INSPIRE: INvestigating Stellar Population In RElics VI - The low-mass end slope of the stellar Initial Mass Function and chemical composition

The INSPIRE project has built the largest sample of ultra-compact massive galaxies (UCMGs) at 0 . 1 < 𝑧 < 0 . 4 and obtained their star formation histories (SFHs). Due to their preserved very old stellar populations, relics are the perfect systems to constrain the earliest epochs of mass assembly in the Universe and the formation of massive early-type galaxies. The goal of this work is to investigate whether a correlation exists between the degree of relicness (DoR), quantifying the fraction of stellar mass formed at 𝑧 > 2, and the other stellar population parameters. We use the Full-Index-Fitting method to fit the INSPIRE spectra to single stellar population (SSP) models. This allows us to measure, for the first time, the low-mass end slope Γ 𝑏 of the IMF, as well as stellar metallicity [M/H], [Mg/Fe], [Ti/Fe] and [Na/Fe] ratios, and study correlations between them and the DoR. Similarly to normal-sized galaxies, UCMGs with larger stellar masses have overall higher metallicities. We found a correlation between the low-mass end of the IMF slope and the DoR, that, however, breaks down for systems with a more extended SFH.An even stronger dependency is found between the IMF and the fraction of mass formed at high-z. At equal velocity dispersion and metallicity, galaxies with a higher DoR have a dwarf-richer IMF than that of low-DoR counterparts. This might indicate that the cosmic epoch and formation mechanisms influence the fragmentation of the star formation cloud and hence might be the explanation for IMF variations detected in massive ETGs.


INTRODUCTION
More than half of the total stellar mass in the Universe is contained in massive early-type galaxies (ETGs, e.g., Renzini 2006, sec.1), which play an essential role in the context of cosmic structure formation and evolution (Blumenthal et al. 1984).They host the oldest populations of stars, thus retaining the memory of the earliest star formation activity.However, in local, giant massive ETGs, this oldest population is contaminated by accreted and/or later formed stars.Luckily, since galaxy interactions and mergers are stochastic phenomena, a small fraction of galaxies avoid the accretion and merging phase and retain their original stellar population and ultra-compact nature, becoming ★ E-mail: michalina.maksymowicz.maciata@gmail.comthe relics of the ancient Universe (Trujillo et al. 2009;Taylor et al. 2010;Valentinuzzi et al. 2010;Poggianti et al. 2013).As relic galaxies can be observed at low redshifts with their structure intact, these 'sealed time-capsules' allow us to study the earliest cosmic epochs with the amount of detail only achievable in the nearby Universe.Studying relic galaxies can therefore help answer many questions about the early Universe and the formation and evolution of massive ETGs.This is the reason why, in the last few years, UCMGs in general, and relics more specifically, received a lot of attention from the scientific community.To date, only a dozen of relics have been confirmed and fully characterised at  ∼ 0 (Trujillo et al. 2014;Ferré-Mateu et al. 2017;Yıldırım et al. 2017;Comerón et al. 2023;Salvador-Rusiñol et al. 2021)  1 .From Hubble Space Telescope highresolution imaging, their morphology and density profiles match those found in high-z passive UCMGs (Trujillo et al. 2014;Ferré-Mateu et al. 2017).From spatially resolved spectroscopy, it has been found that these three relics have large rotation velocities ( ∼ 200 -300 km s −1 ) and very high central stellar velocity dispersion values ( ★ > 300 km s −1 ).Finally, from a stellar population point of view, they have very peaked and extremely high-z star formation histories (SFH) and are populated by stars with super-solar metallicities (Z∼ 0.2-0.3, with strong spatial gradients) and [Mg/Fe], old ages (∼ 13 Gyr), and a bottom-heavy (i.e.dwarf rich) Initial Mass Function (IMF) slope (Martín-Navarro et al. 2015b).
To validate these results, a larger statistical sample is necessary, whilst enlarging the redshift boundaries outside the local Universe.This has been the goal of the INSPIRE Project.Targeting 52 spectroscopically confirmed UCMGs at 0.1 <  < 0.4, INSPIRE has built the largest sample of spectroscopically confirmed UCMGs with measured kinematics and stellar population parameters.Of these, 38 have been classified as relics, as they formed more than 75% of their stellar masses already by  > 2. The survey has been presented in Spiniello et al. (2021a) and Spiniello et al. (2021b, hereafter INSPIRE DR1).In D 'Ago et al. (2023, hereafter INSPIRE DR2), extensive tests were performed on the kinematics, deriving the stellar velocity dispersion within an aperture encapsulating 50% of the light.However, since the spectra are fully seeing-dominated, and the seeing is much larger than the effective radii of the UCMGs, the velocity dispersion values have to be interpreted as lower limits 2 .Finally, the stellar populations analysis of the entire sample has been presented in Spiniello et al. (2023, hereafter INSPIRE DR3).In Ferré-Mateu et al. (2017) it was suggested that three local relics could be ranked from the most extreme to the least extreme, in terms of their morphological and stellar population characteristics, hence following a degree of relicness (DoR).Motivated by this finding, and thanks to the larger statistical sample, INSPIRE was able to quantify such DoR parameter, as a dimensionless number, varying from 0 to 1.The DoR is defined in terms of the fraction of stellar mass formed by  = 2 (assumed to be the end of the first phase, Zolotov et al. 2015), the cosmic time at which a galaxy has assembled 75% of its mass and the final assembly time.According to this, galaxies with a high DoR are the most extreme relics as they assembled their stellar mass at the earliest epochs and very quickly, while low-DoR objects show a non-negligible fraction of later-formed populations and hence a spread in ages and metallicities.It was unambiguously found that at similar stellar masses, objects with a higher DoR have larger stellar metallicity and velocity dispersion values.However, the analysis in DR3 was done while keeping the elemental abundances fixed and assuming a universal IMF, which might not be the best assumption, given the results emerging in the last few years.Indeed, since the work by van Dokkum et al. (2010), increasing evidence has emerged supporting a non-universal IMF varying across galaxies (Treu et al. 2010;Conroy & van Dokkum 2012a;Cappellari et al. 2012Cappellari et al. , 2013;;Tortora et al. 2013;Spiniello et al. 2014Spiniello et al. , 2012Spiniello et al. , 2015b;;La Barbera et al. 2013;Martín-Navarro et al. 2015c) and spatially within single massive objects (Martín-Navarro et al. 2015a;van Dokkum et al. 2017;Parikh et al. 2018;Barbosa et al. 2021b).The reason for these variations is however still debated.
For the three local relics, a bottom-heavy (i.e., with a fraction of low-mass stars being at least a factor of 2 larger than that found in 1 The number depends on the mass and size threshold used to define relics. 2 See Appendix A in Spiniello et al. (2021b).
the Milky Way) IMF has been inferred up to a few effective radii (Martín-Navarro et al. 2015b;Ferré-Mateu et al. 2017;Comerón et al. 2023).For normal-sized massive galaxies, instead, a similarly bottom-heavy IMF, with a fraction of stars with M < 0.5 ⊙ a factor of two larger than the one measured in the Milky Way, is only required in the innermost region (Martín-Navarro et al. 2015a;Sarzi et al. 2018;Parikh et al. 2018;La Barbera et al. 2019;Barbosa et al. 2021a).Here, according to the two-phase formation scenario, the relic component is supposed to dominate the light budget (Navarro-González et al. 2013;Pulsoni et al. 2021;Barbosa et al. 2021a).All these observations can be explained assuming that the IMF might have been bottom-heavy during the early stages of galaxy formation when the Universe was much more dense and richer in hot gas, but only if stars formed through a very intense (SFR ≥ 10 3 M ⊙ yr −1 ) and very short ( ∼ 100 Myr) burst (Chabrier et al. 2014;Smith 2020;Barbosa et al. 2021a).Hence, in UCMGs with high DoR, where the great majority of the stellar mass was formed during the first phase at high-z, we should be able to measure a steep IMF slope even from an integrated spectrum covering a large portion of the galaxy size.Indeed this is what we preliminary found in Martín-Navarro et al. (2023), the fourth paper of the INSPIRE series (hereafter INSPIRE IV).Stacking the UVB+VIS spectra of 5 relics and those of 5 non-relics with very similar velocity dispersion and metallicity values, we measured a systematic difference in the IMF slope which is dwarf-richer for relics.However, stacking spectra from different galaxies, broadened the final IMF probability density distribution (PDF), especially for non-relics, where the single stellar population (SSP) assumption could be less reliable, given the more heterogeneous and extended SFHs.Hence, a larger statistical sample is necessary to confirm this result.This is one of the main goals of this sixth paper of the INSPIRE series.
In this work, we carefully inspect one by one the 52 INSPIRE UCMGs, identifying contamination and bad pixels that could affect line-indices measurements.This allows us to derive the IMF slopes of each single object and study how they relate to their DoR and other stellar population and kinematical parameters.The paper is organised as follows.In Section 2 we introduce the data and explain the methods and analysis of the stellar populations.Section 3 presents the results and correlations between the obtained stellar population parameters.We then focus on the metallicity (Sec 3.1) and the lowmass end of the IMF slope (Sec.3.2) and their correlations with the DoR, as well as with the stellar velocity dispersion and with each other (Sec.3.3).Finally, Section 4 is reserved for discussion, while in Section 5, we present our conclusions and summarise the findings of the paper.

