Geodesic-light-cone coordinates and the Bianchi I spacetimelensing in general [15, 16], to galaxy number counts [17], and to the propagation of ultra-relativistic particles [18]. See

Published in JCAP 06 (2016) 008

Geodesic-light-cone coordinates andthe Bianchi I spacetime

Pierre Fleuryab Fabien Nugierc Giuseppe Fanizzade

aDepartment of Mathematics and Applied Mathematics University of Cape TownRondebosch 7701 Cape Town South AfricabDepartment of Physics University of the Western CapeRobert Sobukwe Road Bellville 7535 South AfricacLeung Center for Cosmology and Particle Astrophysics National Taiwan UniversityNo 1 Sec 4 Roosevelt Road Taipei 10617 Taiwan (ROC)dDipartimento di Fisica Universita di BariVia G Amendola 173 70126 Bari ItalyeIstituto Nazionale di Fisica Nucleare Sezione di BariVia E Orabona n 4 70125 Bari Italy

E-mail pierrefleuryuctacza fnugierntuedutw giuseppefanizzabainfnit

Abstract The geodesic-light-cone (GLC) coordinates are a useful tool to analyse lightpropagation and observations in cosmological models In this article we propose a detailedpedagogical and rigorous introduction to this coordinate system explore its gauge degrees offreedom and emphasize its interest when geometric optics is at stake We then apply theGLC formalism to the homogeneous and anisotropic Bianchi I cosmology More than a simpleillustration this application (i) allows us to show that the Weinberg conjecture accordingto which gravitational lensing does not affect the proper area of constant-redshift surfacesis significantly violated in a globally anisotropic universe and (ii) offers a glimpse into newways to constrain cosmic isotropy from the Hubble diagramar

Contents

1 Introduction 1

2 The GLC coordinate system 2

21 Geometrical construction 2

22 Coordinate basis and expression of the metric 3

23 Gauge freedom 6

231 Relabelling lightcones 6

232 Relabelling light rays 7

233 Reparametrising light rays 7

24 Conformal transformations 8

3 Optics with GLC coordinates 9

31 Light rays 9

32 Light beams 10

321 Sachs basis 10

322 Jacobi matrix 10

323 Angular diameter distance 11

33 Lightcone averages 12

4 Bianchi I and GLC 13

41 The Bianchi I spacetime 13

42 Coordinate transformation 14

421 The conformal trick 14

422 Choosing the Bianchi GLC coordinates 14

423 Mapping between comoving and GLC coordinates 15

424 GLC form of the Bianchi I metric 16

43 Jacobi matrix 17

45 Directional average of the inverse magnification 18

46 Bias and dispersion of the Hubble diagram 20

5 Conclusion 22

1 Introduction

The current standard cosmological model provides an excellent description of the Universein agreement with the great majority of observations This success even allows one to testfundamental physics from particles to gravitation However the outcome of such tests as wellas our understanding of the Universe strongly depend on the reliability of the cosmologicalprinciple which states that our Universe is statistically homogeneous and isotropic [1] It istherefore crucial to independently test those hypotheses

A sensible approach to this issue consists in describing the cosmological spacetime interms of coordinates adapted to observations rather than to peculiar symmetries Examplesof such coordinate systems were proposed in the literature since the late 1930s with Templersquos

ndash 1 ndash

optical coordinates [2] followed in the 1970s by Maartensrsquo observational coordinates [3ndash5]and more recently the geodesic-light-cone (GLC) coordinates [6] Such a method was usedin refs [7 8] to directly reconstruct spacetimersquos metric from observations More specificallythe GLC coordinates were first exploited to perform lightcone averages in a perturbedFriedmann-Lemaıtre-Robertson-Walker (FLRW) spacetime in order to determine the effectof inhomogeneities on the distance-redshift relation [9ndash11] and therefore on the interpretationof the Hubble diagram [12ndash14] GLC coordinates were also recently applied to gravitationallensing in general [15 16] to galaxy number counts [17] and to the propagation of ultra-relativistic particles [18] See also the review [19] by one of us

In contrast with homogeneity less effort was dedicated to testing the isotropy of theUniverse Once the so-called bulk flowmdashie our local motion with respect to the cosmic-microwave-background (CMB) rest framemdashis removed any large-scale anisotropy in theobserved sky shall be interpreted as a deviation with respect to the cosmological principleSuch a signal has been tracked within the CMB [20ndash23] or in supernova (SN) data [24ndash33]so far no significant deviation from global isotropy has been found but the power of thosemethods is limited respectively by cosmic variance or by the sparsity and inhomogeneity ofsky coverage by SN surveys An alternative approach based on the cross-correlation betweenweak-lensing E-modes and B-modes has been proposed recently by refs [34ndash36] and shouldprovide an efficient and independent test of cosmic isotropy

The purpose of the present article is twofold On the one hand we propose a pedagogicalintroduction to the GLC coordinate system with in particular the first bottom-up derivationof the expression of spacetimersquos metric with such coordinates It is also the occasion to exploretheir geometrical properties and gauge degrees of freedom in details On the other handwe use GLC coordinates to determine the optical properties of a simple homogeneous butanisotropic cosmological model namely the Bianchi I spacetime This allows us to analysethe statistical properties of the associated distance-redshift relation whether it is averagedover the sky or over sources We will show in particular that the directional average of thesquared angular distance is significantly affected by anisotropy

The article is organised as follows Section 2 is devoted to the construction of GLCcoordinates and to the exploration of their geometry In sec 3 we review the expression of themain optical quantitiesmdashsuch as the redshift the Jacobi matrix and the angular distancemdashinterms of GLC coordinates We then turn to their application to the Bianchi I spacetime insec 4 where we explicitly check the consistency of our results with the literature and discussthe outcome of lightcone averages We finally conclude and suggest further works in sec 5

2 The GLC coordinate system

The geodesic-light-cone (GLC) coordinate system [6] consists of four coordinates a timecoordinate τ associated with a spacelike foliation of spacetime a null coordinate w associatedwith a null foliation and two angles (θa)a=12 Any spacetime geometry can at least locallybe expressed in terms of those coordinates whose great interest lies in the fact that theydeeply simplify the equation of geometric optics as we shall see in this section

21 Geometrical construction

Suppose spacetime geometry allows the existence of an irrotational congruence of timelikegeodesics that we shall call fundamental observers By virtue of the Frobenius theorem [37]it defines a particular foliation by spacelike hypersurfaces that we choose to label with a

ndash 2 ndash

coordinate τ In other words τ = cst defines one of those hypersurfaces and umicro equiv minuspartmicroτis normal to it We further impose umicroumicro = minus1 so that umicro represents the four-velocity offundamental observers and τ their proper time

Let Lo be the worldline of an arbitrary fundamental observer simply referred to as theobserver in the following Spacetime can be foliated by the set of all past lightcones of Lo

1We choose to label them with a coordinate w so that w = cst defines one such lightconeie the set of all events from which emitted photons are all received simultaneously by theobserver A natural choice for wmdashthough not the only possible onemdashis thus the receptiontime τo of a photon emitted from elsewhere towards the observer Just like for τ partmicrow defines anormal vector to w = cst hypersurfaces which is here proportional to kmicro the wave four-vectorof a photon travelling towards the observer

The above definitions imply that the intersection of two hypersurfaces τ = cst andw = cst is a spatial two-dimensional surface which shall be thought of as a sphere whoseradius is the travel time of a photon for it to reach Lo This surface can be parametrisedby two angles (θa)a=12 we define them so that a null geodesic linking an event E to Lo ischaracterised by θa = cst additionally to the w = cst condition The angles (θa)a=12 aresomehow Lagrangian coordinates for photons a particular choice being the right ascensionand declination of the direction in which the observer actually detects them

All those definitions are summarised in fig 1a An arbitrary event E is thus coordinatedby (i) the simultaneity hypersurface τ it belongs to (ii) the observerrsquos past lightcone w it ispart of and (iii) the particular light ray (θa) which connects it to Lo

Note that the GLC coordinates are unable to describe properly a region of spacetimewith caustics ie where a given event E can be linked with O isin Lo by several distinct lightrays like in strong gravitational lensing systems Indeed in such a case E would be describedby several sets of angular coordinates (θa) This would make GLC coordinates ill-definedbecause not injective We will not consider this case in the remainder of the article

22 Coordinate basis and expression of the metric

Let us now derive the general form of spacetimersquos metric in GLC coordinates For thatpurpose it is convenient to analyse the properties of the coordinate basis vectors partτ partwpartaassociated with τ w θa

By definition partτ points in the direction of the infinitesimal displacement E(τ w θa)rarrEprime(τ + dτ w θa) As illustrated in fig 1b this displacement occurs along the light rayconnecting E to Lo because (w θa) is unchanged thus partτ prop k in other words kmicro prop δmicroτ Theproportionality coefficient is determined by noticing that by definition umicro = minusδτmicro so that

kτ = minusumicrokmicro equiv ω (21)

where ω physically represents the cyclic frequency of the electromagnetic wave associatedwith k as measured by a fundamental observer at E The first basis vector is therefore

partτ = ωminus1k (22)

which has two consequences On the one hand

gττ = g(partτ partτ ) = ωminus2g(kk) = 0 (23)

1A similar construction can be achieved with future lightcones though less sensible from a cosmologicalpoint of view

ndash 3 ndash

Eτ = cst

(a) Definition of the GLC coordinates τ w θaThe curve Lo is the observerrsquos worldline

τ + dτ

Eprimepartτ prop k

(b) Basis vector partτ is tangent to the w θa = cstlines hence parallel to k

τ = cst

Eprime

(c) Basis vector partw is tangent to the τ θa = cstlines It defines a notion of radial direction

Eτ = cst

Eprime

(d) Basis vectors parta are tangent to theτ w θb6=a = cst lines

Figure 1 Geometry of the GLC coordinates and the associated vector basis

as k is a null vector On the other hand

gτa = g(partτ parta) = ωminus1ka = 0 (24)

because kmicro prop partmicrow = δwmicro The other basis vectors partw and parta are depicted in figs 1c 1d The former defines a

notion of radial direction while the latters are tangent to the sphere formed by the intersectionbetween the τ = cst hypersurface and the w = cst lightcone Note that by definition allthree are spatial vectors in the sense that they are orthogonal to u At the present stage it isconvenient to introduce the usual 3+1 decomposition of the wave four-vector

k = ω(u+ d) (25)

where d represents the spatial direction of propagation in the frame of a fundamental observerof the photon described by k It is a unit vector (dmicrodmicro = 1) orthogonal to u It is also

ndash 4 ndash

orthogonal to parta as

g(dparta) = g(ωminus1kparta)minus g(uparta) = 0 + 0 (26)

Since the four vectors partwpart1part2d are all tangent to the τ = cst hypersurface which isthree-dimensional they are not linearly independent so there exist coefficients Υ Ua such that

partw = minusΥdminus Uaparta (27)

EEprime

partwparta

minusd

w + dwLo

Figure 2 View ldquofrom aboverdquo of fig 1c Two lightrays belonging to two successive lightcones w and w +dw but with the same angular parameters θa arerepresented in green The quantities Ua defined ineq (27) quantify the misalignment between partw andthe local spatial direction of light propagation d Non-zero Ua are due to both light bending and the rotationof the observer

As illustrated in fig 2 Ua quantifies the failure of the parametrisation of light rays onsuccessive lightcones to follow their propagation We shall thus call it the photoshift vectorIn terms of Υ Ua the metric reads

gτw = g(partτ partw) = minusΥ (28)

gww = g(partwpartw) = Υ2 + gabUaU b (29)

gwa = g(partwparta) = minusgabU b equiv minusUa (210)

gab = g(partapartb) equiv γab (211)

so that the general expression of the line element in GLC coordinates is finally

ds2 = Υ2dw2 minus 2Υdwdτ + γab(dθa minus Uadw)(dθb minus U bdw) (212)

in agreement with ref [6] Under a matrix form the metric and its inverse in τ w θa

coordinates read

[gGLCmicroν ] =

0 minusΥ 0minusΥ Υ2 + U2 minus(Ub)0 minus(Ua)

T γab

[gmicroνGLC] =

minus1 minusΥminus1 minusΥminus1(U b)minusΥminus1 0 0

minusΥminus1(Ua)T 0 γab

where γab is the inverse of γab in the sense that γacγcb = δab while a T superscript denotesmatrix transposition

The quantities Υ Ua and γab encode the six degrees of freedom of the metric they area priori functions of the four coordinates τ w θa Note however that the definition of γabimposes that it vanishes on Lo In other words the GLC coordinate system is singular on theobserverrsquos worldline just as spherical polar coordinates are singular at their origin

Let us prove this assertion Consider two events E(τ w θa) and Eprime(τ w θa + dθa)corresponding to the simultaneous emission of two photons received simultaneously by theobserver Suppose these events are very close to Lo so that spacetime can be considered flat

ndash 5 ndash

in this region Then the proper distance from E (or Eprime) to Lo is the photonsrsquo travel timeτo minus τ where τo is the reception time So the proper separation between E and Eprime is

d`2 simErarrLo

(τo minus τ)2dΩ2 (214)

which is also by definition of the GLC coordinates γabdθadθb whence γab = O(τo minus τ)2

23 Gauge freedom

In fact the construction presented in sect 21 does not fully define the GLC coordinatesbecause we did not specify (i) how exactly lightcones are labelled and (ii) how the anglesare transferred from one lightcone to another This freedom in the definition of w and θa

allows gauge transformations of the GLC metric ie diffeomorphisms which preserve theform (212) of the metric but change the functional expression of the quantities Υ Ua γab

231 Relabelling lightcones

Suppose one has found GLC coordinates for a given metric and wish to relabel the lightconesas w rarr wprime(w) It is clear that the GLC form of the metric is preserved and that the GLCfunctions change according to

Υrarr Υprime = Υdw

dwprime Ua rarr U primea = Ua

dwprime (215)

The first of the above relation tells us in particular that it is always possible by a redefinitionof w to set Υ to 1 along any curve which crosses at most once each lightcone of Lo

Interestingly imposing this condition on Lo corresponds to the case w = τo wherelightcones are labelled by the reception time of their photons In order to prove this considertwo events E and Eprime separated by dxmicro assumed to be very close to Lo Let t r be such that(t r θa) form a Fermi normal coordinate system [37] about Lomdashnote that t coincides with τon Lo The contraction kmicrodxmicro can then be expressed in two different ways

GLC kmicrodxmicro = kowdw (216)

Fermi kmicrodxmicro = ktdt+ krdr = minusωo(dτ + dr) = minusωodτo (217)

the last equality being illustrated in fig 3 Besides the observed frequency is by definition

ωo = minus(umicrokmicro)o = minus(gwτuτkw)o = minus(kwΥ

which once combined with eqs (216) (217) finally yields

Υ|Lo =dτo

dw (219)

In the following we shall call temporal gauge the choice w = τo The articles which haverecently introduced and exploited the GLC coordinates [6 9ndash13 15ndash17 38] often give theexample of the FLRW geometry in such a way that w is the conformal time ηo of the photonreception by Lo This choice does not correspond to the temporal gauge

ndash 6 ndash

dtdxmicroE

Eprime

Figure 3 The time interval dτo betweenthe receptions of two photons emitted atE and Eprime is equal to the sum of the timeinterval dt and the spatial distance dr be-tween those events This reasoning is onlyvalid in terms of Fermi normal coordinatesin the vicinity of Lo

232 Relabelling light rays

Another gauge freedom concerns the way light rays are labelled by the angles θa and howthis labelling is transferred from one lightcone to another Changing the ray-labelling schemeindeed corresponds to a coordinate transformation of the form θa rarr ϕα(w θa) which preservesthe GLC form of the metric The quantities γab Ua however change according to

γab rarr γαβ = γabpartaϕαpartbϕ

β Ua rarr Uα = Uapartaϕα minus partwϕα (220)

Since ϕα is a function of three coordinates (w θa) the last transformation tells us that it ispossible to impose Uα = 0 on an arbitrary timelike hypersurface homeomorphic to a cylinderaround Lo This was already noticed in ref [38]

A special case is the limit where this hypersurface is actually Lo so that Uα(Lo) = 0Geometrically speaking it means that the labelling of rays locally follows their propagationaround Lo Such a gauge could be baptised photocomoving However it is important to keepin mind that within such a gauge the angles θa do not necessarily have any observationalmeaning namely the directions θa = cst do not necessarily correspond to fixed directions onthe observerrsquos celestial sphere

An observationally more sensible choice is thus the case where θa are regular sphericalangles on the observerrsquos sky at any moment This choice shall be called the observational gaugeIf moreover the observer is not rotating ie if any unit vector pointing in a θa = cst-directionis parallely transported along Lo then we talk about the non-rotating observational gaugeIn the Minkowski or FLRW spacetimes the non-rotating observational gauge has Ua = 0 butas we will see in sec 4 it is not the case for spacetimes where light rays are not straight linessuch as Bianchi I

233 Reparametrising light rays

This paragraph indirectly related to gauge transformations deals with the proportionalitysign in the relation kmicro prop partmicrow In general the proportionality coefficient kw is a function ofall coordinates τ w θa but as we will see below it is always possible to reparametrise the nullgeodesics spanning spacetime so that kw is a pure constant

Let λ be a parameter on light rays forming the lightcones w = cst such that k = ddλFor k to physically correspond to a wave four-vector λ cannot be any parameter but anaffine parameter which means that k satisfies the geodesic equation under its simplest form

ndash 7 ndash

kmicronablamicrokν = 0 Independently of the parametrisation k is null so that we can write

2nablaν(kmicrokmicro) (221)

= kmicronablaνkmicro (222)

= kmicro(partνkw)partmicrow︸︷︷︸0

+kwkmicronablaνpartmicrow︸︷︷︸nablamicropartνw

= kmicronablamicro(kwpartνw)minus (kmicropartmicrokw)partνw (224)

= kmicronablamicrokν minus ω(partτkw)partνw (225)

Therefore λ is an affine parameter iff partτkw = 0In principle the affine parametrisation can differ from one ray to another In order to

visualise what this means consider all the rays forming a lightcone w = cst suppose theirrespective affine parameter is fixed to zero on Lo and consider the set of points correspondingto the (small) value δλ of the affine parameter on each of these rays Those points are thereforeseparated from Lo by

δxmicro = kmicroδλ = minuskwδλΥo

δmicroτ (226)

hence they form a two-dimensional surface of equation w = cst τ = minusΥminus1o (w)kw(w θa)δλ

The affine parametrisation of the different rays can be considered identical if this surface is asphere which is the case iff partakw = 0

A parametrisation of null geodesics such that both the above conditions are fulfilledie such that kw = kw(w) shall be called isotropic affine parametrisation In this case kw ishomogeneous on each lightcones and by virtue of eqs (216) (217) equals

kw = kow = minusωo

dw= minusωoΥo (227)

Besides its only non-zero contravariant component is kτ = minusΥminus1kw = ωoΥoΥ We will seein sec 3 that this makes kτ directly proportional to the redshift 1 + z

Finally a similar reasoning can be applied to the way the parametrisation of nullgeodesics is transferred from one lightcone to another Suppose we want the events at anaffine distance δλ from Lo to be located at the same physical distance from it whatever thelightcone they belong to By virtue of eq (226) this imposes that partw(Υminus1

o kw) = minuspartwωo = 0This choice shall be called static affine parametrisation

