A comparative study of the fundamental metallicity relation The impact of methodology on its observed evolution

Aims. We investigate the influences on the evolution of the Fundamental Metallicity Relation of di ff erent selection criteria. Methods. We used 5487 star-forming galaxies at a median redshift z ≈ 0 . 63 extracted from the VIMOS Public Extragalactic Red-shift Survey (VIPERS) and 143774 comparison galaxies in the local Universe from the GALEX-SDSS-WISE Legacy Catalog. We employed two families of methods: parametric and non-parametric. In the parametric approaches, we compared the Fundamental Metallicity Relation projections plagued by observational biases on di ff erently constructed control samples at various redshifts. Then, the metallicity di ff erence between di ff erent redshifts in stellar mass-star formation rate bins. In the non-parametric approach, we re-lated the metallicity and the normalized specific star formation rate (sSFR). To compare galaxies with the same physical properties, we normalized the median of our samples according to the median sSFR at median redshift z ≈ 0 . 09. Then, the galaxies with the same distance from the star-forming main sequence at their respective redshifts are compared when the sSFR is normalized according to the expected values from their respective star-forming main sequence


Introduction
Stars, embedded in the parent galaxy, produce heavy elements at different stages of their lives.These are released in the gas phase during their evolution and at their death, changing the Catalogue is available at the CDS via anonymous ftp to cdsarc.cds.unistra.fr (130.79.128.5) or via https: //cdsarc.cds.unistra.fr/viz-bin/cat/J/A+A/683/A203metallicity, that is the amount of heavy elements relative to hydrogen and helium of the interstellar medium (ISM).The key processes that impact this abundance during galaxy evolution are inflows, outflows, star formation, and quenching (Lilly et al. 2013;Peng & Maiolino 2014), implying that the gas-phase metallicity can be used to trace the star formation history (SFH) and the evolution of the ISM in galaxies.Measuring the metallicity of galaxies at different epochs constrains and strengthens galaxy formation models (Maiolino & Mannucci 2019), providing information on the early enrichment processes of galaxies and their intergalactic medium (IGM).
The metallicity is strongly correlated to other galaxy properties, in particular to stellar mass (M ).The so-called mass metallicity relation (MZR; see, e.g., Tremonti et al. 2004;Savaglio et al. 2005;Kewley & Ellison 2008), has been studied with a high level of precision using data from, among others, the Sloan Digital Sky Survey (SDSS; see, e.g., Tremonti et al. 2004;Mannucci et al. 2010;Curti et al. 2020), where a positive correlation between metallicity and M was found for star-forming (SF) galaxies.This trend is observed at all redshifts studied so far (Lee et al. 2006;Maiolino et al. 2008;Lamareille et al. 2009;Pérez-Montero et al. 2009;Mannucci et al. 2010;Zahid et al. 2011;Cresci et al. 2019;Curti et al. 2020;Pistis et al. 2022), up to at least z ∼ 3.5 (Maiolino et al. 2008).Understanding the relation between M and metallicity is crucial when establishing a model of a galaxy's evolution (Lequeux et al. 1979).The origin of the MZR is still debated, and different models of galaxy evolution were proposed to explain this relation.To this aim, these models need to incorporate, among other aspects, the outflow of metal-rich gas and the inflow of metal-poor gas (Tremonti et al. 2004;Finlator & Davè 2008;Davè et al. 2010Davè et al. , 2011;;Chisholm et al. 2018), the dependence of star formation efficiency on galaxy mass (so-called downsizing scenario models; Spitoni et al. 2020;Lilly et al. 2013), and an initial mass function (IMF) that varies with galaxy mass (Köppen et al. 2007;De Masi et al. 2018;Lian et al. 2018a,b).
The metallicity also shows an anti-correlation with the star formation rate (SFR), as shown by Mannucci et al. (2010Mannucci et al. ( , 2011)); meaning, more strongly star-forming galaxies contain fewer metals, even for galaxies at fixed M .This anticorrelation is in agreement with models including inflows of metal-poor gas, a process that ignites star formation and dilutes the metallicity of the ISM, and outflows of metal-rich gas: a process that suppresses star formation and removes the metals from the ISM (Lilly et al. 2013;Peng & Maiolino 2014).
The combination of this metallicity-SFR relation and MZR is known as the fundamental metallicity relation (FMR).The FMR has a reduced scatter compared to the MZR and does not show any evolution, at least up to z ∼ 2.5 (Mannucci et al. 2010;Cresci et al. 2019;Curti et al. 2020).The SFR and M are also correlated, which results in a relation known as the main sequence of SF galaxies (MS; Brinchmann et al. 2004;Noeske et al. 2007;Elbaz et al. 2007).The MS is then yet another projection of the FMR, where the observed scatter around the MS can be interpreted as a consequence of the inflows of gas resulting in a period of a burst of star formation (Abramson et al. 2014;Tacchella et al. 2016;Mitra et al. 2017) and of the starvation of the galaxy due to the outflows of gas (Trussler et al. 2020;Brownson et al. 2022).The MS also shows an evolution with redshift (Speagle et al. 2014;Schreiber et al. 2015) with galaxies with a higher SFR at a given M at higher redshifts.Stronger star-forming processes in massive galaxies in the younger Universe may be caused by bigger reservoirs of gas available in a galaxy to form new stars in the earlier cosmic time (Tacconi et al. 2010;Scoville et al. 2016;Kokorev et al. 2021).
While most studies about FMR focus on star-forming galaxies, the empirical metallicity calibration for non-SF galaxies based on nebular lines (Kumari et al. 2019) allowed the authors to expand the study of the FMR toward non-SF galaxies (Kumari et al. 2021).The FMR of non-SF galaxies agrees with models that include enhancement of the star formation due to the infall of metal-poor gas and starvation, which prevent  the infall stopping the dilution process of metals (Kumari et al. 2021).It remains unclear if the FMR agrees with merger-induced starbursts (Grønnow et al. 2015;Bustamante et al. 2020) for this class of galaxies.However, the evolution of the FMR for non-SF galaxies has not yet been studied.Assuming a regulator system in which the SFR is modulated by the gas reservoir (Lilly et al. 2013;Peng & Maiolino 2014) allows the authors to model and reproduce the local FMR (Mannucci et al. 2010) and MZR (Tremonti et al. 2004;Mannucci et al. 2010).To model the theoretical evolution of the FMR, this model considers i) the evolution in cosmic time of the specific SFR (sSFR, defined as the ratio between SFR and M ) corresponding to the halo growth, ii) the gas-phase metallicities over the galaxy population, and iii) the stellar-to-darkmatter mass ratio of the halos.
The comparison of galaxy samples at different redshifts involves many difficulties, including: i) different selection effects (e.g., survey limit, such as the signal-to-noise ratio, S/N, of spectra Pistis et al. 2022); ii) different rest frames observed; iii) different selection criteria for SF galaxies and for estimation of metallicity; iv) different metallicity calibrators based on different sets of emission lines or different calibration methodology (Kewley & Ellison 2008); v) small samples available at higher z (as seen from Table 1); vi) different methods of selecting galaxies for comparison; vii) different methods of comparison (to be addressed in this study).
In order to investigate if the FMR is really fundamental or if it depends on (redshift-variant) methods of comparing the samples, we made use of the unprecedented statistics of the VIMOS Public Extragalactic Redshift Survey (VIPERS) to study and compare, for the first time and with a high level of statistical significance, the MZR and FMR at median z ∼ 0.63 and at median z ∼ 0.09.In this paper, we used a variety of methods  of comparison in order to determine how strong the conclusions are depending on the method used.We applied the following methods: i) a family of parametric methods, based on the direct comparison of different projections of the FMR, ii) a nonparametric method based on Salim et al. (2014Salim et al. ( , 2015)).The nonparametric method is based on the comparison between the metallicity and the normalized sSFR in different M bins.The choice of normalization for the sSFR in the nonparametric method allows us to choose the properties that will be compared between the samples.
The paper is organized as follows.In Sect.2, we describe the main samples with their initial data selection.In Sect.3, we describe the construction of the so-called control samples, that are subsamples with specific properties cross-matched at different redshift ranges.In Sect.4, we present two families of methods to compare low-and intermediate-redshift data.In Sect.5, we present the comparison of the samples.In Sect.6, we analyze the evolution of the MZR and the metallicity-SFR relation.In Sects.7 and 8, we discuss the results and present the final conclusions.The cosmological parameters adopted in this paper are H 0 = 70 km s −1 Mpc −1 ; Ω M = 0.3; Ω Λ = 0.7; we assumed a Chabrier (2003) IMF.

