The Age of the Universe with Globular Clusters: Reducing Systematic Uncertainties
David Valcin, Raul Jimenez, Licia Verde, Jose Luis Bernal, Benjamin D. Wandelt
PPrepared for submission to JCAP
The Age of the Universe with GlobularClusters: reducing systematicuncertainties
David Valcin, a,b
Raul Jimenez, a,c
Licia Verde, a,c
Jos´e Luis Bernal, d Benjamin D. Wandelt e,f,g a ICC, University of Barcelona, Mart´ı i Franqu`es, 1, E08028 Barcelona, Spain b Dept. de F´ısica Qu`antica i Astrof´ısica, University of Barcelona, Mart´ı i Franqu`es 1, E08028Barcelona, Spain c ICREA, Pg. Lluis Companys 23, Barcelona, 08010, Spain. d Department of Physics and Astronomy, Johns Hopkins University, 3400 North CharlesStreet, Baltimore, Maryland 21218, USA e Sorbonne Universit´e, CNRS, UMR 7095, Institut d’Astrophysique de Paris, 98 bis bd Arago,75014 Paris, France. f Sorbonne Universit´e, Institut Lagrange de Paris (ILP), 98 bis bd Arago, 75014 Paris,France. g Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue, 10010, NewYork, NY, USA.E-mail: [email protected], [email protected], [email protected],[email protected], [email protected]
Abstract.
The dominant systematic uncertainty in the age determination of galactic globu-lar clusters is the depth of the convection envelope of the stars. This parameter is partiallydegenerate with metallicity which is in turn degenerate with age. However, if the metal con-tent, distance and extinction are known, the position and morphology of the red giant branchin a color-magnitude diagram are mostly sensitive to the value of the depth of the convectiveenvelope. Therefore, using external, precise metallicity determinations this degeneracy andthus the systematic error in age, can be reduced. Alternatively, the morphology of the redgiant branch of globular clusters color magnitude diagram can also be used to achieve thesame. We demonstrate that globular cluster red giant branches are well fitted by values ofthe depth of the convection envelope consistent with those obtained for the Sun and thisfinding is robust to the adopted treatment of the stellar physics. With these findings, theuncertainty in the depth of the convection envelope is no longer the dominant contributionto the systematic error in the age determination of the oldest globular clusters, reducingit from 0 . .
23 or 0 .
33 Gyr, depending on the methodology adopted: i.e., whether re-sorting to external data (spectroscopic metallicity determinations) or relying solely on themorphology of the clusters’s color-magnitude diagrams. This results in an age of the Universe t U = 13 . +0 . − . (stat . ) ± . . . ) at 68% confidence level, accounting for the formationtime of globular clusters and its uncertainty. An uncertainty of 0.27(0.36) Gyr if added inquadrature. This agrees well with 13 . ± .
02 Gyr, the cosmological model-dependent valueinferred by the Planck mission assuming the ΛCDM model. a r X i v : . [ a s t r o - ph . GA ] F e b ontents α MLT α MLT on the RGB 15B Effect of opacities and nuclear reaction rates 17C Parameter constrains for all GCs 18D Assessing robustness: Tests of microphysics 19
A Bayesian analysis to estimate, as precisely and accurately as possible, the absolute ages ofgalactic globular clusters (GCs) with resolved stellar populations was presented in a recentpaper [1]. The objective of the work in Ref. [1] was to use the age of the oldest GCsto obtain an estimate of the age of the Universe insensitive to cosmology and, in turn,constrain cosmological models. By using the morphology of the color-magnitude diagram(CMD) and not just the luminosity of the main sequence turn off, we showed that the age,distance and metal content could be determined without relying on external data sets. Byusing the extensive set of GC CMDs from the ACS-HST survey, an age for the oldest GCsof t GC = 13 . ± . . ) ± . . ), at 68% confidence level, was obtained. As it isapparent, the uncertainty in the age is dominated by the systematic uncertainty, which inturn dominates the estimate of the age of the Universe (see also [2–4]).The most important “known unknown” contributing to the systematic uncertainty bud-get is the value of the depth of the convection envelope in low mass stars (those around solarmass). This by itself contributes to 60% of the systematic uncertainty (see Table 2 in Ref. [2],the rest of the systematic error budget being due to reaction rates and opacities). The prob-lem is at follows: low mass stars have fully convective and turbulent envelopes (Reynoldsnumber (cid:39) ) and because of this, a full hydro-dynamical solution is prohibitive for a largegrid of stellar models varying parameters like mass, metallicity and age (this can be done– 1 –or a single star, and it is done when modelling the Sun, but cannot -yet- be extended to afull library of stellar models). Instead, one models the gradient of the convective transportby assuming 1D geometry and following a convective cell as it dissolves into the envelope.With this approach, the equations of stellar structure contain five independent differentialequations for five variables and an extra parameter: the mixing length ( α MLT ). The valueof the mixing length parameter has to be obtained from fits to observations. While thereis some recent theoretical progress on matching 3D to 1D models for low mass stars (seee.g., Ref. [13]) which could open the possibility to eliminate the need to empirically calibrate α MLT , this step cannot be avoided at the moment.The standard way to determine the free mixing length parameter is to fit it to theSun and assume this value applies to all stars. This, of course, is an assumption that is notguaranteed to hold for stars in GCs which have very different metallicity than the Sun. Whilethe adopted value for the mixing length parameter does not affect the age determinationdirectly, it indirectly does so via degeneracies with other parameters, most notably metallicity.The approach of Ref. [1] is to propagate a variation of the mixing length parameter over awide range into the systematic error budget for the age, as adopted and motivated by e.g.,Ref. [2]. However, as anticipated in Refs. [1, 8] this does not need to be the case as themixing length can, at least in principle, be constrained from the morphology of the CMD ofGCs. We address this methodology in this article.The rationale behind this approach is simple. As a first step, let us assume that themetallicity of the GC has been determined (for example, via spectroscopic observations). Fora fixed metallicity, the color of the red giant branch (RGB) in a theoretical CMD dependsmostly on the value of α MLT (see Fig. 1). As we will show below, other parameters affectingstellar structure do not modify the color of the RGB as much as α MLT . Hence the spreadin color of the RGB in the CMD of a single GC yields an upper limit to the star-to-starvariations in α MLT (assuming the scatter is solely due to spread in α MLT values). Withoutresorting to external constraints on the GC metallicity, the metallicity determination for eachGC of Ref. [1], obtained assuming a fixed fiducial mixing length parameter value, should beaffected by an (unknown) shift induced by the α MLT choice. Now, if the distance is known,GCs of similar estimated metallicity can be suitably aligned on the theoretical CMD (or anHR diagram). In this case, the dispersion in color of the RGB can be used to constrain the α MLT range, in particular if one assumes that the full scatter is solely induced by α MLT .In this paper we quantify this dispersion and constrain the range of α MLT values. Thissignificantly reduces the systematic uncertainty in the age estimation of GCs, making themixing length contribution to the statistical error budget now subdominant to other system-atics, and propagates into a determination of the age of the Universe with systematic errorsreduced by ∼ Following Ref. [1], we consider the globular clusters from the HST-ACS catalog which wegroup into three metallicity samples (according to the best fit metallicity value): 12 clusterswith [Fe/H] < .
