Two Approaches to Interpretation of Hubble Diagram

Supernovae of type Ia are used as standard candles in modern cosmology, they serve to test cosmological models. Interpretation of the Hubble diagram based on the standard cosmological model led cosmologists to conclusion that the Universe is filled mostly with cosmic dust and mysterious dark energy. In this paper we present exact solutions of the Friedmann equation in standard cosmology and conformal cosmology. The theoretical curves interpolating the Hubble diagram for the latest supernova data are expressed in an analytical form. The functions belong to the class of meromorphic Weierstrass functions. Both approaches describe the modern Hubble diagram with the same accuracy. Physical interpretation from the standpoint of conformal cosmology is preferable, since supernova data are described without using a Λ-term. In the standard cosmology, the Hubble diagram is described by some characteristics: a Hubble parameter H(t), a deceleration q(t), and a jerk j(t). As calculations show, the deceleration parameter q changes its sign during the evolution of the Universe, the j-parameter remains constant. In the modern era, the Universe expands with acceleration, and in the past its acceleration was negative. The change in the sign of acceleration, without a clear physical reason, puzzles cosmologists. It seems obvious to us that to study objects dislocated from us at distances of billions of light years, we should not use the coordinate time customary for work in laboratories, but the conformal time. In conformal coordinates, the behavior of photons is described as in the Minkowski space. The time intervals dt and dη are different, they are related by the scale factor: dt = adη. The conformal luminosity distance is longer than the standard luminosity distance, which is manifested when observing distant stellar objects. As a result, the effective magnitude value — the redshift relationship, on which the Hubble diagram is constructed, will be different. Using the conformal Friedmann equation, we introduce the conformal parameters H(η), q(η), j(η). All parameters remain positive during the evolution of the Universe. The scale factor grows with deceleration. The Universe does not experience a jerk.


Introduction
A type Ia supernova occurs when a white dwarf in a double star system accumulates a mass by accretion that sufficient to overcome the Chandrasekhar limit. The nature of explosion of such a supernova depends on its prehistory insignificantly. Redshifts > 0.1 are large enough to ignore peculiar motions of light sources. Supernovae type Ia, sufficiently bright stars whose absolute luminosity is known with good accuracy, they serve as standard candles for testing cosmological models. Two collaborations The Supernova Cosmology Project and High-z Supernova Search Team compared the results of observations of supernovae with theoretical predictions for luminosity distances as redshift functions. Interpretation of the Hubble diagram on the basis of a standard cosmological model with adjustable cosmological parameters led cosmologists to conclusion that the Universe is filled mainly with dust and, so-called, dark energy -a substance with an equation of state not found in Nature [1][2][3]. Phenomenological approach has not led to an understanding of the state of matter from which the Universe consists.
Conformal cosmological model [4], based on conformal Dirac's variables [5], allows to explain data on supernovae without Λ-term [6][7][8][9]. We show that solutions of the differential Friedmann equation belong to the class of Weierstrass meromorphic functions [10]. Therefore, it is natural to use them to compare predictions of these two approaches [11][12][13]. In conclusion, we show that the difference between distance modulus is interpolated by means of the Chebyshev polynomial of the fourth order with a sufficient degree of accuracy.

