Issue 
A&A
Volume 674, June 2023



Article Number  A209  
Number of page(s)  15  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/202243553  
Published online  22 June 2023 
Covariant formulation of refracted gravity
^{1}
Dipartimento di Fisica, Università di Cagliari, Cittadella Universitaria, 09042 Monserrato, Italy
email: asanna@dsf.unica.it
^{2}
Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Cagliari, Cittadella Universitaria, 09042 Monserrato, Italy
^{3}
Dipartimento di Fisica, Università di Torino, Via P. Giuria 1, 10125 Torino, Italy
email: titos.matsakos@gmail.com; antonaldo.diaferio@unito.it
^{4}
Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy
Received:
15
March
2022
Accepted:
17
April
2023
We propose a covariant formulation of refracted gravity (RG), which is a classical theory of gravity based on the introduction of gravitational permittivity – a monotonic function of the local mass density – in the standard Poisson equation. Gravitational permittivity mimics dark matter phenomenology. The covariant formulation of RG (CRG) that we propose belongs to the class of scalartensor theories, where the scalar field φ has a selfinteraction potential 𝒱(φ) = − Ξφ, with Ξ being a normalization constant. We show that the scalar field is twice the gravitational permittivity in the weakfield limit. Far from a spherical source of density ρ_{s}(r), the transition between the Newtonian and the RG regime appears below the acceleration scale a_{Ξ} = (2Ξ − 8πGρ/φ)^{1/2}, with ρ = ρ_{s} + ρ_{bg} and ρ_{bg} being an isotropic and homogeneous background. In the limit 2Ξ ≫ 8πGρ/φ, we obtain a_{Ξ} ∼ 10^{−10} m s^{−2}. This acceleration is comparable to the acceleration a_{0} originally introduced in MOdified Newtonian Dynamics (MOND). From CRG, we also derived the modified Friedmann equations for an expanding, homogeneous, and isotropic universe. We find that the same scalar field φ that mimics dark matter also drives the accelerated expansion of the Universe. From the stressenergy tensor of φ, we derived the equation of state of a redshiftdependent effective dark energy w_{DE} = p_{DE}/ρ_{DE}. Current observational constraints on w_{DE} and distance modulus data of type Ia supernovae suggest that Ξ has a comparable value to the cosmological constant Λ in the standard model. Since Ξ also plays the same role of Λ, CRG suggests a natural explanation of the known relation a_{0} ∼ Λ^{1/2}. CRG thus appears to describe both the dynamics of cosmic structure and the expanding Universe with a single scalar field, and it falls within the family of models that unify the two dark sectors, highlighting a possible deep connection between phenomena currently attributed to dark matter and dark energy separately.
Key words: gravitation / cosmology: theory / dark matter / dark energy
© The Authors 2023
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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1. Introduction
The current standard Λ cold dark matter (ΛCDM) cosmological model assumes that gravitational interactions are ruled by general relativity (GR); the model relies on the existence of collisionless nonbaryonic CDM, and a positive cosmological constant Λ (Ostriker & Steinhardt 1995). Nonbaryonic dark matter is required to account for the abundance of light elements (Cyburt et al. 2016), the amplitude of the power spectrum of the temperature anisotropies in the cosmic microwave background (CMB; Planck Collaboration IV 2020), and the dynamics of cosmic structures (Clowe et al. 2006; Dodelson et al. 2001; Del Popolo 2014; Akrami et al. 2020; Planck Collaboration I 2020). The cosmological density parameter associated with Λ, with H_{0} being the Hubble constant, accounts for the negative value of the deceleration parameter q_{0} measured with the Hubble diagram of type Ia supernovae (SNeIa; Riess et al. 1998; Perlmutter et al. 1999). The curvature of the Universe, Ω_{k} = Ω_{Λ0} + Ω_{0} − 1, measured from the CMB power spectrum, suggests a null curvature (Planck Collaboration VI 2020).
Although the ΛCDM model agrees with most of the rich observational information currently available, a number of tensions are present both on large cosmic scales and on the scale of galaxies. The Hubble constant H_{0} estimated with the distance ladder in the local Universe (Riess et al. 2016, 2019) is more than 4σ larger than H_{0} inferred from the measurements of the CMB (Verde et al. 2019). The observed abundances of the light elements are consistent with the Big Bang nucleosynthesis scenario, except ^{7}Li, whose abundance is a factor of ∼3 smaller (Mathews et al. 2020). Numerous features of the CMB temperature anistropies are present on large scales (Ade et al. 2016). The probability of some of these features to appear individually is ≲0.1%; the combined probability of the uncorrelated features is ≲0.001% and might represent a serious challenge to the ΛCDM model (Schwarz et al. 2016; Luongo et al. 2022). The lensing amplitude in the CMB power spectra is enhanced compared to ΛCDM expectations (Planck Collaboration IV 2020) and would suggest a positive rather than a null curvature of the Universe (Handley 2021; Di Valentino et al. 2019). A slight tension also appears for the normalization of the power spectrum σ_{8} (e.g. Raveri 2016): the value inferred from the CMB measurements (Planck Collaboration VI 2020) is 2σ larger than the value derived from the tomographic weak gravitational lensing analysis of the KiloDegree Survey (KiDS) imaging data (Hildebrandt et al. 2017).
In addition, the cosmological constant poses a finetuning problem that is theoretically challenging. If we associate Λ with the ground state energy level of the vacuum in quantum field theory, its measured value, Λ ∼ 10^{−12} eV^{4}, appears to be ∼120 orders of magnitude smaller than expected (Weinberg 1989; Padilla 2015). If we associate Λ with the energy scale up to which the standard model of particle physics has been tested, ∼1 TeV, the discrepancy reduces to ∼60 orders of magnitude (Joyce et al. 2015), but it remains severe. The most popular solution to the Λ problem is to suppress Λ in the EinsteinHilbert action and generate the accelerated expansion of the Universe with dark energy, an auxiliary scalar field with proper features. The specific implementation of this idea has generated a large number of different models that may or may not modify Einstein’s equations (see Peebles & Ratra 2003; Copeland et al. 2006; Bamba et al. 2012; Joyce et al. 2015; Amendola et al. 2018, for extensive reviews).
Collisionless CDM poses additional problems on small scales: the corecusp problem in dwarf and disk galaxies, the missing satellite problem, the toobigtofail problem, and the plane of satellite galaxies in the Milky Way and nearby large galaxies (Salucci 2003; Ferrero et al. 2012; BoylanKolchin et al. 2012; GarrisonKimmel et al. 2014; Del Popolo & Le Delliou 2017; Kroupa 2012; de Martino et al. 2020). In addition, some relations, such as the radial acceleration relation or the baryonic Tully–Fisher relation in disk galaxies, would require finely tuned interactions between CDM and baryonic matter (Desmond & Wechsler 2015; Di Cintio & Lelli 2016; Desmond 2017).
