ALMA-IMF. V. Prestellar and protostellar core populations in the W43 cloud complex

Observations of W43-MM1 at 1.3 mm (Band 6) were carried out in Cycle 2 between July 2014 and June 2015 (project #2013.1.01365.S), with the ALMA 12-m array covering baselines ranging from 7.6 m to 1045 m. W43-MM1 was imaged with a $78\arcsec\times$ 53$\arcsec$ (2.1 pc $\times$ 1.4 pc) mosaic composed of 33 fields. The primary beam FWHM is 26.7$\arcsec$ and the maximum detectable scale is $\sim 12\arcsec$. Observations of W43-MM2 and W43-MM3 were carried out between December 2017 and December 2018 as part of the ALMA-IMF Large Program (project #2017.1.01355.L, PIs: Motte, Ginsburg, Louvet, Sanhueza, see Motte et al. 2022) with similar spatial and spectral setup. Mosaics are composed of 27 fields and the maximum detectable scale is $\sim 11\arcsec$. Our observations with the 7-m array configuration are not used in this work. The combined 12-m + 7-m data indeed have higher noise levels than the 12-m only data (see Pouteau et al. 2022) without bringing information significant for this work, whether on the detection of compact cores or the identification of collimated outflows.

Continuum data for the three regions were processed with CASA 5.4 using the data reduction pipeline developed by the ALMA-IMF consortium (described in detail in Ginsburg et al. 2022). In short, the pipeline performs several iterations of cleaning and phase self-calibration using masks of increasing size and decreasing thresholds. We used the multi-scale multi-frequency synthesis (MS-MFS) method of tclean with two Taylor terms and scales of [0,3,9,27] pixels, corresponding to point sources and several larger scales. For the final clean of W43-MM1, an additional scale of 54 pixels was used.

CO (2–1) cubes of W43-MM2&MM3 were also processed using the ALMA-IMF data pipeline, although without self-calibration (see Cunningham et al. 2023 for more detail on line data reduction). We used the multiscale method of tclean with scales of [0,6,18,54] and [0,4,12,24] pixels for W43-MM2 and W43-MM3, respectively, corresponding to point sources and multiples of the beam size. The resulting cubes have a similar beam of about 0.61$\arcsec\times$ 0.52$\arcsec$. The CO cube of W43-MM1 was presented and published by Nony et al. (2020). Parameters of the various continuum images and CO cubes are summarised in Table 1. Continuum has been subtracted from the cubes in the image plane using the imcontsub task of CASA. CO cubes were created after ALMA-IMF data were reprocessed in QA3 to correct the known spectral normalisation issue¹¹1See ALMA ticket https://help.almascience.org/kb/articles/what-errors-could-originate-from-the-correlator-spectral-normalization-and-tsys-calibration.

The ALMA-IMF pipeline provides two different continuum images (see Ginsburg et al. 2022). The first one, called cleanest, is produced from a selection of channels free of line contamination using the findcont routine of CASA. The second one, called bsens, covers the whole frequency setup to favour the best sensitivity possible but can be contaminated by strong line emission, especially that from the CO (2–1) line. The following analysis of the three W43 regions is based on the deep core extraction method on these maps, developed by Pouteau et al. (2022) for W43-MM2&MM3 and applied to W43-MM1.

In order to determine the completeness level of the source extraction, we performed synthetic source extractions using 1.3 mm continuum maps of each region as background images. In W43-MM2&MM3, Pouteau et al. (2022) evaluated a 90% completeness level of 0.8$\,M_{\odot}$ (see their Appendix C), with 113 cores out of 205 above this level.

We performed similar analyses in W43-MM1 and found a 90% completeness level of 1.6$\,M_{\odot}$ (see Appendix C), consistent with the completeness estimation of Motte et al. (2018b). This higher completeness mass could be due to the brighter and more centrally concentrated continuum emission in W43-MM1 compared to W43-MM2&MM3, which produces higher structural noise (see also Section 4.2). Among the 127 cores of the new core catalogue, 68 fall above the 1.6$\,M_{\odot}$ completeness level. 59 of these (87%) are also detected in the cleanest catalogue of Louvet et al. (subm.) and 48 (71%) are in the core catalogue of Motte et al. (2018b) (see also Appendix B). The complete core sample of W43, with masses above $1.6\,M_{\odot}$, has 126 cores (69 cores in W43-MM1 and 57 cores in W43-MM2&MM3). In addition, 56 cores in W43-MM2&MM3 with masses in the [0.8-1.6] $M_{\odot}$ interval are included in the following analysis.

