Gross–Pitaevskii–Poisson Equations from a ξRϕ⁴ Non-Minimal Scalar-Curvature Coupling

Abstract

In cosmological scenarios where the Peccei–Quinn symmetry is broken after inflation, small-scale axion field inhomogeneities can undergo gravitational collapse, leading to the formation of bound structures. The dynamics of these systems are commonly described using cosmological perturbation theory applied to the Einstein–Klein–Gordon equations. In the non-relativistic regime, this description reduces to the Gross–Pitaevskii–Poisson or Schrödinger–Poisson equations, depending on whether axion self-interactions are included. In this work, we extend the axion’s relativistic action by introducing a non-minimal scalar-curvature coupling of the form $\xi R{\varphi }^4$, which effectively induces a gravitationally mediated pairwise interaction. By performing a perturbative expansion and subsequently taking the non-relativistic limit, we derive a modified set of evolution equations governing the early stages of axion structure formation.

1. Introduction

Dark matter (DM) remains as one of the clearest indications of physics beyond the Standard Model (SM), shaping developments across particle theory, astrophysics, and cosmology. The axion, originally just a byproduct of the Peccei–Quinn solution to the strong CP problem in Quantum Chromodynamics (QCD) [1,2], has gained remarkable attention as a DM candidate in recent years in both experimental and theoretical studies [3,4,5]. At energy scales below which QCD non-perturbative effects become significant, the axion can be effectively described by a real scalar field $\varphi$ under the Klein–Gordon action,

$$S=-\int d^{4}x\left({\displaystyle \frac{1}{2}}{\partial }_{\mu }\varphi {\partial }^{\mu }\varphi +V\left(\varphi \right)\right),$$

(1)

with a non-trivial interaction potential $V\left(\varphi \right)$ that can be derived from, for instance, chiral perturbation theory [6]. Since $V\left(\varphi \right)$ is an even function of $\varphi$, it can be expanded in powers of ${\varphi }^2$ around 0, with the quadratic term fixing the axion mass $m_a$. It can be shown that $m_a=\sqrt{\chi }/f_a$, where $\chi$ is the topological susceptibility of QCD and $f_a$ is the axion decay constant. The most precise computation to date yields [5,7]:

$$m_a=5.691\left(51\right)\left({\displaystyle \frac{{10}^{12}\mathrm{GeV}}{f_a}}\right)\mu \mathrm{eV}.$$

(2)

giving a viable DM mass window for the axion of $m_a\in ({10}^{-6}$–${10}^{-3})\mathrm{eV}$ [8]. The higher-order terms of $V\left(\varphi \right)$ encode the axion self-interaction couplings. Since $m_a$ and $f_a$ dictate most of the axion’s properties and interactions, it is practical to express the higher-order terms using these parameters,

$$V\left(\varphi \right)={\displaystyle \frac{1}{2}}{m}_a^2{\varphi }^2+{\left(m_a{f}_a\right)}^2\sum _{n=2}^{\infty }{\displaystyle \frac{{\lambda }_{2n}}{\left(2n\right)!}}{\left({\displaystyle \frac{\varphi }{f_a}}\right)}^{2n},$$

(3)

where ${\lambda }_{2n}$ are dimensionless coupling constants of $\mathcal{O}\left(1\right)$. The ${\mathrm{Z}}_2$ symmetry of $V\left(\varphi \right)$ ensures that the number of axions in a scattering reaction is conserved modulo 2. Among these couplings, the axion-axion interactions are predominant within the pure axion sector. Notably, these pairwise interactions are attractive. The parametrization in Equation (3) indicates that $m_a^2/f_a^2$ acts as a quantum-loop factor. Within the axion DM mass window, this factor lies in $m_a^2/f_a^2\in ({10}^{-52}$–${10}^{-44})$. Such an exceedingly small value emphasizes the negligible effect of quantum-loop corrections, therefore validating the use of classical field theory for the axion description [9].

While most axion detection efforts focus on direct detection experiments, several indirect search strategies have been proposed. These approaches seek to detect astrophysical and cosmological effects arising from the axion field presence. In particular, if axion inhomogeneities survive the inflationary period, localized bound structures can emerge from them. Axion miniclusters, also known as axion minihalos, are gravitationally bound and virialized objects decoupled from the Universe’s expansion that originate from density fluctuations around the time of matter-radiation equality [9,10]. Axion stars, by contrast, are stable bound states where the gravitational attraction is balanced by gradient pressure. These configurations are categorized into two branches: a dilute branch, where stability involves only gravity and kinetic pressure, and a dense branch, where self-interactions become significant beyond a critical mass threshold. In addition, several authors have suggested the existence of self-bound configurations known as axitons [9,10,11].

DM halo regions hosting axion miniclusters may reach mass densities several orders above the halo average. Such pronounced densities could give rise to detectable gravitational lensing effects, offering an avenue to constrain their abundance. Nevertheless, further simulations are required to assess the feasibility of this strategy. An alternative detection strategy relies on magnetometer networks sensitive enough to detect the passage of compact DM objects through the Earth’s vicinity. Further observable signatures may result from head-on collisions of axion stars with other astrophysical objects [5,9,12]. Recently, the James Webb Space Telescope has reported observations supporting the existence of dark stars, a class of early stellar objects composed of hydrogen and helium clouds whose gravitational collapse is delayed by DM heating mechanisms. They are typically modeled using WIMPs or self-interacting DM [13,14]. An analogous concept, termed a Hydrogen axion star, in which dense axion stars serve as the heat source, was proposed in Ref. [15]. Consequently, modeling the evolution and the observable implications of compact axion configurations continues to be a promising DM research avenue.

Since DM is mostly non-relativistic, the axion cosmological role is best described by a non-relativistic effective field theory (NREFT). Several methods have been developed to construct this low-energy limit. Herein, an approach involving an explicit field redefinition that maps the real scalar field to a complex scalar field is adopted [16,17,18,19]. Moreover, an accurate model of axion gravitational collapse must rely on General Relativity (GR). While the unification of Einstein’s theory of gravity and particle physics remains unsolved, curvature effects are often incorporated via the minimal coupling prescription. This approach leads to the Einstein–Klein–Gordon equations that are widely used to characterize macroscopic axion structures. Notably, cosmological perturbation theory is used to study their formation, with the low-energy limit subsequently reached through the aforementioned field redefinition. This results in the Gross–Pitaevskii–Poisson system when self-interactions are present, and the Schrödinger–Poisson equations when they are neglected [4]. From an effective field theory perspective in curved spacetime, scalar fields are generically expected to couple non-minimally to curvature invariants, even if such terms are absent at tree level [20,21]. Radiative corrections and renormalization in a gravitational background typically induce scalar–curvature couplings, motivating the exploration of extensions beyond minimal coupling. In this context, non-minimal interactions may alter the effective gravitational dynamics of scalar condensates without modifying the underlying cosmological background evolution.

