Cosmic Lensing -- A new paradigm for Universe expansion interpretation
CCosmic Lensing - A new paradigm for Universe expansioninterpretation
J. De Vicente-Albendea (cid:63)
Centro de Investigaciones Energ´eticas, Medioambientales y Tecnol´ogicas (CIEMAT),Avda. Complutense 40, E-28040, Madrid, Spain
ABSTRACT
The Einstein’s
General Relativity theory and the
Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) metrics define the main equations that rule the history and future ofthe Universe. The
Standard Model for cosmology collects this knowledge along withsome other elements as cosmological distances. Thus, the link between the theory andobservations is performed through some basic cosmological distances as luminositydistance D L and angular diameter distance D A . In this paper, Cosmic Lensing ( CL ) ispresented. CL is a novel paradigm that demonstrates the inverse square law invariancewithin the FLRW geometry. In this way, CL reveals a new relationship between D L and D A in an expanding universe given by D L = D A (1 + z ), opposed to the relation D L = D A (1 + z ) established about a century ago and assumed by the Standard Model . Inthe same sense, a compatible surface brightness vs luminosity relation —different fromthe one proposed by Tolman— is derived from
Cosmic Microwave Background (CMB)equations. As a consequence, previous cosmology methods and results entrusting onluminosity as observational data must be reviewed. The expansion rate and the relativedensities of the dark components of the Universe, as dark matter and dark energy ,should be revised within the new paradigm.
Key words: theory, distance scale, cosmological parameters, dark energy, dark mat-ter, cosmic background radiation
During the 20th century were established the foundations ofmodern cosmology. The field equations of General Relativ-ity were formulated by Einstein (1915), and its applicationto the Universe stated the basis for further steps. Fried-mann (1922), Lemaˆıtre (1927, 1931), Robertson (1933) andWalker (1937) (FLRW) contributed to FLRW model, whichdescribes the solutions to Einstein’s field equations for anexpanding homogeneous and isotropic universe whose scalefactor varies with time. According to GR and FLRW equa-tions, the evolution and fate of the Universe depends onthe nature of different density components, i.e. radiation,matter, curvature and dark energy. Contemporaneously tothese achievements, observational evidences of the Universeexpansion were found by Hubble (1929); the correlation be-tween redshifts and distances for extragalactic sources wasconsidered the major evidence of the Universe expansion.Different cosmological tests were proposed to probe (cid:63)
E-mail:[email protected] whether the Universe is expanding or remains static. A con-clusive test for Universe expansion is the time dilation ofType Ia supernovae light curves that was suggested by Wil-son (1939). The results obtained by Leibundgut et al. (1996)and Goldhaber et al. (2001) on this test, strongly supportsthe cosmological expansion and argues against alternativeexplanations. In another test, Tolman (1930, 1934) predictedthat in an expanding universe, the surface brightness of a re-ceding source with redshift z will be dimmed by ∼ (1+ z ) − .Consequently to Tolman surface brightness prediction, therelation D L = (1 + z ) D A was established between lumi-nosity distance D L and angular diameter distance D A , rela-tion commonly known as Etherington distance-duality (El-lis (2007)). On the other hand, the discovery of the Cos-mic Microwave Background (CMB) in 1964 (Wilson & Pen-zias (1967)) supports strongly the Universe expansion, andtheoretical CMB considerations apparently agrees to aboveEtherington and Tolman relations.In this paper
Cosmic Lensing , a new paradigm for Uni-verse expansion interpretation is presented. Focusing on ex-pansion,
Cosmic Lensing ( CL ) revises the common live ex- a r X i v : . [ phy s i c s . g e n - ph ] J u l J. De Vicente-Albendea
Figure 1.
