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Is the Transition Redshift a New Cosmological Number?

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Is the Transition Redshift a New Cosmological Number? SLAC-PUB-15400 Is the transition redshift a new cosmological number? J. A. S. Lima1,∗ J. F. Jesus1,† R. C. Santos2,‡ and M. S. S. Gill3,4§ 1Departamento de Astronomia, Universidade de S˜ao Paulo, 05508-900, S˜ao Paulo, SP, Brazil 2Departamento de Ciˆencias Exatas e da Terra, Universidade Federal de S˜ao Paulo, 09972-270 Diadema, SP, Brasil 3Kavli Institute for Particle Astrophysics & Cosmology, Stanford, USA and 4Center for Cosmology and Astro-Particle Physics, The Ohio State University, 191 West Woodruff Avenue, Columbus, OH 43210, USA Observations from Supernovae Type Ia (SNe Ia) provided strong evidence for an expanding ac- celerating Universe at intermediate redshifts. This means that the Universe underwent a dynamic phase transition from deceleration to acceleration at a transition redshift zt of the order unity whose value in principle depends on the cosmology as well as on the assumed gravitational theory. Since cosmological accelerating models endowed with a transition redshift are extremely degenerated, in principle, it is interesting to know whether the value of zt itself can be observationally used as a new cosmic discriminator. After a brief discussion of the potential dynamic role played by the transition redshift, it is argued that future observations combining SNe Ia, the line-of-sight (or “ra- dial”) baryon acoustic oscillations, the differential age of galaxies, as well as the redshift drift of the spectral lines may tightly constrain zt, thereby helping to narrow the parameter space for the most realistic models describing the accelerating Universe. I. INTRODUCTION The correct physical explanation for such a transi- tion is the most profound challenge for cosmology to- The extension of the Hubble diagram to larger dis- day. Within the General Relativistic (GR) paradigm, tances by using observations from supernovae type Ia the simplest manner for explaining such a phenomenon (SNe Ia) as standard candles allowed the history of cos- is by postulating a cosmological constant Λ in the Ein- mic expansion to be probed at much higher accuracy at stein equations. Indeed, anything which contributes to low and intermediate redshifts. Independent measure- a decoupled vacuum energy density also behaves like a ments by various groups indicated that the current ex- cosmological constant. However, the existence of the so- pansion is in fact speeding up and not slowing down, as called cosmological constant and coincidence problems was believed for many decades [1–3]. In other words, by [4], inspired many authors to consider alternative candi- virtue of some unknown mechanism, the expansion of the dates, thereby postulating the existence of an exotic fluid Universe underwent a “dynamic phase transition” whose with negative pressure (in addition to cold dark matter), effect was to change the sign of the universal deceleration usually called dark energy [5]. parameter q(z). Possible theoretical explanations for the present accel- erating stage without dark energy are also surprisingly ∗Electronic address: [email protected] abundant [6, 7]. Even in the framework of GR there are † Electronic address: [email protected] some alternative proposals where the existence of a new ‡Electronic address: [email protected] §Electronic address: [email protected] dark component is not necessary in order to have an ac- Published in arXiv:1205.4688. KIPAC, SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025 Work supported in part by US Department of Energy contract DE-AC02-76SF00515. 2 celerating regime at low redshifts. For instance, many 48 authors have claimed that the fact that the acceleration 44 comes out very close to the beginning of the nonlinear evolution of the contrast density is not just a trivial co- 40 incidence [8]. In this connection, several averaging proce- Deceleration Acceleration 36 dures have been developed in order to take into account Magnitude a possible “back reaction” effect associated with the ex- z (2 ) t 32 istence of inhomogeneities [9]. Another possibility still Union2 - SNe Ia (2010) CDM within the GR framework is that ‘dynamic transition’ 28 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 can be powered uniquely by the gravitationally-induced Redshift creation of cold dark matter particles [10, 11]. The ba- sic idea is that the irreversible process of cosmological FIG. 1: The relative magnitude as a function of the redshift particle creation at the expense of the gravitational field for the SNe Ia sample compiled by Amanullah et al. [3]. The can phenomenologically be described by a negative pres- vertical strip shows the Riess et al. [15] limits on the transi- +0.27 sure and the associated entropy production [12]. Another tion redshift, zt = 0.426−0.089 (at 95% c.l.), based on a kine- possibility is provided by models with interaction in the matic