Due to the inherently attractive nature of axion pairwise interactions, gravitationally bound configurations cannot grow as extensively as they might if the interactions were repulsive. Large-scale objects require some form of repulsive force or pressure to counteract gravitational collapse. Heisenberg’s uncertainty principle provides this support for bosons, although its stabilization impact is weaker than the one provided by Pauli’s exclusion principle in neutron stars. Consequently, the maximum mass of bosonic stars with attractive self-interactions is relatively low [22,23]. In this work, a non-minimal coupling term proportional to $R{\varphi }^4$, where R is the scalar of curvature, is introduced as it provides a simple extension that preserves the scalar field’s self-interaction structure while introducing an effective gravitationally mediated interaction at the level of the condensate. Furthermore, the inclusion of the $R{\varphi }^4$ term may facilitate the formation of more complex axion structures. Since R varies with the local geometry, the non-minimal coupling term can produce either attractive or repulsive effects, potentially enhancing both the stability and mass range of axion stars. The novelty of this work lies in the derivation of a modified non-relativistic framework in which the effects of the non-minimal scalar-curvature coupling are consistently incorporated at the level of the Gross–Pitaevskii–Poisson equations. After establishing the non-relativistic equations relevant to the early-time behavior of structure formation, the Hamiltonian that governs a system of N axions is obtained. This provides a foundation for analyzing the thermodynamics of the axion field during gravitational collapse, as the corresponding partition function can now be computed.

The structure of this manuscript is as follows. In Section 2, we introduce the relativistic action and field equations, including the non-minimal coupling term. In Section 3 and Section 4, we perform the perturbative expansion and derive the corresponding non-relativistic limit, respectively. In Section 5, we present and discuss the resulting modified Gross–Pitaevskii–Poisson equations. Finally, we summarize our results and outline possible directions for future work in Section 6.

2. Gravity Coupling

The simplest way to construct an action that remains invariant under general coordinate transformations is to begin with the flat-spacetime action and to apply the minimal coupling prescription [24,25]:

• Replace all ordinary derivatives ${\partial }_{\mu }$ with covariant derivatives ${\nabla }_{\mu }$.
• Substitute the Minkowski metric ${\eta }_{\mu \nu }$ with the general metric $g_{\mu \nu }$.
• Swap $d^{4}x$ with the invariant volume element $d^{4}x\sqrt{-g}$, where $g=det\left(g_{\mu \nu }\right)$.

Applying this procedure to the axion Lagrangian in Equation (1) yields

$$S_m=-\int d^{4}x\sqrt{-g}\left({\displaystyle \frac{1}{2}}{\nabla }^{\mu }\varphi {\nabla }_{\mu }\varphi +V\left(\varphi \right)\right).$$

(4)

The subscript m indicates that this action corresponds to the matter sector of the system. Meanwhile, the dynamics of the gravitational field are described by the Einstein–Hilbert action1:

$$S_g={\displaystyle \frac{1}{16\pi G}}\int d^{4}x\sqrt{-g}R,$$

(5)

where G is the gravitational constant and R, as mentioned above, the scalar curvature. The total action, $S_{tot}=S_m+S_g$, upon variation with respect to $\delta g^{\mu \nu }$, yields the Einstein’s field equations2:

$$G_{\mu \nu }=8\pi G{T}_{\mu \nu },$$

(6)

where $G_{\mu \nu }=R_{\mu \nu }-{\displaystyle \frac{1}{2}}R{g}_{\mu \nu }$ is the Einstein tensor, $R_{\mu \nu }$ is the Ricci tensor and $T_{\mu \nu }={\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -g_{\mu \nu }\left({\displaystyle \frac{1}{2}}{\partial }^{\alpha }\varphi {\partial }_{\alpha }\varphi +V\left(\varphi \right)\right)$ is the stress-energy tensor. On the other hand, varying $S_{tot}$ with respect to $\delta \varphi$ results in the covariant generalization of the Klein–Gordon equation,

$${\nabla }^{\mu }{\nabla }_{\mu }\varphi -{\displaystyle \frac{\partial V}{\partial \varphi }}=0.$$

(7)

In contrast to the special relativistic case, plane wave solutions and Fourier analysis cannot be generally employed. Instead, it has to be solved on a case-by-case basis depending on the specific symmetries of the metric [24]. When $V\left(\varphi \right)$ includes only the mass term, Equations (6) and (7) are denoted as the Einstein–Klein–Gordon equations.

While minimal coupling provides a first approximation of how curvature affects a physical system, it is not uncommon to go beyond the standard covariant substitutions. These non-minimal couplings arise naturally in various settings, including QFT in curved spacetime, Higgs inflation models, modified gravity theories, and the low-energy effective actions derived from superstring theory [24,26,27,28,29,30]. Any extra modification must be invariant under diffeomorphisms and respect the internal symmetries of the theory. More often than not, the additional terms involve the Riemann tensor $R_{\mu \nu \rho \sigma }$ and its contractions, which ensures that they vanish in the Minkowski spacetime. Since the ${\varphi }^4$ term in $V\left(\varphi \right)$ characterizes the axion pair self-interactions, it is quite natural to include a term proportional to $R{\varphi }^4$ to incorporate gravitationally mediated pair interactions into the framework. Lower-dimensional operators such as $R{\varphi }^2$ correspond to the standard non-minimal coupling and primarily induce curvature-dependent corrections to the effective mass. In contrast, $R{\varphi }^4$ directly modifies the quartic self-interaction structure relevant for condensate dynamics. From an effective field theory perspective, this term constitutes the leading higher-dimensional curvature–matter operator that alters axion pairwise interactions while preserving the Einstein–Hilbert gravitational sector. As a result, the total action becomes

$$\begin{array}{cc}\hfill S_{tot}=& -\int d^{4}x\sqrt{-g}\left({\displaystyle \frac{1}{2}}{\partial }^{\mu }\varphi {\partial }_{\mu }\varphi +V\left(\varphi \right)+\xi R{\varphi }^4\right)+{\displaystyle \frac{1}{16\pi G}}\int d^{4}x\sqrt{-g}R\hfill \\ \hfill =& -\int d^{4}x\sqrt{-g}\left({\displaystyle \frac{1}{2}}{\partial }^{\mu }\varphi {\partial }_{\mu }\varphi +V\left(\varphi \right)\right)+{\displaystyle \frac{1}{16\pi G}}\int d^{4}x\sqrt{-g}f\left(\varphi \right)R,\hfill \end{array}$$

(8)

where $\xi$ is a constant and $f\left(\varphi \right)=1-\left(16\pi G\right)\xi {\varphi }^4$.

This type of non-minimal coupling can be described in two mathematically equivalent frames [25]. In the Jordan frame, $f\left(\varphi \right)$ is retained explicitly in the action. Alternatively, the canonical Einstein-Hilbert action is recovered in the Einstein frame by performing a conformal transformation of the metric,

$${g^{\prime }}_{\mu \nu }=\left(16\pi G\right)f\left(\varphi \right)g_{\mu \nu }.$$

(9)

Cosmological perturbations can be carried out in either frame. Nevertheless, physical observables are more straightforward to interpret in the Jordan frame since the metric is not rescaled. For this reason, Equation (9) is not going to be employed in this work.