Standard Model : Relation between angular diameterdistance ( D A ), comoving distance ( D C ) and luminosity distance ( D L ) for a flat universe. D A is the distance at emission, D C isthe distance at reception and D L account for the distance elon-gation due to Universe Expansion ( ∼ (1 + z )), time dilation andwavelength redshifting ( ∼ (1+ z )). The relation D L = D A (1+ z ) can be deduced from the figure. perience of receding light sources where the flux is dilutedwith distance. Contrary to what is commonly expected, theflux from a receding image in an expanding universe is notdimmed by distance elongation, since the image of the sourceremains focused to the observer from emission to reception.As a consequence, a new luminosity-angular distances rela-tion is established as D L = (1 + z ) D A . In addition, a com-patible surface-brightness relation —different from the onepredicted by Tolman— is obtained from CMB equations,providing and independient derivation of the new paradigm. Cosmic Lensing affects deeply to cosmology and the conclu-sions of many previous studies should be revised.The rest of the paper is organized as follows: in Section2 some basic definitions on the
Standard Model are reviewed.The
Cosmic Lensing paradigm is unveiled in Section 3, pro-viding new relevant cosmological distance equations. Finally,the conclusions are presented in Section 4.
The
Friedmann-Lemaitre-Robertson-Walker (FLRW) met-ric is a solution of Einstein’s field equation of
General Rel-ativity describing a homogeneous and isotropic expandinguniverse given by − c dτ = − c dt + a ( t ) [ dr − kr + r d Ω ] (1)being d Ω = dθ + sin θdφ (2)and where k describes the curvature and a(t) the scale fac-tor responsible of the universe expansion. There is different distance ladders defined on FLRW metric and related to ob-servable measurements. Let one to provide a brief summaryof some distances and its relation to cosmological modelsdescribed by the normalized densities Ω M , Ω r , Ω Λ , Ω k formatter, radiation, cosmological constant and curvature re-spectively (Hogg (1999)). The first Friedmann equation canbe expressed from the Hubble parameter H at any time, andthe Hubble constant H today as˙ a ( t ) a ( t ) = H = H E ( z ) (3)where E ( z ) = (cid:112) Ω K (1 + z ) + Ω Λ + Ω M (1 + z ) + Ω r (1 + z ) (4)By integrating Eq. 1 along with Eq. 3 one can obtain theline of sight comoving distance D C as D C = D H (cid:90) z dz (cid:48) E ( z (cid:48) ) (5)where D H = c/H = 3000 h − Mpc is the
Hubble distance .From the same equations one can get the transverse comov-ing distance D M as D M = D H √ Ω k sinh [ √ Ω k D C /D H ] for Ω k > D c for Ω k = 0 D H √ | Ω k (cid:107) sin [ (cid:112) | Ω k (cid:107) D C /D H ] for Ω k < (6)With respect to observable quantities, the angular diame-ter distance D A is defined as the ratio between the objectphysical size S and its angular size θD A = Sθ (7)The angular diameter distance is related to the transversecomoving distance by D M = D A (1 + z ) (8)where z is the redshift. On the other hand, the luminositydistance defines the relation between the bolometric fluxenergy f received at earth from an object, to its bolometricluminosity L by means of f = L πD L (9)or finding D L D L = (cid:114) L πf (10)The relation between the D L and D M within the StandardModel is given by D L = D M (1 + z ) (11)and taking into account Eq. 8 osmic Lensing - A new paradigm for Universe expansion interpretation D L = D A (1 + z ) (12)where D A is also the distance to the object at time of emis-sion. There are four (1+z) factors affecting to flux energydiminution (Fig. 1). Two come from the elongation of thedistance D A by a factor of (1 + z ) due to universe expan-sion. It is assumed that such elongation dilutes the lumi-nosity by D A (1 + z ) according to the inverse square law.Another factor comes from the time dilation due to universeexpansion that reduces the photon emission/reception rateby (1 + z ) − . The last factor comes from the cosmologi-cal wavelength redshift