Main data samples
We used spectroscopic data from two surveys: VIPERS (0.5 < z < 1.2) and SDSS (0 < z < 0.3).We also made use of the VIMOS VLT Deep Survey (VVDS;Le Fèvre et al. 2013) to validate the spectroscopic measurements performed on the VIPERS sample and to validate the shape of the MZR obtained with VIPERS.The main selection steps are described below, and a more detailed description of this selection can be found in Pistis et al. (2022).Table 2 summarizes the data selection at each step.

VIPERS sample
We built a sample of galaxies at intermediate redshift by crossmatching two catalogs.The first one is the Public Data Release-2 (PDR2) spectroscopic catalog of VIPERS, which contains 88 340 galaxies (Scodeggio et al. 2018).The target selection requires brighter sources than i AB = 22.5 and a pre-selection in the (u − g) and (r − i) color-color diagram is made to remove galaxies at low redshift.The VIPERS has a spectral resolution of R ∼ 250, which makes it possible to examine specific spectroscopic characteristics of galaxies with a wavelength coverage ranging from 5500 to 9500 Å.The data reduction pipeline and redshift quality system are described in Garilli et al. (2014).This catalog contains the redshifts and flags used to define the confidence level down to 99% (3.0 ≤ z flag ≤ 4.5).
This catalog is then supplemented with the new spectroscopic measurements of fluxes and equivalent widths (EWs) of the emission lines of interest in this paper: Hβ, and [O iii] λ5007.The VIPERS spectra are analyzed with the penalized pixel fitting code (pPXF; Cappellari & Emsellem 2004;Cappellari 2017Cappellari , 2023)), which makes it possible to fit both the stellar and gas components via full spectrum fitting.After shifting the observed spectra to the rest frame and masking out the emission lines, the stellar component of the spectra is fit with a linear combination of stellar templates from the MILES library (Vazdekis et al. 2010) after being convolved to the same spectral resolution as the observations.The gas component is then fit with a single Gaussian for each emission line, giving the integrated fluxes and their errors as a result.In order to have a better estimation of the error, the error given by pPXF is then multiplied by the χ 2 red of the fit under the emission line.
In order to estimate the EWs from the pPXF results, it is useful to normalize to one the spectral continuum of each galaxy.We built the normalized spectrum by dividing the best fit by the stellar component of the fit.The spectra are then analyzed with specutils, an astropy package for spectroscopy (Astropy Collaboration 2013, 2018;Price-Whelan et al. 2018) in a ±1.06 full width half maximum (FWHM) range, which is equivalent to a standard deviation of the Gaussian fit of five (Vietri et al. 2022), around the centroid of the emission line to estimate the EW and its uncertainty.
For this new catalog of spectroscopic measurements, we adopted the same flag system used in the VIPERS catalogs (Garilli et al. 2010;Figueira et al. 2022;Pistis et al. 2022) in the form of a four-digit number, xyzt.The x-value is equal to 1 if the difference between the centroid of the fit and the centroid of the observed data is less than or equal to 7 Å (equivalent to 1 pixel on the VIMOS spectrograph); otherwise, its value is 0. The y-value is equal to 1 if the FWHM is in the 7-22 Å range, equivalent to 1-3 pixels of the spectrograph; otherwise, its value is 0. The z-value is equal to 1 if the difference between the peak of the data and the fit is less than 30%; otherwise, its value is 0. Finally, the t-value is equal to 2 if the signal-to-noise ratio (S/N) for the EW is greater than 3.5 or the S/N for the flux is at least 8. Its value is 1 if the S/N for the EW is at least 3 or the S/N for the flux is at least 7; otherwise, its value is 0.
To check the quality of the measurements, we compared our results with a catalog from the VVDS (Lamareille et al. 2009), as both VIPERS and VVDS used the same instrumental configuration of the VIMOS spectrograph (Le Fèvre et al. 2003).We selected all galaxies from the VVDS sample covering the same z and z flag ranges as the VIPERS sample.We also limited the VVDS sample to the same magnitude as VIPERS, i AB < 22.5, A203, page 3 of 22 (Lamareille 2010) for VIPERS (blue) and VVDS (pink) samples.Contours present 1, 2, and 3 standard-deviation levels of the distributions.The black lines are the thresholds defined in Lamareille (2010) to divide different galaxy types (SF, AGN, and LINERs).
removing deeper observations included in VVDS.The resulting catalog has similar characteristics to VIPERS, both from an instrumental and continuum treatment point of view.For the comparison, we selected galaxies from the VIPERS sample with a minimum flag of 1110 for the lines Hβ, [O The VVDS sample was analyzed with the platefit_vimos pipeline (Lamareille et al. 2009), which is the adapted version of platefit used to analyze high-resolution SDSS spectra (Tremonti et al. 2004;Brinchmann et al. 2004).To make the comparison between VIPERS and VVDS consistent, the same treatment of the continuum and stellar part adopted by plate-fit_vimos was used in pPXF, fitting the gas and stellar spectra separately.
Figure 1 shows the 1, 2, and 3 standard deviation levels of the surface density distribution contours in the BPT diagram (Baldwin et al. 1981) for VIPERS and VVDS samples.
Both samples show very compatible distributions.We report further analysis to validate the new line measurements in Appendix A, and the S/N distributions of the main emission lines are given in Appendix B.
The catalog of line measurements obtained by this procedure is then cross-matched with the physical properties (in particular M ) catalog for VIPERS galaxies (Turner et al. 2021) measured via the spectral energy distribution (SED) fitting with the Code Investigating GALaxy Emission (CIGALE; Burgarella et al. 2005;Noll et al. 2009;Boquien et al. 2019).The construction of the catalog reduces the sample to 39 204 galaxies since the catalog with the physical properties is not complete in the W4 field.
To select SF galaxies and exclude LINERS and Seyfert galaxies from the sample, we used the line ratio classification of the BPT diagram.The original line ratio diagram to select different types of active galaxies uses [O iii] λ5007, [N ii] λ6584, [S ii] λ6717, 6731, Hα, and Hβ.Because of the redshift range covered by VIPERS (z > 0.5 for most of its sample) and the VIMOS spectrograph wavelength coverage (5500-9500 Å), the emission lines necessary for the original BPT diagram (Kauffmann et al. 2003) are not detected ([N ii] λ6584 and [S ii] λ6717, 6731), or they are detected for few galaxies (Hα).For this sample, it is necessary to use the "blue" BPT diagram (Lamareille 2010).This version of the BPT diagram can be used for a higher redshift range and uses the [O ii] λ3727, [O iii] λλ4959, 5007, and Hβ spectral lines.The selection of SF galaxies based on the Lamareille (2010) recipe reduces the sample to 6251 galaxies.
The computations of the properties starting from the line fluxes are done via bootstrapping.For each galaxy, we generate the fluxes and attenuation as random numbers with Gaussian distribution centered at the "observed" values and with standard deviations equal to estimated errors.This process is repeated 1000 times for each galaxy.We then calculate all the derived quantities (R 23 , SFRs, and metallicity) as the median and the error as the standard deviation of the distributions generated during the bootstrapping.
To correct the flux of emission lines for attenuation, we used the attenuation in the V-band (A V ; see Appendix C for the estimation of this property) provided by the fit of the SED via the CIGALE code, corrected by the f factor ( f = 0.57; Rodríguez-Muñoz et al. 2022) to pass from stellar to nebular A V , and a Cardelli et al. (1989) attenuation law with R V = 3.1.
The SFR is computed from the [O ii] luminosity (Kennicutt 1998) transformed into a Chabrier (2003) IMF: where L [O ii] is expressed in units of erg s −1 .The [O ii] luminosity was previously corrected for attenuation.
The metallicity is estimated via the calibration of Tremonti et al. (2004) based on the R 23 parameter (Pagel et al. 1979): and where x ≡ log R 23 .This calibration is only valid for the upper branch of the double-valued R 23 abundance relation.We select the upper branch according to the threshold Nagao et al. 2006).This calibration is valid only for 12 + log (O/H) > 8.4.Consequently, we remove galaxies for which we obtained a lower value of 12 + log (O/H).We also remove all galaxies with uncertainties on metallicity (estimated using the bootstrap method) larger than 0.3 dex.After these steps, our final intermediate redshift sample is reduced to 6018 SF galaxies with a median redshift of z ∼ 0.63.
To obtain the sample we used in our analysis, we followed the data selection introduced by Curti et al. (2020).We applied an S/N limit of 15 and 3 for Hα and Hβ spectral lines, respectively, and corrected all emission lines for attenuation using the Balmer decrement, assuming the case B recombination (Hα/Hβ = 2.87; Baker & Menzel 1938) and adopting the Cardelli et al. (1989) law with R V = 3.1, again correcting the A V by the f factor.We limited the SDSS sample to z ≥ 0.027 to include the [O ii] line in the observed spectra and removed all galaxies showing high extinction, meaning, with values of E (B − V) higher than 0.8.After these selections, the crossmatched sample contained 267 922 galaxies.
To have a homogeneous selection for both the SDSS and VIPERS samples, we selected SF galaxies using the same "blue" BPT diagram (Lamareille 2010) as for the case of galaxies at intermediate redshift.As shown in Pistis et al. (2022), the use of this diagram does not significantly bias the analysis of the FMR.The SF selection further reduces the sample to 158 416 sources.
The physical properties are computed with the same bootstrap method and calibrations used for the VIPERS sample.Because the fiber system of the SDSS survey measures the light with different spatial coverage, the SFR needs a correction to take into account the fiber aperture (Hopkins et al. 2003): where u Petro and u fiber are the modified forms of the Petrosian magnitude (Petrosian 1976) and the magnitude measured within the aperture of the spectroscopic fiber, respectively.We also removed all galaxies with 12 + log (O/H) < 8.4 and a metallicity error larger than 0.3 dex.
From this sample, we also removed all sources with log M [M ] < 7, log SFR M /yr < −10 to exclude lowmass galaxies, not present in the VIPERS sample, as well as all remaining passive objects.Finally, we removed galaxies that did not possess a reliable rest-frame blue magnitude by applying a cut at M B > −24 3 .All those selections removed galaxies with bad SED fit results and reduced the total number of local SF galaxies to 155 893 with a median redshift z ∼ 0.09.