0, 11 clusters with − < [Fe/H] < -1.75 and 15 with − . < [Fe/H] < -1.5. One of the clusters in Ref. [1], (NGC6715) shows clear signs of multiple populationsin the RGB, its metallicity is just at the high edge of the range considered in this work andits age determination has a very large error-bar. We exclude this cluster from the present See the discussion in section 2.2 in Ref. [2] and references therein. – 2 – igure 1 . Variation of the HR diagram due to changes in the mixing length parameter α MLT (∆ α =0.1) for a star with a fixed initial mass and metallicity. As we can see, the color (effective temperature)of the RGB is the most sensitive region to the mixing length value, while the sub-giant branch is theleast sensitive part of the HR diagram. analysis, leaving us with a sample of 38 clusters (including it does not change the results inany significant way, due to the large uncertainties on its age).These clusters also have spectroscopically-determined metallicities from Ref. [2] (foronly 16 of the 38 clusters) and [14, 15] (for all 38); the latter determination is complete forour purposes, then it is the main one we use here. The comparison between spectroscopicmetallicity and the metallicities estimated by Ref. [1] is shown in Fig. 2. The comparisonwith Ref. [2] metallicities can be found in Fig. 5, of Ref. [1].In order to compare the clusters with each other and with the stellar tracks, it isnecessary to convert the apparent magnitudes of each star into absolute magnitudes. To dothis we use the best fits obtained in Ref. [1] (see their Table 3, Appendix E) for the absorptionand the distance modulus. In the three panels of Fig. 3, we show the (absolute) CMD of allthe clusters, as if they were all at the same distance (i.e., 10 pc), subdividing the sample inthe three metallicity ranges listed above. Clearly, the RGBs of the different clusters appearnicely aligned in each metallicity interval. When considering the combined distribution ofall the stars from all the GC in each sample, there are many possible contributions to theresulting width of the RGB: photometric errors, errors in the best-fit parameter values (e.g.,distance, absorption) used to generate the plots, errors in metallicity determinations and thespread in metallicity within the selected sample, and the effects of variations in α MLT (whichis the quantity we are interested in).
Several parameters affect the color of the RGB in low-mass stars ( < M (cid:12) ). The mostimportant one is the metallicity content, as can be seen from Fig. 8 in Ref. [1], whereone can appreciate that the color of the RGB is quite insensitive to age, but sensitive to– 3 – igure 2 . Metallicity determination of Ref. [1] vs spectroscopic metallicity determination of Ref. [15]of the 38 clusters in our sample. The 1:1 line guides the eye. The scatter around this relation is σ [Fe / H] = 0 .
09, with no indication of a dependence on metallicity or systematic bias (i.e., a systematicdeviation from the 1:1 line). metallicity. The next leading parameter determining the color of the RGB is the mixinglength ( α MLT ), as we will show below by varying the parameters of the microphysics inthe star and comparing the resulting stellar tracks (see Fig. 1). As recognized by Ref. [2],this parameter dominates the systematic uncertainty when obtaining ages of GCs using theluminosity of the main sequence turn off.Here we explore two approaches to reduce this uncertainty. The first one is based onexternal metallicity determinations, the second one uses only internal information from themorphology of the CMDs of each GC.
We can appreciate in Fig. 2 that there is a good agreement between spectroscopic and CMD-estimated metallicities, with no indication of a dependence on metallicity and no indicationof a systematic bias (i.e., a systematic deviation from the 1:1 line). When a linear fit isperformed, the best-fit line has a slope of 0 . ± .
15 and an intercept of − . ± .
28. Whenforcing the line to have a slope of 1 a possible systematic normalization shift in the metallicitydetermination is − . ± . σ [Fe / H] = 0 . α MLT =1 .
938 fixed a priori, and there is an expected degeneracy between α MLT and metallicity,a systematically incorrect choice of α MLT would have biased the metallicity determination, The value of the α MLT parameter assumed depends on the stellar code used. In Ref. [1], the values reportedwere according to the convention of the codes used there i.e.,
DSED . Here we convert to the convention of the
MESA (and
JimMacD ) codes, see below. For reference the solar value for α MLT are 1.938, 2, 1.4, respectively.This change, however, only amounts to a shift; the relevant quantity for our argument is the interval or rangeadopted, which does not depend on the convention. – 4 – igure 3 . Top panel: Combined CMD of the GCs with metallicities below [Fe/H] < −
2, shifted tobe at the same distance using the best fit distances and absorption from Ref. [1]. Middle and bottompanels: same as top panel but for metallicity ranges as indicated. and hence the age. On the other hand, an incorrect value of α MLT with a cluster-to-clustervariation would induce a scatter in the comparison of Fig. 2, which we estimate to be σ [Fe / H] =0 . α MLT by attributing the full scatter of– 5 –his relation to variations in the mixing length parameter.
When using the full morphology of the CMD, it should be possible to treat α MLT as anadditional model parameter to be constrained by the data, as mentioned in Ref. [1]. Here wedevelop this idea. We start by illustrating the sensitivity of the RGB to stellar parametersand in particular to α MLT using the publicly available 1D stellar structure and evolutioncodes
MESA [5] and the
JimMacD code [6]. These codes compute the 1D equations of stellarstructures and evolve them in time, thus providing the structure of a star and its position andevolution in time in the theoretical CMD for given initial mass and chemical composition.The numerical solution of the stellar structure equations of both codes are the same. Themain difference between the two codes is that
MESA is a modern 1D stellar code that employsnew updates in opacities and nuclear reaction rates. On the other hand, the older versionof
JimMacD that we use adopts different values for opacities and nuclear reaction rates and adifferent formulation of the mixing length formulation. We use these two different codes toillustrate that recent updates in nuclear reaction rates and opacities do not affect our results.We then proceed to constrain the mixing length parameter from the color of the RGB andquantify its spread for the oldest GCs.
