Comparison of physical characteristics of mass and luminosity function of disk systems in barred and unbarred spiral galaxies

Among the most important ways to investigate galaxies' distribution over cosmic time is the luminosity function LF in terms of baryonic disc mass ψ S (M b ), magnitude ∅ 𝑩 (𝑴 𝑩 ) . We have studied an estimate of the baryon mass density in the sample of barred and unbarred spiral-type galaxies from previous literature, which virtually involves, for each class of objects with visible baryon content, an integral over the luminosity of the product of the luminosity function (LF) and the mass-to-light ratio. The multiple regression technique used the statistical software package in our study and results, such as database analysis and graphing software (Statistics Win and Origin Pro programs). According to the statistical analysis, there is a strong positive and very relevant correlation ( ∅ 𝑩 (𝑴 𝑩 ), 𝛙 𝐒 (𝐌 𝒃 ) α M B~1 ), and both barred and unbarred disc spiral galaxies often exhibit M B < -18 mag. The "knee" of the luminosity function for spiral galaxies shows a large cutoff at the baryonic mass of M b >10 10 M ʘ for barred and unbarred spirals. These provide evidence supporting the hypothesis that disc system spirals began to form inside an increased mass threshold. Since the increase of the star's initial mass function with redshift is much more rapid, our findings have indicated that the comoving initial mass function ψ S (M b ) of barred and unbarred galaxies at elevated redshift (z > 0.027 for barred and z > 0.02 for unbarred) appears to be declining compared to the critical universe.


