartment of Physics, Wuppertal University, Gaussstrasse 20, D42119 Wuppertal, GermanyDepartment of PhysicsWuppertal UniversityGaussstrasse 20WuppertalD42119GermanybDeutsches ElektronenSynchrotron DESY, Notkestrasse 85, D22607 Hamburg, GermanyDeutsches ElektronenSynchrotron DESYNotkestrasse 85HamburgD22607GermanycIAS/JSC, Forschungszentrum Jülich, D52425 Jülich, GermanyIAS/JSCForschungszentrum JülichJülichD52425GermanydInstitute for Theoretical Physics, Eötvös University, Pázmány Peter sétany 1/A, H1117 Budapest, HungaryInstitute for Theoretical PhysicsEötvös UniversityPázmány Peter sétany 1/ABudapestH1117HungaryeMTAELTE Lendület Lattice Gauge Theory Research Group, Budapest, HungaryMTAELTE Lendület Lattice Gauge Theory Research GroupBudapestHungaryfUniversity of Regensburg, D93053 Regensburg, GermanyUniversity of RegensburgRegensburgD93053GermanygKavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 931064030, USAKavli Institute for Theoretical PhysicsUniversity of CaliforniaSanta BarbaraCA931064030USAhDepartamento de Física Teórica, Universidad de Zaragoza, Pedro Cerbuna 12, E50009 Zaragoza, SpainDepartamento de Física TeóricaUniversidad de ZaragozaPedro Cerbuna 12ZaragozaE50009SpainiMaxPlanckInstitut für Physik (WernerHeisenbergInstitut), Föhringer Ring 6, D80805 München, GermanyMaxPlanckInstitut für Physik (WernerHeisenbergInstitut)Föhringer Ring 6MünchenD80805Germany⁎Corresponding author.Editor: B. GrinsteinAbstractAxions are one of the most attractive dark matter candidates. The evolution of their number density in the early universe can be determined by calculating the topological susceptibility χ(T) of QCD as a function of the temperature. Lattice QCD provides an ab initio technique to carry out such a calculation. A full result needs two ingredients: physical quark masses and a controlled continuum extrapolation from nonvanishing to zero lattice spacings. We determine χ(T) in the quenched framework (infinitely large quark masses) and extrapolate its values to the continuum limit. The results are compared with the prediction of the dilute instanton gas approximation (DIGA). A nice agreement is found for the temperature dependence, whereas the overall normalization of the DIGA result still differs from the nonperturbative continuum extrapolated lattice results by a factor of order ten. We discuss the consequences of our findings for the prediction of the amount of axion dark matter.KeywordsAxion dark matterQCD on the latticeInstantons1IntroductionOne of the greatest puzzles in particle physics is the nature of dark matter. A prominent particle candidate for the latter is the axion A [1,2]: a pseudo Nambu–Goldstone boson arising from the breaking of a hypothetical global chiral U(1) extension [3] of the Standard Model at an energy scale fA much larger than the electroweak scale.A key input for the prediction of the amount of axion dark matter [4–6] is its potential as a function of the temperature, V(A,T). It is related to the free energy density in QCD, F(θ,T)≡−lnZ(θ,T)/V, via(1)V(A,T)≡−1Vln[Z(θ,T)Z(0,T)]θ=A/fA, where V is the Euclidean space–time volume. Here A(x) is the axion field and fA is the axion decay constant. The axion field has mass dimension one, therefore A(x)/fA is dimensionless and can be interpreted as an x dependent θ value. The angle θ enters the Euclidean QCD Lagrangian via the additional term involving the topological charge density q(x),(2)−iθq(x)≡−iθαs16πϵμνρσFμνa(x)Fρσa(x), with Fμνa being the gluonic field strength and αs≡gs2/(4π) the fine structure constant of strong interactions.On general grounds, the free energy density and thus the axion potential has an absolute minimum at θ=A/fA=0. In fact, this is the reason why in this extension of the Standard Model there is no strong CP problem [3]. The curvature around this minimum determines the axion mass mA at finite temperature,(3)mA2(T)≡∂2V(A,T)∂A2A=0=χ(T)fA2, in terms of the topological susceptibility, i.e. the variance of the θ=0 topological charge distribution,(4)χ(T)≡∫d4x〈q(x)q(0)〉Tθ=0=limV→∞〈Q2〉Tθ=0V, where Q≡∫d4xq(x). Similarly, selfinteraction terms in the potential, e.g. the A4 term occurring in the expansion of the