EXPLORING THE INTERCONNECTEDNESS OF COSMOLOGICAL PARAMETERS AND OBSERVATIONS: INSIGHTS INTO THE PROPERTIES AND EVOLUTION OF THE UNIVERSE

This research aims to investigate the relationship between Confidence Interval, Hubble Parameter, Comoving Distance, and Distance-Volume Relationship, which are important equations in cosmology. The Confidence Interval equation is used to estimate the range of values for the difference between the mean redshift and Hubble parameter. The Hubble Parameter equation is used to measure the expansion rate of the universe, while the Comoving Distance equation is used to calculate the distance between two objects in the expanding universe, and the Distance-Volume Relationship equation is used to calculate the distance between an observer and a cosmic object based on the object's redshift. This study seeks to address several research questions, including the accuracy of estimating parameters using these equations and the potential for developing more precise equations. The study employs cosmological data analysis using the R program to analyze existing data and gain a better understanding of cosmological parameters. The results of this research contribute to our understanding of the nature and evolution of the universe, providing insights into the distribution of matter and the role of dark matter and dark energy in shaping the universe's evolution. By examining the relationship between cosmological parameters, this study enables us to make predictions about cosmic phenomena and improve the accuracy of future measurements. The findings of this research have


INTRODUCTION
The topic of this research is focused on three important equations in cosmology: Confidence Interval, Hubble Parameter, Comoving Distance, and Distance-Volume Relationship. The purpose of this study is to provide a better understanding of the relationship between these cosmological parameters and to provide new insights into the nature and evolution of the universe. The Confidence Interval equation is used to estimate the range of values within which a population parameter is likely to fall based on a sample statistic [1]. In cosmology, this equation is used to estimate the range of values for the difference between the mean redshift and Hubble parameter [2]. By understanding the Confidence Interval equation, researchers can better estimate the values of these important parameters and improve our understanding of the properties of the universe. The Hubble Parameter equation is used to measure the expansion rate of the universe, which is a key factor in determining the age of the universe [3]. By understanding this equation, researchers can gain insights into the history of the universe and the processes that have shaped its evolution. The Comoving Distance equation is used to calculate the distance between two objects in the expanding universe [4]. This equation takes into account the expansion of the universe and provides a more accurate estimate of the distance between objects than traditional methods. By understanding this equation, researchers can better measure the distances between cosmic objects and improve our understanding of the large-scale structure of the universe. The Distance-Volume Relationship equation is used to calculate the distance between an observer and a cosmic object based on the object's redshift [5]. This equation takes into account the expansion of the universe and provides a way to estimate the average density of matter in the universe. By understanding this equation, researchers can gain insights into the distribution of matter in the universe and the role of dark matter and dark energy in shaping the evolution of the universe.
The background for the author's interest in this topic stems from the need to further investigate and understand the relationships between these cosmological parameters. Previous studies have provided valuable insights into these equations individually, but there is still a need to explore their interrelationships and potential implications for our understanding of the universe [6]. Additionally, there may be limitations or discrepancies in previous studies that need to be addressed and improved upon. Therefore, this research aims to contribute to the existing body of knowledge by examining the relationships between these cosmological parameters in a comprehensive manner.
Previous related studies have been conducted on each of these cosmological parameters individually. However, there may be discrepancies or inconsistencies among these studies, and it is important to thoroughly review and analyze the existing literature to identify any gaps or limitations. By doing so, this research can build upon the findings of previous studies and potentially provide new insights or corrections to improve our understanding of these cosmological parameters.
Furthermore, this research will highlight the benefits of the method used in this study compared to previous studies. The approach taken in this research may offer advantages in terms of accuracy, precision, or comprehensiveness in analyzing the relationships between these cosmological parameters. By explaining the benefits of the method used in this study compared to previous studies, the author can provide a justification for the chosen approach and demonstrate the potential contributions of this research to the field of cosmology.

