At present the Universe is considered a general relativistic Friedmann space-time with flat spatial sections, containing more than 70% dark energy and at about 25% of dark matter. Dark energy could be simply a cosmological constant Λ, or quintessence or something entirely different. There is no widely accepted explanations for the nature of any of the dark matter or dark energy (even the existence of the cosmological constant remains unexplained).

An alternative to introducing dark matter would be to modify the law of gravitation, like in MOND 123 and its relativistic generalization 45. These theories are compatible with the Large scale structure of the Universe 678. However in spite of the successes, certain problems were signaled on smaller scales 910111213.

Quite remarkably, supernova data, which in the traditional interpretation yield to the existence of dark energy, can be explained by certain f(R) 1415 or inverse curvature gravity models 16. However the parameter range, in which the latter is in goood agrement with the supernova data, also presents stability problems 1718.

Modifications of the gravitational interaction could also occur by enriching the space-time with extra dimensions. Originally pioneered by Kaluza and Klein, such theories contained compact extra dimensions. The so-called brane-world models, motivated by string/M-theory, containing our observable 4-dimensional universe (the brane) as a hypersurface, were introduced in 192021 and 22, the latter model allowing for a non-compact extra dimension.

The curved generalizations of the model presented in 22 have evolved into a 5-dimensional alternative to general relativity, in which gravity has more degrees of freedom. In contrast with standard model fields, these evolve in the whole 5-dimensional bulk. In this generalized Randall-Sundrum type II (RS) theory, the brane has a tension λ and gravitational dynamics is governed by the 5-dimensional Einstein equation. Its projections to our observable 4-dimensional universe (the brane) are the twice contracted Gauss equation, the Codazzi equation and an effective Einstein equation, the latter being obtained by employing the junction conditions across the brane 23. The effective Einstein equation (for the case of symmetric embedding and no other contribution to the bulk-energy-momentum than a bulk cosmological constant) was first given in a covariant form in 24. Supplementing this by the pull-back to the brane of the bulk energy momentum tensor Π˜ab MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqHGoaugaacamaaBaaaleaacqWGHbqycqWGIbGyaeqaaaaa@30FD@, which is

P a b = 2 κ ˜ 2 3 ( g a c g b d Π ˜ c d ) T F MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqGqbaudaWgaaWcbaGaemyyaeMaemOyaigabeaakiabg2da9maalaaabaGaeGOmaidcciGaf8NUdSMbaGaadaahaaWcbeqaaiabikdaYaaaaOqaaiabiodaZaaadaqadaqaaiabdEgaNnaaDaaaleaacqWGHbqyaeaacqWGJbWyaaGccqWGNbWzdaqhaaWcbaGaemOyaigabaGaemizaqgaaOGafuiOdaLbaGaadaWgaaWcbaGaem4yamMaemizaqgabeaaaOGaayjkaiaawMcaamaaCaaaleqabaGaemivaqLaemOrayeaaaaa@4741@

(with κ˜2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWF6oWAgaacamaaCaaaleqabaGaeGOmaidaaaaa@2F93@ the bulk coupling constant and g_ab the induced metric on the brane) the effective Einstein equation reads 23:

G a b = − Λ g a b + κ 2 T a b + κ ˜ 4 S a b − ℰ a b + P a b . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGhbWrdaWgaaWcbaGaemyyaeMaemOyaigabeaakiabg2da9iabgkHiTiabfU5amjabdEgaNnaaBaaaleaacqWGHbqycqWGIbGyaeqaaOGaey4kaSccciGae8NUdS2aaWbaaSqabeaacqaIYaGmaaGccqWGubavdaWgaaWcbaGaemyyaeMaemOyaigabeaakiabgUcaRiqb=P7aRzaaiaWaaWbaaSqabeaacqaI0aanaaGccqWGtbWudaWgaaWcbaGaemyyaeMaemOyaigabeaakiabgkHiTmrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab+btifnaaBaaaleaacqWGHbqycqWGIbGyaeqaaOGaey4kaSIaeeiuaa1aaSbaaSqaaiabdggaHjabdkgaIbqabaGccqGGUaGlaaa@5BF6@

Here κ² is the brane coupling constant, related to the bulk coupling constant and the brane tension λ as 6κ² = κ˜4 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWF6oWAgaacamaaCaaaleqabaGaeGinaqdaaaaa@2F97@λ, and

