A derivation of the Etherington s distance-duality equation

Transcription

1 A derivation of the Etherington s distane-duality equation Yuri Heymann 1 Abstrat The Etherington s distane-duality equation is the relationship between the luminosity distane of standard andles and the angular-diameter distane. This relationship has been validated from astronomial observations based on the X-ray surfae brightness and the Sunyaev-Zel dovih effet of galaxy lusters. In the present study, we propose a derivation of the Etherington s reiproity relation in the dihotomous osmology. Keywords Etherington; distane duality; dihotomous osmology 1 Introdution The Etherington s distane-duality equation was introdued in 1933 (Etherington 1933). Etherington mentioned this equation was proposed by Tolman as a way to test a osmologial model. Ellis proposed a proof of this equation in the ontext of Riemannian geometry (Ellis 1971, 2007). A quote from Ellis (Ellis 2007): The ore of the reiproity theorem is the fat that many geometri properties are invariant when the roles of the soure and observer in astronomial observations are transposed. This statement is fundamental in the reiproity theorem as shown here in the derivation of the theorem in the dihotomous osmology. While the proof of the Etherington s distane duality in the ontext of Riemannian geometry is tedious, the derivation in the dihotomous osmology is straightforward. As a reminder, the dihotomous osmology (Heymann 2014a,b) onsists of a stati matter universe with an expanding luminous world. One needs to imagine a ube Yuri Heymann 3 rue Chandieu, 1202 Geneva, Switzerland of light expanding, in a spae where galaxies do not reede from eah other. Fortunately, the Etherington s distane-duality equation, whih is a ruial relationship in osmology, an be verified from astronomial observations. While the luminosity distane is measured from supernova observations, the angular-diameter distane is determined from the X-ray surfae brightness and the Sunyaev- Zel dovih effet (Sunyaev & Zel dovih 1972) of galaxy lusters (Silk & White 1978). In (Bernardis et al. 2006), the authors found that the ratio between the two distanes D L for the luminosity distane and D A for the D L D A (1+z) 2 angular-diameter distane, defined as η = is bound to be η = 1.01 ± 0.07 at 68%.l. Similar results were obtained in (Uzan et al. 2004; Nair et al. 2012), where no signifiant violation of the distane-duality relationship was found. In (Gonçalves et al. 2012), the authors tested the osmi distane duality for different galaxy luster samples. The study (Lima et al. 2011) is foused on analytial expressions for the deformation of the distane duality in terms of the osmi absorption parameter. The reiproity theorem is onsidered to be true when photon number is onserved, gravity is desribed by a metri theory with photons traveling on unique null geodesis (Bassett & Kunz 2004). Any violation of the distane duality would be attributed to exoti physis. Following the introdution in setion 1, the distane measurements are derived in setion 2. To derive the Etherington s reiproity theorem in the dihotomous osmology, we first need the distane measurements, whih may be derived from the tired-light paradigm (setion 2.1) or from expanding metris (setion 2.2). Both derivations lead to the same equations. In setion 2.3, we derive the Etherington s distane duality using our distane measurements. In setion 3, we provide a brief explanation of the method used to estimate the angular-diameter distane from X-ray surfae bright-

2 2 ness measurements and the Sunyaev-Zel dovih effet. In setion 4, we present the line of thought of the dihotomous osmology. Finally, we offer our onlusion in setion 5. 2 Derivation of the distane measurements 2.1 Derivation from tired-light paradigm When a photon loses energy during its travel in spae, the wavelength of light is strethed, and beause the number of yles of the light wave is onserved, an expansion of the luminous world is produed. As a onsequene of this strething of light, the veloity of the light wavefront inreases during its travel (Fig. 1). Aording to speial relativity, the speed of light is invariable. Hene, in order to maintain the light wavefront at the speed of light, the model introdues a time ontration between the emission point and the observer. The study (Heymann 2014a) mentions a time dilation; in order to retify this, the model is based on a time ontration in the arrow of time. where is the Hubble onstant. Therefore E(t) = E 0 exp( t), (3) and E(T ) = E 0 exp( T ), (4) where t is the time whih is equal to zero at the time of observation, and T the light travel time of the soure from the observer. A set of two transformations is onsidered: first a time-variable light wavefront to aommodate the expansion of the luminous world, and seond a time ontration to maintain the light wavefront at the speed of light Light wavefront with respet to the soure The light wavefront veloity before time ontration is expressed as follows: v(t) = E emit E(t). (5) where E emit is the photon energy when emitted, and E(t) the photon energy at time t. To maintain the light wavefront at the speed of light, the following time ontration is applied: δt δt = E emit E(t). (6) Fig. 1 Where (a) is the light wavefront without strething, and (b) with strething. We an see that in (b) the light wavefront is going faster than in (a). Considering that photons lose energy as light gets strethed, the following equation is obtained: 1 + z = E(z) E 0, (1) where E(z) is the photon energy when emitted, E 0 is the photon energy at time of observation, and z is the redshift. A simple deay law of the photon energy is onsidered: Ė E =, (2) Hene, the light travel time with respet to the soure is: T = 0 δt 0 δt dt = E emit dt. (7) E(t) Introduing (3) in the previous equation and integrating, we get: T = E ( emit 1 1 E ) 0. (8) E 0 E emit Introduing (1) in the previous equation, we get: T = z, (9) whih is the light travel time measurement for the luminosity distane.