METHOD
The research method used in this discussion is cosmological data analysis using the R program. Cosmological data obtained from various observational techniques are used to calculate and estimate the values of cosmological parameters such as Confidence Intervals, Hubble Parameters, Komoving Distances, and Distance-Volume Relationships. First of all, statistical calculations are used to calculate the confidence interval of the difference between the z and h means. Then, a mathematical equation is used to calculate the value of the Hubble parameter in redshift z by entering the value of the mass fraction of matter in the universe Ω , fraction of dark energy in the universe Ω , fraction of the empty pocket in the universe Ω , Hubble's current constant (H0), and redshift (z). Next, an equation is used to calculate the moving distance between two objects in cosmic space by integrating the velocity equation. Finally, we use the distance-volume relationship to calculate the measured distance from the observer to the cosmic object at any given time located at a certain redshift by including the moving distance, the angle of the solid seen from the observer, and the effect of the redshift that occurs on the light received from the cosmic object. During data analysis, the R program was used to calculate the values of the cosmological parameters and visualize the calculation results. In the R program, integrated mathematical and statistical functions are used to facilitate calculations. In this discussion, the use of the R program makes it possible to perform data analysis quickly and accurately, making it possible to obtain more reliable and accurate results This table is a data table that contains information about Redshift (z), Hubble Parameter (H(z)), and error a ± 1σ from a number of observations. Redshift is a measure of the spectral shift of an astronomical object caused by a shift in the object's light toward red due to movement away from Earth [7]. The Hubble Parameter is a cosmological parameter that describes the expansion rate of the universe [8]. Error a ± 1σ is the measurement error expressed in standard deviation units from the average value of observations. In this table, each row represents a different observation at a different redshift (z), with the corresponding H(z) value and error a ± 1σ.  Variance is the square of the standard deviation and represents the average of the squared differences between each data point and the mean. It provides information about the spread or dispersion of the data. The range of Redshift (z) is 1.895, H(z) is 135.1, and a ± 1σ error is ±80.6. The range is the difference between the highest and lowest values in the data. It provides an indication of the spread of the data across the entire range. The coefficient of correlation between Redshift (z) and H(z) is 0.2761, between Redshift (z) and a ± 1σ error is -0.1505, and between H(z) and a ± 1σ error is 0.0798. The coefficient of correlation measures the strength and direction of the linear relationship between two variables. It ranges from -1 to 1, where 1 indicates a perfect positive correlation, -1 indicates a perfect negative correlation, and 0 indicates no correlation. In this case, a positive correlation between Redshift (z) and H(z) indicates that as Redshift increases, H(z) also tends to increase, while a negative correlation between Redshift (z) and a ± 1σ error indicates an inverse relationship, where an increase in Redshift leads to a decrease in the error. The correlation between H(z) and a ± 1σ error is weak, as the coefficient is close to 0.

Hubble Parameters (H)
The mathematical equation for calculating the value of Hubble parameters (H) is as follows [2]: Where H(z) is the value of the Hubble parameter on redshift z, 0 is the current Hubble constant, Ω is the mass fraction of matter in the universe, Ω is the fraction of the empty pocket in the universe, Ω is the fraction of dark energy in the universe z is redshift, which is the shift in the spectrum of light produced by astronomical objects as it travels through space.
For the derivation of these equations, we can first start with the Friedmann equation which describe the evolution of the universe [10]. The Friedmann equation shows the relationship between the expansion rate of the universe and the density of matter and energy in it [11]. This equation can be written as: Where H is the Hubble parameter G is Newton's gravitational constant ρ is the density of matter and energy in the universe k is the empty pocket constant a is the cosmological factor scale Λ is the cosmological constant (dark energy) [12]. If we plug in the equation for the cosmological factor scale, = 1 (1+ ) , then we can write the Friedmann equation in a simpler form: Where H0 is the current Hubble constant ,Ω is the mass fraction of matter in the universe, Ω is the fraction of the empty pocket in the universe, Ω is the fraction of dark energy in the universe z is redshift.