Molecular outflows are the most common signature of the protostellar phase of cores in the millimetre wavelength range (e.g. Bontemps et al. 1996). Therefore, we used the presence or absence of outflows as the main criterion to identify protostellar cores. In addition to this, most of massive protostellar cores are also associated with compact emission of complex organic molecules, referred to as hot core emission. The presence of hot core emission has been used as a secondary criterion to identify protostellar cores, based on the systematic survey carried out in W43-MM2&MM3 by Bonfand et al. (in prep) and in W43-MM1 by Brouillet et al. (2022).

We assume the protostellar core sample has the same mass completeness as the total core sample (see Section 2.3), because outflows are detected for cores from the lowest to the highest masses (see Tables F2 and F2). In agreement, Nony et al. (2020) suggested that a dense and dynamic environment is the main limiting factor for outflow detection, more than the mass of the driving core. The mass completeness of protostellar core detection is discussed in further detail in Section 4.2. Uncertainties on the characterisation of protostellar cores are taken into account through the uncertainties in outflow detection or attribution (see the distinction between robust and tentative in the following).

Using outflows and hot cores emission, we identified 26.5 $\pm$ 3.5 protostellar cores out of 127 cores in W43-MM1 and 46.5 $\pm$ 4.5 out of 205 cores in W43-MM2&MM3 (see Section 3.1). Cores with neither outflows nor hot core, 100.5 $\pm$ 3.5 cores in W43-MM1 and 158.5 $\pm$ 4.5 in W43-MM2&MM3, are good candidates to be in a preceding evolutionary stage, corresponding to prestellar cores. In the following analysis, one source (#132) is discarded from the sample of prestellar candidates because of its mass, size, and structure (see Section 4.1). Figure 6 displays the cores from both regions of W43 in a mass-to-size diagram. Whereas prestellar cores from W43-MM1 and W43-MM2&MM3 span a similar range of parameters, protostellar cores in W43-MM2&MM3 are detected down to lower masses than those in W43-MM1 (median mass of 1.3$\,M_{\odot}$ versus 8.9$\,M_{\odot}$). This deficit of low-mass protostellar cores in W43-MM1 could be partly explained by a poorer completeness level (1.6$\,M_{\odot}$ versus 0.8$\,M_{\odot}$, see also discussion in Section 4.2). Indeed, when compared above a common completeness level of 1.6$\,M_{\odot}$, protostellar core populations of the two regions have similar median masses of about 9$\,M_{\odot}$ and cannot be statistically distinguished (Kolgomorov-Smirnov (KS) statistic of 0.14 with p-value of 0.94). Similar results are obtained between prestellar core populations (KS statistic of 0.16 with p-value 0.63).

Figure 6 also displays for comparison the mass-size relation of critical Bonnor-Ebert spheres at the median dust temperature of cores ($T_{\rm dust}=23$ K). Following the analysis of the cleanest core catalogue of ALMA-IMF (Motte et al. 2022), we found that 109 cores from the 127 in W43-MM1 (86%) and 119 cores from the 205 in W43-MM2&MM3 (58%) have $M>M_{\rm BE}$ and could be considered as gravitationally bound. These results are in line with the previous observation that cores in W43-MM1 have higher masses compared to W43-MM2&MM3. It has, however, little impact on our analysis, as all cores above the 1.6$\,M_{\odot}$ completeness level in W43-MM1 and W43-MM2&MM3 are bound, as well as 44 out of 56 cores in [0.8-1.6]$\,M_{\odot}$ in W43-MM2&MM3. Furthermore, this evaluation of cores gravitational boundedness should be considered with caution, as Bonnor-Ebert spheres are not the most appropriate models for the massive and dense cores in our observations. The gravitational boundedness of cores will be evaluated with greater precision in subsequent ALMA-IMF studies measuring turbulent support and evaluating the overall energy budget of cores.