free energy density around θ=A/fA=0,(5)V(A,T)=12χ(T)θ2[1+b2(T)θ2+…]θ=A/fA, are determined by higher moments,(6)b2(T)=−〈Q4〉T−3〈Q2〉T212〈Q2〉Tθ=0. These nonperturbative quantities enter in a prediction of the lower bound on the fractional contribution of axions to the observed cold dark matter as follows11This is a rewriting of equation (2.10) of Ref. [7] where we have used χ(0)=(mAfA)2≃3.6×10−5 GeV4 from the chiral Lagrangian to express fA in terms of mA, the zero temperature mass, that is itself a function of Tosc and the ratio χ(T)/χ(0)=(mA(T)/mA)2 through the condition for the onset of the oscillations, mA(Tosc)=3H(Tosc) with H2=8π3GNg⁎R(T)T4/90 the Hubble expansion rate.(7)RA≳105〈θ2〉f(〈θ2〉)GeV3g⁎S(Tosc)Tosc3χ(Tosc)χ(0), where 〈θ2〉 is the variance of the spatial distribution of the initial values of the axion field A/fA before the formation of the dark matter condensate by the misalignment mechanism, which occurs, when the Hubble expansion rate gets of the order of the axion mass, mA(Tosc)=3H(Tosc), i.e. at a temperature(8)Tosc≃50GeVg⁎R1/4(Tosc)(mAμeV)1/2(χ(Tosc)χ(0))1/4. The function f(〈θ2〉) is taking into account anharmonicity effects arising from the selfinteraction terms in V(A,T) and depends on the specific form of the potential. The functions g⁎R(T) and g⁎S(T) denote the effective number of relativistic energy and entropy degrees of freedom, respectively.What is urgently needed for axion cosmology is thus a precise determination of the topological susceptibility and higher moments of the topological charge distribution. In this context, most predictions have been entirely based on the semiclassical expansion of the Euclidean path integral of finite temperature QCD around a dilute gas of instantons – finite action minima of the Euclidean action with unit topological charge – see e.g. Ref. [8] for an early exhausting study and Ref. [9] for a recent update concerning the quark masses. A comparative study of these predictions based on the dilute instanton gas approximation (DIGA) has been carried out in Ref. [10], where also an analysis in terms of a phenomenological instanton liquid model (IILM) [11] is presented. However, up to now, in all DIGA investigations of the topological susceptibility only the oneloop expression in the expansion around the instanton background field was used. This results in a strong renormalization scale dependence and thus large uncertainties which were neglected in the previous DIGA based predictions. In fact, in the temperature range Tosc∼GeV of interest, one expects a large uncertainty in the overall normalization due to the neglection of higher order loop effects, since in this region αs(Tosc) is not small. We will exploit in this letter both the oneloop DIGA result as well as its twoloop renormalization group improved (RGI) version in order to study the theoretical uncertainties arising from higher loop corrections. Most importantly, we compare these predictions with the outcome of our lattice based fully nonperturbative results.Actually, there have been a number of lattice calculations of χ(T) and b2(T) at temperatures below or slightly above the QCD phase transition, mostly in quenched QCD, see e.g. [12–16]. Here we go beyond those lattice calculations. We extend the available temperature range and carry out a controlled continuum extrapolation for this extended range. In addition, note that there has been no quantitative investigation whether and where the lattice results turn into the DIGA results. We will present our new highquality lattice data for the topological susceptibility in quenched QCD (i.e. neglecting the effects of light quarks) and compare them quantitatively to the DIGA result specialized to the case of nf=0 light quarks.2Axion potential coefficients from the latticeOn the lattice, the topological susceptibility is measured on the torus as the second moment of the distribution of the global topological chargeχt=〈Q2〉/V, where Q is any renormalized discretization of the global topological charge, and V is the fourvolume of the lattice. There are a lot of different fermionic and gluonic definitions of Q available. We choose a gluonic definition based on the