Λ = κ 2 2 λ − κ ˜ 2 2 n c n d Π ˜ c d MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqHBoatcqGH9aqpdaWcaaqaaGGaciab=P7aRnaaCaaaleqabaGaeGOmaidaaaGcbaGaeGOmaidaaiab=T7aSjabgkHiTmaalaaabaGaf8NUdSMbaGaadaahaaWcbeqaaiabikdaYaaaaOqaaiabikdaYaaacqWGUbGBdaahaaWcbeqaaiabdogaJbaakiabd6gaUnaaCaaaleqabaGaemizaqgaaOGafuiOdaLbaGaadaWgaaWcbaGaem4yamMaemizaqgabeaaaaa@43BF@

represents a cosmological "constant" which possibly varies due to the normal projection of the bulk energy-momentum tensor (this includes the contribution – Λ˜gab MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqHBoatgaacaiabdEgaNnaaBaaaleaacqWGHbqycqWGIbGyaeqaaaaa@324B@ due to the bulk cosmological constant Λ˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqHBoatgaacaaaa@2E30@). The source term S_ab is quadratic in the brane energy-momentum tensor T_ab:

S a b = 1 4 [ − T a c T b c + 1 3 T T a b − g a b 2 ( − T c d T c d + 1 3 T 2 ) ] , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGtbWudaWgaaWcbaGaemyyaeMaemOyaigabeaakiabg2da9maalaaabaGaeGymaedabaGaeGinaqdaamaadmaabaGaeyOeI0Iaemivaq1aaSbaaSqaaiabdggaHjabdogaJbqabaGccqWGubavdaqhaaWcbaGaemOyaigabaGaem4yamgaaOGaey4kaSYaaSaaaeaacqaIXaqmaeaacqaIZaWmaaGaemivaqLaemivaq1aaSbaaSqaaiabdggaHjabdkgaIbqabaGccqGHsisldaWcaaqaaiabdEgaNnaaBaaaleaacqWGHbqycqWGIbGyaeqaaaGcbaGaeGOmaidaamaabmaabaGaeyOeI0Iaemivaq1aaSbaaSqaaiabdogaJjabdsgaKbqabaGccqWGubavdaahaaWcbeqaaiabdogaJjabdsgaKbaakiabgUcaRmaalaaabaGaeGymaedabaGaeG4mamdaaiabdsfaunaaCaaaleqabaGaeGOmaidaaaGccaGLOaGaayzkaaaacaGLBbGaayzxaaGaeiilaWcaaa@5D38@

and ℰab MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFWesrdaWgaaWcbaGaemyyaeMaemOyaigabeaaaaa@3A48@ is the electric part of the bulk Weyl tensor C˜abcd MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGdbWqgaacamaaBaaaleaacqWGHbqycqWGIbGycqWGJbWycqWGKbazaeqaaaaa@332E@, given as

ℰ a c = C ˜ a b c d n b n d . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFWesrdaWgaaWcbaGaemyyaeMaem4yamgabeaakiabg2da9iqbdoeadzaaiaWaaSbaaSqaaiabdggaHjabdkgaIjabdogaJjabdsgaKbqabaGccqWGUbGBdaahaaWcbeqaaiabdkgaIbaakiabd6gaUnaaCaaaleqabaGaemizaqgaaOGaeiOla4caaa@48A0@

In a cosmological context and suppressing any energy exchange between the brane and the bulk, this latter term generates the so-called dark radiation. Otherwise it can be called a Weyl fluid.

A review of many aspects related to the theories described by the effective Einstein equation (2) can be found in 25. Both early cosmology 26 and gravitational collapse 272829303132 are essentially modified in these theories. There is also possible to replace dark matter with geometric effects in the interpretation of galactic rotation curves, weak lensing and galaxy cluster dynamics 33343536.