Homogeneous star-forming main sequence
To build one of the control samples used in the following part of this work (called a distance control sample; see Sect. 3 and Sect.5.3), we need properties that are derived from the starforming MS.For this purpose, we require a homogeneous definition of the MS at low and intermediate redshift.
To take into account the MS evolution in the wide redshift range of the VIPERS sample (0.5 ≤ z ≤ 0.8), we divide this sample into three redshift bins: 0.5 ≤ z < 0.6, 0.6 ≤ z ≤ 0.7, and 0.7 < z ≤ 0.8.We perform a linear fit of the SFR-M distribution separately in all three redshift bins.In the next step, we perform 2 https://salims.pages.iu.edu/gswlc/ 3 A B magnitude flagged as −99 in the GSWLC-2 catalog indicates a failure in the estimation of the parameter during the fit procedure.another linear fit to the redshift dependence of the MS parameters (slopes and intercepts) based on these three fits.In this way, we obtain the following redshift-dependent MS relation: with α (z) = −0.04z + 0.70 and β (z) = 1.37 z − 6.44, valid in the redshift range 0.5 < z < 0.8.A detailed analysis of the VIPERS MS was published by Pearson et al. (2023).The redshift range explored by SDSS is small enough that a single linear fit of the SFR-M relation is sufficient for our analysis.The SDSS MS relation is given by log SFR SDSS MS (M ) = 0.92 log M − 8.50.( 6) Figure 2 shows the scatter around the MS as a function of M and redshift for VIPERS and SDSS samples, demonstrating a lack of any statistically significant trend with either M and redshift for both samples.
Both samples show a similar standard deviation level of 1σ of the distributions.The low redshift (SDSS) sample has a bigger scatter around the MS compared to the intermediate redshift (VIPERS) sample, especially at high-M and low redshift.It can be caused by contamination with galaxies moving between active and passive stages that are not removed by the blue BPT diagram used to select SF galaxies.Checking the NUVrK diagram (see Fig. 3; Davidzon et al. 2016), we find two remaining passive galaxies in the VIPERS sample and 1780 (∼0.1% of the sample) remaining passive galaxies in the SDSS sample.
Because of the presence of passive galaxies, especially at large M , that are not removed by the data selection, we add an additional cut in sSFR to remove all galaxies below the fifth percentile in the sSFR distribution, reducing the VIPERS sample to 5717 SF galaxies and the SDSS sample to 148 098 SF galaxies.Then, we perform the fit of the MS again, finding for the SDSS sample.These new MS fits result to be consistent with those found by Whitaker et al. (2012), Lee et al. (2015).

Control samples
In order to study FMR evolution and the robustness of the result against different observational biases, two control samples have been built from our main sample of SDSS galaxies (see Sect. 2.2).These are: i) the property-control sample (hereafter p control) with similar M and SFR to VIPERS galaxies; ii) the distance-control sample (hereafter d control) with the same relative distance from the MS as VIPERS.If the FMR is fundamental -the metallicity depends only on the given M and SFR and not the specific stage of evolution for each galaxyand does not evolve with redshift, the differences in metallicity in the FMR and its projections between the VIPERS sample and the four SDSS samples (all selected SF galaxies, p-control and d-control samples) should be independent of each other.If the metallicity differences in the FMR and its projections are independent of the methodology of cross-matching, the differences depend only on the physical properties (M and SFR) of galaxies.
To build the p-control sample, we selected the three closest SDSS galaxies on the M -SFR plane for each VIPERS galaxy up to 0.1 dex.This way, we avoid unbalancing the galaxy distribution toward the region of the MS where SDSS is much denser compared to the VIPERS sample.This selection reduces the pcontrol SDSS sample to 12 053 galaxies (∼8% of the sample) after removing galaxies counted multiple times.
The MS relations for both SDSS and VIPERS were defined in the same homogeneous way, so they can be used to build the d-control sample.To create this sample, we move vertically over the MS defined by the SDSS sample for each VIPERS galaxy.We then simulate the scatter around the local MS for the d-control sample, adding a random number with Gaussian distribution centered at µ = 0 and width σ (standard deviation of SFR in a 0.1 dex width bin around M of VIPERS sample): We then proceed to select a maximum of three of the closest SDSS galaxies in a radius of 0.1 dex in the M -SFR plane.The d-control sample, constructed in this way, is composed of 14 475 SDSS galaxies (∼10% of the sample) with the same distance from the MS as galaxies in the VIPERS sample.
To summarize, we have two main samples: i) the VIPERS sample at intermediate redshift; and ii) the SDSS sample at low redshift, both built using similar selection criteria and consisting of SF galaxies, and two control samples obtained starting from the SDSS sample and selecting subsamples that simulate the properties of the VIPERS galaxy sample at lower redshift; i) the p-control sample (mimic the physical properties M and SFR); and ii) d-control sample (mimic the scatter around the MS).In the parametric method, we use both main and control samples.In the nonparametric method, we use only the main samples.
Starting from the VIPERS sample, we build a subsample by applying the condition to have a stellar masscomplete sample.Following Davidzon et al. (2016), we adopt the mass threshold as log M [M ] = 10.18 for 0.51 < z ≤ 0.65, log M [M ] = 10.47 for 0.65 < z ≤ 0.8, and log M [M ] = 10.66 for 0.8 < z ≤ 0.9.At the same time, we consider the subsample composed of the galaxies not included in the stellar mass-complete sample, calling this VIPERS sample 'not mass-complete'.These two subsamples are used in Sect.6.
General properties of VIPERS and SDSS main samples and subsamples.Figure 4 presents the kernel density estimation (KDE) of the distributions for the main physical features of the two main samples (VIPERS and SDSS), the VIPERS masscomplete sample (following Davidzon et al. 2016) and what is left out of it -called the VIPERS not-mass-complete sample-and the two control samples (p-control and d-control samples).This figure shows the distribution of M , SFR, metallicity, redshift, sSFR, and the distance from the MS.
The VIPERS mass-complete sample results in a distribution that peaks at higher values for M , SFR, and metallicity with respect to the VIPERS sample.The distribution in sSFR results in an excess of more passive galaxies with respect to the VIPERS sample.The VIPERS not-mass-complete sample, on the other hand, results in a distribution peaked at slightly lower values for M , SFR, and metallicity with respect to the VIPERS sample.
The p-control sample shows a shift toward higher mass with respect to the VIPERS sample, while the SFR distribution shows a good agreement with a small excess at smaller SFR with respect to the VIPERS sample.These differences are due to the fact that, during the building of the p-control sample, galaxies with high M and low SFR are preferentially selected due to the higher sampling in SDSS in the bottom right corner of the M -SFR bin because of the shift in the MSs from the SDSS to VIPERS samples.The distribution in metallicity shows a shift between p-control and VIPERS samples, with the p-control sample being metal richer than the VIPERS sample.By construction, the p-control sample shows a higher sSFR and a bigger distance from the MS with respect to the SDSS sample.Again, the distribution in sSFR has an excess at lower sSFR with respect to the VIPERS sample.
The d-control sample instead recovers the distribution in M .The distributions in SFR, metallicity, and sSFR show a shift toward lower values with respect to the SDSS sample.Both control samples have a reduced redshift range showing a narrower distribution with respect to the SDSS sample, and the construction of the control samples removed the high-z end of the SDSS distribution.
Table 3 summarizes the differences of the median values of M , SFR, and sSFR for all control samples with respect to VIPERS.The distributions of the shift from the MS (defined as the SFR difference between the measured value and the MS value, shown in the bottom right panel of Fig. 4) also differ as the d-control sample has a much narrower distribution than the other samples.

