COSMIC HISTORY OF THE INTEGRATED GALACTIC STELLAR INITIAL MASS FUNCTION: A SIMULATION STUDY

The form of the stellar initial mass function (IMF) is considerably debated in the present era, as it describes the nature of stellar populations, the ratio of high-mass to low-mass stars, and influences the dynamical evolution of star clusters, as well as the star formation history of the whole galaxy. It is usually derived using an observed luminosity function together with an assumed mass-to-light ratio for the stars under consideration. IMFs, as suggested by various authors, are generally either of Salpeter type (Salpeter 1955) or consist of either segmented power laws (Scalo 1986) or lognormal type (Chabrier 2003), with a turnover at some characteristic mass m_c. The power-law slope at high masses is probably close to the Salpeter value (Salpeter 1955) with $\frac{{dN}}{{dm}}\propto {m}^{-\alpha }$, and $\alpha =2.35$ with an uncertainty ∼0.3 (Chabrier 2003). There are conflicting views and evidence of the universality of the IMF at the present time, e.g., to some extent there is evidence of mass segregation in young massive clusters (Zwart et al. 2010 and references therein) and some variations in starburst galaxies (Gunawardhana et al. 2011). In contrast, there are bottom-heavy IMFs in massive ellipticals (Ferreras et al. 2013) and no direct evidence for rapid variation of the IMF within the Milky Way disk (Kroupa 2001; Chabrier 2003). Again, there is the possibility of variation of the IMF with time (hence redshift), metallicity, and environment. Larson (2003, 2005) has argued that the characteristic mass is primarily determined by the Jeans mass, which depends on the temperature. Hence, with an increase in temperature (the cosmic microwave background temperature was higher at higher redshift), one might expect that low-mass star formation would be disfavored, resulting in a top-heavy IMF with a temperature scaling with redshift as (1 + z). Therefore, at sufficiently high redshift, mass scales as ${(1+z)}^{\displaystyle \frac{3}{2}}$, increasing the fraction of high-mass stars. Larson (2005) has suggested that at z = 5 the characteristic mass may be higher than the present day's value by an order of magnitude.

It is clear from various recent observations that stars form in embedded clusters (Lada & Lada 2003; Kroupa 2005). These clusters also follow a mass function, which is again a power law, ${\xi }_{\mathrm{ecl}}(M)\propto {M}_{\mathrm{ecl}}^{-\beta }$. This is known as the embedded cluster mass function (hereafter ECMF). The maximum mass of the ECMF, ${M}_{\mathrm{ecl},\mathrm{max}}$, has been found to depend on the star formation rate of the galaxy (Weidner & Kroupa 2004; Bastian 2008) and is given as

$$\begin{eqnarray}&&{M}_{\mathrm{ecl},\mathrm{max}}=8.5\times {10}^{4}\times {\mathrm{SFR}}^{0.75},\end{eqnarray} \tag{ 1 }$$

where the SFR is in ${M}_{\odot }\;{\mathrm{yr}}^{-1}$. Weidner et al. (2013) have suggested a time-dependent IMF for elliptical galaxies to account for an excess of low-mass stars in these galaxies. They modeled the SFR as a function of time. The SFR reaches a maximum initially and then asymptotically reduces to zero with time. They have discussed a two stage star formation scenario in giant elliptical galaxies and have given an alternative hypothesis over time-independent bottom-heavy IMFs in these galaxies. They have proposed that initially there is a strong starburst stage with a top-heavy IMF, which is followed by a prolonged stage with a bottom-heavy IMF. The latter results from many low-mass clouds (i.e., a high value of beta) being formed as a result of the fragmentation of the gaseous component. A similar trend for SFRs has also been found for various elliptical galaxies as a function of redshift (Spaans & Carollo 1997). Various theoretical models for star formation are difficult to test, as current observational results from cosmological studies do not measure IMF slopes and SFRs for individual galaxies, but instead study indirect evidence for whole populations and average the results over the galaxy luminosity function.

Various authors have suggested an IMF changing with redshift (Larson 1998, 2005; van Dokkum 2008). Some have dealt with the resulting form of the IGIMF using various empirical measures of SFR, such as the minimum mass of embedded clusters (Weidner et al. 2010, 2013). But so far no studies have investigated the joint effect of a time-varying IMF together with a time-varying SFR derived from an observed SFR function varying indirectly with redshift on the resulting slopes of the IGIMF. Hence, considering varying IMFs and SFRs, our aim is to study the cosmic history of the galaxy-wide IMF.

For the SFR in Equation (1) so far there are no observational values available for SFRs directly as a function of redshift that can help study the cosmic star formation history. Smit et al. (2012) have computed the SFR function $\varphi (\mathrm{SFR})$ (in ${\mathrm{Mpc}}^{-3}\;{\mathrm{dex}}^{-1}$, which is a Schechter function (Schechter 1976)), using a characteristic SFR parameter denoted by ${\mathrm{SFR}}^{*}$, whose values are given from published studies for redshifts z ∼ 4–7. The other values for z ∼ 0–3 are taken from the literature (Bell et al. 2007; Reddy et al. 2008; Magnelli et al. 2011; Sobral et al. 2012). But ${\mathrm{SFR}}^{*}$ characterizes a particular value of SFR varying with redshift. This excludes many others for z, a few with higher SFRs than ${\mathrm{SFR}}^{*}$, and many with a lower SFR than ${\mathrm{SFR}}^{*}$. To cover up the abovementioned uncertainty we have also computed  the quartile values of SFRs (SFR1, SFR2, SFR3) from an SFR function by converting it to a probability density function (pdf). Thus, through this process galaxies with low, intermediate, and high SFRs are also considered, giving us an overall view of the variation of the cosmic star formation history of galaxies. In this respect, Smit et al.'s (2012) results are adequate for this study. In this regard it should be mentioned that there are conflicting views on the star formation histories of high-redshift galaxies (z ∼ 2–7). Reddy et al. (2012) have computed the SFRs of high-redshift galaxies in the range z ∼ 2–7 from spectral energy distribution (SED) as well as from infrared and ultraviolet imaging (IR+UV). It shows an exponential decrease with different slopes of SFR with time. Duncan et al. (2014) have also computed the star formation histories for high-redshift galaxies using data in the CANDELS GOODS South field and their SFR density estimates are higher than previously observed in this regime. It will be interesting to incorporate all these studies and compare the resulting IGIMF in a future study.

In the present study we have replaced the SFR in Equation (1) with several significant measures of SFR, e.g., SFR1, SFR2, SFR3, and SFR*, which respectively correspond to the first, second, and third quartiles and the characteristic value of ${\mathrm{log}}_{10}$ SFR distributions computed by Smit et al. (2012). The first three measures have been computed from the normalized ${\mathrm{log}}_{10}$ SFR distribution (Smit et al. 2012) as a function of z and the last one is given in Smit et al. (2012) as a function of z. All four values satisfy a tapered power-law function (see Section 2.1). As a result ${M}_{\mathrm{ecl}.\mathrm{max}}$ indirectly becomes a function of z in all four cases. The significance of the above three quartiles and characteristic SFR and their derivations are discussed in detail in Section 2.1.

In Chattopadhyay et al. (2011), the authors consider the random fragmentation of young massive clusters in our Galaxy as well as in external galaxies. They find no correlation between the maximum mass of a star and its embedded cluster mass. The existence of a correlation between these factors affects the star formation history of the parent cloud. Low-mass clouds do not have enough mass to form high-mass stars (Bruzual & Charlot 2003; Larson 2006; Weidner et al. 2007, 2010). The formation of massive stars is possible if they accrete their masses from the surrounding atmosphere. On the other hand, in massive clouds, once the high-mass stars are formed, their ionizing radiation removes the remaining gas (Weidner & Kroupa 2006). This stops the formation of low-mass stars. This fact is reflected in some observations (Weidner et al. 2010 and references therein).

