Properties of blueshifted light rays in quasispherical Szekeres metrics
aa r X i v : . [ g r- q c ] M a r Properties of blueshifted light rays in quasispherical Szekeres metrics
Andrzej Krasi´nski
N. Copernicus Astronomical Centre,Polish Academy of Sciences,Bartycka 18, 00 716 Warszawa, Poland ∗ (Dated:)This paper is a follow-up on two previous ones, in which properties of blueshifted rays wereinvestigated in Lemaˆıtre – Tolman (L–T) and quasispherical Szekeres (QSS) spacetimes. In thepresent paper, an axially symmetric QSS deformation is superposed on such an L–T backgroundthat was proved, in the first paper, to mimic several properties of gamma-ray bursts. The presentmodel makes z closer to − t B ( r ), which is constant in the Friedmannregion, has a gate-shaped hump in the QSS region. Since a QSS island generates stronger blueshiftsthan an L–T island, the BB hump can be made lower – then it is further removed from the observerand implies a smaller observed angular radius of the source. Consequently, more sources can befitted into the sky – all these facts are confirmed by numerical computations. Null geodesics reachingpresent observers from different directions relative to the BB hump are numerically calculated.Patterns of redshift across the image of the source and along the rays are displayed. PACS numbers:Keywords:
I. MOTIVATION AND BACKGROUND
In Lemaˆıtre [1] – Tolman [2] and Szekeres [3, 4]spacetimes, some of the light rays emitted at the BigBang (BB) reach all observers with infinite blue shift(1 + z def = ν e /ν o = 0, where ν e and ν o are frequencies ofthe emitted and observed radiation, respectively). Thisis in contrast to Robertson – Walker spacetimes, whereall light from the BB is observed with z = ∞ [5, 6]. Thequantity z , traditionally called red shift, being negative(and then called blueshift) means that the frequency ob-served is higher than the frequency at the emission point,and z → − ν o → ∞ . The existence of blueshiftsin L–T models was predicted by Szekeres in 1980 [7], ina casual remark without proof, and then confirmed byHellaby and Lake in 1984 [8] by explicit calculation.Two conditions are necessary for infinite blueshift:(1) The BB time at the emission point of the ray musthave nonzero spatial derivative in comoving-synchronouscoordinates (the BB is “nonsimultaneous”).(2) The ray is emitted at the BB in a radial direction.Condition (2) was derived in Ref. [8], but seems to havebeen overlooked by all later authors until Ref. [9], eventhough it follows quite simply from the geodesic equa-tions. The two conditions together seem to be also suf-ficient, but a general proof of their sufficiency still doesnot exist; it is only implied by the full list of separatecases [8] and hinted at by numerical calculations [9, 10].The Szekeres spacetimes [3, 4], in general, have no sym-metry, thus no radial directions. In view of condition (2) ∗ Electronic address: [email protected] it was not clear whether any rays with infinite blueshiftexist in them. This question was addressed in Ref. [11].It was shown that in an axially symmetric quasisphericalSzekeres (QSS) spacetime, z = − z < − along axial rays emitted from the BB.It was also shown, by a blind numerical search, that rayswith 1 + z < .
07, and with similar spatial profiles of z along neighbouring rays, exist in an exemplary fullynonsymmetric QSS model.Since the L–T and Szekeres models have been provento successfully describe several observed features of ourUniverse [12, 13], and they predict a possible existence ofblueshifts, one must thoroughly test the implications ofblueshifts in order to either find a place for them amongthe observed phenomena, or conclude that the BB in thereal Universe must have been simultaneous. With thismotivation, it was shown in Ref. [9] that an L–T regionwith a gate-shaped “hump” on the BB profile matchedinto a Friedmann background can mimic some observedproperties of gamma-ray bursts (GRBs), such as the fre-quency range (0 . × to 1 . × Hz), the existenceof afterglows and the large distances to the sources. Plac-ing several different L–T regions in the same Friedmannbackground would then account for the large number ofpossible sources. However, the model of Ref. [9] wasunsuccessful on two accounts:(1) The gamma-ray flashes and the afterglows lastedfor too long. The model contains a parameter that shouldallow for controlling the durations, but insufficient nu-merical accuracy did not permit actual use of it.(2) The radiation was emitted isotropically insteadof being collimated into narrow beams, as the observedGRBs are supposed to be [14].Also, the model of Ref. [9] left some problems open.The main one was: how small could the humps on theBB profile be made while still generating the right rangeof frequencies of the observed radiation. Ref. [11] was the first step in improving the model ofRef. [9]. It showed by examples that strongly blueshiftedrays in QSS spacetimes exist only along two oppositedirections. That paper also proved that in a QSS modelthe minimum 1 + z is smaller than in an L–T model thathas the same BB profile.The present paper builds upon this last observation.The model considered here is a QSS deformation super-posed on the L–T region of Ref. [9]. Since the QSS de-formation results in a smaller 1 + z at the observer, theminimum value of 1 + z found in Ref. [9] can be achievedwith a lower BB hump. This implies a greater distancebetween the source of radiation and the observer, and asmaller angular diameter of the source seen in the sky.The progress achieved with respect to Ref. [9] is rathermoderate, but this cannot be the ultimate limit of im-provement: the class of BB profiles used here was foundby trial and error (see Sec. XII), and it is impossible thatthe optimal shape could be hit upon in this way.The L–T and Szekeres metrics are solutions of the Ein-stein equations with a dust source, so they cannot ap-ply to the real Universe at such early times when pres-sure cannot be neglected. It is assumed that they mayapply onward from the end of the last-scattering (LS)epoch. The mean mass density at LS, denoted ρ LS , inthe now-standard ΛCDM model is known [9], see Sec.IV. For every past-directed null geodesic in a QSS (orL–T) region, the mass density at the running point is nu-merically calculated. When this density becomes equalto ρ LS , the integration is stopped. Thus, 1 + z be-tween LS and the present time is bounded from below, z LS ≥ z min > −
1. The computational problem is to ar-range the BB profile so that it makes z LS sufficiently nearto − z LS < . × − [9]), but does not lead toperturbations of the CMB radiation larger than obser-vations allow. Among other things, this implies that themodel must be capable of making the angular diameter ofthe radiation sources smaller than the observed diameterof the GRBs (currently ≈ ◦ , see Sec. XI).In Secs. II and III, the subfamily of QSS models em-ployed here is presented. It is an axially symmetric QSSregion matched into a Friedmann background with cur-vature index k = − .