In Section 3.2 we compared properties of cores from two regions of W43, W43-MM1 and W43-MM2&MM3, and showed that, with regard to their mass, cores come from the same parent population. In the following, we analyse together W43-MM1 and W43-MM2&MM3 and compare the mass distributions of the combined populations of protostellar and prestellar cores in W43, 73 $\pm$ 8 cores and 259 $\pm$ 8 cores, respectively. The populations of protostellar and prestellar cores in W43 above a completeness threshold of 1.6$\,M_{\odot}$ are 45 $\pm$ 5 and 80 $\pm$ 5, respectively.

From the lateral panels of Fig. 6, one observes that protostellar cores are significantly more massive and smaller in size than prestellar cores. In detail, the median deconvolved sizes and masses of protostellar cores are 1800 au and 8.9$\,M_{\odot}$, respectively, while those of prestellar cores are 2600 au and 2.7$\,M_{\odot}$. As a consequence, the population of protostellar cores is about 10 times denser than that of prestellar cores, with median volume densities of 3.3 $\times$ 10⁷ cm^-3 versus 4.7 $\times$ 10⁶ cm^-3. Our result that protostellar cores are more massive than prestellar cores is consistent with observations in other regions by Massi et al. (2019) and Kong et al. (2021). However, both studies also find that protostellar cores have similar or slightly larger sizes than prestellar cores, in contradiction to what we observe in W43. This discrepancy could be due to differences in the definition taken by each algorithm to extract a source.

In Fig. 7 we compare the numbers of prestellar and protostellar cores in seven mass intervals, ranging from 0.8 to 110$\,M_{\odot}$. The bins have been constructed to preserve a roughly constant number (28 or 29) of cores in each of the five first bins and with two reference masses, 1.6 and 16$\,M_{\odot}$. The former corresponds to the completeness level of W43-MM1, cores in the two first bins below 1.6$\,M_{\odot}$ are exclusively located in W43-MM2&MM3. The latter delimits the last bin and corresponds to the minimum mass of a core able to form high-mass stars, assuming a 50% core-to-star mass efficiency. In Fig. 7, the fraction of cores which is protostellar, $f_{\rm pro}$ is also represented in each bin. Its uncertainty is derived from that on the count of protostellar cores (see Section 3.1). When all protostellar cores in a bin are labelled as robust, a minimum of $\pm 2$ cores is used to compute the uncertainty. The fraction of protostellar cores stays at a roughly constant level of $f_{\rm pro}\simeq 15-20\%$ in the four first bins, between 0.8 and $\simeq 3\,M_{\odot}$. It increases in the subsequent intermediate and high-mass bins, from $f_{\rm pro}=33\pm 5\%$ in [3.3-7] $\,M_{\odot}$ to $f_{\rm pro}=80\pm 3\%$ in [16-110] $\,M_{\odot}$. The $f_{\rm pro}\simeq 15-20\%$ measured between 0.8 and 3 $\,M_{\odot}$ is comparable to that observed in other regions (e.g. 14$\%$ in a sample of Infrared Dark Clouds (IRDCs) of Li et al. 2020), although the global fraction of protostellar cores, $f_{\rm pro}\simeq 35\%$, is higher.