that decrease the energy of photonsby (1 + z ) − . Therefore, a relevant relation is establishedbetween the angular diameter distance and the luminositydistance in the Standard Model as D L = D A (1 + z ) (13)Equation commonly known as Etherington distance-dualityrelation. According to the standard cosmology (Peebles (1993), Wein-berg et al. (2008)), about 300.000 years after the Big Bang,the Universe was formed by a soup of protons, electrons andphotons. When the temperature of the Universe fell downto 3000K, electrons were linked to protons to form hydro-gen atoms. At this moment, the decoupling was producedand the Universe became transparent to radiation since thescattering between free electrons and photons dropped dras-tically. At this epoch, radiation and matter were in thermalequilibrium and the released radiation formed a perfect backbody. The Cosmic Microwave Radiation detected nowadayshas the same black body feature, and thus it is assumed tobe the relic of this radiation cooled and redshifted due toexpansion.A key question to support this theory is whether theblack body spectrum shape can be maintained uniquely bythe effect of the expansion or it is required some thermal-ization process. Let
Ndν be the number of photons emittedat decoupling at temperature T with photon frequency be-tween ν and ν + dν given by Ndν = 8 πVc ν e hνkT − dν (14)and let N (cid:48) dν (cid:48) be the number of photons measured todayat temperature T’ with photon frequency between ν (cid:48) and ν (cid:48) + dν (cid:48) given by N (cid:48) dν (cid:48) = 8 πV (cid:48) c ν (cid:48) e hν (cid:48) kT (cid:48) − dν (cid:48) (15)Dividing Eq. 14 by Eq. 15 and substituting the relation-ships V (cid:48) /V = (1 + z ) , ν = (1 + z ) ν (cid:48) , dν = (1 + z ) dν (cid:48) and T = (1 + z ) T (cid:48) , all (1+z) factors are cancelled out obtaining Ndν = N (cid:48) dν (cid:48) (16) Thus, the number of photons emitted at decoupling at tem-perature T with photon frequencies between ν and ν + dν ,corresponds to the number of photons measured today attemperature T’ with photon frequencies between ν (cid:48) and ν (cid:48) + dν (cid:48) . Thus, it is plausible the CMB to be the relic ofthe Universe black body radiation at early states. Standard Model
Tolman surface brightnessderived from CMB
Let l be the CMB bolometric energy emitted per unittime per unit surface ( ∼ π luminosity per unit surfacedue to hemispheric solid angle). According to the Stephan-Boltzmann law one has l = σT em (17)where sigma is the Stephan-Boltzmann constant and T em the blackbody temperature at emission. Let µ be the ob-served CMB bolometric energy per unit surface ( ∼ π sur-face brightness due to hemispheric solid angle). Accordingto the Stephan-Boltzmann law one has µ = σT obs (18)where T obs is the blackbody temperature at reception.Dividing both quantities, µl = T obs T em (19)Since the relation between the temperature at observationand emission in the expanding universe is T em = T obs (1 + z ) (20)Equation 19 becomes µl = (1 + z ) − (21)that apparently agrees to Standard Model
Tolman surfacebrightness-luminosity relation. COSMIC LENSING : A NEWCOSMOLOGICAL PARADIGM
The main support for an expanding universe comes fromthe observed redshift of extragalactic sources. The wave-length of light is stretched out by the universe expansionwhile traveling from galaxies to the earth, displacing thewavelength to the red. While much attention has beenpaid to the effect on the light produced by the expansionin the radial direction –driving to the concept of cosmo-logical redshift– a relevant observational property on thetransversal direction remains unnoticed. Let one to unveilthis feature.
J. De Vicente-Albendea
Figure 2.
Flux focusing (Cosmic Lensing): Simplified 2D view ofthe flux focusing effect on a flat universe. The augmented distancetravelled by light rays due to expansion does not dilute the fluxreceived since the rays are always focused to the observer. Onlythe time dilation and wavelength redshifting decrease the flux.