Summarising in temporal gauge (w = τoΥo = 1) and for a static isotropic affineparametrisation the only non-zero covariant component kw of k in GLC coordinates is a pureconstant We choose this gauge for the remainder of this article

24 Conformal transformations

It is well known that null geodesics are unaffected by conformal transformations of spacetimersquosmetric We therefore expect the GLC coordinates of a metric to be essentially preserved byconformal transformations Let Ω be a scalar function and g such that

gmicroν = Ω2gmicroν (228)

Suppose one has found a GLC coordinate system (τ w θa) for g By definition the curvesw θa = cst are null geodesics for g and thus for g as well Therefore w θa are also GLC

ndash 8 ndash

coordinates for g Besides the coordinate τ only needs to be renormalised as partτ = Ωpartτ anddτ = Ωminus1dτ so that gτ τ = g(partτ partτ ) = Ωminus2g(Ωpartτ Ωpartτ ) = gττ = minus1

With such definitions the line element of g reads in terms of (τ w θa)

ds2 = Ωminus2ds2 (229)

= Ωminus2Υ2dw2 minus 2Ωminus1Υdwdτ + Ωminus2γab(dθa minus Uadw)(dθb minus U bdw) (230)

from which we immediately deduce that Υ = ΩΥ and γab = Ω2γab Summarising the GLCcoordinates and metric components are affected by conformal transformations according to

w = w θa = θa dτ = Ωdτ (231)

Υ = ΩΥ Ua = Ua γab = Ω2γab (232)

However if one works within the temporal gauge for g ie with w = τo then an additionaltransformation is required to preserve this gauge with the conformal transformation namelydw = dτo = Ωodτo = Ωodw As a consequence the conformal dictionary becomes

dw = Ωodw θa = θa dτ = Ωdτ (233)

Υ = (ΩΩo)Υ Ua = Ωminus1o Ua γab = Ω2γab (234)

The main interest of such transformations is that the equations of light propagation aresometimes more easily handled in a conformally-transformed metric g such as in the FLRWor Bianchi I cases where the scale factor can be absorbed into Ω the transformation to GLCcoordinates is thus more easily performed in this case The GLC quantities of the originalmetric g are then recovered from the tilded ones using the dictionary of eqs (233) (234)

3 Optics with GLC coordinates

Because they are directly constructed on null geodesics GLC coordinates are particularlyadapted to the analysis of geometric optics in any curved spacetime This section summarisesthe GLC expressions of the main optical quantities In particular subsec 322 extends thediscussion or ref [38] concerning the Jacobi matrix In this section and the next one exceptexplicit mention of the contrary we work in the temporal observational gauge with a staticisotropic affine parametrisation

31 Light rays

By definition in GLC coordinates any light ray is a completely trivial curve with w θa = cstIts wave four-vector reads k = ωpartτ (see subsec 22) where ω is the cyclic frequency measuredby a fundamental observer It can also be expressed as

ω = minusumicrokmicro = minusgτwuτkw = minuskwΥ (31)

where kw is a normalisation constant given our gauge choice The affine parameter λ along aray admits a one-to-one relation with the foliation time τ namely dλ = ωminus1dτ = Υkminus1

w dτ If a light source following a fundamental worldline (us = partτ ) emits a photon towards

the observer then the redshift z of this photon reads

1 + z equiv ωs

Υs (32)

ndash 9 ndash

Of course this expression is only valid for a comoving source and must be corrected if thelatter has a peculiar velocity2

32 Light beams

Let us now turn to infinitesimal light beams described in terms of the Sachs and Jacobimatrix formalisms a detailed description of which can be found in for example ref [39]

321 Sachs basis

The GLC coordinates come with a particular family of observers with respect to which wechoose to define the Sachs basis (sA)A=12 umicros

microA = 0 so that sτA = 0 Besides as the Sachs

basis spans a screen orthogonal to the line of sight (kmicrosmicroA = 0) we also have swA = 0 whence

sA = saAparta (33)

The Sachs basis is an orthonormal basis (zweibein) of the screen space in other wordsg(sA sB) = γabs

bB = δAB Introducing the covariant components sAa equiv g(sAparta) which

also form the inverse of [saA] seen as a matrix the orthonormality relation implies that

γab = δAB sAa s

Bb (34)

For the Sachs basis to be a consistent reference for the orientation of the beamrsquos patternit must be parallely transported as much as possible along the light beam while remainingorthogonal to u and d that is

SmicroνDsνAdλ

= 0 (35)

where Smicroν equiv gmicroν minus dmicrodν + umicrouν = δABsmicroAsνB denotes the screen projector In terms of GLC

components we have Smicroν = δamicroδbνγab so that that the above condition reduces to DsaAdλ = 0

or more explicitly

partτsaA =

2(γacpartτγbc)s

bA (36)

A general method for solving this differential equation has been proposed in ref [38]

322 Jacobi matrix

It has been shown in ref [38] that the Jacobi matrix D which relates the physical sizeand shape of a light beam to its observed angular aperture is directly related to the GLCcomponents of the Sachs basis as

DAB(λlarr λo) = sAa (λ)CaB (37)

where [CaA] denotes a 2times 2 matrix which is independent from λ The initial condition for thederivative of the Jacobi matrix at λ = λo namely (dDdλ)o = 1 implies that the inverse of[CaA] reads

CAa equiv(

dsAadλ

2By lsquopeculiar velocityrsquo we mean here any misalignment between the four-velocitiy of the source with respectto the fundamental flow described by the GLC spatial foliation Note however that if the flow of sources issmooth and irrotational then there always exists a foliation such that peculiar velocities are zero

ndash 10 ndash

Therefore once the expression of the Sachs basis in GLC coordinates is found the Jacobimatrix comes with it for free

We now extend these results of ref [38] and show that the matrix [CaA] as well as itsinverse enjoys a very simple explicit expression For that purpose we are going to constructa particular Sachs basis on Lo from the GLC coordinates themselves Consider a light rayreceived by Lo at O and let (eα)α=03 with e0 = u be a tetrad at this event We definetwo angles (θa) = (θ ϕ) so that the spatial direction of propagation do of the ray at O reads

minus do = cosϕ sin θ e1 + sinϕ sin θ e2 + cos θ e3 (39)

Those angles define observational GLC coordinates which we can then use to define aparticular Sachs basis (sA) at O

(s1)o = minuspartθdo (s2)o =minus1

sin θpartϕdo (310)

Such vectors indeed satisfy all the required orthogonality conditions In the vicinity of Otheir full GLC expressions are therefore

rpartθ +O(r) s2 =

r sin θpartϕ +O(r) (311)

with r equiv τo minus τ Written in a matrix form the above relations then provide the GLCcomponents saA of the Sachs basis

[s1 s2

]=[partθ partϕ

]middot 1

[1 00 1

sin θ

]︸︷︷︸

+O(r) so [sAa ] = r

[1 00 sin θ

]+O(r3) (312)

By computing the derivative of the latter matrix at O using drdλ = ωdrdτ = minusω wefinally obtain

[CAa ] = minusωo

[1 00 sin θ

] [CaA] = minus 1

[1 00 1

sin θ

] (313)

Of course (sA) is not the only possible choice for the Sachs basis any global rotation withrespect to it is admissible too Thus the most general expression of matrix [CaA] is

[CaA] = minus 1

[cosψ minus sinψsinψ cosψ

] [1 00 1

sin θ

] (314)

where ψ isin [0 2π[ In principle the basis could also be reflected but for simplicity wechoose to eliminate this possibility by imposing that the set (minusd s1 s2) has positive determi-nant Besides as already noticed by ref [38] with generic non-observational GLC angularcoordinates (θm) the above expression must be corrected as CmA = (partθmpartθa)CaA

323 Angular diameter distance

The angular diameter distance DA which relates the physical size of a source to the angleunder which it is seen by the observer can be deduced from the Jacobi matrix as [39]

D2A = ω2

o detD (315)

ndash 11 ndash

Again this quantity enjoys a remarkably simple expression in terms of GLC coordinatesUsing the formula (37) for D indeed yields

detD = det[sAa ]times det[CaA] =radic

det[γab]times1

ω2o sin θ

where in the second equality we used both eqs (34) and (314) The GLC expression of theangular diameter distance is therefore

radicγ

sin θ(317)

where γ denotes the determinant of [γab] This expression agrees with the recent GLCliterature see eg ref [40] and references therein Again a non-observational choice forthe angular coordinates can be accounted for by multiplying eq (317) with a Jacobianfactor det[partθmpartθa]

33 Lightcone averages

The GLC coordinates are particularly adapted to the calculation of the average of cosmologicalobservables [6 40] Let S be any scalar observable such as the luminous intensity or thedensity of matter Its directional average ie its average over the observerrsquos celestial sphereweighted by solid angles is naturally defined as

〈S〉Ω equiv1

intS d2Ω (318)

with d2Ω equiv sin θdθdϕ and where the angles θ ϕ are the GLC angles in observational gaugeIn the above equation coordinate w is fixed as it indicates over which lightcone the average isperformed coordinate τ is either fixed to τs if one wishes to average over simultaneous sourceevents or such that τ = τ(zs θ ϕ) if one is interested in an average over events with the sameredshift Directional averages are typically involved in the analysis of the CMB

Another important observational notion of average is the source average typicallyinvolved in the analysis of cosmological data made involving a catalog of sources such asthe Hubble diagram of SNe or large galaxy surveys Contrary to the directional case it isweighted by the number N of sources

〈S〉N equiv1

intS d2N =

intS Σradicγ dθdϕint

Σradicγ dθdϕ

where we introduced the density of sources per unit of proper area of the source surfaceΣ equiv d2Nd2A = d2N(

radicγ dθdϕ) If this average is performed within a redshift bin then Σ is

related to the volumic density of sources ρ as

Σ = ρ

∣∣∣∣partzpartτ∣∣∣∣minus1

= ρΥ2

(partΥ

partτ

)minus1

in temporal gauge (Υo = 1) Note that introducing the expression (317) of the angulardiameter distance in the definition (319) of the source average we get the following identity

〈S〉N =

intS ΣD2

Ad2ΩintΣD2

Ad2Ω=

langS ΣD2

rangΩlang

rangΩ

ndash 12 ndash

4 Bianchi I and GLC

In the literature the GLC coordinate system has already been applied to the second-orderperturbed FLRW spacetime either in Newtonian gauge [9 16 18] Poisson gauge [11 13 40]or synchronous gauge [38 40] but also to the Lemaıtre-Tolman-Bondi spacetime [15] Thislast section is devoted to another non-perturbative application of the GLC coordinates namelythe Bianchi I geometry

41 The Bianchi I spacetime

The Bianchi geometries [41] describe homogeneous but anisotropic cosmological modelsFollowing the Bianchi classification of homogeneous three-dimensional spaces [42] suchspacetimes exist under nine different forms depending on their spatial curvature TheBianchi I spacetime with zero spatial curvature is the simplest one and also the easiest tointerpret it can be seen as the anisotropic extension of a spatially flat FLRW model withthree different scale factors instead of one In other words it describes a homogeneous universewhose expansion is faster in some directions and slower in others Its metric is convenientlyparametrised as

ds2 = minusdt2 + a2(t)γij(t)dxidxj γij(t) equiv e2βi(t)δij (41)

with no summation over i in the definition of γij and where the three βi must sum to zero

3sumi=1

βi = 0 (42)

In eq (41) t denotes cosmic time which physically represents the proper time of fundamental(or comoving) observers following xi = cst worldlines Those curves are easily shown tobe timelike geodesics The quantity a(t) = (det[gij ])

16 is the volumic scale factor whichdescribes the expansion (or contraction) of a 3d domain with constant spatial coordinatesThe βi exponents then quantify the volume-preserving deformations of such a domain

We can also introduce a conformal time η such that dt = adη so that the volumicscale factor factorises in the expression of the metric allowing us to easily perform conformaltransformations This will turn out to be convenient in the next subsections

The anisotropic component of the expansion rate is encoded in the shear rate tensor

σij equiv1

2(γij)

prime (43)

where a prime denotes a derivative with respect to η The indices i j are here conventionallyraised by the inverse γij of γij so that σij = βprimeiδ

ij The trace of its square is usually denoted by

σ2 equiv σijσij =

3sumi=1

(βprimei)2 (44)

For a Bianchi I Universe filled by matter with no anisotropic stress σij decays as aminus2 whilethe analogue of the Friedmann equation governing the evolution of a is

H2 =8πGρ

)+ Ωσ0

)4] (45)

ndash 13 ndash

where H equiv aprimea is the conformal expansion rate G Newtonrsquos constant ρ the density of matterand a 0 subscript indicates that the quantity is evaluated today The matter cosmologicalparameter Ωm0 has the same definition as in standard cosmology while Ωσ0 equiv σ2

0(6H20)

quantifies the departure from isotropy we shall therefore call it the anisotropy parameter

Note finally that as any homogeneous geometry Bianchi I admits three Killing vec-tors (ξi) which turn out to coincide with the comoving coordinate basis ξi = parti Aninteresting consequence is that the scalar product between parti and the tangent vector t of anygeodesic is then a constant [39] This applies in particular to the wave four-vector k of anylight ray so that

ki = g(kparti) = cst (46)

42 Coordinate transformation

Let us now perform the coordinate transformation in order to derive the GLC quantities forthe Bianchi I geometry

421 The conformal trick

As suggested above the volumic scale factor a(t) can be factorised in the expression of themetric by introducing conformal time η This strongly encourages us to perform a conformaltransformation of the metric gmicroν equiv aminus2gmicroν whose line element is

ds2 = minusdη2 +

3sumi=1

[eβi(η) dxi

]2 (47)

and therefore get free from a(t) In the remainder of this subsection we will perform allintermediate calculations working with g In other words we proceed as if we had a = 1 Thefull results including the scale factor will however be recovered from the conformal dictionaryestablished in subsec 24 with Ω = a

422 Choosing the Bianchi GLC coordinates

We choose the GLC spacelike foliation to coincide with the natural foliation of the Bianchi Ispacetime so that the GLC time coordinate is τ = η Besides we work in temporal gaugehence w = ηo We did not specify explicitly which worldline is Lo because Bianchi I ishomogeneous Lo can thus be any fundamental observer eg xi = 0

Regarding angular coordinates θa we work in the observational gauge and define themfrom a tetrad (eα)α=03 following Lo as in sect 322 Such a tetrad can be

e0 = partη ei = eminusβiparti (no summation) (48)

It corresponds to a non-spinning observer The angles (θa) = (θ ϕ) are then defined from theobserved spatial direction of propagation do of an incoming photon as in eq (39) We callDi(θ ϕ) the components of do over the spatial part of the tetrad ie such that

do = Di(θ ϕ) ei = minus sin θ cosϕ e1 minus sin θ sinϕ e2 minus cos θ e3 (49)

ndash 14 ndash

423 Mapping between comoving and GLC coordinates

Let us now establish the explicit mapping

(η x y z) 7rarr (η ηo θ ϕ) (410)

between comoving and (conformal) GLC coordinates This exact step has actually been donein the late 60s by Saunders [43 44] in order to derive the distance-redshift relation in theBianchi I spacetime The easiest way to perform the mapping consists in integrating the wavefour-vector with respect to the affine parameter

int λ

dλprime ki =

int η

dηprime ωminus1gij(η)kj (411)

where we used that dλ = ωminus1dτ = ωminus1dη We then exploit the fact that ki is a constantalong the light ray hence it is equal to its observed value

ki = koi = ωod

oi = ωo g(doparti) = ωoeβi(ηo)Di(θ ϕ) (no summation) (412)

Besides we can use the fact that k is null to express the frequency ω in terms of ki

ω2 = gij kikj =

3sumi=1

[eminusβi(η)ki

]2 (413)

Gathering eqs (411) (412) we finally get the explicit relation between comoving coordinatesand (conformal) GLC coordinates

xi(η ηo θ ϕ) = eβi(ηo)Di(θ ϕ)

int η

dηprime eminus2βi(ηprime)

3sumj=1

[eβj(ηo)minusβj(ηprime)Dj(θ ϕ)

minus12

In order to derive the GLC form of the Bianchi I metric we will need to express dxi

as a function of dηdηodθa For that purpose let us introduce some useful quantities and

relations The key partial derivatives can be formally expressed as

partxi

partη= ωminus1ki = di (415)

partxi

partηo= minusdio +

int η

dηprimepartdi

partηoequiv minusdio + Iiηo

partxi

partθa=

int η

dηprimepartdi

partθaequiv Iia (417)

where we have defined three integrals Iiηo Iia It is not necessary to calculate them explicitly

but it is worth remarking that they are all orthogonal to k in the sense that

kiIiηo

= kiIia = 0 (418)

Indeed as ki = ki is a constant along light rays it can safely enter into the integrals we arethus interested by terms of the form kipartd

i where part denotes a derivative with respect to eitherηo or θa Since gij only depends on η we can write

0 = part(did

= gijpart(didj

)= 2dipartd

i = 2ωminus1kipartdi (419)

so that the integrands of kiIiηo

and kiIia all vanish identically

ndash 15 ndash

424 GLC form of the Bianchi I metric

We now introduce the expression of dxi as a function of GLC coordinates in the line element

ds2 = gmicroνdxmicrodxν (420)

= minusdη2 + gij

[didη + (Iiηo

minus dio)dηo + Iiadθa][djdη + (Ijηo

minus djo)dηo + Ijbdθb]

= gij(Iiηominus dio)(Ijηo

minus djo)dη2o minus 2dik

io dηdηo minus 2gij(d

io minus Iiηo

)Ija dηodθa + gijIiaIjb dθadθb

which by identification with the general form (212) of the metric in GLC coordinates yields

Υ = didio = ωoω (423)

U2 = gij(Iiηominus dio)(Ijηo

minus djo)minus (ωoω)2 (424)

Ua = gij(dio minus Iiηo

)Iia (425)

γab = gijIiaIjb (426)

Reintroducing the scale factor using the conformal dictionary of subsec 24 we finally get thefollowing semi-explicit expressions for the GLC quantities of the Bianchi I spacetime

Υ =a(t)

3sumi=1

[eβi(to)minusβi(t)Di(θ ϕ)

]2minus12

Ua = gij(dio minus Iito

γab = gijIiaIjb

with Iito equiv Iiηoa(to) In order to obtain eq (427) we introduced the expression (412) of ki in

the formula (413) for ω Besides in eq (428) we used that dio = dioao Following eq (32)we conclude that the redshift in a Bianchi I universe reads

1 + z =a(to)

radicradicradicradic 3sumi=1

[eβi(to)minusβi(ts)Di(θ ϕ)

]2 (430)

in agreement with ref [45]Introducing the inverse of [Iia] denoted [Iai ] and defined by the relations

IiaIbj = δba IiaI

aj = Sij equiv δij minus didj (431)

it is easy to show that the photoshift vector reads Ua = Iai (dio minus Iito) It is not possible toget a more explicit expression However a long but straightforward calculation based onexpansions of the various involved quantities around Lo yields