Wilson flow [17,18], which has the correct continuum limit similarly to the fermionic definitions but is numerically a lot cheaper. In particular we evolve our gauge field configurations with the Wilson plaquette action to a flow time t and define the global topological charge as the integral over the clover definition of the topological charge density. This definition gives a properly renormalized observable when the flow time t is fixed in physical units.We use a treelevel Symanzik improved gauge action. Our temperatures range from below Tc up to 4Tc. Here Tc is the critical temperature, which is the quantity used for scale setting. The critical temperatures for different lattice spacings were determined in earlier work [19,20]. For the whole temperature range we keep the spatial lattice size approximately at L=2/Tc. We checked with dedicated high volume runs at 1.5Tc and 2Tc that this volume is sufficiently large to have negligible finite volume corrections on Q2. This is shown in Fig. 1. Our spatial geometry is L×L×2L to enable tests of subvolume methods which will be reported separately – here we only use the full volume. For all temperatures we have three lattice spacings (aT)−1=5,6,8 to be able to perform an independent continuum extrapolation for every temperature. The local heatbath/overrelaxation algorithm is used for the update, one sweep consists of 1 heatbath and 4 overrelaxation steps. We found that the autocorrelation time of the topological charge depends weakly on (aT), i.e. if the temperature is increased by decreasing the lattice spacing. The number of update sweeps between measurements was chosen in accordance with the autocorrelation time. Table 1 lists the simulation points with the number of sweeps.We integrated the Wilson flow numerically to a maximum flowtime of about 8t≈1/(2Tc2) for all temperatures. Fig. 2 gives the dependence of the susceptibility on the flow time for T=2Tc. While in the continuum limit the result is independent of the choice of the flow time t, different choices have very different lattice artefacts. For small flow times the different lattice spacings give very different results. For larger flow times the expected plateau behavior can be observed for each lattice spacing and the lattice artefacts also decrease significantly. The choice of the flow time brings in some arbitrariness into the analysis, however the continuum result should not depend on this choice once t is fixed in physical units. But this is certainly a subleading source of error compared to the statistical error due to the rare topology tunneling events at high temperatures. In this analysis we choose a temperature dependent flow time for the evaluation of Q2 as(9)8t={1/(1.5Tc)2,T<1.51/T2,T≥1.5. For low/high temperatures this means a temperature independent/dependent flow time. This choice is safely in the expected plateau region for all temperatures.The resulting values for the susceptibility are plotted in Fig. 3. This plot also gives the result of a global continuum extrapolation using a set of temperatures and the 6parameter power law ansatz(10)χt=(χ0+χ0′a2)(TT0+T0′a2)b+b′a2, where χ0, T0, and b are fit parameters giving the continuum limit. χ0′, T0′, and b′ are fit parameters describing the deviation from the continuum limit. The power law form of the fit is motivated by the expected high temperature behavior of the susceptibility. For temperatures close to Tc this is only an empirical fit which seems to describe the lattice results quite well. The fit parameter T0 is included as a consistency check and should give 1 in units of Tc. This is satisfied by the fit result. The variation between different choices for the starting temperature of the fit range Tmin/Tc=1.3,1.5,1.7 gives an estimate of the systematic error of the result. The best fit parameters are(11)χ0=0.11(2)(1),b=−7.1(4)(2),T0=1.02(4)(2), where the first error is the statistical, the second is the systematic. The pointwise continuum extrapolation is consistent with the global fit, evidently the latter has smaller errors for large temperatures. Note that though a controlled continuum extrapolation is possible using three lattice