Requiring that $\delta S_{\mathrm{tot}}=0$ under arbitrary variations of $\delta g^{\mu \nu }$, while systematically discarding surface terms as in the Einstein–Hilbert action, yields a modified form of Einstein’s field equations [25],

$$f\left(\varphi \right)G_{\mu \nu }-8\pi G{T}_{\mu \nu }={\nabla }_{\mu }{\nabla }_{\nu }f-g_{\mu \nu }{\nabla }^{\alpha }{\nabla }_{\alpha }f.$$

(10)

Or equivalently,

$$\left({\displaystyle \frac{1}{8\pi G}}-2\xi {\varphi }^4\right){{\mathrm{G}}^{\mu }}_{\nu }-{{\mathrm{T}}^{\mu }}_{\nu }+2\xi {{\mathcal{Z}}^{\mu }}_{\nu }=0,$$

(11)

where

$$\begin{array}{c}{\mathcal{Z}}_{\mu \nu }={\nabla }_{\mu }{\nabla }_{\nu }{\varphi }^4-g_{\mu \nu }{\nabla }^{\alpha }{\nabla }_{\alpha }{\varphi }^4.\end{array}$$

(12)

Meanwhile, varying Equation (8) with respect to $\delta \varphi$ implies that

$${\nabla }^{\mu }{\nabla }_{\mu }\varphi -{\displaystyle \frac{\partial V}{\partial \varphi }}-4\xi R{\varphi }^3=0.$$

(13)

3. The Newtonian Gauge

Cosmological perturbation theory studies the evolution of small deviations from a homogeneous matter distribution within a background metric. It is crucial for characterizing the growth of density perturbations, the temperature anisotropies and polarization patterns of the Cosmic Microwave Background, and the evolution of gravitational waves. While gravitational collapse can be partially described using Newtonian gravity and classical hydrodynamics, e.g., the Jeans instability criterion, such treatments neglect cosmic expansion. Therefore, a fully consistent description of perturbation growth requires GR [25,31]. In the early Universe, perturbations in both the energy density, $\delta \rho$, and the metric, $\delta g_{\mu \nu }$, are small compared to their background values, i.e., $\delta \rho \ll \rho$ and $|\delta g_{\mu \nu }|\ll |g_{\mu \nu }|$. Under these conditions, a linearized theory of perturbations is both accurate and highly effective. However, linear theory ceases to apply once $\delta \rho /\rho \sim 1$, at which point alternative methods, such as large-scale numerical simulations, are required [31].

It is often convenient to express cosmological perturbations in terms of conformal time $\eta$ instead of cosmological time t. The relation between these time coordinates is

$$a\left(\eta \right)d\eta =dt,$$

(14)

where $a\left(\eta \right)$ is the dimensionless scale factor. Hence, the Hubble parameter is

$$H\left(\eta \right)={\displaystyle \frac{1}{a}}{\displaystyle \frac{da}{dt}}={\displaystyle \frac{1}{a^2}}{\displaystyle \frac{da}{d\eta }}.$$

(15)

The Friedmann–Lemaître–Robertson–Walker (FLRW) metric with zero spatial curvature (k = 0) describes a spatially flat, homogeneous, and isotropic expanding Universe,

$$d{s}^2=a^2\left(\eta \right)\left[-d{\eta }^2+{\delta }_{ij}d{x}^{i}d{x}^j\right]=a^2\left(\eta \right){\eta }_{\mu \nu }d{x}^{\mu }d{x}^{\nu }.$$

(16)

A perturbed version of this metric can be written as

$$d{s}^2=g_{\mu \nu }\left(x\right)d{x}^{\mu }d{x}^{\nu }=a^2\left(\eta \right)\left({\eta }_{\mu \nu }+h_{\mu \nu }\left(x\right)\right)d{x}^{\mu }d{x}^{\nu },$$

(17)

where $h_{\mu \nu }\left(x\right)$ encodes the perturbations and, therefore, $|h_{\mu \nu }|\ll 1$. It is useful to use an overbar to denote background quantities, e.g., the background metric ${\overline{g}}_{\mu \nu }\left(\eta \right)=a^2\left(\eta \right){\eta }_{\mu \nu }$. Analogously to the metric tensor, the stress energy tensor $T_{\mu \nu }$ and the Einstein tensor $G_{\mu \nu }$ are decomposed into a homogeneous background and a perturbative portion,

$$\begin{array}{ccc}\hfill T_{\mu \nu }\left(x\right)={\overline{T}}_{\mu \nu }\left(\eta \right)+\delta T_{\mu \nu }\left(x\right),& \hfill & \hfill G_{\mu \nu }\left(x\right)={\overline{G}}_{\mu \nu }\left(\eta \right)+\delta G_{\mu \nu }\left(x\right).\end{array}$$

(18)

Under the minimal coupling prescription, ${\overline{T}}_{\mu \nu }$ and ${\overline{G}}_{\mu \nu }$ satisfy3

$$\begin{array}{ccc}\hfill {{\overline{G}}^{\mu }}_{\nu }=8\pi G{{\overline{T}}^{\mu }}_{\nu },& \hfill & \hfill {\overline{\nabla }}_{\mu }{{\overline{T}}^{\mu }}_{\nu }=0.\end{array}$$

(19)

Meanwhile, the perturbations are described at linear order by

$$\begin{array}{ccc}\hfill \delta {G^{\mu }}_{\nu }=8\pi G\delta {T^{\mu }}_{\nu },& \hfill & \hfill \delta \left({\nabla }_{\mu }{T^{\mu }}_{\nu }\right)=0.\end{array}$$

(20)

Furthermore, the background and perturbative counterparts of Equation (7) must be included. Since the matter content is described by $\varphi \left(x\right)=\overline{\varphi }\left(\eta \right)+\delta \varphi \left(x\right)$, where $\left|\delta \varphi \right|\ll |\overline{\varphi }|$, $\delta {T^{\mu }}_{\nu }$ depends linearly on $\delta \varphi$ and $h_{\mu \nu }$. Meanwhile, $\delta {G^{\mu }}_{\nu }$ includes only terms linear to $h_{\mu \nu }$.

When focusing on the growth of matter perturbations, only two of the ten degrees of freedom in $h_{\mu \nu }$ remain physically relevant, after accounting for gauge freedom. The perturbed metric is written as

$$d{s}^2=a^2\left(\eta \right)\left[-\left(1+2\mathsf{\Phi }\right)d{\eta }^2+\left(1-2\mathsf{\Psi }\right){\delta }_{ij}d{x}^{i}d{x}^j\right],$$

(21)

where $\mathsf{\Phi }\left(x\right)$ and $\mathsf{\Psi }\left(x\right)$ are scalars which follow that $\left|\mathsf{\Phi }\right|,\left|\mathsf{\Psi }\right|\ll 1$. This line element is known in literature as the Newtonian gauge [25,31]. In the absence of anisotropic stress, i.e., when the traceless part of $\delta T_{ij}$ vanishes, the linearized Einstein equations naturally enforce $\mathsf{\Phi }=\mathsf{\Psi }$. An interesting scenario to outline is the evolution of spherically symmetric matter perturbations. In this context, it is advantageous to express the spatial components in spherical coordinates; thereby, the Newtonian gauge becomes

$$d{s}^2=a^2\left(\eta \right)\left[-\left(1+2\mathsf{\Phi }\right)d{\eta }^2+\left(1-2\mathsf{\Psi }\right)\left(d{r}^2+r^{2}d{\theta }^2+r^2{sin}^2\left(\theta \right)d{\phi }^2\right)\right].$$