4. In Sec. IV the parameters of the It is easy to obtain small 1 + z with a high hump on the BB, butthen the radiation source is close to the observer and has a largeangular diameter in the sky. With a lower hump the diametergets smaller, but 1 + z gets larger. Keeping both the diameterand 1 + z sufficiently small is the main difficulty. Private communication in 2015 from Linda Sparke, then atNASA. The 1 ◦ is the current resolution of the detectors ratherthan the true diameter. background model are specified. They are different fromthose of the ΛCDM model [15, 16] – it was convenientto keep them the same as in the earlier papers by thisauthor [9, 10]. In Sec. V, the equations of null geodesicsin the QSS region are presented. In Sec. VI, basic prop-erties of redshift are described, and the conditions for z = − on which z has maximaor minima along axial rays. In Sec. VIII, the numericalparameters of the model used here are adapted to theGRBs of lowest frequency. In Sec. IX, exemplary non-axial plane rays reaching the present observers are nu-merically determined. The observers are placed in threedirections with respect to the QSS region: (I) – in pro-longation of the dipole minimum, (II) – in prolongationof the dipole maximum, and (III) – in prolongation of thedipole equator of the boundary of the QSS region. Foreach observer, the redshift profiles across the image of theradiation source are presented in tables. In Sec. X, red-shift profiles along the nonaxial rays reaching ObserverI are displayed to show that analogues of the ERS existalso along nonaxial directions. In Sec. XI it is estimatedthat ≈ ,
000 radiation sources of Sec. VIII could befitted into the celestial sphere. The necessary and possi-ble improvements of the model are discussed in Sec. XII.Section XIII contains the summary and conclusions.The present paper is a study in the geometry of theQSS spacetimes and in properties of their blueshiftedrays. Also, it introduces methods that can be used infurther refinements of the model. The observed param-eters of the GRBs were used as a beacon pointing theway, but the configuration derived here needs further im-provements before it can be considered a model of a GRBsource; see Sec. XI.Most results of numerical calculations are quoted upto 17 decimal digits. Such precision is needed to capturetime intervals of ≈
10 min at the observer, which is ≈ × − in the units used here, see Sec. III. (The 10 minis a representative time during which GRBs are visibleto the detectors [9].) II. QSS SPACETIMES
The metric of the QSS spacetimes is [3, 4, 6, 17]d s = d t − (Φ , r − Φ E , r / E ) E ( r ) d r − (cid:18) Φ E (cid:19) (cid:0) d x + d y (cid:1) , (2.1) E def = S "(cid:18) x − PS (cid:19) + (cid:18) y − QS (cid:19) + 1 , (2.2) Sections II, IV, V and VII are partly copied from Ref. [11]. P ( r ), Q ( r ), S ( r ) and E ( r ) being arbitrary functions suchthat S = 0 and E ≥ − / r .The source in the Einstein equations is dust ( p = 0)with the velocity field u α = δ α . The surfaces of con-stant t and r are nonconcentric spheres, and ( x, y ) arestereographic coordinates on each sphere. At a fixed r ,they are related to the spherical coordinates by x = P + S cot( ϑ/
2) cos ϕ,y = Q + S cot( ϑ/
2) sin ϕ. (2.3)The functions ( P, Q, S ) determine the centers of thespheres in the spaces of constant t (see illustrations inRef. [11]). Because of the nonconcentricity, the QSSspacetimes, in general, have no symmetry [18].With Λ = 0 assumed, Φ( t, r ) obeysΦ , t = 2 E ( r ) + 2 M ( r )Φ , (2.4)where M ( r ) is an arbitrary function. We consider modelswith E >
0, thenΦ( t, r ) = M E (cosh η − , sinh η − η = (2 E ) / M [ t − t B ( r )] , (2.5)where t B ( r ) is one more arbitrary function; t = t B ( r ) isthe BB time, at which Φ( t B , r ) = 0. We assume Φ , t > κρ = 2 ( M, r − M E , r / E )Φ (Φ , r − Φ E , r / E ) , κ def = 8 πGc . (2.6)This density distribution is a mass dipole superposed ona spherically symmetric monopole [4, 19]. The dipole,generated by E , r / E , vanishes where E , r = 0. The densityis minimum where E , r / E is maximum and vice versa [20].The arbitrary functions must be such that 0 < ρ < ∞ at all t > t B ( r ). The conditions that ensure this are [20]: M, r M ≥ p ( S, r ) + ( P, r ) + ( Q, r ) S ∀ r, (2.7) E, r E > p ( S, r ) + ( P, r ) + ( Q, r ) S ∀ r. (2.8)These inequalities imply [20] M, r M ≥ E , r E , E, r E > E , r E ∀ r. (2.9)The extrema of E , r / E with respect to ( x, y ) are [20] E , r E (cid:12)(cid:12)(cid:12)(cid:12) extreme = ± p ( S, r ) + ( P, r ) + ( Q, r ) S , (2.10)with + corresponding to maximum and − to minimum.In the following, we will call these two loci “dipole max-imum” and “dipole minimum”, respectively. The L–T models follow from the QSS models as thelimit of constant ( P, Q, S ). Then the constant-( t, r )spheres become concentric, and the spacetime becomesspherically symmetric. The Friedmann limit is obtainedwhen
E/M / and t B are constant (in this limit, ( P, Q, S )can be made constant by a coordinate transformation).A QSS spacetime can be matched to a Friedmann space-time across an r = constant hypersurface.Because of p = 0, the QSS models can describe thepast evolution of the Universe no further back than tothe last scattering hypersurface (LSH). See Sec. VIII forinformation on how to determine it in our model. III. THE QSS MODELS CONSIDERED IN THISPAPER
We will consider such QSS spacetimes whose L–T limitis Model 2 of Ref. [9]. The r -coordinate is chosen so that M = M r , (3.1)and M = 1 (kept in formulae for dimensional clarity)[10]. From this point on, the r -coordinate is unique. Thefunction E ( r ), assumed in the form2 E/r = − k = 0 . , (3.2)is the same as in the background Friedmann model.The units used in numerical calculations were intro-duced and justified in Ref. [21]. Taking [22]1 pc = 3 . × km , . × s , (3.3)the numerical length unit (NLU) and the numerical timeunit (NTU) are defined as follows:1 NTU = 1 NLU = 9 . × y = 3 × Mpc . (3.4) B0 B1 x0y0 A0A1x1 y1r = 0 t = tBfrt B0 B1 x0y0 A0A1x1 y1r = 0 t = tBfrt B0 B1 x0y0 A0A1x1 y1r = 0 t = tBfrt B0 B1 x0y0 A0A1x1 y1r = 0 t = tBfrt B0 B1 x0y0 A0A1x1 y1r = 0 t = tBfrt B0 B1 x0y0 A0A1x1 y1r = 0 t = tBfrt B0 B1 x0y0 A0A1x1 y1r = 0 t = tBfrt B0 B1 x0y0 A0A1x1 y1r = 0 t = tBfrt B0 B1 x0y0 A0A1x1 y1r = 0 t = tBfrt
FIG. 1: Parameters of the bang-time profile in the quasispher-ical Szekeres region; see text for explanation.
The BB profile belongs to the same 5-parameter familyas in Ref. [9], see Fig. 1. It consists of two curved arcsand a straight line segment joining them. The upper-leftarc, shown as a thicker line, is a segment of the curve r B + ( t − t Bf − A ) B = 1 , (3.5)where t Bf = − . ≈ − . × years;(3.6)see Sec. IV for comments on this value. The lower-rightarc (also shown as a thicker line) is a segment of theellipse ( r − B − A ) A + ( t − t Bf − A ) A = 1 . (3.7)The straight segment passes through the point ( r, t ) =( B , t Bf + A ) where the full curves (shown as dottedlines) would meet; x determines its slope.The free parameters are A , A , B , B and x . Figure1 does not show the values used in numerical calculations;in particular x and A are greatly exaggerated. Theactual values in Model 2 of Ref. [9] are A B A B x = . . × − . × − (3.8)( A , B and x are dimensionless). This profile will bethe starting point for modifications.The QSS model used here is axially symmetric, with P ( r ) = Q ( r ) = 0 and S ( r ) the same as in Ref. [11]: S = p a + r , (3.9)where a > E = 12 S (cid:0) x + y + S (cid:1) ; (3.10)This S ( r ) obeys (2.7) and (2.8), which, using (3.1) and(3.2), both reduce to 1 /r > S, r /S. (3.11)The equation of the dipole “equator” E , r = 0 is x + y = S ; (3.12)the axis of symmetry is x = y = 0. The extrema of thedipole are, from (2.10) E , r E (cid:12)(cid:12)(cid:12)(cid:12) extreme = ± S, r S . (3.13) It was introduced to keep d t B / d r finite everywhere. At r > r b , where r b = A + B = 0 . , (3.14)the BB profile becomes flat, and the geometry of themodel becomes Friedmannian. See Sec. V for remarkson the choice of coordinates in that region. IV. THE BACKGROUND MODEL
Our Friedmann background is defined by:Λ = 0 , k = − . , t B = t Bf , (4.1)where k is the curvature index and t B is the BB timegiven by (3.6); t = 0 is the present time. The t Bf is theasymptotic value of the function t B ( r ) in the L–T modelthat mimicked accelerating expansion [10]. This differsby ∼ .