The output of
MESA allows us to plot directly the evolution of a star in the theoretical HRdiagram (effective temperature vs log luminosity), but if we want to compare our tracks withthe GCs observations we must transform the luminosity into the magnitudes correspondingto the filters of the HST-ACS catalog (F606W and F814W). The transformation is carriedout in 4 stages:1. convert luminosity to bolometric magnitude using the formula M bol = − . L (cid:63) L where L (cid:63) is the star’s bolometric luminosity in watts and L is the zero point luminosity= 3 . × W ,2. produce bolometric correction tables using the Vega calibration (Calspec Alpha Lyrae) and the atmospheric models of Castelli & Kurucz [9] for various metallicity values, or the Bolometric Corrections code from Casagrande & VandenBerg [10] . This is furtherdiscussed in Appendix A, where we show that the choice of bolometric correction isunimportant for our purpose,3. interpolate bolometric corrections (BC) using effective temperature, surface gravity,and extinction where we assume to have corrected the data for extinction and thereforetake E(B - V) = 0 ,4. transform bolometric magnitude into absolute magnitude M filter = M bol − BC filter where M filter and BC filter are respectively the absolute magnitude and the bolometriccorrection in the desired filter. https://ssb.stsci.edu/cdbs/current calspec/ For the rest of the paper we choose to work with bolometric corrections from Ref. [10]. – 6 – .2.2 Stellar tracks
The CMD of a GC is an isochrone which covers a range of initial masses for the stars.Isochrones are generally more complicated to model than stellar tracks and are usually avail-able for more limited choices of parameters (such as α MLT , opacities, reaction rates, etc.)than stellar tracks. However, it is well known that a stellar track for a fixed mass, corre-sponding to the mass of the main sequence turn off, will approximate very well the parts ofCMD of the GC which we consider here, in particular the upper part of the RGB (wherethe dependence on the mass is very small). The RGB for an isochrone of a fixed age canalways be approximated by a stellar track for a suitable choice of the initial mass. We fixthe age for the isochrone to be 13.32 Gyr and find that M = 0 . M (cid:12) yields the best fit forour purposes. A variation of the value of the initial mass has an effect on an isochrone verysimilar to changing the age, and mainly affects the main sequence and its turn off. This isfurther discussed in Appendix A, especially Fig. 7, but see also Figure 8 in Ref. [1].The isochrone-track agreement is illustrated in Fig. 4. In both panels, the dotted linescorrespond to isochrones, and the solid lines, to stellar tracks for representative values ofmetallicity and mixing length parameter. For the magnitude range ( M (cid:38) α MLT considered here. In our analysis, we also include a cut for magnitudes M (cid:38) −
2; for M < M > − MESA software package [5], evolving astar from pre main-sequence to a luminosity limit of log (L max ) = 3.25, sufficient to comparethe tip of the RGB for different values of the mixing length parameter. Among the variousparameters needed to configure the tracks, the initial mass and metallicity are the two mostimportant.We calculate the stellar tracks for 8 initial values of metallicity spanning a range from Z = 0 . Z = 0 . − . < [Fe / H] < − . to sample the [Fe / H]range of the 38 GCs in our sample (see Table 3 in Ref. [1]). Except for its tip, the RGB variesonly slightly with changes in the metallicity. This is why we argue that, for a metallicityrange comparable to current uncertainties from CMD studies, the scatter around the theRGB may be used to constrain the maximum range of α MLT . The impact of the choice ofmass, metallicity and the stellar model on the stellar tracks and the CMD is explored in moredetail in Appendix A (see Figure 8).Besides adopting the solar abundance scale, we use the same configuration parametersas those presented in Ref. [7] (see table 1 and section 3 for further explanations). All otherparameters have been used with their default values. Table 1 summarizes the value of therelevant parameters used in our study. We also compute stellar tracks with the old stellarcode
JimMacD [6] in order to show the robustness of the position of the RGB to input physicsand how a different modelling of stellar structure doesn’t affect our conclusions. Whilethe inital mass value adopted using the
MESA code is M = 0 . M (cid:12) , this exact value isnot available for JimMacD . Hence, when comparing the two codes directly, we also considerthe closest available value in
JimMacD which is M = 0 . M (cid:12) , and use initial metallicity Recall that [Fe/H] = log ( Z/Z (cid:12) ) where Z (cid:12) = 0 .
02 for
JimMacD and Z (cid:12) = 0 . MESA . – 7 – igure 4 . The upper part of the RGB of an isochrone is very well approximated by the stellar trackfor a suitable choice of mass (and age, but the sensitivity to age is small). For each isochrone the ageis equal to 13.32 Gyr and for each track the mass is equal to 0 . M (cid:12) . In this figure the dotted linescorrespond to isochrones, the solid lines to stellar tracks. The top panel shows the agreement for fewrepresentative values of metallicity; the bottom panel shows that the differences between isochronesand tracks are much smaller than those induced by a change in α MLT of the magnitude of interest. Z = 0 . − MESA code. α MLT
The grid of stellar tracks enables us to estimate how changes in key parameters (metallicityand α MLT ) affect the color of the RGB. For relatively small changes around fiducial values wecan linearize this dependence and report an estimate of d C /dα MLT (where C denotes the colorof the RGB at a given magnitude) and d C /d Z or d C /d [Fe / H], obtained as finite differencesfor few representative magnitudes. These quantites are only indicative, but can help buildphysical intuition about the effect we want to describe. We find that these quantities, asexpected, depend on the magnitude; results are reported in table 2. The color response to– 8 – arameter Value
Initial mass M Z [0.00005, 0.004] ∆ Z = 0 . × Initial metallicity Z (default)Mixing length α MLT [1.2, 2.8], ∆ α = 0 . Table 1 . Values for the stellar parameters used when computing the
MESA stellar tracks. metallicity ( Z ) is linear to a very good approximation, the response does not depend on the∆ Z adopted to compute the derivative, but it shows some dependence on the fiducial choice of α MLT . We find that we can approximately rescale the derivative to different fiducial values as( α MLT /α fid ) d C /d Z | α fid = const. if α fid is around the solar value ( α MLT (cid:12) ∼ Z and [Fe/H] is not linear hence we report ∆C∆[Fe / H] = Z ∆C∆ Z for two represenativemetallicities; using the linearized relation for [Fe/H] is valid only for small shifts. The α MLT dependence on the other hand is not linear, so the linearized approximation is only valid forsmall changes ∆ α < .
1, which is what we adopt here. M = 0 M = − ∆C∆Z | Z =0 .
115 285 ∆C∆Z | Z =0 .
106 297 ∆C∆[Fe / H] | [Fe / H]= − . ∆C∆[Fe / H] | [Fe / H]= − . ∆C∆ α MLT ( Z = 1 . × − ) 0.116 0.202 ∆C∆ α MLT ( Z = 2 . × − ) 0.125 0.225 Table 2 . Response of the RGB color to changes in metallicity and mixing length parameter arounda fiducial model for the stellar track of α = 2. Here M denotes the magnitude in F606W filter andcolor, C, denotes the difference F606W-F814W. The response to α MLT show some dependence on thefiducial metallicity so we report several representative values.