Some of these smallscale tensions may originate either by an inaccurate treatment of the dynamics of CDM and baryonic matter or by the inappropriate properties adopted for the dark matter model. For example, the corecusp problem emerges when we attempt to interpret the observed kinematics of stars in galaxies by assuming that the galaxies are embedded within CDM halos with a Navarro–Frenk–White (NFW) density profile, as predicted by Nbody simulations (Navarro et al. 1997). By dropping this constraint on the dark matter distribution and adopting a dark matter density profile with a flat central core, we can properly describe the stellar kinematics of spirals with a universal rotation curve (Persic et al. 1996; Salucci et al. 2007; Salucci & De Laurentis 2012; Salucci 2018). Indeed, an exponential disk and a dark matter halo described by the Burkert profile with a core (Burkert 1995) excellently describe the rotation curves of five spirals (Gentile et al. 2004), and suggest that spirals and dwarf galaxies lie on the same scaling relation between the core density and the core radius of the dark matter halo (Salucci & Burkert 2000). A number of different effects from the dynamics of CDM or from baryonic physics, including stellar feedback and star formation efficiency, are advocated to generate a central core in the dark matter distribution (de Martino et al. 2020). For example, tidal effects by a massive hosting galaxy might induce dark matter density profiles shallower than the NFW profile in the central regions of satellite halos and might also alleviate the toobigtofail problem (Tomozeiu et al. 2016).
Dropping the hypothesis that dark matter is collisionless and cold might solve some, albeit not all, of the tensions of CDM on the scale of galaxies (Salucci 2019; Salucci & di Paolo 2021). Indeed, weakly interacting massive particles (WIMPs) are the most plausible candidate to make up the collisionless CDM (Bertone et al. 2005). However, attempts to, directly or indirectly, detect these particles have not yet produced unquestionable results (Tanabashi et al. 2018). Alternative dark matter models include warm dark matter, selfinteracting dark matter, QCD axions, and fuzzy dark matter (de Martino et al. 2020). Some of these models have particle counterparts, such as sterile neutrinos or ultralight bosons. The detection of these particles requires direct or indirect experiments different from those conceived for detecting WIMPs (Buckley & Peter 2018). Dark matter particles with distinct peculiar features, such as superfluidity (Berezhiani & Khoury 2015) or gravitational polarization (Blanchet 2007; Blanchet & Heisenberg 2017), can also partly reproduce the phenomenology of galaxies.
Alternatively, the dynamics on the scale of galaxies could be explained by modifying the theory of gravity in the weakfield Newtonian limit without resorting to the existence of dark matter. The idea of MOdified Newtonian Dynamics (MOND) was originally motivated only by the observations of the flat rotation curves of disk galaxies (Milgrom 1983a; Begeman et al. 1991; Begum & Chengalur 2004), and many of the current observational challenges on the scale of galaxies were actually predicted by MOND (Milgrom 1983b,c; Sanders & McGaugh 2002). The universal rotation curves of spirals (Persic et al. 1996) and the universality of the galactic surface density within the radius of the core of a Burkert dark matter profile also appear to be consistent with MOND (Gentile 2008; Gentile et al. 2009), although the debate on the dynamics of dwarf disk galaxies remains vibrant (Corbelli & Salucci 2007; SanchezSalcedo et al. 2013; Di Paolo et al. 2019; Banik et al. 2020; Salucci & di Paolo 2021). Furthermore, the MOND formulation is purely phenomenological and its extension to a covariant formulation has proven to be challenging (Famaey & McGaugh 2012; Milgrom 2015; Skordis & Złośnik 2019; Złośnik & Skordis 2017; Skordis & Złośnik 2021).
Although the problems of the cosmological constant and dark matter are usually considered two separate issues, attempts to unify the two dark sectors in a single framework are numerous. For example, Ferreira et al. (2019) suggest a model where the dark matter is made of two superfluids arising from two distinguishable states separated by a small energy difference: particles of one species can be converted into the other and the interaction between these multistate components of the dark matter can drive cosmic acceleration. Alternatively, in emergent gravity, where both the classical spacetime structure and gravity emerge from an underlying microscopic quantum theory (Sakharov 1991; Padmanabhan 2015; Verlinde 2017), the two dark sectors can be unified when, for example, the dark energy fluid is modelled as a critical BoseEinstein condensate of gravitons (Cadoni et al. 2018a,b; Tuveri & Cadoni 2019; Cadoni et al. 2020).
Attempts to describe both the accelerated expansion of the Universe and the dynamics of cosmic structures without dark matter and dark energy by building a modified theory of gravity are also numerous (see, e.g., Clifton et al. 2012; Nojiri et al. 2017). According to Lovelock’s theorem (Lovelock 1971, 1972), we can build a metric theory of gravity different from GR by, for example, allowing derivatives of the metric tensor higher than second order in the field equations, or introducing other fields in addition to the metric tensor.
Conformal gravity adopts the former route and replaces the EinsteinHilbert action with the contraction of the fourthrank conformal tensor introduced by Weyl (Mannheim & Kazanas 1994). Conformal gravity does not present ghostlike instabilities, that might be common in theories with highorder derivatives (Bender & Mannheim 2008), and is renormalizable (Mannheim 2012). Unfortunately, although conformal gravity successfully reproduces the accelerated expansion of the Universe (Diaferio et al. 2011), the expected abundance of primordial deuterium is orders of magnitudes smaller than observed (Elizondo & Yepes 1994). Moreover, conformal gravity is unable to reproduce the dynamics of galaxy clusters (Horne 2006; Diaferio & Ostorero 2009), and appears to require a finetuning condition to describe the phenomenology of gravitational lensing and the dynamics of disk galaxies (Campigotto et al. 2019).
We can preserve the secondorder field equations by introducing a single scalar field that drives both the accelerated expansion of the Universe and the formation of cosmic structure (e.g. Carneiro 2018). The case of a classical scalar field with a noncanonical kinetic term in its Lagrangian generates the class of Unified Dark Matter models (Bertacca et al. 2010). These models can be a viable alternative to ΛCDM (Camera et al. 2011, 2019) if the effective sound speed is small enough that the scalar field can cluster (Bertacca et al. 2008; Camera et al. 2009).
Here, we contribute to the quest for a unified model of the two dark sectors by proposing a novel scalartensor theory (Fujii & Maeda 2007; Quiros 2019) where the scalar field is responsible for both the dynamics of cosmic structures and the accelerated expansion of the Universe. This scalartensor theory is the covariant formulation of refracted gravity (RG), which is a new phenomenological modified theory of gravity proposed by Matsakos & Diaferio (2016). Refracted gravity appears to reproduce the phenomenology on the scale of galaxies and galaxy clusters by introducing a monotonic function of the local massdensity in the standard Poisson equation, termed gravitational permittivity. Indeed, Cesare et al. (2020) showed that RG properly describes the rotation curves and the vertical velocity dispersion profiles of 30 disk galaxies in the DiskMass Survey (Bershady et al. 2010), and the dynamics of stars and globular clusters in the outer regions of three elliptical galaxies of type E0 (Cesare et al. 2022). Here, we provide a covariant formulation of this nonrelativistic formulation of RG.
Section 2 reviews the relevant features of RG. In Sect. 3, we derive the covariant formulation of RG (CRG, hereafter) in the framework of scalartensor theories, and show that the scalar field can be identified with the gravitational permittivity. In Sect. 4, we consider a homogeneous and isotropic universe and derive the modified Friedmann equations; we show that the scalar field is responsible for the accelerated expansion of the Universe, and derive the equation of state of a redshiftdependent effective dark energy. We conclude in Sect. 5.