spacings, estimating the systematic uncertainty of this extrapolation would require at least one more lattice resolution.In a recent analysis [15] the topological susceptibility was calculated using the techniques [21,22]. The calculation was carried out at two temporal extensions, corresponding to two lattice spacings at each temperature. The exponent b=−5.64(4) was found which differs from our value. Note however, that our temperature range is larger, thus we are closer to the applicability range of the DIGA. Furthermore, the two lattice spacings were not sufficient for a controlled continuum extrapolation thus uncertainties related to this final step are not included in the result of [15].We have also determined the second important coefficient b2 of the axion potential, characterizing its anharmonicity, by measuring the observableb2,t=−〈Q4〉−3〈Q2〉212〈Q2〉. The result is plotted in Fig. 4.3Comparison between lattice and DIGA resultsIn this section, we confront the lattice results with the ones obtained from the DIGA framework. For the sake of completeness let us collect first the available formulas for the latter [23–30]. At very high temperatures, far above the QCD phase transition, it makes sense to infer the θ dependence of QCD from the DIGA, in which the partition function, for any nf, can be written as [24],(12)Z(θ,T)≃∑nI,nI¯1nI!nI¯!ZInI+nI¯(T)exp[iθ(nI−nI¯)], where ZI=ZI¯ is the contribution arising from the expansion of the path integral around a single instanton I (antiinstanton I¯). It follows directly that the potential has the form(13)V(A,T)≃χ(T)(1−cosθ)θ=A/fA, from which one infers(14)b2(T)≃−112. This can be confronted rightaway with our lattice results, cf. Fig. 4. Similar to Ref. [13] we find that the prediction from the DIGA for b2 is reached already at surprisingly low values of T/Tc≳1.The whole temperature dependence of the axion potential arises in the DIGA through the topological susceptibility, which in this case is explicitly given by(15)χ(T)≃ZI(T)+ZI¯(T)V=2∫0∞dρD(ρ)G(πρT), in terms of the instanton size distribution at zero temperature, D(ρ), and a factor G(πρT) taking into account finite temperature effects. The former is known in the framework of the semiclassical expansion around the instanton for small αs(μr)ln(ρμr) and ρmi(μr), where αs is the strong coupling, μr is the renormalization scale and mi(μr) are the running quark masses. To oneloop accuracy, it is given by22For quenched QCD the number of light quarks is nf=0; the general formula can be found explicitly in e.g. Ref. [30], which contains a pioneering confrontation of cooled lattice data on D(ρ) with the twoloop RG improved DIGA result.(16)D(ρ)≡dMS‾ρ5(2παMS‾(μr))6exp(−2παMS‾(μr))×(ρμr)β0[1+O(αMS‾(μr))], with(17)dMS‾=e5/6π2e−4.534122;β0=11.At finite temperature, electric Debye screening prohibits the existence of largescale coherent fields in the plasma, leading to the factor [26,27],(18)G(x)≡exp{−2x2−18A(x)}, with(19)A(x)≃−112ln[1+(πρT)2/3]+α[1+γ(πρT)−3/2]−8, and α=0.01289764 and γ=0.15858, in Eq. (15). This factor cuts off the integration over the size distribution in Eq. (15) at x=πρT∼1 and ensures the validity of the DIGA at large temperatures, at which αs(πT)≪1.Collecting all the factors, the topological susceptibility, in the oneloop DIGA, reads(20)χ(T)≃2dMS‾(πT)4(μrπT)11I(2παMS‾(μr))6×exp(−2παMS‾(μr))[1+O(αMS‾(μr))], with(21)I=∫0∞dxx6G(x)=0.267271.This result, however, still suffers from a sizeable dependence on the renormalization scale μr, reflecting the importance of the neglected twoloop and higher order contributions. In fact, it is tamed by taking into account the ultraviolet part of the twoloop correction. The latter has been calculated in Ref. [29] and shown to have exactly the form that the gauge coupling becomes a parameter running according to the renormalization group (RG). Therefore, the ultimate, all order result for the topological susceptibility becomes independent of μr, for μr→∞. At two loop, the corrections amount to a factor(22)(ρμr)(β1−12β0)αMS‾(μr)/(4π);β1=102, in D(ρ). Therefore, this RG improvement can be taken into account by replacing the factor I in Eq. (20) by(23)I˜=(μrπT)−30αMS‾(μr)/(4π)∫0∞dxx6−30αMS‾(μr)/(4π)G(x). In