One of the first and simplest proposed Friedmann–Robertson–Walker (FRW) cosmological model is the Λ cold dark matter (ΛCDM) universe, which involves Einstein's cosmological constant Λ. This standard model of cosmology, which is also referred to as the concordance model, assumes that the total energy density ρ of the universe is made up of three components, namely matter ρm (baryonic and dark matter), radiation ρr, and dark energy or vacuum energy ρΛ, which produces the necessary gravitational repulsion. In this model, dark energy which has an equation of state (EOS) ωΛ = pΛ/ρΛ = −1, is a property of the space itself and its density ρΛ = −pΛ = Λc4/8πG is constant, such that as the universe expands the constant vacuum energy density will eventually exceed the matter density of the universe which is ever decreasing. The spatially flat ΛCDM model dominated by vacuum energy with ΩΛ ∼ 0.70, with the rest of the energy density being in the form of non-relativistic cold dark matter with Ωm ∼ 0.25 and non-relativistic baryonic matter with Ωb ∼ 0.05, fits observational data reasonably well (Riess et al. 1998; Permutter et al. 1999; Knop et al. 2003; Riess et al. 2004). However, the main problem in this model is the huge difference of about 10120 orders of magnitude between the observed value of the cosmological constant and the one predicted from quantum field theory; known as the cosmological constant problem (Weinberg 1989). Another issue is the so called coincidence problem which expresses the fact that although in this model the matter and dark energy components scale differently with redshift during the evolution of the universe, both components today have comparable energy densities, and it is unclear why we happen to live in this narrow window of time. Besides these main issues, there are other inherent problems faced by the ΛCDM, some of which arose as a result of recent observations that are in disagreement with the model's predictions. For example, in order to account for the general isotropy of the cosmic microwave background (CMB), the standard model invokes an early period of inflationary expansion (Kazanas 1980; Guth 1981; Linde 1982). However, the latest observations by Planck (Planck Collaboration XXIII 2003) indicate that there may be some problems with such an inflationary scenario (Ijjas, Steinhardt & Loeb 2013; Guth, Kaiser & Nomura 2014). It was partly due to these issues of the standard ΛCDM, that during the last decade several alternative dark energy models have been proposed and tested with observations. In these models the dark energy density component ρde is not constant and in most cases ωde = pde/ρde depends on time, redshift, or scale factor. For example in some of these so called dynamical dark energy models, late time inflation is achieved using a variable cosmological term Λ(t) (Ray et al. 2011; Basilakos 2015) sometimes taken in conjunction with a time dependent gravitational constant G(t) (Ray, Mukhopadhyay & Dutta Choudhury 2007; Ibotombi Singh, Bembem Devi & Surendra Singh 2013). Other sources of dark energy include scalar fields such as quintessence (Peebles & Ratra 2003), K-essence (Armendariz-Picon et al. 2001) and phantom fields (Singh, Sami & Dadhich 2003). An alternative approach to the dark energy problem relies on the modification of Einstein's theory itself such that in these alternative theories of gravity, cosmic acceleration is not provided solely by the matter side Tμν of the field equations, but also by the geometry of spacetime. These theories include the scalar-tensor theory with non-minimally coupled scalar fields (Barrow & Parsons 1997; Bertolami & Martins 2000), f(R) theory (Tsujikawa 2008), conformal Weyl gravity (Mannheim 2000) and higher dimensional theories such as the Randall–Sundrum (RS) braneworld model (Randall & Sundrum 1999), and the braneworld model of Dvali–Gabadadze–Porrati (DGP) (Dvali, Gabadadze & Porrati 2000). Over the last few years considerable interest has been shown in the simple FRW linearly expanding (coasting) model in Einstein's theory with a(t) ∝ t, H(z) = H0(1 + z). Like the ΛCDM the total energy density and pressure in this model are expressed in terms of matter, radiation and dark energy components, such that p = ωρ with ρ = ρm + ρr + ρde and p = pr + pde (since pm ≈ 0), but it includes the added assumption ω = −1/3, i.e. the cosmic fluid acting as the source has zero active gravitational mass. So this would definitely exclude a cosmological constant as the source of the dark energy component in this case. The model was first discussed by Kolb (1989) who referred to this zero active mass cosmic fluid as ‘K-matter’. Interest in this model