Uao = minus(gijd

ipartadj)

o (432)

where a dot denotes a derivative with respect to t Note that this quantity vanishes if do

is aligned with a proper axis of g This is in agreement with the discussion illustrated infig 2 Indeed we have seen that a non-zero value of Ua is due to both light deflection and therotation of the observer Here the observer does not spin but light rays are generally bentexcept when they propagate along a proper axis of the expansion

ndash 16 ndash

43 Jacobi matrix

Let us now check the consistency between the expression of the Jacobi matrix obtained fromthe GLC formalism [38] and recent results of the literature [45] In this subsection we keepworking in the conformal geometry g We however drop all the tildes in order to alleviatenotation In ref [45] the Jacobi matrix in a conformal Bianchi I spacetime is found to be

DAB = sAi Iij(sBj)o (433)

where the integral matrix Iij is defined by

Iij equivint η

dηprime ωminus1Sij (434)

with Sij = gij minus didj = δABsiAsjB the spatial components of the screen projector introduced

in subsec 321

In order to connect this result with the GLC quantities that we have just obtained afirst step consists in remarking that as partad

i is orthogonal to di and gij is independent fromthe angles

partadi = Sijpartadj = ωminus1Sijpartakj (435)

therefore

Iia equivint ηo

ηdηprime partad

(int ηo

ηdηprime ωminus1Sij

)partakj equiv Iijpartakj (436)

where we used that ki is conserved along null geodesics

Let us now start from the GLC expression (37) of the Jacobi matrix

DAB = sAa CaB (437)

where the inverse of the constant matrix reads CAa = (dsAa dλ)o Reference [45] has proposedvarious techniques and explicit examples to determine the components siA of the Sachs basiswith respect to the comoving basis parti There is no need to repeat them here Once thosecomponents are found the GLC ones are simply expressed as

sAa = g(sAparta) = g(sA Iiaparti) = sAi I

ia (438)

since (parta)i = partxipartθa = Iia The components of the matrix [CAa ] then read

dsAadλ

partη

∣∣∣∣ηoθϕ

(IiasAi ) (439)

= (partadi)sAi + Iia(partηs

Ai ) (440)

On Lo as η = ηo the second term vanishes so that

CAa = (sAi partaki)o (441)

We now claim that the inverse CaA of the above matrix reads

CaA = Iai Iij(sAj)o (442)

ndash 17 ndash

Indeed their contraction yields

CaACAb = Iai Iij (sAj)o(sAl )o︸︷︷︸

(Sjl)o

(partbkl)o (443)

besides note thatIij(Sjl)o = Iij(gjl minus djdl)o = Iij(gjl)o (444)

since (di)o = ki and Iijkj = 0 Therefore we have

CaACAb = Iai Iij(gjl)o(partbk

l)o (445)

= Iai Iijpartbkj (446)

= Iai Iib (447)

= δab (448)

Plugging the expression (442) of CaA in the GLC expression of the Jacobi matrix finally yieldsexactly eq (433) which concludes the proof

Following eq (317) the angular distance in a Bianchi I spacetime is given by

sin θ

radicradicradicradic( 3sumi=1

e2βi(Ii1)2

)(3sumi=1

e2βi(Ii2)2

)minus

(3sumi=1

e2βiIi1Ii2

where we used that γab = gijIiaIjb Though not so complicated this expression requires to

compute the six different integrals (Iia)i=123a=12 and is therefore less efficient than the alternative

formulaD2

A = a2 Υminus1sumi 6=j 6=`

JiJj(k`)2 (450)

with k` = eβo`D` from eq (412) and

Ji equivint ηo

ηdηprime Υ3e2(βiminusβo

i ) (451)

as found by Saunders in ref [44] and recently rederived in refs [32 45] which involves onlythree different integrals (Ji) The equality between those two expressions for DA is not to beproven as our GLC expression of the Jacobi matrix agrees with the one of ref [45] whichitself agrees with Saundersrsquo formula (450)

45 Directional average of the inverse magnification

A still ongoing debate in cosmology is whether or not gravitational lensing affects the averagedistance-redshift relation that we use to interpret almost all cosmological observations [39]In particular in ref [46] Weinberg argued that there is no such effect in a transparentUniverse a result physically justified by energy conservation Although this claim and itsjustification are inaccuratemdashas already noticed by Bertotti [47] and emphasized later on byseveral authors [48ndash57]mdasha weaker version of it has been formulated by Kibble amp Lieu whoshowed that in a locally inhomogeneous Universe the total area of a constant-affine-parameter

ndash 18 ndash

surface is essentially unaffected by gravitational lensing [58] This is equivalent to saying thatthe directional average of the squared angular distance

langD2

A(λ)rang

Ωis unchanged with respect

to its background value3 If moreover the affine parameter-redshift relation is assumed to beessentially unaffected by the inhomogeneity of the Universe then one obtains the conjecture

langmicrominus1(z)

rangΩequiv

lang[DA(z)

DFLRWA (z)

]2rang

= 1 (453)

This relation has been carefully checked at second-order in the cosmological perturbationtheory by the independent works of Bonvin et al [59] Kaiser amp Peacock [60] and Ben-Dayan et al [12 13] Note however that the non-perturbative numerical analysis of lightpropagation through a random Swiss-cheese model with transparent Lemaıtre-Tolman-Bondiholes performed by Lavinto amp Rasanen [61] exhibited deviations with respect to this rule

In this subsection we show that the effect of anisotropy on cosmological observationsalso violates the unity of

langmicrominus1

rangΩ

For simplicity we consider from now on an axisymmetricBianchi I spacetime with (βi) = (β βminus2β) filled by dust and with no cosmological constantWe then compare the (squared) angular distance-redshift relation D2

A(z) in this anisotropicuniverse to the isotropic case There are however different ways to draw a correspondencebetween a Bianchi I and a FLRW model Are considered here what we believe to be the twomost sensible identifications (i) models with the same volumic expansion rate H0 today or(ii) models with the same matter density ρ0 today In the former case ρBI

0 ρFLRW0 = 1minus Ωσ0

in the latter case HFLRW0 HBI

0 =radic

1minus Ωσ0

The results are plotted in fig 4 for three different values of the anisotropy parameter Ωσ0We see that regardless of the identification scheme between anisotropic and isotropic models(same expansion rate or same density) we get

langmicrominus1

rangΩminus 1 sim Ωσ0 sim σ2 The violation of

eq (453) is therefore of order two in metric perturbations while the analysis of ref [59]suggests that

langmicrominus1

rangΩminus1 = O(4) The fact that Bianchi Imdashwhich can be seen as a perturbation

with respect to FLRW if Ωσ0 1mdashescapes from the eq (453) can be explained by twophenomena First the relation between affine parameter and redshift is much more affected inBianchi I than in a perturbed FLRW model as the impact of anisotropy is cumulative whereasmost of the effects cancel in the statistically homogeneous and isotropic case Second animportant identity leading to eq (453) in a perturbed FLRW universe is that the first-orderconvergence is on average equal to the first-order optical shear

langκ2

rang=langγ2

rang[59] This has

no reason to apply in Bianchi I Both effects combine in a way that eventually violates theconservation of the proper area of iso-z surfaces

These two differences between Bianchi I and FLRW also explain the behaviour oflangmicrominus1

rangΩminus1 with redshift namely positive [

lang(DBI

A )2rang

Ωgt (DFLRW

A )2] at low z and then negativeAt low redshift anisotropic expansion manifests mostly as a modification of the Hubble lawthe ratio between the recession velocity (hence redshift) and distance is not uniform in thesky but depends on the direction in which the source is observed This effect of anisotropy iscompletely equivalent to the presence of a large-scale anisotropic peculiar velocity field withinan otherwise isotropic Universe It has been understood in ref [62] that any peculiar velocity

3The area of an iso-λ surface is indeed

A(λ) =

D2A(λ)d2Ω equiv 4π

langD2

A(λ)rang

Ω (452)

so that A(λ) = AFLRW(λ) is equivalent tolangD2

A(λ)rang

Ω=[DFLRW

A (λ)]2

ndash 19 ndash

Ωσ0=10-1

Ωσ0=10-2

Ωσ0=10-3

00 02 04 06 08 10 12 1410-5

redshift z

|langμ-1rang Ω

Figure 4 Directional average of the inverse magnification in a Bianchi I universe with respectto a FLRW universe defined by microminus1 equiv (DBI

A DFLRWA )2 as a function of redshift z for three

different values of the anisotropy parameter Ωσ0 = 10minus1 10minus2 10minus3 There are two sensibleways to associate a FLRW model with a Bianchi I model either by identifying their Hubbleconstant today H0 (solid lines) or by identifying their matter density today ρ0 (dashed lines)For each curve

langmicrominus1

rangΩminus 1 is first positive and then becomes negative

field generates a positive (Malmquist-like) bias in the sky-averaged distance-redshift relationThe low-z behaviour of

langmicrominus1

rangΩminus 1 is thus in agreement with this general result At larger

redshifts the effect of peculiar velocities decreases progressively dominated by gravitationallensing The fact that

lang[DBI

A (z)]2rang

Ωlt [DFLRW

A (z)]2 in this regime is due to the presence ofWeyl lensing in Bianchi I which tends to focus light (see eg ref [39]) ie magnify imagesand thus reduces the observed angular distance at a given redshift This interpretation is alsoconsistent with the fact that the turnover redshift (when

langmicrominus1

rangΩminus 1 changes sign) is higher

when the Bianchi I and FLRW have the same H0 (solid lines in fig 4) Indeed in this casethe Bianchi I model has a smaller matter density ρBI

0 = (1 minus Ωσ0)ρFLRW0 lt ρFLRW

0 so itsRicci focusing is weaker which counterbalances the presence of Weyl lensing Although thelatter increases faster with redshift and eventually dominates its overall effect is reduced andthe turnover happens at higher z

46 Bias and dispersion of the Hubble diagram

The violation of (453) presented in the previous subsection is mostly a conceptual issue Letus now turn to a more observationally relevant impact of cosmic anisotropy namely the biasand dispersion of the Hubble diagram of SNe ie of the magnitude-redshift relation m(z)Our goal is still essentially illustrative hence we will not try to be realistic and will use thesame setting as in subsec 45 The inclusion of a cosmological constant as well as more

ndash 20 ndash

realistic values for the anisotropy and its sources shall be postponed to future worksWe remind the reader that the apparent magnitude m of a luminous source is defined

from its observed luminosity in such a way that

m = 5 log10DL + cst (454)

where DL is the luminosity distance between the source and the observer related to DA bythe distance-duality relation [39] DL = (1 + z)2DA and where the constant only depends onthe intrinsic properties of the source The Hubble diagram consists in plotting the magnitude(more precisely the distance modulus) of a catalogue of SNe as a function of their redshift Ina homogeneous and isotropic FLRW universe this relation is univocal in particular it doesnot depend on the position of the SN on the sky In a Bianchi I universe however it doesdepend on the direction so that the Hubble diagram gets both biased and spread with respectto the isotropic case

Because the Hubble diagram is constructed from individual sources the relevant statisticsinvolves averages over sources 〈〉N defined in subsec 33 rather than directional averagesThe bias of the Hubble diagram is therefore

∆m(z) equiv 〈mBI(z)〉N minusmFLRW(z) (455)

while its dispersion can be quantified by

σm(z) equivradiclang

m2BI(z)

rangNminus 〈mBI(z)〉2N (456)

Given the Bianchi I model considered here it is straightforward to show that the sourceaverage defined by eq (319) takes the form

〈S〉N (z) =

intd2ΩKN (z θ)Sintd2Ω KN (z θ)

with the integration kernel

KN (z θ) = D2A(z θ)

]3H(z) +

sum3i=1 βi(z)

oiminusβi(z)Di(θ ϕ)

]2sum3i=1

]2minus1

Note that KN does not depend on ϕ because of the axisymmetry of our Bianchi I spacetimeThe behaviours of ∆m and σm are depicted in fig 5 for two different values of the

anisotropy parameter namely Ωσ0 = 10minus2 10minus3 In terms of orders of magnitude the bias is∆m sim Ωσ0 while the dispersion is σm sim 10 Ωσ0 For comparison the typical dispersion of themost recent Hubble diagram [63] is of order 02 mag Note the interesting shrinking of thisdispersion at high redshift due to the fact that the distance-redshift relations DA(z θ) invarious directions θ tend to approximately intersect Such a behaviour is a feature of anisotropywhich cannot be mixed with any other effect such as lensing or peculiar velocities [12] Thiscould be exploited in a component-separation analysis in order to efficiently detect ananisotropy signal from the Hubble diagram with future large SN surveys

It may seem far fetched to try to constrain cosmic anisotropy from the Hubble diagramwhile extremely tight constraints have been put by the lastest analyses of the CMB [22 23]namely Ωσ0 lt 10minus13 (and even lt 10minus20 for scalar and vector anisotropy modes) Howeversuch constraints are based on the analysis of the polarisation of the CMB and thus probe

ndash 21 ndash

Ωσ=-

00 02 04 06 08 10 12 14

redshift z

mBI-mFLRW

Ωσ=-

00 02 04 06 08 10 12 14-006

redshift z

mBI-mFLRW

Figure 5 Bias and dispersion of the Hubble diagram of SNe due to anisotropy with respectto an isotropic model with the same volumic expansion rate H0 today Dashed lines representthe bias ∆m equiv 〈mBI〉N minusmFLRW while solid lines represent ∆mplusmn σm Two values of theanisotropy parameter are considered Ωσ0 = 10minus2 (left panel) and Ωσ0 = 10minus3 (right panel)

primordial anisotropy which decays later on if not sourced This method is much lesssentitive to late-time anisotropy which is a prediction of many phenomenological dark energymodel [64ndash68] bi-gravity theories [69] or backreaction [70] Using low-redshift observationssuch as SNe is therefore complementary to the analyses reported in refs [22 23]

5 Conclusion

The geodesic-light-cone (GLC) coordinate system has been introduced and exploited inseveral previous works in order to facilitate the analysis of cosmological observations in aninhomogeneous universe [6 9ndash13 15ndash17] In this article we presented the first bottom-upconstruction of those coordinates explicitly derived the expression of the GLC metric andexplored its gauge degrees of freedom in details We also reminded the reader the formof the equations of geometric optics in terms of the GLC coordinates and clarified thegeneral expression of the Jacobi matrix by providing a simple interpretation of the constantmatrix [CaA]

The second part of the article was dedicated to the application of GLC coordinates tothe homogeneous and anisotropic Bianchi I cosmological model We derived the expression ofthe associated metric and checked the consistency of its optical predictions with the literatureWe finally exploited our results to analyse the statistical properties of the distance-redshiftrelation of an anisotropic universe On the one hand we showed that the directional averageof the inverse magnification with respect to a FLRW model differs from 1

langmicrominus1(z)

rangΩ6= 1 in

other words the Weinberg conjecture stating that the area of an iso-z surface is unaffectedby gravitational lensing is violated in a Bianchi I spacetime In that respect the effect ofanisotropy is thus different from the one of inhomogeneity as in the latter case the inversemagnification is shown to be unity when averaged over the sky [12 13 59 60] On the otherhand we illustrated the effect of anisotropy on the Hubble diagram which displays a bias onthe order of Ωσ0 mag and a dispersion on the order of 10 Ωσ0 mag The particular behaviourof this dispersion as a function of the redshift quite distinct from the effects of inhomogeneityarises as an interesting property to extract any anisotropy signal from future SN observations

This last point clearly calls for more observationally-oriented follow-up works It could becomplemented by other lensing observables such as the cosmic shear generated by anisotropic

ndash 22 ndash

expansion Furthermore the theoretical results presented here should facilitate the analysisof light propagation through anisotropic and inhomogeneous cosmological modelsmdashsuch asperturbed Bianchi I spacetimesmdashin the same spirit of what has been already done withperturbed FLRW spacetimes Finally the whole analysis presented here could be repeatedfor the other types of Bianchi models

Acknowledgements

PF and GF thank Giovanni Marozzi for stimulating debates on the GLC coordinates duringthe early stages of this project PF thanks Chris Clarkson for clarifying discussions on averagesin gravitational lensing and we all warmy thank Chris Clarkson Roy Maartens GiovanniMarozzi Syksy Rasanen and Gabriele Veneziano for valuable comments on a preliminaryversion of this article PFrsquos research is funded by the South African National ResearchFoundation The financial assistance of the National Research Foundation (NRF) towards thisresearch is hereby acknowledged Opinions expressed and conclusions arrived at are those ofthe author and are not necessarily to be attributed to the NRF FNrsquos research was supportedby the project GLENCO (FP7 Ideas Grant Agreement n 259349) during the first stages ofthis project His research is now supported by the Leung Center for Cosmology and ParticleAstrophysics (LeCosPA) of National Taiwan University GFrsquos work is supported by theresearch grant ldquoTheoretical Astroparticle Physicsrdquo number 2012CPPYP7 under the programPRIN 2012 funded by the Ministero dellrsquoIstruzione Universita e della Ricerca (MIUR) and bythe Italian Istituto Nazionale di Fisica Nucleare (INFN) through the ldquoTheoretical AstroparticlePhysicsrdquo project GF also wish to thank the University of Geneva for its hospitality duringpart of the elaboration of this document

References

[1] P Peter and J Uzan Primordial Cosmology Oxford Graduate Texts OUP Oxford 2013

[2] G Temple New Systems of Normal Co-ordinates for Relativistic Optics Royal Society of LondonProceedings Series A 168 (Oct 1938) 122ndash148

[3] R Maartens Idealised observations in relativistic cosmology PhD thesis University of CapeTown 1980

[4] R Maartens and D R Matravers Isotropic and semi-isotropic observations in cosmologyClassical and Quantum Gravity 11 (Nov 1994) 2693ndash2704

[5] G F R Ellis S D Nel R Maartens W R Stoeger and A P Whitman Ideal observationalcosmology PhysRep 124 (1985) 315ndash417

[6] M Gasperini G Marozzi F Nugier and G Veneziano Light-cone averaging in cosmologyFormalism and applications JCAP 1107 (2011) 008 [arXiv11041167]

[7] H L Bester J Larena P J van der Walt and N T Bishop Whatrsquos Inside the ConeNumerically reconstructing the metric from observations JCAP 1402 (2014) 009[arXiv13121081]

[8] H L Bester J Larena and N T Bishop Towards the geometry of the Universe from dataMNRAS 453 (Nov 2015) 2364ndash2377 [arXiv150601591]

[9] I Ben-Dayan M Gasperini G Marozzi F Nugier and G Veneziano Backreaction on theluminosity-redshift relation from gauge invariant light-cone averaging JCAP 1204 (2012) 036[arXiv12021247]

ndash 23 ndash

[10] I Ben-Dayan G Marozzi F Nugier and G Veneziano The second-order luminosity-redshiftrelation in a generic inhomogeneous cosmology JCAP 1211 (2012) 045 [arXiv12094326]

[11] G Marozzi The luminosity distance-redshift relation up to second order in the Poisson gaugewith anisotropic stress Class Quant Grav 32 (2015) no 4 045004 [arXiv14061135][Corrigendum Class Quant Grav32179501(2015)]

[12] I Ben-Dayan M Gasperini G Marozzi F Nugier and G Veneziano Do stochasticinhomogeneities affect dark-energy precision measurements Phys Rev Lett 110 (2013) 021301[arXiv12071286]

[13] I Ben-Dayan M Gasperini G Marozzi F Nugier and G Veneziano Average and dispersion ofthe luminosity-redshift relation in the concordance model JCAP 1306 (2013) 002[arXiv13020740]

[14] I Ben-Dayan R Durrer G Marozzi and D J Schwarz The value of H0 in the inhomogeneousUniverse Phys Rev Lett 112 (2014) 221301 [arXiv14017973]