Introduction
Bars and spirals share morphological identifications, they play featured roles within galaxies' dynamics.Bars are often known as significant components driving galactic dynamics, while spirals are believed to have potential outcomes influenced by the presence of bars.Many ordinary galaxies lack bars while still showing spiral features.The significance of bars and spirals in galaxies' long-term evolution cannot be excessive.Barred galaxies can be regarded as the peak of galaxy morphology, organized by highly clear structures.In such galaxies, the presence of a bar represents a prominent disturbance.On the other hand, galaxies devoid of these driving mechanisms typically do not exhibit long-term evolution.These include unbarred galaxies featuring classical bulges and minimal or absent global spiral patterns.Both barred and unbarred galaxies may show outer rings, typically located at roughly double the radius of the inner disk.Unbarred galaxies typically possess intrinsic pivotal ratios of approximately 0.85 for their inner disks.Bars, on the other hand, present greater extension, with typical Abstract Among the most important ways to investigate galaxies' distribution over cosmic time is the luminosity function LF in terms of baryonic disc mass ψ S (Mb), magnitude ∅  (  ) .We have studied an estimate of the baryon mass density in the sample of barred and unbarred spiral-type galaxies from previous literature, which virtually involves, for each class of objects with visible baryon content, an integral over the luminosity of the product of the luminosity function (LF) and the mass-to-light ratio.The multiple regression technique used the statistical software package in our study and results, such as database analysis and graphing software (Statistics Win and Origin Pro programs).According to the statistical analysis, there is a strong positive and very relevant correlation (∅  (  ),   (  )α MB ~1), and both barred and unbarred disc spiral galaxies often exhibit MB < -18 mag.The "knee" of the luminosity function for spiral galaxies shows a large cutoff at the baryonic mass of Mb >10 10 Mʘ for barred and unbarred spirals.These provide evidence supporting the hypothesis that disc system spirals began to form inside an increased mass threshold.Since the increase of the star's initial mass function with redshift is much more rapid, our findings have indicated that the comoving initial mass function ψ S (Mb) of barred and unbarred galaxies at elevated redshift (z > 0.027 for barred and z > 0.02 for unbarred) appears to be declining compared to the critical universe.
Published Online First: March, 2024 https://doi.org/10.21123/bsj.2024.25401P-ISSN: 2078-8665 -E-ISSN: 2411-7986 Baghdad Science Journal axial ratios of approximately 0.2, but only a smaller portion, typically less than one-third of the disk, participates in forming bars [1][2][3] .In a modern analysis involving H-band (1.65 μm) images of 186 bright galaxies, 56% were visually categorized as 'strongly barred,' 16% as 'weakly barred,' and 27% as 'unbarred in the near-infrared 4,5 .Bars in spiral galaxies exhibit different morphologies and nonelliptical shapes.These elongated linear structures within galaxies' central regions result from disc instabilities and the redistribution of angular momentum within the disc.A typical normal bar in an early-type galaxy consists of two segments: a broad inner region and narrower ends.Barred galaxies and those dominated by bulges tend to be fixed in denser environments compared to their unbarred counterparts, which have predominantly disk-like structures 6 .
Comprehending the intricacies of galaxy formation and evolution remains in its early stages, and a comprehensive framework that fully elucidates the processes governing how galaxies emerge and transform into their current structures has yet to be fully elucidated.A classical approach involves leveraging the statistical characteristics of galaxy properties across various galaxy populations 7,8 .In many realms of astronomy and cosmology, a commonly employed tool for scrutinizing the statistical attributes of object populations is the analysis of luminosity and mass functions.Mass functions (MFs) associated with stars and gas play a key role in studying the physical means that guide the formation and evolution of galaxies.These functions depict an essential cornerstone of improving theories through experimental observations 9,10 .During the Big Bang, approximately 5% of the generated mass materialized as normal baryonic matter composed of neutrons and protons.Of this fraction, approximately 10% eventually assemble to form star-shaped galaxies or various gaseous states, which can contain molecular, atomic, or ionized states [11][12][13][14] .
Salucci and Persic 15 computed the baryonic mass function, defined as ψ s (Mbayronic), for disc system galaxies.They utilized reliable luminosity functions and baryonic mass-to-light ratios in their analysis.Their findings revealed that the baryonic mass function of disc systems can be approximated as a power law ψ s (Mbayronic) α Mbayronic -1/2 , where this relationship holds from 10 8 to 2x10 11 solar masses (Mʘ).However, a distinct abrupt cut-off point is observed due to the scarcity of objects beyond this range.Shankar, et al 16 derived the Baryonic Mass Function (BMF) for galactic structures by examining their inner kinematics.They ensure its cosmological significance by demonstrating its universal character.Their research establishes a connection between a galaxy's baryonic mass and its virial (total, dark) mass.Generally, this study underscored the influence of baryonic feedback mechanisms, such as supernova explosions and quasar activity, in linking dark and ordinary matter within galaxies.In a prior investigation conducted by Read, and Neil 17 measured the baryonic mass function of galaxies, providing insights into how baryonic mass is distributed among different types of galaxies (e.g., spiral or elliptical) and various size categories, their study developed in the conclusion that a significant portion of the Universe's baryons exist outside of galaxies, likely residing within the warm/hot intergalactic medium (WHIM).In 2008, Kevin et al 18 presented estimates of low-mass stars' luminosity and mass functions, drawing from the SDSS/2MASS data catalog.Their investigation revealed that the logarithmically binned mass function best conforms to a log-normal distribution with a peak at Mc = 0.29, encompassing a 90% confidence interval ranging from Mc = 0.20 to 0.50.This translates to linearly binned mass functions peaking between 0.27 Mʘ and 0.12 Mʘ.
Trachternach et al 19 investigated the relationship between baryonic mass and maximum rotation velocity using the baryonic Tully-Fisher relation "BTF".Their concluded results found the significance of the Baryonic Tully-Fisher (BTF) relation as a fundamental link connecting the visual, baryonic matter, and dark matter masses within galaxies.Amanda et al 20 initiated a study of the galaxy cosmological mass function "GCMF" using a semi-empirical relativistic system, employing observational data drawn from current galaxy redshift surveys.Their research outcomes corroborated that the "GCMF" adheres to the theoretical expectations derived from cold dark matter models, wherein less massive objects are formed first, followed by the emergence of more massive counterparts.
Hunt et al 21

Sample collections and databases
This work is established on the data group gathered from the NASA/IPAC Extragalactic Database (NED) surveys [22][23][24] , which is supported by the National Aeronautics and Space Administration and operated by the California Institute of Technology, such as Hubble luminosity distance (dL) of the Cosmic Microwave Background "CMB", and the velocity-integrated intensity SCO (for the transition J=1-0) in the unit (Jy.km/s).The apparent magnitude (Btc) in the blue band is corrected by (galactic extinction "ag", internal extinction "ai" and kcorrection "ak"), absolute magnitude MB in the blue band, which is computed from the correct magnitude Btc and the distance modulus μdz from the redshift or μ0 redshift-independent (for nearby galaxies MB=Btc-μ0, and a more distant galaxies MB=Btc-μdz) , and the neutral hydrogen HI flux SHI in 21-cm line profile ( 21 = −2.5    + 17.4) appointed from the HyperLEDA database, which is the Observatoire de Lyon "France" and the Special Astrophysical Observatory "Russia" 25,26 .Furthermore, the data that has been involved in this research was collected from our previous study 27 .
The names of the barred and unbarred spiral galaxies, redshift z, and their morphological type, and these parameters are offered in Tables (1 and 2) 27 .