3 Light wavefront with respet to the observer The light wavefront veloity before time ontration is expressed as follows: v(t) = E 0 E(t). (10) To maintain the light wavefront at the speed of light, the following time ontration is applied: δt 0 δt = E 0 E(t). (11) Hene, the light travel time with respet to the observer is: T 0 = 0 δt 0 0 δt dt = E 0 dt. (12) E(t) Introduing (3) in the previous equation and integrating, we get: T 0 = 1 (1 exp( T )). (13) Introduing (4) in the previous equation, we get: T 0 = 1 ( 1 E ) 0. (14) E emit Introduing (1) in the previous equation, we get: T 0 = 1 z (1 + z), (15) whih is the light travel time measurement for the atual distane. 2.2 Derivation from expanding metris In the dihotomous osmology, the luminous world is expanding; therefore, we an derive the distane measurements using expanding metris Luminosity distane The luminosity distane is the distane measured from the luminosity of standard andles. Supernovae Ia are onsidered stanard andles, meaning they all have the same absolute brightness when they explode. From their apparent brightness, we an dedue the luminosity distane, beause the brightness diminishes proportionally to the inverse of the distane squared. The formula used to measure the luminosity distane is the distane modulus equation. For onsidering a photon travelling away from the enter of a supernova, the luminosity distane is alulated as follows: dr L dt = + r L, (16) where r L is the luminosity distane, the Hubble onstant, and the speed of light. By integrating this equation between 0 and T, we get: r L = (exp( T ) 1). (17) Beause da dt = a, we get dt = da a, where a is the sale fator. In addition, the relationship between the sale fator and the redshift is given by the osmologial redshift equation (1 + z) = 1 a, where the sale fator is equal to one at present time. Hene, the light travel time versus redshift is as follows: T = r L = 1 1/(1+z) da a = 1 ln(1 + z). (18) Equations (17) and (18) yield: z. (19) whih is idential to (9) with r L = T Eulidean distane A measurement of the distane is obtained by alulating the orresponding distane if there were no expansion, whih we all the Eulidean distane. Let us introdue y to this distane measurement. By onsidering a photon moving towards the observer, we get: dy dt = + y. (20) By setting time zero at a referene T b in the past, we get: t = T b T ; therefore, dt = dt (where T is the light travel time when looking at a soure into the past). Hene:

4 4 dy dt = y, (21) with boundary ondition y(t = 0) = 0. Integrating this equation between 0 and T, we get: y = y = (1 exp( T )). (22) By substitution of (18) into (22), we get: z (1 + z), (23) whih is idential to (15) with y = T Etherington s distane duality From (19) and (23), we get: r L = (1 + z)y. (24) The angular-diameter distane d A of an objet is defined in terms of x, the objet s atual size, and θ, the angular size of the objet as viewed from earth. The equation is as follows: d A = x θ. (25) Beause of the expansion of the luminous world, the apparent size of elestial objets is strethed by a fator (1+z), and the apparent angular size is inreased by the same fator. Hene, the relationship between the atual distane y and the angular-diameter distane is as follows: y = (1 + z)d A. (26) Equations (24) and (26) yield: r L = (1 + z) 2 d A, (27) whih is the Etherington s distane-duality relationship. We have just derived the Etherington s reiproity theorem. 3 Method It is worthwhile to provide a brief explanation of the method used to validate the Etherington s distaneduality equation based on astronomial observations. The luminosity distane is measured from supernova observations using the distane modulus and is related to the redshift. The hallenge is to measure the angulardiameter distane beause we dont know the atual size of astronomial objets. This has been done for galaxy lusters using X-ray surfae brightness measurements and the Sunyaev-Zel dovih effet. Galaxy lusters ontain large quantities of hot and ionized gas at temperatures between 10 to 100 megakelvins. This hot gas radiates in the X-ray domain through bremsstrahlung, or radiation produed by the deeleration of a harged partile when defleted by another harged partile. This intra-luster gas distorts the osmi mirowave bakground radiation (CMBR) through the so-alled Sunyaev-Zel dovih effet: inverse Compton interation of photons whih reeive an energy boost when olliding with high energy free eletrons. This dereases the CMBR brightness at low frequenies but inreases it at high frequenies. The drop in temperature or brightness of the CMBR spetrum in the Rayleigh-Jeans region due to the Sunyaev-Zeldovih effet is a funtion of eletron temperature and density. The X-ray surfae brightness is a funtion of the volume of the luster and eletron temperature and density. Using both measures, we an therefore eliminate the eletron density term and estimate the size of the luster. Finally, we ompute the angular-diameter distane using the size of the luster and angular size as shown in (25). The details of the method and quantitative aspets are desribed in (Birkinshaw et al. 1991; Inagaki et al. 1995). 4 Interpretation The dihotomous osmology is in line with the shool of thought of the Greek philosopher Demoritus. Born around 460 B.C., Demoritus was a materialist philosopher disiple of Leuippus. Both held that everything is omposed of atoms, the smallest partile of a substane, whih interat with eah other and lie in empty spae. In the dihotomous osmology there is no need for dark energy or other exoti substanes, and the universe onsists of atoms and vauum. The dihotomous osmology is in ontradition with the big bang theory. In the big bang theory the universe is expanding, whereas in the dihotomous osmology the universe is stati. The three pillars of the big bang are respetively, the expansion of the universe aording to Hubbles law, the disovery of the mirowave

5 5 bakground radiation, and the relative abundanes of light elements. The dihotomous osmology hallenges the first pillar of the big bang. A onsequene of our theory is that the age of the universe is indefinite. In the big bang theory, the age of the universe, is defined by the moment when all the universe was onfined in one point - the big bang singularity, whih is estimated to have ourred around 13.7 billion years ago. In the dihotomous osmology, we annot define a beginning of time. Hubble time, whih is the inverse of the Hubble onstant, beomes the maximum distane that light an travel in the universe. 5 Conlusion The Etherington s distane-duality equation, whih relates the luminosity distane of standard andles to the angular-diameter distane, is a ruial relationship in osmology. Although the Etherington s reiproity theorem is onsidered to be peuliar to osmologial models based on Riemannian geometry, in the present study we propose a new derivation of this relationship in the dihotomous osmology. This derivation is straightforward and follows naturally from the dihotomous osmology. Today, the Etherington s reiproity theorem is onsidered established and has been verified using astronomial observations based on X-ray surfae brightness and the Sunyaev-Zel dovih effet of galaxy lusters.

6 6 Referenes Bassett, B. A., & Kunz, M., 2004, Physial Review D, 69, Bernardis, F., Giusarma, E., & Melhiorri, A. 2006, International Journal of Modern Physis D, 15, Birkinshaw, M., Hughes, J. P., & Arnaud, K. A., 1991, The astrophysial Journal, 379, Ellis, G. F. R. 1971, Proeedings of the 47th International Shool of Physis Enrio Fermi, edited by R. K. Sahs (Aademi Press, New York and London), 15, Ellis, G. F. R. 2007, Gen. Relativ. Gravit., 39, Etherington, I. M. H. 1933, Philos. Mag., 15, Gonçalves, R. S., Holanda, R. F. L., & Alaniz, J. S. 2012, Monthly Notie Letters of the Royal Astronomial Soiety, 420, L43-L47 Heymann, Y. 2014(a), Progress in Physis, 10(3), Heymann, Y. 2014(b), Progress in Physis, 10(4), Inagaki, Y., Suginohara, T., & Suto, Y., 1995, Publiations of the Astronomial Soiety of Japan, 47, Lima, J. A. S., Cunha, J. V., & Zanhin, V. T. 2011, The Astrophysial Journal Letters, 742, L26 Nair, R., Jhingan, S., & Jain, D. 2012, preprint, arxiv: astro-ph/ Silk, J., & White, S. D. M. 1978, The Astrophysial Journal Letters, 226, L103 Sunyaev, R. A., & Zel dovih, Ya. B. 1972, Comm. Astrophys. Spae Phys., 4, Uzan, J.-P., Aghanim, N., & Mellier, Y., 2004, Physial Review D, 70, This manusript was prepared with the AAS LATEX maros v5.2.