[15] G Fanizza and F Nugier Lensing in the geodesic light-cone coordinates and its (exact)illustration to an off-center observer in Lemaıtre-Tolman-Bondi models JCAP 1502 (2015)no 02 002 [arXiv14081604]

[16] G Fanizza M Gasperini G Marozzi and G Veneziano A new approach to the propagation oflight-like signals in perturbed cosmological backgrounds JCAP 1508 (2015) no 08 020[arXiv150602003]

[17] E Di Dio R Durrer G Marozzi and F Montanari Galaxy number counts to second order andtheir bispectrum JCAP 1412 (2014) 017 [arXiv14070376] [ErratumJCAP1506no06E01(2015)]

[18] G Fanizza M Gasperini G Marozzi and G Veneziano Time of flight of ultra-relativisticparticles in a realistic Universe a viable tool for fundamental physics arXiv151208489

[19] F Nugier The Geodesic Light-Cone Coordinates an Adapted System for Light-Signal-BasedCosmology arXiv150807464

[20] A Pontzen and A Challinor Bianchi model CMB polarization and its implications for CMBanomalies MNRAS 380 (Oct 2007) 1387ndash1398 [arXiv07062075]

[21] Planck Collaboration P A R Ade N Aghanim M Arnaud M Ashdown J AumontC Baccigalupi A J Banday R B Barreiro N Bartolo and et al Planck 2015 results XVIIIBackground geometry and topology ArXiv e-prints (Feb 2015) [arXiv150201593]

[22] D Saadeh S M Feeney A Pontzen H V Peiris and J D McEwen A framework for testingisotropy with the cosmic microwave background ArXiv e-prints (Apr 2016)[arXiv160401024]

[23] D Saadeh S M Feeney A Pontzen H V Peiris and J D McEwen How isotropic is theUniverse ArXiv e-prints (May 2016) [arXiv160507178]

[24] T S Kolatt and O Lahav Constraints on cosmological anisotropy out to z = 1 from Type Iasupernovae MNRAS 323 (May 2001) 859ndash864 [astro-ph0008041]

[25] D J Schwarz and B Weinhorst (An)isotropy of the Hubble diagram comparing hemispheresAampA 474 (Nov 2007) 717ndash729 [arXiv07060165]

[26] T Koivisto and D F Mota Anisotropic dark energy dynamics of the background andperturbations JCAP 6 (June 2008) 18 [arXiv08013676]

[27] M Blomqvist J Enander and E Mortsell Constraining dark energy fluctuations with supernovacorrelations JCAP 10 (Oct 2010) 18 [arXiv10064638]

[28] I Antoniou and L Perivolaropoulos Searching for a cosmological preferred axis Union2 dataanalysis and comparison with other probes JCAP 12 (Dec 2010) 12 [arXiv10074347]

ndash 24 ndash

[29] J Colin R Mohayaee S Sarkar and A Shafieloo Probing the anisotropic local Universe andbeyond with SNe Ia data MNRAS 414 (June 2011) 264ndash271 [arXiv10116292]

[30] B Kalus D J Schwarz M Seikel and A Wiegand Constraints on anisotropic cosmicexpansion from supernovae AampA 553 (May 2013) A56 [arXiv12123691]

[31] R-G Cai Y-Z Ma B Tang and Z-L Tuo Constraining the anisotropic expansion of theUniverse Phys Rev D 87 (June 2013) 123522 [arXiv13030961]

[32] T Schucker A Tilquin and G Valent Bianchi I meets the Hubble diagram MNRAS 444 (Nov2014) 2820ndash2836 [arXiv14056523]

[33] B Javanmardi C Porciani P Kroupa and J Pflamm-Altenburg Probing the Isotropy ofCosmic Acceleration Traced By Type Ia Supernovae ApJ 810 (Sept 2015) 47[arXiv150707560]

[34] C Pitrou J-P Uzan and T S Pereira Weak lensing B modes on all scales as a probe of localisotropy Phys Rev D 87 (Feb 2013) 043003 [arXiv12036029]

[35] C Pitrou T S Pereira and J-P Uzan Weak lensing by the large scale structure in a spatiallyanisotropic universe Theory and predictions Phys Rev D 92 (July 2015) 023501[arXiv150301125]

[36] T S Pereira C Pitrou and J-P Uzan Weak-lensing $B$-modes as a probe of the isotropy ofthe universe ArXiv e-prints (Mar 2015) [arXiv150301127]

[37] E Poisson A relativistrsquos toolkit the mathematics of black-hole mechanics 2004

[38] G Fanizza M Gasperini G Marozzi and G Veneziano An exact Jacobi map in the geodesiclight-cone gauge JCAP 1311 (2013) 019 [arXiv13084935]

[39] P Fleury Light propagation in inhomogeneous and anisotropic cosmologies PhD thesis Institutdrsquoastrophysique de Paris Universite Pierre et Marie Curie Paris 6 2015 arXiv151103702

[40] F Nugier Lightcone Averaging and Precision Cosmology PhD thesis Universite Pierre et MarieCurie Paris 6 2013 arXiv13096542

[41] G F R Ellis and M A H MacCallum A class of homogeneous cosmological modelsCommunications in Mathematical Physics 12 (June 1969) 108ndash141

[42] L Bianchi Sugli spazii a tre dimensioni che ammettono un gruppo continuo di movimenti (onthe spaces of three dimensions that admit a continuous group of movements) Soc Ital Sci MemMat 11 (1898) 267

[43] P T Saunders Observations in homogeneous model universes Month Not R Astron Soc 141(1968) 427

[44] P T Saunders Observations in some simple cosmological models with shear Month Not RAstron Soc 142 (1969) 213

[45] P Fleury C Pitrou and J-P Uzan Light propagation in a homogeneous and anisotropicuniverse ArXiv e-prints (Oct 2014) [arXiv14108473]

[46] S Weinberg Apparent luminosities in a locally inhomogeneous universe ApJL 208 (Aug 1976)L1ndashL3

[47] B Bertotti The Luminosity of Distant Galaxies Royal Society of London Proceedings Series A294 (Sept 1966) 195ndash207

[48] N Mustapha B A Bassett C Hellaby and G F R Ellis Shrinking 2 The Distortion of thearea distance redshift relation in inhomogeneous isotropic universes Class Quant Grav 15(1998) 2363ndash2379 [gr-qc9708043]

[49] G F R Ellis B A Bassett and P K S Dunsby Lensing and caustic effects on cosmologicaldistances Class Quant Grav 15 (1998) 2345ndash2361 [gr-qc9801092]

ndash 25 ndash

[50] G F R Ellis and D M Solomons Caustics of compensated spherical lens models Class QuantGrav 15 (1998) 2381ndash2396 [gr-qc9802005]

[51] K Enqvist M Mattsson and G Rigopoulos Supernovae data and perturbative deviation fromhomogeneity JCAP 0909 (2009) 022 [arXiv09074003]

[52] N Mustapha C Hellaby and G F R Ellis Large scale inhomogeneity versus source evolutionCan we distinguish them observationally Mon Not Roy Astron Soc 292 (1997) 817ndash830[gr-qc9808079]

[53] K Enqvist Lemaitre-Tolman-Bondi model and accelerating expansion Gen Rel Grav 40 (2008)451ndash466 [arXiv07092044]

[54] J Garcia-Bellido and T Haugboelle Looking the void in the eyes - the kSZ effect in LTB modelsJCAP 0809 (2008) 016 [arXiv08071326]

[55] K Bolejko M-N Celerier and A Krasinski Inhomogeneous cosmological models Exactsolutions and their applications Class Quant Grav 28 (2011) 164002 [arXiv11021449]

[56] Sundell Peter and Mortsell Edvard and Vilja Iiro Can a void mimic the Λ in ΛCDM JCAP1508 (2015) 037 [arXiv150308045]

[57] Lavinto Mikko and Rasanen Syksy and Szybka Sebastian J Average expansion rate and lightpropagation in a cosmological Tardis spacetime JCAP 1312 (2013) 051 [arXiv13086731]

[58] T W B Kibble and R Lieu Average Magnification Effect of Clumping of Matter ApJ 632(Oct 2005) 718ndash726 [astro-ph0412275]

[59] C Bonvin C Clarkson R Durrer R Maartens and O Umeh Cosmological ensemble anddirectional averages of observables JCAP 7 (July 2015) 40 [arXiv150401676]

[60] N Kaiser and J A Peacock On the Bias of the Distance-Redshift Relation from GravitationalLensing arXiv150308506

[61] M Lavinto and S Rasanen CMB seen through random Swiss Cheese JCAP 10 (Oct 2015) 057[arXiv150706590]

[62] N Kaiser and M J Hudson Kinematic bias in cosmological distance measurement MNRAS 454(Nov 2015) 280ndash286 [arXiv150201762]

[63] M Betoule R Kessler J Guy J Mosher D Hardin R Biswas P Astier P El-HageM Konig S Kuhlmann J Marriner R Pain N Regnault C Balland B A Bassett P JBrown H Campbell R G Carlberg F Cellier-Holzem D Cinabro A Conley C B DrsquoAndreaD L DePoy M Doi R S Ellis S Fabbro A V Filippenko R J Foley J A FriemanD Fouchez L Galbany A Goobar R R Gupta G J Hill R Hlozek C J Hogan I M HookD A Howell S W Jha L Le Guillou G Leloudas C Lidman J L Marshall A Moller A MMourao J Neveu R Nichol M D Olmstead N Palanque-Delabrouille S Perlmutter J LPrieto C J Pritchet M Richmond A G Riess V Ruhlmann-Kleider M SakoK Schahmaneche D P Schneider M Smith J Sollerman M Sullivan N A Walton and C JWheeler Improved cosmological constraints from a joint analysis of the SDSS-II and SNLSsupernova samples AampA 568 (Aug 2014) A22 [arXiv14014064]

[64] M Bucher and D N Spergel Is the dark matter a solid Phys Rev D 60 (1999) 043505[[astro-ph9812022]]

[65] D Mota J Kristiansen T Koivisto and N Groeneboom Constraining Dark EnergyAnisotropic Stress Mon Not R Astron Soc 382 (2007) 793ndash800 [[arXiv07080830]]

[66] T Koivisto and D F Mota Accelerating Cosmologies with an Anisotropic Equation of StateAstrophys J 679 (2008) 1ndash5 [arXiv07070279]

[67] S A Appleby and E V Linder Probing Dark Energy Anisotropy Phys Rev D 87 (2013)023532 [[arXiv12108221]]

ndash 26 ndash

[68] S Appleby R Battye and A Moss Constraints on the anisotropy of dark energy Phys Rev D81 (2010) 081301 [[arXiv09120397]]

[69] T Damour I I Kogan and A Papazoglou Nonlinear bigravity and cosmic acceleration PhysRev D 66 (2002) 104025 [[hep-th0206044]]

[70] G Marozzi and J-P Uzan Late time anisotropy as an imprint of cosmological backreactionPhys Rev D 86 (2012) 063528 [[arXiv12064887]]

ndash 27 ndash

1 Introduction



23 Gauge freedom






31 Light rays

32 Light beams

321 Sachs basis

322 Jacobi matrix



4 Bianchi I and GLC







43 Jacobi matrix




5 Conclusion

Contents

1 Introduction 1

2 The GLC coordinate system 2

21 Geometrical construction 2

22 Coordinate basis and expression of the metric 3

23 Gauge freedom 6

231 Relabelling lightcones 6

232 Relabelling light rays 7

233 Reparametrising light rays 7

24 Conformal transformations 8

3 Optics with GLC coordinates 9

31 Light rays 9

32 Light beams 10

321 Sachs basis 10

322 Jacobi matrix 10

33 Lightcone averages 12

4 Bianchi I and GLC 13

41 The Bianchi I spacetime 13

42 Coordinate transformation 14

421 The conformal trick 14

422 Choosing the Bianchi GLC coordinates 14

423 Mapping between comoving and GLC coordinates 15

424 GLC form of the Bianchi I metric 16

43 Jacobi matrix 17

45 Directional average of the inverse magnification 18

46 Bias and dispersion of the Hubble diagram 20

5 Conclusion 22

1 Introduction

The current standard cosmological model provides an excellent description of the Universein agreement with the great majority of observations This success even allows one to testfundamental physics from particles to gravitation However the outcome of such tests as wellas our understanding of the Universe strongly depend on the reliability of the cosmologicalprinciple which states that our Universe is statistically homogeneous and isotropic [1] It istherefore crucial to independently test those hypotheses

A sensible approach to this issue consists in describing the cosmological spacetime interms of coordinates adapted to observations rather than to peculiar symmetries Examplesof such coordinate systems were proposed in the literature since the late 1930s with Templersquos

ndash 1 ndash

ndash 2 ndash

ndash 3 ndash

Eτ = cst

τ + dτ

Eprimepartτ prop k

τ = cst

Eprime

Eτ = cst

Eprime

k = ω(u+ d) (25)

ndash 4 ndash

EEprime

partwparta

minusd

w + dwLo

coordinates read

[gGLCmicroν ] =

T γab

[gmicroνGLC] =

ndash 5 ndash

d`2 simErarrLo

23 Gauge freedom

dwprime (215)

Υ|Lo =dτo

dw (219)

ndash 6 ndash

dtdxmicroE

Eprime

ndash 7 ndash

δmicroτ (226)

kw = kow = minusωo

ndash 8 ndash

w = w θa = θa dτ = Ωdτ (231)

31 Light rays

1 + z equiv ωs

Υs (32)

ndash 9 ndash

32 Light beams

321 Sachs basis

sA = saAparta (33)

γab = δAB sAa s

Bb (34)

SmicroνDsνAdλ

= 0 (35)

or more explicitly

partτsaA =

2(γacpartτγbc)s

bA (36)

322 Jacobi matrix

CAa equiv(

dsAadλ

ndash 10 ndash

rpartθ +O(r) s2 =

[s1 s2

]=[partθ partϕ

]middot 1

[1 00 1

sin θ

]︸︷︷︸

+O(r) so [sAa ] = r

[1 00 sin θ

]+O(r3) (312)

[CAa ] = minusωo

[1 00 sin θ

] [CaA] = minus 1

[1 00 1

sin θ

] (313)

[CaA] = minus 1

] [1 00 1

sin θ

] (314)

D2A = ω2

o detD (315)

ndash 11 ndash

det[γab]times1

ω2o sin θ

radicγ

sin θ(317)

〈S〉Ω equiv1

intS d2Ω (318)

〈S〉N equiv1

intS d2N =

Σradicγ dθdϕ

Σ = ρ

= ρΥ2

(partΥ

partτ

)minus1

〈S〉N =

intS ΣD2

Ad2ΩintΣD2

Ad2Ω=

langS ΣD2

rangΩlang

rangΩ

ndash 12 ndash

4 Bianchi I and GLC

3sumi=1

βi = 0 (42)

σij equiv1

2(γij)

prime (43)

3sumi=1

(βprimei)2 (44)

H2 =8πGρ

)+ Ωσ0

)4] (45)

ndash 13 ndash

0(6H20)

ds2 = minusdη2 +

3sumi=1

[eβi(η) dxi

]2 (47)

ndash 14 ndash

int λ

dλprime ki =

int η

ki = koi = ωod

ω2 = gij kikj =

3sumi=1

[eminusβi(η)ki

]2 (413)

int η

3sumj=1

minus12

partxi

partηo= minusdio +

int η

dηprimepartdi

partxi

partθa=

int η

dηprimepartdi

kiIiηo

= kiIia = 0 (418)

0 = part(did

= gijpart(didj

)= 2dipartd

ndash 15 ndash

= minusdη2 + gij

[didη + (Iiηo

io minus Iiηo

)Iia (425)

Υ =a(t)

3sumi=1

]2minus12

γab = gijIiaIjb

1 + z =a(to)

]2 (430)

IiaIbj = δba IiaI

Uao = minus(gijd

ipartadj)

o (432)

ndash 16 ndash

43 Jacobi matrix

Iij equivint η

in subsec 321

therefore

Iia equivint ηo

ηdηprime partad

(int ηo

DAB = sAa CaB (437)

ia (438)

dsAadλ

partη

∣∣∣∣ηoθϕ

(IiasAi ) (439)

Ai ) (440)

ndash 17 ndash

(Sjl)o

(partbkl)o (443)

l)o (445)

= Iai Iib (447)

= δab (448)

sin θ

e2βi(Ii1)2

)(3sumi=1

e2βi(Ii2)2

)minus

(3sumi=1

e2βiIi1Ii2

formulaD2

JiJj(k`)2 (450)

Ji equivint ηo

i ) (451)

ndash 18 ndash

langD2

A(λ)rang

langmicrominus1(z)

rangΩequiv

lang[DA(z)

DFLRWA (z)

]2rang

= 1 (453)

langmicrominus1

rangΩ

0 =radic

1minus Ωσ0

langmicrominus1

langκ2

rang=langγ2

rang[59] This has

lang(DBI

A )2rang

Ωgt (DFLRW

A(λ) =

langD2

A(λ)rang

Ω (452)

A(λ)rang

Ω=[DFLRW

A (λ)]2

ndash 19 ndash

Ωσ0=10-1

Ωσ0=10-2

Ωσ0=10-3

00 02 04 06 08 10 12 1410-5

redshift z

|langμ-1rang Ω

langmicrominus1

lang[DBI

A (z)]2rang

Ωlt [DFLRW

langmicrominus1

ndash 20 ndash

m = 5 log10DL + cst (454)

m2BI(z)

〈S〉N (z) =

]3H(z) +

sum3i=1 βi(z)

]2sum3i=1

]2minus1

ndash 21 ndash

Ωσ=-

00 02 04 06 08 10 12 14

redshift z

mBI-mFLRW

Ωσ=-

00 02 04 06 08 10 12 14-006

redshift z

mBI-mFLRW

5 Conclusion

langmicrominus1(z)

rangΩ6= 1 in

ndash 22 ndash

Acknowledgements

References

ndash 23 ndash

ndash 24 ndash

ndash 25 ndash

ndash 26 ndash

ndash 27 ndash

1 Introduction



23 Gauge freedom






31 Light rays

32 Light beams

321 Sachs basis

322 Jacobi matrix



4 Bianchi I and GLC







43 Jacobi matrix




5 Conclusion

ndash 2 ndash

ndash 3 ndash

Eτ = cst

τ + dτ

Eprimepartτ prop k

τ = cst

Eprime

Eτ = cst

Eprime

k = ω(u+ d) (25)

ndash 4 ndash

EEprime

partwparta

minusd

w + dwLo

coordinates read

[gGLCmicroν ] =

T γab

[gmicroνGLC] =

ndash 5 ndash

d`2 simErarrLo

23 Gauge freedom

dwprime (215)

Υ|Lo =dτo

dw (219)

ndash 6 ndash

dtdxmicroE

Eprime

ndash 7 ndash

δmicroτ (226)

kw = kow = minusωo

ndash 8 ndash

w = w θa = θa dτ = Ωdτ (231)

31 Light rays

1 + z equiv ωs

Υs (32)

ndash 9 ndash

32 Light beams

321 Sachs basis

sA = saAparta (33)

γab = δAB sAa s

Bb (34)

SmicroνDsνAdλ

= 0 (35)

or more explicitly

partτsaA =

2(γacpartτγbc)s

bA (36)

