Development of Zeldovich's approach for cosmological distances measurement in the Friedmann Universe

We present our development of Zeldovich's ideas for the measurement of the cosmological angular diameter distance (ADD) in the Friedmann Universe. We derive the general differential equation for the ADD measurement which is valid for an open, spatially-flat and closed universe, and for any stress energy tensor. We solve the mentioned equations in terms of quadratures in a form suitable for further numerical investigations for the present universe filled by radiation, (baryonic and dark) matter and dark energy. We perform the numerical investigation in the absence of radiation, and show the strong dependence ADD on the filling of the cone of light rays (CLR). The difference of the empty and totally filled CLR may reach 600-700 Mps. for the redshift $f\simeq 3$.


Introduction
In the present article we are going to reconsider the issue about cosmological distances measurement in cosmology. Commonly used methods by astronomers are collected in the well cited review [1] and it may be considered as a good introduction in this topic. The angular diameter distance and the luminosity one are known for being the most useful in practice. These two distances are connected by the following relation [2]: where d l is the luminosity distance, d a the angular diameter distance, and f the redshift. This relation gives us feasibility to concentrate our study on the angular diameter distance. As a rule the derivation of the angular diameter distance [1] is undertaken for the homogeneous Universe [3], i.e. all matter of the Universe is distributed homogeneously by the assumption. This assumption is valid for the volumes of order about 500 Mps as the side of a cube and it does not reflect the real situation in the case of the distance measurement when the beam of light is propagating through a generally small volume.
Let us briefly define the problem of cosmological angular diameter distance (ADD) measurement. The definition of ADD which is valid in Euclidean space is usually extended to a curved space [4] with the formula Here z is the linear size of the object and φ angular size of the object (figure 1). In the Friedmann-Robertson-Walker geometry (5) we can find the linear size of a distant object using the metric (5). For example, in the spatially-flat universe, where k = 0, one can obtain where a e is the scale factor at the time of emission. Combining the expressions above it is easy to obtain (commonly used by astronomers) the equation for the angular diameter distance: To obtain the result (4) we used the FRW metric (5) corresponding to the homogeneous universe. Therefore if we suggested that our universe is not homogeneous in the cone of light rays (CLR) of ADD measurement, then we must take into account local inhomogeneities inside or close to the CLR. What will the ADD measurement be if we take into account a local inhomogeneity? As far as we know, the first man who was dealing with this question was Zeldovich (1964). He presented the solution for "the homogeneous in the mean Universe" in [5]. It is interesting to mention that in the monograph [4] Novikov and Zeldovich noted that during the Symposium in Burokan (1966), American astronomers reported that familiar ideas were declared by R. Feinman. Recently in the paper [6] it was also confirmed that R. Feinman pondered over the same problem, and suggested as well to consider a zero stress-energy tensor inside the light cone ‡.
Considering a congruence of null geodesics we will use the following terminology proposed by Penrose [7]: • The Ricci focusing is the focusing due to the gravitational effect of intrinsic mass inside the light cone; • The Weil focusing is the focusing due to inhomogeneous clumps of matter along the null geodesics path.
Let us remind the main stream of Zeldovich's original ideas represented in the papers [5], [8]. In the article [5] Zeldovich introduced "a homogeneous in the mean universe" and he analyzed the effect of the local non-uniformity of the matter-dominated spatially-flat Friedmann universe on the angular and luminosity distances measurement. It was found for the ADD under suggestion that there were negligible amount of matter inside the light cone and it was possible to neglect the gravitational effect of that matter. The method applied is the integration of numerous lensing deflections due to the intrinsic mass of a light cone.
The solutions of ADD measurement for a nonhomogeneous nonflat universe were found in the paper by Dashevskii and Zeldovich [8]. In this paper they also presented the method for describing Ricci focusing through momentums of a photon. Later on we will explain in detail this approach where we will use this method in our study. In the work by Dashevskii and Slysh [9] the analytical solution for a closed matter-dominated Universe for a partly filled light cone was found. They also derived the differential equation on the linear distance between the two rays emitted by the outer points of the object. Let us mention here that this equation does not contain the dark energy component and it may not be applied for ADD measurement now from the current observational data.
Let us note that Zeldovich's original ideas were used afterward in the series of papers by Dyer and Roeder [10], [11], [12]. Their papers are well cited. Let us make a brief comment about these papers.
In the first paper [10] they used Sachs' equations [13] and obtained the result which was found in [5]. In the second paper [12] they obtained a differential equation similar to the equation from [9] which was valid only for the Universe without dark energy, because the energy momentum tensor was set up as T µν = diag(ρ, 0, 0, 0). The solution of this equation proves the result from [9]. In the third paper [11] the result was found for the Swiss Cheese Universe.