In contrast, recent observations by Maschberger & Clarke (2008) and Parker & Goodwins (2007) have included several examples of low-mass clusters containing high-mass stars. Corbelli et al. (2010) found that for YMCs of M33, such a strict correlation does not exist. Moreover, unresolved binaries play an important role. Elmegreen (2006) argued that clusters are built stochastically: the large amount of molecular gas present in the star formation region allows high-mass stars to form even in a low-SFR region, i.e., the entire range of masses (0.01 ${M}_{\odot }$ to 150 ${M}_{\odot }$) is possible even in a low-SFR region. Andrews et al. (2013) studied the dwarf starburst galaxy and found no such correlation between the maximum mass of a star and cluster mass. Furthermore, previous observations included very small numbers of YMCs ($\sim {10}^{5}-{10}^{6}\;{M}_{\odot }$) ( ∼10%; Weidner et al. 2013), for which any such correlation is difficult to predict.

Cerviño et al. (2013) have argued that simulated sampling does not contradict a possible ${m}_{\mathrm{max}}-{M}_{\mathrm{ecl}}$ correlation and it depends on the star formation process and the assumed definition of a stellar cluster. Thus, considering all aspects, we have not assumed any such correlation but only the scenario that massive clouds have a general tendency to form massive stars, and have taken the minimum and maximum mass of stars to be $0.1\;{M}_{\odot }$ (Hass & Ander 2010) and 150 ${M}_{\odot }$, respectively.

In the present work, we have considered the resulting integrated galactic stellar mass function (IGIMF) as a function of redshift due to the random fragmentation of embedded clusters of various masses present in that parent galaxy. Section 2 describes the model with model parameters. Section 3 describes the method. Sections 4 and 5 present our results and conclusion.

In the present model, the star formation scenario of a galaxy has been considered. The component of a galaxy that forms stars consists of molecular clouds and each cloud, under gravitational instability, undergoes hierarchical fragmentation (Hoyle 1953), giving rise to a number of fragments of various masses. These fragments ultimately form stars, resulting in what we call the stellar IMF of these star clusters embedded into the molecular clouds. For simplicity we have assumed that the IMF in each parent cloud has the same distributional form and thus is a segmented power law of the form

$$\begin{eqnarray}\mathrm{Let},{\xi }_{\mathrm{IMF}}(M)=\displaystyle \frac{{dN}}{{dm}}=\left\{\begin{array}{ll}{{Am}}^{-{\alpha }_{1,\mathrm{IMF}}}; & {m}_{\mathrm{min}}\lt m\leqslant {m}_{{\rm{c}}}\\ {{Bm}}^{-{\alpha }_{2,\mathrm{IMF}}}; & {m}_{{\rm{c}}}\lt m\leqslant {m}_{\mathrm{max}}\end{array}\right.,\end{eqnarray} \tag{ 2 }$$

where m_min and m_max are the minimum and maximum masses of the stars, and m_c is the characteristic mass at which the turnover occurs. The values of A and B are calculated as follows.

Since ${\xi }_{\mathrm{IMF}}$ is a probability density function, we have the normalization condition,

$$\begin{eqnarray}&&{\displaystyle \int }_{{m}_{\mathrm{min}}}^{{m}_{\mathrm{max}}}{\xi }_{\mathrm{IMF}}(M){dm}=1.\end{eqnarray} \tag{ 3 }$$

Also, the IMF is continuous at m_c. Hence the continuity condition gives ${{Am}}_{{\rm{c}}}^{-{\alpha }_{1,\mathrm{IMF}}}={{Bm}}_{{\rm{c}}}^{-{\alpha }_{2,\mathrm{IMF}}}$, i.e.,

$$\begin{eqnarray}&&A={{Bm}}_{{\rm{c}}}^{{\alpha }_{1,\mathrm{IMF}}-{\alpha }_{2,\mathrm{IMF}}}.\end{eqnarray} \tag{ 4 }$$

Then from Equation (3)

$$\begin{eqnarray*}&&{\displaystyle \int }_{{m}_{\mathrm{min}}}^{{m}_{{\rm{c}}}}{{Am}}^{-{\alpha }_{1,\mathrm{IMF}}}{dm}+{\displaystyle \int }_{{m}_{{\rm{c}}}}^{{m}_{\mathrm{max}}}{{Bm}}^{-{\alpha }_{2,\mathrm{IMF}}}{dm}=1.\end{eqnarray*}$$

Substituting the value of A from Equation (4) we get

$$\begin{eqnarray*}&&B=\displaystyle \frac{1}{[\displaystyle \frac{{m}_{{\rm{c}}}^{{\alpha }_{1,\mathrm{IMF}}-{\alpha }_{2,\mathrm{IMF}}}}{(1-{\alpha }_{1,\mathrm{IMF}})}({m}_{{\rm{c}}}^{1-{\alpha }_{1,\mathrm{IMF}}}-{m}_{\mathrm{min}}^{1-{\alpha }_{1,\mathrm{IMF}}})}\end{eqnarray*}$$

$$\begin{eqnarray}&&+\;\displaystyle \frac{1}{1-{\alpha }_{2,\mathrm{IMF}}}({m}_{\mathrm{max}}^{1-{\alpha }_{2,\mathrm{IMF}}}-{m}_{{\rm{c}}}^{1-{\alpha }_{2,\mathrm{IMF}}})].\end{eqnarray} \tag{ 5 }$$

Then using B from Equation (5), A is found from Equation (4). The representative values of ${\alpha }_{1,\mathrm{IMF}}$ and ${\alpha }_{2,\mathrm{IMF}}$ are chosen as 1.25 (the maximum value is 1.25 in a low-mass regime; Bastian et al. 2010) and 2.35 (Salpeter 1955), respectively.

The values of m_min and m_max are chosen as $0.1\;{M}_{\odot }$ and $150\;{M}_{\odot }$, respectively (Zinnecker & Yorke 2007). The value of the characteristic mass at z = 0 is taken as m_c $(z=0)$ = $0.3\;{M}_{\odot }$ (Larson 2005). Since we have assumed a top-heavy IMF with increasing redshift, the characteristic mass is given by

$$\begin{eqnarray}&&{m}_{{\rm{c}}}=D{(1+z)}^{\frac{3}{2}},\end{eqnarray} \tag{ 6 }$$