322 Jacobi matrix

CAa equiv(

dsAadλ

ndash 10 ndash

rpartθ +O(r) s2 =

[s1 s2

]=[partθ partϕ

]middot 1

[1 00 1

sin θ

]︸︷︷︸

+O(r) so [sAa ] = r

[1 00 sin θ

]+O(r3) (312)

[CAa ] = minusωo

[1 00 sin θ

] [CaA] = minus 1

[1 00 1

sin θ

] (313)

[CaA] = minus 1

] [1 00 1

sin θ

] (314)

D2A = ω2

o detD (315)

ndash 11 ndash

det[γab]times1

ω2o sin θ

radicγ

sin θ(317)

〈S〉Ω equiv1

intS d2Ω (318)

〈S〉N equiv1

intS d2N =

Σradicγ dθdϕ

Σ = ρ

= ρΥ2

(partΥ

partτ

)minus1

〈S〉N =

intS ΣD2

Ad2ΩintΣD2

Ad2Ω=

langS ΣD2

rangΩlang

rangΩ

ndash 12 ndash

4 Bianchi I and GLC

3sumi=1

βi = 0 (42)

σij equiv1

2(γij)

prime (43)

3sumi=1

(βprimei)2 (44)

H2 =8πGρ

)+ Ωσ0

)4] (45)

ndash 13 ndash

0(6H20)

ds2 = minusdη2 +

3sumi=1

[eβi(η) dxi

]2 (47)

ndash 14 ndash

int λ

dλprime ki =

int η

ki = koi = ωod

ω2 = gij kikj =

3sumi=1

[eminusβi(η)ki

]2 (413)

int η

3sumj=1

minus12

partxi

partηo= minusdio +

int η

dηprimepartdi

partxi

partθa=

int η

dηprimepartdi

kiIiηo

= kiIia = 0 (418)

0 = part(did

= gijpart(didj

)= 2dipartd

ndash 15 ndash

= minusdη2 + gij

[didη + (Iiηo

io minus Iiηo

)Iia (425)

Υ =a(t)

3sumi=1

]2minus12

γab = gijIiaIjb

1 + z =a(to)

]2 (430)

IiaIbj = δba IiaI

Uao = minus(gijd

ipartadj)

o (432)

ndash 16 ndash

43 Jacobi matrix

Iij equivint η

in subsec 321

therefore

Iia equivint ηo

ηdηprime partad

(int ηo

DAB = sAa CaB (437)

ia (438)

dsAadλ

partη

∣∣∣∣ηoθϕ

(IiasAi ) (439)

Ai ) (440)

ndash 17 ndash

(Sjl)o

(partbkl)o (443)

l)o (445)

= Iai Iib (447)

= δab (448)

sin θ

e2βi(Ii1)2

)(3sumi=1

e2βi(Ii2)2

)minus

(3sumi=1

e2βiIi1Ii2

formulaD2

JiJj(k`)2 (450)

Ji equivint ηo

i ) (451)

ndash 18 ndash

langD2

A(λ)rang

langmicrominus1(z)

rangΩequiv

lang[DA(z)

DFLRWA (z)

]2rang

= 1 (453)

langmicrominus1

rangΩ

0 =radic

1minus Ωσ0

langmicrominus1

langκ2

rang=langγ2

rang[59] This has

lang(DBI

A )2rang

Ωgt (DFLRW

A(λ) =

langD2

A(λ)rang

Ω (452)

A(λ)rang

Ω=[DFLRW

A (λ)]2

ndash 19 ndash

Ωσ0=10-1

Ωσ0=10-2

Ωσ0=10-3

00 02 04 06 08 10 12 1410-5

redshift z

|langμ-1rang Ω

langmicrominus1

lang[DBI

A (z)]2rang

Ωlt [DFLRW

langmicrominus1

ndash 20 ndash

m = 5 log10DL + cst (454)

m2BI(z)

〈S〉N (z) =

]3H(z) +

sum3i=1 βi(z)

]2sum3i=1

]2minus1

ndash 21 ndash

Ωσ=-

00 02 04 06 08 10 12 14

redshift z

mBI-mFLRW

Ωσ=-

00 02 04 06 08 10 12 14-006

redshift z

mBI-mFLRW

5 Conclusion

langmicrominus1(z)

rangΩ6= 1 in

ndash 22 ndash

Acknowledgements

References

ndash 23 ndash

ndash 24 ndash

ndash 25 ndash

ndash 26 ndash

ndash 27 ndash

1 Introduction



23 Gauge freedom






31 Light rays

32 Light beams

321 Sachs basis

322 Jacobi matrix



4 Bianchi I and GLC







43 Jacobi matrix




5 Conclusion

ndash 3 ndash

Eτ = cst

τ + dτ

Eprimepartτ prop k

τ = cst

Eprime

Eτ = cst

Eprime

k = ω(u+ d) (25)

ndash 4 ndash

EEprime

partwparta

minusd

w + dwLo

coordinates read

[gGLCmicroν ] =

T γab

[gmicroνGLC] =

ndash 5 ndash

d`2 simErarrLo

23 Gauge freedom

dwprime (215)

Υ|Lo =dτo

dw (219)

ndash 6 ndash

dtdxmicroE

Eprime

ndash 7 ndash

δmicroτ (226)

kw = kow = minusωo

ndash 8 ndash

w = w θa = θa dτ = Ωdτ (231)

31 Light rays

1 + z equiv ωs

Υs (32)

ndash 9 ndash

32 Light beams

321 Sachs basis

sA = saAparta (33)

γab = δAB sAa s

Bb (34)

SmicroνDsνAdλ

= 0 (35)

or more explicitly

partτsaA =

2(γacpartτγbc)s

bA (36)

322 Jacobi matrix

CAa equiv(

dsAadλ

ndash 10 ndash

rpartθ +O(r) s2 =

[s1 s2

]=[partθ partϕ

]middot 1

[1 00 1

sin θ

]︸︷︷︸

+O(r) so [sAa ] = r

[1 00 sin θ

]+O(r3) (312)

[CAa ] = minusωo

[1 00 sin θ

] [CaA] = minus 1

[1 00 1

sin θ

] (313)

[CaA] = minus 1

] [1 00 1

sin θ

] (314)

D2A = ω2

o detD (315)

ndash 11 ndash

det[γab]times1

ω2o sin θ

radicγ

sin θ(317)

〈S〉Ω equiv1

intS d2Ω (318)

〈S〉N equiv1

intS d2N =

Σradicγ dθdϕ

Σ = ρ

= ρΥ2

(partΥ

partτ

)minus1

〈S〉N =

intS ΣD2

Ad2ΩintΣD2

Ad2Ω=

langS ΣD2

rangΩlang

rangΩ

ndash 12 ndash

4 Bianchi I and GLC

3sumi=1

βi = 0 (42)

σij equiv1

2(γij)

prime (43)

3sumi=1

(βprimei)2 (44)

H2 =8πGρ

)+ Ωσ0

)4] (45)

ndash 13 ndash

0(6H20)

ds2 = minusdη2 +

3sumi=1

[eβi(η) dxi

]2 (47)

ndash 14 ndash

int λ

dλprime ki =

int η

ki = koi = ωod

ω2 = gij kikj =

3sumi=1

[eminusβi(η)ki

]2 (413)

int η

3sumj=1

minus12

partxi

partηo= minusdio +

int η

dηprimepartdi

partxi

partθa=

int η

dηprimepartdi

kiIiηo

= kiIia = 0 (418)

0 = part(did

= gijpart(didj

)= 2dipartd

ndash 15 ndash

= minusdη2 + gij

[didη + (Iiηo

io minus Iiηo

)Iia (425)

Υ =a(t)

3sumi=1

]2minus12

γab = gijIiaIjb

1 + z =a(to)

]2 (430)

IiaIbj = δba IiaI

Uao = minus(gijd

ipartadj)

o (432)

ndash 16 ndash

43 Jacobi matrix

Iij equivint η

in subsec 321

therefore

Iia equivint ηo

ηdηprime partad

(int ηo

DAB = sAa CaB (437)

ia (438)

dsAadλ

partη

∣∣∣∣ηoθϕ

(IiasAi ) (439)

Ai ) (440)

ndash 17 ndash

(Sjl)o

(partbkl)o (443)

l)o (445)

= Iai Iib (447)

= δab (448)

sin θ

e2βi(Ii1)2

)(3sumi=1

e2βi(Ii2)2

)minus

(3sumi=1

e2βiIi1Ii2

formulaD2

JiJj(k`)2 (450)

Ji equivint ηo

i ) (451)

ndash 18 ndash

langD2

A(λ)rang

langmicrominus1(z)

rangΩequiv

lang[DA(z)

DFLRWA (z)

]2rang

= 1 (453)

langmicrominus1

rangΩ

0 =radic

1minus Ωσ0

langmicrominus1

langκ2

rang=langγ2

rang[59] This has

lang(DBI

A )2rang

Ωgt (DFLRW

A(λ) =

langD2

A(λ)rang

Ω (452)

A(λ)rang

Ω=[DFLRW

A (λ)]2

ndash 19 ndash

Ωσ0=10-1

Ωσ0=10-2

Ωσ0=10-3

00 02 04 06 08 10 12 1410-5

redshift z

|langμ-1rang Ω

langmicrominus1

lang[DBI

A (z)]2rang

Ωlt [DFLRW

langmicrominus1

ndash 20 ndash

m = 5 log10DL + cst (454)

m2BI(z)

〈S〉N (z) =

]3H(z) +

sum3i=1 βi(z)

]2sum3i=1

]2minus1

ndash 21 ndash

Ωσ=-

00 02 04 06 08 10 12 14

redshift z

mBI-mFLRW

Ωσ=-

00 02 04 06 08 10 12 14-006

redshift z

mBI-mFLRW

5 Conclusion

langmicrominus1(z)

rangΩ6= 1 in

ndash 22 ndash

Acknowledgements

References

ndash 23 ndash

ndash 24 ndash

ndash 25 ndash

ndash 26 ndash

ndash 27 ndash

1 Introduction



23 Gauge freedom






31 Light rays

32 Light beams

321 Sachs basis

322 Jacobi matrix



4 Bianchi I and GLC







43 Jacobi matrix




5 Conclusion

Eτ = cst

τ + dτ

Eprimepartτ prop k

τ = cst

Eprime

Eτ = cst

Eprime

k = ω(u+ d) (25)

ndash 4 ndash

EEprime

partwparta

minusd

w + dwLo

coordinates read

[gGLCmicroν ] =

T γab

[gmicroνGLC] =

ndash 5 ndash

d`2 simErarrLo

23 Gauge freedom

dwprime (215)

Υ|Lo =dτo

dw (219)

ndash 6 ndash

dtdxmicroE

Eprime

ndash 7 ndash

δmicroτ (226)

kw = kow = minusωo

ndash 8 ndash

w = w θa = θa dτ = Ωdτ (231)

31 Light rays

1 + z equiv ωs

Υs (32)

ndash 9 ndash

32 Light beams

321 Sachs basis

sA = saAparta (33)

γab = δAB sAa s

Bb (34)

SmicroνDsνAdλ

= 0 (35)

or more explicitly

partτsaA =

2(γacpartτγbc)s

bA (36)

322 Jacobi matrix

CAa equiv(

dsAadλ

ndash 10 ndash

rpartθ +O(r) s2 =

[s1 s2

]=[partθ partϕ

]middot 1

[1 00 1

sin θ

]︸︷︷︸

+O(r) so [sAa ] = r

[1 00 sin θ

]+O(r3) (312)

[CAa ] = minusωo

[1 00 sin θ

] [CaA] = minus 1

[1 00 1

sin θ

] (313)

[CaA] = minus 1

] [1 00 1

sin θ

] (314)

D2A = ω2

o detD (315)

ndash 11 ndash

det[γab]times1

ω2o sin θ

radicγ

sin θ(317)

〈S〉Ω equiv1

intS d2Ω (318)

〈S〉N equiv1

intS d2N =

Σradicγ dθdϕ

Σ = ρ

= ρΥ2

(partΥ

partτ

)minus1

〈S〉N =

intS ΣD2

Ad2ΩintΣD2

Ad2Ω=

langS ΣD2

rangΩlang

rangΩ

ndash 12 ndash

4 Bianchi I and GLC

3sumi=1

βi = 0 (42)

σij equiv1

2(γij)

prime (43)

3sumi=1

(βprimei)2 (44)

H2 =8πGρ

)+ Ωσ0

)4] (45)

ndash 13 ndash

0(6H20)

ds2 = minusdη2 +

3sumi=1

[eβi(η) dxi

]2 (47)

ndash 14 ndash

int λ

dλprime ki =

int η

ki = koi = ωod

ω2 = gij kikj =

3sumi=1

[eminusβi(η)ki

]2 (413)

int η

3sumj=1

minus12

partxi

partηo= minusdio +

int η

dηprimepartdi

partxi

partθa=

int η

dηprimepartdi

kiIiηo

= kiIia = 0 (418)

0 = part(did

= gijpart(didj

)= 2dipartd

ndash 15 ndash

= minusdη2 + gij

[didη + (Iiηo

io minus Iiηo

)Iia (425)

Υ =a(t)

3sumi=1

]2minus12

γab = gijIiaIjb

1 + z =a(to)

]2 (430)

IiaIbj = δba IiaI

Uao = minus(gijd

ipartadj)

o (432)

ndash 16 ndash

43 Jacobi matrix

Iij equivint η

in subsec 321

therefore

Iia equivint ηo

ηdηprime partad

(int ηo

DAB = sAa CaB (437)

ia (438)

dsAadλ

partη

∣∣∣∣ηoθϕ

(IiasAi ) (439)

Ai ) (440)

ndash 17 ndash

(Sjl)o

(partbkl)o (443)

l)o (445)

= Iai Iib (447)

= δab (448)

sin θ

e2βi(Ii1)2

)(3sumi=1

e2βi(Ii2)2

)minus

(3sumi=1

e2βiIi1Ii2

formulaD2

JiJj(k`)2 (450)

Ji equivint ηo

i ) (451)

ndash 18 ndash

langD2

A(λ)rang

langmicrominus1(z)

rangΩequiv

lang[DA(z)

DFLRWA (z)

]2rang

= 1 (453)

langmicrominus1

rangΩ

0 =radic

1minus Ωσ0

langmicrominus1

langκ2

rang=langγ2

rang[59] This has

lang(DBI

A )2rang

Ωgt (DFLRW

A(λ) =

langD2

A(λ)rang

Ω (452)

A(λ)rang

Ω=[DFLRW

A (λ)]2

ndash 19 ndash

Ωσ0=10-1

Ωσ0=10-2

Ωσ0=10-3

00 02 04 06 08 10 12 1410-5

redshift z

|langμ-1rang Ω

langmicrominus1

lang[DBI

A (z)]2rang

Ωlt [DFLRW

langmicrominus1

ndash 20 ndash

m = 5 log10DL + cst (454)

m2BI(z)

〈S〉N (z) =

]3H(z) +

sum3i=1 βi(z)

]2sum3i=1

]2minus1

ndash 21 ndash

Ωσ=-

00 02 04 06 08 10 12 14

redshift z

mBI-mFLRW

Ωσ=-

00 02 04 06 08 10 12 14-006

redshift z

mBI-mFLRW

5 Conclusion

langmicrominus1(z)

rangΩ6= 1 in

ndash 22 ndash

Acknowledgements

References

ndash 23 ndash

ndash 24 ndash

ndash 25 ndash

ndash 26 ndash

ndash 27 ndash

1 Introduction



23 Gauge freedom






31 Light rays

32 Light beams

321 Sachs basis

322 Jacobi matrix



4 Bianchi I and GLC







43 Jacobi matrix




5 Conclusion

EEprime

partwparta

minusd

w + dwLo

coordinates read

[gGLCmicroν ] =

T γab

[gmicroνGLC] =

ndash 5 ndash

d`2 simErarrLo

23 Gauge freedom

dwprime (215)

Υ|Lo =dτo

dw (219)

ndash 6 ndash

dtdxmicroE

Eprime

ndash 7 ndash

δmicroτ (226)

kw = kow = minusωo

ndash 8 ndash

w = w θa = θa dτ = Ωdτ (231)

31 Light rays

1 + z equiv ωs

Υs (32)

ndash 9 ndash

32 Light beams

321 Sachs basis

sA = saAparta (33)

γab = δAB sAa s

Bb (34)

SmicroνDsνAdλ

= 0 (35)

or more explicitly

partτsaA =

2(γacpartτγbc)s

bA (36)

322 Jacobi matrix

CAa equiv(

dsAadλ

ndash 10 ndash

rpartθ +O(r) s2 =

[s1 s2

]=[partθ partϕ

]middot 1

[1 00 1

sin θ

]︸︷︷︸

+O(r) so [sAa ] = r

[1 00 sin θ

]+O(r3) (312)

[CAa ] = minusωo

[1 00 sin θ

] [CaA] = minus 1

[1 00 1

sin θ

] (313)

[CaA] = minus 1

] [1 00 1

sin θ

] (314)

D2A = ω2

o detD (315)

ndash 11 ndash

det[γab]times1

ω2o sin θ

radicγ

sin θ(317)

〈S〉Ω equiv1

intS d2Ω (318)

〈S〉N equiv1

intS d2N =

Σradicγ dθdϕ

Σ = ρ

= ρΥ2

(partΥ

partτ

)minus1

〈S〉N =

intS ΣD2

Ad2ΩintΣD2

Ad2Ω=

langS ΣD2

rangΩlang

rangΩ

ndash 12 ndash

4 Bianchi I and GLC

3sumi=1

βi = 0 (42)

σij equiv1

2(γij)

prime (43)

3sumi=1

(βprimei)2 (44)

H2 =8πGρ

)+ Ωσ0

)4] (45)

ndash 13 ndash

0(6H20)

ds2 = minusdη2 +

3sumi=1

[eβi(η) dxi

]2 (47)

ndash 14 ndash

int λ

dλprime ki =

int η

ki = koi = ωod

ω2 = gij kikj =

3sumi=1

[eminusβi(η)ki

]2 (413)

int η

3sumj=1

minus12

partxi

partηo= minusdio +

int η

dηprimepartdi

partxi

partθa=

int η

dηprimepartdi

kiIiηo

= kiIia = 0 (418)

0 = part(did

= gijpart(didj

)= 2dipartd

ndash 15 ndash

= minusdη2 + gij

[didη + (Iiηo

io minus Iiηo

)Iia (425)

Υ =a(t)

3sumi=1

]2minus12

γab = gijIiaIjb

1 + z =a(to)

]2 (430)

IiaIbj = δba IiaI

Uao = minus(gijd

ipartadj)

o (432)

ndash 16 ndash

43 Jacobi matrix

Iij equivint η

in subsec 321

therefore

Iia equivint ηo

ηdηprime partad

(int ηo

DAB = sAa CaB (437)

ia (438)

dsAadλ

partη

∣∣∣∣ηoθϕ

(IiasAi ) (439)

Ai ) (440)

ndash 17 ndash

(Sjl)o

(partbkl)o (443)

l)o (445)

= Iai Iib (447)

= δab (448)

sin θ

e2βi(Ii1)2

)(3sumi=1

e2βi(Ii2)2

)minus

(3sumi=1

e2βiIi1Ii2

formulaD2

JiJj(k`)2 (450)

