The New Wedge-Shaped Hubble Diagram of 398 SCP Supernovae According to the Expansion Center Model
** * *THE NEW WEDGE-SHAPED HUBBLE DIAGRAM OF 398 SCP SUPERNOVAEACCORDING TO THE EXPANSION CENTER MODELECM paper IX by Luciano Lorenzi54th Annual Meeting of the Italian Astronomical Society - Naples 2010”
L’Astronomia italiana: prospettive per la prossima decade”
ABSTRACT
Following the successful dipole test on 53 SCP SNe Ia presented atSAIt2004 in Milan, this 9th contribution to the ECM series beginning in1999 in Naples (43th SAIt meeting:”Revolutions in Astronomy”) deals withthe construction of the new wedge-shaped Hubble diagram obtained with398 supernovae of the SCP Union Compilation (Kowalski et al. 2008) byapplying a calculated correlation between SNe Ia absolute blue magnitude M B and central redshift z , according to the expansion center model. TheECM distance D of the Hubble diagram ( cz versus D ) is computed as theratio between the luminosity distance D L and z . Mathematically D results to be a power series of the light-space r run inside the expandingcosmic medium or Hubble flow; thus its expression is independent of thecorresponding z . In addition one can have D = D ( z, h ) from the ECMHubble law by using the h convention with an anisotropic H X .It is proposed to the meeting that the wedge-shape of this new Hubblediagram be confirmed independently as mainly due to the ECM dipoleanisotropy of the Hubble ratio cz/D . . Introduction After the successful test of the expansion center model (Lorenzi 2004) carried out on 53 high-redshift Type Ia supernovae from the Supernova Cosmology Project (SCP: Perlmutter et al. 1999or P99; Knop et al. 2003 or K03), here is presented the ECM construction of the new wedge-shapedHubble diagram obtained by data from the SCP Union Compilation (Kowalski et al. 2008). Inparticular this large ”Union” sample reports redshifts and blue magnitudes of 398 SNe Ia, or of307 SNe Ia after selection cuts, including the distant supernovae recently observed with HST.Let us remark that the cited papers I-II-III-IV-V-VI-VII-VIII are those of the author’s refer-ences: Lorenzi 1999 →
2. Distances from the ECM equation
The new Hubble law (59) of paper I˙ r = r · ( H + ∆ H ) − R ∆ H cos γ (1), after substituting H = H + ∆ H, ∆ H, R with the formulas (37)(39) from paper I, becomes the
ECM ˙ r equation (from eq. (3) in paper II) of the nearby Universe, expressed in Hubble units(H.u.) as follows ˙ r = H · r (cid:1) x − x (cid:2) (cid:3) q (1 − x ) x cos γ (cid:4) (2)with x = 3 H rc < q = − H R c cos γ = sin δ V C sin δ + cos δ V C cos δ cos( α − α V C )being K R = a = − H q Specifically γ is the angle between the direction ( α V C ≈ h , δ V C ≈ +30 ) of the huge voidcenter (Bahcall & Soneira 1982), also called the expansion center or Big Bang central point (Lorenzi1989-91-93), distant R from the Local Group (LG) at our epoch and that ( α, δ ) of the observedouter galaxy/group/cluster/supernova at a distance r from LG, with the nearby Universe radialvelocity ˙ r corrected only by the standard vector (Sandage & Tammann 1975a)(Lorenzi: paper I).Of course ∆ H = 0 in eq. (1) should give the original Hubble law: ˙ r = H r .2ere it should be noted that the above equation (2) allows us to define at least three differentcosmic distances, the following r , D and D L , which in practice have approximately the same valueonly for the very nearby Universe. r : Distance as light-space