Methods of comparison
In order to compare samples at different redshifts (from z ∼ 0.8, VIPERS, to z ∼ 0, SDSS), we apply two families of methods to study the FMR.The first is parametric: a) the study of the projections of the FMR and b) the metallicity difference in the M -SFR bin.The second is nonparametric: the study of the normalized metallicity-sSFR relation in different mass bins.The idea of the parametric method is to infer information about the FMR via the median projections on different planes.Here, we studied i) the MZR (Tremonti et al. 2004;Savaglio et al. 2005;Mannucci et al. 2010;Curti et al. 2020); ii) the metallicity-SFR relation; iii) the metallicity-sSFR relation; and iv) the projection of minimum scatter (Mannucci et al. 2010).The study of the projections of the FMR has the problem of being affected by biases introduced by the observations and by the data selection (Pistis et al. 2022), especially the MZR and the metallicity-SFR relation.In this method, it is necessary to cross-match the samples in order to compare galaxies with specific properties.This, in turn, tends to introduce additional selection effects to the samples.A second parametric method infers information by studying the metallicity difference between samples in M -SFR bins projected over the MS.This method compares galaxies with similar physical properties directly.
The nonparametric method (Salim et al. 2014(Salim et al. , 2015) ) studies the relation between the normalized sSFR and the metallicity.
Because of the use of the sSFR, this method is independent of the simple shift of M and/or SFR resulting from different techniques of estimation of those physical values; for example, if different samples have physical properties derived using different IMFs, the sSFR automatically takes into account the shift.This method has the advantage of using the projection of the FMR that is less affected by biases (Pistis et al. 2022).The original method (Salim et al. 2014(Salim et al. , 2015) ) defines the normalized sSFR as where log sSFR is the average or median sSFR in the mass bin of the sample at low redshift, for both samples.We also divide the sample into four M bins centered at log M [M ] = 9.5, 10.0, 10.5, and 11.0 with a bin width equal to 0.5 dex.This normalization allows the comparing of galaxies with the same M and SFR.
In this work, we decided to use a second normalization defined as where log sSFR MS is the "local" MS at the redshift of each sample.This normalization makes it possible to compare galaxies with the same relative distance from the MS, giving information about the processes that generate the scatter of the MS itself.
The anticorrelation between metallicity and sSFR has been interpreted in terms of gas accretion from the IGM and circumgalactic medium (CGM; Mannucci et al. 2010;Curti et al. 2020;Kumari et al. 2021).The accreted gas dilutes the gas metallicity and enhances the star formation.However, the large scatter of sSFR per fixed metallicity means that galaxies with the most significant offsets from the MS are not always those with fewer metals with respect to their MS counterparts.This implies the complexity of physical mechanisms in galaxy evolution, for example, environmental effects (shock-heated gas in overdensities cannot cool down efficiently and galaxies become metal-rich A203, page 7 of 22 Galaxies with negative scatter (δ log sSFR < 0) with respect to the MS most probably undergo a quenching process in their recent history (e.g., via depletion or outflows of gas; Ciesla et al. 2016Ciesla et al. , 2018)), while galaxies with a positive scatter (δ log sSFR > 0) with respect to the MS experienced a recent SFR enhancement (burst; e.g., via a merging event or inflows of gas; Elbaz et al. 2018).
In order to account for these processes, we divide the samples into the subsamples of galaxies above the MS (δ log sSFR > 0) and below the MS (δ log sSFR < 0).By studying the slope of the normalized (according to the MS) metallicity-sSFR relation as a function of the mass, we can infer the impact of processes that enhance or quench the SFR on the metallicity during galaxy evolution.
In the following sections (Sects.5 and 6), we study the median relation on different planes, following the binning scheme described in Table 4.
For a fair comparison with the literature, we use the same bin width used in Curti et al. (2020) when binning with a constant bin width.We used a different method of binning the properties of SDSS in the range covered by this sample but not by the VIPERS sample.We only keep bins with more than 25 galaxies (following Curti et al. 2020).
We defined two errors based on i) the distribution of the population inside the bin σ dist from the 84th and 16th percentiles (equivalent to 68% of the population inside each bin) and ii) the error on the median σ med = σ dist / √ N (N is the number of galaxies in the bin).In the following analysis, we use σ med to draw conclusions about the possible evolution of the considered relations, while σ dist provides us with information about their scatter.

Comparison of FMR between different redshift ranges
In this section, we present the results of the comparison of the FMR of the main samples at median z ∼ 0.09 (SDSS), median z ∼ 0.63 (VIPERS), and the control samples.In Sect.5.1, we report the results using the parametric method.In Sect.5.3, we report the results using the nonparametric method.

Parametric method I: FMR projections with control samples
We proceed with the comparison of the samples at median z ∼ 0.09 (SDSS), median z ∼ 0.63 (VIPERS), and the SDSSbased control samples via the parametric method.Figure 5 shows the projections of the FMR on the M , SFR, sSFR, and the plane of minimum scatter µ α = log M − α log SFR (Mannucci et al. 2010) planes.

MZR
The MZR (Fig. 5, upper left panel) of galaxies at intermediate redshift shows lower metallicities at a given M with respect to galaxies at low redshift.This shift is statistically significant with respect to σ med showing an evolution of the MZR with the redshift.However, a large scatter of both populations should be noted.As seen from the upper left panel of Fig. 5, the separation between them is below 1σ dist of both samples.The samples at both low and intermediate redshifts show a similar scatter, given by σ dist , suggesting a lack of evolution of the scatter itself.This implies that the detection or nondetection of the evolution of MZR may be sensitive to source selection in small samples.
The p-control sample shows a similar MZR to the VIPERS sample, while the d-control sample follows the same MZR as the SDSS sample.However, the p-control sample does not show a strong flattening as the VIPERS sample at low M (log M [M ] 9.75).This flattening could be due to the lack of observation or selection of high-SF galaxies at low M , due to the VIPERS sample being mass-complete only above log M [M ] 10.50.The same flattening is observed in the VVDS sample (Lamareille et al. 2009;Pérez-Montero et al. 2009), which has the same survey depth as VIPERS.This counter-intuitive bias, which results in a flattening at low M of the MZR, needs to be explored in more detail.

Metallicity-SFR relation
The metallicity-SFR relation (Fig. 5, upper right panel) of galaxies at intermediate redshift shows lower metallicities at a given SFR with respect to galaxies at low redshift.As the case of the MZR, the metallicity-SFR relation shows a statistically significant shift with respect to σ med , showing an evolution of the relation with the redshift.The difference between low and intermediate redshift decreases at high SFR.Again, a large scatter of both populations should be noted.As seen from the upper right panel of Fig. 5, the separation between them is below 1σ dist of both samples.The samples at both low and intermediate redshifts show a similar scatter, given by σ dist , suggesting a lack of evolution of the scatter itself.
The p control shows a metallicity-SFR relation closer to that of the VIPERS sample.The process of cross-matching removes metal-rich galaxies from the SFR interval of the VIPERS sample , metallicity versus sSFR (bottom left), and metallicity versus log M − α log SFR (bottom right, inside this panel, the scatter as a function of α is reported) for VIPERS (blue dots), SDSS (orange dots), SDSS p-control (green dash-dotted line), and SDSS d-control (red dotted line).The shaded areas show the 1σ dist, while the black error bars show the 1σ med for the metallicity.We report the number of galaxies in each sample in the legend.The vertical black solid line indicates the M (log M [M ] = 9.75) below which the MZR at intermediate redshift flattens.The vertical black dashed line in the MZR plane shows the most conservative mass limit for completeness (log M [M ] = 10.47 for 0.65 < z <= 0.8) in the redshift range observed by VIPERS (Davidzon et al. 2016).For each sample, we report the number of galaxies in the legend.
when the control samples are built.However, the p-control sample does not get as close as the VIPERS sample to the SDSS sample at high SFR.At the same time, the d-control sample shows a weaker correlation between metallicity and SFR with respect to the SDSS sample.In Appendix D, we report the effects due to different calibrations for SFRs.