Ji equivint ηo

i ) (451)

ndash 18 ndash

langD2

A(λ)rang

langmicrominus1(z)

rangΩequiv

lang[DA(z)

DFLRWA (z)

]2rang

= 1 (453)

langmicrominus1

rangΩ

0 =radic

1minus Ωσ0

langmicrominus1

langκ2

rang=langγ2

rang[59] This has

lang(DBI

A )2rang

Ωgt (DFLRW

A(λ) =

langD2

A(λ)rang

Ω (452)

A(λ)rang

Ω=[DFLRW

A (λ)]2

ndash 19 ndash

Ωσ0=10-1

Ωσ0=10-2

Ωσ0=10-3

00 02 04 06 08 10 12 1410-5

redshift z

|langμ-1rang Ω

langmicrominus1

lang[DBI

A (z)]2rang

Ωlt [DFLRW

langmicrominus1

ndash 20 ndash

m = 5 log10DL + cst (454)

m2BI(z)

〈S〉N (z) =

]3H(z) +

sum3i=1 βi(z)

]2sum3i=1

]2minus1

ndash 21 ndash

Ωσ=-

00 02 04 06 08 10 12 14

redshift z

mBI-mFLRW

Ωσ=-

00 02 04 06 08 10 12 14-006

redshift z

mBI-mFLRW

5 Conclusion

langmicrominus1(z)

rangΩ6= 1 in

ndash 22 ndash

Acknowledgements

References

ndash 23 ndash

ndash 24 ndash

ndash 25 ndash

ndash 26 ndash

ndash 27 ndash

1 Introduction



23 Gauge freedom






31 Light rays

32 Light beams

321 Sachs basis

322 Jacobi matrix



4 Bianchi I and GLC







43 Jacobi matrix




5 Conclusion

d`2 simErarrLo

23 Gauge freedom

dwprime (215)

Υ|Lo =dτo

dw (219)

ndash 6 ndash

dtdxmicroE

Eprime

ndash 7 ndash

δmicroτ (226)

kw = kow = minusωo

ndash 8 ndash

w = w θa = θa dτ = Ωdτ (231)

31 Light rays

1 + z equiv ωs

Υs (32)

ndash 9 ndash

32 Light beams

321 Sachs basis

sA = saAparta (33)

γab = δAB sAa s

Bb (34)

SmicroνDsνAdλ

= 0 (35)

or more explicitly

partτsaA =

2(γacpartτγbc)s

bA (36)

322 Jacobi matrix

CAa equiv(

dsAadλ

ndash 10 ndash

rpartθ +O(r) s2 =

[s1 s2

]=[partθ partϕ

]middot 1

[1 00 1

sin θ

]︸︷︷︸

+O(r) so [sAa ] = r

[1 00 sin θ

]+O(r3) (312)

[CAa ] = minusωo

[1 00 sin θ

] [CaA] = minus 1

[1 00 1

sin θ

] (313)

[CaA] = minus 1

] [1 00 1

sin θ

] (314)

D2A = ω2

o detD (315)

ndash 11 ndash

det[γab]times1

ω2o sin θ

radicγ

sin θ(317)

〈S〉Ω equiv1

intS d2Ω (318)

〈S〉N equiv1

intS d2N =

Σradicγ dθdϕ

Σ = ρ

= ρΥ2

(partΥ

partτ

)minus1

〈S〉N =

intS ΣD2

Ad2ΩintΣD2

Ad2Ω=

langS ΣD2

rangΩlang

rangΩ

ndash 12 ndash

4 Bianchi I and GLC

3sumi=1

βi = 0 (42)

σij equiv1

2(γij)

prime (43)

3sumi=1

(βprimei)2 (44)

H2 =8πGρ

)+ Ωσ0

)4] (45)

ndash 13 ndash

0(6H20)

ds2 = minusdη2 +

3sumi=1

[eβi(η) dxi

]2 (47)

ndash 14 ndash

int λ

dλprime ki =

int η

ki = koi = ωod

ω2 = gij kikj =

3sumi=1

[eminusβi(η)ki

]2 (413)

int η

3sumj=1

minus12

partxi

partηo= minusdio +

int η

dηprimepartdi

partxi

partθa=

int η

dηprimepartdi

kiIiηo

= kiIia = 0 (418)

0 = part(did

= gijpart(didj

)= 2dipartd

ndash 15 ndash

= minusdη2 + gij

[didη + (Iiηo

io minus Iiηo

)Iia (425)

Υ =a(t)

3sumi=1

]2minus12

γab = gijIiaIjb

1 + z =a(to)

]2 (430)

IiaIbj = δba IiaI

Uao = minus(gijd

ipartadj)

o (432)

ndash 16 ndash

43 Jacobi matrix

Iij equivint η

in subsec 321

therefore

Iia equivint ηo

ηdηprime partad

(int ηo

DAB = sAa CaB (437)

ia (438)

dsAadλ

partη

∣∣∣∣ηoθϕ

(IiasAi ) (439)

Ai ) (440)

ndash 17 ndash

(Sjl)o

(partbkl)o (443)

l)o (445)

= Iai Iib (447)

= δab (448)

sin θ

e2βi(Ii1)2

)(3sumi=1

e2βi(Ii2)2

)minus

(3sumi=1

e2βiIi1Ii2

formulaD2

JiJj(k`)2 (450)

Ji equivint ηo

i ) (451)

ndash 18 ndash

langD2

A(λ)rang

langmicrominus1(z)

rangΩequiv

lang[DA(z)

DFLRWA (z)

]2rang

= 1 (453)

langmicrominus1

rangΩ

0 =radic

1minus Ωσ0

langmicrominus1

langκ2

rang=langγ2

rang[59] This has

lang(DBI

A )2rang

Ωgt (DFLRW

A(λ) =

langD2

A(λ)rang

Ω (452)

A(λ)rang

Ω=[DFLRW

A (λ)]2

ndash 19 ndash

Ωσ0=10-1

Ωσ0=10-2

Ωσ0=10-3

00 02 04 06 08 10 12 1410-5

redshift z

|langμ-1rang Ω

langmicrominus1

lang[DBI

A (z)]2rang

Ωlt [DFLRW

langmicrominus1

ndash 20 ndash

m = 5 log10DL + cst (454)

m2BI(z)

〈S〉N (z) =

]3H(z) +

sum3i=1 βi(z)

]2sum3i=1

]2minus1

ndash 21 ndash

Ωσ=-

00 02 04 06 08 10 12 14

redshift z

mBI-mFLRW

Ωσ=-

00 02 04 06 08 10 12 14-006

redshift z

mBI-mFLRW

5 Conclusion

langmicrominus1(z)

rangΩ6= 1 in

ndash 22 ndash

Acknowledgements

References

ndash 23 ndash

ndash 24 ndash

ndash 25 ndash

ndash 26 ndash

ndash 27 ndash

1 Introduction



23 Gauge freedom






31 Light rays

32 Light beams

321 Sachs basis

322 Jacobi matrix



4 Bianchi I and GLC







43 Jacobi matrix




5 Conclusion

dtdxmicroE

Eprime

ndash 7 ndash

δmicroτ (226)

kw = kow = minusωo

ndash 8 ndash

w = w θa = θa dτ = Ωdτ (231)

31 Light rays

1 + z equiv ωs

Υs (32)

ndash 9 ndash

32 Light beams

321 Sachs basis

sA = saAparta (33)

γab = δAB sAa s

Bb (34)

SmicroνDsνAdλ

= 0 (35)

or more explicitly

partτsaA =

2(γacpartτγbc)s

bA (36)

322 Jacobi matrix

CAa equiv(

dsAadλ

ndash 10 ndash

rpartθ +O(r) s2 =

[s1 s2

]=[partθ partϕ

]middot 1

[1 00 1

sin θ

]︸︷︷︸

+O(r) so [sAa ] = r

[1 00 sin θ

]+O(r3) (312)

[CAa ] = minusωo

[1 00 sin θ

] [CaA] = minus 1

[1 00 1

sin θ

] (313)

[CaA] = minus 1

] [1 00 1

sin θ

] (314)

D2A = ω2

o detD (315)

ndash 11 ndash

det[γab]times1

ω2o sin θ

radicγ

sin θ(317)

〈S〉Ω equiv1

intS d2Ω (318)

〈S〉N equiv1

intS d2N =

Σradicγ dθdϕ

Σ = ρ

= ρΥ2

(partΥ

partτ

)minus1

〈S〉N =

intS ΣD2

Ad2ΩintΣD2

Ad2Ω=

langS ΣD2

rangΩlang

rangΩ

ndash 12 ndash

4 Bianchi I and GLC

3sumi=1

βi = 0 (42)

σij equiv1

2(γij)

prime (43)

3sumi=1

(βprimei)2 (44)

H2 =8πGρ

)+ Ωσ0

)4] (45)

ndash 13 ndash

0(6H20)

ds2 = minusdη2 +

3sumi=1

[eβi(η) dxi

]2 (47)

ndash 14 ndash

int λ

dλprime ki =

int η

ki = koi = ωod

ω2 = gij kikj =

3sumi=1

[eminusβi(η)ki

]2 (413)

int η

3sumj=1

minus12

partxi

partηo= minusdio +

int η

dηprimepartdi

partxi

partθa=

int η

dηprimepartdi

kiIiηo

= kiIia = 0 (418)

0 = part(did

= gijpart(didj

)= 2dipartd

ndash 15 ndash

= minusdη2 + gij

[didη + (Iiηo

io minus Iiηo

)Iia (425)

Υ =a(t)

3sumi=1

]2minus12

γab = gijIiaIjb

1 + z =a(to)

]2 (430)

IiaIbj = δba IiaI

Uao = minus(gijd

ipartadj)

o (432)

ndash 16 ndash

43 Jacobi matrix

Iij equivint η

in subsec 321

therefore

Iia equivint ηo

ηdηprime partad

(int ηo

DAB = sAa CaB (437)

ia (438)

dsAadλ

partη

∣∣∣∣ηoθϕ

(IiasAi ) (439)

Ai ) (440)

ndash 17 ndash

(Sjl)o

(partbkl)o (443)

l)o (445)

= Iai Iib (447)

= δab (448)

sin θ

e2βi(Ii1)2

)(3sumi=1

e2βi(Ii2)2

)minus

(3sumi=1

e2βiIi1Ii2

formulaD2

JiJj(k`)2 (450)

Ji equivint ηo

i ) (451)

ndash 18 ndash

langD2

A(λ)rang

langmicrominus1(z)

rangΩequiv

lang[DA(z)

DFLRWA (z)

]2rang

= 1 (453)

langmicrominus1

rangΩ

0 =radic

1minus Ωσ0

langmicrominus1

langκ2

rang=langγ2

rang[59] This has

lang(DBI

A )2rang

Ωgt (DFLRW

A(λ) =

langD2

A(λ)rang

Ω (452)

A(λ)rang

Ω=[DFLRW

A (λ)]2

ndash 19 ndash

Ωσ0=10-1

Ωσ0=10-2

Ωσ0=10-3

00 02 04 06 08 10 12 1410-5

redshift z

|langμ-1rang Ω

langmicrominus1

lang[DBI

A (z)]2rang

Ωlt [DFLRW

langmicrominus1

ndash 20 ndash

m = 5 log10DL + cst (454)

m2BI(z)

〈S〉N (z) =

]3H(z) +

sum3i=1 βi(z)

]2sum3i=1

]2minus1

ndash 21 ndash

Ωσ=-

00 02 04 06 08 10 12 14

redshift z

mBI-mFLRW

Ωσ=-

00 02 04 06 08 10 12 14-006

redshift z

mBI-mFLRW

5 Conclusion

langmicrominus1(z)

rangΩ6= 1 in

ndash 22 ndash

Acknowledgements

References

ndash 23 ndash

ndash 24 ndash

ndash 25 ndash

ndash 26 ndash

ndash 27 ndash

1 Introduction



23 Gauge freedom






31 Light rays

32 Light beams

321 Sachs basis

322 Jacobi matrix



4 Bianchi I and GLC







43 Jacobi matrix




5 Conclusion

δmicroτ (226)

kw = kow = minusωo

ndash 8 ndash

w = w θa = θa dτ = Ωdτ (231)

31 Light rays

1 + z equiv ωs

Υs (32)

ndash 9 ndash

32 Light beams

321 Sachs basis

sA = saAparta (33)

γab = δAB sAa s

Bb (34)

SmicroνDsνAdλ

= 0 (35)

or more explicitly

partτsaA =

2(γacpartτγbc)s

bA (36)

322 Jacobi matrix

CAa equiv(

dsAadλ

ndash 10 ndash

rpartθ +O(r) s2 =

[s1 s2

]=[partθ partϕ

]middot 1

[1 00 1

sin θ

]︸︷︷︸

+O(r) so [sAa ] = r

[1 00 sin θ

]+O(r3) (312)

[CAa ] = minusωo

[1 00 sin θ

] [CaA] = minus 1

[1 00 1

sin θ

] (313)

[CaA] = minus 1

] [1 00 1

sin θ

] (314)

D2A = ω2

o detD (315)

ndash 11 ndash

det[γab]times1

ω2o sin θ

radicγ

sin θ(317)

〈S〉Ω equiv1

intS d2Ω (318)

〈S〉N equiv1

intS d2N =

Σradicγ dθdϕ

Σ = ρ

= ρΥ2

(partΥ

partτ

)minus1

〈S〉N =

intS ΣD2

Ad2ΩintΣD2

Ad2Ω=

langS ΣD2

rangΩlang

rangΩ

ndash 12 ndash

4 Bianchi I and GLC

3sumi=1

βi = 0 (42)

σij equiv1

2(γij)

prime (43)

3sumi=1

(βprimei)2 (44)

H2 =8πGρ

)+ Ωσ0

)4] (45)

ndash 13 ndash

0(6H20)

ds2 = minusdη2 +

3sumi=1

[eβi(η) dxi

]2 (47)

ndash 14 ndash

int λ

dλprime ki =

int η

ki = koi = ωod

ω2 = gij kikj =

3sumi=1

[eminusβi(η)ki

]2 (413)

int η

3sumj=1

minus12

partxi

partηo= minusdio +

int η

dηprimepartdi

partxi

partθa=

int η

dηprimepartdi

kiIiηo

= kiIia = 0 (418)

0 = part(did

= gijpart(didj

)= 2dipartd

ndash 15 ndash

= minusdη2 + gij

[didη + (Iiηo

io minus Iiηo

)Iia (425)

Υ =a(t)

3sumi=1

]2minus12

γab = gijIiaIjb

1 + z =a(to)

]2 (430)

IiaIbj = δba IiaI

Uao = minus(gijd

ipartadj)

o (432)

ndash 16 ndash

43 Jacobi matrix

Iij equivint η

in subsec 321

therefore

Iia equivint ηo

ηdηprime partad

(int ηo

DAB = sAa CaB (437)

ia (438)

dsAadλ

partη

∣∣∣∣ηoθϕ

(IiasAi ) (439)

Ai ) (440)

ndash 17 ndash

(Sjl)o

(partbkl)o (443)

l)o (445)

= Iai Iib (447)

= δab (448)

sin θ

e2βi(Ii1)2

)(3sumi=1

e2βi(Ii2)2

)minus

(3sumi=1

e2βiIi1Ii2

formulaD2

JiJj(k`)2 (450)

Ji equivint ηo

i ) (451)

ndash 18 ndash

langD2

A(λ)rang

langmicrominus1(z)

rangΩequiv

lang[DA(z)

DFLRWA (z)

]2rang

= 1 (453)

langmicrominus1

rangΩ

0 =radic

1minus Ωσ0

langmicrominus1

langκ2

rang=langγ2

rang[59] This has

lang(DBI

A )2rang

Ωgt (DFLRW

A(λ) =

langD2

A(λ)rang

Ω (452)

A(λ)rang

Ω=[DFLRW

A (λ)]2

ndash 19 ndash

Ωσ0=10-1

Ωσ0=10-2

Ωσ0=10-3

00 02 04 06 08 10 12 1410-5

redshift z

|langμ-1rang Ω

langmicrominus1

lang[DBI

A (z)]2rang

Ωlt [DFLRW

langmicrominus1

ndash 20 ndash

m = 5 log10DL + cst (454)

m2BI(z)

〈S〉N (z) =

]3H(z) +

sum3i=1 βi(z)

]2sum3i=1

]2minus1

ndash 21 ndash

Ωσ=-

00 02 04 06 08 10 12 14

redshift z

mBI-mFLRW

Ωσ=-

00 02 04 06 08 10 12 14-006

redshift z

mBI-mFLRW

5 Conclusion

langmicrominus1(z)

rangΩ6= 1 in

ndash 22 ndash

Acknowledgements

References

ndash 23 ndash

ndash 24 ndash

ndash 25 ndash

ndash 26 ndash

ndash 27 ndash

1 Introduction



23 Gauge freedom






31 Light rays

32 Light beams

321 Sachs basis

322 Jacobi matrix



4 Bianchi I and GLC







43 Jacobi matrix




5 Conclusion

w = w θa = θa dτ = Ωdτ (231)

31 Light rays

1 + z equiv ωs

Υs (32)

ndash 9 ndash

32 Light beams

321 Sachs basis

sA = saAparta (33)

γab = δAB sAa s

Bb (34)

SmicroνDsνAdλ

= 0 (35)

or more explicitly

partτsaA =

2(γacpartτγbc)s

bA (36)

322 Jacobi matrix

CAa equiv(

dsAadλ

ndash 10 ndash

rpartθ +O(r) s2 =

[s1 s2

]=[partθ partϕ

]middot 1

[1 00 1

sin θ

]︸︷︷︸

+O(r) so [sAa ] = r

[1 00 sin θ

]+O(r3) (312)

[CAa ] = minusωo

[1 00 sin θ

] [CaA] = minus 1

[1 00 1

sin θ

] (313)

[CaA] = minus 1

] [1 00 1

sin θ

] (314)

D2A = ω2

o detD (315)

ndash 11 ndash

det[γab]times1

ω2o sin θ

radicγ

sin θ(317)

〈S〉Ω equiv1

intS d2Ω (318)

〈S〉N equiv1

intS d2N =

Σradicγ dθdϕ

Σ = ρ

= ρΥ2

(partΥ

partτ

)minus1

〈S〉N =

intS ΣD2

Ad2ΩintΣD2

Ad2Ω=

langS ΣD2

rangΩlang

rangΩ

ndash 12 ndash

4 Bianchi I and GLC

3sumi=1

βi = 0 (42)

σij equiv1

2(γij)

prime (43)

3sumi=1

(βprimei)2 (44)

H2 =8πGρ

)+ Ωσ0

)4] (45)

ndash 13 ndash

0(6H20)

ds2 = minusdη2 +

3sumi=1

[eβi(η) dxi

]2 (47)

ndash 14 ndash

int λ

dλprime ki =

int η

ki = koi = ωod

ω2 = gij kikj =

3sumi=1

[eminusβi(η)ki

]2 (413)

int η

3sumj=1

minus12

partxi

partηo= minusdio +

int η

dηprimepartdi

partxi

partθa=

int η

dηprimepartdi

kiIiηo

= kiIia = 0 (418)

0 = part(did

= gijpart(didj

)= 2dipartd

ndash 15 ndash

= minusdη2 + gij

[didη + (Iiηo

io minus Iiηo

)Iia (425)