In 1976 Weinberg [14] showed that the summation of gravitational deflection caused by individual clumps of matter is equal to the effect caused by the homogeneous distribution of the same mass. This paper reduced the interest of the community to effects of inhomogeneity to the distance measurement and It provides a strong criticism of the Swiss cheese universe model. Nonetheless, Weinberg's results are in agreement with Zeldovich ideas. Indeed the matter enclosed inside a light cone for the homogeneous in the mean Universe may be considered as a homogeneous distribution (of a small density). What is the problem with the direct application of the original solutions of Zeldovich for calculating of cosmological distances? The problem is in the fact that these solutions were obtained in the absence of dark energy ‡ As a light cone in this paper we define the cone of null geodesics congruence. and dark matter, i.e. for the Universe which contains only baryonic matter. This is controversial in our current understanding of the Universe. Therefore in the present paper we are going to extend Zeldovich's original ideas for solving this problem in a general form.
Let us mention the series of papers by Alcock and Anderson where they discuss the problem of distance measurement. In the first paper [15] they emphasized on the importance of correct cosmological distance measurement for calculating the Hubble constant through the gravitational lensing. In the second paper [16] they presented an original method for the distance measurement; the so called "effective distance". The problem with their approach is that all our cosmological and astronomical theories were developed for angular diameter (luminosity) distance. That's why we are not ready to apply "effective distance" methods and we are going to keep commonly accepted definitions of fundamental concepts.
The ideas of Dyer and Roeder were developed by Kantowski [17], who constructed an analysis of differential equations and their solutions for the Swiss cheese universe. In the paper [18] the solutions for various inhomogeneous cosmological models were found. We would like to mention the impact of Schneider in development of Dyer and Roeder ideas in the following papers [19], [20].
Why are we going to reconsider previous results in the distance measurements? From our point of view, the main problem of these results is that they were obtained for an inhomogeneous Universe. Our understanding is that this assumption is too strong, because it is already proved that globally Universe mass distribution is homogeneous. Thus we assume that the Universe evolves like a homogeneous universe, but if we measure the distance to an object in the space the effect of Ricci focusing becomes weaker. The reason is in fact that the density inside the light cone is smaller than the critical density of the Universe. Summing up the discussion above we can state that the Zeldovich model of the Universe (the homogeneous in the mean Universe) is very suitable for describing observations in the present Universe.
Let us explain our last thesis in detail. First of all, let us discuss the method of Dyer and Roeder. They are starting from Sachs' equations [13], which are a version of the Raychaudhuri equation for null geodesics [21]. It should be noted that the Raychaudhuri equation is more general than the Friedmann equation [22] and it already includes the effects of gravitational lensing. Starting from these mentioned works by Dyer and Roeder, physicists are using (proposed by Penrose [7]) the Weil and Ricci focusing for calculating ADD.
For the Friedmann universe the Weil tensor is equal to zero (W iklm = 0) [22]. In the case of small clumps of the Swiss cheese universe Dyer and Roeder showed [23] that effects of the Weil focusing on every clump can be approximated by the Ricci focusing. This proposition proves Weinberg's results [14]. If we will follow by Zeldovich's ideas for the ADD measurement, then our universe remains Friedmann universe and there are no concentrated clumps of matter on the line of sight. Thus the effect of the Weil focusing can be neglected. To take into account the concentrated clumps of matter we suggest one to use the well developed gravitational lensing theory. How will we hold the value of the Ricci focusing in cases of the ADD measurement? We will follow the Zeldovich and Dashevskii ideas which were published in [8], i.e. we use the fraction of perpendicular and longitudinal components of the photon.
It should be noted that the interest of the scientific community with regards to the problem of distance measurement in an inhomogeneous Universe does not end. For example, in the paper by Bolejko [24] the author once again emphasises the importance of the effects of inhomogeneities in cosmology. We should also mention that our idea of developing Zeldovich's original ideas for cosmological distance measurement is not new. In the paper by Kayser et al [25] the differential equation presented in the paper [9] was used with the aim to find the cosmological distance in the Universe filled with dark energy. Unfortunately, as it was mentioned before, the differential equation obtained by Dashevskii and Slysh for the Universe did not account for dark energy and could not be applied for ΛCDM model. It is easy to check by using standard cosmological parameters and assume a filled light cone that the equations obtained by Kayser give wrong results in this case. In our paper we had obtained general differential equations for ADD measurement, which could be used for finding analytical solutions for empty and filled light cones. We present the numerical solution for a partly filled light cone in the end of this paper.