Observations
This work uses the sample of 52 UCMGs from the INSPIRE project, with data collected as part of an ESO Large Programme (LP, ID: 1104.B-0370, PI: C. Spiniello).The programme started in P104 (October 2019) and was completed in March 2023, delivering high SNR spectra (20 ≤ SNR ≤ 80 per Å), from the UVB to the NIR, with the X-Shooter spectrograph (XSH, Vernet et al. 2011).We refer the reader to INSPIRE DR1 and INSPIRE DR3 for a comprehensive description of the sample selection and characteristics.
In this paper, we use the one-dimensional (1D) INSPIRE spectra extracted at the R50 radius.This radius is obtained considering the surface brightness profiles of the 2D spectra (for each single observation block) and integrating them up to the aperture that encapsulates 50% of the total light.However, because ground-based observations are seeing-limited, and the seeing is much larger than the galaxies' size, the aperture contains a mix of light from inside and outside the real effective radius.We refer the readers to INSPIRE DR1 for a more detailed description of the 1D extraction, which however plays an almost negligible role in the relic confirmation.
Moreover, we only consider the rest-framed and telluric-corrected combined UVB+VIS spectra smoothed at a final resolution of FWHM= 2.51Å, matching that of the SSP models we use for the fitting.Our choice of not including the NIR is two-fold.First, the telluric contamination on the spectra in this wavelength range is much larger than the one affecting the VIS.Secondly, a detailed treatment of several individual element abundances, currently not included in the MILES models, is critical to properly fit the observed strengths of infrared absorption features, especially regarding Carbon-sensitive indices, which are systematically underestimated by the current SSP models (Eftekhari et al. 2022).
The characteristics of the INSPIRE, including coordinates, redshift, DoR, velocity dispersion values, and SNR inferred from the UVB and the VIS spectra, are reported in Table 1.The SNR of the spectra cover a wide range and it is generally higher in the optical, as expected for red galaxies with evolved stellar populations.In the last two columns of the table, we list the stellar masses and size of the 52 objects, taken from Tortora et al. (2018) and Scognamiglio et al. (2020) 3 .