6% from ( − T ), where T is the age of the Universegiven by the Planck satellite team [15] T = 13 . × y = 0 .
141 NTU . (4.2)The density at the last scattering time is [9] κρ LS = 56 . × (NLU) − . (4.3)This value follows from the model of the cosmologicalrecombination process [23–25] and is independent of theafter-recombination model. With (4.1), ρ LS implies theredshift relative to the present time1 + z bLS = 952 . . (4.4)This differs by ∼ .
7% from the ΛCDM value [15, 16] z LS = 1090 . (4.5)The present temperature of the CMB radiation is directlymeasured, so if (4.4) were taken for real, the temperatureof the background radiation at emission would be ∼ ∼ V. NULL GEODESICS IN THE AXIALLYSYMMETRIC QSS SPACETIMES
In an axially symmetric QSS metric, x and y can bechosen such that P = Q = 0; then x = y = 0 is thesymmetry axis [26, 27]. However, the loci x = ∞ and y = ∞ are coordinate singularities (they are at the pole ofthe stereographic projection), and numerical integrationof nonaxial geodesics breaks down on crossing those sets.Therefore, we introduce the new coordinates ( ϑ, ϕ ) by x = S b cot( ϑ/
2) cos ϕ, y = S b cot( ϑ/
2) sin ϕ, (5.1)where S b is S at the Szekeres/Friedmann boundary: S b def = S ( r b ) = p a + r b . (5.2)This changes (2.1) and (2.2) tod s = d t − N d r E ( r ) − (cid:18) Φ F (cid:19) (cid:0) d ϑ + sin ϑ d ϕ (cid:1) , (5.3) F def = S b S (1 + cos ϑ ) + S S b (1 − cos ϑ ) , N def = Φ , r − Φ F , r / F . (5.4)The dipole equator F , r = 0 is now at cot( ϑ eq /
2) =
S/S b (so ϑ eq = π/ r = r b we have F = 1 and ( ϑ, ϕ ) become thespherical coordinates with the origin at r = 0.Along a geodesic we denote (cid:0) k t , k r , k ϑ , k ϕ (cid:1) def = d( t, r, ϑ, ϕ )d λ , (5.5)where λ is an affine parameter. The geodesic equationsfor (5.3) – (5.4) ared k t d λ + N N , t E ( k r ) + ΦΦ , t F h(cid:0) k ϑ (cid:1) + sin ϑ ( k ϕ ) i = 0 , (5.6)d k r d λ + 2 N , t N k t k r + (cid:18) N , r N − E, r E (cid:19) ( k r ) + 2 S, r sin ϑ Φ S F N k r k ϑ − Φ(1 + 2 E ) F N h(cid:0) k ϑ (cid:1) + sin ϑ ( k ϕ ) i = 0 , (5.7)d k ϑ d λ + 2 Φ , t Φ k t k ϑ − S, r sin ϑ N S Φ(1 + 2 E ) ( k r ) + 2 N Φ k r k ϑ + F , ϑ F h − (cid:0) k ϑ (cid:1) + sin ϑ ( k ϕ ) i − cos ϑ sin ϑ ( k ϕ ) = 0 , (5.8)d k ϕ d λ + 2 Φ , t Φ k t k ϕ + 2 N Φ k r k ϕ + 2 (cid:20) cos ϑ sin ϑ − F , ϑ F (cid:21) k ϑ k ϕ = 0 . (5.9)The geodesics determined by (5.6) – (5.9) are null when (cid:0) k t (cid:1) − N ( k r ) E ( r ) − (cid:18) Φ F (cid:19) h(cid:0) k ϑ (cid:1) + sin ϑ ( k ϕ ) i = 0 . (5.10)Note that k ϕ ≡ ϑ ≡ ϑ ≡ π (axial rays) are solutions of (5.8).To calculate k r on nonaxial null geodesics, Eq. (5.10)will be used, which is insensitive to the sign of k r . Anumerical program for integrating the set { (5.6), (5.8) –(5.10) } will have to change the sign of k r wherever k r reaches zero. There exist no null geodesics on which k ϕ ≡ ϑ has any constant value other than 0 or π . This followsfrom (5.8): Suppose k ϕ ≡ k ϑ = 0 ata point. Then, if sin ϑ = 0, the third term in (5.8) willbe nonzero (because | S Φ(1 + 2 E ) | < ∞ , S, r = 0 from(3.9), N 6 = 0 from no-shell-crossing conditions [20] and k r = 0 from (5.10)), and so d k ϑ / d λ = 0. Consequently,in the axially symmetric case the only analogues of radialdirections are ϑ = 0 and ϑ = π . The fact reported under(6.4) below is consistent with this.The coefficient 1 / Φ in (5.8) and (5.9) becomes infiniteat r = 0, where Φ = 0 [9], but all the suspicious-lookingterms are in fact finite there [11]. In the present paperthe only geodesics running through r = 0 will be the axialones, on which (5.8) and (5.9) are obeyed identically.Let the subscript o refer to the observation point. Onpast-directed rays k t < k to = − . (5.11)Then, from (5.10) we have (cid:0) k ϑo (cid:1) + sin ϑ ( k ϕo ) ≤ (cid:18) F o Φ o (cid:19) ; (5.12)the equality occurs when the ray is tangent to a hyper-surface of constant r at the observation event, k ro = 0.On the boundary r = r b between the QSS and Fried-mann regions the coordinates on both sides must coin-cide. Thus, for the Friedmann region one must use themetric (5.3) with t B = t Bf given by (3.6) ( E has theFriedmann form (3.2) everywhere). The metric then be-comes Friedmann with no further limitation on S . Butfor correspondence with Ref. [9], we choose the coordi-nates in the Friedmann region so that S = p a + r b = S b . (5.13)Then, F = 1 and ( ϑ, ϕ ) are the spherical coordinatesthroughout the Friedmann region. VI. THE REDSHIFT IN AXIALLY SYMMETRICQSS SPACETIMES
Along a ray emitted at P e and observed at P o z = ( u α k α ) e ( u α k α ) o , (6.1)where u α are the four-velocities of the emitter and ofthe observer, and k α is the affinely parametrised tangentvector field to the ray [5]. In our case, both u α = δ α ,and then (6.1) simplifies to 1 + z = k et /k ot . If the affineparameter is rescaled so that (5.11) holds, then1 + z = − k et . (6.2)Equation (5.9) has the first integral: k ϕ sin ϑ Φ / F = J , (6.3)where J is constant. When (6.3) is substituted in (5.10),the following results:( k t ) = N ( k r ) E + (cid:18) Φ F (cid:19) (cid:0) k ϑ (cid:1) + (cid:18) J F sin ϑ Φ (cid:19) . (6.4)Equations (6.4) and (6.2) show that for rays emitted atthe BB, where Φ = 0, the observed redshift is infinitewhen J = 0. A necessary condition for infinite blueshift(1 + z o = 0) is thus J = 0, so(a) either k ϕ = 0, i.e. the ray proceeds in the hyper-surface of constant ϕ ,(b) or ϑ = 0 , π along the ray ( J / sin ϑ → ϑ → , π by (6.3)).Condition (b) appears to be also sufficient, but this hasbeen demonstrated only numerically in concrete exam-ples of QSS models ([11] and Sec. VIII here).Consider a ray proceeding from event P to P andthen from P to P . Denote the redshifts acquired in theintervals [ P , P ], [ P , P ] and [ P , P ] = [ P , P ] ∪ [ P , P ]by z , z and z , respectively. Then, from (6.1)1 + z = (1 + z ) (1 + z ) . (6.5)In particular, for a ray proceeding to the past from P to P , and then back to the future from P to P :1 + z = 11 + z . (6.6) VII. THE EXTREMUM REDSHIFT SURFACE