ADD in the Friedmann universe: general approach
Let us start from the definition of the angular diameter distance (2) which is generally exploited in astronomy. As it was mentioned berfore it is the fraction of linear and angular sizes of the object. In astronomy we generally deal with spherical bodies, which is to say, that we can simpify our task by considering the diameter of the object (z in (2)) instead of it's area. As we are discussing a congruence from a distance object of radial null geodesics which are crossed at a point of observation (center of our spherical coordinate system), the angle φ between the boundary points of a diameter z remains constant by its defintion (2).
Let us choose a spherical coordinate system [t, r, θ, φ] with an observer placed in the center of it. To calculate the changing of the linear size z during the travel of the light beems, we have to account for the following effects: (i) Expansion of the universe.
Let us remind ourselves that in the Friedmann Universe all matter is involved with the Hubble current and in the comoving coordinates (with the Earth observer in the center) the spatial coordinates of all particles are not changingẋ i = 0 [26]. Therefore we have to consider not the object itself, but the photons which are moving along null radial geodesics. In this context z will be the distance between two light beams from the end points of the object at the time of emmision. Evidently z will be variable.
Our next task is to derive the differential equation on z which allows us to take into account the effects arising when measuring distances in Friedmann universe.
For the Friedmann-Robertson-Walker (FRW) metric we will use two forms where f (r) = sin r, r, sinh r or k=1,0,-1 for a closed, spatially-flat, open universe respectively. For the sake of simplicity we choose θ = π 2 . Let us analyze, with radial null geodesics dφ = 0, then the metric (5) is reduced to Now we define new coordinates where η = dt a is a conformal time. Then we have u = const for ongoing geodesics (in relation to the observer), and v = const for ingoing ones. Since we are interested in ingoing geodesics we choose v = const or in terms of cosmic time t The co-vector field should be a null one and has to satisfy the geodesics equation. From the metric (5) it is easy to find k in α = (− 1 a , −1, 0, 0). Then rising the index α, we obtain It is easy to check that k α k α = 0. Now we will find the affine parameter for k α in from the geodesic equation By substituting (10) to (11) we obtain 1 a 2 From the equation (12) we can find the relation for the affine parameter Using the obtained relation (13) we can calculate the expansion Θ [21].
For this purpose we substitute (10) to (14). The result is We can also use the geometrical interpretation of the expansion Θ It's easy to see, a congruence's cross-sectional area in the chosen metric (5) is where z being the diameter of the cross-section. Our task now is to represent Θ through z -the distance between two neighbour beams (one from the beginning of the object and another from the end of the object). Further we will interpret z as the diameter of the cross-section if we assume rotational symmetry in the FRW metric. For the sake of simplicity let the first light beam will propagate along the axis φ = 0. As we noted in the begining of this section, first of all we want to find a differential equation for z in the Friedmann universe. From the definition of the ADD in the curved space-time (for FRW metric (5) d a = af (r)) we have the distance between the two rays z = aφf (r) (18) As it was mentioned earlier we chose the equatorial plane for simplicity (the coordinate θ = π 2 ). Substituting (18) and (17) into (16) one can find az and using (15) For the derivative along the path, using (8), we find To exclude f ′ (r) f (r) in (19) we should determineż by differentiating (18) Finally we obtain the desirable expression Then by substituting this expression into (19) yields the equation Taking into account that f ′′ (r) f (r) = k, we obtain a generalization of the equlation (20) for an arbitrary curvaturez The initial conditions we represent in the following way The initial conditions are derived from the fact that the distance at the point of observation between two beams from the object is equal to zero and the change in the velocity of mentioned distance equals to the angle of observation by the definition, and the fact that c = 1.
Our following task is to derive the expression for Ricci focusing in the Friedmann Universe. To this end let us approach the work in [8], and restore the result for the sake of completeness without essential changes.