Stellar population models
We use the MILES SSP models developed and described in Vazdekis et al. (2015) with BaSTI theoretical isochrones (Pietrinferni et al. 2004(Pietrinferni et al. , 2006)).The MILES SSPs are based on the MILES empirical stellar library4 (Sánchez-Blázquez et al. 2006;Falcón-Barroso et al. 2011), covering the wavelength range [3525-7500] Å.The SSPs cover a broad range of stellar ages, ranging from 30 Myr to 14 Gyr, and sampled at logarithmic steps, total stellar metallicities in the range −2.27 <[M/H]< +0.40, two values of the [/Fe] ratios (solar and super-solar, +0.4 dex) and a suite of stellar IMF slopes.For this paper, we use the Bimodal IMF parametrisation defined in Vazdekis et al. (1996).This is described by two power-law regimes with a break at 0.6 M ⊙ , where the low-mass slope is fixed at 1.3 (Salpeter 1955), while the high-mass slope (Γ  ) is free to vary between 0.3 and 3.5.
Since degeneracies exist between variation in the IMF slope, especially at its low-mass end, and single elemental abundances (e.g., Spiniello et al. 2014), we need to take this into account.In particular, Sodium (Na) and Titanium (Ti) absorption features have been extensively used to measure the IMF in massive ETGs (Conroy & van Dokkum 2012b;Ferreras et al. 2013;Spiniello et al. 2012Spiniello et al. , 2014Spiniello et al. , 2015a;;Parikh et al. 2018;Sarzi et al. 2018;La Barbera et al. 2019).To model the variation of Na and Ti, we use response functions of these two elements, computed using the SSP models of Conroy & van Dokkum (2012a), hereafter CvD.We follow similar prescriptions to these given in Spiniello et al. (2015c).In particular, the response functions are computed by simply taking the ratio between two CvD spectra, with the same age, same IMF, and metallicity, one with solar abundances and one with different [Na/Fe] or [Ti/Fe].In this way, we are able to isolate the effect of changing the elemental abundance from the effect of changing other stellar population parameters.Specifically, in this case, we compute response functions for a 13.5 Gyr population, which is sufficiently similar to the expectations for our sample given the weak age dependence of elemental abundance corrections (e.g.Vazdekis et al. 2015).The conclusions presented here will not change if a slightly younger age (e.g.10-11 Gyr) would have been used to obtain the response functions and this is enough to draw conclusions on the elemental abundances in relics.

Full-Index Fitting
The core of the modelling in this work revolves around the Full-Index Fitting method (FIF, Martín-Navarro et al. 2019, 2021), which is a hybrid approach between a more standard line-strength analysis and full-spectrum fitting.Instead of calculating the equivalent widths of key absorption features or fitting every pixel across a wide wavelength range, this method fits every pixel within the selected key absorption features to the models, after normalising the continuum using the index definition.This hybrid approach has lower SNR requirements than pure line-strength, reduced computational time than full-spectrum fitting (Martín-Navarro et al. 2019), and allows to focus on well-tested and studied spectral regions, where the information about specific stellar population properties is concentrated.
We use FIF in combination with the Penalised Pixel-fitting software5 (pPXF; Cappellari & Emsellem 2004;Cappellari 2017Cappellari , 2023)), already used in previous INSPIRE publications.Specifically, we first run pPXF on the UVB+VIS combined and smoothed spectra, computing the stellar velocity dispersion, as well as the light-weighted stellar age (in logarithmic scale) and metallicity, and a first guess of the IMF slope.We note that we have customised the publicly available version of the code to let it constrain also the slope of the IMF, but we keep elemental abundance ratios fixed to solar and we do not fit for them.Moreover, we stress to the readers in INSPIRE DR3 we performed two separate runs, one for kinematics and one for stellar population constraints.Here instead, we perform a single run.We, therefore, believe that, although we obtain the stellar velocity dispersion as a bi-product, we believe that the values computed in DR3 are more trustable, as the pPXF code was optimised for kinematics (e.g., using an additive polynomial and performing tests on the systematics and random errors).Hence we list the DR3 values in Table 1 and plot and use these values throughout the paper.We nevertheless show in Appendix B that the agreement between the two measurements is good (see the rightmost panel of Fig. B1).
Following the same line of thoughts, we did not re-derive the DoR here based on the new ages and metallicities computed with pPXF.Indeed, we did not perform any regularisation during the fit (see INSPIRE DR3 for details), which often revealed signs of younger ages, when there, as in the case of systems with low DoR.We, therefore, believe that the DoR computed in INSPIRE DR3 is more correct and use that for the remainder of the paper.
At this point, we input the best-fitting age and the velocity dispersion values to the FIF routine as fixed values and obtain an estimate for the stellar metallicity, Mg, Na and Ti abundances, and the IMF Table 1.The INSPIRE sample.From left to right we give the coordinates, redshift, DoR, stellar velocity dispersion, SNR (per Å) in the UVB and VIS arms, the stellar mass inferred from SED fitting in the  bands, and the median effective radius in kpc (computed from the measurements in ,  and  bands).These two last quantities are taken from Tortora et al. (2018) and Scognamiglio et al. (2020), while all the others are taken from INSPIRE DR3.Finally, in the last column, we report whether the galaxy is in the Golden Sample, as defined in Sec.In principle, the larger the number of indices fitted, the better the constraining power should be (Spiniello et al. 2014).However, some of the optical spectral indices for several galaxies in the INSPIRE sample are contaminated by residuals of skylines and bad pixels, rendering them unfit to use.Hence, we need to select the minimum number of indices that allow us to break the degeneracy between the stellar population parameters while maximising the number of galaxies without any signs of contamination.After extensive testing (see Appendix A), we selected 6 optical indices, namely Mgb, Fe5270, Starting from these six indices, and checking galaxy by galaxy whether the spectra present bad pixels and contaminated regions affecting the calculation of the equivalent width, we define a Golden Sample of 39 galaxies, for which none of the six selected features is contaminated (see last column of Table 1).
In Appendix A, we present a series of tests aimed at checking whether the results of the FIF depend on the spectral indices chosen to be fitted to the models, and which parameters are let free in the run.The main conclusion of these tests is that the only two significant differences in the inferred stellar population parameters arise when i) changing the Mg index used for the fit (from Mgb to Mg2), or ii) letting the age free to vary, adding a Balmer line to constrain it.However, importantly, the inference on the IMF is run-independent and therefore very robust and the results are almost completely unchanged for extreme relics, while a larger variation is found for galaxies with lower DoR.