Comoving distance r(z)
This equation is used in cosmology to calculate the comoving distance or the distance between two objects in cosmic space [13]. This formula is obtained through the integration of the velocity equation. First, we start by considering Hubble's law, which states that the speed of an object in cosmic space is proportional to its distance from the observer, as follows: where v is the velocity of the object at a distance r from the observer and H(z) is the Hubble parameter, which is the expansion rate of the universe at a given time. We can turn this equation into a more useful form by multiplying both sides by the scale factor, a, so we get: with a as the scale factor and ( ) as the rate of change of the scale factor. Next, we take advantage of the fact that the speed of an object at a certain distance can be expressed as the rate of change of distance in time. In this case, we can write: We can then solve this equation for dr: We can divide both sides by r and perform integration on both sides from the initial time to the present time, with integration limits from 0 to z. The result is:  ) is the comoving distance, which is the distance between any two objects in cosmic space that takes into account the expansion of the universe, where as a is the cosmological integral, which gives information about how fast the universe is expanding at any given time.

Distance-volume relation (Dv(z))
The distance-volume relation in cosmology is a mathematical relationship used to calculate the measured distance (expressed in units of volume) from an observer to a cosmic object at a specific redshift (z) during a given time in the universe [14]. The derivation of this equation involves several basic concepts in cosmology, such as the general theory of relativity, redshift, and the Hubble constant [15]. First, we need to know the definition of a measured distance in cosmology. In cosmology, measured distances are expressed in units of volume (usually megaparsec 3 ) and are defined as: where c is the speed of light, H(z) is the Hubble constant at redshift z, and r(z) is the moving distance from the observer to the cosmic object at redshift z. The moving distance is a fixed distance in cosmological space that expands with time. At the present time (z = 0), the moving distance is equal to the physical distance (which can be measured in meters). To get a simpler equation, we can manipulate the above equation by eliminating factors ( ⋅ ( )) 2 , so we get: We can also use the definition of the commoving distance in cosmology, ie: where the integral is calculated from redshift 0 (current time) to redshift z (earlier time in cosmological history). Substituting r(z) in the above equation, we get:

Luminosity Distance Equation D M (z)
In cosmology, the comoving distance r(z) is calculated from the integration of the cosmic scale factor a from the moment (a=1) to the redshift moment z, as follows: where c is the speed of light, H(z) is the Hubble parameter at redshift z, and the integral is calculated from redshift 0 (current) to redshift z. Sound horizon rs is defined as the physical distance traveled by acoustic vibrations in the plasma during the last redshift before the photons escape from the plasma, which is expressed in Mpc units [16].
In cosmology, the comoving distance and the sound horizon are usually expressed in Mpc/h, where h is the current Hubble constant in km/s/Mpc [17]. By combining the definitions of comoving distance and sound horizon, the DM(z) equation can be derived as follows: In this last equation, the c/rs factor can be calculated using the sound horizon value given in Mpc/h units and the c factor expressed in km/s units. The DM(z) equation is used to calculate the angular distance comoving diameter DM at redshift z, which is the distance between two objects in Mpc/h units, where the positions of these objects are observed from the earth [18].

Angular Diameter Distance D A (z)
The derivation of the equation starts from the definition of the angular distance θ in radian units as , where r is the physical distance between the two objects and is the angular diameter distance. The physical distance r can be expressed as = ⋅ (1 + ), where is the comoving distance which is the physical distance between two objects at the same time in moving coordinates [13]. Therefore, the angular distance θ can be expressed as: The above equation can be changed in such a way that DA can be isolated. First of all, the ratio between these two distances is defined as the scale factor: By using this definition, the equation can be produced: In this equation form, the scale factor a can be expressed in units of comoving distance DM, so that DA can be expressed as: = ⋅ (26) This last equation is an equation , which shows that the angular diameter distance ( ) can be calculated from the comoving distance ( ) which is known.