2. Nonrelativistic refracted gravity
The phenomenological RG is based on the modified Poisson equation (hereafter RG equation; Matsakos & Diaferio 2016)
where Φ is the gravitational potential, G the gravitational constant, and ρ the density of ordinary matter. The function ϵ is the gravitational permittivity, according to the mathematical similarity with the term on the lefthand side of the Poisson equation that describes electric fields in matter. We emphasize that no other parallels have been drawn with electrodynamics. As a starting hypothesis, the permittivity was prescribed to be a monotonic function of the local mass density, that is ϵ(ρ). However, ρ was only chosen because is the simplest scalar field characterizing the matter distribution in the weakfield regime; ϵ could in principle depend on other local quantities, for example the trace of the energymomentum tensor or the entropy.
On scales where the visible mass can accurately explain the observed dynamics according to Newtonian gravity, for example on the scale of stars, ϵ must be constant and equal to 1 in order to recover the standard Newtonian Poisson equation, ∇^{2}Φ = 4πGρ. Therefore, we can adopt the form of the permittivity
where ρ_{thr} is the density threshold that sets the transition between the Newtonian regime and the modified gravity regime;^{1} 0 < ϵ_{v} < 1 is the permittivity of the vacuum.
The presence of ϵ in the Poisson equation, Eq. (1), has two main effects: (1) in lowdensity regions, it generates a stronger gravitational field than in Newtonian gravity; and (2) it bends the gravitational field lines in regions where ∇Φ and ∇ϵ are not parallel. The former effect is trivially seen in the simple case ϵ = const < 1, when ρ < ρ_{thr}: Eq. (1) becomes ∇^{2}Φ = 4πGρ/ϵ, and shows that the resulting field is equivalent to a Newtonian field originating from a larger effective mass density ρ/ϵ, or a larger gravitational constant G/ϵ. When ∇Φ and ∇ϵ are not parallel, the field lines are refracted; in other words, the field lines change their direction when they cross the isosurfaces of ϵ at an angle different from π/2. This effect can also generate nonNewtonian phenomena in regions where the density is larger than ρ_{thr}, because the redirection of the field lines has nonlocal consequences. For example, when we consider only ordinary matter as the gravitational source, RG predicts flat rotation curves in disk galaxies even in regions where ρ > ρ_{thr} (Cesare et al. 2020). The redirection of the field lines is also expected to be responsible for the mass discrepancy in galaxy clusters (Matsakos & Diaferio 2016). In these two studies, ϵ(ρ) is assumed to be a smooth step function, with the density threshold ρ_{thr} in the range 10^{−27}–10^{−24} g cm^{−3}.
3. Refracted gravity as a scalartensor theory
We present the fundamental equations of CRG in Sect. 3.1, and the relation of CRG with the Horndeski theory and the CRG screening mechanisms in Sects. 3.2 and 3.3, respectively. In Sect. 3.4, we derive the weak field limit of CRG.
3.1. Fundamental equations of CRG
The family of scalartensor (TeS) theories, with the scalar field φ nonminimally coupled to the rank2 tensor field g_{μν}, can be derived from the action^{2}
where −g is the determinant of g_{μν}, R = g^{μν}R_{μν} is the Ricci scalar, R_{μν} is the Ricci tensor, g^{μν} is the inverse metric, ∇_{μ} is the covariant derivative, ∇^{α}φ∇_{α}φ ≡ g^{αβ}∇_{α}φ∇_{β}φ, and ℒ_{m} is the matter Lagrangian density, with ψ_{m} being the matter fields (Faraoni 2004). The potential 𝒱(φ) and the general differentiable function of the scalar field 𝒲(φ) parametrize the family of TeS theories. When φ → 1, the TeS action reduces to the standard EinsteinHilbert action with a cosmological constant equal to −𝒱(1). Hereafter, for the sake of simplicity, we consider the φdependence of 𝒲 and 𝒱 implicit.
Varying Eq. (3) with respect to the metric yields the ten modified Einstein field equations in the Jordan frame (JF)^{3}^{4}
where □ ≡ g^{μν}∇_{μ}∇_{ν} is the d’Alembertian operator (e.g. Quiros et al. 2016). Varying Eq. (3) with respect to the scalar field yields the equation for φ
We define CRG by setting
We note that the original BransDicke theory – with a constant 𝒲 of order unity and a zero potential – is ruled out by postNewtonian expansions and solar system experimental tests, which give the constraint 𝒲 ≳ 40 000 (Faraoni 2004; Clifton et al. 2012), and by recent results from CMB observations (e.g. Li et al. 2013; Avilez & Skordis 2014). However, these constraints seem to be weaker on large cosmological scales and can be avoided by adding a selfinteraction potential (Hrycyna et al. 2014; Quiros 2019), which we define as
with Ξ being a constant parameter.
With the definitions of Eqs. (6) and (7), the modified field equations and the equation for φ, in the JF, become
By using Eq. (9) to simplify Eq. (8), and by using Eq. (8), contracted with g^{μν}, to simplify the resulting Eq. (9), we obtain the following CRG equations:
We derive the stress–energy tensor for the scalar field by recasting Eq. (8) as
where refers to the matter/energy stress–energy tensor, and refers to the scalarfield stress–energy tensor, whose explicit form is^{5}
We use the stress–energy tensor of φ to derive an effective dark energy in Sect. 4.3.
3.2. Relation to Horndeski theories
Scalartensor theories belong to the wider family of Horndeski models (for reviews on the topic, see, e.g., Joyce et al. 2015; Kobayashi 2019), and, consequently, CRG can also be identified as a special case of Horndeski theories. The detection of GW170817 and GRB170817A (Kase & Tsujikawa 2019; Wang et al. 2017; Sakstein & Jain 2017) has greatly constrained the Horndeski theory space. The currently allowed Lagrangian is (Noller & Nicola 2019; Creminelli & Vernizzi 2017; Sakstein & Jain 2017, and references therein)
where 𝒢_{i}’s are functions of the scalar field φ and X ≡ ∇^{α}φ∇_{α}φ is the kinetic term. By comparing Eq. (14) with the Lagrangian of Eq. (3), and by ignoring the factor 1/(16πG), we infer that 𝒢_{4} ↔ φ, , and 𝒢_{3} ↔ 0pagination ^{6}. Therefore, CRG is expected to inherit the important properties of the viable Horndeski theories, including the absence of ghostlike degrees of freedom, namely the Ostrogradski instabilities (Gleyzes et al. 2015).
3.3. Screening mechanism of CRG
General relativity accurately describes the gravitational interactions on the scale of stars and smaller scales, including the strong gravitational field regime. Therefore, any attempt to modify the theory of gravity by adding new degrees of freedom must provide a screening mechanism to avoid detectable discrepancies in the local tests of gravity (Joyce et al. 2015).
The screening mechanism depends on the local mass density and/or the local gravitational potential. It is thus convenient to study the screening mechanism in the Einstein frame (EF), where the scalar field is minimally coupled to both gravity and the matter fields. The advantage of the EF is that the field equations have a manifestly GRlike form^{7} and computations can be performed more easily. However, the presence of the extra coupling between the scalar and matter fields alters rods and clocks, and makes the identification of physical observables harder.