The difference between prestellar and protostellar core populations also results in markedly distinct CMFs. Two cumulative representations of the CMF for the W43 core populations are shown in Figs. 8 and 10. In addition, CMFs for the individual regions W43-MM1 and W43-MM2&MM3 are shown in Fig. E18. In agreement with the above analysis, the protostellar CMF extends up to much higher masses than the prestellar CMF, 109 versus 37$\,M_{\odot}$, respectively. In Fig. 8, histograms comparing the counts of the prestellar, protostellar, and total core populations are represented and fitted with the polyfit method (least square, Sciutto 1989). In Fig. 10, normalised distributions (complementary cumulative distribution functions, C-CDFs) are represented. For the fit, we applied the Maximum Likelihood Estimator (MLE) method of Clauset et al. (2009) implemented in the powerlaw python package (Alstott et al. 2014). In this method, estimations of the power-law index $\alpha_{\rm fit}$ are performed for a given $x_{\rm min}$, the minimum value in the data to include in the fit, which can be provided by the user or set as a free parameter. When $x_{\rm min}$ is set at the completeness level, $x_{\rm min}=1.6\,M_{\odot}$, the high-mass slopes of the prestellar and protostellar CMFs are $\alpha_{\rm fit,pre}=-1.46\pm 0.17$ and $\alpha_{\rm fit,pro,1}=-0.58\pm 0.08$, respectively. When $x_{\rm min}$ is set as a free parameter, the couple of parameters ($x_{\rm min}$, $\alpha_{\rm fit}$) which minimises the KS distance between the data and the fit is provided. For the prestellar CMF, the optimum ($x_{\rm min}$, $\alpha_{\rm fit}$) is ($1.3\,M_{\odot}$, -1.43). The optimum $x_{\rm min}$ is slightly lower than the completeness and the associated slope is in very good agreement with the previous evaluation, $\alpha_{\rm fit,pre}=-1.46$. In contrast, for the protostellar distribution, the optimum $x_{\rm min}$ is found to be 8.2$\,M_{\odot}$, a mass much larger than the completeness level. The MLE slope on [8.2-109]$\,M_{\odot}$, $\alpha_{\rm fit,pro,2}=-1.22\pm 0.23$ is much steeper than the slope on [1.6-109]$\,M_{\odot}$, $\alpha_{\rm fit,pro,1}=-0.58\pm 0.08$, and therefore closer to the prestellar and Salpeter slopes. This result indicates that the protostellar CMF of W43, in its cumulative form, is poorly represented by a single power-law from 1.6 to 109 $\,M_{\odot}$. Interestingly, the protostellar CMF in its differential form is relatively flat, as suggested by the representation of Fig. 7, which supports its top-heavy character.

We also applied the bootstrap procedure, which provides a robust measurements of the most likely CMF power-law index and its uncertainty. In detail, we built $N=3000$ synthetic set of mass generated from a random draw with discount of the measured core masses. The high-mass end of the $N$ associated CMF are then fitted using the MLE method of Alstott et al. (2014). The bootstrapping probability density function of the $N$ fitted slopes is shown in Fig. 10 for the prestellar and protostellar core populations. Two type of uncertainties are included in the procedure. The uncertainties associated with the estimation of the core mass are accounted by drawing according to a Gaussian law the mass of each core in a [$M_{\rm min}$ - $M_{\rm max}$] interval. The maximum and minimum masses of each core, $M_{\rm max}$ and $M_{\rm min}$ respectively, are computed from its measured flux, estimated temperature and dust opacity, plus or minus their associated $1\sigma$ uncertainties (see Sect. 4.2 of Pouteau et al. 2022). The uncertainties associated with the sample incompleteness are accounted by allowing $x_{\rm min}$ to uniformly vary by $\pm 0.2\,M_{\odot}$ from $1.6\,M_{\odot}$, the 90% completeness level. The histograms of the bootstrapping probability density functions are fitted by exponentially modified Gaussians (EMG) with a negative skewness. Their peak $\alpha_{\rm BS}$ and the asymmetrical 1$\sigma$ uncertainties are also reported on Fig. 10. For the prestellar CMF, the bootstrap slope $\alpha_{\rm BS,pre}=-1.46_{-0.19}^{+0.12}$ is identical to the fitted slope and compatible at 1$\sigma$ with the Salpeter slope, -1.35. The bootstrap slope of the protostellar CMF, $\alpha_{\rm BS,pro}=-0.64_{-0.07}^{+0.09}$, is also consistent at 1$\sigma$ with the fitted slope $\alpha_{\rm fit,pro,1}=-0.58\pm 0.08$. It deviates from the Salpeter and prestellar slopes by more than 3$\sigma$. The protostellar CMF fitted with a single power law above the completeness level is therefore significantly top-heavy. From the tests reported above, we nevertheless caution that the latter term should be understood as an over-abundance of cores at high masses rather than a statistical description of a power-law behaviour.