Υ =a(t)

3sumi=1

]2minus12

γab = gijIiaIjb

1 + z =a(to)

]2 (430)

IiaIbj = δba IiaI

Uao = minus(gijd

ipartadj)

o (432)

ndash 16 ndash

43 Jacobi matrix

Iij equivint η

in subsec 321

therefore

Iia equivint ηo

ηdηprime partad

(int ηo

DAB = sAa CaB (437)

ia (438)

dsAadλ

partη

∣∣∣∣ηoθϕ

(IiasAi ) (439)

Ai ) (440)

ndash 17 ndash

(Sjl)o

(partbkl)o (443)

l)o (445)

= Iai Iib (447)

= δab (448)

sin θ

e2βi(Ii1)2

)(3sumi=1

e2βi(Ii2)2

)minus

(3sumi=1

e2βiIi1Ii2

formulaD2

JiJj(k`)2 (450)

Ji equivint ηo

i ) (451)

ndash 18 ndash

langD2

A(λ)rang

langmicrominus1(z)

rangΩequiv

lang[DA(z)

DFLRWA (z)

]2rang

= 1 (453)

langmicrominus1

rangΩ

0 =radic

1minus Ωσ0

langmicrominus1

langκ2

rang=langγ2

rang[59] This has

lang(DBI

A )2rang

Ωgt (DFLRW

A(λ) =

langD2

A(λ)rang

Ω (452)

A(λ)rang

Ω=[DFLRW

A (λ)]2

ndash 19 ndash

Ωσ0=10-1

Ωσ0=10-2

Ωσ0=10-3

00 02 04 06 08 10 12 1410-5

redshift z

|langμ-1rang Ω

langmicrominus1

lang[DBI

A (z)]2rang

Ωlt [DFLRW

langmicrominus1

ndash 20 ndash

m = 5 log10DL + cst (454)

m2BI(z)

〈S〉N (z) =

]3H(z) +

sum3i=1 βi(z)

]2sum3i=1

]2minus1

ndash 21 ndash

Ωσ=-

00 02 04 06 08 10 12 14

redshift z

mBI-mFLRW

Ωσ=-

00 02 04 06 08 10 12 14-006

redshift z

mBI-mFLRW

5 Conclusion

langmicrominus1(z)

rangΩ6= 1 in

ndash 22 ndash

Acknowledgements

References

ndash 23 ndash

ndash 24 ndash

ndash 25 ndash

ndash 26 ndash

ndash 27 ndash

1 Introduction



23 Gauge freedom






31 Light rays

32 Light beams

321 Sachs basis

322 Jacobi matrix



4 Bianchi I and GLC







43 Jacobi matrix




5 Conclusion

32 Light beams

321 Sachs basis

sA = saAparta (33)

γab = δAB sAa s

Bb (34)

SmicroνDsνAdλ

= 0 (35)

or more explicitly

partτsaA =

2(γacpartτγbc)s

bA (36)

322 Jacobi matrix

CAa equiv(

dsAadλ

ndash 10 ndash

rpartθ +O(r) s2 =

[s1 s2

]=[partθ partϕ

]middot 1

[1 00 1

sin θ

]︸︷︷︸

+O(r) so [sAa ] = r

[1 00 sin θ

]+O(r3) (312)

[CAa ] = minusωo

[1 00 sin θ

] [CaA] = minus 1

[1 00 1

sin θ

] (313)

[CaA] = minus 1

] [1 00 1

sin θ

] (314)

D2A = ω2

o detD (315)

ndash 11 ndash

det[γab]times1

ω2o sin θ

radicγ

sin θ(317)

〈S〉Ω equiv1

intS d2Ω (318)

〈S〉N equiv1

intS d2N =

Σradicγ dθdϕ

Σ = ρ

= ρΥ2

(partΥ

partτ

)minus1

〈S〉N =

intS ΣD2

Ad2ΩintΣD2

Ad2Ω=

langS ΣD2

rangΩlang

rangΩ

ndash 12 ndash

4 Bianchi I and GLC

3sumi=1

βi = 0 (42)

σij equiv1

2(γij)

prime (43)

3sumi=1

(βprimei)2 (44)

H2 =8πGρ

)+ Ωσ0

)4] (45)

ndash 13 ndash

0(6H20)

ds2 = minusdη2 +

3sumi=1

[eβi(η) dxi

]2 (47)

ndash 14 ndash

int λ

dλprime ki =

int η

ki = koi = ωod

ω2 = gij kikj =

3sumi=1

[eminusβi(η)ki

]2 (413)

int η

3sumj=1

minus12

partxi

partηo= minusdio +

int η

dηprimepartdi

partxi

partθa=

int η

dηprimepartdi

kiIiηo

= kiIia = 0 (418)

0 = part(did

= gijpart(didj

)= 2dipartd

ndash 15 ndash

= minusdη2 + gij

[didη + (Iiηo

io minus Iiηo

)Iia (425)

Υ =a(t)

3sumi=1

]2minus12

γab = gijIiaIjb

1 + z =a(to)

]2 (430)

IiaIbj = δba IiaI

Uao = minus(gijd

ipartadj)

o (432)

ndash 16 ndash

43 Jacobi matrix

Iij equivint η

in subsec 321

therefore

Iia equivint ηo

ηdηprime partad

(int ηo

DAB = sAa CaB (437)

ia (438)

dsAadλ

partη

∣∣∣∣ηoθϕ

(IiasAi ) (439)

Ai ) (440)

ndash 17 ndash

(Sjl)o

(partbkl)o (443)

l)o (445)

= Iai Iib (447)

= δab (448)

sin θ

e2βi(Ii1)2

)(3sumi=1

e2βi(Ii2)2

)minus

(3sumi=1

e2βiIi1Ii2

formulaD2

JiJj(k`)2 (450)

Ji equivint ηo

i ) (451)

ndash 18 ndash

langD2

A(λ)rang

langmicrominus1(z)

rangΩequiv

lang[DA(z)

DFLRWA (z)

]2rang

= 1 (453)

langmicrominus1

rangΩ

0 =radic

1minus Ωσ0

langmicrominus1

langκ2

rang=langγ2

rang[59] This has

lang(DBI

A )2rang

Ωgt (DFLRW

A(λ) =

langD2

A(λ)rang

Ω (452)

A(λ)rang

Ω=[DFLRW

A (λ)]2

ndash 19 ndash

Ωσ0=10-1

Ωσ0=10-2

Ωσ0=10-3

00 02 04 06 08 10 12 1410-5

redshift z

|langμ-1rang Ω

langmicrominus1

lang[DBI

A (z)]2rang

Ωlt [DFLRW

langmicrominus1

ndash 20 ndash

m = 5 log10DL + cst (454)

m2BI(z)

〈S〉N (z) =

]3H(z) +

sum3i=1 βi(z)

]2sum3i=1

]2minus1

ndash 21 ndash

Ωσ=-

00 02 04 06 08 10 12 14

redshift z

mBI-mFLRW

Ωσ=-

00 02 04 06 08 10 12 14-006

redshift z

mBI-mFLRW

5 Conclusion

langmicrominus1(z)

rangΩ6= 1 in

ndash 22 ndash

Acknowledgements

References

ndash 23 ndash

ndash 24 ndash

ndash 25 ndash

ndash 26 ndash

ndash 27 ndash

1 Introduction



23 Gauge freedom






31 Light rays

32 Light beams

321 Sachs basis

322 Jacobi matrix



4 Bianchi I and GLC







43 Jacobi matrix




5 Conclusion

rpartθ +O(r) s2 =

[s1 s2

]=[partθ partϕ

]middot 1

[1 00 1

sin θ

]︸︷︷︸

+O(r) so [sAa ] = r

[1 00 sin θ

]+O(r3) (312)

[CAa ] = minusωo

[1 00 sin θ

] [CaA] = minus 1

[1 00 1

sin θ

] (313)

[CaA] = minus 1

] [1 00 1

sin θ

] (314)

D2A = ω2

o detD (315)

ndash 11 ndash

det[γab]times1

ω2o sin θ

radicγ

sin θ(317)

〈S〉Ω equiv1

intS d2Ω (318)

〈S〉N equiv1

intS d2N =

Σradicγ dθdϕ

Σ = ρ

= ρΥ2

(partΥ

partτ

)minus1

〈S〉N =

intS ΣD2

Ad2ΩintΣD2

Ad2Ω=

langS ΣD2

rangΩlang

rangΩ

ndash 12 ndash

4 Bianchi I and GLC

3sumi=1

βi = 0 (42)

σij equiv1

2(γij)

prime (43)

3sumi=1

(βprimei)2 (44)

H2 =8πGρ

)+ Ωσ0

)4] (45)

ndash 13 ndash

0(6H20)

ds2 = minusdη2 +

3sumi=1

[eβi(η) dxi

]2 (47)

ndash 14 ndash

int λ

dλprime ki =

int η

ki = koi = ωod

ω2 = gij kikj =

3sumi=1

[eminusβi(η)ki

]2 (413)

int η

3sumj=1

minus12

partxi

partηo= minusdio +

int η

dηprimepartdi

partxi

partθa=

int η

dηprimepartdi

kiIiηo

= kiIia = 0 (418)

0 = part(did

= gijpart(didj

)= 2dipartd

ndash 15 ndash

= minusdη2 + gij

[didη + (Iiηo

io minus Iiηo

)Iia (425)

Υ =a(t)

3sumi=1

]2minus12

γab = gijIiaIjb

1 + z =a(to)

]2 (430)

IiaIbj = δba IiaI

Uao = minus(gijd

ipartadj)

o (432)

ndash 16 ndash

43 Jacobi matrix

Iij equivint η

in subsec 321

therefore

Iia equivint ηo

ηdηprime partad

(int ηo

DAB = sAa CaB (437)

ia (438)

dsAadλ

partη

∣∣∣∣ηoθϕ

(IiasAi ) (439)

Ai ) (440)

ndash 17 ndash

(Sjl)o

(partbkl)o (443)

l)o (445)

= Iai Iib (447)

= δab (448)

sin θ

e2βi(Ii1)2

)(3sumi=1

e2βi(Ii2)2

)minus

(3sumi=1

e2βiIi1Ii2

formulaD2

JiJj(k`)2 (450)

Ji equivint ηo

i ) (451)

ndash 18 ndash

langD2

A(λ)rang

langmicrominus1(z)

rangΩequiv

lang[DA(z)

DFLRWA (z)

]2rang

= 1 (453)

langmicrominus1

rangΩ

0 =radic

1minus Ωσ0

langmicrominus1

langκ2

rang=langγ2

rang[59] This has

lang(DBI

A )2rang

Ωgt (DFLRW

A(λ) =

langD2

A(λ)rang

Ω (452)

A(λ)rang

Ω=[DFLRW

A (λ)]2

ndash 19 ndash

Ωσ0=10-1

Ωσ0=10-2

Ωσ0=10-3

00 02 04 06 08 10 12 1410-5

redshift z

|langμ-1rang Ω

langmicrominus1

lang[DBI

A (z)]2rang

Ωlt [DFLRW

langmicrominus1

ndash 20 ndash

m = 5 log10DL + cst (454)

m2BI(z)

〈S〉N (z) =

]3H(z) +

sum3i=1 βi(z)

]2sum3i=1

]2minus1

ndash 21 ndash

Ωσ=-

00 02 04 06 08 10 12 14

redshift z

mBI-mFLRW

Ωσ=-

00 02 04 06 08 10 12 14-006

redshift z

mBI-mFLRW

5 Conclusion

langmicrominus1(z)

rangΩ6= 1 in

ndash 22 ndash

Acknowledgements

References

ndash 23 ndash

ndash 24 ndash

ndash 25 ndash

ndash 26 ndash

ndash 27 ndash

1 Introduction



23 Gauge freedom






31 Light rays

32 Light beams

321 Sachs basis

322 Jacobi matrix



4 Bianchi I and GLC







43 Jacobi matrix




5 Conclusion

det[γab]times1

ω2o sin θ

radicγ

sin θ(317)

〈S〉Ω equiv1

intS d2Ω (318)

〈S〉N equiv1

intS d2N =

Σradicγ dθdϕ

Σ = ρ

= ρΥ2

(partΥ

partτ

)minus1

〈S〉N =

intS ΣD2

Ad2ΩintΣD2

Ad2Ω=

langS ΣD2

rangΩlang

rangΩ

ndash 12 ndash

4 Bianchi I and GLC

3sumi=1

βi = 0 (42)

σij equiv1

2(γij)

prime (43)

3sumi=1

(βprimei)2 (44)

H2 =8πGρ

)+ Ωσ0

)4] (45)

ndash 13 ndash

0(6H20)

ds2 = minusdη2 +

3sumi=1

[eβi(η) dxi

]2 (47)

ndash 14 ndash

int λ

dλprime ki =

int η

ki = koi = ωod

ω2 = gij kikj =

3sumi=1

[eminusβi(η)ki

]2 (413)

int η

3sumj=1

minus12

partxi

partηo= minusdio +

int η

dηprimepartdi

partxi

partθa=

int η

dηprimepartdi

kiIiηo

= kiIia = 0 (418)

0 = part(did

= gijpart(didj

)= 2dipartd

ndash 15 ndash

= minusdη2 + gij

[didη + (Iiηo

io minus Iiηo

)Iia (425)

Υ =a(t)

3sumi=1

]2minus12

γab = gijIiaIjb

1 + z =a(to)

]2 (430)

IiaIbj = δba IiaI

Uao = minus(gijd

ipartadj)

o (432)

ndash 16 ndash

43 Jacobi matrix

Iij equivint η

in subsec 321

therefore

Iia equivint ηo

ηdηprime partad

(int ηo

DAB = sAa CaB (437)

ia (438)

dsAadλ

partη

∣∣∣∣ηoθϕ

(IiasAi ) (439)

Ai ) (440)

ndash 17 ndash

(Sjl)o

(partbkl)o (443)

l)o (445)

= Iai Iib (447)

= δab (448)

sin θ

e2βi(Ii1)2

)(3sumi=1

e2βi(Ii2)2

)minus

(3sumi=1

e2βiIi1Ii2

formulaD2

JiJj(k`)2 (450)

Ji equivint ηo

i ) (451)

ndash 18 ndash

langD2

A(λ)rang

langmicrominus1(z)

rangΩequiv

lang[DA(z)

DFLRWA (z)

]2rang

= 1 (453)

langmicrominus1

rangΩ

0 =radic

1minus Ωσ0

langmicrominus1

langκ2

rang=langγ2

rang[59] This has

lang(DBI

A )2rang

Ωgt (DFLRW

A(λ) =

langD2

A(λ)rang

Ω (452)

A(λ)rang

Ω=[DFLRW

A (λ)]2

ndash 19 ndash

Ωσ0=10-1

Ωσ0=10-2

Ωσ0=10-3

00 02 04 06 08 10 12 1410-5

redshift z

|langμ-1rang Ω

langmicrominus1

lang[DBI

A (z)]2rang

Ωlt [DFLRW

langmicrominus1

ndash 20 ndash

m = 5 log10DL + cst (454)

m2BI(z)

〈S〉N (z) =

]3H(z) +

sum3i=1 βi(z)

]2sum3i=1

]2minus1

ndash 21 ndash

Ωσ=-

00 02 04 06 08 10 12 14

redshift z

mBI-mFLRW

Ωσ=-

00 02 04 06 08 10 12 14-006

redshift z

mBI-mFLRW

5 Conclusion

langmicrominus1(z)

rangΩ6= 1 in

ndash 22 ndash

Acknowledgements

References

ndash 23 ndash

ndash 24 ndash

ndash 25 ndash

ndash 26 ndash

ndash 27 ndash

1 Introduction



23 Gauge freedom






31 Light rays

32 Light beams

321 Sachs basis

322 Jacobi matrix



4 Bianchi I and GLC







43 Jacobi matrix




5 Conclusion

4 Bianchi I and GLC

3sumi=1

βi = 0 (42)

σij equiv1

2(γij)

prime (43)

3sumi=1

(βprimei)2 (44)

H2 =8πGρ

)+ Ωσ0

)4] (45)

ndash 13 ndash

0(6H20)

ds2 = minusdη2 +

3sumi=1

[eβi(η) dxi

]2 (47)

ndash 14 ndash

int λ

dλprime ki =

int η

ki = koi = ωod

ω2 = gij kikj =

3sumi=1

[eminusβi(η)ki

]2 (413)

int η

3sumj=1

minus12

partxi

partηo= minusdio +

int η

dηprimepartdi

partxi

partθa=

int η

dηprimepartdi

kiIiηo

= kiIia = 0 (418)

0 = part(did

= gijpart(didj

)= 2dipartd

ndash 15 ndash

= minusdη2 + gij

[didη + (Iiηo

io minus Iiηo

)Iia (425)

Υ =a(t)

3sumi=1

]2minus12

γab = gijIiaIjb

1 + z =a(to)

]2 (430)

IiaIbj = δba IiaI

Uao = minus(gijd

ipartadj)

o (432)

ndash 16 ndash

43 Jacobi matrix

Iij equivint η

in subsec 321

therefore

Iia equivint ηo

ηdηprime partad

(int ηo

DAB = sAa CaB (437)

ia (438)

dsAadλ

partη

∣∣∣∣ηoθϕ

(IiasAi ) (439)

Ai ) (440)

ndash 17 ndash

(Sjl)o

(partbkl)o (443)

l)o (445)

= Iai Iib (447)

= δab (448)

sin θ

e2βi(Ii1)2

)(3sumi=1

e2βi(Ii2)2

)minus

(3sumi=1

e2βiIi1Ii2

formulaD2

JiJj(k`)2 (450)

Ji equivint ηo

i ) (451)

ndash 18 ndash

langD2

A(λ)rang

langmicrominus1(z)

rangΩequiv

lang[DA(z)

DFLRWA (z)

]2rang

= 1 (453)

langmicrominus1

rangΩ

0 =radic

1minus Ωσ0

langmicrominus1

langκ2

rang=langγ2

rang[59] This has

lang(DBI

A )2rang

Ωgt (DFLRW

A(λ) =

langD2

A(λ)rang

Ω (452)

A(λ)rang

Ω=[DFLRW

A (λ)]2

ndash 19 ndash

Ωσ0=10-1

Ωσ0=10-2

Ωσ0=10-3

00 02 04 06 08 10 12 1410-5

redshift z

|langμ-1rang Ω

langmicrominus1

lang[DBI

A (z)]2rang

Ωlt [DFLRW

langmicrominus1

ndash 20 ndash

m = 5 log10DL + cst (454)

m2BI(z)

〈S〉N (z) =

]3H(z) +

sum3i=1 βi(z)

]2sum3i=1

]2minus1

ndash 21 ndash

Ωσ=-

00 02 04 06 08 10 12 14

redshift z

mBI-mFLRW

Ωσ=-

00 02 04 06 08 10 12 14-006

redshift z

mBI-mFLRW

5 Conclusion

langmicrominus1(z)

rangΩ6= 1 in

ndash 22 ndash

Acknowledgements

References

ndash 23 ndash

ndash 24 ndash

ndash 25 ndash

ndash 26 ndash

ndash 27 ndash

1 Introduction



23 Gauge freedom






31 Light rays

32 Light beams

321 Sachs basis

322 Jacobi matrix



4 Bianchi I and GLC







43 Jacobi matrix




5 Conclusion

0(6H20)

ds2 = minusdη2 +

3sumi=1

[eβi(η) dxi

]2 (47)

ndash 14 ndash

int λ

dλprime ki =

int η

ki = koi = ωod

ω2 = gij kikj =

3sumi=1

[eminusβi(η)ki

]2 (413)

int η

3sumj=1

minus12

partxi

partηo= minusdio +

int η

dηprimepartdi

partxi

partθa=

int η

dηprimepartdi

kiIiηo

= kiIia = 0 (418)