(22)

In the non-minimal coupling framework proposed in this work, the linearization must be applied to Equations (11) and (13). The $\xi$-dependent perturbations are linear to both $\delta \varphi$ and $h_{\mu \nu }$. In standard linear perturbation theory, a direct computation of ${G^{\mu }}_{\nu }$ can be done by retaining only the first-order perturbative terms through the calculation. This procedure involves using the background metric ${\overline{g}}_{\mu \nu }$ to raise and lower indices. However, a simplified index manipulation of ${{\mathcal{Z}}^{\mu }}_{\nu }$ may lead to overlooked contributions. To avoid such omissions, all relevant geometric quantities were derived without any perturbative approximation, leaving the linearization as a final step. The complete expressions are presented in Appendix A. The components of the linearized Einstein tensor are:

$$\begin{array}{cc}\hfill a^2{G^{\eta }}_{\eta }=& -3a^2{H}^2\left(1-2\mathsf{\Phi }\right)+6aH{\partial }_{\eta }\mathsf{\Psi }-2{\nabla }^2\mathsf{\Psi },\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill {G^{\eta }}_r=-{G^r}_{\eta }=& -{\displaystyle \frac{2}{a^2}}{\partial }_r\left(aH\mathsf{\Phi }+{\partial }_{\eta }\mathsf{\Psi }\right),\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill {G^{\eta }}_{\theta }=-r^2{G^{\theta }}_{\eta }=& -{\displaystyle \frac{2}{a^2}}{\partial }_{\theta }\left(aH\mathsf{\Phi }+{\partial }_{\eta }\mathsf{\Psi }\right),\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill {G^{\eta }}_{\phi }=-r^2{sin}^2\left(\theta \right){G^{\phi }}_{\eta }=& -{\displaystyle \frac{2}{a^2}}{\partial }_{\phi }\left(aH\mathsf{\Phi }+{\partial }_{\eta }\mathsf{\Psi }\right),\hfill \\ \hfill a^2{G^r}_r=& {\rm Y}\left(1-2\mathsf{\Phi }\right)+2{\partial }_{\eta }^2\mathsf{\Psi }+2aH{\partial }_{\eta }\left(\mathsf{\Phi }+2\mathsf{\Psi }\right)\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill & +\left({\nabla }^2-{\partial }_r^2\right)\left(\mathsf{\Phi }-\mathsf{\Psi }\right),\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill {G^r}_{\theta }=r^2{G^{\theta }}_r=& {\displaystyle \frac{r}{a^2}}{\partial }_r\left({\displaystyle \frac{{\partial }_{\theta }\left(\mathsf{\Psi }-\mathsf{\Phi }\right)}{r}}\right),\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill {G^r}_{\phi }=r^2{sin}^2\left(\theta \right){G^{\phi }}_r=& {\displaystyle \frac{r}{a^2}}{\partial }_r\left({\displaystyle \frac{{\partial }_{\phi }\left(\mathsf{\Psi }-\mathsf{\Phi }\right)}{r}}\right),\hfill \\ \hfill a^2{G^{\theta }}_{\theta }=& {\rm Y}\left(1-2\mathsf{\Phi }\right)+2{\partial }_{\eta }^2\mathsf{\Psi }+2aH{\partial }_{\eta }\left(\mathsf{\Phi }+2\mathsf{\Psi }\right)\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill & +\left({\nabla }^2-{\displaystyle \frac{{\partial }_r}{r}}-{\displaystyle \frac{{\partial }_{\theta }^2}{r^2}}\right)\left(\mathsf{\Phi }-\mathsf{\Psi }\right),\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill {G^{\theta }}_{\phi }={sin}^2\left(\theta \right){G^{\phi }}_{\theta }=& {\displaystyle \frac{sin\left(\theta \right)}{a^2{r}^2}}{\partial }_{\theta }\left({\displaystyle \frac{{\partial }_{\phi }\left(\mathsf{\Psi }-\mathsf{\Phi }\right)}{sin\left(\theta \right)}}\right),\hfill \\ \hfill a^2{G^{\phi }}_{\phi }=& {\rm Y}\left(1-2\mathsf{\Phi }\right)+2{\partial }_{\eta }^2\mathsf{\Psi }+2aH{\partial }_{\eta }\left(\mathsf{\Phi }+2\mathsf{\Psi }\right)\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill & +\left({\partial }_r^2+{\displaystyle \frac{{\partial }_r}{r}}+{\displaystyle \frac{{\partial }_{\theta }^2}{r^2}}\right)\left(\mathsf{\Phi }-\mathsf{\Psi }\right),\hfill \end{array}$$

(32)

where ${\rm Y}\left(\eta \right)=a^2{H}^2-{\displaystyle \frac{2}{a}}{\displaystyle \frac{d^{2}a}{d{\eta }^2}}$. Besides retaining only terms linear in $\mathsf{\Phi }$ and $\mathsf{\Psi }$, it is necessary to exclude all contributions dependent on the metric perturbations that are proportional to the spatial derivatives of $\varphi$. Upon decomposing $\varphi \left(x\right)=\overline{\varphi }\left(\eta \right)+\delta \varphi \left(x\right)$, such terms are clearly of second order and thus negligible. This consideration is particularly relevant when computing the linearized components of ${{\mathcal{Z}}^{\mu }}_{\nu }$,