The possible modifications of gravitational dynamics are even more versatile in the so-called induced gravity models. These can be regarded as brane-world models enhanced with the first quantum-correction arising from the interaction of the brane matter with bulk gravity. The induced gravity correction couples to the 5-dimensional Einstein-Hilbert action with the coupling constant γκ˜2/κ2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFZoWzcuWF6oWAgaacamaaCaaaleqabaGaeGOmaidaaOGaei4la8Iae8NUdS2aaWbaaSqabeaacqaIYaGmaaaaaa@34F1@. The simplest of such models, the DGP model was introduced in 37. This model however suffers from linear instabilities (ghost modes in the perturbations), as shown for de Sitter branes 383940. The ghost modes withstand even the introduction of a second brane 41. Generalizations of the DGP model are discussed covariantly in 42 and 43 when the embedding is symmetric, and in 44 when it is asymmetric. In these models the role of the effective Einstein equation (2) is taken by a more complicated equation (see for example Eq. (29) of 44), which contains the square of the Einstein tensor G_ab. This implies that in certain sense the degree of nonlinearity of the theory is squared. In a cosmological setup the square root of this equation can be taken, leading to a set of modified Friedmann and Raychaudhuri equations, which however contain a sign ambiguity ε = ±1 due to the involved square root. These are called the BRANE1 [DGP(-)] branch for ε = -1 and BRANE2 [DGP(+)] for ε = 1 in the terminology of 42 [or 45, respectively]. Both the original Randall-Sundrum type II model and the DGP model are contained as special subcases. Notably, the BRANE2 branch contains cosmological models which self-accelerate at late-times. We give in Fig 1 a diagram containing a classification of these theories and how they emerge as different limits from each other.

In this paper we discuss analytically the luminosity distance – redshift relation in various generalized Randall-Sundrum type II brane-world models described by Eq. (2). Our analytical approach can enhance the confrontation of these models with current and most notably, with future supernova observations. We note that recently analytical results have been given in Ref. 46 for a wide class of phantom Friedmann cosmologies too, in terms of elementary and Weierstrass elliptic functions.

In section 2 we review the notion of luminosity distance, its relation with the redshift and how these can be measured independently. This section was included mainly for didactical purposes.

In section 3 we review the modification of this relation in the Randall-Sundrum type II brane-world scenario. These include the introduction of the parameters Ω_λ and Ω_d which can be traced back to the source terms S_ab and ℰab MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFWesrdaWgaaWcbaGaemyyaeMaemOyaigabeaaaaa@3A48@ of the modified Einstein equation (2). The other cosmological parameters are Ω_ρ, representing (baryonic and dark) matter and Ω_Λ. We do not include bulk sources in the analysis, with the notable exception of a bulk cosmological constant.

Section 4 contains the derivation of the analytic expression for the luminosity distance – redshift relation for the brane-worlds which are closest to the original Randall-Sundrum scenario 22, thus with no cosmological constant (Randall-Sundrum fine-tuning). The generic expression (35) of the luminosity distance derived here is given in terms of elementary functions and elliptic integrals of the first and second kind. From this most generic case we take the subsequent limits: Ω_d = 0 (subsection 4.2), Ω_λ = 0 (subsection 4.3); and both Ω_d = Ω_λ = 0, this being the general relativistic Einstein-de Sitter case (subsection 4.4).

Such models however could not allow for late-time acceleration, therefore in section 5 we discuss the luminosity distance – redshift relation for brane-worlds with Λ. First we present in subsection 5.1 a class of models, for which the luminosity distance can be given in terms of elementary functions alone. These models are characterized by an extremely low value of the brane tension, thus are in conflict with various constraints on brane-world models.

Next, in subsection 5.2 we discuss brane-worlds for which the brane-characteristic contributions Ω_λ and Ω_d represent small perturbations. This is a good assumption as observational evidences suggest that general relativity is a sufficiently accurate theory of the universe, and as such the deviations from it could not be very high, at least at late-times. We give analytical expressions in terms of both elementary functions and elliptic integrals of the first and second kind for the luminosity distance, to first order accuracy in the chosen small parameters of the model. Some of the most lengthy computations needed in order to achieve the result are presented in the Appendix.

Section 6 contains the concluding remarks.

Throughout the paper c = 1 was employed.

The Friedmann-Lemaître-Robertson-Walker (FLRW) metric

d s F L R W 2 = − d τ 2 + a 2 ( τ ) [ d r 2 1 − k r 2 + r 2 ( d θ 2 + sin ⁡ 2 θ d ϕ 2 ) ] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGKbazcqWGZbWCdaqhaaWcbaGaemOrayKaemitaWKaemOuaiLaem4vaCfabaGaeGOmaidaaOGaeyypa0JaeyOeI0IaemizaqgcciGae8hXdq3aaWbaaSqabeaacqaIYaGmaaGccqGHRaWkcqWGHbqydaahaaWcbeqaaiabikdaYaaakiabcIcaOiab=r8a0jabcMcaPmaadmaabaWaaSaaaeaacqWGKbazcqWGYbGCdaahaaWcbeqaaiabikdaYaaaaOqaaiabigdaXiabgkHiTiabdUgaRjabdkhaYnaaCaaaleqabaGaeGOmaidaaaaakiabgUcaRiabdkhaYnaaCaaaleqabaGaeGOmaidaaOGaeiikaGIaemizaqMae8hUde3aaWbaaSqabeaacqaIYaGmaaGccqGHRaWkcyGGZbWCcqGGPbqAcqGGUbGBdaahaaWcbeqaaiabikdaYaaakiab=H7aXjabdsgaKjab=v9aQnaaCaaaleqabaGaeGOmaidaaOGaeiykaKcacaGLBbGaayzxaaaaaa@636B@