First of all the distance r in eq. (1) and (2) represents the light-space run with constant speed c inside the expanding ”cosmic medium” (CM hereafter) or Hubble flow. In particular such a CMflow refers to the motion of galaxies running away from the Big Bang central point, with radialvelocity ˙ R = HR (cf. papers I, V and VIII). Let us rewrite the light-space r formulation. r = − c ( t − t ) with − c = δrδt (3)In eq. (3) t is a constant representing our epoch, which is also represented by r = 0; at t the light emitted at an epoch t reaches the observer at rest in the local Hubble flow, whichnow is more rarefied like the CM; δr is the infinitesimal CM space covered by the light duringan infinitesimal δt of the light travel-time from the past. To all intents and purposes the sourcedistance r of eq. (3) may be considered to be equal to that of the source at the emission epoch t . However the cosmic medium is expanding, while light speed c remains constant with respect tothe local cosmic medium, as follows: λ = cT ⇒ dλ = cdT ⇒ λ + ∆ λ = c ( T + ∆ T ) ⇒ λ = cT .In other terms the travelling light has two speeds, the former being c inside CM, the latter thatof the supporting expanding CM or Hubble flow. The observed velocity of this expanding CM isthe derivative of r to t , with dt /dt ≡ λ /λ assumed, as shown in papers V and VIII, sections 4.7and 2.1 respectively. That dr/dt results to be c ∆ λ/λ , that is ˙ r = cz . By introducing z in eq. (2)we obtain the dimensionless ECM z equation (eq. (22) of paper V or eq. (13) of paper VI oreq. (6) of paper VII) z = x (cid:1) x − x (cid:2) (cid:3) q (1 − x ) x cos γ (cid:4) (4)where we must specify r → ⇒ x → ⇒ z → ⇒ t = t ; H = H ; R = R ; K = K ; a = a ; q = q ; r = D = D L cos γ = 0 ⇒ z = z ( x ) ≡ z ⇒ x = x ( z ) ⇒ r = r ( z )In fact the value cos γ = 0 in eq. (4) leads naturally to another important convention, thatis the introduction of z to represent the central redshift , which must not be confused with z ( t ) = 0. 3oreover the previous eq. (4), with the H and q values obtained within the nearby Universe(paper II: H = 69 . ± . km s − Mpc − ; q ∼ = − . numerical calculus of x , that is of the light-space r as a function of theobserved z and γ , as follows x = x ( z, cos γ ) = 3 H r/c ⇒ r = c · x ( z, cos γ )3 H = r ( z, cos γ ) (5)Note that cos γ = 0 gives to z = 0 . x = 0 .
5, that is r ∼ = 716 M pc . D : Distance in the Hubble diagram and the h convention In 1975 Sandage & Tammann published a paper (S&T: Paper V) in which an accurate datalisting of nearby galaxies (mean depth of ∼ M pc ) was tabled and reported in a famous wedge-shaped Hubble diagram, where the Hubble ratios appeared scattered between ∼
30 and ∼ kms − M pc − . Another wedge-shaped velocity-distance diagram, with different symbols for differentmethods and a covered distance depth of about 200 Mpc , is that of Rowan-Robinson (1988); herethe Hubble constant appears to lie in the range 50 − km s − M pc − , with a current best valuein the middle of this range.Such a wedge feature of the original Hubble diagram is well represented by eq. (2) and (4). Infact, after putting D = r · (cid:1) x − x (cid:2) (6)we can transform eq. (2) into the ECM Hubble law cz = [ H − a ∗ ( x ) cos γ ] · D = [ H − a X ( x, cos γ )] · D = H X · D (7)being a ∗ ( x ) = a · (1 − x ) / (1 + x ) X ( x, cos γ ) = cos γ · (1 − x ) / (1 + x )where both x and X are dimensionless variables (cf. paper V and VI); hence the above eq. (7)contains an anisotropic angular coefficient , that is H X = H (1 − a H X ) (8)As in the very nearby Universe in practice x →
0, here eq. (7) gives a ∗ (cid:9) a , that is H X ( γ = 0 ) ∼ = 57 km s − M pc − and H X ( γ = 180 ) ∼ = 83 km s − Mpc − with a ∼ = 12 . km s − Mpc − (paper V: section 4.6). 4he MacLaurin Series applied to (6) and the ECM Hubble law (7) give D both in terms of apower series of the light-space r , D = r + 2 3 H c r + 2 9 H c r + ... (9)and as a function of z , that is the ratio between the central velocity cz and the constant H : D = cz H = D ( z ) (10)The eqs. (6)(9)(10) represent the distance D of the wedge-shaped Hubble diagram of eq. (7).At the same time the ECM Hubble law (7) is able to substantiate the powerful h convention (Zeilik & Smith 1988) for large-scale surveys of radial distance D in H.u., by using avariable h = h ( X ) tied to the ECM apparent anisotropy (cf. paper