approach [16–19]. It will be argued here that the tran- dark sector, as happens, for instance, in decaying vacuum sition redshift will become accessible by future observations, cosmologies [13], as well as, in many variants of coupled and, as such, it may play the role of a primary cosmological parameter. dark energy models [14]. In this paper we advocate a different approach based can be thought as a new cosmic parameter even in the on the simple existence of a transition redshift (zt) as context of the ΛCDM model. In section IV, the transi- required by the SNe Ia data. Its leitmotiv is summa- tion redshit is discussed as a new cosmological number rized in the caption of Figure 1. In our view, due to in the sense of Sandage. The possibility to access it from the recent advances of astronomical observations such a independent observations is also discussed in the corre- quantity defining the transition between a decelerating to sponding subsections. Finally, in the conclusion section an accelerating stage will become a powerful cosmologi- we summarize the basic results. cal probe. In particular, it is argued that future obser- vations combining SNe Ia, the line-of-sight (or “radial”) II. TRANSITION REDSHIFT IN FRW Baryon Acoustic Oscillations (BAO), the differential ages GEOMETRIES of galaxies (DAG), as well as the redshift drift of the spec- In what follows we restrict our attention to the class tral lines (RDSL) will constrain zt, thereby helping to narrow the parameter space for the most realistic models of spacetimes described by the FRW line element (unless describing the accelerating Universe. explicitly stated we set c = 1): dr2 ds2 = dt2 − a2(t) + r2(dθ2 + sin2θdφ2) , (1) The article is structured as follows. In Sect. II, we 1 − kr2 discuss the general problem of the transition redshift for where a(t) is the scale factor and k is the curvature con- different cosmologies. In Sect. III, we discuss how it stant, which can be −1, 0, or 1, for a spatially open, flat 3 or closed Universe, respectively. Although inflationary ΛCDM model which has only one dynamic free parame- models and recent observations from CMB favor a spa- ter, namely, the vacuum energy density. It seems to be tially flat Universe, we shall not restrict ourselves to this consistent with all the available observations provided case. that the vacuum energy density is fine tuned to fit the In this background, the Einstein Field Equations data (ΩΛ ∼ 0.7). However, even considering that the (EFE) and the decelerating parameter, q, can be writ- addition of extra fields explain the late time accelerating ten as: stage and other complementary observations [20–23], the need of (yet to be observed) dark energy component with 2 a˙ k 8πG unusual properties is certainly a severe hindrance. + = ρT , (2) a a2 3 For the sake of simplicity, next section we focus our attention on the ΛCDM model and its predicted transi- a¨ 4πG = − (ρ +3p ), (3) tion redshift. The main aim is to show how to built a a 3 T T complementary space parameter based on the transition aa¨ 1 a¨ redshift as a basic quantity. Further, it will be discussed q(z) ≡− = − , (4) a˙ 2 H2 a how such an approach may be useful to discriminate the where ρT and pT are the total energy density and pres- realistic accelerating world models proposed in the liter- sure of the mixture and H(t) =a/a ˙ is the Hubble param- ature. eter. Note that the acceleration equation (3) does not de- pend explicitly on the curvature, and, similarly, the same III. TRANSITION REDSHIFT IN ΛCDM happens with the transition redshift (zt) since it is implic- MODELS itly defined by the condition q(zt)=¨a(zt) = 0. It is also worth noticing that all kinematic approaches developed The late time observed Universe in ΛCDM models is in the literature [16–19] point to a transition redshift in composed almost completely of pressureless matter (con- the past, that is, at intermediate redshifts (z < 1). The t sisting of dark and normal baryonic matter components), importance of such a result comes from the fact that it is and a negative pressure cosmological constant energy independent of any dark energy models as well as of the density, since the radiation contribution at low redshifts underlying gravity theory. is just ∼ 10−5 of the total energy density. Following As widely known, the first SNe Ia analyses were done standard lines, we write deceleration parameter as assuming a constant Λ for the dark energy component. However, due to the coincidence and cosmological con- 1 a¨ (1 + z) dH(z) stant problems several candidates for dark energy were q(z) ≡− = − 1, (5) H2 a H(z) dz proposed in the literature. At present, beyond the cos- with the Hubble parameter assuming the form below mological constant there is a plethora of relativistic dark energy candidates capable to explain the late time ac- 3 2 1/2 H(z)= H0 ΩM (1 + z) +ΩΛ +Ωk(1 + z) ) , (6) celerating stage, and, as such, the space parameter of the basic observational quantities is rather degenerate.
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