describes a homogeneous and isotropic universe. Here τ is cosmological time, (r, θ, ϕ) are comoving coordinates, a is the scale factor and k = 0, ±1 the curvature index. The proper radial distance is defined as ar. A useful alternative form of the FLRW metric is

d s F L R W 2 = − d τ 2 + a 2 ( τ ) [ d χ 2 + r 2 ( χ ; k ) ( d θ 2 + sin ⁡ 2 θ d ϕ 2 ) ] , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGKbazcqWGZbWCdaqhaaWcbaGaemOrayKaemitaWKaemOuaiLaem4vaCfabaGaeGOmaidaaOGaeyypa0JaeyOeI0IaemizaqgcciGae8hXdq3aaWbaaSqabeaacqaIYaGmaaGccqGHRaWkcqWGHbqydaahaaWcbeqaaiabikdaYaaakiabcIcaOiab=r8a0jabcMcaPiabcUfaBjabdsgaKjab=D8aJnaaCaaaleqabaGaeGOmaidaaOGaey4kaSIaemOCai3aaWbaaSqabeaacqaIYaGmaaGccqGGOaakcqWFhpWycqGG7aWocqWGRbWAcqGGPaqkcqGGOaakcqWGKbazcqWF4oqCdaahaaWcbeqaaiabikdaYaaakiabgUcaRiGbcohaZjabcMgaPjabc6gaUnaaCaaaleqabaGaeGOmaidaaOGae8hUdeNaemizaqMae8x1dO2aaWbaaSqabeaacqaIYaGmaaGccqGGPaqkcqGGDbqxcqGGSaalaaa@64FD@

with

r = r ( χ ; k ) = { sin ⁡ χ , k = 1 , χ , k = 0 , sinh ⁡ χ , k = − 1. , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGYbGCcqGH9aqpcqWGYbGCcqGGOaakiiGacqWFhpWycqGG7aWocqWGRbWAcqGGPaqkcqGH9aqpdaGabeqaauaabeqadmaaaeaacyGGZbWCcqGGPbqAcqGGUbGBcqWFhpWyaeaacqGGSaalaeaacqWGRbWAcqGH9aqpcqaIXaqmcqGGSaalaeaacqWFhpWyaeaacqGGSaalaeaacqWGRbWAcqGH9aqpcqaIWaamcqGGSaalaeaacyGGZbWCcqGGPbqAcqGGUbGBcqGGObaAcqWFhpWyaeaacqGGSaalaeaacqWGRbWAcqGH9aqpcqGHsislcqaIXaqmcqGGUaGlaaaacaGL7baacqGGSaalaaa@5869@

χ being an other comoving radial coordinate.

If a photon stream emitted by an astrophysical light source travel without collisions, the number of photons dN_γ from a comoving elementary volume of the 6-dimensional phase space (x→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWG4baEgaWcaaaa@2E37@, p→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGWbaCgaWcaaaa@2E27@) is conserved 47. Thus the phase space density

f ( t , x → , p → ) = d N d 3 x → d 3 p → = d N ω 2 d τ   d A   d ω   d Ω MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGMbGzcqGGOaakcqWG0baDcqGGSaalcuWG4baEgaWcaiabcYcaSiqbdchaWzaalaGaeiykaKIaeyypa0ZaaSaaaeaacqWGKbazcqWGobGtaeaacqWGKbazdaahaaWcbeqaaiabiodaZaaakiqbdIha4zaalaGaemizaq2aaWbaaSqabeaacqaIZaWmaaGccuWGWbaCgaWcaaaacqGH9aqpdaWcaaqaaiabdsgaKjabd6eaobqaaGGaciab=L8a3naaCaaaleqabaGaeGOmaidaaOGaemizaqMae8hXdqNaeeiiaaIaemizaqMaemyqaeKaeeiiaaIaemizaqMae8xYdCNaeeiiaaIaemizaqMaeuyQdCfaaaaa@55B7@