II, secion 1.2) and the correct z , obtained after subtracting from the observed heliocentric redshift the kinematic component dueto the entire motion of the Sun with respect to the Hubble flow traced by the CMB (Lorenzi 1993,1999a, 2008, 2009). So we confirm the following useful formula: D = cz km s − M pc − h − with h = H X km s − Mpc − (11) D L : Luminosity distance and correlated absolute magnitude M Papers V and VI have empirically confirmed the ECM even for Abell clusters of Richness 3 andType Ia supernovae from SCP, up to a light-space distance r of ∼ M pc . Here the luminositydistance D L has been successfully represented by the following ECM D C multiple formula D C = D (1 + z ) = xc H (cid:1) x − x (cid:2) (1 + z ) = cz (1 + z ) H (1 − a H X ) − (12)Consequently, with D C ≡ D L assumed, the distance D of the Hubble diagram can be simplyinferred from the position D = D L z (13)when one knows the absolute magnitude M , that is M = m − D L −
25 (14)By combining the canonic eq. (14) in H.u. with (12) and (13), we can obtain the
ECM M equation, written in a double form: M = m − (cid:5) xc H (cid:1) x − x (cid:2) (1 + z ) (cid:6) −
25 (15)5 = m − cz (1 + z )] + 5 log H + ∆ −
25 (16)In (14)(15)(16) m and z are the observed magnitude and redshift within the Hubble flow; in(15) x = x ( z, cos γ ) from eq. (4); ∆ of eq. (16) results to be a power series of X ( x, cos γ ), asfollows ∆ = 5 log(1 − a H X ) = − a H log e · ( X + a H X a H X ..... ) (17)Eq. (16) can be simplified by introducing the central redshift z corresponding to z of eq. (4)with cos γ = 0, that is H X = H . In this case, being X ≡ ⇒ ∆ = 0 ⇒ z = z ⇒ m = m ( z ) ⇒ D L = D L ( m ) (18), we also obtain the ECM M ( z ) equation , in the form M = m − cz (1 + z )] + 5 log H −
25 (19)
3. Construction of the ECM Hubble diagram of 398 SCP supernovae
The main aim of the present work is the application of the above formulae to the largest availablesample of homogeneous datasets. The SCP ”Union” SNe Ia compilation holds such a sample,bringing together data from 414 SNe (Kowalski et al 2008: Table 11) drawn from 13 independentdatasets, of which 398 SNe have both the required redshifts z and blue magnitudes m max B listed,while a wide subsample of 307 SNe Ia pass usability cuts. Note that here the redshifts z arereferred to the CMB; hence they include the correction due to the standard motion of the LocalGroup, without taking into account the ECM 3K dipole able to generate a fictitious vector v f (Lorenzi 1993, 1999a, 2008). As the involved correction to z is about 0 .
001 on average, the z ofthe distant supernovae in effect do not suffer an imprecise correction; it is different for the verynearby SNe, whose redshifts in the Hubble diagram should be corrected only for the Sun’s velocityinside the Local Group (by the standard vector of S&T (1975)), because our LG belongs to a largelocal cosmic flow also running almost in the same direction (cf. p. 19 of paper I).On the whole the present analysis aims directly to construct the ECM Hubble diagram, ofcourse without using cos γ , but showing in any case that the diagram’s wedge-shape is due to theECM dipole anisotropy. A further and crucial confirmation of the model is expected by introducingthe supernova astronomical coordinates, that is to say cos γ , into the ECM analysis both for SCPUnion (Kowalski et al. 2008) and the SCP ”Union2” shown at 2010 AAS (Rubin et al. 2010).6 .1 Search for a correlation between M B and central redshift z Initially the ECM Hubble law (7) was tested over the 398 SCP supernovae (Kowalski et al. 2008:Table 11), by assuming H as the average Hubble ratio, that is H = (cid:10) H X (cid:11) = (cid:10) czD (cid:11) (20)The procedure, based on the mean eq. (20) in H.u. with D derived from (13) and D L =10 . m max B − M B ) − , was applied to five large z bins of the Hubble flow, precautionally excludingthe nearby SNe with z < .