Metallicity-sSFR relation
The metallicity-sSFR relation (Fig. 5, bottom left panel) shows smaller differences between low and intermediate redshifts.The difference between SDSS and VIPERS samples decreases at high sSFR, with vipers having a higher metallicity at a given sSFRs.Again, this difference is significant with respect to σ med , showing an evolution of the metallicity-sSFR relation.Again, a large scatter of both populations should be noted.As seen from the bottom left panel of Fig. 5, the separation between them is below the 1σ dist of both samples.The samples at both low and intermediate redshifts show a similar scatter, given by σ dist , suggesting a lack of evolution of the scatter itself.
All control samples follow the same metallicity-sSFR relation as the SDSS sample with a small shift toward lower metallicity around log sSFR yr −1 = 9.The differences between the SDSS and all control samples are negligible in the metallicity-sSFR relation with respect to σ dist .

Projection of minimum scatter
For the projection of minimum scatter (defined in Mannucci et al. 2010) of the FMR, both control samples follow the same relation as the SDSS sample.
The projection of minimum scatter shows smaller differences between low and intermediate redshifts.Again, this difference is significant with respect to σ med , especially at high µ α values.Again, a large scatter of both populations should be noted.As seen from the bottom right panel of Fig. 5, the separation between them is below 1σ dist of both samples.The samples at both low and intermediate redshifts show a similar scatter, given by σ dist , suggesting a lack of evolution of the scatter itself.

Parametric method I: Surface of the fundamental metallicity relation
Figure 6 shows the surfaces of the FMR for both SDSS and VIPERS samples.The shapes of the two surfaces agree with each other at high M .
Metallicity difference in M -SFR bins.Figure 7 presents the metallicity difference between SDSS and VIPERS samples for each M -SFR bin.This is the most direct comparison of the FMR between samples.
The average difference in metallicity between all SDSSbased and VIPERS samples is listed in Table 5, where σ VIPERS is the average standard deviation in metallicity inside the M -SFR bin of the VIPERS sample.
The small changes in the ∆ log (O/H) values for the control samples are mainly due to the limited area in the MS observed in these subsamples with respect to the main sample.
We find an average metallicity difference ∆ log (O/H) = 1.78 σ med VIPERS .The majority of bins have a metallicity difference ∆ log (O/H) between ±3σ med,VIPERS .If we apply a stronger selection on Hβ (S /N > 5), the difference is reduced to ∆ log (O/H) ∼ 1.15 σ med VIPERS .

Nonparametric method
In this subsection, we compare the main samples at median z ∼ 0.09 (SDSS) and median z ∼ 0.63 (VIPERS) using the nonparametric method.First, we normalize the sSFR of both samples by the median sSFR of the low-redshift sample to assure that we compare galaxies with both the same M and SFR.
Inside each mass bin, we divided the sample in 0.15 dexwide bins in ∆ log sSFR.Then, we estimated the median in each bin and we estimate the errors according to σ dist and σ med Fig. 8 shows the relations for the main samples.
The difference between the two samples increases with the M .This increasing difference with the M is in agreement with the differences in metallicity among the SDSS (full, p, and d control) samples (as shown in Fig. 7).The small differences in the low-mass bins can be dominated by the flattening at low mass in the MZR of the VIPERS sample.Then, we normalize the sSFR of each sample by the value of their "local" MS.This way, we compare galaxies with the same relative distance to the MS.We proceeded in the same way as the nonparametric method with the division in the same M bins.In each mass bin, we divided the sample in 0.15 dex-wide bins in δ log sSFR.Then, we estimated the median in each bin and we estimated the errors according to σ dist and σ med .Figure 9 shows bigger differences between the samples compared to the normalization with respect to the median value.
To analyze the processes that lead the galaxies to move around the MS, we study the slope of the normalized metallicity-sSFR relation as a function of the M for galaxies above (δ log sSFR > 0) and below (δ log sSFR < 0) the MS.Table 6 reports the values of the slope.
Figure 10 shows the slope of the normalized metallicity-sSFR relation as a function of the M .
For both samples and both positions with respect to the MS, a negative slope indicates processes that are diluting the metals in the ISM while quenching or enhancing the SFR of the galaxies.
Above the MS (δ log sSFR > 0), we find stronger dilution effects from processes enhancing the SFR at low redshift for smaller M .The dilution effects remain almost constant in the whole mass range, explored here at low redshift, while more massive galaxies at intermediate redshift have a stronger dilution than the less massive galaxies.The difference between low and intermediate redshifts decreases with the M .Below the MS (δ log sSFR < 0), we find a consistent relation within uncertainties between low and intermediate redshifts with less dilution of the metals increasing the M .Again, the differences between samples reduce as the M increases, with the exception of the last M bin.Below the MS, the dilution of metals remains stronger at intermediate redshift in comparison to the low redshift for the whole range of the M .

Evolution of the MZR, metallicity-SFR relation, and FMR
Being particularly careful to homogenize the property estimations of both samples, we want to statistically quantify the evolution (within the uncertainties) of the MZR, metallicity-SFR relation up to z ∼ 0.8 (Fig. 5 upper panels), and FMR.The systematic trend of the VIPERS sample having lower metallicities than the SDSS at almost all the examined mass ranges is visible, but given the scatter of both samples, it is significant only at the level of σ med .
A possible reason for this apparent weak evolution of the scatter in the MZR and metallicity-SFR relation could be A203, page 10 of 22  (Davidzon et al. 2016).In the left panel, the M -SFR bins are color-coded according to the metallicity difference in dex.In the right panel, the bins are color-coded according to the metallicity difference normalized by the σ med VIPERS .
Table 5.Average (over bins) differences in metallicity between all SDSS-based and VIPERS samples (as in Fig. 7) in M -SFR bins.