We can then proceed to estimate (approximately, given the linearization assumption im-plicitly made when computing derivatives by finite differences) what change ∆ α is needed tokeep the color of the RGB unchanged under a change in metallicity ∆ [Fe / H] . This is visualizedin Fig. 5 where one can directly appreciate that a change in metallicity of ± ∆ [Fe / H] = 0 . ± ∆ α (cid:46) .
04. This upper limit in the required shift in α MLT provides a conservative estimateof the uncertainty in this parameter introduced by its degeneracy with [Fe/H]. This is inbroad agreement with the values reported in Tab. 2 if one keeps in mind that the results inthe table are approximated because evaluated at a fixed magnitude value implicitly assuminglinear dependence of the color on the parameters and that in practice the overall effect shouldbe seen as a suitably weighted average shift over the magnitude range M >
0. In section 4– 9 – igure 5 . Response of the color of the RGB to a change in metallicity and a change in α MLT fordiscrete representative values. The top panels show the full RGB range while the bottom panels area zoom in around magnitude M (solid horizontal line). This illustrates that to keep the color of theRGB unchanged for a small change in metallicity ∆ [Fe / H] = 0 .
09, (very close to the scatter evaluatedin sec. 3.1, Fig. 2) around the fiducial ([Fe / H] = − .
75 and α MLT = 2), the corresponding change in α MLT is given by ∆ α / ∆ [Fe / H] (cid:39) − .
4. This is the value we adopt, as it corresponds to that of theeffective metallicity of our GC sample. we conservatively adopt dα MLT /d [Fe / H] = − . α MLT . In order to select only stars in the RGB for each cluster we define a band of color aroundthe best fit obtained in Ref. [1]. We choose a value of ∆C = 0.06 large enough to include allthe stars in the red giant branch and narrow enough to remove most of the stars belongingto the horizontal branch. We also define a magnitude cut, M , corresponding to the startof the RGB. The collection of stars selected for all clusters in the low metallicity sample isshown in Figure 6 where the 12 low metallicity GCs are plotted on top of each other. Indeedthe best fit α MLT can be biased and the dispersion σ α can be increased by the dispersionin color induced either by outliers or by misclassified stars belonging to the horizontal orasymptotic branch. As the number of stars decreases when we move towards the brightestmagnitudes, we define two additional magnitudes cuts to study the dispersion in α MLT . Thefirst one M = M − . M = M − . M − .
0. A value ofthe scatter in the color of the RGB changing across different magnitude cuts would indicateoutlier contamination. Because the correspondence isochrone-stellar tracks is less precisefor M >
0, we adopt results using the M cut and report the M cut results only in theAppendices. – 10 – igure 6 . The different cuts in luminosity used to define the spread computed in Table 3 and Table 4.The gray points represents all the stars in the CMD of the combined 12 GCs of the [Fe / H] < We compute the RGB dispersion for each individual cluster and the combination of all theGC in each of the three metallicity samples. If we assume that the photometric dispersion isGaussian around the isochrone (as argued in Ref. [1]), we can define the dispersion of colorby measuring the color distance to the fit: σ color = (cid:118)(cid:117)(cid:117)(cid:116) N (cid:88) i=1 (C istar − C ifit ) (3.1)where C istar and C ifit respectively correspond to the color of a given star (index i ) and thecolor of the track at the same magnitude, and N is the number of stars in the magnitudeinterval considered. As we compute the dispersion for brighter magnitude cuts ( M and M ), the number of stars N decreases and the distribution becomes sometimes dominatedby Poisson noise for individual clusters. Therefore, we set a limit of N = 10 under which thedispersion is not computed. To estimate the scatter in the mixing length the ideal approachwould be to perform a Bayesian analysis with the mixing length as a free parameter (akinto the approach of Ref. [1]). However stellar grids are defined only for a specific value of α MLT ; recomputing full grids of different mixing lengths and would be very computationallyexpensive and is beyond the scope of this paper. We proceed instead as follows. We start bymapping the response of the RGB track to discrete changes in α MLT (see Appendix B).A (small) shift in the α MLT value might happen when matching tracks with isochrones.We then perform a least square fit between stellar tracks for different values of the mixinglength parameter and the isochrone obtained from the best fit parameters of Ref. [1] for eachGC. First we select all the evolutionary equivalent points of the isochrone for the magnituderange considered in this work. Then we interpolate all the computed tracks on the samemagnitude interval. Finally we compare the tracks and the isochrone at each EEPs magnitudeand do a least square fit for α MLT . This yields for each cluster a “best match” α MLT since it– 11 –s obtained by comparing two different models (track to isochrone). For the combination ofall the clusters within a metallicity sample, since a best fit isochrone is not available we donot compare tracks to the best fit isochrone, instead we perform a fit to all the stars in theCMD for the full sample following the same fitting procedure as in Ref. [1] but using tracksas the theoretical model instead of isochrones and varying only α MLT . We refer to this asbest fit α MLT since it is obtained via a model to data comparison.To compute the scatter for each cluster, each star in the RGB is assigned a value of themixing length parameter, α iMLT , obtained by linear interpolation of the values correspondingto the two closest tracks at the same magnitude, and a corresponding shift S i as the differencebetween the interpolated α iMLT and the corresponding best match α MLT .The dispersion in α MLT is then given by σ α = (cid:118)(cid:117)(cid:117)(cid:116) N (cid:88) i=1 (S i ) . (3.2)For each sample we take the mean of the scatters of the individual clusters in the sample.The results can be seen in in Table 3, but more detailed results on a cluster by cluster basisare reported in Appendix C, Table 4. The considerations of Sec. 3.1 indicate that the adopted fiducial value for α MLT adopted byRef. [1] is not biased. The findings of Sec. 3.3 and Fig. 5 yield an estimate for ∆ α / ∆ [Fe / H] = − . α MLT required to compensate a change in metallicity in order to keepthe color of the RBG unchanged) around the fiducial value for α MLT and for the effectivevalue of [Fe/H] and magnitude cut. In Sec. 3.1 the scatter between spectroscopic and CMD-estimated metallicity is estimated to be σ [Fe / H] = 0 . σ spec . met α = 0 .