Cosmic Lensing : FLRW inverse square law
The cosmological principle in an expanding universe requiresthat the effect of the expansion on light rays be homoge-neous and isotropic both on radial and transversal direc-tions. Consequently, the scale factor a(t) of FLRW metric—responsible of the universe expansion— affects to both theradial and transversal directions (Eq. 1). The application ofthe scale factor to the light in the radial axis drives to differ-ent known observational effects: the redshift of photons, thestretching of the radial distance by a factor ∼ (1 + z ) andthe time dilation of events. With respect to the transversaldirection, the angular diameter distance ( D A ) is properlydefined as the ratio between the size of the source and theangle subtended by the source. Note also that, accordingto this definition, the apparent image size (i.e. subtendedangle) remains unaltered to the observer from the emissionto reception in spite of the expansion. This fact has deepconsequences currently unnoticed.According to General Relativity, light rays follow nullgeodesics where dτ = 0. Substituting this value in Eq. 1,light rays follow the equation c dt = a ( t ) [ dr − kr + r d Ω ] (22)Let one set the origin of coordinates at the observer O,and consider a extended cosmological object (galaxy) ini-tially located at a distance D A from O at time of emission t e (Fig. 2). Assume for a moment we live in a static uni-verse (a(t)=1 ∀ t). According to Eq. 22, the galaxy lightrays that will arrive to O in the future are those pointinginitially towards the observer at time of emission t e . Theserays will maintain the same direction up to arriving to theobserver, i.e. θ = cte , dθ = 0, φ = cte , dφ = 0 in Eq. 2, andhence d Ω = 0 (leaving apart non pure cosmological effects as gravitational lensing or astrophysical events). Substitut-ing d Ω = 0 in Eq. 22, the light rays that will arrive to theobserver meet c dt = a ( t ) dr − kr (23)The same reasoning can be applied to an expandinguniverse responding to any scaling function a(t). The galaxylight rays that will arrive in the future to the observer arethose pointing initially to the observer at time of emission.Those rays will maintain the same direction all the timein spite of expansion. The unique effect of expansion is thescaling (i.e. preserving the angles) of the light rays trajec-tories with respect to the static universe. The same numberof photons arrive to the observer in static as in expandinguniverses with respect to the euclidean inverse square law.The difference in the flux energy received by the observerbetween static and expanding universes comes from Eq. 23,i.e. from FLRW geometry. Integrating this equation from t e = 0 we have (cid:90) t cdta ( t ) = (cid:90) D M dr √ − kr (24)From this equation it is clear the dependence of D M on thetime dilation by the factor δt (cid:48) = (cid:82) t dt/a ( t ). Therefore, con-trary to what is assumed in Eq. 11 within the StandardModel , the luminosity distance D L is well represented by D M since it already accounts in its metric for the luminos-ity dilution due to the effects of time dilation, and hencethose coming from wavelength redshift (no additional (1+z)factors are required). Therefore, we can express the CosmicLensing equations as D L = D M (25)and D L = D A (1 + z ) (26)Thus, once the light rays are emitted, any dilution inthe flux energy received by the observer with respect to astatic universe, comes uniquely from the wavelength redshiftdue to the increment on the path length (Eq. 27), which isalso responsible of time dilation. Both effects are integratedin D M value through FLRW metric. D M = λ e λ o D A (27)The phenomenon can also be explained from the light-cone produced by the bundle of rays from the galaxy tothe observer (Fig. 2). The scaling of the light-cone pro-duced by the universe expansion, preserves the light-coneshape and hence the angle θ subtended by the imaging ofthe galaxy. The angular size conservation of the image whileuniverse expands, maintains the image focused on the ob-server preventing from flux dilution. The focusing of theimages towards the observer, produces a high increment onthe flux received from the galaxy image with respect to theexpected by the Standard Model . Thus, the received flux by osmic Lensing - A new paradigm for Universe expansion interpretation Figure 3.
Cosmic lensing: Effect of the expanding universe onthe images of an extended source (flat universe). The image of thesource remains focused towards the observer all along the time itis travelling to the earth. the observer (Eq. 28) resemble the one received in a non-expanding universe, but dimmed only by the time dilationand wavelength redshifting due to the path elongation. f = L πD L = L πD M = L πD A (1 + z ) (28)We can see that the inverse square law is now preservedin the FLRW geometry. Note that the results presented arederived for any scale factor function a(t). Therefore, takinginto account Eq. 3 and Eq. 4, the results are valid for anyfeasible combination of Ω M , Ω r , Ω Λ , Ω k . Standard Model) to the observer (CosmicLensing ) Let one first to analyse the inverse-square law representedby Fig. 1 for the universe within the Standard Model. Asource stated at the origin emits a bundle of rays towardsthe detector located at a distance D A . According the dis-tance