The transition from the JF to the EF can be performed with the conformal transformation of the metric (Clifton et al. 2012), namely a lengthscale transformation^{8}
where the tilde indicates quantities computed in the EF and . From the JF action of Eq. (3), the corresponding action in the EF is
where Ψ is the scalar field related to the scalar field in the JF by Ψ = ∓lnφ, is the Ricci scalar, A(Ψ)≡φ^{−1/2} is the coupling between the scalar field and matter, and 𝒰(Ψ) is the potential (Clifton et al. 2012)
The potential 𝒰(Ψ) = 2Ξe^{−Ψ} is the runaway potential of the chameleon cosmology (Clifton et al. 2012; Joyce et al. 2015; Khoury & Weltman 2004). This potential can be studied by considering the nonrelativistic equation of motion for the scalar field (see, e.g., Clifton et al. 2012, and references therein)
where V_{eff} is the effective potential. In particular, d^{2}V_{eff}/dΨ^{2} can be interpreted as an effective mass of the scalar field, m_{Ψ}, which reads as
In regimes of large densities ρ ≫ e^{−2Ψ}Ξ/2πG, the scalar field becomes increasingly more massive and hence the fifth force mediated by it has an increasingly shorter range; in other words, in this regime the presence of the scalar field is effectively screened. Conversely, in regimes of small densities ρ ≪ e^{−2Ψ}Ξ/2πG, m_{Ψ} gets smaller, the force mediated by Ψ has an increasingly longer range, and the field is free to propagate^{9}. The density scale separating the two regimes thus depends on both Ψ and Ξ. This result is consistent with the assumption, in the phenomenological RG, that the gravitational permittivity depends on the local density, and suggests a relation between ϵ and Ψ, or equivalently φ. In addition, this result indicates that the density scale depends on Ξ, which is a constant universal value independent of the local environment. In Sect. 4.3 and Appendix E, we find that Ξ plays the role of the cosmological constant Λ in the standard cosmological model. Therefore, we expect that the local value of the scalar field Ψ (or φ), rather than Ξ, plays the major role in setting the density threshold for the screening mechanism.
3.4. The weakfield limit of CRG
As in standard GR, we take the static weakfield limit of the metric g_{μν}^{10}, which can be expanded around the Minkowski metric η_{μν} as
where h≪1 and ∂_{μ}h≪1. We write the metric as
where Φ and U are two potentials, and we ignore terms of order 𝒪(Φ^{2})∼𝒪(U^{2}). These expressions enable the calculation of the lefthand side of Eqs. (10) and (11) up to terms of order 𝒪(Φ)∼𝒪(U) (see Appendix A for the detailed calculations).
For the righthand side of Eqs. (10) and (11), we consider a static nonrelativistic fluid: the only nonzero component of the energymomentum tensor is T_{00} ≃ ρ, and thus its trace is T ≃ −ρ. With these assumptions, the 00component of the field equations reduces to
We thus recover the RG equation if we identify the scalar field with twice the permittivity: φ = 2ϵ. In the Newtonian regime, we have a constant scalar field, namely ∇φ = 0, and thus we recover the standard Poisson equation for φ = 2.
3.4.1. The gravitational field of a spherical source immersed in a homogeneous background
In Appendix A.1, we compute the gravitational field generated by a spherical source immersed in a homogeneous background with density ρ_{bg}. The source is described by a density profile ρ_{s}(r) decreasing with r. We estimate the field in the limit ρ_{s}(r)≫ρ_{bg}, close to the source, and ρ_{s}(r)≪ρ_{bg}, at large distance from the source.
At small distances, we find
with the scalar field
At large distances from the source, the scalar field φ and the gravitational field dΦ/dr satisfy the implicit relation
with ρ(r) = ρ_{s}(r)+ρ_{bg}. This relation sets the acceleration scale
In regions where dΦ/dr ≫ a_{Ξ}, the gravitational field dΦ/dr ≃ −(1/2)dlnφ/dr has a similar dependence of the field at small distances (Eqs. (25) and (26)). In regions where dΦ/dr ≪ a_{Ξ}, the RG acceleration deviates from the Newtonian acceleration.
This result resembles the starting hypothesis of MOND, that introduces the acceleration scale a_{0} to separate the Newtonian from the modified gravity regimes. Moreover, in Sect. 4, we find Ξ ∼ Λ. We thus find that a_{Ξ} ∼ 10^{−10} m s^{−2}, in the limit 2Ξ ≫ 8πGρ/φ, that occurs at large distances from the source. The existence of this acceleration scale appears in a number of observations on the scale of galaxies (see, e.g., de Martino et al. 2020; McGaugh 2020; Chae et al. 2020; McGaugh et al. 2016, 2018; Merritt 2020). Indeed, the dynamics of disk and elliptical galaxies in the lowacceleration regime is described in the RG framework without requiring the existence of dark matter (Cesare et al. 2020, 2022). Nevertheless, some of the phenomenology predicted by MOND in the lowacceleration regime, set by an acceleration scale a_{0} independent of the source, like the radial acceleration relation (McGaugh et al. 2016), appears to be inconsistent with the rotation curves of dwarf disk spirals and lowsurface brightness galaxies (Di Paolo et al. 2019; SantosSantos et al. 2020). These tensions might suggest that the acceleration scale could indeed depend on the source, as it happens for a_{Ξ} in CRG.
The connection between a_{Ξ} and Ξ, in the limit 2Ξ ≫ 8πGρ/φ, is similar to the connection between a_{0} and Λ in MOND: a_{0} ∼ Λ^{1/2}. A number of different sensible arguments have been suggested for the interpretation of the latter relation (Milgrom 1989, 1999; Famaey & McGaugh 2012; Milgrom 2020). In the CRG context, a_{Ξ} ∼ (2Ξ)^{1/2} emerges naturally.
3.4.2. The scalar field φ for a spherical source immersed in a constant background
According to the results of Appendix A.1, the scalar field φ is positive and broadly in the range [0, 2] consistently with the RG ansatz ϵ ∈ [0, 1] for the permittivity. In the limit ρ_{bg} ≪ ρ_{s}, we have
whereas in the limit ρ_{bg} ≫ ρ_{s},
These limits are broadly in agreement with the smooth step function considered for ϵ = ϵ(ρ) in previous RG studies (Cesare et al. 2022, 2020; Matsakos & Diaferio 2016). In those studies, the local mass density was found to be a good proxy of the transition between the Newtonian and the RG regimes.
4. The homogeneous and isotropic universe in CRG
We derive the basic equations of a homogeneous and isotropic universe in Sect. 4.1 and we solve these equations for a spatially flat universe in Sect. 4.2. We derive the equation of state of the effective dark energy in Sect. 4.3 and discuss the evolution of the scalar field in Sect. 4.4.
4.1. Basic equations
The covariant formulation of RG enables the description of a homogeneous and isotropic universe that can be described by the Friedmann–Lemaître–Robertson–Walker (FLRW) metric
where a(t) is the scale factor, k is the spatial curvature, and (t, r, θ, ϕ) are the comoving coordinates. By treating the content of the Universe as a perfect fluid, Eq. (10) readily yields the modified Friedmann equations (see Appendix B)
and
The equation for the scalar field, Eq. (11), reduces to (see Appendix B)
In the JF, the stress–energy tensor is covariantly conserved (Faraoni 2004). We assume the standard equation of state
with w = 0 and w = 1/3 describing the dust and radiation components, respectively. The dependence of the matter or radiation density on time t, or equivalently on the scale factor a(t), is thus
with ρ_{0} the mean density of the component at the present time, t_{0}, when a(t_{0}) = a_{0} = 1.