Testing the SSP assumption
We remind the readers that pPXF uses a series of SSPs to derive the star formation history of a galaxy.For extreme relics, when the entire totality of the stellar mass has been formed quickly at very early cosmic time, it is expected that the code only uses a low number of very old SSPs.At decreasing DoR, the fraction of mass formed during the first phase of the formation scenario becomes smaller and the star formation history more extended.Hence the number of SSP models, with different ages and metallicities, used for the fit 6 The indices definition, including also blue and red band-passes are given in Table C1, in Appendix C increases.This also means that the assumption that the star formation history can be reconstructed by combining SSP is less solid.In this section, we investigate the spread in age that pPXF attributes to each galaxy and how this correlates with the DoR obtained in INSPIRE DR3.We believe that this number represents a qualitative way to measure the validity of the SSP assumption: the larger the number of age bins is, the worse an SSP approximation is for a given galaxy.
Figure 2 presents how the DoR correlates with the maximum sum of the weights of 5 neighbouring bins of the age-distribution histogram (sum 5 ).To compute this number, we start by taking all the non-zero weights that pPXF has attributed to the array of SSP models (with different ages and metallicities).Then, marginalising over metallicity, we produce an age-distribution histogram.From it, we calculate the sum of weights attributed by pPXF to every five neighbouring age-bins (corresponding to 2.5 Gyr, given that the models span an age grid of 0.5 Gyr), and check which group of five holds the greatest sum of their weights.As expected a clear correlation is found.Moreover, we also note that a plateau is reached for the sum of weights > 0.95.In the right panel of the same figure, we sketch the age distribution for different age bins.The threshold of 0.95 corresponds to a very peaked distribution of ages above 9 Gyr, without any second peak at lower redshift (younger ages).Below this number, the age histograms start to be broader and peaks at younger ages appear and therefore the SSP-assumption becomes less valid.We note that even though the sum 5 strongly correlated with the DoR for similar, intermediate DoR (0.4-0.6), different galaxies can have different sum 5 .In the remainder of this paper we will use this threshold to derive relations between stellar population parameters that might hold only for stars formed through a quick star formation episode at early cosmic time.We note that the threshold choice is arbitrary and different relations could be found based on different definitions, however we believe it to be informative to check whether the correlations hold especially for the most strict age-distribution constraints.Furthermore, we acknowledge that the sum of the weights expresses a concept which is very similar to that measured with the DoR, but we note that the latter depends on the fitting assumptions within pPXF, which are different between this paper and the INSPIRE DR3 (see Appendix B for more details).

STELLAR POPULATION ANALYSIS RESULTS
This Section presents the main results of the FIF analysis in terms of stellar population parameters and their relation with the DoR, obtained in INSPIRE DR3.A direct comparison with the results obtained in DR3 is presented in Appendix B. When not otherwise specified, we will focus on results obtained from the fiducial run, which are listed in Table 2.
Figure 3 summarises the results of the FIF analysis and also shows how the stellar population parameters correlate with the DoR, the stellar mass, size and velocity dispersion of the INSPIRE objects.In particular, the only two parameters that seem to correlate with the DoR are metallicity and velocity dispersion, as already hinted for in INSPIRE DR3.More specifically, relics, and especially extreme relics have systematically larger stellar velocity dispersion than nonrelics.For the metallicity, there is a large scatter but, all the extreme relics have super-solar [M/H] and a hint for a correlation can be seen.This will be further investigated in Section 3.1.No correlation is found instead between the DoR and the considered elemental abundances (Mg, Na and Ti)7 .We caution the reader that the estimates for [Na/Fe], and especially [Ti/Fe], could be biased both by the SNR of the data and by the uncertainties in the models.This is especially true for the TiO molecular bands, since the line lists feeding the response functions from the CvD models are incomplete, and these broad features also depend on other elements, such as Carbon.Nevertheless, we need to fit for them in order to break the IMF-elemental abundance degeneracy (e.g., Spiniello et al. 2015c).
and the [Mg/Fe], estimated via line-indices.However, as already reported in Barbosa et al. (2021b), the estimates obtained via different techniques do not match.
Another interesting result is that while a scatter exists in the sizes and masses of the entire sample, although they are selected to be similarly ultra-compact and massive, the most extreme relics cluster around R e ∼ 1 kpc and ∼ 1-1.5 ×10 11 M ⊙ .Nevertheless, we stress that only with higher spatial resolution data (e.g., from space or using AO-supported ground telescopes) we can obtain a more precise and robust estimate of the size of these objects. .FIF fitted metallicity against the DoR, colour-coded by the maximum sum of weights of 5 neighbouring bins of the age-distribution, quantifying how good the single-stellar population assumption is.Diamonds represent galaxies from the Golden Sample, while dots show the galaxies with contaminated spectral indices.Golden Sample galaxies with sum 5 > 0.95 are marked by dark border.The lines represent the linear fits for the three samples.
Table 3. Linear fits to the metallicity versus the DoR (Fig. 4) for different sub-samples of galaxies.The last two columns report the Pearson correlation coefficient () and its p-value.