Sample
∆ Notes.The differences are expressed in absolute units, in units of σ dist VIPERS , and in units of σ med VIPERS .The difference found in this work is about half of the difference found in Pistis et al. (2022).This change is due to the different catalog of line measurements used.
observational biases in the VIPERS sample; a particular reason is the fact that the sample is not mass-complete, meaning, with increasing z we lose less bright, and consequently less massive, galaxies.To check if this small evolution between SDSS and VIPERS samples results from the VIPERS lower redshift galaxies dominating the MZR and metallicity-SFR relation of the whole sample, we conducted a series of tests, described below.
To check the impact of the mixture of galaxies at different redshifts, we split the VIPERS sample into two redshift bins (with a threshold at the central redshift z = 0.65).
Figure 11 shows the MZR and the metallicity-SFR relation for different VIPERS-based sub-samples.Any difference in the MZR (upper panel in Fig. 11) between the redshift sub-samples is negligible, well below the statistical σ dist of the measurements.The MZR for the mass-complete sub-sample results at higher metallicities with respect to the VIPERS sample while the not mass-complete sub-sample follows with better agreement the VIPERS sample.
The metallicity-SFR relation (bottom panel in Fig. 11) of the two redshift subsamples shows bigger differences than the MZR, with the high-z subsample having higher metallicity (likely because of it being dominated by more massive galaxies) and showing a stronger anticorrelation.Also, the mass-complete subsample shifts toward higher metallicity (also likely because it is dominated by higher mass galaxies) showing a much stronger anticorrelation (see also Appendix D).The not-mass-complete subsample follows the same relation as the VIPERS sample up to log SFR M yr −1 ∼ 1.25 when it shifts toward lower metal-licity with respect to the VIPERS sample.However, all these differences are not statistically significant compared to the σ dist of the metallicity.
Regarding the total FMR, the role of a mass-complete sample is the one with the major impact on the results compared to the methodology applied in this study.Figure 12 shows the metallicity difference between SDSS and the (not) mass-complete VIPERS samples.These are also reported in Tables 7 and 8. Once a mass-complete sample is required, the metallicity difference ∆ log (O/H) σ med VIPERS (Table 7) is reduced by half with respect to the VIPERS SF sample (Table 5).This can be due to the simple fact that we are now comparing a much smaller area of the FMR (median values of log M [M ] = 10.46,log SFR M yr −1 = 1.42, and 12 + log (O/H) = 8.83, and much narrower distributions than the VIPERS sample), removing a large area at low M .The average metallicity difference at low M can be dominated by the observation bias that allows us to see only the brightest low-M galaxies.However, without taking into account a stellar mass-complete sample, the comparison is not as reliable as once the condition is applied.
In order to quantify whether SDSS and VIPERS are statistically different, we perform a Kolmogorov-Smirnov (KS) test.We divide both samples in the same M or SFR bins and perform the KS test between the distributions of the samples within each bin. Figure 13 shows the resulting p value as a function of the M and SFR.
Only the three highest M , the fourth and fifth to last, and the lowest SFR bins have a p value ≥0.05, meaning the  (Salim et al. 2014(Salim et al. , 2015) ) between VIPERS (blue dots) and SDSS (orange dots) samples.The shaded show the 1σ dist while the black error bars show the 1σ med for the metallicity.Mass bins are centered on the values indicated in each panel and are 0.5 dex wide.We also report the number of galaxies for both samples in each mass bin.
probability that the two samples are drawn by the same distribution is statistically significant (≥95%).Figure 13 also shows the probability of having the samples drawn from the same distribution in M and SFR ranges.Here, for the metallicity-SFR relation, the samples are statistically equivalent only for log SFR M yr −1 ∼ 1.75.For the MZR, the samples are statistically equivalent only for log M [M ] ≥ 10.4.From the point of view of the evolution of the MZR and metallicity-SFR relation, only the high M end of the MZR is comparable between low and intermediate redshifts, while the metallicity-SFR is statistically not comparable in the whole range explored.
We also compare the metallicity distributions in M -SFR bins in order to compare the FMR at low and intermediate redshifts by KS-test. Figure 14 shows the scatter in the M -SFR plane color-coded according to the p value resulting from the KS test.
The majority of bins of the surface explored by the VIPERS sample show a p value ≤10 −4 .For the majority of the bins, the two samples are drawn from different distributions.Even removing the area where the MZR at intermediate redshift flattens at low M , the majority of bins still have really low values of the p value.Also, the evolution of the FMR is statistically significant between low and intermediate redshifts.If we again apply a more strict data selection to Hβ (S /N > 5), we cannot accept the null hypothesis of the KS test (data drawn from the same distribution); the number of bins for which the null hypothesis can be accepted is greatly reduced.
Figure 15 shows the comparison between the MZR for the VIPERS and the SDSS samples with different fits reported in the literature (Tremonti et al. 2004;Savaglio et al. 2005;Mannucci et al. 2010;Huang et al. 2019;Curti et al. 2020).
We also report the fits of the MZR of studies claiming to measure the evolution of the MZR (Savaglio et al. 2005;Huang et al. 2019).The shape of the MZR for both samples agrees with the literature at high-M , but the tail at low-M seems to be much too flatter in the VIPERS sample.
In the same plot, the MZR of the VVDS wide (i AB < 22.5) and deep (i AB < 24.0) fields are also reported.These VVDS  (Salim et al. 2014(Salim et al. , 2015) ) normalized according to MS between VIPERS (blue dots) and SDSS (orange dots) samples.The shaded areas show the 1σ dist, while the black error bars show the 1σ med for the metallicity.Mass bins are centered on the values indicated in each panel and are 0.5 dex wide.We also report the number of galaxies for both samples in each mass bin.Table 6.Slope of δ log sSFR-metallicity relation for δ log sSFR < 0 and δ log sSFR > 0.  2013).Since we only use the VVDS sample here to validate the shape of the MZR, we do not recompute the physical properties using the values found in the catalog (Lamareille et al. 2009).The MZR of the VIPERS sample follows the same shape as the VVDS samples showing a flattening at the low-M tail.This behavior happens for log M [M ] < 10.0 where the VIPERS survey is not mass complete, while the completeness of the VVDS sample is ensured in the whole magnitude range of the spectroscopic survey (McCracken et al. 2003;Le Fèvre et al. 2013).

Discussion
In this paper, we studied the FMR at low redshift (SDSS data, median z ∼ 0.09) and intermediate redshift (VIPERS data, median z ∼ 0.63) using two different methods: i) the study of the FMR projections also using direct cross-matching between samples at different redshifts; II) the nonparametric approach, which compares the metallicity versus the normalized sSFR at different M bins, cross-matching galaxies according to the normalization of the sSFR.We aimed to study the influence of the methods on the conclusion about the evolution of the FMR.At the same time, we decided to check for the presence of observational biases that A203, page 13 of 22 were not taken into account by cross-matching the catalogs at low and intermediate redshifts (Pistis et al. 2022).In this section, we discuss the results of the methods used for the comparison of the FMR between different redshift ranges (Sect.7.1), the comparison of the samples (Sect.7.2), and the evolution of the MZR and the metallicity-SFR relation (Sect.7.3).

Methods of comparison
In the study using the parametric method, we built two control samples with the following characteristics: i) cross-matched physical properties (M and SFR, p-control sample); ii) a reproduction of the relative distance from the MS of the VIPERS sample (d-control sample).We used two different normalizations in the nonparametric method: i) one with respect to the median sSFR of the sample at low redshift allowing us to compare galaxies with the same physical properties (M and SFR), and ii) one with respect to the sSFR value from the fit of the MS allowing us to compare galaxies with the same relative distance from the MS at different  (Davidzon et al. 2016).The shaded areas show the 1σ dist while the black errorbars show the 1σ med for the metallicity.For each sample, we report the number of galaxies in the legend.
redshift.This last method allows us to study the processes that move galaxies around the MS producing the intrinsic scatter of the MS itself via enhancement of the SFR or starvation of the galaxy.We find that i) The parametric method has the problem of inferring information about the FMR surface from the study of its median projections.In fact, these are expected to evolve compared to the whole surface.Moreover, in order to compare samples with specific properties, it is necessary to cross-match the specific properties.ii) Moreover, the nonparametric method has the advantage of being mostly independent of bias.Changing the normalization, this method allows us to compare galaxies with the same physical properties (M and SFR) or galaxies with the same relative distance from the MS between low and intermediate redshift.Positive points include the fact that the nonparametric method is simpler to use than the various projections with the advantage of directly comparing galaxies with similar physical properties or relative distance from the MS without the necessity to A203, page 14 of 22 Table 7.Average (over bins) differences in metallicity between all SDSS-based and VIPERS mass-complete samples (as in Fig. 12) in M -SFR bins.

Sample
∆ Notes.The differences are expressed in absolute units, in units of σ dist VIPERS , and in units of σ med VIPERS .
Table 8.Average (over bins) differences in metallicity between all SDSS-based and VIPERS non-mass-complete samples (as in Fig. 12) in M -SFR bins.(dots) and SFR bins for the metallicity-SFR relation (diamonds).The bottom panel is the same as the top panel, but we report the product of the sequence for X > x where X is the M or the SFR and x is the value on the x-axis.The dashed horizontal line (y = 0.05) shows the threshold for the 95% confidence level of the KS test.cross-match the catalogs.They are also independent of the biases that could be introduced by observation and data selection.The nonparametric method, using the normalization from the MS, also gives information about the processes that lead to a drop in or an enhancement of the SFR.
Disadvantages include the fact that the parametric method needs to be taken with a lot of caution as it can be affected by biases introduced by observation and data selection as the FMR projections are sensitive to these kinds of biases (combining the M and SFR reduces the effects due to biases).Since the FMR projections are expected to evolve compared to the FMR itself, it can be difficult to deduce information on the whole FMR starting from its projections.To compare specific properties between different samples, it is necessary to cross-match these properties.