where D is determined from the condition that at z = 0, m_c = $0.3\;{M}_{\odot }$ (Larson 2005). The choice of the above relation is not arbitrary but has a strong physical ground. The influence of temperature on the Jeans mass (Jeans 1902) is a very well-known phenomenon. Larson (1998, 2005) has suggested that the characteristic turnover mass may be primarily determined by the thermal Jeans mass, which is strongly influenced by temperature ($\sim {T}^{3/2}$) at fixed density. Hence it is expected that the environment, where heating occurs through far infrared radiation, disfavors the formation of low-mass stars. Such extreme environments really occur in the super clusters at the center of the Milky Way. Some young super clusters at the center of M82 really appear to have a top-heavy IMF (e.g., Rieke et al. 1993; McCrady et al. 2003), along with those at the center of our Galaxy (Stolte et al. 2005; Maness et al. 2007). The mass functions in these super clusters also have the additional traits of complex dynamical phenomena, which make them top-heavy over time (McCrady et al. 2005; Kim et al. 2006; Harayama et al. 2008). At the initial stage of star formation in giant as well as in dwarf galaxies, the star formation occurs in a "burst" rather than through a continuous process (Steidel et al. 1996; Blain et al. 1999; Lacey et al. 2008). This means that the IMF becomes more and more top-heavy at redshifts 1–3 and beyond. Also, at high redshift the metallicity was lower in star forming clouds. Thus initially the cooling process was not efficient, which may lead to an extremely top-heavy IMF for the first-generation stars (Abel et al. 2002; Bromm et al. 2002). Hence the IMF may depend on redshift. The CMB temperature plays a significant role in increasing the temperature of the medium, which scales as (1 + z). Beyond z ∼ 2, the CMB temperature exceeds the temperature of the Galactic molecular clouds (Evans et al. 2001; Tafalla et al. 2004). Hence it can be speculated that the characteristic mass ${m}_{{\rm{c}}}\sim {T}^{3/2}$ at fixed density varies with redshift as ${(1+z)}^{3/2}$ at high redshift, leading to a top-heavy IMF (Larson 1998). The effect becomes more pronounced when pressure is taken into account and Larson (2005) has shown that at z = 5, m_c becomes higher by an order of magnitude than its present value. Direct evidence of a top-heavy IMF at high redshift is very rare, though there are a few observations, e.g., blue rest-frame ultraviolet colors of galaxies at z ∼ 6 may imply a top-heavy IMF (Stanway et al. 2005). Tumlinson (2007) finds that the properties of carbon-enhanced metal poor stars in our Galaxy are best explained by the relatively large number of stars in the mass range 1–8 ${M}_{\odot }$ at high redshift.

The maximum mass of the embedded cluster ${M}_{\mathrm{ecl},\mathrm{max}}$ has been assumed to be a function of SFR and indirectly becomes a function of redshift as discussed in Section 1 and Equation (1). The ECMF is assumed to be

$$\begin{eqnarray}&&{\xi }_{\mathrm{ecl}}(M)=\displaystyle \frac{{dN}}{{{dM}}_{\mathrm{ecl}}}={{EM}}_{\mathrm{ecl}}^{-\beta },{M}_{\mathrm{ecl},\mathrm{min}}\leqslant {M}_{\mathrm{ecl}}\leqslant {M}_{\mathrm{ecl},\mathrm{max}},\end{eqnarray} \tag{ 7 }$$

where ${M}_{\mathrm{ecl},\mathrm{min}}$ is the minimum mass of the embedded cluster. The value of index β  is around 2 (Zhang & Fall 1999; de Grijs et al. 2003 McCrady & Graham 2007). Some studies also suggest flatter slopes like 1.8 (Dowell et al. 2008). The mass spectrum of giant molecular clouds shows β ∼ 1.7 (Rosolowsky 2005). In the present work we have considered β ranging from 2 to 2.6. The lower limit of the embedded cluster is considered to be a parameter, having values of 500 and 1000 ${M}_{\odot }$, respectively. The value of the constant E in Equation (7) is determined by assuming galaxy masses of $5\times {10}^{9}$ ${M}_{\odot }$, $5\times {10}^{10}$ ${M}_{\odot }$, and $5\times {10}^{11}$ ${M}_{\odot }$, respectively, as representative values of dwarf, intermediate, and giant galaxies, whose 30% masses have been exhausted due to star formation (Verschueren & Hensberge1982; Lada & Hensberge 1984; Elmegreen & Clemens 1985).

Then the IGIMF as a function of fragment mass m and redshift z is the collection of all the IMFs of all the parent clusters (Vanbeveren 1982; Kroupa & Weidner 2003; Weidner & Kroupa 2005), which is

$$\begin{eqnarray}&&{\xi }_{\mathrm{IGIMF}}(m,z)={\displaystyle \int }_{{M}_{\mathrm{ecl},\mathrm{min}}}^{{M}_{\mathrm{ecl},\mathrm{max}}}{\xi }_{\mathrm{IMF}}(M)\ {\xi }_{\mathrm{ecl}}(M)\;{{dM}}_{\mathrm{ecl}}.\end{eqnarray} \tag{ 8 }$$

All the values of the parameters considered are listed in Table 1.

Table 1.  Initial Values of the Parameters

Parameter	Value
${\alpha }_{1,\mathrm{IMF}}(z=0)$	1.25
${\alpha }_{2,\mathrm{IMF}}(z=0)$	2.35
Galaxy masses	$5\times {10}^{9}\;{M}_{\odot }$, $5\times {10}^{10}\;{M}_{\odot }$, $5\times {10}^{11}\;{M}_{\odot }$
mmin	$0.1\;{M}_{\odot }$
mmax	$150\;{M}_{\odot }$
${m}_{{\rm{c}},\mathrm{IMF}}(z=0)$	$0.3\;{M}_{\odot }$
${M}_{\mathrm{ecl},\mathrm{min}}$	500,1000 $({M}_{\odot })$
β	2.0, 2.4, 2.6
z	0.1, 0.2, 0.8, 1.5, 2.2, 3.8, 5.0, 5.9, 6.8
Efficiency	$30\%$

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2.1. Various Measures of SFR

To compute various measures of the SFR as a function of redshift (z) we start with the SFR function φ (in ${\mathrm{Mpc}}^{-3}\;{\mathrm{dex}}^{-1}$) derived by Smit et al. (2012), which is

$$\begin{eqnarray}&&\varphi (\mathrm{SFR})d\mathrm{SFR}={\varphi }_{\mathrm{SFR}}^{*}{(\displaystyle \frac{\mathrm{SFR}}{{\mathrm{SFR}}^{*}})}^{{\alpha }_{\mathrm{SFR}}}\mathrm{exp}(-\displaystyle \frac{\mathrm{SFR}}{{\mathrm{SFR}}^{*}})\displaystyle \frac{d\mathrm{SFR}}{{\mathrm{SFR}}^{*}},\end{eqnarray} \tag{ 9 }$$

where ${\varphi }_{\mathrm{SFR}}^{*}$, ${\alpha }_{\mathrm{SFR}}$, and SFR* are various Scechter parameters given as a function of z in Tables 2 and 3 of Smit et al. (2012).

Then the  ${\mathrm{log}}_{10}\varphi$(SFR)'s function is

$$\begin{eqnarray}&&{\mathrm{log}}_{10}\varphi (\mathrm{SFR})\ d{\mathrm{log}}_{10}\mathrm{SFR}\\ &&\quad =\;[{\mathrm{log}}_{10}({\varphi }_{\mathrm{SFR}}^{*})+{\alpha }_{\mathrm{SFR}}\ {\mathrm{log}}_{10}(\displaystyle \frac{\mathrm{SFR}}{{\mathrm{SFR}}^{*}})\\ &&\qquad -\;\displaystyle \frac{\mathrm{SFR}}{{\mathrm{SFR}}^{*}}{\mathrm{log}}_{10}e-{\mathrm{log}}_{10}{\mathrm{SFR}}^{*}]d{\mathrm{log}}_{10}\mathrm{SFR}.\end{eqnarray} \tag{ 10 }$$

As a first step we convert ${\mathrm{log}}_{10}\varphi$(SFR) to a density function at each z, dividing by

$$\begin{eqnarray*}&&T={\int }_{0}^{{\mathrm{log}}_{10}{\mathrm{SFR}}_{\mathrm{max}}}{\mathrm{log}}_{10}\varphi (\mathrm{SFR})\ d\mathrm{log}(\mathrm{SFR}),\end{eqnarray*}$$

where ${\mathrm{log}}_{10}$ SFR_max is the maximum value of ${\mathrm{log}}_{10}\mathrm{SFR}$ at a particular z, taken from Figure 2 of Smit et al. (2012) for z = 4, 5, 6, 7. For other values of z the values of ${\mathrm{log}}_{10}$ SFR_max were found by plotting the function. The lower boundary is not strictly zero and it also includes negative values (Figure 2 of Smit et al. 2012), but the number of observations for negative values of ${\mathrm{log}}_{10}$ SFR decreases and for z ∼ 6, 7 (see. Table 1 of Smit et al. 2012) it is just one. So the negative part of the  ${\mathrm{log}}_{10}$ SFR contribution is insignificant compared to the  positive part on the basis of observational range. Therefore we limited our study of ${\mathrm{log}}_{10}$ SFR from 0 to ${\mathrm{log}}_{10}$ SFR_max due to a lack of observational points for the negative part and we have worked with the available observational range.

