Issue 
A&A
Volume 556, August 2013



Article Number  A13  
Number of page(s)  2  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/201322088  
Published online  17 July 2013 
Research Note
A curious relation between the flat cosmological model and the elliptic integral of the first kind
^{1} Charles University in Prague, Faculty of Mathematics and Physics, Astronomical Institute, V Holešovičkách 2, 18000 Prague 8, Czech Republic
email: meszaros@cesnet.cz
^{2} Institute of Basic Science, Natural Sciences Campus, Sungkyunkwan University, Engineering Building 2, 2066 Seoburo, Jangangu, Suwon, 440746 Gyeonggido, Korea
email: ripa.jakub@gmail.com
Received: 14 June 2013
Accepted: 16 June 2013
Context. The dependence of the luminosity distance on the redshift has a key importance in the cosmology. This dependence can well be given by standard functions for the zero cosmological constant.
Aims. The purpose of this article is to present a similar relation also for the nonzero cosmological constant, if the universe is spatially flat.
Methods. A definite integral was used.
Results. The integration ends in the elliptic integral of the first kind.
Conclusions. The result shows that no numerical integration is needed for the nonzero cosmological constant if the universe is spatially flat.
Key words: cosmology: theory
© ESO, 2013
1. Introduction
The dependence of the luminosity distance d_{L}(z) on the redshift z is a key formula in cosmology (Carroll et al. 1992). It is given by three independent cosmological parameters: by two omegaparameters Ω_{M}, Ω_{Λ}, and by the Hubble constant H_{0}. Its relation to the socalled propermotion distance is given by d_{PM}(z)(1 + z) = d_{L}(z) (Weinberg 1972).
One has (Carroll et al. 1992)
In this equation c is the speed of light in vacuum, and it holds Ω_{k} + Ω_{M} + Ω_{Λ} = 1. The notation “sinn” means the standard function sinh for Ω_{k} > 0, and sin for Ω_{k} < 0, respectively. If Ω_{k} = 0, then one simply has an integration: (2)In addition, from the physical point of view, in both equations it must be Ω_{M} > 0. (The case Ω_{M} = 0 can serve as a limit, but Ω_{M} < 0 is fully unphysical.) On the other hand, Ω_{Λ} can have both signs, but the observations of the past two decades strongly disfavour negative values (see, e.g., Perrett et al. 2012, and references therein).
In the special case of Ω_{Λ} = 0, the integral in Eq. (1) can be given by the socalled Mattigformula (Mattig 1958) for any Ω_{M} > 0. The formula can then easily be used in cosmological applications (see, e.g., Mészáros 2002). For Ω_{Λ} ≠ 0 the integral in Eqs. (1), (2) is usually solved numerically^{1}.
In this note we show that the integral on the righthand side of Eq. (2) can be solved analytically also for Ω_{Λ} ≠ 0.
2. Integration
We rewrite the righthand side of Eq.(2) into the form (3)where z ≥ 0, Ω_{M} > 0 and Ω_{M} + Ω_{Λ} = 1. Introducing the substitution x = (1 + z)^{1}, we obtain (4)where 0 < x_{0} = (1 + z_{0})^{1} ≤ 1.
In this section we consider only the case Ω_{M} < 1, i.e. Ω_{Λ} = 1 − Ω_{M} > 0. Introducing a further substitution y = (Ω_{Λ}/Ω_{M})^{1/3}x, we obtain (5)where y_{1} = (Ω_{Λ}/Ω_{M})^{1/3}x_{0} and y_{2} = (Ω_{Λ}/Ω_{M})^{1/3}. The two limits are nonnegative with y_{1} ≤ y_{2}.
We aim to find the primitive function for (6)where y ≥ 0. The following substitution helps: (7)There is a onetoone correspondence between α and y. For y = 0 one has α = 0, and for y → ∞ one has , i.e. α_{∞} = 105.54°. If y increases in the interval [0,∞), α is also increasing in the interval [0,α_{∞}). Hence, this substitution is welldefined. Conversely, one obtains (8)and (9)Using these two formulas, we curiously obtain (10)The righthand side of Eq. (10) is the function in the elliptic integral of the first kind (Gradshteyn et al. 2007) with (11)where 0 < m < 1, as it should be in an elliptic integral.
To calculate the definite integral (12)in Eq. (5) for nonnegative y_{1} ≤ y_{2}, one can write (13)After this one should use the formula with α from Eq. (10) and determine the integration limits in variable α. The substitution from Eq. (7) gives for y = 0 the value α = 0. This means
that – using α – the lower limits in both definite integrals are zeros. The upper limits from y_{1} and y_{2}, respectively, are also calculable analytically and unambiguously from Eq. (7) via the arccos function. One obtains α_{1} and α_{2}, respectively, as upper limits in the integrals. It must be α_{2} > α_{1}. One should only clarify that, of course, if it were α_{2} from the interval (π/2,α_{∞}), then the first elliptic integral itself should be given by a sum of two integrals: in one integral the limits should be 0 and π/2, and in the second one the limits should be (π − α_{2}) and π/2. Both integrals must give positive values. If it were also α_{1} > π/2, then one should proceed similarly in the second integral, too. In any case, the integral I(Ω_{M},x_{0}) is easily obtainable from standard elliptic integrals of the first kind.
3. Remarks
The integral in Eq. (5) is presented by Gradshteyn et al. (2007) (formula 3.166.22). Moreover, Carroll et al. (1992) reported that the integral of Eq. (2) can also be solved analytically. Paál et al. (1992) described similar efforts. But, on the other hand, we found nothing in the literature about the nonnumerical integration of Eq. (2) using the elliptic integrals. Therefore, the substitution given by Eqs. (7), (8) is new and original.
For the sake of completeness it should still be added that integral of Eq. (4) can be solved also for Ω_{Λ} < 0 and Ω_{M} > 1. One obtains (up to a constant) the formula d, which is also integrable (see the formula 3.166.23 of Gradshteyn et al. 2007).
4. Conclusion
We have proven that the integral on the righthand side of Eq. (2) can also be solved analytically using the elliptic integral of the first kind.
Acknowledgments
We wish to thank M. Křížek for the useful discussions and comments on the manuscript. This study was supported by the OTKA Grant K77795, by the Grant Agency of the Czech Republic Grant P209/10/0734, by the Research Program MSM0021620860 of the Ministry of Education of the Czech Republic, and by Creative Research Initiatives (RCMST) of MEST/NRF.
References
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