Comparison of FMR between different redshift ranges
Being particularly careful to homogenize the samples at different redshifts, the expected evolution of the MZR (Fig. 5, upper  left panel) is statistically significant compared with the σ med on metallicity.This evolution is confirmed by the KS test (Sect.6, Fig. 13).The metallicity difference between SDSS-based and VIPERS samples (Table 5) in M -SFR bins does not show any particular variation for different SDSS-based samples, but it shows a systematic increase with the M , which is dominated A203, page 16 of 22 by the flattening at low M present in the sample at intermediate redshift.
The study of the slopes in the normalized (by the value expected by the MS) metallicity-sSFR relation (Fig. 9) shows an increasing difference between samples at intermediate and low redshift with M .The same shift in the M was also found by Salim et al. (2015) at z ∼ 2.3 based on Steidel et al. (2014) from the Keck Baryonic Structure Survey (KBSS).
The study of the slopes in the normalized (by the value expected from the MS) metallicity-sSFR relation (Fig. 9) allows us to study the processes that move the galaxies around the MS.Galaxies above the MS (δ log sSFR > 0, Fig. 10 upper panel) undergo processes enhancing the SFR.From the point of view of gas inflow, the small dilution (shallower slope) at intermediate redshift suggests an advanced stage of evolution of the infalling gas, with metallicity closer to the ISM.The small metallicity difference between ISM and infalling gas can be explained by assuming a less processed ISM or a more processed infalling gas.The first scenario does not seem to be accurate since the metallicity of galaxies at intermediate redshift is not (statistically significantly) different from those at median z ∼ 0.09.According to the hierarchical model of galaxy formation, the merging rate increases with redshift assuming closer galaxies inside the clusters.This already "metal-rich" gas can be expelled into the intracluster medium (ICM) beforehand, suggesting environmental effects.This also suggests that the assumption of the pristine nature of the infalling gas is not always true.
The difference in metal dilution between the VIPERS and SDSS samples is more prominent at low-M , but such an effect is less prominent towards higher M .It has been shown that metallicity dilution can be increasingly significant in the case of gaseous mergers (both major and minor, e.g., Ellison et al. 2013).Bustamante et al. (2020) showed that mergers cause a large scatter of FMR and the trends continue to the post-merger stage.Similar conclusions were obtained by theoretical studies that utilize idealized hydro simulations (e.g., Bustamante et al. 2018).Although a detailed analysis of galaxy merger impact on estimated FMR is out of the scope of the present paper, the fact that the galaxy merging rate rises with redshift (e.g., Ventou et al. 2017) may partly explain the difference between the slopes inferred from VIPERS and SDSS data.
Below the MS (δ log sSFR < 0; Fig. 10, bottom panel), the intermediate redshift sample shows, compared to the lowredshift sample, ongoing processes that are more significantly diluting the metals in the whole M range.Again, the difference at different redshifts decreases with increasing M .In this case, the situation is inverted at low M , with the low redshift sample having weaker dilution of the metals.
On the one hand, it can reflect the dilution-starvation scenario proposed in recent studies (e.g., more recent star formation dilutes metals more efficiently, while suppression of fresh gas leads to enhanced metallicity and lower SFR, Kumari et al. 2021).On the other hand, among the processes that can happen to quench galaxy SFR (δ log sSFR < 0), there is the socalled dry merger (Bell et al. 2006;Khochfar & Silk 2009).These dry merging events are characterized by a low amount of gas, suggesting the participation of older galaxies already in an advanced stage of evolution, and they occur for massive galaxies (log M [M ] ≥ 10.4).The reduction in dilution, corresponding to a shallower slope, can be explained by dry merging events.From the point of view of outflows instead, a more negative slope at small M suggests a higher efficiency in removing the metals from the ISM with the main production of metals in the bulge of the or bigger amount of gas expelled from the galaxy itself (starvation).
Our study does not aim to quantify dust masses of the different samples used, mainly due to the lack of infrared detections for the majority of these galaxies.However, dust content in galaxies plays a major role in the evolution of the ISM, and a driver of the SFR.Higher redshift galaxies tend to have larger dust reservoirs (e.g., Takeuchi et al. 2005;Whitaker et al. 2017).Additionally, galaxies with higher M have larger dust masses (Beeston et al. 2018).Metals can be converted into dust during the complex evolution of the ISM.Therefore, Fig. 10 can be seen as a metal depletion process by the current content of dust.In fact, for actively SF galaxies (δ log sSFR > 0), the similar depletion efficiency of metals at higher M at different redshifts, can potentially be explained by the supposed larger dust content towards the higher M .For lower M , metal depletion of VIPERS galaxies is weaker than that of SDSS one.
For the less SF galaxies (δ log sSFR < 0), the metal depletion efficiency at low redshift is weaker than at higher redshift for the whole range of M .The difference is likely to be driven by a more efficient dust-to-metal ratio and a higher fraction of available cold gas (De Vis et al. 2019).Indeed, very recent studies of dust-to-metal co-evolution in galaxies at intermediate redshifts (z < 0.7) found that the conversion of metals to dust can be efficient even in evolved systems with old stellar ages (Donevski et al. 2023).Our finding displayed in Fig. 10 qualitatively agrees with this scenario.
All the aforementioned physical aspects of galaxies are responsible for metal depletion at different redshift ranges.Even though dark matter and galaxy environment are responsible for shaping the evolution of galaxies, dust is crucial in the interaction between the ISM components.The metallicity plays an important role in dust content and consequently dust attenuation (Shivaei et al. 2020;Casasola et al. 2022;Pantoni et al. 2021;Hamed et al. 2023a), along with the dependence on the environment (Hamed et al. 2023b).
The reduced change between different redshifts at higher M can be explained by the dark matter halo bias.This bias leads to a faster evolution of massive galaxies.For the same reason, the MZR and FMR flatten at high M .

Evolution of the MZR and metallicity-SFR relation
Looking for the most comparable and homogeneous measurements of the galaxy properties, we observe a small shift between the low redshift and intermediate redshift samples.This evolution is not statistically significant with respect to the σ med on metallicity (∆ log (O/H) < 2 σ med VIPERS in M -SFR bins).However, we do not observe any evolution of the scatter around the relation with respect to the σ dist from intermediate and low redshift.
Once the intermediate sample is divided into two redshift bins, 0.48 < z < 0.65 and 0.65 < z < 0.80, the MZR within the redshift bins overlaps entirely with the MZR using the whole sample (Fig. 11 upper panel).Once the stellar masscompleteness is taken into account, the VIPERS mass-complete subsample follows the same MZR as the main VIPERS sample.Instead, the metallicity-SFR relation is slightly more sensitive to both redshift and mass-completeness.The differences remain within the σ dist in metallicity but not within the σ med .
From the point of view of the FMR, the metallicity difference is reduced by about half.Then, stellar mass-completeness is not an important property of the catalog in the MZR, but A203, page 17 of 22 it is an important property in the metallicity-SFR relation or FMR.The stellar mass-completeness affects the metallicity-SFR relation by changing the correlation coefficient by ∼−0.3 and reducing the difference between different SFR calibrations (see Appendix D).The stellar mass-completeness condition reduces the mass range explored by the VIPERS sample around the position of the peak of the SDSS sample.The SFR and metallicity distributions are shifted toward higher values once the condition is applied.This is reflected in the FMR with a smaller difference between SDSS and VIPERS samples (see Sect. 6).Previous studies (Mannucci et al. 2010;Cresci et al. 2012;Maier et al. 2015;Salim et al. 2015;Gao et al. 2018;Sanders et al. 2021;Strom et al. 2022;Curti et al. 2023) did not take into account the stellar mass-completeness of the catalogs.The evolution of the FMR can be scaled down once this property of the catalogs is considered.Also, the KS tests performed on the distribution of the samples at low and intermediate redshift within M , SFR, and M -SFR bins confirm the evolution of the MZR and FMR (Sect. 6,Figs. 13 and 14).