Then the cumulative distribution function (c.d.f.) of ${\mathrm{log}}_{10}\mathrm{SFR}$ distribution is given by

$$\begin{eqnarray*}&&\displaystyle \frac{1}{T}{\int }_{0}^{{\mathrm{log}}_{10}\mathrm{SFR}}{\mathrm{log}}_{10}\varphi (\mathrm{SFR})\ d{\mathrm{log}}_{10}\mathrm{SFR}\end{eqnarray*}$$

$$\begin{eqnarray*}&&=\;\displaystyle \frac{1}{T}{\displaystyle \int }_{0}^{{\mathrm{log}}_{10}\mathrm{SFR}}[{\mathrm{log}}_{10}({\varphi }_{\mathrm{SFR}}^{*})+{\alpha }_{\mathrm{SFR}}\ {\mathrm{log}}_{10}(\displaystyle \frac{\mathrm{SFR}}{{\mathrm{SFR}}^{*}})\end{eqnarray*}$$

$$\begin{eqnarray}&&-\;\displaystyle \frac{\mathrm{SFR}}{{\mathrm{SFR}}^{*}}{\mathrm{log}}_{10}e-{\mathrm{log}}_{10}{\mathrm{SFR}}^{*}]d{\mathrm{log}}_{10}\mathrm{SFR}.\end{eqnarray} \tag{ 11 }$$

In Equation (11) when the LHS is 0.25 then the corresponding value of ${\mathrm{log}}_{10}\mathrm{SFR}$ is ${\mathrm{log}}_{10}$ SFR1, i.e., the first quartile. At this point 75% of the galaxies have ${\mathrm{log}}_{10}\mathrm{SFR}\gt {\mathrm{log}}_{10}$ SFR1 and the remaining 25% of the galaxies have ${\mathrm{log}}_{10}\mathrm{SFR}\leqslant {\mathrm{log}}_{10}$ SFR1 at a particular z. Similarly, we have values of ${\mathrm{log}}_{10}$ SFR1 for different z for different values of ${\mathrm{log}}_{10}{\varphi }_{\mathrm{SFR}}^{*}$, ${\alpha }_{\mathrm{SFR}}$, and ${\mathrm{log}}_{10}$ SFR ${}^{*}$ at different z given in Tables 2 and 3 of Smit et al. (2012). We fit the values of ${\mathrm{log}}_{10}$ SFR1 at different z by a tapered power-law function of the form ${\mathrm{log}}_{10}\mathrm{SFR}1\propto {z}^{-\gamma }[1-{e}^{{(-z/\delta )}^{x}}]$, where γ, δ, and x are constants. We repeat the above process for values of c.d.f. as 0.5 and 0.75 and we get ${\mathrm{log}}_{10}$ SFR2 and ${\mathrm{log}}_{10}$ SFR3 as a function of z. The fitted tapered power-law functions against ${\mathrm{log}}_{10}$ SFR1, ${\mathrm{log}}_{10}$ SFR2, ${\mathrm{log}}_{10}$ SFR3, and ${\mathrm{log}}_{10}$ SFR${}^{*}$ are shown in Figures 1–4 along with their p-values. We have not only fitted the tapered power law, but also performed the goodness of fit test for which the p-values are much higher (more than 0.25). Hence we can accept the null hypothesis (the tapered power law is a suitable curve). The significance of constructing these quartile points and ${\mathrm{log}}_{10}$ SFR* as a function of z is that we will have a clear view of the SFRs for most of the galaxies varying with redshift. SFR2 is the most representative measure because ${\mathrm{log}}_{10}$ SFR2 is the median value of the ${\mathrm{log}}_{10}\mathrm{SFR}$ distribution.

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To generate a sample of embedded cluster masses from a power law with a given range of values, the power law can be considered a truncated Pareto distribution. For this we have used the standard method of inverting the c.d.f. Let X be a random variable with pdf f(x) and c.d.f. F(x), where

$$\begin{eqnarray}&&F(x)={\displaystyle \int }_{-\infty }^{x}f(x)\;{dx},\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray*}&&{\displaystyle \int }_{-\infty }^{\infty }f(x)\;{dx}=1.\end{eqnarray*}$$

We know that the c.d.f. F(x) follows uniform distribution over the range (0,1). Hence a simulated value x of X can be obtained by solving the equation F(x) = r, where r is a random fraction. Thus one simulated value is given by x = ${F}^{-1}$(r). Corresponding to n choices of r, we will have n values of x giving a simulated sample of size n. Of course the above method is valid when the inverse function of F exists, which is true in the present case. To generate the value of X, it is necessary to know the parameters of the Pareto distribution, which are the constants in the power laws already known from physical considerations. In the present work, in Equation (7) the lower limit of the cluster mass is taken as ${M}_{\mathrm{ecl},\mathrm{min}}$ instead of $-\infty$, with f(x) = ${\xi }_{\mathrm{ecl}}(M)$ for sampling cluster masses.

The method of generating samples of stellar masses from a segmented power law (truncated Pareto distribution) is as follows.

$$\begin{eqnarray}\mathrm{Let},{\xi }_{\mathrm{IMF}}(M)=\displaystyle \frac{{dN}}{{dm}}=\left\{\begin{array}{ll}{{Am}}^{-{\alpha }_{1,\mathrm{IMF}}}; & {m}_{\mathrm{min}}\lt m\leqslant {m}_{{\rm{c}}}\\ {{Bm}}^{-{\alpha }_{2,\mathrm{IMF}}}; & {m}_{{\rm{c}}}\lt m\leqslant {m}_{\mathrm{max}}\end{array}\right.,\end{eqnarray} \tag{ 13 }$$

where A and B are constants to be determined by Equations (4) and (5). Now, to generate samples from the above power laws, we use a conditional c.d.f. defined as follows:

$$\begin{eqnarray*}&&{F}_{1}(M)=P(X\lt m| {m}_{\mathrm{min}}\lt x\lt {m}_{{\rm{c}}})=\displaystyle \frac{F(M)-F({m}_{\mathrm{min}})}{F({m}_{{\rm{c}}})-F({m}_{\mathrm{min}})}\end{eqnarray*}$$

$$\begin{eqnarray}&&{m}_{\mathrm{min}}\lt m\lt {m}_{{\rm{c}}}.\end{eqnarray} \tag{ 14 }$$

In the same way,

$$\begin{eqnarray*}&&{F}_{2}(M)=P(X\lt m| {m}_{{\rm{c}}}\lt x\lt {m}_{\mathrm{max}})=\displaystyle \frac{F(M)-F({m}_{{\rm{c}}})}{F({m}_{\mathrm{max}})-F({m}_{{\rm{c}}})}\end{eqnarray*}$$

$$\begin{eqnarray}&&{m}_{{\rm{c}}}\lt m\lt {m}_{\mathrm{max}}.\end{eqnarray} \tag{ 15 }$$