0 = part(did

= gijpart(didj

)= 2dipartd

ndash 15 ndash

= minusdη2 + gij

[didη + (Iiηo

io minus Iiηo

)Iia (425)

Υ =a(t)

3sumi=1

]2minus12

γab = gijIiaIjb

1 + z =a(to)

]2 (430)

IiaIbj = δba IiaI

Uao = minus(gijd

ipartadj)

o (432)

ndash 16 ndash

43 Jacobi matrix

Iij equivint η

in subsec 321

therefore

Iia equivint ηo

ηdηprime partad

(int ηo

DAB = sAa CaB (437)

ia (438)

dsAadλ

partη

∣∣∣∣ηoθϕ

(IiasAi ) (439)

Ai ) (440)

ndash 17 ndash

(Sjl)o

(partbkl)o (443)

l)o (445)

= Iai Iib (447)

= δab (448)

sin θ

e2βi(Ii1)2

)(3sumi=1

e2βi(Ii2)2

)minus

(3sumi=1

e2βiIi1Ii2

formulaD2

JiJj(k`)2 (450)

Ji equivint ηo

i ) (451)

ndash 18 ndash

langD2

A(λ)rang

langmicrominus1(z)

rangΩequiv

lang[DA(z)

DFLRWA (z)

]2rang

= 1 (453)

langmicrominus1

rangΩ

0 =radic

1minus Ωσ0

langmicrominus1

langκ2

rang=langγ2

rang[59] This has

lang(DBI

A )2rang

Ωgt (DFLRW

A(λ) =

langD2

A(λ)rang

Ω (452)

A(λ)rang

Ω=[DFLRW

A (λ)]2

ndash 19 ndash

Ωσ0=10-1

Ωσ0=10-2

Ωσ0=10-3

00 02 04 06 08 10 12 1410-5

redshift z

|langμ-1rang Ω

langmicrominus1

lang[DBI

A (z)]2rang

Ωlt [DFLRW

langmicrominus1

ndash 20 ndash

m = 5 log10DL + cst (454)

m2BI(z)

〈S〉N (z) =

]3H(z) +

sum3i=1 βi(z)

]2sum3i=1

]2minus1

ndash 21 ndash

Ωσ=-

00 02 04 06 08 10 12 14

redshift z

mBI-mFLRW

Ωσ=-

00 02 04 06 08 10 12 14-006

redshift z

mBI-mFLRW

5 Conclusion

langmicrominus1(z)

rangΩ6= 1 in

ndash 22 ndash

Acknowledgements

References

ndash 23 ndash

ndash 24 ndash

ndash 25 ndash

ndash 26 ndash

ndash 27 ndash

1 Introduction



23 Gauge freedom






31 Light rays

32 Light beams

321 Sachs basis

322 Jacobi matrix



4 Bianchi I and GLC







43 Jacobi matrix




5 Conclusion

int λ

dλprime ki =

int η

ki = koi = ωod

ω2 = gij kikj =

3sumi=1

[eminusβi(η)ki

]2 (413)

int η

3sumj=1

minus12

partxi

partηo= minusdio +

int η

dηprimepartdi

partxi

partθa=

int η

dηprimepartdi

kiIiηo

= kiIia = 0 (418)

0 = part(did

= gijpart(didj

)= 2dipartd

ndash 15 ndash

= minusdη2 + gij

[didη + (Iiηo

io minus Iiηo

)Iia (425)

Υ =a(t)

3sumi=1

]2minus12

γab = gijIiaIjb

1 + z =a(to)

]2 (430)

IiaIbj = δba IiaI

Uao = minus(gijd

ipartadj)

o (432)

ndash 16 ndash

43 Jacobi matrix

Iij equivint η

in subsec 321

therefore

Iia equivint ηo

ηdηprime partad

(int ηo

DAB = sAa CaB (437)

ia (438)

dsAadλ

partη

∣∣∣∣ηoθϕ

(IiasAi ) (439)

Ai ) (440)

ndash 17 ndash

(Sjl)o

(partbkl)o (443)

l)o (445)

= Iai Iib (447)

= δab (448)

sin θ

e2βi(Ii1)2

)(3sumi=1

e2βi(Ii2)2

)minus

(3sumi=1

e2βiIi1Ii2

formulaD2

JiJj(k`)2 (450)

Ji equivint ηo

i ) (451)

ndash 18 ndash

langD2

A(λ)rang

langmicrominus1(z)

rangΩequiv

lang[DA(z)

DFLRWA (z)

]2rang

= 1 (453)

langmicrominus1

rangΩ

0 =radic

1minus Ωσ0

langmicrominus1

langκ2

rang=langγ2

rang[59] This has

lang(DBI

A )2rang

Ωgt (DFLRW

A(λ) =

langD2

A(λ)rang

Ω (452)

A(λ)rang

Ω=[DFLRW

A (λ)]2

ndash 19 ndash

Ωσ0=10-1

Ωσ0=10-2

Ωσ0=10-3

00 02 04 06 08 10 12 1410-5

redshift z

|langμ-1rang Ω

langmicrominus1

lang[DBI

A (z)]2rang

Ωlt [DFLRW

langmicrominus1

ndash 20 ndash

m = 5 log10DL + cst (454)

m2BI(z)

〈S〉N (z) =

]3H(z) +

sum3i=1 βi(z)

]2sum3i=1

]2minus1

ndash 21 ndash

Ωσ=-

00 02 04 06 08 10 12 14

redshift z

mBI-mFLRW

Ωσ=-

00 02 04 06 08 10 12 14-006

redshift z

mBI-mFLRW

5 Conclusion

langmicrominus1(z)

rangΩ6= 1 in

ndash 22 ndash

Acknowledgements

References

ndash 23 ndash

ndash 24 ndash

ndash 25 ndash

ndash 26 ndash

ndash 27 ndash

1 Introduction



23 Gauge freedom






31 Light rays

32 Light beams

321 Sachs basis

322 Jacobi matrix



4 Bianchi I and GLC







43 Jacobi matrix




5 Conclusion

= minusdη2 + gij

[didη + (Iiηo

io minus Iiηo

)Iia (425)

Υ =a(t)

3sumi=1

]2minus12

γab = gijIiaIjb

1 + z =a(to)

]2 (430)

IiaIbj = δba IiaI

Uao = minus(gijd

ipartadj)

o (432)

ndash 16 ndash

43 Jacobi matrix

Iij equivint η

in subsec 321

therefore

Iia equivint ηo

ηdηprime partad

(int ηo

DAB = sAa CaB (437)

ia (438)

dsAadλ

partη

∣∣∣∣ηoθϕ

(IiasAi ) (439)

Ai ) (440)

ndash 17 ndash

(Sjl)o

(partbkl)o (443)

l)o (445)

= Iai Iib (447)

= δab (448)

sin θ

e2βi(Ii1)2

)(3sumi=1

e2βi(Ii2)2

)minus

(3sumi=1

e2βiIi1Ii2

formulaD2

JiJj(k`)2 (450)

Ji equivint ηo

i ) (451)

ndash 18 ndash

langD2

A(λ)rang

langmicrominus1(z)

rangΩequiv

lang[DA(z)

DFLRWA (z)

]2rang

= 1 (453)

langmicrominus1

rangΩ

0 =radic

1minus Ωσ0

langmicrominus1

langκ2

rang=langγ2

rang[59] This has

lang(DBI

A )2rang

Ωgt (DFLRW

A(λ) =

langD2

A(λ)rang

Ω (452)

A(λ)rang

Ω=[DFLRW

A (λ)]2

ndash 19 ndash

Ωσ0=10-1

Ωσ0=10-2

Ωσ0=10-3

00 02 04 06 08 10 12 1410-5

redshift z

|langμ-1rang Ω

langmicrominus1

lang[DBI

A (z)]2rang

Ωlt [DFLRW

langmicrominus1

ndash 20 ndash

m = 5 log10DL + cst (454)

m2BI(z)

〈S〉N (z) =

]3H(z) +

sum3i=1 βi(z)

]2sum3i=1

]2minus1

ndash 21 ndash

Ωσ=-

00 02 04 06 08 10 12 14

redshift z

mBI-mFLRW

Ωσ=-

00 02 04 06 08 10 12 14-006

redshift z

mBI-mFLRW

5 Conclusion

langmicrominus1(z)

rangΩ6= 1 in

ndash 22 ndash

Acknowledgements

References

ndash 23 ndash

ndash 24 ndash

ndash 25 ndash

ndash 26 ndash

ndash 27 ndash

1 Introduction



23 Gauge freedom






31 Light rays

32 Light beams

321 Sachs basis

322 Jacobi matrix



4 Bianchi I and GLC







43 Jacobi matrix




5 Conclusion

43 Jacobi matrix

Iij equivint η

in subsec 321

therefore

Iia equivint ηo

ηdηprime partad

(int ηo

DAB = sAa CaB (437)

ia (438)

dsAadλ

partη

∣∣∣∣ηoθϕ

(IiasAi ) (439)

Ai ) (440)

ndash 17 ndash

(Sjl)o

(partbkl)o (443)

l)o (445)

= Iai Iib (447)

= δab (448)

sin θ

e2βi(Ii1)2

)(3sumi=1

e2βi(Ii2)2

)minus

(3sumi=1

e2βiIi1Ii2

formulaD2

JiJj(k`)2 (450)

Ji equivint ηo

i ) (451)

ndash 18 ndash

langD2

A(λ)rang

langmicrominus1(z)

rangΩequiv

lang[DA(z)

DFLRWA (z)

]2rang

= 1 (453)

langmicrominus1

rangΩ

0 =radic

1minus Ωσ0

langmicrominus1

langκ2

rang=langγ2

rang[59] This has

lang(DBI

A )2rang

Ωgt (DFLRW

A(λ) =

langD2

A(λ)rang

Ω (452)

A(λ)rang

Ω=[DFLRW

A (λ)]2

ndash 19 ndash

Ωσ0=10-1

Ωσ0=10-2

Ωσ0=10-3

00 02 04 06 08 10 12 1410-5

redshift z

|langμ-1rang Ω

langmicrominus1

lang[DBI

A (z)]2rang

Ωlt [DFLRW

langmicrominus1

ndash 20 ndash

m = 5 log10DL + cst (454)

m2BI(z)

〈S〉N (z) =

]3H(z) +

sum3i=1 βi(z)

]2sum3i=1

]2minus1

ndash 21 ndash

Ωσ=-

00 02 04 06 08 10 12 14

redshift z

mBI-mFLRW

Ωσ=-

00 02 04 06 08 10 12 14-006

redshift z

mBI-mFLRW

5 Conclusion

langmicrominus1(z)

rangΩ6= 1 in

ndash 22 ndash

Acknowledgements

References

ndash 23 ndash

ndash 24 ndash

ndash 25 ndash

ndash 26 ndash

ndash 27 ndash

1 Introduction



23 Gauge freedom






31 Light rays

32 Light beams

321 Sachs basis

322 Jacobi matrix



4 Bianchi I and GLC







43 Jacobi matrix




5 Conclusion

(Sjl)o

(partbkl)o (443)

l)o (445)

= Iai Iib (447)

= δab (448)

sin θ

e2βi(Ii1)2

)(3sumi=1

e2βi(Ii2)2

)minus

(3sumi=1

e2βiIi1Ii2

formulaD2

JiJj(k`)2 (450)

Ji equivint ηo

i ) (451)

ndash 18 ndash

langD2

A(λ)rang

langmicrominus1(z)

rangΩequiv

lang[DA(z)

DFLRWA (z)

]2rang

= 1 (453)

langmicrominus1

rangΩ

0 =radic

1minus Ωσ0

langmicrominus1

langκ2

rang=langγ2

rang[59] This has

lang(DBI

A )2rang

Ωgt (DFLRW

A(λ) =

langD2

A(λ)rang

Ω (452)

A(λ)rang

Ω=[DFLRW

A (λ)]2

ndash 19 ndash

Ωσ0=10-1

Ωσ0=10-2

Ωσ0=10-3

00 02 04 06 08 10 12 1410-5

redshift z

|langμ-1rang Ω

langmicrominus1

lang[DBI

A (z)]2rang

Ωlt [DFLRW

langmicrominus1

ndash 20 ndash

m = 5 log10DL + cst (454)

m2BI(z)

〈S〉N (z) =

]3H(z) +

sum3i=1 βi(z)

]2sum3i=1

]2minus1

ndash 21 ndash

Ωσ=-

00 02 04 06 08 10 12 14

redshift z

mBI-mFLRW

Ωσ=-

00 02 04 06 08 10 12 14-006

redshift z

mBI-mFLRW

5 Conclusion

langmicrominus1(z)

rangΩ6= 1 in

ndash 22 ndash

Acknowledgements

References

ndash 23 ndash

ndash 24 ndash

ndash 25 ndash

ndash 26 ndash

ndash 27 ndash

1 Introduction



23 Gauge freedom






31 Light rays

32 Light beams

321 Sachs basis

322 Jacobi matrix



4 Bianchi I and GLC







43 Jacobi matrix




5 Conclusion

langD2

A(λ)rang

langmicrominus1(z)

rangΩequiv

lang[DA(z)

DFLRWA (z)

]2rang

= 1 (453)

langmicrominus1

rangΩ

0 =radic

1minus Ωσ0

langmicrominus1

langκ2

rang=langγ2

rang[59] This has

lang(DBI

A )2rang

Ωgt (DFLRW

A(λ) =

langD2

A(λ)rang

Ω (452)

A(λ)rang

Ω=[DFLRW

A (λ)]2

ndash 19 ndash

Ωσ0=10-1

Ωσ0=10-2

Ωσ0=10-3

00 02 04 06 08 10 12 1410-5

redshift z

|langμ-1rang Ω

langmicrominus1

lang[DBI

A (z)]2rang

Ωlt [DFLRW

langmicrominus1

ndash 20 ndash

m = 5 log10DL + cst (454)

m2BI(z)

〈S〉N (z) =

]3H(z) +

sum3i=1 βi(z)

]2sum3i=1

]2minus1

ndash 21 ndash

Ωσ=-

00 02 04 06 08 10 12 14

redshift z

mBI-mFLRW

Ωσ=-

00 02 04 06 08 10 12 14-006

redshift z

mBI-mFLRW

5 Conclusion

langmicrominus1(z)

rangΩ6= 1 in

ndash 22 ndash

Acknowledgements

References

ndash 23 ndash

ndash 24 ndash

ndash 25 ndash

ndash 26 ndash

ndash 27 ndash

1 Introduction



23 Gauge freedom






31 Light rays

32 Light beams

321 Sachs basis

322 Jacobi matrix



4 Bianchi I and GLC







43 Jacobi matrix




5 Conclusion

Ωσ0=10-1

Ωσ0=10-2

Ωσ0=10-3

00 02 04 06 08 10 12 1410-5

redshift z

|langμ-1rang Ω

langmicrominus1

lang[DBI

A (z)]2rang

Ωlt [DFLRW

langmicrominus1

ndash 20 ndash

m = 5 log10DL + cst (454)

m2BI(z)

〈S〉N (z) =

]3H(z) +

sum3i=1 βi(z)

]2sum3i=1

]2minus1

ndash 21 ndash

Ωσ=-

00 02 04 06 08 10 12 14

redshift z

mBI-mFLRW

Ωσ=-

00 02 04 06 08 10 12 14-006

redshift z

mBI-mFLRW

5 Conclusion

langmicrominus1(z)

rangΩ6= 1 in

ndash 22 ndash

Acknowledgements

References

ndash 23 ndash

ndash 24 ndash

ndash 25 ndash

ndash 26 ndash

ndash 27 ndash

1 Introduction



23 Gauge freedom






31 Light rays

32 Light beams

321 Sachs basis

322 Jacobi matrix



4 Bianchi I and GLC







43 Jacobi matrix




5 Conclusion

m = 5 log10DL + cst (454)

m2BI(z)

〈S〉N (z) =

]3H(z) +

sum3i=1 βi(z)

]2sum3i=1

]2minus1

ndash 21 ndash

Ωσ=-

00 02 04 06 08 10 12 14

redshift z

mBI-mFLRW

Ωσ=-

00 02 04 06 08 10 12 14-006

redshift z

mBI-mFLRW

5 Conclusion

langmicrominus1(z)

rangΩ6= 1 in

ndash 22 ndash

Acknowledgements

References

ndash 23 ndash

ndash 24 ndash

ndash 25 ndash

ndash 26 ndash

ndash 27 ndash

1 Introduction



23 Gauge freedom






31 Light rays

32 Light beams

321 Sachs basis

322 Jacobi matrix



4 Bianchi I and GLC







43 Jacobi matrix




5 Conclusion

Ωσ=-

00 02 04 06 08 10 12 14

redshift z

mBI-mFLRW

Ωσ=-

00 02 04 06 08 10 12 14-006

redshift z

mBI-mFLRW

5 Conclusion

langmicrominus1(z)

rangΩ6= 1 in

ndash 22 ndash

Acknowledgements

References

ndash 23 ndash

ndash 24 ndash

ndash 25 ndash

ndash 26 ndash

ndash 27 ndash

1 Introduction



23 Gauge freedom






31 Light rays

32 Light beams

321 Sachs basis

322 Jacobi matrix



4 Bianchi I and GLC







43 Jacobi matrix




5 Conclusion

Acknowledgements

References

ndash 23 ndash

ndash 24 ndash

ndash 25 ndash

ndash 26 ndash

ndash 27 ndash

1 Introduction



23 Gauge freedom






31 Light rays

32 Light beams

321 Sachs basis

322 Jacobi matrix



4 Bianchi I and GLC







43 Jacobi matrix




5 Conclusion

ndash 24 ndash

ndash 25 ndash

ndash 26 ndash

ndash 27 ndash

1 Introduction



23 Gauge freedom






31 Light rays

32 Light beams

321 Sachs basis

322 Jacobi matrix



4 Bianchi I and GLC







43 Jacobi matrix




5 Conclusion

ndash 25 ndash

ndash 26 ndash

ndash 27 ndash

1 Introduction



23 Gauge freedom






31 Light rays

32 Light beams

321 Sachs basis

322 Jacobi matrix



4 Bianchi I and GLC







43 Jacobi matrix




5 Conclusion

ndash 26 ndash

ndash 27 ndash

1 Introduction



23 Gauge freedom






31 Light rays

32 Light beams

321 Sachs basis

322 Jacobi matrix



4 Bianchi I and GLC







43 Jacobi matrix




5 Conclusion

ndash 27 ndash

1 Introduction



23 Gauge freedom






31 Light rays

32 Light beams

321 Sachs basis

322 Jacobi matrix



4 Bianchi I and GLC







43 Jacobi matrix




5 Conclusion

Documents

Geodesic-light-cone coordinates and the Bianchi I spacetimelensing in general [15, 16], to galaxy number counts [17], and to the propagation of ultra-relativistic particles [18]. See