Friedmann equation in classical cosmology
In the standard cosmological model the Friedmann equation is used for fitting SNe Ia data. It ties a scale of the Universe ( ) with density of matter .
Here is Newton's constant, is a sign of curvature of a space, a dot denotes a derivative with respect to coordinate time . In generic case, the equation (1) is represented in the following form where the variable is given as a ratio of a scale ( ) to a modern one 0 = 1: is a redshift of spectral lines (observed variable), 0 = ℎ·10 5 m/s/Mpc, ℎ = 0.72±0.08 -Hubble constant. In the right hand of the Friedmann equation (2) Ω are partial densities of, correspondingly, Λ-term, curvature of the space, dust-like matter, radiation, stiff-state matter. For distant sources with > 1 the interpretation of the cosmological redshift as a Doppler shift is not valid [14]. An equation of continuitẏ with an equation of state of matter = , which connects the density and the pressure , yields the dependence of the density on the scale factor. So, for interstellar dust = 0: ∼ −3 ; for radiation = /3: ∼ −4 ; for contribution from Λ-term = − : ∼ Λ; for stiff state of matter = : ∼ −6 . The data of modern astronomical observations are fitted using cosmological parameters [2]: Ω Λ = 0.72, Ω M = 0.28. A solution of the Friedmann equation (2) with such parameters is presented in analytical form The second derivative of the scale factor is Hence, in the modern era the Universe expands with acceleration, since 2Ω Λ > Ω M ; in the past, its acceleration was negative¨< 0. This change of the sign of acceleration without a clear physical cause is noted by cosmologists. From the solution (4), taking into account the relation between the scale factor and the redshift (3), it follows ageredshift relationship The age 0 of the modern Universe in the coordinate time can be obtained by setting = 0 in (6) If we know the redshift of a certain galaxy, how do we find the coordinate distance to it? Since for the rays of light the spacetime interval is zero we have the relation between intervals of space and time d = − ( )d , and, using the notation introduced above (3) ≡ / 0 , Substituting now the Friedmann equation (2) into (8) we get an integral where we denote the ratio of partial densities as 4 3 ≡ Ω M /Ω Λ . The integral is calculated using the inverse Weierstrass ℘-function [12] and one obtains coordinate distanceredshift relationship The resulting formula is expressed in terms of the Weierstrass ℘-function [10], satisfying the differential equation where ℘( ) = , ℘ ′ ( ) = 0, = 1, 2, 3 -three roots of a cubic polynomial on the right-hand side of the differential equation (12) Invariants of the Weierstrass function are the following the discriminant of a cubic polynomial is negative: ∆ ≡ 3 2 − 27 2 3 < 0. In astronomy, the method of determining distances to ultra-distant objects is based on measuring their luminosity. The radiation power of an object (a star or a galaxy) is called its absolute luminosity. The flux density ℓ, i.e., the radiation power per unit area, is called its visible luminosity. In Euclidean geometry they are connected by the formula where is the distance from us to the radiation object. In the second century BC the Greek astronomer Hipparchus classified the stars visible to the naked eye into six classes according to their brightness. The brightest stars were assigned the first magnitude, and barely visible -the sixth magnitude. According to Norman Pogson in 1856 it was decided that the luminosity of the objects of the first stellar magnitude is a hundred times greater than the luminosity of the objects of the sixth stellar magnitude [14], i.e., where ℓ 0 and 0 are some relevant luminosities. With the creation of photomultipliers at the beginning of the XX-th century, the factors ℓ 0 and 0 were fixed. We express from (14) the apparent stellar magnitude and the absolute stellar magnitude of the object using the decimal logarithms Then we express from (15) the distance modulus ( − ) through the distance to the radiation object, using (13) For performing calculations, the factors ℓ 0 and 0 in (14) are chosen in such a way that the distance is measured in megaparsecs where ℳ = 25.
In the Friedman-Robertson-Walker cosmology, by analogy with the formula for the distance in Euclidean geometry (13), we determine a luminosity distance to a star object In standard cosmology, the luminosity distance ( ) is related to the coordinate distance [2]: Here it was taken into account that the area of the sphere around the luminous object passing through the Earth increases, the frequency of photons decreases during their motion. Substituting the formula for the coordinate distance (11) into (18), we get an analytical expression for luminosity distance Modern observational cosmology is based on the Hubble diagram. Effective stellar magnitude -redshift relationship is used to test cosmological theories ( in megaparsecs) [2]. Here ( ) is the apparent magnitude, is its absolute magnitude, and ℳ = 25 is a constant.