Statistical procedure
The link between several independent or predictor variables and an associated or criterion variable is explored further in multivariate regression.One dependent (criterion, endogenous) variable is associated with many independent (predictor, exogenous) variables in an orderly multiple regression study, computed using the multiple regression module.The multivariate regression equation is extremely significant generally (for an explanation of statistical significance testing, see Elementary Basics).So, it is possible to "predict" impoverishment better than what might have been anticipated by pure chance independently given the variables that are not dependent.An indicator of the relationship between any numbers of variables is a correlation.Interval scales should be used as a minimum for measurements.However, additional correlation coefficients are available to accommodate different kinds of datasets.The range of coefficients of correlation extends from -1 to +1.A perfectly negative correlation is denoted by a value of -1, whereas a perfect positive correlation is expressed by a value of +1 26,27 .The associations between the corresponding independent and dependent variables after accounting, for all other factors are called partial correlations.After accounting for all independent variables, it is the correlation between the leftovers.The partial relationship shows each independent variable's distinctive contribution to the dependent variable's estimation.The statistical software programs (such as Statistics Win and Origin Pro) that we utilized for analyzing databases and plotting our investigation and outcomes were employed in the multiple regression techniques.

The mass and luminosity function of spiral galaxies
The luminosity function (LF) is a fundamental tool in examining galaxy evolution.A challenge in determining the LF lies in assembling a sample of galaxies located at the same distance.Therefore, investigating clusters of galaxies, where all galaxies share a common distance, becomes valuable, albeit at the cost of focusing on a specific region of the cosmos 17 .Current cosmological research is driven by a quest to comprehend the origins and development of the structures that have given rise to the galaxies we observe today [28][29][30][31] .The LF, represented as Φ(L) and measured in units of ergs s −1 Mpc −3 , stands as a pivotal tool for exploring the distribution of galaxies across cosmic epochs 10 .
The calculation of the Universe's baryon mass density essentially entails integrating, for each class of objects with visible baryon content, the product of the luminosity function (LF), denoted as Φ(L), luminosity (L), and the mass-to-light ratio for the baryon component, Mb/L.This is expressed as 32,33 : where "dN" is the surveyed number of galaxies within a luminosity range [L, L+dL], and T represents spiral galaxies, clusters, and superclusters.
The overall distribution of galaxies on a global scale can be approximated using the Schechter luminosity function (LF).Initially derived in the form of a mass function during investigations into structure formation and evolution, this function is generally expressed in the following form 10,[32][33][34] : Here  * is characteristic luminosity, (L) is a galaxy luminosity, the normalisation coefficient for the density in the unit (h 3 Mpc - 3) is (∅ * ), the incline of the luminosity function for small stars (L) is , and (∅) is the total number of galaxies per unit volume in (Mpc -3 ) per unit luminosity function 33 .
The disk system fraction, which refers to the percentage of stars in a star cluster with system disks, is often used to study the formation and evolution of protoplanetary disks.This fraction can be influenced by factors such as the star cluster's age and the system disk's mass distribution.The disk system's primary mass function "DMF" can greatly affect the disk fragment time variability 35 .To understand this, we can calculate the initial luminosity function ψ S (MB), which tells us how many stars of magnitude M in the interval ΔMB have formed in spiral galaxies per pc 3 during their existence.The disk systems mass function is defined as the fractional number of disks with mass between ψ S (Mb) and ψ S (Mb) dMb 15