Sample
Linear

Correlation between stellar metallicity and DoR
In Figure 4 we focus on the correlation between [M/H] and the DoR, colour-coding the data points by the number of bins defined above, to assess the validity of the SSP assumption.
A very large spread in metallicity is observed for objects with a low DoR, which also corresponds to a larger number of different ages that pPXF uses for the fit.Indeed, for objects that pass the threshold having sum 5 > 0.95, the scatter is reduced and a clear direct correlation between the DoR and [M/H] is found.This correlation seems to hold, although with larger scatter both considering all galaxies in the Golden Sample or even all the INSPIRE objects.We use a trimmed least squares linear fit to draw the correlation and show the resulting fits in Figure 4. We also list them in Table 3 for all the galaxies in the Golden Sample, only these with sum 5 > 0.95, and all the INSPIRE objects.Finally, we compute, for each fit, the Pearson coefficient and its p-value to assess how probable a correlation between the stellar metallicity and the DoR is.As expected, the strongest correlation is found for the sub-group of relics with sum 5 > 0.95, for which the totality of the mass was formed during the first phase of the formation scenario.

Correlation between the IMF slope and the DoR
In this section, we focus on the stellar IMF and the correlation between its low-mass end slope and the DoR, which is shown in Figure 5.Although with a non-negligible scatter, objects with a higher  DoR are better fitted with a dwarf-richer IMF.Also in this case, we apply a least trimmed squares linear fitting to the data and report the results in Table 4.The strongest correlation is found for the Golden Sample galaxies with a very peaked age distribution (sum 5 > 0.95), however, a significant correlation is found also when considering all the galaxies.All the extreme relics, i.e.DoR> 0.7, are inconsistent with having a Milky-Way IMF and require a steeper slope.However, we note that there are objects with a DoR as low as ∼ 0.3 that fit with an IMF slope equally steep than objects with a higher DoR.This result is fully consistent with the results in INSPIRE IV in which we found a larger scatter in the IMF slope of non-relics than that of relics, which might be caused by the fact that the SSP assumption does not hold well for galaxies with a more extended SFH, that could have stellar populations with different IMFs (and age and [M/H]).Furthermore, it is important to stress that data and modelling systematics might play a role, increasing the scatter.Indeed, although we carefully checked each individual spectrum, we are aware that the SNR is, in some cases, towards the lower limit for which FIF produces trustable results.
To better understand why some galaxies with a relatively low DoR have a bottom-heavy IMF, we have considered each of the three addenda that went into the definition of the DoR (from INSPIRE DR3) separately.We remind the readers that the DoR is defined as the sum of the fraction of stellar mass assembled at  = 2 (i.e.,   ★

𝑡BB=3
), the inverse of the cosmic time at which 75% of the stellar mass was in place ( 75 ), and the inverse of the final assembly time (i.e. the time at which a galaxy has formed the 100% of its stellar mass), re-scaled to the age of the Universe at the redshift of each galaxy ([ uni −  fin ]/ uni ).The relation between the IMF slope and each of these three quantities is plotted in Figure 6, with the same symbology used in other figures.Interestingly, the strongest relation is found with the   ★ BB=3 : there are no points with Γ  > 2 for   ★ BB=3 < 0.6.Hence, UCMGs have a dwarf-rich IMF only when they have formed at least 60% of their stellar mass at  > 2.
In conclusion, there is a clear dependency of the IMF slope from the cosmic assembly time (second-left panel of Figure 6), especially for objects with a very peaked SFH and very old ages.The IMF is dwarf-richer than the Milky-Way one for stellar populations formed at  > 2, i.e., during the first phase of the two-phase formation scenario.