Consider a null geodesic that stays in the surface { ϑ, ϕ } = { π, constant } ; it obeys (5.8) and (5.9) identi-cally. On it, k r = 0 at all points because with k ϑ = k ϕ =0 the geodesic would be timelike wherever k r = 0, so r can be used as a parameter. Assume the geodesic is past-directed so that (6.2) applies. Using (6.2) and changingthe parameter to r , we obtain from (5.6)d z d r = N N , t E k r . (7.1)Since N 6 = 0 from no-shell-crossing conditions [20] and k r = 0, the extrema of z on such a geodesic occur where N , t ≡ Φ , tr − Φ , t F , r / F = 0 . (7.2)In deriving (7.2), ϑ = π was assumed, but ϕ was anarbitrary constant. Thus, the set in spacetime definedby (7.2) is 2-dimensional; it is the Extremum RedshiftSurface (ERS) [11].From (2.4) and (3.2) we obtainΦ , t = r r M r Φ − k, (7.3)Φ , tr = r M r Φ − k + M r Φ t B,r . (7.4) Using (7.3), (7.4) and (5.4) with ϑ = π , Eq. (7.2) be-comes r M r Φ − k (cid:18) − r S, r S (cid:19) = − M r Φ t B,r . (7.5)To avoid shell crossings, t B,r < r > so the right-hand side of (7.5) is non-negative.The left-hand side is positive with S given by (3.9). Using(2.5) for Φ, remembering that k < χ def = sinh ( η/
2) (7.6)we obtain from (7.5) χ + χ = − k (cid:20) rt B,r M (1 − rS, r /S ) (cid:21) . (7.7)With k <
0, (7.7) is solvable for χ at any r , since itsleft-hand side is independent of r and can vary from 0 to+ ∞ while the right-hand side is non-negative.Note that where t B,r = 0, Eqs. (7.7) and (7.6) imply χ = η = 0, i.e. at those points the ERS is tangent tothe BB. Also, the ERS is tangent to the BB at r = 0unless d t B / d r −→ r → ∞ . (This would imply d ρ/ d r −→ r → ∞ ,an infinitely thin peak in density at r = 0 – an unusualconfiguration, but not a curvature singularity [28].) Themodel considered here will have t B,r = 0 at r = 0.In the limit S, r = 0, (7.7) reproduces the equation ofthe Extremum Redshift Hyper surface (ERH) of Ref. [10].Equation (7.7) was derived for null geodesics proceed-ing along ϑ = π , where F , r / F = S, r /S >
0. With S given by (3.9) we have F = 1 / (1 − rS, r /S ) = ( r/a ) + 1 > , (7.8)so, at a given r , the ERS has a greater η (and so a greater t − t B ) than the corresponding ERH of the L–T model.Also, the extrema of z along the dipole maximum occurat a greater χ (and thus greater t − t B ) when a is smaller.This will be illustrated by Fig. 2 in the next section.Conversely, for a ray proceeding along the dipole min-imum axis (where ϑ = 0), the factor F is replaced by F = 1 / (1 + rS, r /S ) = a + r a + 2 r < , (7.9)and so the ERS has a smaller t − t B than the ERH inL–T. Also here, a smaller a has a more pronounced effect.Extrema of redshift also exist along directions otherthan ϑ = 0 and ϑ = π , as will be demonstrated bynumerical examples in Sec. X, but a general equationdefining their loci remains to be derived. Refs. [20] and [6] did not spell out the condition r >
VIII. A GENERALISED MODEL 2 OF REF. [9]
Along each past-directed null geodesic, the mass den-sity is calculated using (2.5) – (2.6). As explained in Sec.IV, in any model the density at the LSH must be thesame as in (4.3). So, the instant of crossing the LSH isthat where the density becomes equal to (4.3).The starting point for this paper is Model 2 of Ref.[9], whose functions M ( r ), E ( r ) and t B ( r ) are given by(3.1), (3.2) and (3.5) – (3.8). In that model, the strongestblueshift between the LSH and the present epoch was1 + z maxb = 1 . × − . (8.1)It was calculated by the rule (6.5). The first factor,1 + z ols2 = 1 . × − , (8.2)was the blueshift between the LSH and r = 0, achievedon a path that will be called “Ray A”. The second factor,1 + z po2 = 126 . . (8.3)was the redshift between r = 0 and the present epoch ona path going off from the same initial point as Ray A,but to the future; it will be called “Ray B”.On Model 2, axially symmetric QSS deformationsgiven by P = Q = 0, (3.9) and (3.10) are superposed.Numerical experiments with rays proceeding along ϑ = π were done to improve on (8.1) as much as possible. Asexplained under (7.8), smaller a increases the region un-der the ERS. So, with the parameters of (3.8), a wasgradually changed from 10 through 1, 1/10, 10 − , 10 − to 10 − . For each a the quantity t (0) − t B (0) def = ∆ t c (8.4)was chosen such as to obtain a minimum 1 + z betweenthe LSH and r = 0. This led to smaller 1 + z on Ray Aonly down to a = 0 . a still smaller, the rayeither flew over the BB hump and crossed the LSH in theFriedmann region with a large z > z >