… … … … … … … … … 13
A good fit for the disk systems mass function ψ s,B in the blue band is 15 : Hence, Mb is the baryonic disk mass, which is the sum of the stellar mass and the gas mass 19,27 : The stellar mass is calculated through 19 : where Υ * , is the stellar population mass-to-light ratio in the Blue-band, the Sun's magnitude in the blue range is "5.48", while MB is a star's absolute magnitude in the blue band, it is clear that we are talking about stellar emission generated in the 4400 A 0 region of the spectrum, which is considered to be the "blue" regime.The relationship between a galaxy's apparent magnitude, distance, and distance modulus determines its absolute magnitude.As a result, the stellar mass depends on apparent magnitude, distance, and stellar mass-to-light proportion.In this work, the mass-to-light ratios in the blue band for the stellar mass populations of barred and unbarred spiral galaxies are adopted to be Υ * , =1.4 Mʘ/Lʘ 36 .
The second term contributing to the baryonic disk systems mass is gaseous disc mass Mgas, atomic and molecular gas masses used to compute the total gas mass, which is given by 27,[36][37][38][39] : Helium's impact is considered to have a constant of 1.33.The mass-to-light proportions Ύ* essentially include whatever dark gas emission (whether molecular or ionized gases), provided that it increases alongside M* 37,38 .Ionized gas was first discovered using ground-breaking H-emission measurements, but it can be hard to calculate its mass since the HI/HII evolution has an acute point.The current study regards the HII component as baryonic dark matter that is yet unspecified 15,[40][41][42][43] .Accordingly, the mass of neutral atomic hydrogen MHI is traced by an emission of 21 cm and is calculated through the followin equation 38,[44][45][46][47] : Where dL is the luminosity distance in unit (Mpc) collected from NASA/IPAC Extragalactic Database (NED), and SHI is observed frame velocity integrated flux at νHI=1.4GHz.Here, the number of HI atoms 48 , with a spontaneous emission rate of AHI, an emitted photon energy hνHI, and mH = 1.673533× 10 −27 kg.
According to HI measurements, atomic gas often dominates Mg in disk spiral galaxies.Given a CO-to-H2 transformation component, which could differ from galaxy to galaxy based on metallicity or other features, the molecular hydrogen gas mass could be approximated from CO data.Low-mass, metal-poor disk galaxies frequently exhibit undiscovered carbon monoxide "CO" emissions 37,40 .Fortunately, molecules often only make up a small portion of their dynamics and motion (fewer than ten per cent of Mb) 40,42 .Thus, the molecular gas mass MH2 was calculated using the succeeding Eq. 20 if the flux is available at the carbon monoxide line 12 CO ( J=1-0) in our sample for the barred and unbarred spirals 49- 51 .
The luminosity of the galaxies in the blue-optical band (λ=4400A 0 ) in unit blue solar luminosity (LB,ʘ) using an absolute magnitude of the Sun in the blue band MB,ʘ = +5.48m is computed by the method 27,52 :