Luminosity Distance D L (z)
In order to derive this equation, we must first understand the concepts of apparent magnitude and absolute magnitude [19]. Apparent magnitude refers to the brightness of a light source as we observe it on earth, while absolute magnitude refers to the brightness of a light source when placed at a specified standard distance [20]. We can use the equations for apparent magnitude and absolute magnitude to calculate the distance between the light source and the observer. This equation is known as the distance modulus equation, and is given by: where m is the apparent magnitude, M is the absolute magnitude, and d is the distance between the light source and the observer in parsecs. We can solve this equation for d and get: To calculate the luminosity distance, we must take into account the redshift of the light source. Redshift refers to the phenomenon where the spectral lines of a light source are shifted towards longer wavelengths, appearing "redder," due to the expansion of the universe. The farther the light source is from us, the greater redshift. In cosmology, the distance that is often used is the comoving distance, which is the distance that follows the expansion of the universe [21]. The moving distance is expressed in units , that is, the radius of the particle horizon at the beginning of the universe [22]. Therefore, we can calculate the luminosity distance in units . In this equation, ( ) has been calculated before. To count ( ), we can use the luminosity distance equation: where r is the moving distance, a is the cosmological scale factor, and z is the redshift. The cosmological factor scale, a, is the ratio of the size of the universe at a given time to the size of the universe at another time [23]. The scale of this factor is time dependent and is usually expressed as a function of time, a(t). In cosmology, we work with the FLRW (Friedmann-Lemaître-Robertson-Walker) cosmological model, which assumes that the universe is homogeneous and isotropic [24]. In this model, the cosmological factor scale is expressed as: Where 0 is the Hubble constant, Ω , Ω , Ω , and Ω . As for each is a cosmological parameter for radiation density, matter density (including dark matter)

Parameter cosmological universe calculation results
The calculation results will be made in the data table which contains information about several cosmological parameters calculated from observations and cosmological modeling at the time of observation. The parameters will include: • z: redshift, namely the magnitude of the redshift of the observed object • r: comoving distance, namely the physical distance between these objects in megaparsec units (Mpc) adjusted for the expansion of the universe • DV: sound size, namely the size of the volume at redshift z which is calculated from the combination of cosmological parameters • H: Hubble parameter, namely the expansion rate of the universe at the time of observation in units of km/s/Mpc • DM: luminosity distance, namely the physical distance between these objects in megaparsec units (Mpc) which is calculated using the object's intrinsic luminosity • DA: angular distance, namely the physical distance between these objects in megaparsec units (Mpc) which is calculated using the observed angle of the object DL: redshift-adjusted luminosity distance, namely the physical distance between these objects in megaparsec units (Mpc) calculated using the object's intrinsic luminosity and z-adjusted redshift.
This data table is the result of this calculation and can be used for cosmological research, especially in testing and comparing cosmological models. By observing the values of these parameters at different redshifts, we can evaluate the fit between the existing cosmological model and the actual observations, as well as improve and enhance the cosmological model, shown by TABLE 2.