0. Nointermediate value of ∆ t c led to z < a = 0 .
001 was1 + z = 8 . × − .In the next experiments, the slope of the straight seg-ment of the BB profile was gradually decreased, i.e x wasincreased from 2 × − through 1 × − to 1 × − ,with the other parameters unchanged. For each value of x , the ∆ t c leading to the smallest 1 + z was determined.The best result achieved at this stage was1 + z = 6 . × − . (8.5)Varying A , B , B , and lowering the degree of (3.5) to4 and to 2, led to nothing better than (8.5). So, thisis taken as the best improvement over the L–T modelachieved using an axially symmetric QSS deformation. Figure 2 shows Ray A, with 1 + z given by (8.5),and the corresponding ERS and BB profiles. Curve 1is the ERH profile of Model 2 from Ref. [9], and Curve2 is the ERS profile with a = 10 − . As stated above,smaller a gives more space under the ERS, but when toosmall it creates a discontinuity in z that prevents z < -0.13946-0.13944-0.13942-0.1394-0.13938-0.13936-0.13934-0.13932-0.1393 0 0.005 0.01 0.015 0.02ERS ERSthe ray the ray + BB + ERS BBBBBB + ERS 12t r FIG. 2: The Big Bang profile and the axial ray with the small-est 1 + z in the Szekeres model. The short horizontal strokesare at the ends of the straight BB segment. The dot showswhere the ray hits the LSH. See text for more explanation. The ERS profile has two branches on each side of r =0, so some rays will intersect it four times and z alongthem will have two local maxima and two local minima.Examples will appear in Sec. X.On Ray B, the upward 1 + z is1 + z = 6 . × − . (8.6)Thus, total (1 + z ) between LSH and now is1 + z = 1 + z z = 1 . × − . (8.7)This fits the lowest-frequency GRBs, for which [9]1 + z max ≈ . × − , (8.8)with a wider margin than (8.1), so the BB hump can nowbe lowered to yield (1 + z ) closer to (8.8). The easiestway to do this is to decrease B (see Fig. 1). Then ∆ t c isfine-tuned to make (1+ z ols ) on Ray A as small as possible(1 + z ols gets larger when B gets smaller, so there is alimit on decreasing B ). The B that allows sufficientlysmall (1 + z ) ols is B = 0 . , (8.9)and then the smallest 1 + z on Ray A is1 + z ols3 = 1 . × − . (8.10)For Ray B corresponding to Ray A of (8.10) (proceed-ing along the dipole minimum), the 1 + z between r = 0and the present epoch is1 + z = 7 . × − , (8.11)so (1 + z ) between LSH and now along Rays A and B is1 + z = 1 + z ols3 z = 1 . × − , (8.12)and the present observer is at r = r obs = 0 . . (8.13)This is larger than r O2 = 0 . t c = 0 . . (8.14)The corresponding results for rays propagating in theopposite direction, i.e. along the dipole minimum be-tween the LSH and r = 0 (Ray C), and along the dipolemaximum between r = 0 and the observer (Ray D), areas follows. The best value of 1 + z on Ray C is1 + z = 1 . × − , (8.15)achieved with∆ t c = 0 . . (8.16)Then, 1 + z calculated toward the future along the dipolemaximum is1 + z = 7 . × − . (8.17)So, the 1 + z between the LSH and the present time is1 + z = 1 + z z = 2 . × − . (8.18)The present time was reached by the ray at e r obs = 0 . . (8.19)Thus, on this ray 1 + z is larger while r obs is smaller.In each case the numerical calculation overshot thepresent time. For the ray that produced (8.11) and(8.13), the value of t at the endpoint was t end 1 = 5 . × − NTU , (8.20)and for the ray that produced (8.17) and (8.19) it was t end 2 = 9 . × − NTU . (8.21) IX. NONAXIAL PLANE RAYS
So far, rays crossing the symmetry axis of the t = con-stant spaces in the metric (5.3) – (5.4) were considered.Now, we will consider nonaxial rays ( ϑ will no longer be0 or π all along the ray) propagating in a hypersurface of constant ϕ . By (6.3), J = 0 along them, and theyobey (5.9) identically. Because of axial symmetry of themodel, the image will be the same for every ϕ .We will consider pencils of rays flying through thevicinity of the BB hump shown in Fig. 2 and reachingthe present observer situated in three locations:Observer I: At( t, r, ϑ ) I = ( t end 1 , r obs , , (9.1)with r obs given by (8.13). This is the endpoint of Ray B.Observer II: At( t, r, ϑ ) II = ( t end 2 , e r obs , π ) , (9.2)with e r obs given by (8.19). This is the endpoint of Ray D.Observer III: At( t, r, ϑ ) III = (0 , r p , π/ , where r p = ( r obs + e r obs ) / . (9.3)The ϑ III is at the dipole equator on the boundary of theSzekeres region. One ray reaching Observer III will have ϑ = π/ t d λ = k t , (9.4)d k t d λ = − N N , t E ( k r ) − ΦΦ , t F (cid:0) k ϑ (cid:1) , (9.5)d ϑ d λ = k ϑ , (9.6)d k ϑ d λ = − , t Φ k t k ϑ + sin ϑS, r N S Φ(1 + 2 E ) ( k r ) − N Φ k r k ϑ + sin ϑ (cid:0) S − S b (cid:1) SS b F (cid:0) k ϑ (cid:1) , (9.7)d r d λ = k r , (9.8) k r = ± √ E N p ξ,ξ def = (cid:0) k t (cid:1) − (cid:18) Φ k ϑ F (cid:19) . (9.9)The initial values for ( t, r, ϑ ) will be at the observer po-sitions specified above, the initial value for k t is (5.11),and the rays will be calculated backward in time fromthere. With k ϕ = 0, Eq. (5.12) reduces to (cid:0) k ϑo (cid:1) ≤ (cid:18) F o Φ o (cid:19) . (9.10)As before, the equality occurs when k ro = 0.For observers in the Friedmann region, F o = 1, asexplained under Eq. (5.13). For Observer I Φ o was cal-culated by the program that found (8.11); it is(Φ o ) obs 1 = 0 . . (9.11)The angle α between two rays at an observer can becalculated as follows. The direction of a ray is determinedby the unit spacelike vector given by [6] n α = u α − k α k ρ u ρ , (9.12)where k α is the tangent vector to the ray and u α isthe velocity vector of the observer; n α u α = 0. Since g αβ n α n β = −
1, the angle between two directions obeyscos α = − g αβ n α n β . (9.13)Since u α = δ α everywhere, and k = − n α at the observer are n αo = (cid:0) , k ro , k ϑo , k ϕo (cid:1) . (9.14)Using (9.9), (5.3) and assuming k ϕo = 0 we then obtainfor the angle α RS between rays R and Scos α RS = s − (cid:18) k ϑRo Φ o F o (cid:19) × s − (cid:18) k ϑSo Φ o F o (cid:19) + k ϑRo k ϑSo (cid:18) Φ o F o (cid:19) . (9.15)Both k ϑo must obey (9.10), so | cos α RS | ≤ α RS obeying (9.15) exists.When Ray R is axial ( k ϑRo = 0), and the observer liesin the Friedmann region where F o = 1, (9.15) becomescos α RS = q − (cid:0) k ϑSo Φ o (cid:1) = ⇒ sin α RS = k ϑSo Φ o . (9.16)This equation can be used to estimate the angular ra-dius of a radiation source in the sky; then α is the anglebetween the direction of the central ray (going along thesymmetry axis for Observers I and II) and the direction ofthe ray that grazes the edge of the source. The latter canbe approximately determined in numerical experiments.The redshift in the Friedmann background between theLSH and the present time, calculated numerically alonga null geodesic is1 + z b = 951 . . (9.17)This differs slightly from (4.4), which was calculated from1 + z bLS = R now / R LS , where R is the Friedmann scalefactor, and also from 1 + z comp = 951 . k r for each ray follows from (9.9) after the valueof k ϑo is chosen. At all initial points, k r <
0, but ξ was monitored along each ray, and when it went down to orbelow zero, the sign of k r was reversed. A. Rays reaching Observer I
Table I lists the parameters of exemplary nonradialrays received by Observer I, with the angular radii cal-culated by (9.16). The angular radius of the whole BBhump (Ray 9 in the table) here is somewhat smaller thanthe 1 . ◦ in the L–T/Friedmann model of Ref. [9].Decreasing this radius was one of the aims of replacingthe L–T region with Szekeres. TABLE I: Parameters of nonaxial rays reaching Observer I.For Ray 9, k ϑo = 0 . k ϑo Angular radius ( ◦ ) 1 + z at LSH0 0.000001 2 . × − In Figs. 3 and 4 the coordinates are X = − r cos ϑ, Y = r sin ϑ. (9.18)Figure 3 shows the projections of the rays from Table I ona surface of constant t along the flow lines of the dust ina neighbourhood of the QSS region. Figure 4 is a closeupview on the vicinity of the BB hump. The dotted circle isat r = r b , the r -coordinate of the edge of the BB hump.The cross marks the center r = 0 of the dotted circle; thearrow on the horizontal arm of the cross in Fig. 4 pointsin the direction of the Szekeres dipole maximum. Thelarge dots in Fig. 3 mark the points where the rays in-tersect the LSH. The endpoints of the rays are where thenumerical calculation determined their crossing the BB.Figures 3 and 4 are nearly the same as the correspondingones for the L–T/Friedmann model in Ref. [9]; there areonly small quantitative differences between them. They Note that ξ < k ϕ = 0 by(5.10). But it can happen because of numerical inaccuracy. If ξ < n , then for this step it was replaced by ( − ξ ); thenit should begin to grow. Along some rays the sign reversals of ξ in a vicinity of the smallest r had to be done many times . -0.07-0.06-0.05-0.04-0.03-0.02-0.0100.010.02 -0.04 -0.02 0 0.02 0.04 0.06 1234 5 6 7 89XY-0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 FIG. 3:
Upper panel:
Projections of the rays listed in TableI on a surface of constant t along the flow lines of the cosmicdust. Observer I is at X ≈ − . , Y = 0 beyond the leftmargin of the figure. The large dots mark the intersectionsof the rays with the last-scattering hypersurface. The dottedcircle has the radius r = A + B , where the BB hump hasits edge. More explanation in the text. Lower panel:
Ray9 shown all the way between Observer I and the BB.