Correlation between the IMF slope and other parameters
Two of the most accredited relations involving global IMF variation are the IMF slope-velocity dispersion (Cappellari et al. 2012(Cappellari et al. , 2013;;Ferreras et al. 2013;La Barbera et al. 2013;Spiniello et al. 2014) and the IMF-metallicity relation (Martín-Navarro et al. 2015c).In Figure 7, we check whether we are able to reproduce them with the INSPIRE UCMGs.The least-square linear fits to the data, together with the Pearson coefficient () and its p-value, are reported in Table 5.
A strong correlation is found with the  ★ , as shown by the least square fitting to the data (left panel, grey lines), both all the galaxies (solid) and only these belonging to the Golden Sample (dashed).The linear relation is even steeper than that found from the literature on normal-sized ETGs covering a similar range in  ★ (La Barbera et al. 2013, magenta line, andSpiniello et al. 2014, cyan line).However, we also note that for the same value of  ★ , we find a very large range of IMF slopes, but that objects with a higher DoR have systematically steeper IMF slopes.
The correlation of the IMF slope with the [M/H] is instead much weaker than the one reported in Martín-Navarro et al. (2015c).Furthermore, also in this case, we note that at equal metallicity, galaxies with a higher DoR have a dwarf-richer IMF.
Hence we conclude that even though clear and solid global relations have been reported (La Barbera et al. 2013;Spiniello et al. 2014;McDermid et al. 2014;Martín-Navarro et al. 2015c), which seem to hold also for UCMGs, there might not be a casual connection between the IMF slope and the stellar velocity dispersion and metallicity.Or, at least, these two factors are not the only drivers of the variations in the low-mass end of the IMF.Hints from the fact that the local value of the stellar velocity dispersion is not the main driver behind dwarf-to-giant ratio variations was already suggested by Martín-Navarro et al. (2015b) for the local relic NGC1277.
We speculate that the relations might arise from the fact that by selecting a galaxy with a larger velocity dispersion (i.e. a more massive galaxy), or with a metal richer population, the chances to find a compact progenitor in its centre (or a relic) are higher (Pulsoni et al. 2021).And that the low-mass end of the IMF slope might be steeper for it.This is for instance the case of NGC 3311, the central galaxy of the Hydra-I cluster.From very high SNR spatially resolved spectra, Barbosa et al. (2021b) found a bottom-heavy IMF slope in the galaxy's centre, where the stellar velocity dispersion was as low as ∼ 150 km s −1 .In this region, the stars are incredibly old and metal-rich.The IMF becomes instead Milky-Way-like moving away from the centre, where the  ★ rises up to ∼ 400 km s −1 , and both age and metallicity drop.) and total stellar metallicity (right, computed with FIF).The INSPIRE galaxies are colour-coded by their degree of relicness (top row) and sum 5 (bottom row).The lines show linear fits to the data, from this work (grey) and from literature papers, as reported in the legend and in the third and fourth block of Table 5. Symbols are like in other figures.
Table 5. Linear fits to the IMF slope versus the stellar velocity dispersion (first block) and the metallicity (second block) for different sub-samples of galaxies.The lines are shown in Figure 7.The last two columns report the Pearson correlation coefficient () and its p-value.

Sample
Linear fit

DISCUSSION
The results presented in this paper unambiguously demonstrate that a correlation exists between the low-mass end of the IMF, the stellar metallicity and the cosmic epoch of star formation.In particular, we found that, among the 52 INSPIRE UCMGs, the objects with the highest DoR, i.e. those that formed the totality of their stellar masses at  > 2 and did not experience any other star formation episode, i.e. extreme relics, are metal richer and have a dwarf-rich IMF.This result could seem in strong disagreement with the latest findings from JWST data, suggesting that the IMF might be top-heavy, i.e. giant richer, at very high redshift (Cameron et al. 2023;Trinca et al. 2023;Woodrum et al. 2023).However, we note that, in principle, the high-and low-mass ends of the IMF are not necessarily coupled.Here below we highlight a couple of possible scenarios that would allow us to reconcile the different observations.
In the context of the integrated galaxy-wide stellar initial mass functions (IGIMFs) paradigm (Kroupa 1995;Kroupa & Weidner 2003), all stars form in groups or embedded star clusters, and the most massive star that can be formed in them depends on the mass and the metallicity of the star cluster (Weidner et al. 2010;Dabringhausen & Kroupa 2023).The IGIMF will then be the combination of all the stars that formed in the different star clusters of a galaxy (Kroupa & Weidner 2003).Furthermore, recent simulations on the first, metal-free stars suggest that the fraction of massive stars directly depends on the gas temperature of star-forming clouds: the higher the temperature the larger the mass of the produced stars (Abel et al. 2002;Fukushima et al. 2020).Also, the top-heaviness of the IMF depends on the metallicity of the gas (the lower the metallicity the more top-heavy the IMF would be, Fukushima et al. 2020) and on the background radiation intensity (the higher the intensity the more top-heavy the IMF would be, Chon et al. 2022).Hence, one possibility is that the IMF could be top-heavy in low metallicity and dense gas environments (Dabringhausen et al. 2012;Dib 2023) while it becomes progressively bottom-heavy as the metallicity of the environment increases (Chabrier et al. 2014), at any redshifts.However, we do not find a strong dependency of the low-mass end of the IMF on the measured stellar metallicity.
Another interesting possibility, instead, is to allow for a timeevolution of the IMF slope (Vazdekis et al. 1996(Vazdekis et al. , 1997;;Weidner et al. 2013;Ferreras et al. 2015).A first and quick phase with a top-heavy IMF occurs at a very high redshift ( ≥ 5).Then, the very massive ( > 50 ⊙ ) giant stars rapidly die (less than 10 Myr, Yusof et al. 2013), polluting the interstellar medium with metals.At this point, if the conditions of the local environment (i.e., gas temperature, pressure and density) are extreme, and a violent starburst happens, fragmentation of the star-formation clouds becomes easier (Chabrier et al. 2014) and therefore a large number of dwarf stars are produced, transforming the IMF into a bottom-heavy one.In fact, according to theoretical works (e.g.Hennebelle & Chabrier 2008), a larger fraction of low-mass cores in very dense, hot, and highly turbulent environments is a direct consequence of the enhanced gas compression by highly turbulent motions and of the shorter free-fall times for the collapsing over-dense region (Chabrier et al. 2014).This scenario also naturally explains why relics are metal-richer than nonrelics, as they formed in a metal-rich gas.Finally, stars are produced following a canonical Milky-Way-like IMF slope for the more timeextended standard star formation occurring during the second phase of the two-phase formation scenario.