04 (the superscrip stresses that this is computed resorting to spectroscopicmetallicity data).Without resorting to external data sets, we can proceed empirically. The spread incolor (color scatter) of the RGB for each GC, is generated by a combination of effects,the dominant one being measurement and photometric errors, as well as all other stellarparameters variations, which are subdominant. If the color scatter is attributed solely tochanges in mixing length parameter it can be used to provide a conservative estimate of starto star variations in α MLT . This statement assumes that measurement and photometric errorsand variations of stellar parameters are all random and uncorrelated. In principle, if differentsources of scatter are suitably (anti) correlated, this would not necessarily be conservativeestimate. We deem this possibility very unlikely. Combining GC of similar metallicity, thescatter around the RGB also accounts for possible cluster-to-cluster variations as RGBs ofclusters of similar metallicities are mostly affected by α MLT . In this case there is an additionalcontribution to the scatter arising from the fact that to convert the observed CMD to theabsolute one we have used the best-fit values of absorption and distance, which may beaffected by their own measurement errors.Results are reported in table 4. First note that the scatter in color ∼ .
02 is muchsmaller than the initial color cut of ∆C = 0 .
06, which clips only the tails of the color distri-bution beyond 3 σ s and thus confirming that the initial cut does not affect the estimate of the– 12 – Fe / H] < − . ,
12 clustersMetallicity α ( M ) best fit α ( M ) best match σ α ( M ) σ α ( M ) Z = 0 . Z = 0 . Z = 0 . Z = 0 . − . < [Fe / H] < − . ,
11 clustersMetallicity α ( M ) best fit α ( M ) best match σ α ( M ) σ α ( M ) Z = 0 . Z = 0 . Z = 0 . Z = 0 . − . < [Fe / H] < − . ,
15 clustersMetallicity α ( M ) best fit α ( M ) best match σ α ( M ) σ α ( M ) Z = 0 . Z = 0 . Z = 0 . Z = 0 . Table 3 . Summary of the mixing length best fit and best match for the M cut and dispersion foreach of the metallicity samples (full information for individual clusters is available in appendix C).Each sample spans a range in metallicities, and we report results for several fiducial metallicity valuescovering the range. The values closer to the effective [Fe/H] metallicity of the sample are flagged bythe asterisks. The best fit value for α MLT depends very weekly on metallicity (both the metallicity ofthe sample and the adopted fiducial metallicity), the scatter does not show any significant dependenceon metallicity. scatter. In Appendix C we report the results for the dispersion in color and in mixing lengthparameter for each cluster and for each of our three sub-samples combined. We note that allGCs of similar metallicity have a similar value of the dispersion. Finally, Table 3 reports themixing length best fit and dispersion for each of the metallicity samples. The best fit valuefor α MLT depends very weekly on metallicity (both the metallicity of the sample and theadopted fiducial metallicity), the scatter does not show any significant dependence on metal-licity. The scatter for each metallicity sample (which include (cid:38)
10 clusters) is comparablewith the individual cluster scatter indicating that there is not additional cluster-to-clustervariation. We adopt a, suitably weighted combined scatter across the three metallicity sam-ples of σ CMD α = 0 .
15 as a conservative estimate of the α MLT scatter estimated from the CMDof the clusters in the sample. Recall that we have attributed the full color scatter of the RGBto α MLT , when the color scatter include contributions from measurement errors, photometricerrors as well as variations of all other model parameters.A suite of tests ensuring the robustness of these results to several commonly adoptedassumptions is presented in the appendices. In particular Appendix A discusses the impactof chosen fiducial values of mass and metallicity; Appendix B compares different stellarcodes and Appendix D tests the effects of microphysics modelling. In summary, there is noindication that stars in old GCs have values for the depth of the convection envelope differentfrom those obtained for the Sun. For the 38 GC considered here we find that the preferredvalue for the mixing length parameter is α MLT = 1 . ± .
04 (or ± .
15) with (without)resorting to spectroscopic metallicity determinations, where the reported error is estimatedconservatively. – 13 –
Conclusions and implications for the age of the Universe
An estimate of the age of the Universe from the age of the oldest globular clusters waspresented in Ref. [1]: the age of the oldest clusters being t GC = 13 . ± . . ) ± . . )Gyr, and the inferred age of the Universe t U = 13 . +0 . − . (stat . ) ± . . ) Gyr. The dominantcontribution to the error on this quantity is due to the systematic uncertainty in the depthof the convention envelope (the mixing length parameter) accounting for 0.3 (i.e. 60%) ofthe 0.5 systematic error budget. Ref. [1] adopted a range in α MLT corresponding to the fullrange considered in Ref. [2] which in the convention of this paper corresponds to 2∆ α = 0 . α MLT , in order to provide a more realistic estimate of the uncertainty on thisparameter. We have shown that the range used in Ref. [1] include values that do not fit theobserved properties of the GCs in our sample. After studying the degeneracy between α MLT and metallicity, we have estimated an upper limit for the uncertainty of α MLT for our sample: σ spec . met α = 0 .
04 or σ CMD α = 0 .
15 (depending whether using external spectroscopic metallicitydeterminations or not). It is interesting to note that recently, Ref. [11] performed a Bayesiancalibration of the mixing length parameter α using mock and real data of the Hyades opencluster and found an average value (cid:104) α (cid:105) = 2 . ± .
05. This result is in good agreement withour findings.With this reduction of the dominant systematic contribution to the age determination(from 0.3 to 0.13 or 0.034 Gyr), the mixing length parameter cease to be the dominantcontribution to the uncertainty; now the leading systematic uncertainties are due to nuclearreaction rates and opacities.Thanks to the reduction in the systematic error budget achieved in this work, we con-clude that the age of the oldest globular clusters is t GC = 13 . ± . . ) ± . . . )Gyr, which corresponds to an age of the Universe of t U = 13 . +0 . − . (stat . ) ± . . . )Gyr., an uncertainty of 0.27(0.36) Gyr if statistical and systematic errors are added in quadra-ture. This determination of the age of the Universe is cosmological-model agnostic, in thesense that it does not depend in any significant way on the cosmological model adopted, and isin good agreement with the cosmological model-dependent determination of t U = 13 . ± . z ∼ Acknowledgments
We thank the stellar modelers for making their stellar models publicly available. This work issupported by MINECO grant PGC2018-098866-B-I00 FEDER, UE. LV acknowledges supportby European Union’s Horizon 2020 research and innovation program ERC (BePreSySe, grantagreement 725327). JLB is supported by the Allan C. and Dorothy H. Davis Fellowship. Thework of BDW is supported by the Labex ILP (reference ANR-10-LABX-63) part of the IdexSUPER, received financial state aid managed by the Agence Nationale de la Recherche, aspart of the programme Investissements d’avenir under the reference ANR-11-IDEX-0004-02;and by the ANR BIG4 project, grant ANR-16-CE23-0002 of the French Agence Nationale To propagate exactly an error in α MLT into an error in age one would have to reproduce the full calculationsof Ref. [2], which goes well beyond the scope of this paper. Ref. [2] shows that ∆ α = ± .