fact, exploiting the twoloop RG improvement, the μr dependence is heavily reduced, as is obvious from Fig. 5, where we have used as a natural renormalization scale(24)μr=κ/ρm=κπT/1.2, with ρm being approximately the maximum of the integrand of I˜, and varied the remaining free parameter κ between 0.6 and 2. The renormalization scale dependence appears to be highly reduced in the regimes >3Tc and <Tc. However, in an intermediate region, ∼Tc–2Tc, it is comparable in size to the one at oneloop.We present in Table 2 the powerlaw behavior predicted by the twoloop RGI DIGA at various temperatures, which can be compared to the fit (11) to the continuum lattice result. As far as the overall normalization of the DIGA result for χ is concerned, one still expects a large uncertainty in the temperature range available from the lattice. In fact, at these temperatures, αs is not small, see Table 3. Apart from the ultraviolet part, there will be a finite part of the twoloop correction which will affect mainly the overall normalization of χ and will depend on the temperature only logarithmically. Unfortunately, this finite part is not known, yet. Therefore, when comparing to the continuum extrapolated lattice results, we allow a multiplicative factor K to account for this uncertainty, i.e. we absorb the remaining higher loop uncertainties by replacing(25)[1+O(αMS‾(μr=κπT/1.2))]→K(T/Tc) in Eq. (20) and the corresponding twoloop RGI expression. Clearly, the Kfactor should approach unity at very large T/Tc.Fig. 6 nicely illustrates the agreement between the DIGA and the lattice result, if a Kfactor of order ten is included.33K factors of order ten to fifty are not uncommon even at nexttoleading order in ordinary perturbative QCD, see e.g. Ref. [32]. More precisely, fitting the lattice continuum data with the rescaled DIGA expression in the temperature range T/Tc≥1, one finds(26)K=8.9±0.7,at 95%CL, while a fit in the temperature range T/Tc≥2 yields(27)K=7.9±3.3,at 95%CL. Apparently, in the temperature range accessible to the lattice, the higher order corrections to the prefactor of the DIGA are still appreciable, but there are indications of a trend that the Kfactor gets smaller, as expected, towards larger values of T/Tc.The Kfactor strongly depends on the value of Tc//ΛMS‾(nf=0): it reduces to one for Tc//ΛMS‾(nf=0)≃1.03. However, the latter value is about 3 sigma below the central value determined in Ref. [20], Tc//ΛMS‾(nf=0)≃1.26(7).4ConclusionsThis paper presents lattice and DIGA calculations of the θ and temperature dependence of the free energy density of QCD in the quenched limit, i.e. for infinite quark masses. In the lattice approach the temperature ranges from 0.9Tc to 4Tc and thus significantly extends former results. The comparison of the quenched continuum extrapolated lattice data and the DIGA framework has led to two highly nontrivial findings. The first one is the observation that the exponent of the temperature dependence of χ(T) is correctly described by the DIGA, even all the way down to Tc (at least within our error bars), cf. Fig. 7 (top). The second nontrivial finding is about the prefactor of this temperature dependence: the oneloop based DIGA prefactor is off by a large factor of about ten (we call this a Kfactor, a familiar notion in perturbative QCD predictions). This is expected, since in the considered temperature range the strong coupling is still sizeable. However, such a Kfactor uncertainty has not been accounted for in any of the previous studies of the axion dark matter abundance, e.g. Refs. [8,9]. Nevertheless, there is a piece of physics information, which essentially fixes this Kfactor. As expected, the topological susceptibility changes only a little from T=0 to T=Tc and drops further on for T>Tc. Actually, the quenched data is very well described if one takes χ(T)/χ(0)=1 at T/Tc∼1 in order to fix the Kfactor44We obtain K=11.7−3.1+5.5 by taking χ(Tc)/χ(0)=1, where the errors come from varying κ in the (0.5,2) interval, and K=7.9±3.3, by taking χ(T)/χ(0)=1 at T/Tc=0.92, for κ=1. and then uses the exponent of the DIGA framework for the rest of the temperature