We use the method of inversion to draw samples using these two conditional c.d.f.s. First, when ${m}_{\mathrm{min}}\lt m\lt {m}_{{\rm{c}}}$, we draw a random sample, say ${u}_{1}$, from a uniform distribution, i.e., U(0, 1). and equate it to

$$\begin{eqnarray}&&{F}_{1}(M)=\displaystyle \frac{F(M)-F({m}_{\mathrm{min}})}{F({m}_{{\rm{c}}})-F({m}_{\mathrm{min}})}={u}_{1}.\end{eqnarray} \tag{ 16 }$$

So, inverting it we get the expression for the sample m as

$$\begin{eqnarray}m & = & \left[{u}_{1}\times ({m}_{{\rm{c}}}^{(1-{\alpha }_{1,\mathrm{IMF}})}-{m}_{\mathrm{min}}^{(1-{\alpha }_{1,\mathrm{IMF}})})\right.\\ & & {\left.+{m}_{\mathrm{min}}^{(1-{\alpha }_{1,\mathrm{IMF}})}\right]}^{\displaystyle \frac{1}{(1-{\alpha }_{1,\mathrm{IMF}})}}.\end{eqnarray} \tag{ 17 }$$

Thus when ${u}_{1}=0,m={m}_{\mathrm{min}}$ and when ${u}_{1}=1,m={m}_{{\rm{c}}}$.

Similarly, when ${m}_{{\rm{c}}}\lt m\lt {m}_{\mathrm{max}}$, we draw a random sample, say ${u}_{2}$, from a uniform distribution, i.e., U(0, 1), and equate it to

$$\begin{eqnarray}&&{F}_{2}(M)=\displaystyle \frac{F(M)-F({m}_{{\rm{c}}})}{F({m}_{\mathrm{max}})-F({m}_{{\rm{c}}})}={u}_{2}.\end{eqnarray} \tag{ 18 }$$

So, inverting it we get the expression for the sample m as

$$\begin{eqnarray}&&m={[{u}_{2}\times ({m}_{\mathrm{max}}^{(1-{\alpha }_{2,\mathrm{IMF}})}-{m}_{{\rm{c}}}^{(1-{\alpha }_{2,\mathrm{IMF}})})+{m}_{{\rm{c}}}^{(1-{\alpha }_{2,\mathrm{IMF}})}]}^{\displaystyle \frac{1}{(1-{\alpha }_{2,\mathrm{IMF}})}}.\end{eqnarray} \tag{ 19 }$$

Thus when ${u}_{2}=0,m={m}_{{\rm{c}}}$ and when ${u}_{2}=1,m={m}_{\mathrm{max}}$.

We simulate from ${F}_{1}(M)$ as long as the total mass of the embedded cluster is equal to the mass in the low-mass regime (see ${m}_{\mathrm{min}}\lt m\lt {m}_{{\rm{c}}}$) and then we simulate from ${F}_{2}(M)$ for the high-mass regime (see ${m}_{{\rm{c}}}\lt m\lt {m}_{\mathrm{max}}$). The mass fractions for each embedded cluster in the low- and high-mass regimes are computed at the beginning for different m_c.

In the present work we have simulated random samples from various segmented power-law distributions as follows.

(i) First we simulate a sample of embedded cluster masses following the normalized power law given in Equation (7), where the maximum mass is computed at any particular z following Equation (1) for different SFRs (SFR1, SFR2, SFR3, and SFR*). The simulation is continued as long as the total mass of the embedded cluster is less than or equal to 30% of the total mass of the galaxy.

(ii) Second, for each mass of a parent cluster we simulate a sample of stellar masses following the segmented power-law distributions (as discussed before) given in Equation (2) at a particular value of z so that the value of the characteristic mass, m_c, is prefixed at that value of z (refer to Equation (6)). Each time a stellar mass is simulated, the total mass of the previous stellar masses is checked with the total mass of the embedded cluster and as soon as it exceeds the mass of the parent cluster, the simulation is stopped.

(iii) Finally, the mass spectrum of all simulated stellar masses from all the parent clusters of the galaxy is computed and fitted by segmented power laws to give the resulting form of the IGIMF.

(iv) The above procedure is performed at various SFRs (SFR1, SFR2, SFR3, SFR*), redshifts z, ${M}_{\mathrm{ecl},\mathrm{min}}$, and β, respectively.

Tables 2–9 and Figures 5–11 show the resulting IGIMF slopes for stars in a galaxy which consists of segmented power laws with slopes, ${\alpha }_{1,\mathrm{IGIMF}}$ in a low-mass regime ,and ${\alpha }_{2,\mathrm{IGIMF}}$ in a high-mass regime for various values of SFR, β, ${M}_{\mathrm{ecl},\mathrm{min}}$, and redshift z = 0.1 to z = 6.8, respectively. The following observations are envisaged.

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Table 2.  IGIMF and IMF Slopes with Varying z and ${\mathrm{Mecl}}_{\mathrm{min}}$ at β = 2, 2.4 for SFR1

$\beta =2.0$
$z=0.1$						$z=0.2$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	0.381	−0.05	1.475	0.224	1.25	2.35	0.434	0.161	1.367	0.282
1000	1.25	2.35	0.381	0.072	1.431	0.178	1.25	2.35	0.434	−0.303	1.315	0.282
$z=0.8$						$z=1.5$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	0.797	0.235	1.415	0.562	1.25	2.35	1.304	0.178	1.492	0.891
1000	1.25	2.35	0.797	0.305	1.404	0.355	1.25	2.35	1.304	0.095	1.384	0.891
$z=2.2$						$z=3.8$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	1.889	0.315	1.555	1.412	1.25	2.35	3.471	0.168	1.358	2.239
1000	1.25	2.35	1.889	0.224	1.528	1.122	1.25	2.35	3.471	0.505	1.293	2.239
$z=5.0$						$z=5.9$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	4.849	0.237	1.474	4.467	1.25	2.35	5.981	0.522	1.499	11.22
1000	1.25	2.35	4.849	0.354	1.211	4.467	1.25	2.35	5.981	0.261	1.416	4.467
$z=6.8$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	7.189	0.348	1.409	11.22	...	...	...	...	...	...
1000	1.25	2.35	7.189	0.312	1.145	7.079	...	...	...	...	...	...
$\beta =2.4$
$z=0.1$						$z=0.2$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	0.381	0.654	1.421	0.224	1.25	2.35	0.434	0.254	1.409	0.282
1000	1.25	2.35	0.381	0.645	1.535	0.224	1.25	2.35	0.434	0.335	1.462	0.282
$z=0.8$						$z=1.5$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	0.797	0.205	1.359	0.562	1.25	2.35	1.304	0.163	1.362	1.122
1000	1.25	2.35	0.797	0.226	1.461	0.562	1.25	2.35	1.304	0.204	1.455	0.891
$z=2.2$						$z=3.8$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	1.889	0.255	1.369	1.412	1.25	2.35	3.471	0.344	1.432	2.818
1000	1.25	2.35	1.889	0.215	1.243	1.412	1.25	2.35	3.471	0.294	1.311	4.467
$z=5.0$						$z=5.9$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	4.849	0.185	1.367	3.548	1.25	2.35	5.981	0.183	1.347	4.467
1000	1.25	2.35	4.849	0.239	1.531	2.818	1.25	2.35	5.981	0.141	1.443	4.467
$z=6.8$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	7.189	0.291	1.312	7.079	...	...	...	...	...	...
1000	1.25	2.35	7.189	0.249	1.281	7.079	...	...	...	...	...	...