Ray 0 is not included in the figures because, at theirscale, it would coincide with the Y = 0 axis. It is includedin the table in order to show how 1+ z LSH abruptly jumpsfrom the near-zero value (8.12) on an axial ray to a largepositive value on a ray that is only slightly nonaxial.The redshifts initially increase with the viewing an-gle. The maximum z LSH is achieved on Ray 8 insidethe image of the source, not at its edge, and it is largerthan the background (9.17). The same thing happenedin the L–T/Friedmann model [9], and will again occurfor Observers II and III further in this paper. Ray 9 justgrazes the world-tube r = r b , and z LSH on it is close to(9.17). Its k ϑo was determined by trial and error: for eachray the program that calculated its path determined theminimum r def = r cl along it, and Ray 9 is the one where r cl − r b = 0 . r = r b . The change issharper on the second intersection with r = r b where theray is closer to the BB. When the rays travel over the BBhump further from its edge the deflections are smaller.The angle of deflection depends on the interval of t that the ray spends near the edge of the BB hump. Ray 1meets r = r b nearly head-on and does not strongly changedirection on first encounter. On second encounter, it is -0.015-0.01-0.00500.0050.010.015-0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.021 12 233 44 556 67 78 89 9XY FIG. 4: The region near the BB hump in Fig. 3. The arrowon the horizontal arm of the cross points in the direction ofthe dipole maximum. More explanation in the text. closer to the BB and is forced to bend around more.The other rays meet the r = r b surface at smaller an-gles than Ray 1, so they stay near it for longer times. ForRays 3, 4 and 5, this causes a much stronger deflectionthan for Ray 1. For Rays 6 – 8, another effect prevails:they fly farther from the axis, so they approach the BB atlarger t − t B and stay over it for a shorter time; thereforethe deflection angle decreases again. Ray 9 does not enterthe Szekeres region but only touches it, so it propagatesalmost undisturbed as in the Friedmann region.Figures 3 and 4 show only those rays for which k ϑo > k ϑo < ϑ = 0 is the axisof symmetry, the image will be the same for every ϕ , soone should imagine the complete collection of constant- ϕ null geodesics by rotating Figs. 3 and 4 around the ϑ = 0axis. B. Rays reaching Observer II
Table II and Fig. 5 are analogues of Table I and Fig.4 for Observer II. The analogue of Ray n from Table Iis Ray 10 + n in Table II. The k ϑo are the same as inTable I, with the exception of Ray 19 – see below for anexplanation. The angular radii are slightly smaller herebecause Φ o for Observer II is slightly smaller than (9.11):(Φ o ) obs 2 = 0 . . (9.19)But at the level of precision used in the tables, the angu-lar radii for Rays 11 – 18 are the same as those for Rays1 – 8. The analogue of Ray 0 is not included.1 TABLE II: Parameters of rays reaching Observer IIRay k ϑo z at LSH11 0.0005 358.2298917448562712 0.002 388.8085398078382013 0.005 408.0647651749574714 0.009 504.7918344887468215 0.012 620.0851187241804616 0.02 885.0297235747242417 0.03 970.7064414472338318 0.035 982.1381744629547919 0.04205 951.83804564661989 -0.015-0.01-0.00500.0050.010.015-0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.01511 12 131415 16 171819111213 14 15161718 19 XY FIG. 5: The analogue of Fig. 4 for Observer II, who is at X ≈ . , Y = 0 beyond the right margin of the figure. SeeTable II for the parameters of the rays. Ray 19 grazes the edge of the Szekeres region – so its k ϑ determines the angular radius of the whole source by(9.16). Since r obs is smaller here, the angular radius forRay 19 is larger than for Ray 9; it is α II = 0 . ◦ . (9.20)The values of 1 + z LSH in Table II are different fromthose in Table I, but the general pattern is the same: z LSH initially increases with the viewing angle, achieves amaximum inside the image of the source, then decreasesto the background value at its edge. The maximum isachieved at the same k ϑo as before, on Ray 18. C. Rays reaching Observer III
Observer III, unlike Observers I and II, is not locatedon the axis of symmetry, so the (past-directed) rays goingoff from her position with k ϑo < k ϑo >
0. Therefore, these two groupsof rays are shown in separate tables and separate figures.Table III and Fig. 6 contain the rays for which k ϑo ≤ k ϑo >
0. The set ofvalues of (cid:12)(cid:12) k ϑo (cid:12)(cid:12) is the same as in Table I and Fig. 4. Theanalogues of Ray n from Table I are Ray 20 + n in TableIII and Ray 30 + n in Table IV. TABLE III: Parameters of rays with k ϑo ≤ k ϑo z at LSH20 0.0 342.2996485543710621 -0.0005 350.6455833705118722 -0.002 337.4930838065238823 -0.005 361.1911333148372624 -0.009 470.6218970252115225 -0.012 629.7293711023639826 -0.02 900.5613827935025027 -0.03 971.4183800080751328 -0.035 982.3036326381213729 -0.0425 951.83650139022654 The value of Φ o here is between the previous ones,(Φ o ) obs 3 = 0 . , (9.21)while t o = 0 does not differ significantly from (8.20) and(8.21), so the angular radii would also be intermediate;they are not listed in the tables.The most conspicuous difference from the previouscases is in Ray 20, which proceeds along ϑ = π/ ϑ onentry to the Szekeres region, and bends oppositely to allother rays on leaving it. Rays 21 and 22 get deflected sostrongly that they cross the line ϑ = π/ , π/ ϑ = 0 , π lines just before leavingthe Szekeres region. Beginning with Ray 23, the pathsof the rays become similar (though different in numericaldetail) to the corresponding ones for Observers I and II.The pattern of 1 + z LSH across the image of the sourcehere is different from those for Observers I and II: withdecreasing k ϑo < r = 0,but with 0 = ϑ = π , so it will not have z = − r = 0 were numerically2 -0.02-0.015-0.01-0.00500.0050.010.015-0.015 -0.01 -0.005 0 0.005 0.01 0.01520 212223242526 272829 20 21 222324 2526272829 XY 20 212223242526 272829 20 21 222324 2526272829 XY FIG. 6: The analogue of Fig. 4 for Observer III, who is at X = 0 , Y ≈ .