dependence in the high temperature plasma phase, cf. Fig. 7 (top).Unquenched lattice simulations with physical quark masses are about twotothree orders of magnitude more CPU intensive than quenched ones. In addition one expects much smaller topological susceptibilities and larger cutoff effects. If one combines all these cost factors one ends up with a very CPU demanding project. Thus, it is of extreme importance to give an estimate of the ranges in temperature and topological susceptibility of interest for axion cosmology.This can be done by exploiting the two highly nontrivial observations we discussed above: we set χ(Tc)/χ(0)=1 and use the exponent suggested by the DIGA framework, which is readily available also for full QCD, for the temperature dependence.55Qualitatively, both the approximate constancy of χ(T)/χ(0) for T/Tc≲1 as well as the rapid fall off for T/Tc≳1 have been observed already in pioneering lattice studies in full QCD [12,14]. Our finding is shown in the lower panel of Fig. 7, where also the result with the DIGA prefactor and that of the interacting instanton liquid model are depicted. Fortunately, the estimated value of the Kfactor does not affect very strongly the extraction of the values of the axion mass of interest for axion dark matter, because of the steepness of the falloff of χ(T), cf. Fig. 7 (bottom).There are two important messages, which dominantly influence any future lattice study. The first one is that one needs somewhat larger T/Tc values than in the quenched case making the calculation quite CPU demanding. The other important message is that using dynamical QCD one has to determine about one order of magnitude smaller values of χ(T)/χ(0) than in the quenched case. Since already χ(0) is far less in the unquenched case than in the quenched one, the smallness of the needed χ(T)/χ(0) makes the task even more CPU demanding. The planning of any future unquenched project should consider the estimates listed above.AcknowledgementsThe authors wish to thank G. Moore for useful discussions. Computations were performed on JUQUEEN at FZJülich and on GPU clusters at Wuppertal and Budapest. This project was funded by the DFG grant SFB/TR55. by OTKA under grant OTKANF104034 and by the National Science Foundation under Grant No. NSF PHY1125915. S.D.K. is funded by the “Lendület” program of the Hungarian Academy of Sciences (LP201244/2012). M.D. acknowledges support from the Studienstiftung des Deutschen Volkes and J.R. from the Ramon y Cajal Fellowship 201210597 from the Spanish Ministry of Economy and Competitivity.References[1]S.WeinbergA new light boson?Phys. Rev. Lett.40197822322610.1103/PhysRevLett.40.223[2]F.WilczekProblem of strong P and T invariance in the presence of instantonsPhys. Rev. Lett.40197827928210.1103/PhysRevLett.40.279[3]R.D.PecceiH.R.QuinnCP conservation in the presence of instantonsPhys. Rev. Lett.3819771440144310.1103/PhysRevLett.38.1440[4]J.PreskillM.B.WiseF.WilczekCosmology of the invisible axionPhys. Lett. B120198312713210.1016/03702693(83)906378[5]L.F.AbbottP.SikivieA cosmological bound on the invisible axionPhys. Lett. B120198313313610.1016/03702693(83)90638X[6]M.DineW.FischlerThe not so harmless axionPhys. Lett. B120198313714110.1016/03702693(83)906391[7]P.AriasD.CadamuroM.GoodsellJ.JaeckelJ.RedondoA.RingwaldWISPy cold dark matterJ. Cosmol. Astropart. Phys.1206201201310.1088/14757516/2012/06/013arXiv:1201.5902[8]M.S.TurnerCosmic and local mass density of invisible axionsPhys. Rev. D33198688989610.1103/PhysRevD.33.889[9]K.J.BaeJ.H.HuhJ.E.KimUpdate of axion CDM energyJ. Cosmol. Astropart. Phys.0809200800510.1088/14757516/2008/09/005arXiv:0806.0497[10]O.WantzE.P.S.ShellardAxion cosmology revisitedPhys. Rev. D82201012350810.1103/PhysRevD.82.123508arXiv:0910.1066[11]O.WantzE.P.S.ShellardThe topological susceptibility from grand canonical simulations in the interacting instanton liquid model: chiral phase transition and axion massNucl. Phys. B829201011016010.1016/j.nuclphysb.2009.12.005arXiv:0908.0324[12]B.AllesM.D'EliaA.Di GiacomoTopological susceptibility in full QCD at zero and finite temperaturePhys. Lett. B483200013914310.1016/S03702693(00)005566arXiv:heplat/0004020[13]C.BonatiM.D'EliaH.PanagopoulosE.VicariChange