4.2. A spatially flat universe: Analytic solution
Here, we solve the field equations, Eqs. (32)–(34), in the special case of a spatially flat universe with k = 0. We assume that the universe only contains baryonic matter with density ρ_{b} and negligible pressure, p = 0, namely w = 0 in the equation of state, Eq. (35).
After substituting Eq. (32) into Eq. (33) and dividing by H^{2} ≡ (ȧ/a)^{2}, Eqs. (32)–(34) become
The solution of the above system of coupled equations together with the equation of the mass–energy conservation, Eq. (36), determines the time evolution of both the scalar field and the scale factor.
We simplify the field equations by introducing the modified cosmological parameter
where Ω_{b} = ρ_{b}/ρ_{cr} ≡ ρ_{b}(8πG/3H^{2}) is the density parameter associated with the homogeneous baryonic density ρ_{b}. The density parameter Ω is analogous to the total matter density parameter of the standard model, that includes both baryonic and dark matter; in CRG, the gravitational role of the baryonic matter density is amplified by the factor 2/φ, unlike the standard model, where the baryonic and the dark matter components simply add up.
Similarly, we define
The parameter Ω_{Ξ} is analogous to the standard cosmological parameter Ω_{Λ} = Λ/3H^{2}, with Λ the cosmological constant in the GR field equations; therefore, in CRG, Ξ exactly plays the role of Λ in the standard model. At t = t_{0}, the values of the two parameters Ω and Ω_{Ξ} are and .
We further introduce two deceleration parameters, q and q_{φ}, related to the scale factor and the scalar field, respectively:
With the above definitions, the field equations, Eqs. (37)–(39), become
It is convenient to define the new quantity
The field equations, Eqs. (43)–(45), now become
Combining the first two equations yields
namely, with the definitions of q and ζ,
This equation can be integrated to obtain φ = 𝒞/(a^{2}ȧ), with 𝒞 an integration constant. Its value can be found from the boundary conditions a(t_{0}) = 1, H(t_{0}) = H_{0}, and φ(t_{0}) = φ_{0}. We obtain 𝒞 = H_{0}φ_{0} and thus
Moreover, since 3Ha^{3} = da^{3}/dt, we can write the scalar field as
This expression shows that the scalar field φ is inversely proportional to the rate of the variation of the volume of the universe.
As shown in Appendix C, adding up Eqs. (47) and (49) yields two solutions for dH/dt:
each solution being a first order differential equation for H. Hereafter, we refer to the solution corresponding to the upper (+) sign as CRG+, and to the lower (−) sign as CRG–, respectively. The integration of Eq. (54) yields
with the upper and lower signs corresponding to CRG+ and CRG–, respectively. The integration constant, C_{1}, can be found from the boundary condition at t = t_{0}:
Furthermore, the scale factor can be derived by integrating Eq. (55)
where
and C_{2} is an integration constant, whose value can be fixed by the condition a_{0} = 1:
with
Inverting Eq. (60) yields
which implies that the cosmological parameters Ω_{0} and Ω_{Ξ0} satisfy the relation
The parameter Ω_{0} must be positive, because it is proportional to the mass–energy density ρ_{b0}, according to Eq. (40); therefore, the only physically viable solutions of these inequalities are
The parameter Ξ must thus be positive.
The first test of the viability of CRG is its ability to properly describe the Hubble diagram of the observed SNeIa at high redshift. Using the scale factor in Eq. (57), the luminosity distance can be computed exactly, for both signs appearing in Eq. (57) (see Appendix D). As in the standard cosmological model, the values of the cosmological parameters can be set by modelling the SNIa Hubble diagram (see Appendix E).
4.3. The equation of state of the effective dark energy
Equation (13) is the general expression of the stressenergy tensor of the scalar field. By using the FLRW metric of Eqs. (31), (13) becomes
If we consider to be in the form of a perfect fluid, , the 00 and ijcomponents of Eq. (64) lead to the effective density and pressure associated with the scalar field:
In order to derive the density and pressure associated with the effective dark energy in CRG, we can combine Eq. (32) with Eq. (34), and Eq. (33) with Eq. (34), to rewrite the modified Friedmann Eqs. (32)(33) as
By comparing these equations with those of a general TeS theory with a nonminimal coupling between the scalar field and the metric,
where φ_{0} is the value of the scalar field computed at the present epoch^{11}, we derive the general expressions
where ρ = ρ_{m} + ρ_{r} and p = p_{r} = ρ_{r}/3 are the density and pressure associated with matter, ρ_{m}, and radiation, ρ_{r} and p_{r}, respectively (Frusciante & Perenon 2020).
Based on these definitions, we can evaluate the parameter associated with the equation of state of the effective dark energy, w_{DE} = p_{DE}/ρ_{DE}. Its analytical expression is reported in Eq. (F.4) of Appendix F. At the present epoch, for H = H_{0} and a = a_{0} = 1, we obtain
where the upper and lower signs refer to the CRG+ and the CRG– solutions, respectively.
The value which best fits the observational data is w_{DE} = −1 (Planck Collaboration VI 2020), which is consistent with the cosmological constant of the ΛCDM model. By assuming Ω_{0} ∼ 0.3, we find Ω_{Ξ0} ∼ 0.64, which is only admitted by CRG– and is near the upper bound of Eq. (63).
Values of w_{DE} slightly different from −1 can nonetheless be accommodated by the data (e.g. Amendola & Tsujikawa 2015; Copeland et al. 2006; Frusciante & Perenon 2020; Wen et al. 2018; Capozziello et al. 2006; Gerardi et al. 2019). Observational constraints on the equation of state of the effective dark energy generally depend on the model used to describe its effects (but see Gerardi et al. 2019, for a modelindependent reconstruction). We can gain some insight by adopting the parametrization w_{DE}(z) = w_{0} + w_{a}z/(1 + z) (Chevallier & Polarski 2001; Linder 2003). At the present epoch z = 0, this parametrization only depends on w_{0}, and it is therefore sufficiently general to account for a broad range of dark energy models, either with w_{DE} < −1 (phantom models) or w_{DE} ≥ −1 (Amendola & Tsujikawa 2015; Frusciante & Perenon 2020; Copeland et al. 2006; Sola & Stefancic 2005; Bean & Melchiorri 2002). Baryonic acoustic oscillations (BAO), SNIa and CMB observational data (Hazra et al. 2015; Wen et al. 2018) constrain w_{0} to be approximately in the range w_{0} ∈ [ − 1.18, − 0.85]. For Ω_{0} = 0.3, this range of w_{0} yields, by using Eq. (73), Ω_{Ξ0} ∈ [0.62, 0.65] for the CGR– solution; for the CRG+ solution, the allowed range of w_{0} is limited to w_{0} < −1.1765 and it thus only admits Ω_{Ξ0} ≃ 0.65.
These values of Ω_{Ξ0} close to the upper limit (1 + Ω_{0})/2 (Eq. (63)) are consistent with the constraints we obtain from the SNIa data that we investigate in Appendix E. Future observations aimed at investigating the largescale structure of the Universe and the evolution of dark energy (Amendola et al. 2018) will further constrain w_{0}, and hence the value of Ξ.