$$\begin{array}{cc}\hfill a^2{{\mathcal{Z}}^{\eta }}_{\eta }=& 3{\partial }_{\eta }{\varphi }^4\left(aH\left(1-2\mathsf{\Phi }\right)-{\partial }_{\eta }\mathsf{\Psi }\right)-{\nabla }^2{\varphi }^4,\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill {{\mathcal{Z}}^{\eta }}_r=-{{\mathcal{Z}}^r}_{\eta }=& {\displaystyle \frac{1}{a^2}}\left(aH{\partial }_r{\varphi }^4+{\partial }_r\mathsf{\Phi }{\partial }_{\eta }{\varphi }^4-{\partial }_{\eta }{\partial }_r{\varphi }^4\right),\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill {{\mathcal{Z}}^{\eta }}_{\theta }=-r^2{{\mathcal{Z}}^{\theta }}_{\eta }=& {\displaystyle \frac{1}{a^2}}\left(aH{\partial }_{\theta }{\varphi }^4+{\partial }_{\theta }\mathsf{\Phi }{\partial }_{\eta }{\varphi }^4-{\partial }_{\eta }{\partial }_{\theta }{\varphi }^4\right),\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill {{\mathcal{Z}}^{\eta }}_{\phi }=-r^2{sin}^2\left(\theta \right){{\mathcal{Z}}^{\phi }}_{\eta }=& {\displaystyle \frac{1}{a^2}}\left(aH{\partial }_{\phi }{\varphi }^4+{\partial }_{\phi }\mathsf{\Phi }{\partial }_{\eta }{\varphi }^4-{\partial }_{\eta }{\partial }_{\phi }{\varphi }^4\right),\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill a^2{{\mathcal{Z}}^r}_r=& {\partial }_{\eta }{\varphi }^4\left(aH\left(1-2\mathsf{\Phi }\right)-{\partial }_{\eta }\mathsf{\Phi }-2{\partial }_{\eta }\mathsf{\Psi }\right)+{\partial }_{\eta }^2{\varphi }^4\left(1-2\mathsf{\Phi }\right)\hfill \\ \hfill & -\left({\nabla }^2-{\partial }_r^2\right){\varphi }^4,\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill {{\mathcal{Z}}^r}_{\theta }=r^2{{\mathcal{Z}}^{\theta }}_r=& {\displaystyle \frac{r}{a^2}}{\partial }_r\left({\displaystyle \frac{{\partial }_{\theta }{\varphi }^4}{r}}\right),\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill {{\mathcal{Z}}^r}_{\phi }=r^2{sin}^2\left(\theta \right){{\mathcal{Z}}^{\phi }}_r=& {\displaystyle \frac{r}{a^2}}{\partial }_r\left({\displaystyle \frac{{\partial }_{\phi }{\varphi }^4}{r}}\right),\hfill \\ \hfill a^2{{\mathcal{Z}}^{\theta }}_{\theta }=& {\partial }_{\eta }{\varphi }^4\left(aH\left(1-2\mathsf{\Phi }\right)-{\partial }_{\eta }\mathsf{\Phi }-2{\partial }_{\eta }\mathsf{\Psi }\right)+{\partial }_{\eta }^2{\varphi }^4\left(1-2\mathsf{\Phi }\right)\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill & -\left({\nabla }^2-{\displaystyle \frac{{\partial }_r}{r}}-{\displaystyle \frac{{\partial }_{\theta }^2}{r^2}}\right){\varphi }^4,\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill {{\mathcal{Z}}^{\theta }}_{\phi }={sin}^2\left(\theta \right){{\mathcal{Z}}^{\phi }}_{\theta }=& {\displaystyle \frac{sin\left(\theta \right)}{a^2{r}^2}}{\partial }_{\theta }\left({\displaystyle \frac{{\partial }_{\phi }{\varphi }^4}{sin\left(\theta \right)}}\right),\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill a^2{{\mathcal{Z}}^{\phi }}_{\phi }=& {\partial }_{\eta }{\varphi }^4\left(aH\left(1-2\mathsf{\Phi }\right)-{\partial }_{\eta }\mathsf{\Phi }-2{\partial }_{\eta }\mathsf{\Psi }\right)+{\partial }_{\eta }^2{\varphi }^4\left(1-2\mathsf{\Phi }\right)\hfill \\ \hfill & -\left({\partial }_r^2+{\displaystyle \frac{{\partial }_r}{r}}+{\displaystyle \frac{{\partial }_{\theta }^2}{r^2}}\right){\varphi }^4.\hfill \end{array}$$

(42)

Moreover, the stress-energy tensor equals

$${T^{\mu }}_{\nu }=g^{\mu \alpha }{\partial }_{\alpha }\varphi {\partial }_{\nu }\varphi +{\delta }_{\nu }^{\mu }\left[{\displaystyle \frac{1}{2a^2}}\left(1-2\mathsf{\Phi }\right){\left({\partial }_{\eta }\varphi \right)}^2-V\left(\varphi \right)\right].$$

(43)

At last, $\varphi \left(x\right)=\overline{\varphi }\left(\eta \right)+\delta \varphi \left(x\right)$ has to be substituted into ${T^{\mu }}_{\nu }$ and ${{\mathcal{Z}}^{\mu }}_{\nu }$, followed by a separation of background and perturbed Einstein equations. This decomposition extends to Equation (13). Nonetheless, a transition to the non-relativistic regime is warranted before further proceeding.

4. The Non-Relativistic Limit

The authors of Ref. [17] developed a formalism to study the non-relativistic limit within a minimal coupled FLRW metric perturbed solely by scalar modes. Later, this framework was extended in Ref. [18] to handle arbitrary curved spacetimes and to include self-interactions. In a similar spirit to Refs. [16,19], who focus on the flat spacetime case, the low-energy regime is obtained by mapping the real field $\varphi$ to a complex field $\psi$ through an exact transformation. The derivation in the Minkowski spacetime was facilitated by a non-local operator in the field redefinition. However, this method becomes less effective in curved spacetime, as it leads to an equation of motion that differs from the Schrödinger equation [17,18]. Hence, the relation between $\varphi$ and $\psi$ now reads:

$$\begin{array}{cc}\hfill \varphi \left(x\right)& ={\displaystyle \frac{1}{\sqrt{2m_a}}}\left(e^{-i{m}_{a}t}\psi \left(x\right)+e^{i{m}_{a}t}{\psi }^{\ast }\left(x\right)\right),\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill {\partial }_t\varphi \left(x\right)& =-i\sqrt{{\displaystyle \frac{m_a}{2}}}\left(e^{-i{m}_{a}t}\psi \left(x\right)-e^{i{m}_{a}t}{\psi }^{\ast }\left(x\right)\right),\hfill \end{array}$$

(45)

which is no longer a canonical field transformation. Although it may seem natural to express the above relations using conformal time $\eta$ instead of cosmic time t, doing so introduces additional terms that complicate the effective description. It follows directly from the above expressions that

$$e^{-imt}{\partial }_t\psi +e^{imt}{\partial }_t{\psi }^{\ast }=0.$$

(46)

To simplify the notation, it is convenient to define

$$V_{\mathrm{int}}\left(\varphi \right)=V\left(\varphi \right)-{\displaystyle \frac{1}{2}}{m}_a^2{\varphi }^2+\xi R{\varphi }^4,$$

(47)

which collects all self-interaction terms. Moreover, Equation (13) can be rewritten as

$${\displaystyle \frac{1}{\sqrt{-g}}}{\partial }_{\mu }\left(\sqrt{-g}{g}^{\mu \nu }{\partial }_{\nu }\varphi \right)={\displaystyle \frac{\partial V_{\mathrm{int}}}{\partial \varphi }}.$$

(48)

Upon expanding the index sum,

$$\begin{array}{cc}\hfill -g^{\eta \eta }{\partial }_{\eta }^2\varphi =& {\displaystyle \frac{1}{\sqrt{-g}}}{\partial }_{\mu }\left(\sqrt{-g}{g}^{\mu \eta }\right){\partial }_{\eta }\varphi +{\displaystyle \frac{1}{\sqrt{-g}}}{\partial }_{\mu }\left(\sqrt{-g}{g}^{\mu i}\right){\partial }_i\varphi +g^{ij}{\partial }_i{\partial }_j\varphi \hfill \\ \hfill & +2g^{i\eta }{\partial }_i{\partial }_{\eta }\varphi -m_a^2\varphi -{\displaystyle \frac{\partial V_{\mathrm{int}}}{\partial \varphi }}.\hfill \end{array}$$

(49)