CONCLUSIONS
In this paper, the sixth of the INvestigating Stellar Populations In RElics (INSPIRE, Spiniello et al. 2021a,b) series, we have presented an extensive stellar populations analysis on the entire INSPIRE sample made of 52 ultra-compact (R e < 2 kpc) massive ( ★ > 6 × 10 10  ⊙ ) galaxies (UCMGs) at 0.1 <  < 0.4.Of these, 38 were confirmed as relics in INSPIRE DR3, as they formed more than 75% of their stellar mass by the end of the first phase of the formation scenario.At the time of writing, this is the largest catalogue of spectroscopically confirmed relics known in the nearby Universe.
In INSPIRE DR3 the stellar population ages and metallicities were inferred fitting UVB+VIS spectra, joined and convolved to a common final resolution of FWHM = 2.51Å, in the wavelength range [3500-7000]Å.The [Mg/Fe] abundances were instead obtained directly from line index analysis, with the Mg  -Fe24 index-index plot.The IMF slope was kept fixed to a Kroupa-like one with Γ  = 1.3 and all other elemental abundances were assumed to be solar.Here, instead, we have used the Full-Index Fitting (FIF) technique to infer stellar metallicities, [Mg/Fe], [Ti/Fe] and [Na/Fe] ratios, and IMF slope from the same spectra.This much more flexible approach, combined with a careful visual inspection of each single spectrum, allowed us to investigate whether a correlation exists between the stellar population parameters and the degree of relicness.We have been also able, for the first time, to infer the low-mass end of the IMF slope on individual galaxies.Moreover, we have tested for which systems the SSP assumption holds, by quantifying the spread in age between all the stellar models used by the pPXFfit and assesses whether the derived relation depends upon this quantity.
In particular, we found: • a clear correlation between the velocity dispersion and the DoR, confirming the results already presented in previous INSPIRE papers: at equal stellar mass relics have larger  ★ than non-relics; • a linear relation between the stellar metallicity and the DoR, although with a large scatter.The slope of the relation is much steeper considering only extreme relics for which the SSP assumption certainly holds; • a correlation between the IMF slope and the DoR, with a scatter that increases when the stellar population fit requires a larger number of SSP models with different ages.Specifically, all UCMGs with DoR> 0.7, i.e. extreme relics, require an IMF which is dwarf-richer than that of the Milky Way, with slopes of Γ  ≥ 1.8.For UCMGs with lower DoR, the spread in the IMF slope becomes very large, consistent with what was found in INSPIRE IV.
• considering the three components forming the DoR, there are no UCMGs with Γ  > 2 when the fraction of stellar mass formed by  ∼ 2 is lower than 60%; • a correlation between the slope of the IMF and the stellar velocity dispersion, which is even steeper than the ones from the literature; • a very weak correlation (or no correlation) between the IMF slope and the stellar metallicity; • that at equal velocity dispersion and/or metallicity, galaxies with a higher DoR have a dwarf-richer IMF; • no correlation between the DoR and the other stellar population parameters we infer ([Mg/Fe], [Na/Fe], [Ti/Fe]).
In conclusion, our data supports a scenario whereby an excess of dwarf stars might originate from the first phase of the mass assembly at high-z when the density and temperature of the Universe were higher, and thus when fragmentation might have been easier (Chabrier et al. 2014).However, to reconcile the very recent findings at very high-z from JWST, we speculate that the IMF might vary with cosmic time (Vazdekis et al. 1996(Vazdekis et al. , 1997;;Weidner et al. 2013;Ferreras et al. 2015).An incredibly quick top-heavy phase at very high-z (> 5), that produces very metal-poor stars of ∼ 100 M ⊙ .These stars quickly die and pollute the interstellar medium with metals.Then, a large number of dwarfs is produced in regions of high density and temperature, through fast starbursts up to  ∼ 2, while stars are distributed with a Milky-Way-like IMF slope if they formed in the second phase of the two-phase formation scenario, under less extreme temperature and density conditions.ii) we have plotted the histogram distribution of the indices computed on all the models varying all the parameters, and have flagged systems for which a given index falls outside this distribution, iii) we have produced as many index-index plots as possible, creating large model grids with changing resolution (in km s −1 ) and elemental abundance.
The result of this latter test is visualised in Figure A1.Here we show four representative index-index plots (top: Mg2 versus the two Fe lines, bottom: NaD vs the two TiO lines) and a series of grids predicted by the MILES SSP models.For each panel, if a galaxy falls outside the model grid, it is highlighted with a black circle and flagged as 'contaminated'.We note that we have performed these tests considering a larger set of indices but still arising from the same chemical elements (H𝛽G,H𝛽,H𝛽0,Mg2,Mgb5117,Fe5015,Fe5270,Fe5335,Fe5709,NaD,H𝛿A,TiO1,TiO2,TiO3,TiO4).This was done in order to select, for each element the least-contaminated index.
Indeed, inspecting by eye each of the spectra, we found that for 40/52 UCMGs, the six indices mentioned above are clean from residual sky-lines and bad pixels.From these, however, we exclude two more objects that have a non-negligible percentage (≥ 25%) of stars younger than 1 Gyrs.This is because in these two cases, the assumption of an SSP model is not a valid one.The final sample of 39 uncontaminated, old galaxies has thus been denoted as the Golden Sample.All the plots presented in the main body of the paper always show them as diamonds, while the objects for which one or more indices are contaminated are plotted as circles.
At this point, as a further test for the robustness of our results, we have made several modelling runs with different combinations of spectral indices and different setups (e.g., not fitting for [Na/Fe] or/and not fixing for the age and hence including or not Balmer lines).All the runs are summarised in Table A1.
Firstly (Runs 1-6), we investigated whether the results would differ greatly by excluding/including other spectral indices from the fiducial ones.In particular, in Run 1 and 3 we have excluded the NaD index and not fitted for Na abundance, which did not lead to any notable changes in the modelled values.We note that we do not test the case without TiO indices and fixing the [Ti/Fe] to solar because the TiO indices are the best gravity-sensitive indicators and we need to include them in order to have a solid constraint on the IMF slope, which is the main goal of this paper.In Run 2 we have used an alternative definition of the Mgb spectral feature (Mg2 instead of the more classical Mgb5177, see Table C1 for the index definition), which led to larger metallicities at the cost of lower Na abundance, as seen in Figure A2.These changes are systematic but are smaller for more extreme relics.Importantly, the IMF slopes inferred from