4.4. Additional remarks on the evolution of the scalar field
By using Eqs. (36), (40), and (41) for a matterdominated universe (p = 0), we can write the field equation of the scalar field, Eq. (52), as
The righthand side of this equation is negative at the present time t_{0}, for Ω_{0} = 0.3 and Ω_{Ξ0} = 0.65. In other words, the contribution of Ω_{Ξ0} dominates over Ω_{0}. However, the Ξ term plays a decreasingly important role at increasing redshifts with the righthand side of the equation above becoming positive, by adopting H(t) from Eq. (55), at times t earlier than
In the radiationdominated era, however, the righthand side of Eq. (34) is zero, and therefore Ξ may become an important source term in the scalar field equation.
Big Bang nucleosynthesis could potentially constrain the value of Ξ and, more generally, the scalar potential. Scalartensor theories, and specifically BransDickelike models, provide a timevarying effective gravitational constant. This feature alters the rate of expansion of the universe and therefore its temperature, which both contribute to regulate the primordial nucleosynthesis. In particular, the observed abundances of ^{2}H, ^{3}He, and ^{7}Li set a bound on the present matter density, whereas the abundance of ^{4}He constrains the rate of change of φ (Arai et al. 1987). For TeS theories without a selfinteraction potential, there is an attractor mechanism towards GR (Damour & Pichon 1999) as the scalar field, and thus G, remains approximately constant during the radiationdominated epoch. Therefore, in these models, the nucleosynthesis of light elements occurs similarly to the standard cosmological scenario, but with a different expansion rate (Damour & Gundlach 1991; Casas et al. 1992a,b; Serna et al. 1992; Clifton et al. 2005). The primordial nucleosynthesis in the presence of a selfinteraction potential – as in CRG – has been extensively studied in the literature (see, e.g., Arai et al. 1987; Uzan 2003; Larena et al. 2007; Coc et al. 2006; Iocco et al. 2009; Clifton et al. 2012, and references therein), also as a possible solution to the problem of the ^{7}Li abundance. An extensive analysis of whether CRG is consistent with the observed abundances of light elements is left to future work.
5. Conclusion
Refracted gravity was originally introduced to describe the dynamics of galaxies and galaxy systems without the aid of dark matter. The contribution of dark matter to the gravitational field is mimicked by the gravitational permittivity ϵ(ρ), a monotonic function of the mass density ρ (Matsakos & Diaferio 2016). Here, we propose a covariant extension of RG, CRG, and show that it belongs to the family of scalartensor theories, thus inheriting all their general properties. A scalartensor theory is specified by the selfinteracting potential 𝒱(φ) of the scalar field φ and by the general differentiable function 𝒲(φ) appearing in the Lagrangian density. For CRG, we adopt 𝒱(φ) = − Ξφ, with Ξ a normalization constant, and 𝒲(φ) = − 1. In the weakfield limit, this theory correctly reduces to the original phenomenological RG and identifies the gravitational permittivity with the scalar field, with φ = 2ϵ.
When used to describe an expanding and homogeneous universe, the scalar field φ is also responsible for the observed accelerated expansion of the Universe. The cosmological density associated with the cosmological constant in the standard model, , is now replaced by the CRG parameter . It follows that the normalization constant Ξ plays the role of the cosmological constant Λ of the standard model. Moreover, the Hubble diagram of highredshift SNeIa and current observational constraints on the parameter w_{DE} of the equation of state of the effective dark energy, that we derive from the stress–energy tensor associated with the scalar field, suggest that, in a universe with a flat geometry, Ω_{Ξ0} ∼ (1 + Ω_{0})/2, with Ω_{0} the ratio between the baryonic matter and the homogeneous scalar field at the present time. The parameter Ω_{0} can be identified with Ω_{0} = 0.31 of the standard model (Planck Collaboration VI 2020). We find at the 90% confidence level for the CRG+ solution. It thus follows that Ξ and Λ have comparable values.
In addition, in the weakfield limit, Ξ sets an acceleration scale, (2Ξ)^{1/2} ∼ 10^{−10} m s^{−2}, below which RG deviates from Newtonian gravity and appears to describe the dynamics of disk and elliptical galaxies without the aid of dark matter (Cesare et al. 2020, 2022). This acceleration scale is indeed present in real systems (Chae et al. 2020; McGaugh et al. 2018) and is comparable to the acceleration a_{0} introduced in MOND (Milgrom 1983a,b,c). Therefore, being Ξ ∼ Λ, the known relation a_{0} ∼ Λ^{1/2} (Milgrom 1999, 2020) naturally appears in CRG.
CRG provides a connection between phenomena generally attributed to dark matter and dark energy separately and it thus belongs to the family of modified gravity models that connect the two dark sectors within a unified scenario. The same property is currently shared by other models, including some f(R) theories (Sotiriou & Faraoni 2010), “quartessence” theories (Brandenberger et al. 2019) or generalized Chaplygin gas models (Bento et al. 2002; Zhang et al. 2006).
Assessing if the evolution of the universe described by CRG is consistent with observations requires extensive additional work. For example, the power spectrum of the temperature anisotropies of the CMB and the power spectrum of matter density perturbations are required to constrain the Lagrangian density and its parameters (see, e.g., Noller & Nicola 2019; Huterer et al. 2015). Similarly, the analysis of the evolution of the density perturbations, at least in the linear perturbation theory (Di Porto & Amendola 2008; Bueno Sanchez & Perivolaropoulos 2011; Pace et al. 2014; Kofinas & Lima 2017), and of the role of the scalar field and its perturbations on structure formation are of crucial importance to test the viability of CRG.
Finally, since the scalar field drives the accelerated expansion of the Universe, we need to further investigate the connection of CRG with current dark energy models (Pettorino et al. 2005; Capozziello et al. 2006; Frusciante & Perenon 2020). Specifically, we find that the parameter w_{DE} of the equation of state of the effective dark energy depends on redshift, unlike the standard cosmological model. When tested with measures from the upcoming Euclid mission (Amendola et al. 2018), these predictions could discriminate CRG from current dark energy models. We plan to tackle all these issues in future work.
Matsakos & Diaferio (2016) adopt the designation critical density ρ_{c} rather than threshold density ρ_{thr}. Here, we prefer to adopt the latter, to avoid any confusion with the critical density of the Universe, that is used in modelling a homogeneous and isotropic universe. In addition, here ρ_{thr} is explicitly related to the value below which ϵ deviates from unity.
We adopt the sign conventions of Weinberg (1972), with the Ricci tensor given by , and the standard Einstein equations G_{μν} = −8πGT_{μν}.
We choose to work in the JF rather than in the Einstein frame (EF) because, in the JF, the scalar field is interpreted as a modification to the gravitational field (the lefthand side of the field equations) rather than a modification to the stressenergy tensor as in the EF (the righthand side of the field equations). In the weakfield limit, this modification is consistent with the Poisson equation of the nonrelativistic formulation of RG (Eq. (1)), which contains modifications to the field, and not to the source term. As we show in Sect. 3.4, this formulation naturally leads to the identification of the scalar field with the permittivity.
There is actually an ambiguity in the definition of , because Eq. (8) can also be written as , where and simply differ by a factor φ. This ambiguity was studied in Bellucci & Faraoni (2002), where the definition in Eq. (12) is referred to as the effective coupling approach, whereas the latter definition leading to is called the mixed approach. Here, we adopt the former definition, which is analogous to the identification of the scalar field φ in the BransDicke theory with the inverse of an effective gravitational constant G_{eff} ∼ G/φ.