of a photon stream is constant in time. Here ω denotes the frequency of the photons, dA and dΩ stand for the elementary area normal to the direction of propagation and for the elementary solid angle around the direction of propagation, respectively (see Fig 2). Eq. (9) holds true for any kind of cosmological evolution, provided d³x→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWG4baEgaWcaaaa@2E37@ ∝ dτdA and d³p→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGWbaCgaWcaaaa@2E27@ ∝ ω²dωdΩ are valid for the photons 47. The luminosity of the source is MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqedmvETj2BSbqee0evGueE0jxyaibaieYdOi=BI8qipeYdI8qiW7rqqrFfpeea0xe9LqFf0xc9q8qqaqFn0dXdHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOaa@2D2A@ = dE_em/dt_em (total energy produced in unit time; the suffix em refers to emission).

A telescope detects the photon flux ℱ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFXeIraaa@3786@ = dE_rec/dτ_rec/A_M (the suffix rec refers to reception). This is the energy detected during unit time on the telescope mirror surface A_M. (The surface A_M is understood to be perpendicular to the incident light stream.)

From their definition, one can easily find a relation between ℱ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFXeIraaa@3786@ and MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqedmvETj2BSbqee0evGueE0jxyaibaieYdOi=BI8qipeYdI8qiW7rqqrFfpeea0xe9LqFf0xc9q8qqaqFn0dXdHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOaa@2D2A@:

ℱ A M ℒ = d E r e c / d τ r e c d E e m / d τ e m . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=ftigjabdgeabnaaBaaaleaacqWGnbqtaeqaaaGcbaGae8NeHWeaaiabg2da9maalaaabaGaemizaqMaemyrau0aaSbaaSqaaiabdkhaYjabdwgaLjabdogaJbqabaGccqGGVaWlcqWGKbaziiGacqGFepaDdaWgaaWcbaGaemOCaiNaemyzauMaem4yamgabeaaaOqaaiabdsgaKjabdweafnaaBaaaleaacqWGLbqzcqWGTbqBaeqaaOGaei4la8IaemizaqMae4hXdq3aaSbaaSqaaiabdwgaLjabd2gaTbqabaaaaOGaeiOla4caaa@581A@

As the energy of the photon stream in the comoving elementary phase space volume is dE = ħω dN, from Eq. (9) we find

d E r e c d E e m = A M A t o t ( ω r e c ω e m ) 3 d ω r e c d ω e m d A r e c d A e m d τ r e c d τ e m . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabdsgaKjabdweafnaaBaaaleaacqWGYbGCcqWGLbqzcqWGJbWyaeqaaaGcbaGaemizaqMaemyrau0aaSbaaSqaaiabdwgaLjabd2gaTbqabaaaaOGaeyypa0ZaaSaaaeaacqWGbbqqdaWgaaWcbaGaemyta0eabeaaaOqaaiabdgeabnaaBaaaleaacqWG0baDcqWGVbWBcqWG0baDaeqaaaaakmaabmaabaWaaSaaaeaaiiGacqWFjpWDdaWgaaWcbaGaemOCaiNaemyzauMaem4yamgabeaaaOqaaiab=L8a3naaBaaaleaacqWGLbqzcqWGTbqBaeqaaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaeG4mamdaaOWaaSaaaeaacqWGKbazcqWFjpWDdaWgaaWcbaGaemOCaiNaemyzauMaem4yamgabeaaaOqaaiabdsgaKjab=L8a3naaBaaaleaacqWGLbqzcqWGTbqBaeqaaaaakmaalaaabaGaemizaqMaemyqae0aaSbaaSqaaiabdkhaYjabdwgaLjabdogaJbqabaaakeaacqWGKbazcqWGbbqqdaWgaaWcbaGaemyzauMaemyBa0gabeaaaaGcdaWcaaqaaiabdsgaKjab=r8a0naaBaaaleaacqWGYbGCcqWGLbqzcqWGJbWyaeqaaaGcbaGaemizaqMae8hXdq3aaSbaaSqaaiabdwgaLjabd2gaTbqabaaaaOGaeiOla4caaa@76FD@

Here we have used that from the isotropy of the FLRW universe dΩ_rec = dΩ_em and we integrate the first to the solid angle encompassing the mirror surface, the second to the whole solid angle (cf. the definitions of E_rec, E_em). In Eq. (11) A_tot represents the proper area of a sphere centered in the light source and containing the reception point on its surface, at the time of reception.