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Table 3.  IGIMF and IMF Slopes with Varying z and ${\mathrm{Mecl}}_{\mathrm{min}}$ at β = 2.6 for SFR1

$\beta =2.6$
$z=0.1$						$z=0.2$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	0.381	0.072	1.383	0.224	1.25	2.35	0.434	0.495	1.388	0.282
1000	1.25	2.35	0.381	0.01	1.477	0.224	1.25	2.35	0.434	0.579	1.493	0.282
$z=0.8$						$z=1.5$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	0.797	0.281	1.388	0.562	1.25	2.35	1.304	0.286	1.392	0.891
1000	1.25	2.35	0.797	0.083	1.468	0.708	1.25	2.35	1.304	0.424	1.409	0.891
$z=2.2$						$z=3.8$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	1.889	0.242	1.461	1.412	1.25	2.35	3.471	0.319	1.357	2.239
1000	1.25	2.35	1.889	0.218	1.404	1.778	1.25	2.35	3.471	0.256	1.366	2.818
$z=5.0$						$z=5.9$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	4.849	0.288	1.433	4.467	1.25	2.35	5.981	0.342	1.335	7.079
1000	1.25	2.35	4.849	0.055	1.386	4.467	1.25	2.35	5.981	0.322	1.421	4.467
$z=6.8$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	7.189	0.345	1.423	7.079	...	...	...	...	...	...
1000	1.25	2.35	7.189	0.324	1.446	7.079	...	...	...	...	...	...

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Table 4.  IGIMF and IMF Slopes with Varying z and ${\mathrm{Mecl}}_{\mathrm{min}}$ at β = 2, 2.4 for SFR2

$\beta =2.0$
$z=0.1$						$z=0.2$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	0.381	0.429	1.415	0.224	1.25	2.35	0.434	0.019	1.437	0.282
1000	1.25	2.35	0.381	0.312	1.321	0.224	1.25	2.35	0.434	−0.054	1.404	0.282
$z=0.8$						$z=1.5$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	0.797	0.129	1.351	0.562	1.25	2.35	1.304	0.307	1.389	0.891
1000	1.25	2.35	0.797	0.293	1.408	0.562	1.25	2.35	1.304	0.073	1.488	0.708
$z=2.2$						$z=3.8$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	1.889	0.129	1.364	1.122	1.25	2.35	3.471	0.449	1.174	2.818
1000	1.25	2.35	1.889	0.046	1.245	1.412	1.25	2.35	3.471	0.099	1.214	1.778
$z=5.0$						$z=5.9$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	4.849	0.249	1.351	3.548	1.25	2.35	5.981	0.323	1.398	7.079
1000	1.25	2.35	4.849	0.348	1.298	3.548	1.25	2.35	5.981	0.137	1.392	5.623
$z=6.8$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	7.189	0.241	1.202	7.079	...	...	...	...	...	...
1000	1.25	2.35	7.189	0.377	1.223	7.079	...	...	...	...	...	...
$\beta =2.4$
$z=0.1$						$z=0.2$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	0.381	0.306	1.404	0.224	1.25	2.35	0.434	0.150	1.358	0.282
1000	1.25	2.35	0.381	0.286	1.448	0.282	1.25	2.35	0.434	0.235	1.321	0.282
$z=0.8$						$z=1.5$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	0.797	0.307	1.400	0.562	1.25	2.35	1.304	0.184	1.378	0.891
1000	1.25	2.35	0.797	0.379	1.446	0.562	1.25	2.35	1.304	0.170	1.432	0.891
$z=2.2$						$z=3.8$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	1.889	0.383	1.365	1.778	1.25	2.35	3.471	0.186	1.351	2.818
1000	1.25	2.35	1.889	0.253	1.392	1.412	1.25	2.35	3.471	0.027	1.264	2.238
$z=5.0$						$z=5.9$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	4.849	0.304	1.420	3.548	1.25	2.35	5.981	0.335	1.333	5.623
1000	1.25	2.35	4.849	0.167	1.425	5.623	1.25	2.35	5.981	0.313	1.495	5.623
$z=6.8$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	7.189	0.219	1.320	5.623	...	...	...	...	...	...
1000	1.25	2.35	7.189	0.167	1.433	5.623	...	...	...	...	...	...

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Table 5.  IGIMF and IMF Slopes with Varying z and ${\mathrm{Mecl}}_{\mathrm{min}}$ at β = 2.6 for SFR2

$\beta =2.6$
$z=0.1$						$z=0.2$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	0.381	0.388	1.433	0.282	1.25	2.35	0.434	−0.404	1.400	0.224
1000	1.25	2.35	0.381	0.195	1.456	0.224	1.25	2.35	0.434	0.645	1.448	0.282
$z=0.8$						$z=1.5$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	0.797	0.213	1.417	0.562	1.25	2.35	1.304	0.212	1.371	1.122
1000	1.25	2.35	0.797	−0.049	1.403	0.708	1.25	2.35	1.304	0.291	1.446	1.122
$z=2.2$						$z=3.8$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	1.889	0.173	1.413	1.412	1.25	2.35	3.471	0.284	1.406	1.412
1000	1.25	2.35	1.889	0.298	1.394	1.412	1.25	2.35	3.471	0.221	1.333	2.818
$z=5.0$						$z=5.9$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	4.849	0.218	1.412	4.467	1.25	2.35	5.981	0.241	1.385	5.623
1000	1.25	2.35	4.849	0.207	1.333	3.548	1.25	2.35	5.981	0.329	1.264	4.467
$z=6.8$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	7.189	0.273	1.391	7.079	...	...	...	...	...	...
1000	1.25	2.35	7.189	0.238	1.283	7.079	...	...	...	...	...

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Table 6.  IGIMF and IMF Slopes with Varying z and ${\mathrm{Mecl}}_{\mathrm{min}}$ at β = 2, 2.4 for SFR3

$\beta =2.0$
$z=0.1$						$z=0.2$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	0.381	0.428	1.369	0.282	1.25	2.35	0.434	0.008	1.428	0.282
1000	1.25	2.35	0.381	−0.795	1.414	0.178	1.25	2.35	0.434	−0.236	1.429	0.282
$z=0.8$						$z=1.5$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	0.797	0.188	1.316	0.562	1.25	2.35	1.304	0.134	1.401	0.562
1000	1.25	2.35	0.797	0.276	1.485	0.447	1.25	2.35	1.304	0.279	1.302	1.122
$z=2.2$						$z=3.8$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	1.889	0.203	1.421	1.412	1.25	2.35	3.471	−0.117	1.232	3.548
1000	1.25	2.35	1.889	−0.072	1.314	1.412	1.25	2.35	3.471	0.273	1.512	2.818
$z=5.0$						$z=5.9$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	4.849	0.269	1.385	3.548	1.25	2.35	5.981	0.313	1.398	7.079
1000	1.25	2.35	4.849	0.382	1.376	3.548	1.25	2.35	5.981	0.446	1.375	5.623
$z=6.8$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	7.189	0.178	1.497	7.079	...	...	...	...	...	...
1000	1.25	2.35	7.189	0.419	1.325	7.079	...	...	...	...	...	...
$\beta =2.4$
$z=0.1$						$z=0.2$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	0.381	0.009	1.491	0.224	1.25	2.35	0.434	0.343	1.393	0.282
1000	1.25	2.35	0.381	0.395	1.416	0.224	1.25	2.35	0.434	0.023	1.362	0.282
$z=0.8$						$z=1.5$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	0.797	0.315	1.349	0.447	1.25	2.35	1.304	0.226	1.386	0.708
1000	1.25	2.35	0.797	0.321	1.407	0.562	1.25	2.35	1.304	0.133	1.501	1.122
$z=2.2$						$z=3.8$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	1.889	0.227	1.361	1.412	1.25	2.35	3.471	0.130	1.315	2.818
1000	1.25	2.35	1.889	0.462	1.380	1.778	1.25	2.35	3.471	0.076	1.300	3.548
$z=5.0$						$z=5.9$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	4.849	0.219	1.372	3.548	1.25	2.35	5.981	0.215	1.322	5.623
1000	1.25	2.35	4.849	−0.076	1.315	4.467	1.25	2.35	5.981	0.131	1.367	5.623
$z=6.8$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	7.189	0.337	1.420	5.623	...	...	...	...	...	...
1000	1.25	2.35	7.189	0.268	1.349	7.079	...	...	...	...	...	...