Results and Discussion
This work investigated the multiple regression analysis of values of parameters on the graph for a Schechter function that fits the luminosity function in baryonic disk mass ψ S (Mb) or in magnitude terms ∅  (  ) as an expression of absolute magnitude at the blue-band range baryonic mass and luminosity blue-optical LB,ʘ, calculated in the blue spectrum of the Sun MB,ʘ = +5.48m for barred and unbarred spiral galaxies.Our analysis makes use of the technique created in 15 .This technique has been effectively used for overseeing the subtraction of background information.The empirical information is boosted even more by considering the spiral luminosity function, which is constructed from the physical characteristics of the objects instead of their spectral properties.Furthermore, at MB < -18 mag., both the luminosity function ψ S (Mb)and the magnitude measure ∅  (  ) ignore low-surface luminosity galaxies, representing only a tiny proportion of the total population of disc systems at these luminosities.
The statistical study revealed fairly robust and highly significant relationships between (∅  (  )-M B) with a very robust partial correlation coefficient (R ~0.88) and a very high probability (P ≤10-7) for unbarred spirals with a slope Log ∅  (  )α MB 0.9 , while this relation showed good partial correlation (R ~ 0.7) for barred galaxies with a slope Log ∅  (  )α MB 0.77 , and displays an extremely strong connection with a correlation coefficient (R~ 1) in the relationship Log ψ S (Mb)-MB with linear regression slope Log ψ S (Mb)α MB 0.97 for barred and Log ψ S (Mb)α MB 0.95 for unbarred spirals.The slope of the luminosity distribution seems to have a typical magnitude, MB < -18 mag, as it steepens as it approaches the more faint end.The luminosity characteristics continue to be at over several magnitudes higher on this scale before decreasing rapidly at the bright ends MB < -18 mag., which is typically found in both barred and unbarred disk system galaxies.The mass function exhibits a severe cut-off at the greatest masses, mainly Mb ≈ 6 x 10 10 Mʘ for barred and Mb ≈ 3 x 10 10 Mʘ for unbarred spirals, the "knee" of the spiral luminosity functions.These lend support to the theory that disk system spirals originated within a higher mass threshold, Mbmax ≈ 4x10 11 Mʘ, Mbmin ≈ 4x10 9 Mʘ for barred, and Mbmax ≈ 2x10 11 Mʘ, Mbmin ≈5x10 9 Mʘ for unbarred, owing to the impossibility of a greater baryonic mass to cool quickly enough to settle into a disk during a Hubble timescale.Considering that the emitting brightness of stellar discs of certain masses has a substantial spread caused by variances in the disk's stellar populations in general, this strong cut-off broadens the luminosity function (Fig. 2).
In the current study, it has been also explored the relationships between baryonic mass (Mb) disk spiral systems and luminosity function in baryonic disk mass ψ S (Mb) or magnitude units ∅  (  ).The results additionally indicated a very strong partial correlation between the associations of Log ψ S (Mb) and LogMb, with a tight correlation coefficient (R) of -0.96 and a slope line of ~ -1 for barred disc galaxies; furthermore, the relationship between Log ∅  (  ) and LogMb has R-value of -0.7 with a regression value of (Log ∅  (  ) α LogMb -0.76 ) for barred disc galaxies, the probability of a very robust chance (P ≤10-7) for two cases.According to statistics, we infer that Log ψ S (Mb) and LogMb have a linear connection (Slope ~ -1).Our analysis additionally demonstrates the presence of a very close relationship between the logarithmic scales ψ S (Mb) and Mb with a partial coefficient correlation (R ≈ -1) and the relationship between Log ∅  (  ) and LogMb with an R-value of -0.86 and a regression coefficient of (Log ∅  (  )α LogMb -0.9 ) for unbarred galaxies with Slope ~ -1 ( see Fig. 2).The general form of the luminosity function varies regularly as baryonic mass increases.All of the evidence shows that when baryonic mass increases, the proportion of the initial luminosity function decreases.The structure, nevertheless, transforms when centrals are taken into account, demonstrating that centrals are the predominant component of lower-mass baryonic.We observe that for baryons with Mb > 10 10 Mʘ in both cases barred and unbarred, the overall fit significantly underestimates the luminous ends of the luminosity function.This is caused by using a constant Mb for all masses (gases + stars).It is possible to see bars in the disk system spiraling on just one side of the nucleus, with the other half being either dim or missing.One often observes regions of increasing luminosity in loops or super relationships, which are almost certainly the locations of star formation at the ends of bars.The existence of a bar influences a variety of processes that take place in disk spiral galaxies.Keep in mind the major role that bars play in the numerous phenomena seen in disk galaxies.For barred spiral galaxies, hydrogen gas flows toward the galaxy's Centre, causing a burst of stars that results in a gradual decline.In contrast, in unbarred galaxies, the generation of star fraction increases steadily and continuously.Unbarred galaxies have a substantially lower active nucleus illumination than barred galaxies, and their inner regions include a smaller amount of gas that can be influenced by interaction.Baghdad Science Journal 0.95 for unbarred).For barred galaxies, there is a substantial connection between Log ∅  (  ) and Log LB,ʘ, with an acceptable partial correlation (R ~ -0.7) and a perfect importance probability (P ≤ 10 -7 ) and a slope line of ~ -0.8, and for unbarred spirals, there is a robust correlation between Log ∅  (  ) and Log LB,ʘ, with R ~-0.87 and a slope line of -0.9.
We study disk systems with luminosities ranging from LB max ≈ 3x10 11 LB,ʘ to LB min ≈ 2x10 9 LB,ʘ for barred, LB max ≈ 10 11 LB,ʘ and LB min ≈ 3x10 9 LB,ʘ for unbarred.In the case of an energy bandwidth, Fig. 3 depicts the development of the comoving galaxies' luminosity distribution with luminosity blue-optical.
For both barred and unbarred disk galaxies, we can observe that the knee of the luminosity function shifted in earlier times in favor of higher luminosities.As a consequence, the connection that exists between the amount of initial mass function ψ S (Mb), the number density of stars with absolute magnitude ∅  (  ) and the blue luminosity of our sample galaxies barred and unbarred are complicated, unstable, and dependent on a wide range of external as well as internal variables, involving the environment, illumination, construction, and the place of creation of stars operation, accordingly.We additionally analyze the evolution of the initial mass function ψ S (Mb)with redshift z, as shown in Fig. 4. The initial mass functions have distinct structures primarily determined by the history of the star's creation.In the past, the luminosity function in baryonic disk mass declined because the mass fraction inside galaxies decreased and their star population reduced.The growth of the stellar initial mass function with redshift is substantially slower, implying that the diminution of the comoving initial mass function of barred and unbarred galaxies at high redshift is less than for the crucial universe (z > 0.027 for barred and z > 0.02 for unbarred).Its reduction in past times is attributable to two factors.Initially, we look at baryons with a high-temperature T > 10 4 K 0 "cutoff caused by poor cooling", resulting in a reduced initial mass fraction there was less mass held in profoundly likely drilling in earlier times.In addition, at higher redshifts, galaxies have a higher gas/star content proportion (smaller time-scale t0 α (1+z ) -3/2 ), implying that most of the mass is in the state of the gas.