The f(R) models of gravity are also obtained from the Lagrangian of Eq. (14) by setting 𝒢_{3} = 0. This result is expected because of the equivalence between the BransDicke theory and f(R) gravity (Sotiriou & Faraoni 2010).
Despite their GRlike form, the field equations in the EF are not those of standard gravity. For example, in the vacuum solution of the EF, the scalar field acts as an additional gravitational source. Moreover, the extra coupling between the scalar and matter fields introduces both deviations from the geodesic motions of freefalling particles and a stressenergy tensor which is not covariantly conserved (Faraoni 2004; Clifton et al. 2012).
Since a direct measurement of an absolute scale is not possible, experiments are unable to distinguish between the EF and the JF frames. These frames represent two different realizations of the same theory, and physical observables must be equivalent in the two frames (see, e.g., Sotiriou et al. 2008; Postma & Volponi 2014).
A theorem guarantees that in TeS theories endowed with a potential 𝒰 that satisfies the condition d^{2}𝒰(Ψ)/dΨ^{2} > 0, like CRG (Eq. (17)), black holes in vacuum are equivalent to GR black holes (Bekenstein 1995; Sotiriou & Faraoni 2012; Cruz et al. 2017). In CRG, black holes embedded in environment with density sufficiently small to make the screening mechanism ineffective, might in principle develop scalar hair. However, in extended theories of gravity, black holes or compact objects with scalar hair remain viable and their existence can be tested with gravitational wave observations (Sotiriou 2015; Berti et al. 2015; Cardoso et al. 2016; Brito et al. 2017; Barack et al. 2019; Maggiore et al. 2020).
Acknowledgments
We thank Mariano Cadoni for useful discussions and an anonymous referee whose insightful comments contributed to correct and clarify some relevant aspects of the presentation of our results. An early version of this work was the Master thesis in Physics at the University of Torino of APS. We acknowledge partial support from the INFN grant InDark and the Italian Ministry of Education, University and Research (MIUR) under the Departments of Excellence grant L.232/2016. This research has made use of NASA’s Astrophysics Data System Bibliographic Services.
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Appendix A: Derivation of the weakfield limit
Adopting the static weakfield metric components of Eqs. (21)(23), the Christoffel symbols and the resulting components of the Ricci tensor are
The ∇_{μ}∇_{ν}φ components are
and
For a static nonrelativistic fluid, whose pressure p is negligible compared to its density ρ, p ≪ ρ, the equation u^{α}u_{α} = −1 implies u_{0} = ( − g_{00})^{1/2}; therefore, the components of the energymomentum tensor T_{μν} = ρu_{μ}u_{ν} are
and its trace is
Inserting these relations into Eqs. (10) and (11), we obtain the system of equations
Equation (A.16) derives from the 00component: it coincides with the modified Poisson equation of RG when we identify φ/2 with the permittivity ϵ. Equations (A.17) and (A.18) derive from the contracted ijcomponents and the scalar field equation, respectively. Given a density distribution ρ, this system of equations can be solved for the three unknowns, Φ, U, and φ.
A.1. The gravitational field of a spherical source immersed in a background of constant density
For sphericallysymmetric systems, the exact solution of Eq. (A.16) is
where ρ(r) = ρ_{s}(r)+ρ_{bg}, with ρ_{s}(r) being the density profile of the source in a homogeneous background with density ρ_{bg}, and m(< r) is the enclosed mass within radius r. Newtonian gravity is recovered when φ = 2 (i.e. ϵ = 1). For sphericallysymmetric systems, Eqs. (A.17) and (A.18) are less straightforward to solve than Eq. (A.16). In the following, we make a few approximations to derive the generic behaviour of the gravitational field dΦ/dr and the scalar field φ in the two limits of small and large distances from the spherical source.
A.1.1. The gravitational field at small distances from the spherical source
In the Newtonian regime where ρ_{s} ≫ ρ_{bg}, we may approximate φ ≃ 2 − φ_{1}, where φ_{1}≪1. By ignoring all the terms of the order , 𝒪(φ_{1})𝒪(Φ), and 𝒪(φ_{1})𝒪(U), Eq. (A.18) yields
where we also ignored the terms containing Ξ: as we show in Sect. 4, we can identify Ξ with the cosmological constant of the standard model and, due to its small value, we can thus safely ignore these terms in the weakfield limit.
Solving Eq. (A.20) yields the scalar field
If the spherical source has finite size R and finite mass m, Eq. (A.21) yields, beyond R,
and Eq. (A.19) reads as follows:
In the vacuum ρ_{bg} = 0, and the RG field reduces to the Newtonian field for φ = 2, in agreement with Eq. (A.16), and it is larger than the Newtonian field for φ < 2.
A.1.2. The gravitational field at large distances from the spherical source
If the mass density profile of the extended spherical source decreases with increasing r, at sufficiently large distances from the source, the background density ρ_{bg} approximates the mean density of the Universe, assuming that the source is isolated. We can thus treat the contribution of the source, ρ_{s}, as a small perturbation to the total density field ρ = ρ_{bg} + ρ_{s}, with ρ_{s} ≪ ρ_{bg}. We can similarly assume that the scalar field reaches a mean cosmic value φ_{bg} = const and write the scalar field as φ = φ_{bg} + φ_{1}, with φ_{1} ≪ φ_{bg}. In the limit ρ_{s} → 0 and φ_{1} → 0, Eqs. (A.17) and (A.18) simplify to
We can derive how the scalar field approaches the cosmic constant value φ_{bg} by assuming that Eq. (A.24) already holds at distances where φ is not yet asymptotically constant, i.e. ∇^{2}U ≃ 2πGρ/φ. By adding up Eqs. (A.17) and (A.18), and using Eq. (A.16), we obtain
In spherical symmetry, we can recast this equation as
which yields an implicit expression for the gravitational field at large distances from the source. Hereafter, we consider the above equation with the minus sign, so that dφ/dr < 0.
We now explore the two limits of the large and small gravitational field dΦ/dr at large distances from the source. In the limit dΦ/dr ≫ (2Ξ − 8πGρ/φ)^{1/2}, Eq. (A.27) reduces to dφ/dr ∝ −2φ dΦ/dr, a dependence broadly approaching the result derived in the previous section for small distances from the source (Eqs. A.19 and A.21).
In the limit dΦ/dr ≪ (2Ξ − 8πGρ/φ)^{1/2}, Eq. (A.27) reduces to
which shows that φ = const when φ_{bg} = 4πGρ/Ξ, and thus ρ = ρ_{bg}.
The relevant result of this analysis is that the transition between the two regimes of large and small gravitational field, dΦ/dr, takes place at the acceleration scale
We discuss this result in Sect. 3.4.1.