Due to cosmological evolution the elementary area dA changes as a² and the frequency of the light is redshifted during cosmic expansion, ω ∝ 1/a 47. In the cosmological evolution of the comoving elementary phase space volume element dω changes accordingly: dω ∝ 1/a. Therefore

ℱ ℒ = 1 A t o t ( a a 0 ) 2 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=ftigbqaaiab=jrimbaacqGH9aqpdaWcaaqaaiabigdaXaqaaiabdgeabnaaBaaaleaacqWG0baDcqWGVbWBcqWG0baDaeqaaaaakmaabmaabaWaaSaaaeaacqWGHbqyaeaacqWGHbqydaWgaaWcbaGaeGimaadabeaaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiabikdaYaaakiabcYcaSaaa@4786@

where a₀ is the present value of the scale factor, and a is understood to be the scale factor at emission time. In the FLRW universe the proper area of a sphere with comoving radius r_em is Atot=4πa02rem2, MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGbbqqdaWgaaWcbaGaemiDaqNaem4Ba8MaemiDaqhabeaakiabg2da9iabisda0GGaciab=b8aWjabdggaHnaaDaaaleaacqaIWaamaeaacqaIYaGmaaGccqWGYbGCdaqhaaWcbaGaemyzauMaemyBa0gabaGaeGOmaidaaOGaeiilaWcaaa@3F84@ and the redshift z is defined as

1 + z = a 0 a e m . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqaIXaqmcqGHRaWkcqWG6bGEcqGH9aqpdaWcaaqaaiabdggaHnaaBaaaleaacqaIWaamaeqaaaGcbaGaemyyae2aaSbaaSqaaiabdwgaLjabd2gaTbqabaaaaOGaeiOla4caaa@389B@

The luminosity distance d_L is defined as in Euclidean geometry:

d L ( z ) : = ( ℒ 4 π ℱ ) 1 / 2 = a 0 r e m ( 1 + z ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGKbazdaWgaaWcbaGaemitaWeabeaakiabcIcaOiabdQha6jabcMcaPiabcQda6iabg2da9maabmaabaWaaSaaaeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFsectaeaacqaI0aaniiGacqGFapaCcqWFXeIraaaacaGLOaGaayzkaaWaaWbaaSqabeaacqaIXaqmcqGGVaWlcqaIYaGmaaGccqGH9aqpcqWGHbqydaWgaaWcbaGaeGimaadabeaakiabdkhaYnaaBaaaleaacqWGLbqzcqWGTbqBaeqaaOGaeiikaGIaeGymaeJaey4kaSIaemOEaONaeiykaKIaeiOla4caaa@5567@

This definition is rigorous as long as we are dealing with the (homogeneous and isotropic) FLRW universe (irrespective of the value of the curvature index k) and the radius of a sphere is measured in the proper distance ra (the FLRW metric (6) guarantees that the surface of a sphere with radius ra is 4πa²r²).

According to Eq. (7) the comoving coordinate r_em can be written in terms of an other radial comoving coordinate χ_em (representing the location of the source):

d_L (z) ≡ a₀ (1 + z) MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqedmvETj2BSbqee0evGueE0jxyaibaieYdOi=BI8qipeYdI8qiW7rqqrFfpeea0xe9LqFf0xc9q8qqaqFn0dXdHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOaa@2D2A@ (χ_em; k).

Disregarding possible deflections by perturbations of the FLRW universe, a light ray follows radial null geodesics of the FLRW metric, characterized by dχ = dτ /a(τ) = da/a²H. Here H = a˙ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGHbqygaGaaaaa@2E00@/a is the Hubble parameter. Then

χ e m = χ ( a e m ) = ∫ a e m a 0 d a a a ˙ = ∫ a e m a 0 d a a 2 H ( a ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFhpWydaWgaaWcbaGaemyzauMaemyBa0gabeaakiabg2da9iab=D8aJjabcIcaOiabdggaHnaaBaaaleaacqWGLbqzcqWGTbqBaeqaaOGaeiykaKIaeyypa0Zaa8qmaeaadaWcaaqaaiabdsgaKjabdggaHbqaaiabdggaHjqbdggaHzaacaaaaaWcbaGaemyyae2aaSbaaWqaaiabdwgaLjabd2gaTbqabaaaleaacqWGHbqydaWgaaadbaGaeGimaadabeaaa0Gaey4kIipakiabg2da9maapedabaWaaSaaaeaacqWGKbazcqWGHbqyaeaacqWGHbqydaahaaWcbeqaaiabikdaYaaakiabdIeaijabcIcaOiabdggaHjabcMcaPaaaaSqaaiabdggaHnaaBaaameaacqWGLbqzcqWGTbqBaeqaaaWcbaGaemyyae2aaSbaaWqaaiabicdaWaqabaaaniabgUIiYdGccqGGUaGlaaa@5D0C@