35 yields a∆ age = ± .
3. Here we simply rescale the age uncertainty according to our newly determined α MLT uncertainty. – 14 –e la Recherche. The Center for Computational Astrophysics is supported by the SimonsFoundation.
Figure 7 . Effect of varying mass for a star with a fixed metallicity and mixing length. The greypoints represent the CMD of all the clusters in the low metallicity sample. The lines are stellar tracksfor an age of 13.32 Gyr corresponding to different initial masses. If the tracks are interpreted asisochrones such a spread in mass would correspond to a (widely unrealistic) range in age from 11to 30 Gyr. Note that the RGB morphology is very insensitive to mass (and age). The α MLT valueadopted here is 1.9 as it is the closest in our grid to the
DSED code solar value.
A Impact of mass, metallicity and α MLT on the RGB
As mentioned in section 3.2.2, mass and metallicity are key parameters for our purpose ofconstraining the mixing length parameter. It is possible to compare the effect of mass (fora track) to the effect of age (for an isochrone). The initial mass of the star influences thetime spent on the main sequence but affects very little the red giant branch. In figure 7 weexplore a range of masses from M = 0 .
65 to 0 . M (cid:12) with ∆ M = 0.05.Metallicity affects the color of the RGB: an increase in metallicity will result in a tiltof the RGB towards higher C. In Figure 8 we show the effect of varying metallicity for atrack with a fixed mass and mixing length, compared to a change in mixing length for fixedmass. This figure illustrates that for a metallicity interval comparable with that of each ofour samples, the mixing length is the determining parameter in the color of the RGB.The value of the α MLT parameter is stellar code dependent. The conversion betweendifferent code-conventions is just a shift, while the relevant quantity for our argument hereis the interval or range adopted which is convention-independent. In Ref. [1] we used theisochrones from the
DSED model for a solar value for α MLT and here stellar tracks froman independent code. An even slightly incorrect conversion would result into adopting anincorrect fiducial α MLT value and possibly an over-estimate of the color scatter. To map theresponse of the RGB to changes in mixing length parameter values, we compute tracks fordiscrete values of α MLT (from 1.2 to 2.8 in steps of 0.1) as illustrated in Figure 9 where thetracks are centered around α MLT = 2. – 15 – igure 8 . Effect of varying metallicity for a track with a fixed mass and mixing length, compared toa change in mixing length for fixed mass. Top panel: Grey points: the combined CMD of the sampleof 12 clusters with metallicity Fe/H < −
2. Solid lines tracks color-coded by metallicity for a rangespanning the low metallicity sample. Blue lines: effects of changes in α MLT . Bottom pane: as for toppanel but for the intermediate metallicity sample.
Bolometric corrections are used to transform from observed colors to theoretical effec-tive temperature and viceversa. These corrections, if not accurate enough, can lead to anadditional systematic uncertainty. To quentify this effect, we use two different sets of bolo-metric corrections [9, 10]. As it is shown in Fig. 9 both lead to the same transformation ofcolors to effective temperature. We therefore do not propagate any additional uncertaintydue to bolometric corrections. – 16 – igure 9 . Stellar tracks are shown for a star with initial mass 0.80 M (cid:12) and Z = 0 . α = 2 . The spread of the RGB roughlycorresponds to ∆ α ∼ .
1. Top panel: Using the Casagrande & VandenBerg [10] bolometric correction,Bottom panel: Using bolometric correction computed from the Castelli & Kurucz 2003 [9] atmosphericspectra. on the relevant part of the RGB the two plots are virtually indistinguishable.
B Effect of opacities and nuclear reaction rates
Uncertainties in opacities and nuclear reaction rates also contribute to the systematic errorbudget on age determinations, but are expected to be sub-dominant compared to the effectof the mixing length parameter. Here we illustrate this by resorting to available outputs of adifferent stellar code where these quantities are different from those assumed in
MESA . In thiscase we chose the old version (1996) of the
JimMacD code and stellar tracks [6], which usesvery different stellar opacities, boundary conditions and mixing-length formulation.
JimMacD tracks are available for initial mass 0.8 M (cid:12) (hence we compare directly with MESA for thisvalue of the mass) and for a coarse grid of α MLT values (recall that the solar value for α MLT for
JimMacD is 1.4, which is however not available on the provided grid). This is shown in– 17 – igure 10 . Response of the RGB to changes in α MLT across different codes. As before, the graypoints correspond to all the stars in the CMD of the 12 clusters in the low metallicity sample. Theblack solid line corresponds to
MESA track for α MLT = 2, M = 0 . M (cid:12) , Z = 0 . α MLT = 1 . α = ± .
35 around the solarvalue. The solid lines in colors correspond to
JimMcD tracks for the same mass and metallicities andfor shifts from the solar value as indicated in the legend. The poor fit at the very tip of the RGB isdue to the lack of cool opacities in the code unlike the modern
MESA stellar code.
Fig. 10. Note that the
JimMacD code fails to fit the tip of the RGB because of the lack of coldopacities, not available at that time. Where this is not a relevant effect, tracks for solar valuesof the mixing langth parameters agree well, and the response of the RGB color to changesin α MLT is very similar across the two codes: we find that for M = 0 and for the same mass0.8 M (cid:12) , metallicity and α MLT step sampled by
JimMacD tracks, ∆C / ∆ α MLT = − .
12 and − .
13 for
MESA and
JimMacD respectively. We therefore conclude that the differences do notsignificantly bias the RGB color or consequently the recovery of an α MLT value consistentwith solar, further supporting the robustness of the results reported in the main text.