The scarcity of high-mass prestellar cores in various surveys of clumps suggests that their lifetime is, at most, very short (see Motte et al. 2018a). Prestellar cores on the verge of collapse, that is when their turbulent and/or magnetic supports have been dissipated, could live for as short as one free-fall time (Galván-Madrid et al. 2007; Bovino et al. 2021). The present study of W43 also reveals very few high-mass prestellar core candidates. Out of the 18 cores more massive than 16$\,M_{\odot}$, only four are not associated with any outflow or hot core signature (see Section 3.3 and Fig. 7). Here we examine these four more closely. The first, MM1#132 ($M=101\,M_{\odot}$) is probably not a high-mass prestellar core. Its large size (FWHM^dec=5000 au) and lower density (2.4 $\times 10^{7}$ cm^-3) compared to other cores of similar mass suggest that this source is not an individual, gravitationally bound cloud structure. Higher-resolution continuum images obtained using the longest baseline of the data (Brouillet, private com.) have confirmed this structure is not centrally concentrated. From its position at the immediate neighbourhood of the W43-MM1 centre (see Fig. E17), source MM1#132 could be rather part of gas inflows.

In contrast, core MM1#6, with a mass $M=37\pm 5\,M_{\odot}$ and a deconvolved size of FWHM^dec=1800 au, was once considered to be an excellent high-mass prestellar core candidate. Previously characterised³³3In Nony et al. (2018), it has a mass of $M=56\pm 9\,M_{\odot}$ within a deconvolved size of FWHM^dec=1300 au. by Nony et al. (2018), it is located at the south-western tip of the main MM1 filament, in a region where outflow confusion is limited. Molet et al. (2019), however, concluded from a detailed chemical characterisation of core MM1#6 that it could be at the beginning of the protostellar phase, casting some doubt on its true prestellar nature. In addition, two cores with lower masses are good prestellar candidates. Core MM1#134 ($M=21\,M_{\odot}$ and FWHM^dec=2400 au) is located near the massive core MM1#2 (see Fig. E17) and was not found earlier by Motte et al. (2018b), most likely because of the confusion between filaments and cores in this area. Core MM2&MM3#22 ($M=18\,M_{\odot}$ and FWHM^dec=3200 au) lies in the central part of the MM2 subregion (see Fig. 4). In summary, we conclude that at most two to three massive cores in W43-MM1 and W43-MM2&MM3 could be prestellar. This likely represents a statistic for the entire W43 molecular cloud complex, since the three studies regions - MM1, MM2 and MM3 - are its most massive fragments.

Before discussing the physical interpretations of our results, we fist consider some possible biases and discuss the mass completeness of our outflow detections. The comparison led in Section 3.2 revealed a significant deficit of low-mass protostellar cores in W43-MM1 compared to W43-MM2&MM3. In the [0.8-1.6]$\,M_{\odot}$ interval, 11 $\pm$ 1 cores out of 56 (20 $\pm$ 2 %) are protostellar in W43-MM2&MM3 a fraction similar to that measured in the subsequent bins, between 1.6 and 3$\,M_{\odot}$. In the same mass interval, a single core out of 34 (3%) has been characterised as protostellar in W43-MM1. This could result from a large number of misclassified protostellar cores, or from a large fraction of spurious low-mass sources which are currently accounted in the total prestellar population. The first hypothesis is unlikely because the identification of outflows was carried out using the same methods in the two regions and with similarly complex CO lines (see below). The second hypothesis is more likely to play a role. The completeness of core extraction has been found to be worse in W43-MM1 than in W43-MM2&MM3 (90% level of 1.6$\,M_{\odot}$ and 0.8$\,M_{\odot}$, respectively), and contamination of the core sample by spurious sources is likely more problematic in the former region. Indeed, as illustrated by the comparison in Appendix B, low-mass core detection is affected by the quality of the interferometric map. The continuum emission is more centrally concentrated and brighter in W43-MM1 compared to W43-MM2&MM3, which produces brighter sidelobes. It likely prevents for good detections of weakest sources and possibly produces more spurious low-mass prestellar core candidates. Finally, the low number of low-mass protostellar cores in W43-MM1 compared to W43-MM2&MM3 could also result from a different star-formation history (see Section 4.3).