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Table 7.  IGIMF and IMF Slopes with Varying z and ${\mathrm{Mecl}}_{\mathrm{min}}$ at β = 2.6 for SFR3

$\beta =2.6$
$z=0.1$						$z=0.2$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	0.381	−0.313	1.391	0.224	1.25	2.35	0.434	0.325	1.419	0.282
1000	1.25	2.35	0.381	−0.017	1.393	0.224	1.25	2.35	0.434	0.052	1.500	0.282
$z=0.8$						$z=1.5$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	0.797	0.402	1.462	0.562	1.25	2.35	1.304	0.383	1.444	0.891
1000	1.25	2.35	0.797	0.385	1.488	0.562	1.25	2.35	1.304	0.358	1.384	0.891
$z=2.2$						$z=3.8$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	1.889	0.246	1.366	1.412	1.25	2.35	3.471	0.266	1.328	2.818
1000	1.25	2.35	1.889	0.307	1.370	0.891	1.25	2.35	3.471	0.294	1.426	2.818
$z=5.0$						$z=5.9$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	4.849	0.337	1.342	4.467	1.25	2.35	5.981	0.295	1.351	5.623
1000	1.25	2.35	4.849	0.258	1.351	3.548	1.25	2.35	5.981	0.306	1.409	5.623
$z=6.8$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	7.189	0.279	1.381	5.623	...	...	...	...	...	...
1000	1.25	2.35	7.189	0.187	1.506	7.079	...	...	...	...	...	...

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Table 8.  IGIMF and IMF Slopes with Varying z and ${\mathrm{Mecl}}_{\mathrm{min}}$ at β = 2, 2.4 for ${\mathrm{SFR}}^{*}$

$\beta =2.0$
$z=0.1$						$z=0.2$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	0.381	−0.265	1.478	0.224	1.25	2.35	0.434	0.532	1.371	0.224
1000	1.25	2.35	0.381	0.644	1.375	0.224	1.25	2.35	0.434	0.354	1.362	0.224
$z=0.8$						$z=1.5$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	0.797	0.050	1.502	0.708	1.25	2.35	1.304	0.055	1.448	0.891
1000	1.25	2.35	0.797	0.494	1.449	0.708	1.25	2.35	1.304	0.093	1.242	0.891
$z=2.2$						$z=3.8$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	1.889	0.021	1.527	1.778	1.25	2.35	3.471	0.121	1.368	1.778
1000	1.25	2.35	1.889	0.256	1.465	1.412	1.25	2.35	3.471	0.318	1.395	2.239
$z=5.0$						$z=5.9$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	4.849	0.117	1.359	4.467	1.25	2.35	5.981	0.152	1.347	4.467
1000	1.25	2.35	4.849	0.472	1.335	4.467	1.25	2.35	5.981	0..373	1.368	4.467
$z=6.8$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	7.189	0.215	1.381	7.079	...	...	...	...	...	...
1000	1.25	2.35	7.189	0.145	1.564	7.079	...	...	...	...	...	...
$\beta =2.4$
$z=0.1$						$z=0.2$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	0.381	0.115	1.474	0.224	1.25	2.35	0.434	0.439	1.409	0.282
1000	1.25	2.35	0.381	−0.037	1.415	0.224	1.25	2.35	0.434	0.529	1.363	0.282
$z=0.8$						$z=1.5$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	0.797	0.221	1.448	0.447	1.25	2.35	1.304	0.331	1.381	0.891
1000	1.25	2.35	0.797	0.076	1.412	0.562	1.25	2.35	1.304	0.218	1.446	0.891
$z=2.2$						$z=3.8$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	1.889	0.135	1.307	1.778	1.25	2.35	3.471	0.210	1.340	2.818
1000	1.25	2.35	1.889	0.329	1.332	1.412	1.25	2.35	3.471	0.191	1.412	3.548
$z=5.0$						$z=5.9$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	4.849	0.226	1.392	4.467	1.25	2.35	5.981	−0.022	1.339	4.467
1000	1.25	2.35	4.849	0.281	1.274	4.467	1.25	2.35	5.981	0.149	1.329	4.467
$z=6.8$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	7.189	0.330	1.373	7.079	...	...	...	...	...	...
1000	1.25	2.35	7.189	0.234	1.342	5.623	...	...	...	...	...	...

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Table 9.  IGIMF and IMF Slopes with Varying z and ${\mathrm{Mecl}}_{\mathrm{min}}$ at β = 2.6 for ${\mathrm{SFR}}^{*}$

$\beta =2.6$
$z=0.1$						$z=0.2$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	0.381	0.150	1.442	0.224	1.25	2.35	0.434	0.241	1.416	0.282
1000	1.25	2.35	0.381	0.211	1.416	0.224	1.25	2.35	0.434	0.240	1.418	0.282
$z=0.8$						$z=1.5$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	0.797	0.156	1.412	0.562	1.25	2.35	1.304	0.196	1.370	0.891
1000	1.25	2.35	0.797	0.260	1.426	0.562	1.25	2.35	1.304	0.224	1.463	1.122
$z=2.2$						$z=3.8$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	1.889	0.366	1.378	1.412	1.25	2.35	3.471	0.238	1.484	3.548
1000	1.25	2.35	1.889	0.295	1.487	1.412	1.25	2.35	3.471	0.276	1.261	2.818
$z=5.0$						$z=5.9$
${\mathrm{Mecl}}_{\mathrm{min}}$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$	${\alpha }_{1,\mathrm{IMF}}$	${\alpha }_{2,\mathrm{IMF}}$	mc	${\alpha }_{1,\mathrm{IGIMF}}$	${\alpha }_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	4.849	0.319	1.258	4.467	1.25	2.35	5.981	0.299	1.339	4.467
1000	1.25	2.35	4.849	0.344	1.523	5.623	1.25	2.35	5.981	0.314	1.281	5.623
$z=6.8$
${\mathrm{Mecl}}_{\mathrm{min}}$	$\alpha {}_{1,\mathrm{IMF}}$	$\alpha {}_{2,\mathrm{IMF}}$	mc	$\alpha {}_{1,\mathrm{IGIMF}}$	$\alpha {}_{2,\mathrm{IGIMF}}$	${m}_{{\rm{c}}\prime }$
500	1.25	2.35	7.189	0.158	1.512	7.079	...	...	...	...	...	...
1000	1.25	2.35	7.189	0.192	1.402	7.079	...	...	...	...	...	...