Meanwhile, from Equation (45), it is straightforward to verify that

$$\begin{array}{c}\hfill \psi \left(x\right)=\sqrt{{\displaystyle \frac{m_a}{2}}}{e}^{i{m}_{a}t}\left(\varphi \left(x\right)+{\displaystyle \frac{i}{a{m}_a}}{\partial }_{\eta }\varphi \left(x\right)\right).\end{array}$$

(50)

Deriving this equality with respect to $\eta$ leads to

$$\begin{array}{cc}\hfill i{\partial }_{\eta }\psi =& -{\displaystyle \frac{a{m}_a}{2}}\left(\psi +e^{2i{m}_{a}t}{\psi }^{\ast }\right)-{\displaystyle \frac{i}{2a}}{\displaystyle \frac{da}{d\eta }}\left(\psi -e^{2i{m}_{a}t}{\psi }^{\ast }\right)-{\displaystyle \frac{1}{a\sqrt{2m_a}}}{e}^{i{m}_{a}t}{\partial }_{\eta }^2\varphi .\hfill \end{array}$$

(51)

Finally, replacing Equation (49) into Equation (51) results in

$$\begin{array}{c}\hfill i{g}^{\eta \eta }{\partial }_{\eta }\psi +\mathcal{D}\psi +e^{2i{m}_{a}t}{\mathcal{D}}^{\ast }{\psi }^{\ast }+{\displaystyle \frac{e^{i{m}_{a}t}}{a\sqrt{2m_a}}}{\displaystyle \frac{\partial V_{\mathrm{int}}}{\partial \varphi }}=0,\end{array}$$

(52)

where $\mathcal{D}$ is the operator

$$\begin{array}{cc}\hfill \mathcal{D}\psi =& {\displaystyle \frac{m_a}{2a}}\left(a^2{g}^{\eta \eta }+1\right)\psi +{\displaystyle \frac{i}{2a}}{\displaystyle \frac{da}{d\eta }}{g}^{\eta \eta }\psi +{\displaystyle \frac{i}{2}}{\displaystyle \frac{1}{\sqrt{-g}}}{\partial }_{\mu }\left(\sqrt{-g}{g}^{\mu \eta }\right)\psi +i{g}^{i\eta }{\partial }_i\psi \hfill \\ \hfill & -{\displaystyle \frac{1}{2a{m}_a\sqrt{-g}}}{\partial }_{\mu }\left(\sqrt{-g}{g}^{\mu i}\right){\partial }_i\psi -{\displaystyle \frac{1}{2a{m}_a}}{g}^{ij}{\partial }_i{\partial }_j\psi .\hfill \end{array}$$

(53)

If all instances of $\eta$ in Equation (52) were changed to t, the same equation of motion that was derived in Ref. [17] would be obtained. However, the definition of $\mathcal{D}$ does differ by a factor of a. Nonetheless, the same symbol is retained for the sake of notational simplicity.

Similarly to the flat spacetime case, the rapidly oscillating modes of $\psi$ exert a non-trivial back-reaction to the dominant, slowly-varying component. This feedback is represented by the last two terms of Equation (52). As demonstrated in Refs. [16,17,18,19], the fast oscillatory modes induce corrections to the effective potential of $\psi$. Interestingly, the contribution responsible for axion pair-interactions remains unaffected. Therefore, if the axion interactions are reduced to

$$\begin{array}{c}\hfill V_{\mathrm{int}}\left(\varphi \right)={\displaystyle \frac{{\lambda }_4}{4!}}{\displaystyle \frac{m_a^2{\varphi }^4}{f_a^2}}+\xi R{\varphi }^4\equiv {\displaystyle \frac{{\overline{\lambda }}_4}{4!}}{\varphi }^4+\xi R{\varphi }^4,\end{array}$$

(54)

the high-frequency contributions can be neglected as a first approximation. For a complete treatment of their impact, see Refs. [17,18]. Consequently, the equation of motion becomes

$$\begin{array}{c}\hfill i{g}^{\eta \eta }{\partial }_{\eta }\psi +\mathcal{D}\psi +{\displaystyle \frac{1}{8a{m}_a^2}}\left({\overline{\lambda }}_4+24\xi R\right)\psi {\left|\psi \right|}^2=0.\end{array}$$

(55)

Upon inserting the Newtonian gauge metric, multiplying by $g_{\eta \eta }$, and linearizing,

$$\begin{array}{cc}\hfill i{\partial }_{\eta }\psi & =a\left(m_a\mathsf{\Phi }-{\displaystyle \frac{3i}{2}}H\right)\psi +{\displaystyle \frac{i}{2}}{\partial }_{\eta }\left(\mathsf{\Phi }+3\mathsf{\Psi }\right)\psi -{\displaystyle \frac{{\nabla }^2\psi }{2a{m}_a}}\hfill \\ \hfill & +\left({\displaystyle \frac{a{\overline{\lambda }}_4}{8m_a^2}}+{\displaystyle \frac{9\xi }{a{m}_a^2}}(a^2{H}^2-{\rm Y})\right)\psi {\left|\psi \right|}^2\hfill \\ \hfill & -{\displaystyle \frac{6\xi }{a{m}_a^2}}\left(3aH{\partial }_{\eta }\left(\mathsf{\Phi }+3\mathsf{\Psi }\right)+3{\partial }_{\eta }^2\mathsf{\Psi }+{\nabla }^2\mathsf{\Phi }-2{\nabla }^2\mathsf{\Psi }\right)\psi {\left|\psi \right|}^2,\hfill \end{array}$$

(56)

where the ${\overline{\lambda }}_4\mathsf{\Phi }$ term was neglected as ${\overline{\lambda }}_4={\lambda }_4{m}_a^2/f_a^2\ll 1$ for $m_a^2/f_a^2\in ({10}^{-52}$–${10}^{-44})$. The background dynamics are obtained by substituting $\psi \left(x\right)=\overline{\psi }\left(\eta \right)$ and setting the metric perturbations to zero,

$$\begin{array}{cc}\hfill i{\partial }_{\eta }\overline{\psi }=& -{\displaystyle \frac{3i}{2}}aH\overline{\psi }+\left({\displaystyle \frac{a{\overline{\lambda }}_4}{8m_a^2}}+{\displaystyle \frac{9\xi }{a{m}_a^2}}(a^2{H}^2-{\rm Y})\right)\overline{\psi }{\left|\overline{\psi }\right|}^2.\hfill \end{array}$$

(57)