has been revived recently after it was noted (Melia 2003) that in the standard model the radius of the gravitational horizon Rh(t0) (also known as the Hubble radius) is equal to the distance ct0 that light has travelled since the big bang, with t0 being the current age of the universe. In the ΛCDM this equality is a peculiar coincidence because it just happens at the present time t0. It was then proposed (Melia 2007, 2009; Melia & Shevchuk 2012) that this equality may not be a coincidence at all, and should be satisfied at all cosmic time t. This was done by an application of Birkhoff's theorem and its corollary, which for a flat universe allows the identification of the Hubble radius Rh with the gravitational radius Rh = 2GM/c2, given in terms of the Misner–Sharp mass $M = (4\pi /3)R_{{\rm h}}^{3}(\rho /c^2)$ (Misner & Sharp 1964). The added assumption of a zero active gravitational mass ρ + 3p = 0 implies (Melia & Shevchuk 2012) that Rh = ct or H = 1/t for any cosmic time t. This linear model became known as the Rh = ct universe. Unlike the ΛCDM/ωCDM which contains at least the three parameters H0, Ωm and ωde, the Rh = ct model depends only on the sole parameter H0, so that for example the luminosity distance used to fit Type Ia supernova data (Melia 2009) is given by the simple expression dL = (1 + z)Rh(t0)ln (1 + z). Also while the ΛCDM would need inflation to circumvent the well-known horizon problem, the Rh = ct universe does not require inflation. One should also point out that the condition Rh = ct is also satisfied by other linear models such as the Milne universe (Milne 1933), which however has been refuted by observations. Unlike the Rh = ct model discussed here, the Milne universe is empty (ρ = 0) and with a negative spatial curvature (k = −1). As a result of these properties its luminosity distance is given by $d_{L}^{\rm {Milne}} = R_{{\rm h}}(t_0)(1 + z)\sinh [\ln (1+z)]$, and it was shown that this is not consistent with observational data (Melia & Shevchuk 2012). In the last few years the Rh = ct universe received a lot of attention when it was shown (Melia & Maier 2013; Wei, Wu & Melia 2013, 2014a, 2014b, 2015; Melia, Wei & Wu 2015) that it is actually favoured over the standard ΛCDM (and its variant ωCDM with ω ≠ −1) by most observational data. This claim has been contested by Bilicki & Seikel (2012) and Shafer (2015) who argued that measurement of H(z) as a function of redshift and the analysis of Type Ia supernovae favoured the ΛCDM over the Rh = ct universe. However, this was later contested by Melia & McClintock (2015) who showed that the Rh = ct was still favoured when using model-independent measurements that are not biased towards a specific model. Others (see for example van Oirschot, Kwan & Lewis 2010; Lewis & van Oirschot 2012; Mitra 2014) have also criticized the model itself, particularly the validity of the EOS ω = −1/3 (Lewis 2013). These and other criticisms have been addressed by Bikwa, Melia & Shevchuk (2012); Melia (2012) (see also Melia 2015 and references therein.) As pointed out above the Rh = ct model would still require a dark energy component ρde, albeit not in the form of a cosmological constant. So the obvious question at this point would be: what are the possible sources for this component that together with the matter and radiation components will give the required total EOS, ω = −1/3? The purpose of this paper is to answer this question by discussing the various possible sources of dark energy that are consistent with this EOS. Since the radiation component ρr at the present time t0 is insignificant (at least for the ΛCDM with which this model has been compared) we assume that the total energy density ρ = ρde + ρm and the total pressure p = pde (pm ≈ 0), as is normally done in the other alternative dynamical dark energy models found in the literature. So in the next three sections we examine three possibilities for the source of dark energy in the Rh = ct model, namely a variable cosmological term Λ(t), a non-minimally coupled scalar field in Brans–Dicke theory which is equivalent to a variable gravitational constant G(t), and finally quintessence represented by a minimally coupled scalar field ϕ. We show that although the first two sources are consistent with the model, they are both unphysical, which leaves the third source of quintessence as the viable source of dark energy in the Rh = ct universe. Results are then discussed in the Conclusion. Unless otherwise noted we use units such that G = c = 1.
Citation: Ijjas, Steinhardt & Loeb 2013
Citation function: Motivation; Motivation
Citation: Singh, Sami & Dadhich 2003
Citation function: Background; Background