889 above the upper edge of the figure. Only therays with k ϑo ≤ integrated for the same kind of Szekeres dipole (but witha different BB profile and with a = 0 .
1) – only thoseproceeding along ϑ = 0 , π had z ≈ − TABLE IV: Parameters of rays with k ϑo > k ϑo z at LSH31 0.0005 360.7950451323331432 0.002 381.9347098601747933 0.005 453.3427191991163534 0.009 576.9043243424865835 0.012 682.5285710947960136 0.02 895.4567701630637737 0.03 970.7262894708446838 0.035 982.1676188474651839 0.0425 951.83650139022654 For rays with k ϑo > z LSH is similarto that for Observer II: there is only the maximum, onRay 38. However, the values of 1 + z LSH differ, some ofthem substantially, from their counterparts in Table II.The paths of the rays are similar to those for ObserversI and II, but the angle of deflection is smaller for each rayhere. Also, the rays bend away from the X = 0 axis nearthe Y = 0 line – this effect was not visible for Observer -0.02-0.015-0.01-0.00500.0050.010.015 -0.015 -0.01 -0.005 0 0.005 0.01 0.01531 32 33 34 353637 38393132333435 36 3738 39XY 31 32 33 34 353637 38393132333435 36 3738 39XY 31 32 33 34 353637 38393132333435 36 3738 39XY FIG. 7: The rays that have k ϑo > I and barely noticeable for Observer II.
X. REDSHIFT PROFILES ALONG NONAXIALNULL GEODESICS
FIG. 8: The z ( r ) relation for Ray 3 from Fig. 3 and itscharacteristic branches. Observer I is at r ≈ . The z -profiles along Rays 1 – 6 and 9 are shown in Figs.8 and 9; they are similar to those in the L–T/Friedmannmodel [9]. They show that analogues of the ERS (callthem ERS’) exist also along nonaxial rays. Figure 8shows the z ( r ) relation for Ray 3 in a neighbourhoodof r = r b ; it is a key to reading Fig. 9. In segment (a)of the ray, z increases from 0 at the observer to a local3 FIG. 9: The z ( r ) relations for Rays 1 – 6 and 9 from Fig. 3.Along Rays 1 – 6, z has two maxima and two minima, whichpoints to the existence of an analogue of the Extremum Red-shift Surface along them. Ray 9 does not enter the blueshift-generating region, so z is increasing all along it. maximum at r ≈ r b , where the (past-directed) ray in-tersects the outer branch of the ERS’ for the first time.Then, in segment (b), z decreases to a local minimumat a slightly smaller r , where the ray intersects the innerbranch of the ERS’ for the first time. Further along theray, in segment (c), z increases until it reaches the secondlocal maximum at the second intersection of the ray withthe inner branch of the ERS’. Then, in segment (d), z decreases up to the second intersection of the ray withthe outer branch of the ERS’, where it achieves its secondand last local minimum. From then on, in segment (e), z keeps increasing up to ∞ achieved at the BB.Along Rays 1 and 2 in Fig. 9, the second minimum of z is smaller than the first maximum, so those z ( r ) curvesself-intersect. XI. FITTING THE RADIATION SOURCES INTHE CELESTIAL SPHERE
Imagine a radiation source to be a disk on the celestialsphere of angular radius ϑ . How many such disks wouldfit into the celestial sphere at the same time?An equivalent question is, how many non-overlappingcircles of a given radius can be drawn on a sphere of agiven radius? A rough answer would be obtained by di-viding the surface area of the sphere by the surface areainside the circle. But this would be an overestimate –the circles cannot cover the sphere completely. A betterapproximation is to inscribe each circle into a quadrangleof arcs of great circles on the sphere. Such figures can-not cover the sphere either, but this method takes intoaccount some of the area outside the circles. Details ofthe calculation are presented in the Appendix. The area of the sphere divided by the area of the quadrangle is N = π arcsin (cid:0) sin ϑ (cid:1) . (11.1)Taking ϑ = 0 . ◦ , the current resolution of the GRBdetectors (see footnote 2), we obtain N . ≈ , . (11.2)With ϑ = 0 . ◦ of Table I, we obtain N . ≈ , . (11.3)Finally, with ϑ = 0 . ◦ , as in (9.20), we obtain N . ≈ , . (11.4)It is instructive to compare these numbers with thenumber of GRBs detected in observations. This authorwas not able to get access to a definitive answer, but hereis an estimate based on partial information. The BATSE(Burst and Transient Source Explorer) detector, whichworked in the years 1991 – 2000, discovered 2704 GRBs[29] (it was de-orbited in 2000 [30]). Assuming the samerate of new discoveries, 8112 GRBs should have beendetected between 1991 and now – still fewer than (11.4).When the angular radius is divided by f , the numberof possible sources in the sky should be multiplied by f . Equation (11.1) approximately confirms this, sincefor small ϑ we have sin ϑ ≈ ϑ ≈ arcsin ϑ .) XII. POSSIBLE AND NECESSARYIMPROVEMENTS OF THE MODEL
The model presented here accounts for the lowest fre-quency of the radiation in the observed GRBs (the modelof highest-frequency GRBs was discussed in Ref. [9]).The angular radius of the radiation sources seen by thepresent observer is twice as large as the current observa-tions allow (nearly 1 ◦ in the model vs. 0 . ◦ – the resolu-tion of the GRB detectors; see footnote 2). In order todecrease this angle, the BB hump that emits the radiationshould be made narrower or lower; in the second case itwould be further away from the observer seeing the high-frequency flash. The BB profile chosen in this paper can-not be the limit of improvement. The first attempt to ex-plain the GRBs using a cosmological blueshift resulted ina model [31] whose hump had the height A + B = 0 . A + B = 0 . ≈
206 times andthe width 7.2 times. The result of such a blind searchcannot be the best possible. In particular, other classesof shapes of the BB hump should be tried.To get small 1 + z , the BB profile should be such thatthe blueshifted ray spends as much time as possible trav-eling above the LSH but below the ERS. As follows from(7.7) and (7.8), the room under the ERS becomes larger4when d t B / d r is larger and when a is smaller. The prob-lem with small a was described in Sec. VIII, but it mightbe overcome using a greater numerical precision. A largerd t B / d r tends to make the BB hump higher. In order tokeep the hump acceptably low, the large d t B / d r has tobe limited to a short interval of r – this is where the steepslope of the hump in Fig. 2 came from.A serious limitation is the fact mentioned in Sec. VIIthat the ERS is tangent to the BB at r = 0. If thiscould be overcome, the rays would stay in the blueshift-generating region (below the ERS) for a longer time in-terval, and so the required 1 + z range could be achievedwith a lower or narrower hump.Further optimizations are possible. For example,the function E ( r ) here has the Friedmann shape (3.2)throughout the Szekeres region – obviously one shouldcheck what happens when it has other shapes. Fried-mann backgrounds other than the one of Sec. IV shouldbe tested. Szekeres dipoles other than (3.9) should alsobe tested, in particular non-axially-symmetric ones. Car-rying out such tests is laborious – it involves finding, bynumerical shooting, the minimum of a function of severalvariables (in this paper these were 7 variables: the fivein (3.8), the a of (3.9) and the ∆ t c of (8.4)).Similar to the L–T model of Ref. [9], the modelpresented here implies too-long durations for the high-frequency flashes and for their afterglows. This is be-cause, in axially symmetric models, once the observerand the source are placed on the symmetry axis, theystay there forever – the source does not drift [32–34].The only changes of the observed frequency and inten-sity may then occur because the observer receives raysemitted from different points of the BB hump along thesame line of sight, so the changes occur on the cosmolog-ical time scale and are much slower than in the observedGRBs (see Ref. [9] for the numbers).A nonsymmetric Szekeres model offers a new possibil-ity. In such a model there also exist two opposite direc-tions along which radiation is strongly blueshifted [11].However, the cosmic drift [32–34] will cause an observerwho was initially in the path of one of those preferredrays to be off it after a while. The time scale of this pro-cess should be short, as a consequence of the very largedistance between the source and the observer and of thediscontinuous change from blueshift to redshift as soonas the strongly blueshifted ray misses the observer.One solution of the duration problem has already beentested, and will be submitted for publication soon. Ifthere is another QSS region between the radiation sourceand the observer, then the cosmic drift in the interveningQSS region will cause the highest-frequency ray to missthe observer after 10 minutes or less. This satisfactorilysolves the problem of the duration of the high-frequencyflash, but not the problem of the duration of the after-glow. The latter still awaits solution. XIII. SUMMARY AND CONCLUSIONS