Next, the non-relativistic field is split as $\psi \left(x\right)=\overline{\psi }\left(\eta \right)+\delta \psi \left(x\right)$ with $\left|\delta \psi \right|\ll |\overline{\psi }|$. To eliminate ambiguity in the decomposition, it is common to impose that the spatial average of $\delta \psi$ vanishes [17]. Inserting this ansatz into Equation (56) and subtracting the background evolution yields

$$\begin{array}{cc}\hfill i{\partial }_{\eta }\left(\delta \psi \right)=& a{m}_a\mathsf{\Phi }\overline{\psi }-{\displaystyle \frac{3i}{2}}aH\delta \psi +{\displaystyle \frac{i}{2}}{\partial }_{\eta }\left(\mathsf{\Phi }+3\mathsf{\Psi }\right)\overline{\psi }-{\displaystyle \frac{{\nabla }^2\left(\delta \psi \right)}{2a{m}_a}}\hfill \\ \hfill & +\left({\displaystyle \frac{a{\overline{\lambda }}_4}{8m_a^2}}+{\displaystyle \frac{9\xi }{a{m}_a^2}}(a^2{H}^2-{\rm Y})\right)\left({\overline{\psi }}^2\delta {\psi }^{\ast }+2{\left|\overline{\psi }\right|}^2\delta \psi \right)\hfill \\ \hfill & -{\displaystyle \frac{6\xi }{a{m}_a^2}}\left(3aH{\partial }_{\eta }\left(\mathsf{\Phi }+3\mathsf{\Psi }\right)+3{\partial }_{\eta }^2\mathsf{\Psi }+{\nabla }^2\mathsf{\Phi }-2{\nabla }^2\mathsf{\Psi }\right)\overline{\psi }{\left|\overline{\psi }\right|}^2.\hfill \end{array}$$

(58)

Afterwards, the non-relativistic limit of the modified Einstein equations has to be obtained. Inserting Equation (45) into Equation (43) leads to the following non-zero components of the stress-energy tensor:

$$\begin{array}{cc}\hfill {T^{\mu }}_{\mu }=& -m_a{\left|\psi \right|}^2-{\displaystyle \frac{{\overline{\lambda }}_4}{16m_a^2}}{\left|\psi \right|}^4+m_a\mathsf{\Phi }{\left|\psi \right|}^2,\hfill \end{array}$$

(59)

$$\begin{array}{cc}\hfill {T^{\eta }}_r=-{T^r}_{\eta }=& {\displaystyle \frac{i}{2a}}\left(\psi {\partial }_r{\psi }^{\ast }-{\psi }^{\ast }{\partial }_r\psi \right),\hfill \end{array}$$

(60)

$$\begin{array}{cc}\hfill {T^{\eta }}_{\theta }=-r^2{T^{\theta }}_{\eta }=& {\displaystyle \frac{i}{2a}}\left(\psi {\partial }_{\theta }{\psi }^{\ast }-{\psi }^{\ast }{\partial }_{\theta }\psi \right),\hfill \end{array}$$

(61)

$$\begin{array}{cc}\hfill {T^{\eta }}_{\phi }=-r^2{sin}^2\left(\theta \right){T^{\phi }}_{\eta }=& {\displaystyle \frac{i}{2a}}\left(\psi {\partial }_{\phi }{\psi }^{\ast }-{\psi }^{\ast }{\partial }_{\phi }\psi \right),\hfill \end{array}$$

(62)

$$\begin{array}{cc}\hfill {T^r}_r={T^{\theta }}_{\theta }={T^{\phi }}_{\phi }=& -{\displaystyle \frac{{\overline{\lambda }}_4}{16m_a^2}}{\left|\psi \right|}^4-m_a\mathsf{\Phi }{\left|\psi \right|}^2.\hfill \end{array}$$

(63)

It is also necessary to express ${{\mathcal{Z}}^{\mu }}_{\nu }$ in terms of $\psi$. Using Equations (45) and (46), it is possible to show that

$$\begin{array}{ccccc}\hfill {\partial }_{\eta }{\varphi }^4\approx 0,& \hfill & \hfill {\partial }_{\eta }{\partial }_i{\varphi }^4\approx 0,& \hfill & \hfill {\partial }_{\eta }^2{\varphi }^4\approx {\displaystyle \frac{6ia}{m_a}}{\left|\psi \right|}^2\left(\psi {\partial }_{\eta }{\psi }^{\ast }-{\psi }^{\ast }{\partial }_{\eta }\psi \right).\end{array}$$

(64)

Additionally, any term involving a double spatial derivative of ${\varphi }^4$ is evaluated by replacing ${\varphi }^4\approx {\displaystyle \frac{3}{2m_a^2}}{\left|\psi \right|}^4$. It is advantageous to use Equation (56) to eliminate the ${\partial }_{\eta }\psi$ dependence inside ${\partial }_{\eta }^2{\varphi }^4$. This leads to

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }_{\eta }^2{\varphi }^4}{a^2}}\left(1-2\mathsf{\Phi }\right)\approx & {\displaystyle \frac{{3\left|\psi \right|}^2}{a^2{m}_a^2}}\left(\psi {\nabla }^2{\psi }^{\ast }+{\psi }^{\ast }{\nabla }^2\psi \right)-12\mathsf{\Phi }{\left|\psi \right|}^4\hfill \\ \hfill & -\left({\displaystyle \frac{3{\overline{\lambda }}_4}{2m_a^3}}+{\displaystyle \frac{108\xi }{a^2{m}_a^3}}\left(a^2{H}^2-{\rm Y}\right)\left(1-2\mathsf{\Phi }\right)\right){\left|\psi \right|}^6\hfill \\ \hfill & +{\displaystyle \frac{72\xi }{a^2{m}_a^3}}\left(3aH{\partial }_{\eta }\left(\mathsf{\Phi }+3\mathsf{\Psi }\right)+3{\partial }_{\eta }^2\mathsf{\Psi }+{\nabla }^2\mathsf{\Phi }-2{\nabla }^2\mathsf{\Psi }\right){\left|\psi \right|}^6.\hfill \end{array}$$

(65)

When ${{\mathcal{Z}}^{\mu }}_{\nu }$ is inserted into Equation (11), the last three terms of the above expression act as corrections to the axion potential. Given the current simplification of pairwise interactions, the ${\left|\psi \right|}^6$ terms are disregarded because their influence is incomplete without including the fast-oscillating corrections and the ${\varphi }^6$ term of $V\left(\varphi \right)$.

The background modified Einstein equations are obtained by replacing $\psi \left(x\right)=\overline{\psi }\left(\eta \right)$ and $\mathsf{\Phi }\left(x\right)=\mathsf{\Psi }\left(x\right)=0$ in Equation (11). The only non-trivial equations are the $\eta \eta$ component,

$${\displaystyle \frac{3H^2}{8\pi G}}-m_a|\overline{\psi }{|}^2-\left({\displaystyle \frac{{\overline{\lambda }}_4}{16m_a^2}}+{\displaystyle \frac{9\xi H^2}{m_a^2}}\right){\left|\overline{\psi }\right|}^4=0,$$

(66)

and the diagonal components of the spatial sector,

$${\displaystyle \frac{1}{8\pi G}}{\displaystyle \frac{{\rm Y}}{a^2}}+\left({\displaystyle \frac{{\overline{\lambda }}_4}{16m_a^2}}-{\displaystyle \frac{3\xi {\rm Y}}{a^2{m}_a^2}}\right){\left|\overline{\psi }\right|}^4=0.$$

(67)

While Equations (66) and (67) govern the evolution of a, they reflect a simplified scenario in which only $\overline{\psi }$ contributes to the energy content. A realistic cosmological model must include radiation, baryonic matter, dark matter, and dark energy. In practice, a is treated as a known function, obtained from observationally supported cosmological models.