By employing Eq. (13) the radial variable χ can also be expressed in terms of an integral over the redshift as

χ e m ( z ) = 1 a 0 ∫ 0 z d z ′ H ( z ′ ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFhpWydaWgaaWcbaGaemyzauMaemyBa0gabeaakiabcIcaOiabdQha6jabcMcaPiabg2da9maalaaabaGaeGymaedabaGaemyyae2aaSbaaSqaaiabicdaWaqabaaaaOWaa8qmaeaadaWcaaqaaiabdsgaKjqbdQha6zaafaaabaGaemisaGKaeiikaGIafmOEaONbauaacqGGPaqkaaaaleaacqaIWaamaeaacqWG6bGEa0Gaey4kIipakiabc6caUaaa@45BA@

which completes the definition (15) of the luminosity distance d_L in terms of the redshift z.

Differentiating Eq. (15) with χ given by Eq. (17) with respect to z gives

1 H ( z ) = [ 1 − k d L 2 ( z ) a 0 2 ( 1 + z ) 2 ] − 1 / 2 d d z [ d L ( z ) 1 + z ] , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@5CB1@

therefore if independent measurements of d_L and z are available for a set of light sources, the Hubble-parameter H(z) and in consequence the cosmological dynamics can be determined.

From the combined measurements of the large-scale structure of the Universe 4849 and of the structure of the cosmic microwave background 50 the conclusion was reached that the space geometry has flat spatial sections. Therefore in what follows we consider k = 0. Then the luminosity distance-redshift relation becomes

d L ( z ) = ( 1 + z ) ∫ 0 z d z ′ H ( z ′ ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGKbazdaWgaaWcbaGaemitaWeabeaakiabcIcaOiabdQha6jabcMcaPiabg2da9iabcIcaOiabigdaXiabgUcaRiabdQha6jabcMcaPmaapedabaWaaSaaaeaacqWGKbazcuWG6bGEgaqbaaqaaiabdIeaijabcIcaOiqbdQha6zaafaGaeiykaKcaaaWcbaGaeGimaadabaGaemOEaOhaniabgUIiYdGccqGGUaGlaaa@454A@

In practice, the function d_L(z) is conveniently measured with distant supernovae of type Ia. The luminosity is evaluated by photometry, while the redshift from spectroscopic analysis of the host galaxy.

Each cosmological model has its own prediction for the shape of the function d_L(z) [see Eq. (15) with χ given by Eq. (17) for generic k, or Eq. (19) for k = 0]. This is how the measured d_L(z) data turn into a cosmological test.

We consider FLRW branes with k = 0 and brane cosmological constant Λ, embedded symmetrically. The bulk is the Vaidya-anti de Sitter space-time with cosmological constant Λ˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqHBoatgaacaaaa@2E30@, and it contains bulk black holes with masses m on both sides of the brane. The black hole masses can change if the brane radiates into the bulk. An ansatz comparable with structure formation has been advanced in 51 for the Weyl fluid m/a⁴ for the case when the brane radiates, m = m₀a^α, where m₀ is a constant and α = 2, 3. For α = 0 the Weyl fluid is known as dark radiation and then the bulk space-time becomes Schwarzschild-anti de Sitter. The brane tension and the two cosmological constants are inter-related as

2 Λ = κ 2 λ + κ ˜ 2 Λ ˜ . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqaIYaGmcqqHBoatcqGH9aqpiiGacqWF6oWAdaahaaWcbeqaaiabikdaYaaakiab=T7aSjabgUcaRiqb=P7aRzaaiaWaaWbaaSqabeaacqaIYaGmaaGccuqHBoatgaacaiabc6caUaaa@3AD9@

The Friedmann equation gives the Hubble parameter to Λ, m, the scale factor a and the matter energy density ρ on the brane:

H 2 = Λ 3 + κ 2 ρ 3 ( 1 + ρ 2 λ ) + 2 m 0 a 4 − α . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGibasdaahaaWcbeqaaiabikdaYaaakiabg2da9maalaaabaGaeu4MdWeabaGaeG4mamdaaiabgUcaRmaalaaabaacciGae8NUdS2aaWbaaSqabeaacqaIYaGmaaGccqWFbpGCaeaacqaIZaWmaaWaaeWaaeaacqaIXaqmcqGHRaWkdaWcaaqaaiab=f8aYbqaaiabikdaYiab=T7aSbaaaiaawIcacaGLPaaacqGHRaWkdaWcaaqaaiabikdaYiabd2gaTnaaBaaaleaacqaIWaamaeqaaaGcbaGaemyyae2aaWbaaSqabeaacqaI0aancqGHsislcqWFXoqyaaaaaOGaeiOla4caaa@4B05@