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(i) As z increases, ${\alpha }_{2,\mathrm{IGIMF}}$ systematically rises up to z ∼ 2 and then starts falling. It again rises around z ∼ 4 and runs down around z ∼ 6. The effect is more pronounced for β = 2 and 2.4. For β = 2.6 the rise and fall are comparatively small (see Figures 5–7). We have also tested for the equality of means of ${\alpha }_{2,\mathrm{IGIMF}}$ values over β and ${M}_{\mathrm{ecl},\mathrm{min}}$ = $500\;{M}_{\odot }$ and 1000 ${M}_{\odot }$, respectively, for SFR2, e.g., by  the MANOVA test (Multivariate Analysis for Variance). The test has been rejected in all cases (p-values are 0.0278 and 0.0508, respectively, which are very small). This might be explained as follows. Though from z = 0 to z = 0–2 the SFRs and hence ${M}_{\mathrm{ecl},\mathrm{max}}$ are increasing (see Equation (1)), due to the low temperature of the ambient medium the Jeans mass does not favor the formation of massive stars. That is why ${\alpha }_{2,\mathrm{IGIMF}}$ is taking higher values, i.e., steeper slopes for z = 0–2. But gradually due to the rise of temperature of the medium with increasing z, the formation of massive stars predominates even for a comparatively lower but still moderate SFR and hence for moderate ${M}_{\mathrm{ecl},\mathrm{max}}$. This favors the formation of massive stars, which makes ${\alpha }_{2,\mathrm{IGIMF}}$ lower for z ∼ 2–4. The effect becomes reduced due to the rapid fall of the SFR at very high z (see z ∼ 4–6, Figures 1–4) increasing ${\alpha }_{2,\mathrm{IGIMF}}$ indices again.

(ii) As β increases, changes in the rising and falling of ${\alpha }_{2,\mathrm{IGIMF}}$ become faster. The effect is very pronounced for $\beta \sim 2-2.4$. This is because as β increases the number of low-mass clusters becomes higher compared to the number of high-mass clusters. So the abovementioned effect becomes accentuated due to the steepening of the mass function of embedded clusters (Figure 6). The statistical test has shown rejection of the null hypothesis.

(iii) ${\alpha }_{2,\mathrm{IGIMF}}$ becomes flatter as ${M}_{\mathrm{ecl},\mathrm{min}}$ increases when β is low. This is because when β is low, the number of low-mass clusters decreases and as a result massive star formation is favored compared to low-mass stars. This results in the flattening of the slopes, ${\alpha }_{2,\mathrm{IGIMF}}$ in a high-mass regime. For higher values of β (> 2), the number of low-mass clusters increases, which disfavors the formation of massive stars and ${\alpha }_{2,\mathrm{IGIMF}}$ becomes steeper as a result.

(iv) ${\alpha }_{2,\mathrm{IGIMF}}$ is always flatter than the IMF slopes. This might be the joint effect of various SFRs as well as the increasing temperature of the environment with increasing redshifts. Up to z ∼ 2 the temperature of the ambient medium is lower compared to the higher redshift zone. Hence Jeans masses are lower. But at the same time the SFR is increasing to its maximum ${M}_{\mathrm{ecl},\mathrm{max}}$, which is favored over the formation of massive stars compared to the IMF. On the other hand, for z > 2, Jeans masses are higher and SFRs gradually decrease, lowering values of ${M}_{\mathrm{ecl},\mathrm{max}}$, hence increasing low-mass stars. Somehow the joint effect of these two phenomena is responsible for a resulting flatness of ${\alpha }_{2,\mathrm{IGIMF}}$. This is consistent with some observational results (Finoguenov et al. 2003; Alonso-Herrero et al. 2004; Nayakshin & Sunyaev 2005; Loewenstein 2006) that indicate the IGIMF is top-heavy (i.e., massive stars form in large numbers compared to less massive stars) when SFR ≥100 ${M}_{\odot }\;{\mathrm{yr}}^{-1}$. The above trend is also in good agreement with the Galactic and M31 bulge (Ballero et al. 2007), as well as Wilkin et al. (2011) for present-day mass density from cosmological star formation history. The trend for decreasing slope with increasing SFR has also been found by Gunawardhana et al. (2011) for a sample of 40,000 galaxies. For $z\gt 2$, though the SFR decreases and formation of massive clouds are not favored, massive stars are still produced in some optimum zone due to the increase in temperature of the medium so that ${m}_{{\rm{c}}}^{\prime }$ is shifted toward higher mass (i.e., a top-heavy mass spectrum with steeper slope; Figures 9–11).

(v) The characteristic mass m_c of the stellar initial mass function differs from the characteristic mass, ${m}_{{\rm{c}}\prime }$, of the integrated galactic mass function. Generally ${m}_{{\rm{c}}}\geqslant {m}_{{\rm{c}}\prime }$.

(vi) The abovementioned effects are similar for various measures of the SFR though there are small variations. The measured SFR1 is the first quartile, i.e., 25% of the galaxies have $\mathrm{SFR}\leqslant \mathrm{SFR}1$ and 75% of the galaxies have $\mathrm{SFR}\gt \mathrm{SFR}1$, i.e., we can say that SFR1 is representative of low SFRs, which is the main characteristic of dwarf galaxies. On the other hand, SFR3 is representative of high SFRs, which characterizes giant galaxies. In this regard SFR2 is the measure of the average SFRs of galaxies. Now in dwarf galaxies due to their low SFRs,  the formation of massive stars is not favorable in large numbers. Thus it is more likely that ${\alpha }_{2,\mathrm{IGIMF}}$ for SFR1 is rather steeper than that of ${\alpha }_{2,\mathrm{IGIMF}}$ for SFR2, followed by ${\alpha }_{2,\mathrm{IGIMF}}$ for SFR3. It is clear from Tables 2 to 4 that ${\alpha }_{2,\mathrm{IGIMF}}$ for SFR1 > ${\alpha }_{2,\mathrm{IGIMF}}$ for SFR2 in 72%–78% cases for β = 2–2.6 and ${\alpha }_{2,\mathrm{IGIMF}}$ for SFR2 > ${\alpha }_{2,\mathrm{IGIMF}}$ for SFR3 in 50%–60% cases for β = 2–2.6. SFR* is the point of the SFR function where the function levels off from exponential to a shallower power law, i.e., from this point the SFR does not vary much to the left, i.e., for low SFRs. So SFR* is a  somewhat representative value for low SFRs, i.e., of less massive galaxies. ${\alpha }_{2,\mathrm{IGIMF}}$ for SFR* > ${\alpha }_{2,\mathrm{IGIMF}}$ for SFR2 in 50%–67% cases for $\beta =2$–2.6. Therefore SFR1/SFR^*, SFR2, and SFR3 might be representative  of the star formation histories for dwarf, intermediate, and giant galaxies, respectively, and hence the selection of the masses of the galaxies is appropriate.

For the first time the nature of an observed SFR has been investigated (see SFR1, SFR2, SFR3, and SFR*) as a function of redshift rather than using SFR as a parameter (Weidner & Kroupa 2004) for various types of galaxies (dwarf, intermediate, and giant), together with a top-heavy stellar IMF increasing with redshift (see Equation (6)). This aids the study of the cosmic star formation history in galaxies under the combined effect of both varying IMFs and SFRs. A Monte Carlo simulation method is used for its simplicity of computation to find the resulting IGIMF. It is found that up to a redshift of z ∼ 2, the galactic mass function becomes steeper compared to a flatter one for $z\gt 2$, followed again by a steeper one around ${\text{}}z\sim 6$. This is due to the joint effect of the distribution of SFR as a function of z and the temperature of the ambient medium. The galactic mass function is affected by the embedded cluster mass spectrum. The effect is faster for a steeper one. It is also influenced by the minimum mass of the parent cluster, e.g., ${\alpha }_{2,\mathrm{IGIMF}}$ becomes flatter as ${M}_{\mathrm{ecl},\mathrm{min}}$ increases when β is low.

The authors are very much thankful for the referee's suggestions, which improved the quality of the manuscript to a great extent. T.C. wishes to thank the Department of Science and Technology (DST), India, for awarding her a major research project for the work. The authors are also grateful to Soumita Modak for her help.