between the source and the detector increses due toexpansion, the flux is diluted according to the inverse squarelaw (in addition to time dilation and wavelength redshift).The key point of the fault of this model comes from consid-ering a bundle of rays departing from a point source ratherthan from an extended source. In this situation, the sourcetransversal image information is missing in spite it is alsoaffected by the expansion. While it has not impact in staticenvironments as shows the Gauss’s theorem, the situationchanges in an expanding universe. The effect of workingwith extended sources in an expanding universe can be clar-ified by changing the point of view from the source to theobserver. Fig. 3 shows how the apparent image becomesfocused to the observer regardless the redshift and henceindependently of the expansion rate. The focusing of theimage flux towards the observer while travelling to earth,deactivate any luminosity dilution in flux beyond the dropdue to the distance between the source and target at time of emission ( ∼ /D A ). Therefore, contrary to what is expectedby the Standard Model, the expansion of the universe doesnot dilute the flux due to distance elongation. Nevertheless,the universe expansion does drop the energy of photons, andhence the flux decreases as ∼ (1 + z ) − due to time dilationand wavelength redshift. Cosmic Lensing surface brightness-luminosityrelation derived from CMB
In section 2.3, the Tolman surface brightness-luminosityrelation was derived from CMB considerations. Let one torevise this derivation taking into account the differential sur-face elements at each epoch, i.e. dS (at emission) and dS (cid:48) (at reception).Let dl/dS be the CMB bolometric energy emitted perunit time per unit surface ( ∼ π luminosity per unit surfacedue to hemispheric solid angle). According to the Stephan-Boltzmann law one has dldS = σT em (29)where σ is the Stephan-Boltzmann constant and T em theblackbody temperature at emission. Let dµ/dS (cid:48) be the ob-served CMB bolometric energy per unit surface ( ∼ π sur-face brightness due to hemispheric solid angle). Accordingto the Stephan-Boltzmann law one has dµdS (cid:48) = σT obs (30)where T obs is the blackbody temperature at reception.Dividing both quantities, dµdl = T obs T em dS (cid:48) dS (31)The relation between the differential surface elements at ob-servation and emission is given by dS (cid:48) = dS (1 + z ) (32)Taking into account the relation between temperatures (Eq.20) and differential surface elements (Eq. 32), Equation 31becomes dµdl = (1 + z ) − (33)that provides the Cosmic Lensing surface brightness-luminosity relation.
Cosmic Lensing - Etherington equation agreement
The goal of Etherington (1933) paper was to relate distancescomputed from apparent size ∆ (cid:48) (i.e. D A ) to distances com-puted from apparent brightness ∆ (i.e. D L ), assuming theexistence of a redshift from the source to the observer. Therelation obtained by Etherington (1933)-Eq. 23, is repro-duced here J. De Vicente-Albendea ∆ = (1 + δλλ )∆ (cid:48) = (1 + z )∆ (cid:48) (34)That is, the original Etherington equation agrees perfectlywith the Cosmic Lensing luminosity-angular distancerelation (Eq. 26). Even more, among the conclusions ofthis equation, Etherington wrote:“(ii) For spherical objects of the same absolute size andbrightness, moving variously at varying distances from theobserver, the ratio of the apparent diameter to the squareroot of the apparent brightness is proportional to δλ/λ .”That is, for objects of the same absolute size S and bright-ness L, the ratio of the apparent diameter θ to the squareroot of apparent brightness √ f is proportional to (1+z).Thus, from Eq. 7 and Eq. 10 one has D L D A = √ LS √ π θ √ f = (1 + z ) (35)that again agree with Cosmic Lensing.Unfortunalety, Eherington equation has long been mis-interpreted, an unnecessary (1+z) factor has been added tohis equation (probably by confusion with the mainsteam), ina formula commonly known as Etherington distance-dualityrelation (Eq. 13). The
Standard Model compile the current knowledge of theUniverse based on Einstein equations and FLRW metric,along with the definition of basic cosmological distances re-lated by fundamental laws as the distance-duality and the
Tolman surface brightness relations.In this paper, the
Cosmic Lensing paradigm is pre-sented.
Cosmic Lensing unveil an unready effect of extra-galactic images in an expanding universe: the flux focus-ing . The flux focusing produces an amplification on the re-ceived flux from extragalactic sources by a factor (1 + z ) with respect to what is expected by the Standard Model .The phenomenon can be explained from the inverse squarelaw preservation within the FLRW geometry. As a conse-quence, new equations –different from distance duality and
Tolman surface brigthnes relations – are established to reflectthe luminosity-angular cosmological distances relationship.
Cosmic Lensing affects deeply to the equations used todetermine the cosmological parameters (Hubble constant,dark matter, dark energy, etc.) and hence to the Uni-verse evolution. A revision of cosmological methods and re-evaluation of data within the new paradigm are required.
ACKNOWLEDGEMENTS
Funding support for this work was provided by the Au-tonomous Community of Madrid through the projectTEC2SPACE-CM (S2018/NMT-4291).