In the matter dominated era the brane is dominated by dust, obeying the continuity equation

ρ ˙ + 3 H ρ = 0 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFbpGCgaGaaiabgUcaRiabiodaZiabdIeaijab=f8aYjabg2da9iabicdaWiabcYcaSaaa@35FA@

which gives ρ ~ a^-3. We introduce the following dimensionless quantities:

Ω_tot = Ω_Λ + Ω_ρ + Ω_λ + Ω_d,

Ω ρ = κ 2 ρ 0 3 H 0 2 , Ω λ = κ 2 ρ 0 2 6 λ H 0 2 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaGaeuyQdC1aaSbaaSqaaGGaciab=f8aYbqabaGccqGH9aqpdaWcaaqaaiab=P7aRnaaCaaaleqabaGaeGOmaidaaOGae8xWdi3aaSbaaSqaaiabicdaWaqabaaakeaacqaIZaWmcqWGibasdaqhaaWcbaGaeGimaadabaGaeGOmaidaaaaakiabcYcaSaqaaiabfM6axnaaBaaaleaacqWF7oaBaeqaaOGaeyypa0ZaaSaaaeaacqWF6oWAdaahaaWcbeqaaiabikdaYaaakiab=f8aYnaaDaaaleaacqaIWaamaeaacqaIYaGmaaaakeaacqaI2aGncqWF7oaBcqWGibasdaqhaaWcbaGaeGimaadabaGaeGOmaidaaaaakiabcYcaSaaaaaa@4DFD@

Ω d = 2 m 0 a 0 4 − α H 0 2 , Ω Λ = Λ 3 H 0 2 . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaGaeuyQdC1aaSbaaSqaaiabdsgaKbqabaGccqGH9aqpdaWcaaqaaiabikdaYiabd2gaTnaaBaaaleaacqaIWaamaeqaaaGcbaGaemyyae2aa0baaSqaaiabicdaWaqaaiabisda0iabgkHiTGGaciab=f7aHbaakiabdIeainaaDaaaleaacqaIWaamaeaacqaIYaGmaaaaaOGaeiilaWcabaGaeuyQdC1aaSbaaSqaaiabfU5ambqabaGccqGH9aqpdaWcaaqaaiabfU5ambqaaiabiodaZiabdIeainaaDaaaleaacqaIWaamaeaacqaIYaGmaaaaaOGaeiOla4caaaaa@4932@

The subscript 0 denotes the present value of the respective quantities. In terms of these notations the Friedmann equation becomes

H 2 = H 0 2 [ Ω Λ + Ω ρ a 0 3 a 3 + Ω d a 0 4 − α a 4 − α + Ω λ a 0 6 a 6 ] . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGibasdaahaaWcbeqaaiabikdaYaaakiabg2da9iabdIeainaaDaaaleaacqaIWaamaeaacqaIYaGmaaGcdaWadaqaaiabfM6axnaaBaaaleaacqqHBoataeqaaOGaey4kaSIaeuyQdC1aaSbaaSqaaGGaciab=f8aYbqabaGcdaWcaaqaaiabdggaHnaaDaaaleaacqaIWaamaeaacqaIZaWmaaaakeaacqWGHbqydaahaaWcbeqaaiabiodaZaaaaaGccqGHRaWkcqqHPoWvdaWgaaWcbaGaemizaqgabeaakmaalaaabaGaemyyae2aa0baaSqaaiabicdaWaqaaiabisda0iabgkHiTiab=f7aHbaaaOqaaiabdggaHnaaCaaaleqabaGaeGinaqJaeyOeI0Iae8xSdegaaaaakiabgUcaRiabfM6axnaaBaaaleaacqWF7oaBaeqaaOWaaSaaaeaacqWGHbqydaqhaaWcbaGaeGimaadabaGaeGOnaydaaaGcbaGaemyyae2aaWbaaSqabeaacqaI2aGnaaaaaaGccaGLBbGaayzxaaGaeiOla4caaa@5CC8@

In particular at present time this gives Ω_tot = 1. Then the radial coordinate (16) becomes

χ e m = 1 H 0 ∫