At distances where φ is not yet constant, namely φ(r) = φ_{bg} + φ_{1}(r) and ρ(r) = ρ_{bg} + ρ_{s}(r), we can write Eq. (A.28) as
Solving Eq. (A.30) for φ requires an assumption on the form of ρ_{s}(r). For example, we can assume that the density profile ρ_{s}(r) of the source drops exponentially, i.e. ρ_{s}(r) = ρ_{edge}e^{−r/redge}, where r_{edge} is a characteristic scale of the source and ρ_{edge} is an appropriate normalization constant. In this case, φ_{1}(r) = φ_{edge}e^{−r/redge}, with φ_{edge} = 4πGρ_{edge}/Ξ, is a solution of Eq. (A.30) if we ignore higherorder terms. We thus obtain the scalar field
Appendix B: Derivation of the modified Friedmann equations and of the scalar field equation
Based on the FLRW metric, Eq. (31), the Christoffel symbols are^{12}
The components of the Ricci tensor are
and the three terms ∇_{μ}∇_{ν}φ, ∇^{α}∇_{α}φ, and ∇_{μ}φ∇_{ν}φ are
From the stressenergy tensor of a perfect fluid, T_{μν} = (ρ+p)u_{μ}u_{ν} + pg_{μν}, we get
By combining the above results, the timetime component of the modified Einstein field equations, Eq. (10), is
which gives the first modified Friedmann equation
From the rrcomponent of Eq. (10),
we obtain the second modified Friedmann equation
The θθ and ϕϕcomponents reduce to the same expression.
The last equation is the scalar field equation
which becomes
Appendix C: Derivation of the time derivative of the Hubble parameter [Eq. (54)]
Adding up Eqs. (47) and (49) yields
By using the definitions of q and ζ, Eqs. (42) and (46), together with Eq. (52), the above equation reduces to
With Eqs. (52) and (36) and the expression , we can recast Eq. (40) as
and similarly (Eq. 41),
Equation (C.2) thus becomes
and, by adding 3ȧ^{2}/a^{2} on both sides, we obtain
The righthand side can be rewritten as
which leads to the diffential equation for H
Appendix D: Analytical derivation of the luminosity distance
The general definition of the luminosity distance is
We compute its exact form by using the solution of the scale factor in Eq. (57), which we rewrite here as
where the upper and lower signs correspond to the CRG+ and CRG– solutions, respectively.
Inverting Eq. (D.2) yields
where we defined
Differentiating Eq. (D.3) yields
The luminosity distance in Eq. (D.1) then becomes
If we change variable by defining
the luminosity distance becomes
with
Equation (D.8) corresponds to two equations, according to the sign of the parameter A defined in Eq. (D.4).
Appendix E: Estimate of Ξ from SNIa data
We now use the expression of the luminosity distance D_{L} derived in the previous section and the SNIa data from the Supernova Cosmology Project Union 2.1 Compilation (Suzuki et al. 2012) to infer constraints on the value of Ω_{Ξ0}. We compute the quantity
where μ_{i} = m_{i} − M is the observed distance modulus, μ(Ω_{0}, Ω_{Ξ0}, z_{i}) = 25 + 5log_{10}(D_{L}/Mpc) is the expected distance modulus at each SNIa redshift z_{i}, σ_{μi} is the uncertainty on each measured distance modulus, and the sum is over the N = 580 SNeIa of the sample. For estimating D_{L}, we adopt H_{0} = 67.7 km s^{−1} Mpc^{−1}.
Figure E.1 shows that the minimum values of occur along the relation Ω_{Ξ0} ∼ (1 + Ω_{0})/2, namely close to the upper limit of Ω_{Ξ0} (Eq. 63). In other words, Ω_{Ξ0} is fully set by the ratio between the baryonic content of the Universe and the homogeneous scalar field φ at the present time (Eq. 40). The result Ω_{Ξ0} ∼ (1 + Ω_{0})/2 supports the constraints on Ω_{Ξ0} that we derived from the estimates of the dark energy parameter w_{0} discussed in Sect. 4.3.
Fig. E.1. Function , the difference between (Eq. E.1) and its minimum value, for the CRG+ (left panel) and the CRG– (right panel) solutions. Under the assumption of Gaussian random errors, the 68%, 90%, 95%, and 99% confidence regions of the two parameters are the area where Δ(Ω_{0}, Ω_{Ξ0}) < 2.3, 4.6, 6.2, and 9.2, respectively. The 90% confidence regions are thus 0.43 + 0.44Ω_{0} ≲ Ω_{Ξ0} ≤ 0.5(1 + Ω_{0}) for the CRG+ solution and max{0, ( − 0.26 + 0.63Ω_{0})} ≲ Ω_{Ξ0} ≤ 0.5(1 + Ω_{0}) for the CRG– solution. In the two panels, there are no solutions in the white top left area. For the CRG+ solution shown in the left panel, for an easier comparison with the CRG– solution shown in the right panel, Δ(Ω_{0}, Ω_{Ξ0}) is not reported in the white bottom right area where Δ > 7. 
The CRG+ model yields more stringent constraints than the CRG– model. For CRG+, the 90% confidence limit for Ω_{Ξ0} is 0.43 + 0.44Ω_{0} ≲ Ω_{Ξ0} ≤ 0.5(1 + Ω_{0}). Adopting the value Ω_{0} = 0.31 (Planck Collaboration VI 2020), we thus find at the 90% confidence level. For CRG–, the 90% confidence limit is substantially wider: max{0, ( − 0.26 + 0.63Ω_{0})} ≲ Ω_{Ξ0} ≤ 0.5(1 + Ω_{0}). At this confidence level we find for Ω_{0} = 0.31. Figure E.2 compares the SNIa data in the Hubble diagram with the two CRG solutions. For the adopted values Ω_{0} = 0.31 and Ω_{Ξ0} = 0.65, the data are unable to distinguish between the two solutions.
Fig. E.2. Hubble diagram of the SNIa data (open circles with error bars) and the CRG+ (solid line) and CRG– (dashed line) solutions. For the two curves, we adopt H_{0} = 67.7 km s^{−1} Mpc^{−1}, Ω_{0} = 0.31, and Ω_{Ξ0} = 0.65. 
Appendix F: The equation of state of the effective dark energy
The equation of state of the effective dark energy can be calculated by inserting Eqs. (65)(66) into Eqs. (71)(72). For a dustdominated universe, we can use Eqs. (52) and (54) to calculate φ, , and the following quantities
The combination of these results, together with the two definitions of the cosmological parameters Eqs. (40)(41), yields
The upper and lower signs refer to the CRG+ and the CRG– solutions, respectively. We use this equation in Sect. 4.3.
All Figures
Fig. E.1. Function , the difference between (Eq. E.1) and its minimum value, for the CRG+ (left panel) and the CRG– (right panel) solutions. Under the assumption of Gaussian random errors, the 68%, 90%, 95%, and 99% confidence regions of the two parameters are the area where Δ(Ω_{0}, Ω_{Ξ0}) < 2.3, 4.6, 6.2, and 9.2, respectively. The 90% confidence regions are thus 0.43 + 0.44Ω_{0} ≲ Ω_{Ξ0} ≤ 0.5(1 + Ω_{0}) for the CRG+ solution and max{0, ( − 0.26 + 0.63Ω_{0})} ≲ Ω_{Ξ0} ≤ 0.5(1 + Ω_{0}) for the CRG– solution. In the two panels, there are no solutions in the white top left area. For the CRG+ solution shown in the left panel, for an easier comparison with the CRG– solution shown in the right panel, Δ(Ω_{0}, Ω_{Ξ0}) is not reported in the white bottom right area where Δ > 7. 

In the text 
Fig. E.2. Hubble diagram of the SNIa data (open circles with error bars) and the CRG+ (solid line) and CRG– (dashed line) solutions. For the two curves, we adopt H_{0} = 67.7 km s^{−1} Mpc^{−1}, Ω_{0} = 0.31, and Ω_{Ξ0} = 0.65. 

In the text 
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