C Parameter constrains for all GCs
We double check that the α MLT that best fits the RGB is consistent with the adopted fiducialvalue of Ref. [1] and consequently the fiducial adopted here too. An incorrect fiducial α MLT value would yield to biases and possibly an over-estimate of the color scatter. Table 4 reportsfor each cluster the color scatter as a function of magnitude cut, the best match α MLT , thescatter in this quantity as a function of magnitude cut and fiducial assumed metallicity. Thetable also report the best fit α MLT for the combination of the GC in each sample and therecomputed scatter with respect to this quantity rather than the best match α MLT . Thedifferences, however are unimportant.The tables presented here compement Table 3 in the main text. The best fit α MLT iswell consistent with the solar value adopted by
DSED ( α MLT = 1 . ample 1, 12 clusters with [Fe/H] < Z = 0 . Z = 0 . Z = 0 . Z = 0 . σ color ( M ) σ color ( M ) α best match σ α ( M ) σ α ( M ) σ α ( M ) α best match σ α ( M ) σ α ( M ) σ α ( M ) α best match σ α ( M ) σ α ( M ) σ α ( M ) α best match σ α ( M ) σ α ( M ) σ α ( M )NGC 2298 0.0231 0.0236 1.8 0.22 0.19 N/A 1.9 0.22 0.16 N/A 1.9 0.22 0.17 N/A 2 0.21 0.16 N/ANGC 4590 0.0198 0.0158 1.9 0.21 0.14 N/A 2 0.19 0.13 N/A 2 0.2 0.13 N/A 2.1 0.18 0.14 N/ANGC 4833 0.018 0.0196 1.8 0.16 0.15 0.09 1.9 0.15 0.14 0.08 1.9 0.16 0.14 0.09 2 0.16 0.14 0.08NGC 5053 0.0155 0.0195 1.9 0.15 0.14 N/A 2 0.18 0.16 N/A 2 0.16 0.15 N/A 2.1 0.21 0.18 N/ANGC 6341 0.0172 0.0175 1.9 0.17 0.15 0.05 2 0.16 0.16 0.08 2 0.16 0.15 0.04 2 0.16 0.15 0.04NGC 6397 0.0254 0.0277 1.9 0.21 0.2 N/A 1.9 0.22 0.22 N/A 1.9 0.24 0.24 N/A 2 0.22 0.2 N/ANGC 6426 0.0239 0.0287 1.9 0.23 0.24 N/A 2 0.22 0.21 N/A 2 0.22 0.22 N/A 2.1 0.21 0.2 N/ANGC 6779 0.0241 0.0258 1.9 0.21 0.18 0.09 1.9 0.22 0.19 0.11 2 0.2 0.17 0.09 2 0.21 0.18 0.1NGC 7078 0.0219 0.0204 2 0.19 0.15 0.08 2 0.2 0.16 0.08 2.1 0.18 0.15 0.08 2.1 0.18 0.15 0.07NGC 7099 0.0268 0.0211 1.9 0.25 0.16 0.11 1.9 0.27 0.18 0.14 2 0.22 0.14 0.09 2 0.23 0.14 0.12Palomar 15 0.0285 0.0284 1.9 0.27 0.21 N/A 2 0.24 0.17 N/A 2 0.25 0.18 N/A 2 0.26 0.2 N/ATerzan 8 0.0169 0.0189 1.9 0.17 0.14 N/A 1.9 0.17 0.14 N/A 2 0.18 0.15 N/A 2 0.17 0.14 N/AAll 12 GCs 0.0226 0.0223 1.89 0.2 0.17 0.08 1.95 0.2 0.17 0.1 1.98 0.2 0.17 0.08 2.03 0.2 0.17 0.08GC name σ color ( M ) σ color ( M ) α best fit σ α ( M ) σ α ( M ) σ α ( M ) α best fit σ α ( M ) σ α ( M ) σ α ( M ) α best fit σ α ( M ) σ α ( M ) σ α ( M ) α best fit σ α ( M ) σ α ( M ) σ α ( M )All 12 GCs 0.0226 0.0223 1.9 0.21 0.17 0.1 1.9 0.22 0.19 0.13 2 0.2 0.16 0.1 2 0.2 0.17 0.12Sample 2, 11 clusters with -2.0 < [Fe/H] < -1.75 Z = 0 . Z = 0 . Z = 0 . Z = 0 . σ color ( M ) σ color ( M ) α best match σ α ( M ) σ α ( M ) σ α ( M ) α best match σ α ( M ) σ α ( M ) σ α ( M ) α best match σ α ( M ) σ α ( M ) σ α ( M ) α best match σ α ( M ) σ α ( M ) σ α ( M )Arp 2 0.0164 0.0156 1.8 0.16 0.13 N/A 1.9 0.15 0.11 N/A 1.9 0.15 0.11 N/A 2 0.16 0.12 N/ANGC 4147 0.0155 0.0151 1.8 0.14 0.12 N/A 1.8 0.14 0.12 N/A 1.9 0.16 0.11 N/A 1.9 0.15 0.11 N/ANGC 5024 0.0173 0.0193 1.8 0.17 0.16 0.07 1.9 0.16 0.14 0.05 1.9 0.16 0.14 0.06 2 0.16 0.14 0.06NGC 5466 0.0154 0.0159 1.8 0.14 0.12 N/A 1.9 0.14 0.11 N/A 1.9 0.13 0.11 N/A 1.9 0.13 0.11 N/ANGC 6093 0.0212 0.022 1.8 0.21 0.16 0.08 1.8 0.21 0.16 0.1 1.9 0.2 0.14 0.08 1.9 0.2 0.15 0.09NGC 6101 0.0181 0.0215 1.8 0.15 0.15 0.07 1.8 0.16 0.16 0.1 1.9 0.16 0.14 0.06 1.9 0.16 0.14 0.08NGC 6144 0.027 0.0308 1.8 0.24 0.21 N/A 1.8 0.25 0.22 N/A 1.8 0.26 0.24 N/A 1.9 0.22 0.19 N/ANGC 6254 0.0226 0.0219 1.8 0.24 0.15 0.08 1.9 0.21 0.14 0.09 1.9 0.22 0.14 0.08 1.9 0.22 0.14 0.08NGC 6535 0.0201 0.0155 1.8 0.21 0.09 N/A 1.9 0.17 0.07 N/A 1.9 0.17 0.07 N/A 1.9 0.18 0.08 N/ANGC 6541 0.0216 0.0208 1.9 0.19 0.14 0.07 1.9 0.2 0.14 0.07 2 0.18 0.14 0.08 2 0.18 0.13 0.07NGC 6809 0.0177 0.02 1.8 0.15 0.15 N/A 1.8 0.15 0.15 N/A 1.8 0.15 0.15 N/A 1.9 0.17 0.14 N/AAll 11 GCs 0.0205 0.0217 1.81 0.18 0.14 0.07 1.85 0.18 0.14 0.08 1.89 0.18 0.14 0.07 1.93 0.18 0.13 0.08GC name σ color ( M ) σ color ( M ) α best fit σ α ( M ) σ α ( M ) σ α ( M ) α best fit σ α ( M ) σ α ( M ) σ α ( M ) α best fit σ α ( M ) σ α ( M ) σ α ( M ) α best fit σ α ( M ) σ α ( M ) σ α ( M )All 11 GCs 0.0205 0.0217 1.8 0.2 0.16 0.09 1.9 0.19 0.14 0.07 1.9 0.19 0.14 0.07 1.9 0.19 0.15 0.09Sample 3, 15 clusters with -1.75 < [Fe/H] < -1.5 Z = 0 . Z = 0 . Z = 0 . Z = 0 . σ color ( M ) σ color ( M ) α best match σ α ( M ) σ α ( M ) σ α ( M ) α best match σ α ( M ) σ α ( M ) σ α ( M ) α best match σ α ( M ) σ α ( M ) σ α ( M ) α best match σ α ( M ) σ α ( M ) σ α ( M )IC4499 0.0186 0.0215 1.9 0.16 0.14 N/A 1.9 0.16 0.14 N/A 1.9 0.16 0.15 N/A 2 0.16 0.13 N/ANGC 3201 0.0241 0.0248 1.8 0.18 0.14 0.08 1.8 0.17 0.13 0.06 1.8 0.18 0.13 0.07 1.9 0.2 0.14 