05; hence, once M B or a resulting H ∼ = 70 H.u. are fixed, the valueof H or M B follow. Conditions and results of that first check are listed below, in Table 1. Table 1 z bins N (cid:10) z (cid:11) H = H ( M B = − . M B = M B ( H ∼ = 70)0 . ≤ z ≤ . . − . . ≤ z ≤ .
75 197 0 . − . . ≤ z ≤ .
552 308 0 . − . . ≤ z ≤ . . − . . ≤ z ≤ .
25 67 0 . − . The strong variation of the H value in the 4 th column, corresponding to the assumed M B = − . M B . On the other hand the constant value of H gives to SNe Ia a variableintrinsic luminosity, which clearly increases with depth or central redshift, according to the ECM.Owing to the clear result in Table 1 and in order to construct a correlation between M B and thecentral redshift z according to (19), the same ” z bins” procedure has been applied to a normalECM M equation , that is eq. (16) with H = 70 H.u. and (cid:10) ∆ (cid:11) = 0 assumed, as follows (cid:10) M B (cid:11) = (cid:10) m max B (cid:11) − (cid:10) log [ cz (1 + z )] (cid:11) + 5 log H −
25 (21)In this case the check is more useful than the previous one, first of all because eq. (21) givesdirectly (cid:10) M B (cid:11) with its standard deviation; furthermore the ECM eq. (21) seems to be statisticallypowerful, if the z scattering due to unsuitable corrections of Sun kinematics in the CMB is assumedto be neutralized like ∆ by the normal point, apart from any anisotropies of SNe Ia distributionin the sky plus a H imprecision of about ± . M B values correlated to the central redshift z of eq. (19). Thus all the available SCP7Ne Ia of the Union 2008, 91 nearby SNe with z ≤ .
05 included, have been taken into account.Table 2 in the Appendix lists the results of the mean; it reports 30 normal points, including allthe 398 SNe listed in Table 11 of the 2008 SCP paper (Kowalski et al. 2008) and correspondingto 30 ” z bins”. In particular the first 5 columns of Table 2 hold numerical values derived fromthe observed z and m max B listed within the above SCP Union 2008; the values referring to each z bin are in the order: z range, number N of the SNe included in the normal point; unweighedmathematical mean (cid:10) m max B (cid:11) of the observed SN blue magnitudes m max B ; absolute magnitude (cid:10) M B (cid:11) resulting from the normal ECM M equation (21) applied to the bin, with H = 70 H.u. assumed;standard deviation s of the least square fitting carried out on the bin. The 6 th column of Table 2reports the mathematical mean (cid:10) z (cid:11) of the observed redshifts of the z bin, while the last columns,7 th , 8 th , 9 th , include three different distance values, corresponding to an assumed central redshift z ≡ (cid:10) z (cid:11) with H = 70 H.u.. These are in the order: value of the dimensionless variable x = x ( z ),inferred as in (5) from the ECM z equation (4); value in M pc of the light-space distance r = r ( z )connected to the x value by x = 3 H r/c according to procedure (5) applied to eq. (2) or (4);value in M pc of the distance D of the wedge-shaped Hubble diagram, obtained with eq. (6) or(10), that is as the function D = D ( z ) of the central redshift through x ( z ).The resulting normal points, plotted in Figure 1 as (cid:10) M B (cid:11) versus (cid:10) z (cid:11) , clearly point to a fittedtrend line, whose equation formally should give for any z its M B as a function of the centralredshift z , or the corresponding distance D = cz /H as in Figure 2, if z is assumed to refer tothe line fitting the (cid:10) z (cid:11) points. The line equation below, M B ( z ) = A + A z + A z = d + d D + d D = M B ( D ) (22), with A ∼ = − . A ∼ = − . A ∼ = +0 . d = A ; d = A H /c ; d = A H /c ,follows from the automatic fitting.In the same way, according to procedure (5) applied to eq. (2) or (4) with cos γ = 0, thatis with z = z , an alternative plot of normal points (cid:10) M B (cid:11) versus x = x ( z ) or r = r ( z ) canbe constructed; it appears in Figure 3 and Figure 4, where the fitted trend line appears betterrepresented by a third degree equation, that is M B ( x ) = B + B x + B x + B x = C + C r + C r + C r = M B ( r ) (23), with B ∼ = − . B ∼ = − . B ∼ = − . B ∼ = − .