RESULT AND DISCUSSION
The data provided above  The TABLE 2 given is cosmological observational data used to measure cosmological parameters, namely parameters that describe the nature and evolution of the universe. Each row in the table represents an observable cosmic object, and each object has several associated parameters. In cosmology, these parameters are interrelated and provide information about properties of the universe such as age, density and composition. For example, the Hubble parameter (H) is used to measure the age of the universe, while the DV parameter is used to measure the average density of the universe [25]. In addition, changes in these parameters can provide clues about the cause and effect of cosmological phenomena such as the accelerating expansion of the universe and dark energy [26]. A cosmology parameter heatmap is a visualization that displays the intensity or values of cosmology parameters in a matrix form based on the level of observation or measurement [27]. This heatmap is used to identify patterns or trends in data and visualize the relative differences between the values of cosmology parameters in the analyzed dataset [28]. From a physics perspective, this heatmap can depict the relationship or correlation between cosmology parameters present in the data. For example, if there is a positive correlation between two parameters, then brighter areas on the heatmap will indicate higher values for those two parameters at a particular observation [29]. Conversely, if there is a negative correlation, then darker areas on the heatmap will indicate lower values for those two parameters at a particular observation. Theoretical support can aid in identifying differences between observational and theoretical values in the cosmology parameter heatmap. By incorporating theoretical values or frameworks, the heatmap can provide insights into the agreement or discrepancy between observational data and theoretical predictions in cosmology research. This can help researchers validate or refine existing cosmological models and theories based on the observed data, and further our understanding of the Universe's fundamental properties and evolution. In this discussion, we will discuss three important equations in cosmology: Confidence Interval, Hubble Parameter, Komoving Distance and Distance-Volume Relationship. First, we'll cover Confidence Intervals. The calculation results show that the difference between the mean z and h is estimated to be between -105.8591 and -105.0449 with a 95% confidence interval. This shows that the greater the z value, the greater the h value.
Next, we'll cover Hubble Parameters. The mathematical equation for calculating the value of the Hubble parameter at redshift z is based on EQUATION 1. Where H0 is the Hubble current constant, m  is the mass fraction of matter in the universe,   is the fraction of dark energy in the universe, k  is the fraction of the empty pocket in the universe, and z is the redshift.
Then, we will discuss the Komoving Distance. This equation is used in cosmology to calculate the comoving distance or the distance between two objects in cosmic space. This equation is obtained by integration of the velocity equation and can be written as r(z) = c * integral from 0 to z in a/H(a) da. In this equation, c is the constant of the speed of light, r(z) is the moving distance, which is the distance between two objects in cosmic space that takes into account the expansion of the universe, and the cosmological integral gives information about how fast the universe is expanding at any given time. Next we will discuss the Distance-Volume Relationship. This relationship in cosmology is used to calculate the measured distance (measured in units of volume) from an observer to a cosmic object at any given time that lies at a certain redshift (z). Distance measured in cosmology is expressed in units of volume and is defined as ( ) = ((1+ ) 2 ⋅ 2 ( )⋅ΔΩ) . Where c is the speed of light, r(z) is the moving distance from the observer to the cosmic object at redshift z, deltaOmega is the solid angle as seen by the observer, and the factor (1 + ) 2 takes into account the redshift effect that occurs on the incident light. received from the cosmic object.
Next discussion in The TABLE 2 provided contains cosmological observation data that is used to measure cosmological parameters, which describe the nature and evolution of the universe. Each row in the table represents a cosmic object that has been observed, and each object has several related parameters. In cosmology, these parameters are interconnected and provide information about the properties of the universe, such as its age, density, and composition. For example, the Hubble parameter (H) is used to measure the age of the universe, while the DV parameter is used to measure the average density of the universe. In addition, changes in these parameters can provide clues about the causes and effects of cosmological phenomena such as the acceleration of the universe's expansion and dark energy. Cosmological observations are made using various techniques, including telescopes, satellites, and ground-based experiments. The data collected from these observations is then used to build models and theories about the universe. By comparing these models with observational data, cosmologists can test the validity of their theories and gain new insights into the workings of the universe.
One of the key challenges in cosmology is the measurement of cosmological parameters with high precision. This requires the use of advanced technologies and techniques, such as gravitational lensing, which can distort the light from distant objects to reveal information about the distribution of matter in the universe. Another challenge is understanding the relationship between the various cosmological parameters and how they affect each other. This requires the development of mathematical models and simulations that can accurately capture the complex interactions between these parameters. In conclusion, the table provided contains valuable information about the properties of the universe and the interconnectedness of cosmological parameters. By analyzing this data, cosmologists can gain new insights into the workings of the universe and improve our understanding of its origins and evolution.

CONCLUSION
This research paper provides statistical values and correlations for a set of variables used to measure cosmological parameters, such as the age, density, and composition of the universe. The paper presents a cosmological parameter heatmap that visualizes the intensity or values of cosmology parameters in a matrix form, which helps to identify patterns or trends in data and to visualize the relative differences between the values of cosmology parameters in the | analyzed dataset. Theoretical support can be added to the parameter heatmap by incorporating a theoretical framework or theoretical values of cosmology parameters as references. The paper also discusses three important equations in cosmology, namely the Confidence Interval, Hubble Parameter, Komoving Distance, and Distance-Volume Relationship. These equations are used to calculate the value of the Hubble parameter at redshift z, the comoving distance, and the measured distance from an observer to a cosmic object at any given time that lies at a certain redshift (z). The research paper aims to validate or refine existing cosmological models and theories based on observed data and to further our understanding of the Universe's fundamental properties and evolution.