In Ref. [11], existence and properties of blueshifts inexemplary simple quasispherical Szekeres models were in-vestigated. Using that knowledge, in the present paper itwas investigated whether a QSS mass dipole superposedon a L–T background would allow better mimicking ofgamma-ray bursts by cosmological blueshifting than inRef. [9], where pure L–T models were used.The axially symmetric QSS model was introduced inSecs. II and III. The QSS region is matched to anegative-spatial-curvature Friedmann background (Sec.IV), chosen for correspondence with earlier papers bythis author [10, 21]. After presenting definitions and pre-liminary information in Secs. V, VI and VII, in Sec.VIII the parameters of the QSS model are chosen suchthat at present the highest frequency of the blueshiftedradiation agrees with the lowest frequency of the ob-served GRBs (this agreement requires that the blueshiftbetween the last scattering and the present time obeys1 + z ≤ . × − [9]). The introduction of the Szek-eres dipole has the consequence that the required 1 + z is achieved with a lower hump in the BB profile, which isthus at a greater distance from the observer than in theL–T model. In Sec. IX, the paths of nonaxial light raysreaching three different present observers are presented.The observers are placed in prolongation of the mass-dipole maximum axis, of the dipole minimum axis, andof the dipole equator. The distributions of the observedredshift across the image of the source are different foreach observer, and the angular radii of the source arebetween 0 . ◦ and 0 . ◦ . This is nearly twice asmuch as the current GRB observations allow, but themodel has the potential to be improved (see Sec. XII).In Sec. X, the redshift profiles along nonaxial rays werecalculated in order to show that extrema of redshift alsoexist along them. In Sec. XI it was estimated that withthe angular radii of the radiation sources being between0 . ◦ and 0 . ◦ , approximately 11,000 such sourcescould be simultaneously fitted into the sky of the presentobserver. Finally, possible further improvements in themodel were discussed in Sec. XII.The models of generating the high-frequency radiationflashes discussed here and in Ref. [9] are subject to twokinds of tests:1. In the future, the observers should be able to resolvethe fuzzy disks they now see as GRB sources (see footnote2), and measure the distribution of radiation frequenciesand intensities across them. Then it will be possible tocompare those distributions with model predictions. Amodel that would predict such a distribution correctlycould then be used to get information about the sources.2. If the gamma flashes are generated simultaneouslywith the CMB radiation, as proposed here and in Ref.[9], then they are observed now as short-lived becauseits source comes into and out of the observer’s view, buthas existed there since the last-scattering epoch. In thiscase, the central high-frequency ray should be surrounded5by rays with positive redshifts smoothly blending withthe CMB background at the edge of the source image,as shown in tables in Sec. IX. But if a source of theradiation flash lies later than the last scattering, then itis independent of the CMB. It should black out all CMBrays within some angle around the central ray, and theredshift profile across the image of the source would notneed to continuously match the CMB at the edge.This author does not wish to question the validity ofthe GRB models proposed so far. The motivation forthis work was this: history of science teaches us thatif a well-tested theory predicts a phenomenon, then theprediction has to be taken seriously and checked againstexperiments and observations. Since general relativityclearly predicts that some of the light generated duringlast scattering might reach us with strong blueshift, con-sequences of this prediction have to be worked out andsubmitted to tests. In trying to accommodate blueshifts,the suspicion fell on the GRBs because it is generallyagreed that at least some of their sources lie billions ofyears to the past from now [14]. The BB humps discussedhere would lie about twice as far, at ≈ . Appendix A: How many circles of a given radius canbe drawn on a sphere of a given radius?
Imagine a circle K drawn on a sphere S of radius a and a cone that intersects S along K and has its vertexat the center of S; see Figs. 10 and 11. Let the openingangle of the cone be ϑ . Now imagine a square pyramidcircumscribed on this cone. The pyramid intersects Salong the curvilinear quadrangle shown in thicker linesin Fig. 10. The part of S inside the quadrangle has thesurface area 8 times the surface area inside the curvilineartriangle ABC; see also Fig. 12.Suppose the center of the sphere is at x = y = z = 0,so the equation of the sphere is x + y + z = a , andthe axis of the cone goes along the z axis. The metric ofthe sphere in the ( x, y ) coordinates isd x + d y + d z = (A1) (cid:0) a − y (cid:1) d x + 2 xy d x d y + (cid:0) a − x (cid:1) d y a − x − y , and so the surface element of the sphere is √ g d x d y = a p a − x − y d x d y. (A2)The side AC of the triangle lies in the plane x = 0, and y on it changes from 0 to a sin ϑ . The side AB lies in theplane y = x . The y -coordinate of the point B is y B = a sin ϑ p ϑ , (A3) -1-0.500.51 -1 -0.5 0 0.5 1 a K A BC xy
FIG. 10: View from the z > a K A ϑ o COE y z FIG. 11: Projection of the cone and of the pyramid from Fig.10 on the x = 0 coordinate plane. The lines OE and OC areintersections of the cone (and of the faces of the pyramid)with the plane of the figure; the letters K, A and C have thesame meaning as in Fig. 10. as is easy to calculate knowing that this point lies simul-taneously on the sphere x + y + z = a , in the plane y = x and in the plane z = y cot ϑ that contains theright face of the pyramid. The auxiliary point D has thesame y -coordinate as B. The arc BC (which is part ofthe intersection of the right face of the pyramid with thesphere) obeys the equation x = s a − y sin ϑ = x BC ( y ) . (A4)The surface area of the triangle ABC is thus S ABC = Z y B d y Z y a p a − x − y d x A BCDA BCD
FIG. 12: A sketch to calculate the surface area of the triangleABC on a sphere. See text. + Z a sin ϑ y B d y Z x BC ( y )0 a p a − x − y d x (A5)= Z y B a arcsin y p a − y ! d y + Z a sin ϑ y B a arcsin x BC ( y ) p a − y ! d y. (A6)The two integrals in (A6) are S I = a ϑ sin ϑ p ϑ − a arcsin (cid:0) sin ϑ (cid:1) , (A7) S II = − a ϑ sin ϑ p ϑ + a arcsin (cid:0) sin ϑ (cid:1) . (A8) So, the area of the triangle ABC is a arcsin (cid:0) sin ϑ (cid:1) ,and the area of the quadrangle in Fig. 10 is S quad = 4 a arcsin (cid:0) sin ϑ (cid:1) . (A9)(When ϑ = π/
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