006 and C = B ; C = 3 B H /c ; C = 9 B H /c ; C = 27 B H /c , again obtained from the automatic fitting. Here the curve8grees with that found in paper VI for 33 SNe Ia of K03 (cf. paper VI-integral version: Fig. 5),however with a systematic shift of about 0 . The above equation (22), that expresses M B ( D ), has a crucial role in the construction of the SNeIa Hubble diagram, which requires the distance D to combine with the observed redshift as cz . Infact it is now possible to extract numerically just the distance D from the canonic eq. (14), thatbecomes the following: d D + d D + d = m max B − D (1 + z )] −
25 (24)The numerical solution point by point of eq. (24), here applied to 398 SNe Ia with z and m max B listed in Table 11 of the SCP Union Compilation (Kowalski et al. 2008), gives the value of D .Once found, one can infer numerically also the corresponding values of the distance indicator x and the light-space distance r .Finally, two resulting wedge-shaped Hubble diagrams in H.u. are obtained by plotting cz versus D for the 398 SNe, in Figure 5, and the 307 SNe passing usability cuts, in Figure 6. Herewe look at the Hubble diagram of the Deep Universe. Table 3abcdefghi in the Appendix lists thevalues in H.u. of D (3 rd column) and cz (2 nd column) of 249 SNe Ia (1 st column: Name), lying inthe distance range 800 M pc < D <
M pc , from the 307 SNe selected by the SCP Union.
4. ECM analysis of the wedge-shaped Hubble diagram
The diagrams in Fig. 5 and Fig. 6 have a wedge shape, whose amplitude with increasing depth isvery large, indeed. In order to verify the accordance with the model, it is necessary to compare theobserved wedge shape to the calculated one. In practice we should carry out an (O-C) procedure.That has been done by calculating the wedge amplitude foreseen by the ECM Hubble law (7)through a numerical simulation of the maximum scattering of cz around the central value cz .To this end, let us analyse the observed Hubble flow or CM as follows: z = λ − λ e λ e = T e T − cz = cz + c ∆ z (26)∆ z = − T e T (cid:1) ∆ T T + ∆ T (cid:2) (27)9s the ECM gives cz = H D c ∆ z = − a DX (28), after fixing x , then r and D , solely the dimensionless X varies owing to the variation of cos γ between 1 and −
1. In this case: − (1 − x ) / (1 + x ) ≤ X ≤ (1 − x ) / (1 + x ) (29) c ∆ z = c ∆(∆ z ) = − a D ∆ X (30) r = cx H cz = cx (cid:1) x − x (cid:2) D = cx H (cid:1) x − x (cid:2) (31)(∆ X ) max = 2(1 − x ) / (1 + x ) (32) c | ∆ z | max = a D (∆ X ) max (33)Now we add to the above formulae the following ones, which practically, being based solelyon the eqs. (25)(26)(27), are unaffected by the ECM (cf. section 6 of paper VII).∆ T = ∆ T D cos γ (34)∆ z z = − ∆ T T + ∆ T (35) γ = 0 ⇒ ∆ z z = − ∆ T T ≡ v f c ⇒ v f = c ∆ z z (36)The last equation of (36) gives the value of the fictitious velocity v f observed towards the ex-pansion center. Its value in H.u., obtained within the ECM , is listed in the 7 th column of Table 4,while the previous 5 columns present the other simulated values in H.u. of r, cz , D, (∆ X ) max , c | ∆ z | max from the above eqs. (31)(32)(33) corresponding to the x value of the first column. The last row ofTable 4, where x = 0 . ECM values extrapolated to the CMB , whosefictitious velocity results to be only of the order − km s − . Curiously this value, that is inaccordance with the observed 3K anisotropy and a local cosmic flow of about 530 km s − (cf.Lorenzi 1993, 1999a, 2008), is the same of the nearby Universe at D ∼ = 20 M pc , while at D ≈ M pc the corresponding value of v f reaches a maximum of about − km s − .On the whole, the ECM simulation is able to reproduce the variable wedge-shape of the Hubblediagram at different depths, as summarized in the table below.10 able 4 x r cz D (∆ X ) max c | ∆ z | max v f . . .