Let us define the angle ψ between the rays by the formula where (18) and (8) were used. The first ray propagates along the axis of the coordinates. Let us write the angle ψ as the ratio of the perpendicular component of the momentum of the second quantum q to its longitudinal component h Since |q| ≤ h the total momentum P is proportional to the frequency of the quantum, and it is equal to h. From the redshift formula the total momentum P is given by where K = ω(t 0 )a(t 0 ) is a constant. Form (25) we obtain the expression for the transverse component q in the case of propagation in the homogeneous universe while for the derivative of this component along the path, using (8), we find (for closed universe, f (r) = sin r) that or for arbitrary curvature k = {−1, 0, 1} Thus we can rewrite (21) using (27) z Let us resume ourselves that we derived the general differential equation for the Friedmann universe where the part, responsible for Ricci focusing, is represented by the separate term. We want to attract your attention on the difference of the resulting differential equation from that obtained by Dashevsky and Slysh [9]. The result obtained by them is valid for the special case -a Friedmann Universe with Ω 0 = Ω M = 1. Our equation (28) is valid for any type of Friedmann universe: open, spatially-flat or closed universe, and for any expression and form of the stress energy tensor. Thus our equation is valid even for modified theories of gravitation if the Friedmann Universe is being considered.

Method of ADD calculation
Let us investigate "homogeneous" and "homogeneous in the mean universe". We will study the Friedmann Universe filled by a perfect fluid (matter, radiation) with vacuum energy. We will consider the method of the ADD calculation for such a universe.
Let us start with the homogeneous universe. The dynamics of the universe can be described by the Einstein-Friedmann equationṡ Using these equations in (27) we can obtain the important relatioṅ Thus the equation (28) transforms tö z −ȧ aż + 4πGz(p + ρ) = 0 (32) We can find the solutions for (32) independent of the form of the scale factor a(t) and compatible with the initial conditions (22) One can easily check these solutions by direct substitution of the solutions (33) into (32).
As an example let us check the case for k = 1. So we choose the solution One can find the first derivativė and the second derivativë Inserting the derivatives above into (32) one can obtain Using (29) and (30) one can make sure that (37) is the identity. The same procedure can be performed for k = 0 and k = −1. Thus (33) is the solution for the homogeneous universe, and the angular diameter distance can be calculated with the formulae: This equation can be solved for matter (p = 0), radiation (p = ρ/3), and vacuum energy (p = −ρ Λ = const). The expression for the mixture of them is where the present energy densities in the vacuum, non-relativistic matter, and relativistic matter are, respectively .
(41) § It shoud not be confused with f(r) used in the section 2 Using (41), equation (29) can be represented in the form ,f -redshift. Thus following (38) we can present the resulting formulae for ADD in the form: Let us mention that the result (43) is in agreement with widely used formulae represented in [1]. We should mention, that (38) is valid for any modified gravitational theory, where the "Einstein-Friedmann" equations can be written in the form a a = ξ(t) (44) Here ξ(t) and ψ(t) -arbitrary functions. Let us turn our attention to the investigation of "homogeneous in the mean universe". The description of the universe homogeneous in the mean contains two propositions: (1) we have homogeneous distribution of matter along the whole universe; (2) the interaction between matter and light rays is negligible in standard observations.
The proposition(2) means, that for this type of the universe we haveq = 0 and equation (28)  The solution of (46) can be presented in the form compatible with the initial conditions (22) The same expression can be found fromq = 0 directly. One can check (47) by substituting the derivativesż into (46). Inserting these derivatives into (46) one will obtain the identity. Thus the expression for the angular diameter distance for any curvature in the homogeneous in the mean universe is Note that the result is valid for any modified gravity theory when the Friedmann Universe is under work.
If we consider the ΛCDM model, the ADD formula (49), with the help of (42), transforms to The obtained expression for ADD coincides with the solution found by Zeldovich and Dashevsky [5], [8], when Ω Λ = Ω R = 0.

Conclusions
We extended Zeldovich's ideas for ADD measurements in two directions. Firstly, the generalization of the ADD formula from a closed to spatially-flat and open Friedmann Universes. Secondly, we proposed not only empty and filled cone of light rays (CLR), but also the partly filled CLR. These main results are represented by the differential equation (28) which allows us to separate effects of expansion of the homogeneous universe and Ricci focusing for congruence of radial null geodesics. It was shown, that this equation consists of a classical solution for the homogeneous Friedmann Universe and, as a special case, it reduces to equations obtained by Zeldovich et al . The solution of (28) was presented in quadratures in the form suitable for further numerical analysis.
The numerical solution for the partly filled CLR was obtained. From this solution (figure 2) it became evident that the standard ADD measurement (in the Universe filled by (dark and baryonic) matter and dark energy) may be applied for an object with redshift f not more than 0.5. For objects with f > 0.5 the influence of CLR filling became crucial. For example, on the redshift f = 3 the ADD may have the difference of about 600-700 Mps. for empty and totally filled CLR.
These results can help astronomers to improve their calculations, where ADD is involved. We plan to present extension of this method to gravitational lensing, supernova data analysis in the next publication.