0.08NGC 5139 0.0192 0.0211 1.9 0.17 0.14 0.06 1.9 0.17 0.14 0.07 1.9 0.17 0.14 0.09 2 0.18 0.14 0.06NGC 5272 0.0157 0.0202 1.7 0.15 0.14 0.08 1.8 0.13 0.13 0.05 1.8 0.13 0.13 0.06 1.8 0.14 0.13 0.07NGC 5286 0.0221 0.0228 1.9 0.2 0.16 0.07 1.9 0.21 0.17 0.09 2 0.18 0.15 0.06 2 0.18 0.15 0.07NGC 5986 0.0268 0.0285 1.8 0.23 0.17 0.07 1.8 0.23 0.19 0.08 1.9 0.22 0.16 0.08 1.9 0.22 0.16 0.07NGC 6218 0.0161 0.0141 1.7 0.15 0.07 N/A 1.7 0.15 0.07 N/A 1.8 0.12 0.08 N/A 1.8 0.12 0.07 N/ANGC 6584 0.0173 0.0194 1.8 0.16 0.14 0.05 1.8 0.17 0.15 0.06 1.9 0.14 0.13 0.06 1.9 0.14 0.13 0.05NGC 6656 0.0233 0.0243 1.9 0.19 0.14 0.07 1.9 0.2 0.15 0.1 2 0.17 0.13 0.05 2 0.17 0.13 0.07NGC 6681 0.0206 0.0214 1.8 0.19 0.15 0.09 1.9 0.18 0.14 0.1 1.9 0.18 0.13 0.09 1.9 0.18 0.14 0.09NGC 6752 0.0174 0.0204 1.7 0.17 0.12 0.11 1.8 0.14 0.11 0.08 1.8 0.14 0.11 0.09 1.8 0.15 0.11 0.11NGC 6934 0.017 0.0211 1.8 0.15 0.14 0.08 1.8 0.15 0.14 0.06 1.8 0.16 0.14 0.06 1.9 0.14 0.13 0.08NGC 7006 0.0197 0.0258 1.8 0.17 0.18 0.09 1.9 0.17 0.17 0.1 1.9 0.17 0.17 0.09 1.9 0.17 0.17 0.09NGC 7089 0.0176 0.0184 1.8 0.18 0.14 0.05 1.9 0.16 0.13 0.06 1.9 0.16 0.13 0.05 1.9 0.16 0.13 0.05Ruprecht 106 0.0223 0.0202 1.9 0.22 0.13 N/A 1.9 0.23 0.14 N/A 2 0.18 0.11 N/A 2 0.19 0.11 N/AAll 15 GCs 0.021 0.0251 1.81 0.18 0.14 0.07 1.85 0.17 0.14 0.08 1.89 0.16 0.13 0.07 1.91 0.17 0.13 0.07GC name σ color ( M ) σ color ( M ) α best fit σ α ( M ) σ α ( M ) σ α ( M ) α best fit σ α ( M ) σ α ( M ) σ α ( M ) α best fit σ α ( M ) σ α ( M ) σ α ( M ) α best fit σ α ( M ) σ α ( M ) σ α ( M )All 15 GCs 0.021 0.0251 1.8 0.19 0.16 0.1 1.8 0.19 0.17 0.12 1.9 0.18 0.15 0.1 1.9 0.18 0.15 0.1 Table 4 . Dispersion in color and mixing length for the 4 fiducial adopted metallicity values. Theresults are given for each cluster in each of the sub-samples and all combined. For the combined samplewe report both the means of the best match α MLT values and the best-fit α MLT . For magnitude binswith less than 10 stars, the dispersion α is not computed (N/A). The color dispersion is not affectedby the choice of initial metallicity. D Assessing robustness: Tests of microphysics
Throughout this work, we have assumed a fixed α MLT through the stellar tracks: this valuedoes not change in itself the duration of the lifetime of the star. The lifetime of low-massstars to helium flash is mostly dependent on mass and metallicity. In fact, the lifetime inGyr to the He flash can be fitted with 5% precision by the formula (adopting a helium massfraction Y = 0 . t = − . − .
691 log(
M/M (cid:12) ) + 11 .
327 log(1 . − Z ) + 0 .
870 log(0 . Z ) (D.1)Beside the mixing length and the metallicity, which, as we have seen, determine thecolor of the RGB to leading and next-to-leading order, other parameters can also affect the– 19 – igure 11 . The effect on the RGB of (extreme) changes of other microphysics parameters besidesthe mixing length. The three panels are subsequent zoom ins in the relevant part of the CMD. Notethat the effect is much smaller than that of α MLT , and well below the intrinsic broadening of the RGB(gray points) and thus below star-to-star color variations. color of the RGB, although with a weaker dependence: initial mass, initial helium massfraction, overshooting of the convection depth, mass loss, rotational mixing and element– 20 –iffusion. These are kept fixed at their fiducial values in our main analysis, here we quantifytheir effect. We also explore changing the mixing length theory to the Cox formulation [17] .We compute stellar tracks by varying a single parameter at the time while keeping the otherparameters fixed to the fiducial configuration (see section 3.2.2) with a mixing length value α = 1 . M = 0 . Y = 0 .
04, are heavily disfavouredby the data as these parameters also change the main sequence and the MSTO and with suchshifts models do not fit this part of the CMD. We also test different models of convection(Henyey and Cox [17, 18]), the implementation or not of overshoot, and with or withoutLedoux criterion (for more information see Ref. [6] and references therein). The value for themass loss Reimers parameter η = 0 . M and in the initial helium mass fraction ∆ Y have amuch bigger effect than all other changes, which are much more subtle, and small comparedto the intrinsic broadening of the RGB. Unlike α LMT , the other parameters of the mixinglength theory do modify the stellar tracks but maintaining the color of the RGB unaltered.We also illustrate the effect of ∆ α = 0 .
35 , the value very conservatively adopted by [1], and∆ α = 0 . .
04 as estimated by the two approaches presented in the main text. Theseconsiderations indicate that the results reported are robust to any possible changes in keystellar parameters.
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