998 25 − . .
865 700 10 1 .
982 252 − . .
46 1400 20 1 .
964 499 − . .
93 2800 40 1 .
931 981 − . .
51 4200 60 1 .
900 1448 − . .
29 5600 80 1 .
871 1901 − . .
34 7000 100 1 .
844 2342 − . . .
295 16445 − . . .
079 27407 − . . . − . . − . . − . . − . . − .
99 1413 19687373 281248 0 . − . . (cid:9) . × (cid:9) . × . (cid:9) . × − D ≤ M pc and the latter to the Deep Universe with D ≤ M pc .We obtain two simulated ECM Hubble diagrams in H.u., where the plotted points correspondingto each tabled D are the central cz , the upper cz + c | ∆ z | max and the lower cz − c | ∆ z | max . Fromtheir comparison, it results clearly that the wedge amplitude has to decrease with depth, until H X → H for x →
5. Conclusions
This paper validates the wedge-shaped Hubble diagram predicted by the espansion center model.In fact the diagrams of the Deep Universe in Fig. 5 and Fig. 6 are in good accordance withthat simulated in Fig. 8, as the observed amplification is certainly due to various sources ofbackground noise. Therefore this 9 th ECM contribution, based above all on a large sample of SCP11ata obtained in space with HST, is further confirmation of the cosmic expansion center, followingthe ground-based astronomical proof collected in about half a century.At this point one cannot desist from pressing the international astronomical community topronounce itself once and for all on the subject. In conclusion the author extends an invitationto all astronomers to analyse independently the presented new wedge-shaped Hubble diagram, inorder to confirm, or confute, the dipole anisotropy of the Hubble ratio cz/D at any cosmic depth.12
Fig. 1: - SNe Ia absolute B magnitudes versus central redshift based on the ECM from SCP Union data -20-19.8-19.6-19.4-19.2-19-18.8-18.6 0 0.2 0.4 0.6 0.8 1 1.2
EFERENCESBahcall, N.A. and Soneira, R.M. 1982, ApJ 262, 419Knop, R.A. et al. 2003, ApJ 598, 102 (K03)Kowalski, M. et al. 2008, arXiv:0804.4142v1 [astro-ph] 25 Apr 2008 → range N (cid:10) m max B (cid:11) (cid:10) M B (cid:11) s (cid:10) z (cid:11) x r Dz ≤ .
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848 0 . . . . z ≤ .
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741 0 . . . . z ≤ .
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352 0 . . . . ≤ z ≤ . − . ± .
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330 0 . . . . ≤ z ≤ .
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316 0 . . . . ≤ z ≤ .
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379 1 . . . z ≥ .
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378 1 . . . z ≥ . . − . ± .
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407 1 . . . cz D cz D cz D cz D cz D cz D cz D d058 174779 2630d084 155592 2877d085 120217 1580d087 101930 1212d089 130710 1550d093 108825 1175d097 130710 1550d117 92636 1605d149 102529 1413e029 99531 1722e108 140603 1550e132 71650 1186e136 105527 1994e138 183473 3463e140 189169 2228e147 193366 2189e148 128611 1700e149 148997 1855f011 161588 2264f041 168184 1968f076 122915 1472f096 123515 2205f216 179576 2874 Name cz D f231 185572 2335f235 126512 1525f244 161888 2276f308 120217 2242g005 65355 986g050 189769 1959g052 114821 1474g055 90537 2830g097 101930 1481g120 152894 1720g133 126213 2340g142 119617 2874g160 147798 1884g240 205957 2143h283 150496 2587h300 205957 2307h319 148397 1858h323 180775 2411h342 126213 1517h359 104328 1831h363 63856 1445h364 103129 1084k396 81244 1242 Name cz Dcz D