For the combined sample of cores in W43, we showed in Section 3.3 that the fraction of protostellar cores increases with mass, from $f_{\rm pro}\simeq 15-20\%$ between 0.8 and 3 $\,M_{\odot}$ to $f_{\rm pro}=80\pm 3\%$ above 16$\,M_{\odot}$. This result could be affected by the completeness of protostellar core detection, that is the limits of our ability to detect outflows from low-mass cores. Outflow momentum and mass are known to be well correlated with the mass of the driving core (e.g. Bontemps et al. 1996; Beuther et al. 2002; Maud et al. 2015). To test whether the sensitivity limit of CO observations might prevent robust detection of the fainter outflows, we defined and measured a detection intensity for the outflow lobes associated with the lowest mass cores (see Appendix D). We found it to be significantly above 5$\sigma$ and conclude that sensitivity is not limiting the detection of low-mass outflows in W43. Outflows detection is more likely limited by the complexity of CO observations. First, we are relatively insensitive to outflows emitting only at low velocity, where the CO spectra suffer from strong self-absorption or contamination (about 10$\,\rm km\,s^{-1}$ from the central velocity, see Fig. 3). This effect, however, is not expected to affect preferentially low-mass outflows. Then, Nony et al. (2020) pointed out environmental effects as a possible bias for outflow detection. Outflows in the central part of the protocluster, with high gas densities and complex dynamics, are more difficult to identify and attribute because they are smaller and more prone to confusion with other lobes. In regions such as W43-MM1 and W43-MM2 with strong mass segregation

Most of the CMFs studies focusing on prestellar cores have been conducted in nearby ($d<$ 700 pc) regions. These have found good agreements between the CMFs high-mass ends and the Salpeter slope (e.g. Motte et al. 1998; Polychroni et al. 2013; Könyves et al. 2015, 2020; Massi et al. 2019). Among the few studies that compared the prestellar and protostellar populations, Enoch et al. (2008) used a combined sample of cores in three clouds of the Gould Belt and found a prestellar CMF with a high-mass end slope consistent with Salpeter ($\alpha=-1.3\pm 0.4$) and a protostellar CMF that is flatter ($\alpha=-0.8$ with constant $T=15$ K) and extends to higher masses. In Vela C, Massi et al. (2019) found a starless CMF mostly consistent with the canonical IMF with some indications that the protostellar CMF could be flatter. As for studies at larger distances with ALMA, few have separated protostellar from prestellar cores. In a combined survey of 12 IRDCs, Sanhueza et al. (2019) found a prestellar CMF slope with a high-mass end slightly flatter than Salpeter (-1.17 $\pm$ 0.1).

In this work, we study prestellar and protostellar core populations in a single, distant molecular complex with strong high-mass star formation activity. In Section 3.3, we showed that the prestellar CMF in W43 is significantly different from the protostellar CMF. The former does not cover the complete mass range of stars expected to form in W43 and its high-mass end is compatible with the canonical Salpeter slope. In contrast, the protostellar CMF is significantly top-heavy, up to core masses of 100$\,M_{\odot}$. We also showed that the protostellar core population has a higher median mass compared to the prestellar core population (median mass of 8.9$\,M_{\odot}$ and 2.7$\,M_{\odot}$, resp.), and that the fraction of cores with protostellar activity increases with mass.

In the following, we first assume that the statistics performed in space reconcile with temporal statistics (i.e. the hypothesis of ergodicity). In that regard, the measured prestellar core population is assumed to be representative of the ”parent” prestellar core population, which evolved into the measured protostellar core population. If the total mass reservoir available to form a star is mainly determined by the mass of its prestellar core, as assumed in ”core fed” models, one would expect the mass of a protostellar core to be lower than that of the prestellar core at the time of protostar formation, ($M_{\rm pre,end}$). Denoting $\dot{M}_{\rm out}$ as the mass loss rate of the gas core, through outflows or accretion onto the protostar, the mass of an isolated protostellar core would indeed evolve as $M_{\rm pro}(t)=M_{\rm pre,end}-\dot{M}_{\rm out}\,t$, with $t=0$ corresponding to the beginning of the first collapse. The existence of protostellar cores more massive than prestellar cores thus implies that cores are able to grow in mass during the protostellar phase through inflow and accretion from their environment, which can be included in the previous equation with an additional term $\dot{M}_{\rm core}$:

To observe a core at a given time $t_{1}$ such that $M_{\rm pro}(t_{1})>M_{\rm pre,end}$, the core mass growth must be on average larger than the core mass loss ($\dot{M}_{\rm core}>\dot{M}_{\rm out}$) during the core lifetime. Continuous core mass growth during the protostellar phase has also been proposed by Kong et al. (2021) to account for their observations, as well as in the analytical core evolution models developed by Hatchell & Fuller (2008) to explain the lack of high-mass prestellar cores in Perseus. Such processes align with clump-fed models (see e.g. Wang et al. 2010; Smith et al. 2009; Vázquez-Semadeni et al. 2019).

We note that Eq. 1 can easily be generalised to describe the mass evolution of prestellar cores, introducing $M_{\rm pre,ini}$ as the initial mass and assuming that the core mass growth processes represented in $\dot{M}_{\rm core}$ are similar in the prestellar and protostellar phases:

In addition, the flattening of slope between the prestellar and protostellar CMFs implies that $\dot{M}_{\rm core}$ depends on the core mass, and more precisely that high-mass cores are able to grow in mass more strongly than low-mass cores. This is in line with observations showing a correlation between mass and infall rate (see e.g. on clumps scale, Yue et al. 2021).

Beyond the discussion of possible mechanisms explaining the mass evolution of a single core from the prestellar to the protostellar stage, our results have also implications for the time evolution of core populations. From their study of the spatial variations of star formation in W43-MM2&MM3, Pouteau et al. (2023) proposed that subregions undergoing a burst of star-formation, such as the centre of W43-MM2, could be associated with top-heavy CMFs, while others more quiescent would show Salpeter-like distributions. Following this approach, our results suggest that the population of protostellar cores which shows a top-heavy CMF is growing in mass during a burst of star formation. After this burst, the star-formation activity is expected to decrease and enter in a more quiescent state. In this context, the measured prestellar core population could evolve to the protostellar stage preserving a Salpeter-like CMF. Similarly, the measured protostellar population could have inherited its top-heavy mass distribution from an alike prestellar CMF, that is, the parent prestellar core population could have been top-heavy compared to the current one. This would however contravene the hypothesis of ergodicity, as the observed prestellar core population would no longer be representative of the parent prestellar core population.

Our results have implications for the issue of the origin of the stellar IMF. After having constituted the main paradigm in star formation for decades , the universality of the IMF is now under serious debate, both from Galactic and extra-galactic studies. In the Milky Way, various claims of top-heavy IMFs have emerged in young massive clusters in the vicinity of the Galactic centre (Hußmann et al. 2012; Lu et al. 2013; Hosek et al. 2019). At the same time, several ALMA observations of top-heavy CMFs have been reported in distant, high-mass star-forming regions (see, e.g. Motte et al. 2018b; Kong 2019; Pouteau et al. 2022). Therefore, although simultaneous observations of CMF and IMF in the same region are challenging (see however Takemura et al. 2021), we could reasonably expect that in some regions neither the CMF nor the IMF have a Salpeter slope. Interestingly, recent analytical models for the IMF obtain high-mass end slopes flatter than -1.35, such as -1 (Ballesteros-Paredes et al. 2015) or -0.75 (Lee & Hennebelle 2018), depending on the equation of state. Predicting the IMF shape from a CMF is a complicated question that involves various parameters such as the core-to-star mass efficiency, fragmentation, and time evolution. Some of these effects have been tested by Pouteau et al. (2022), that suggested that the IMF resulting from the measured CMF in W43-MM2&MM3 could remain top-heavy. Our results also support that the current protostellar core population will produce stars with a top-heavy IMF. As discussed above, however, future populations of protostellar cores could evolve with a Salpeter-like CMF and produce stars with a canonical IMF. Therefore, the shape of the final IMF of W43, built from the accumulation of various episodes of star formation over time, will depend on the complex interplay between various processes in time and space.