Friedmann equation in conformal cosmology
Interpretation of the Hubble diagram, based on a conformal cosmological model with parameters Ω rigid = 0.755, Ω M = 0.245, yields the same qualitative approximation as the standard cosmological model with parameters Ω Λ = 0.72, Ω M = 0.28 [9]. A parameter Ω rigid corresponds to a stiff state of matter, when the energy density is equal to the pressure = [15,16] that is happened under a nucleosynthesis regime in stars. We write out conformal Friedmann equation [4] with using meaningful conformal partial parameters, discarding insignificant contributions The right-hand side of the equation (20) includes the densities of the state of matter ( ) in accordance with their conformal weights; in the left, the prime denotes the derivative with respect to the conformal time. After introducing the dimensionless variable ≡ / 0 , the conformal Friedmann equation (20) takes the form where one root in the cubic polynomial on the right-hand side (21) is real, while the other ones are complex conjugated The invariants of the cubic polynomial are as following: where ℋ 0 is a conformal Hubble constant. The conformal Hubble parameter is related to the usual parameter as ℋ ≡ ( / 0 ) . Then we introduce a new variable by the rule The Weierstrass -function satisfies the quasi-periodicity conditions [10] ( + 2 ) = ( ) + 2 ( ), where and ′ are semi-periods of the function ℘( ). The conformal Friedmann equation (21) is integrated and we obtain the conformal age -redshift relationship in explicit form The equation written out in integral form is known in cosmology as the Hubble law. An explicit formula for conformal age of the Universe 0 can be obtained by putting in (23) = 0: The interval of the coordinate conformal distance is equal to the interval of the conformal time d = d , so we can represent (23) as the conformal distance -redshift relationship.
In conformal coordinates, the behavior of photons is exactly the same as in the where ( ) is a coordinate distance. For photons d / = −1, so we get an explicit dependence: luminosity distance -redshift relationship The effective magnitude -redshift relationship in the conformal cosmology has the form

Comparative analysis
According to the conformal cosmological model, the conformal quantities are physical observables. Pearson's criterion 2 was applied in [9] for statistical fitting of results on supernovae Ia [2]. The contribution from the component corresponding to the rigid state of matter rigid in the conformal model replaced the contribution from the Λ-term in the standard model. In the rigid state of matter, its energy density is equal to the pressure. As a result of the analysis, the conclusion was drawn: the best fitting of the conformal model was not inferior to the best fitting of the standard model [9].
The curves for the two models, according to (26) and (19), are shown in Fig. 1. The difference between the curves: effective magnitude value -redshift -predictions of models (26) and (19), is demonstrated in Fig. 2. A slight difference between the curves, within the error of observations, is manifested in the early and late stages of the evolution of the Universe [12]. In the standard cosmology, the following characteristics are introduced to describe the Hubble diagram: Hubble parameter, a deceleration and a jerk [2] ( ) ≡ + As we see, the deceleration parameter (29) changes its sign during the evolution of the Universe at the inflection point -parameter remains constant. We define analogous parameters for the conformal cosmology We calculate the conformal parameters using the conformal Friedmann equation (20). Hubble parameter changes from 1 to (Ω rigid − Ω M /2); hence, the scale factor grows with deceleration; the jerk parameter ( ) = 3Ω rigid Ω rigid + Ω M 3 > 0 changes from 3 to 3Ω rigid . Dimensionless parameters ( ) and ( ) remain positive during the evolution. The Universe does not undergo, during its evolution, the so-called jerkartifact of approach of the standard cosmological model.

Conclusions
We present exact solutions of the Friedmann equation in standard cosmology and conformal cosmology. The theoretical curves interpolating the Hubble diagram for the latest supernova data are expressed in an analytical form. The functions belong to the class of meromorphic Weierstrass functions. Both approaches describe the modern Hubble diagram with the same accuracy. We introduce conformal parameters describing the Hubble diagram. All parameters remain positive during the evolution of the Universe. 17 Сверхновые типа Ia используются как стандартные свечи в современной космологии, служат для проверки космологических моделей. Интерпретация диаграммы Хаббла на основе стандартной космологической модели привела космологов к заключению, что Вселенная заполнена в основном космической пылью и загадочной тёмной энергией.