Conclusions
We checked if the so-called unified or fundamental relations on metallicity stand at different redshifts and under various selection criteria and methods of comparison.Our analysis focused on the FMR and its behavior at low (SDSS data, median z ∼ 0.09) and intermediate redshifts (VIPERS data, median z ∼ 0.63).We found a systematic shift in metallicity at median redshift z ∼ 0.63 compared to the local Universe.The average metallicity difference in M -SFR bins is ∆ log (O/H) ∼ 1.8 σ med VIPERS , which is statistically significant according to the performed KS-test.Despite the fact that the small difference can be dominated by systematic uncertainties in the calculation of SFR and metallicities (e.g., the f factor , the attenuation parameter, the extinction law, the used SFH in SED fitting), we minimized the systematics by computing the physical properties most homogeneously.This study highlights an earlier evolution of the FMR than expected.Future surveys, for example, the Euclid space mission, will be able to confirm this evolution once the uncertainties can be reduced.
In mass-complete samples, the metallicity difference is reduced to ∆ log (O/H) ∼ 1 σ med VIPERS .This conclusion may be influenced by the limited parameter space spanned, as this represents the lower stellar mass galaxies where the difference is father out from the analysis.A careful reading of the results and their underlying selection criteria is crucial in studies of the mass-metallicity and fundamental-metallicity relations.Another effect of having a mass-complete sample is the change in the value of the correlation coefficient between metallicity and SFR when some calibration (e.g., based on Hβ or [O ii] once the metallicity correction is applied) are used.
The reduction of metal dilution of galaxies below the MS (δ log sSFR < 0) at high M suggests processes of dry mergers at work.Instead, above the MS (δ log sSFR > 0) an inflow of metal-rich gas with metallicity close to the ISM is suggested at intermediate redshift.On average, the difference in metal dilution is decreasing with stellar mass at larger redshifts.
When studying the mass-metallicity and fundamentalmetallicity relations, we recommend using the nonparametric approach, which provides similar results compared to parametric prescriptions; yet, it is easier to use and the given results are easy to interpret.The nonparametric methodology provides a convenient way to compare physical properties, with a smaller impact on observational selection biases.

Fig. 2 .
Fig.2.Scatter around MS as a function of M (upper panels) and redshift (bottom panels) for VIPERS (blue) and SDSS (orange) samples.The gray contours show the scatter around the MS for the samples before the cut in sSFR.Contours show the 1σ, 2σ, and 3σ levels of the distributions.
Fig. 4. Kernel density estimations (KDEs) of M (upper left panel), SFR (upper mid panel), metallicity (upper right panel), redshift (bottom left panel), sSFR (bottom mid panel), and the SFR difference with respect to the corresponding MS value (bottom right panel) for the VIPERS (blue solid line), VIPERS mass-complete (gray dotted line), VIPERS non-mass-complete (olive dash-dotted line), SDSS (orange solid line), SDSS p-control (green dash-dotted line), and SDSS d-control (red dotted line) samples.

Fig. 5 .
Fig. 5. Four projections of FMR: MZR (top left), metallicity-SFR relation (top right), metallicity versus sSFR (bottom left), and metallicity versus log M − α log SFR (bottom right, inside this panel, the scatter as a function of α is reported) for VIPERS (blue dots), SDSS (orange dots), SDSS p-control (green dash-dotted line), and SDSS d-control (red dotted line).The shaded areas show the 1σ dist, while the black error bars show the 1σ med for the metallicity.We report the number of galaxies in each sample in the legend.The vertical black solid line indicates the M (log M [M ] = 9.75) below which the MZR at intermediate redshift flattens.The vertical black dashed line in the MZR plane shows the most conservative mass limit for completeness (log M [M ] = 10.47 for 0.65 < z <= 0.8) in the redshift range observed by VIPERS(Davidzon et al. 2016).For each sample, we report the number of galaxies in the legend.

Fig. 7 .
Fig. 7. Metallicity differences ∆ log (O/H) between SDSS and VIPERS samples.Contours show the 1σ, 2σ, and 3σ levels of the MS distributions for the VIPERS sample.The vertical black solid line indicates the mass below which the MZR at intermediate redshift flattens.The vertical black dashed line shows the most constrictive mass limit for completeness (log M [M ] = 10.47 for 0.65 < z <= 0.8) in the redshift range observed by VIPERS(Davidzon et al. 2016).In the left panel, the M -SFR bins are color-coded according to the metallicity difference in dex.In the right panel, the bins are color-coded according to the metallicity difference normalized by the σ med VIPERS .

Fig. 8 .
Fig. 8.Comparison in nonparametric method(Salim et al. 2014(Salim et al. , 2015) ) between VIPERS (blue dots) and SDSS (orange dots) samples.The shaded show the 1σ dist while the black error bars show the 1σ med for the metallicity.Mass bins are centered on the values indicated in each panel and are 0.5 dex wide.We also report the number of galaxies for both samples in each mass bin.

Fig. 9 .
Fig. 9. Comparison in nonparametric method(Salim et al. 2014(Salim et al. , 2015) ) normalized according to MS between VIPERS (blue dots) and SDSS (orange dots) samples.The shaded areas show the 1σ dist, while the black error bars show the 1σ med for the metallicity.Mass bins are centered on the values indicated in each panel and are 0.5 dex wide.We also report the number of galaxies for both samples in each mass bin.
-matched by the catalogs used byLamareille et al. (2009) and Le Fèvre et al. (

Fig. 10 .
Fig. 10.Slope of relation between metallicity and δ log sSFR as a function of the M above (top) and below (bottom) MS for VIPERS (blue solid line) and SDSS (orange solid line) samples.The vertical black dashed line shows the most conservative mass limit for completeness (log M [M ] = 10.47 for 0.65 < z <= 0.8) in the redshift range observed by VIPERS(Davidzon et al. 2016).

Fig. 11 .
Fig. 11.MZR (top) and the metallicity-SFR relation (bottom) for full VIPERS (blue dots), VIPERS low redshift (purple dashed line), VIPERS high-z (brown dash-dotted line), and VIPERS mass-complete (gray dotted line).The vertical black solid line indicates the mass below which the MZR at intermediate redshift flattens.The vertical black dashed line shows the most constrictive mass limit for completeness (log M [M ] = 10.47 for 0.65 < z <= 0.8) in the redshift range observed by VIPERS(Davidzon et al. 2016).The shaded areas show the 1σ dist while the black errorbars show the 1σ med for the metallicity.For each sample, we report the number of galaxies in the legend.

Fig. 12 .
Fig. 12. Metallicity differences ∆ log (O/H) between SDSS and VIPERS mass-complete samples (top row) and between SDSS and VIPERS nonmass-complete samples (bottom row).Contours show the 1σ, 2σ, and 3σ levels of the MS distributions for the VIPERS subsamples.In the left panel, the M -SFR bins are color-coded according to the metallicity difference in unit of dex.In the right panel, the bins are color-coded according to the metallicity difference normalized by the σ med VIPERS .

Fig. 13 .
Fig. 13.Results of KS test between VIPERS and SDSS samples in order to check evolution of MZR and metallicity-SFR relation.The top panel shows the p value of the KS test performed in M bins for the MZR (dots) and SFR bins for the metallicity-SFR relation (diamonds).The bottom panel is the same as the top panel, but we report the product of the sequence for X > x where X is the M or the SFR and x is the value on the x-axis.The dashed horizontal line (y = 0.05) shows the threshold for the 95% confidence level of the KS test.

Fig. 14 .
Fig. 14. Results of KS-test between VIPERS and SDSS samples in order to check evolution of FMR.The scatter plot is color-coded according to the p value resulting from the KS test in M -SFR bins.The color bar shows the two colors chosen above and below the threshold for the 95% confidence level of the KS test.The vertical black solid line indicates the mass below which the MZR at intermediate redshift flattens.The vertical black dashed line shows the most constrictive mass limit for completeness (log M [M ] = 10.47 for 0.65 < z <= 0.8) in the redshift range observed by VIPERS (Davidzon et al. 2016).

Fig. 15 .
Fig. 15.Comparison of MZR for VIPERS (blue dots), SDSS (orange dots), VVDS wide (dash-dotted pink line), and VVDS deep (dotted forest-green line) samples together with the fit functions in the literature.The shaded areas show the 1σ dist, while the black error bars show the 1σ med for the metallicity.The median redshifts are reported for the samples.

Table 1 .
Samples used in different studies of the FMR.
Notes.Bold values: Higher statistics with respect to previous studies.References.(1)Mannucciet al.

Table 2 .
Steps of data selection and size of VIPERS and SDSS samples after each selection step.
Notes.Selection on S/N on Hα was applied only for the SDSS as this line is not visible in the VIPERS spectra.The redshift range of the VIPERS sample (0.48 < z < 0.8) is a natural consequence of the requirement of having the lines Hβ, [O ii] λ3727, [O iii ] λ4959, and [O iii] λ5007 in the spectral range.In the SDSS, we do not use the redshift or the line flags, but the spectral quality is assured by the high S/N of Hα and we limit the redshift at z ≥ 0.027 to include [O ii] doublet in the spectra.

Table 3 .
Differences of the median values of M , SFR, and sSFR between VIPERS and control samples.

Table 4 .
Binning scheme for VIPERS and SDSS (main, p-control, and d-control)samples according to the property to bin.
Notes.The differences are expressed in absolute units, in units of σ dist VIPERS , and in units of σ med VIPERS .