Conclusion
This study estimated the luminosity function in terms of baryonic disc mass ψ S (Mb), magnitude ∅  (  ), baryonic disc mass and luminosity blue-optical.Our major conclusions may be summed up as: On a range of magnitudes, the luminosity features are still many magnitudes higher before sharply declining at the brilliant ends.Both barred and unbarred disc system galaxies generally have MB < -18 mag.The "knee" of the spiral brightness function, Mb ≈ 6 x 10 10 Mʘ for barred spirals and Mb ≈ 3 x 10 10 Mʘ for unbarred spirals, is where the mass function shows a significant cut-off value.The general fit seems to severely undervalue the luminous ends of the luminosity function for baryons with Mb > 10 10 Mʘ in both barred and unbarred configurations.This results from using a fixed Mb value for all abundance gases and stars.The knee of the luminosity operation has shifted earlier to support greater brightness for both barred and unbarred disc galaxies.As a result, the relationship between the quantity of the initial mass function ψ S (Mb), the number density of stars with absolute magnitude ∅  (  ), and the blue luminosity of our sample galaxies, barred and unbarred, is complex, unsteady and reliant on a wide range of external as well as internal variables, involving the surroundings, lighting, development, and the location of the stars' formation, respectively.Finally, it implied that the comoving initial mass function ψ S (Mb) of barred and unbarred galaxies at high redshift (z > 0.027 for barred and z > 0.02 for unbarred) is less diminished than for the crucial universe because the increase of the star initial mass function with redshift is significantly quicker.Galaxies with larger redshifts have a larger star-togas ratio (short time scale t0 α 1/(1+z ) 3/2 ), indicating that most of the mass is in gaseous form.

Fig. 1
Fig. 1 depicts the Schechter function supplied to the luminosity function in absolute magnitude ∅  (  ) as solid blue lines.The dashed red lines represent the Schechter function fitting to the mass function ψ S (Mb).To compute the Schechter function for each barred and unbarred spiral Hubble classification, right down to MB < -18, with an exact estimation of the faint-end slope and global normalization.The empirical information is boosted even more by considering the spiral luminosity function, which is constructed from the physical characteristics of the objects instead of their spectral properties.Furthermore, at MB < -18 mag., both the luminosity function ψ S (Mb)and the magnitude measure ∅  (  ) ignore low-surface luminosity galaxies, representing only a tiny proportion of the total population of disc systems at these luminosities.

Figure 1 .
Figure 1.On the right: (Log ψ S (Mb) and ∅  (  )) as a function of blue absolute magnitude (MB) for unbarred spiral galaxies.On the left: (Log ψ S (Mb) and ∅  (  )) as a function of (MB) for barred spiral galaxies.

Figure 2 .
Figure 2. On the right: (Log ψ S (Mb) and ∅  (  ))as a function of the baryonic mass disk system (Mb) for unbarred spiral galaxies.On the left: (Log ψ S (Mb) and ∅  (  )) as a function of baryonic mass disk system (Mb) for barred spiral galaxies Fig.3describes the relationship between the initial mass function ψ S (Mb)and the number density of stars with absolute magnitude ∅  (  ) and luminosity blue-optical LB,ʘ determined in the blue part of these galaxies, which shows the existence of a very significant relation Log ψ S (Mb)-Log LB,ʘ with a negative correlation coefficient (R ~ -0.93) for barred spirals, whereas an extremely high correlation (R ~ -1) for the unbarred spiral galaxies with a significantly steep probability (P ≤10-7  ) in the two cases.It is very important to note that the slope of the line is linear and equal to minus one ((Log ψ S (Mb) α Log LB,ʘ -0.97 for barred and Log ψ S (Mb) α Log LB,ʘ -

Figure 3 .
Figure 3. On the right: (Log ψ S (Mb) and ∅  (  )) as a function of blue luminosity (Log LB) for unbarred spiral galaxies.On the left: (Log ψ S (Mb) and ∅  (  )) as a function of (Log LB) for barred spiral galaxies.

Figure 4 .
Figure 4. On the right: Log ψ S (Mb) as a function of redshift (z) for unbarred spiral galaxies.On the left: Log ψ S (Mb) as a function of redshift (z) for barred galaxies. .