APPENDIX C: SPECTRAL INDICES DEFINITION
In Table C1 we provide the index definition, along with the blue and red continuum bandpasses for all the indices used in this paper.We also give the reference to the paper where each index was defined and used for the first time.[4895.125−4957.625] [5154.125−5196.625] [5301.125−5366.125]mag Trager et al. (1998) Mgb5177 [5142.625−5161.375] [5160.125−5192.625] [5191.375−5206.375

Figure 1 .
Figure 1.The six indices constituting our fiducial set: Mgb, Fe5270 and Fe5335 (top line) and NaD, TiO1 and TiO2 (bottom line).For each index, the top panel shows the index (grey shaded region) and blue and red bands (blue shaded regions) around which the flux is normalised.The black lines correspond to model spectra of solar metallicity [M/H] = 0, [Mg/Fe] = 0, [Ti/Fe] = -0.3,[Na/Fe] = -0.3 and IMF slope of Γ B = 1.3.The bottom panels show the relative change in the spectrum after varying different stellar population parameters: the IMF slope was varied by ΔΓ B = 1, the metallicity by 0.4 dex and the different abundance ratios by 0.6 dex each.

Figure 2 .
Figure 2. The degree of relicness against the maximum sum of the weights held by 5 neighbouring bins in the age distribution histograms.Dashed line marks the 0.95 threshold of the sum of weights.On the right-most panel a sample of age-distributions is presented for a quick reference.

Figure 3 .
Figure 3. Stellar population analysis results.The data points are colour-coded by the DoR, computed in DR3.The only two parameters that show a correlation with it are the stellar metallicity and stellar velocity dispersion, which are both larger for relics than for non relics.Moreover, the most extreme relics (red points) seem to cluster around log(R e ) ∼ 0 kpc and M ★ ∼ 1-1.5 ×10 11 M ⊙ .

Figure 4
Figure 4. FIF fitted metallicity against the DoR, colour-coded by the maximum sum of weights of 5 neighbouring bins of the age-distribution, quantifying how good the single-stellar population assumption is.Diamonds represent galaxies from the Golden Sample, while dots show the galaxies with contaminated spectral indices.Golden Sample galaxies with sum 5 > 0.95 are marked by dark border.The lines represent the linear fits for the three samples.

Figure 5 .
Figure 5.The slope of the IMF against the DoR, colour-coded by the maximum sum of weights of 5 neighbouring bins of the age-distribution.Symbols and lines are as in previous figures.

Table 4 .
Linear fits to the IMF slope versus the DoR (Fig 5) for different sub-samples of galaxies.The last two columns report the Pearson correlation coefficient () and its p-value.

Figure 6 .
Figure 6.The slope of the IMF against the DoR (left) and the three quantities that were used to define it, and the distribution of Γ  inferred for the 52 UCMGs (histogram in the rightmost panel).The datapoints are colour-coded by the maximum sum of weights help by 5 neighbouring bins of the age-distribution.Diamonds represent galaxies from the Golden sample, while dots mark the galaxies with contaminated spectral indices.

Figure 7 .
Figure7.Stellar IMF slope versus stellar velocity dispersion (left, computed in DR3) and total stellar metallicity (right, computed with FIF).The INSPIRE galaxies are colour-coded by their degree of relicness (top row) and sum 5 (bottom row).The lines show linear fits to the data, from this work (grey) and from literature papers, as reported in the legend and in the third and fourth block of Table5.Symbols are like in other figures.

Figure A1 .
Figure A1.Examples of four index-index plots showing the SSP model grid and the measurements for the INSPIRE galaxies colour-coded by their resolution/velocity dispersion.Systems that have no contaminated indices (Golden sample) are marked by crosses, while systems that have at least one contaminated index are marked by a dot.The black circles mark the systems that were eliminated on each panel based on the corresponding indices.A typical error-bar is reported in the bottom right corner of each panel.

Figure A2 .
Figure A2.Comparison between the stellar population parameters obatained using the two different spectral index definitions for Magnesium (Mg2 and Mg5177).The data points are colour-coded by the DoR.

Figure A3 .
Figure A3.Histograms of the different results of all 19 FIF runs (blue) and the runs from 1 to 4 only (orange) and the fiducial fit.Each panel shows results for a different stellar population parameter.The majority of the results, as well as the overall distribution, is consistent with the fiducial fit results (within the errors, grey area), confirming the robustness of our fitting.

Figure B1 .
Figure B1.Comparison between the DR3 results and the ones obtained in this paper with pPXF, colour-coded by the DoR.As in previous figures, galaxies in the Golden Sample are shown as diamonds, although we note that these results are obtained full-spectrum fitting, which should overcome the problem of having contamination within one or more index bandpass.

Table C1 .
Indices bandpasses, units in which they are measured, and papers from which they have been taken.