of θ dependence in 4D SU(N) gauge theories across the deconfinement transitionPhys. Rev. Lett.11025201325200310.1103/PhysRevLett.110.252003arXiv:1301.7640[14]V.G.BornyakovE.M.IlgenfritzB.V.MartemyanovV.K.MitrjushkinM.MüllerPreusskerTopology across the finite temperature transition studied by overimproved cooling in gluodynamics and QCDPhys. Rev. D8711201311450810.1103/PhysRevD.87.114508arXiv:1304.0935[15]E.BerkowitzM.I.BuchoffE.RinaldiLattice QCD input for axion cosmologyPhys. Rev. D923201503450710.1103/PhysRevD.92.034507arXiv:1505.07455[16]R.KitanoN.YamadaTopology in QCD and the axion abundancearXiv:1506.00370[17]M.LuscherProperties and uses of the Wilson flow in lattice QCDJ. High Energy Phys.08201007110.1007/JHEP08(2010)071arXiv:1006.4518M.LuscherJ. High Energy Phys.03201409210.1007/JHEP03(2014)092Erratum[18]M.LuscherP.WeiszPerturbative analysis of the gradient flow in nonabelian gauge theoriesJ. High Energy Phys.02201105110.1007/JHEP02(2011)051arXiv:1101.0963[19]G.CellaG.CurciA.VicereB.VignaThe SU(3) deconfining phase transition with Symanzik actionPhys. Lett. B333199445746010.1016/03702693(94)901678arXiv:heplat/9405018[20]S.BorsanyiG.EndrodiZ.FodorS.D.KatzK.K.SzaboPrecision SU(3) lattice thermodynamics for a large temperature rangeJ. High Energy Phys.07201205610.1007/JHEP07(2012)056arXiv:1204.6184[21]L.Del DebbioH.PanagopoulosE.Vicaritheta dependence of SU(N) gauge theoriesJ. High Energy Phys.08200204410.1088/11266708/2002/08/044arXiv:hepth/0204125[22]S.DurrZ.FodorC.HoelblingT.KurthPrecision study of the SU(3) topological susceptibility in the continuumJ. High Energy Phys.04200705510.1088/11266708/2007/04/055arXiv:heplat/0612021[23]G.'t HooftComputation of the quantum effects due to a fourdimensional pseudoparticlePhys. Rev. D1419763432345010.1103/PhysRevD.18.2199.3G.'t HooftPhys. Rev. D181978219910.1103/PhysRevD.14.3432Erratum[24]C.G.CallanJr.R.F.DashenD.J.GrossToward a theory of the strong interactionsPhys. Rev. D171978271710.1103/PhysRevD.17.2717[25]C.W.BernardGauge zero modes, instanton determinants, and QCD calculationsPhys. Rev. D191979301310.1103/PhysRevD.19.3013[26]R.D.PisarskiL.G.YaffeThe density of instantons at finite temperaturePhys. Lett. B97198011011210.1016/03702693(80)905596[27]D.J.GrossR.D.PisarskiL.G.YaffeQCD and instantons at finite temperatureRev. Mod. Phys.5319814310.1103/RevModPhys.53.43[28]M.LuscherA semiclassical formula for the topological susceptibility in a finite space–time volumeNucl. Phys. B205198248310.1016/05503213(82)903716[29]T.R.MorrisD.A.RossC.T.SachrajdaHigher order quantum corrections in the presence of an instanton background fieldNucl. Phys. B255198511510.1016/05503213(85)901312[30]A.RingwaldF.SchremppConfronting instanton perturbation theory with QCD lattice resultsPhys. Lett. B459199924925810.1016/S03702693(99)006826arXiv:heplat/9903039[31]K.G.ChetyrkinB.A.KniehlM.SteinhauserDecoupling relations to O (alphas**3) and their connection to lowenergy theoremsNucl. Phys. B5101998618710.1016/S05503213(97)006494arXiv:hepph/9708255[32]M.RubinG.P.SalamS.SapetaGiant QCD Kfactors beyond NLOJ. High Energy Phys.09201008410.1007/JHEP09(2010)084arXiv:1006.2144[33]M.KawasakiK.SaikawaT.SekiguchiAxion dark matter from topological defectsPhys. Rev. D916201506501410.1103/PhysRevD.91.065014arXiv:1412.0789[34]Y.AokiG.EndrodiZ.FodorS.D.KatzK.K.SzaboThe order of the quantum chromodynamics transition predicted by the standard model of particle physicsNature443200667567810.1038/nature05120arXiv:heplat/0611014[35]Y.AokiZ.FodorS.D.KatzK.K.SzaboThe QCD transition temperature: results with physical masses in the continuum limitPhys. Lett. B6432006465410.1016/j.physletb.2006.10.021arXiv:heplat/0609068[36]Y.AokiS.BorsanyiS.DurrZ.FodorS.D.KatzS.KriegK.K.SzaboThe QCD transition temperature: results with physical masses in the continuum limit IIJ. High Energy Phys.06200908810.1088/11266708/2009/06/088arXiv:0903.4155[37]S.BorsanyiZ.FodorC.HoelblingS.D.KatzS.KriegC.RattiK.K.SzaboIs there still any Tc mystery in lattice QCD? Results with physical masses in the continuum limit IIIJ. High Energy Phys.09201007310.1007/JHEP09(2010)073arXiv:1005.3508