The perturbed form of Equation (11) is obtained by inserting $\psi \left(x\right)=\overline{\psi }\left(\eta \right)+\delta \psi \left(x\right)$, followed by the removal of their background counterparts. Notably, $\mathsf{\Phi }\left(x\right)$ and $\mathsf{\Psi }\left(x\right)$ are not dynamical variables but instead serve as auxiliary fields. Hence, the perturbed Einstein equations act as constraints rather than equations of motion. These ligatures are best written in terms of

$$\begin{array}{cc}\hfill {\mathcal{R}}_{\delta \psi }& =\overline{\psi }\delta {\psi }^{\ast }+{\overline{\psi }}^{\ast }\delta \psi ,\hfill \end{array}$$

(68)

$$\begin{array}{cc}\hfill {\mathcal{I}}_{\delta \psi }& =i\left(\overline{\psi }\delta {\psi }^{\ast }-{\overline{\psi }}^{\ast }\delta \psi \right),\hfill \end{array}$$

(69)

$$\begin{array}{cc}\hfill 8\pi \mathcal{G}\left(\eta \right)& ={\left({\displaystyle \frac{1}{8\pi G}}-{\displaystyle \frac{3\xi |\overline{\psi }{|}^4}{m_a^2}}\right)}^{-1}.\hfill \end{array}$$

(70)

The $\eta \eta$ component is

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{4\pi \mathcal{G}}}\left(3a^2{H}^2\mathsf{\Phi }+3aH{\partial }_{\eta }\mathsf{\Psi }-{\nabla }^2\mathsf{\Psi }\right)-a^2{m}_a{\left|\overline{\psi }\right|}^2\mathsf{\Phi }\hfill \\ \hfill & +a^2\left[m_a+{\displaystyle \frac{|\overline{\psi }{|}^2}{m_a^2}}\left({\displaystyle \frac{{\overline{\lambda }}_4}{8}}+{\displaystyle \frac{6\xi }{a^2}}\left(3a^2{H}^2-{\nabla }^2\right)\right)\right]{\mathcal{R}}_{\delta \psi }=0.\hfill \end{array}$$

(71)

The temporal-spatial components can be combined into

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{4\pi \mathcal{G}}}\nabla \left(aH\mathsf{\Phi }+{\partial }_{\eta }\mathsf{\Psi }\right)=& {\displaystyle \frac{6\xi aH|\overline{\psi }{|}^2}{m_a^2}}\nabla {\mathcal{R}}_{\delta \psi }-{\displaystyle \frac{a}{2}}\nabla {\mathcal{I}}_{\delta \psi }.\hfill \end{array}$$

(72)

The trace of the spatial sector equals

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{8\pi \mathcal{G}}}\left({\partial }_{\eta }^2\mathsf{\Psi }+{\displaystyle \frac{1}{3}}{\nabla }^2(\mathsf{\Phi }-\mathsf{\Psi })+aH{\partial }_{\eta }\left(\mathsf{\Phi }+2\mathsf{\Psi }\right)-{\rm Y}\mathsf{\Phi }\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{a}^2{m}_a{\left|\overline{\psi }\right|}^2\mathsf{\Phi }-12\xi a^2{\left|\overline{\psi }\right|}^4\mathsf{\Phi }+{\displaystyle \frac{a^2{\left|\overline{\psi }\right|}^2}{m_a^2}}\left({\displaystyle \frac{{\overline{\lambda }}_4}{16}}-{\displaystyle \frac{\xi }{a^2}}\left(3{\rm Y}-{\nabla }^2\right)\right){\mathcal{R}}_{\delta \psi }=0.\hfill \end{array}$$

(73)

The non-diagonal space Einstein equations are

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{8\pi \mathcal{G}}}{\partial }_r\left({\displaystyle \frac{{\partial }_{\theta }\left(\mathsf{\Phi }-\mathsf{\Psi }\right)}{r}}\right)=& {\displaystyle \frac{6\xi |\overline{\psi }{|}^2}{m_a^2}}{\partial }_r\left({\displaystyle \frac{{\partial }_{\theta }{\mathcal{R}}_{\delta \psi }}{r}}\right),\hfill \end{array}$$

(74)

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{8\pi \mathcal{G}}}{\partial }_r\left({\displaystyle \frac{{\partial }_{\phi }\left(\mathsf{\Phi }-\mathsf{\Psi }\right)}{r}}\right)=& {\displaystyle \frac{6\xi |\overline{\psi }{|}^2}{m_a^2}}{\partial }_r\left({\displaystyle \frac{{\partial }_{\phi }{\mathcal{R}}_{\delta \psi }}{r}}\right),\hfill \end{array}$$

(75)

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{8\pi \mathcal{G}}}{\partial }_{\theta }\left({\displaystyle \frac{{\partial }_{\phi }\left(\mathsf{\Phi }-\mathsf{\Psi }\right)}{sin\left(\theta \right)}}\right)=& {\displaystyle \frac{6\xi |\overline{\psi }{|}^2}{m_a^2}}{\partial }_{\theta }\left({\displaystyle \frac{{\partial }_{\phi }{\mathcal{R}}_{\delta \psi }}{sin\left(\theta \right)}}\right).\hfill \end{array}$$

(76)

Imposing $a=1$, $\xi =0$, $\mathsf{\Phi }=\mathsf{\Psi }$, and ${\partial }_{\eta }\mathsf{\Phi }=0$ in Equations (56) and (71) reduces the system to the Gross-Pitaevskii-Poisson equations. Further omitting the axion self-interactions yields the Schrödinger–Poisson equations. Both prescriptions are often used to model the non-relativistic dynamics of gravitationally interacting axions. Thus, the equations derived in this work can be viewed as a systematic refinement of these models.

It is noteworthy that if $\delta \psi \left(x\right)$ remains spherically symmetric throughout its entire evolution, Equations (74)–(76) naturally enforce $\mathsf{\Psi }=\mathsf{\Phi }$, which mirrors an absence of anisotropic stress. Under this simplification, Equation (58) reduces to

$$\begin{array}{cc}\hfill i{\partial }_{\eta }\left(\delta \psi \right)=& a{m}_a\mathsf{\Phi }\overline{\psi }-{\displaystyle \frac{3i}{2}}aH\delta \psi +2i{\partial }_{\eta }\mathsf{\Phi }\overline{\psi }-{\displaystyle \frac{{\nabla }^2\left(\delta \psi \right)}{2a{m}_a}}\hfill \\ \hfill & +\left({\displaystyle \frac{a{\overline{\lambda }}_4}{8m_a^2}}+{\displaystyle \frac{9\xi }{a{m}_a^2}}(a^2{H}^2-{\rm Y})\right)\left({\overline{\psi }}^2\delta {\psi }^{\ast }+2{\left|\overline{\psi }\right|}^2\delta \psi \right)\hfill \\ \hfill & -{\displaystyle \frac{6\xi }{a{m}_a^2}}\left(12aH{\partial }_{\eta }\mathsf{\Phi }+3{\partial }_{\eta }^2\mathsf{\Phi }-{\nabla }^2\mathsf{\Phi }\right)\overline{\psi }{\left|\overline{\psi }\right|}^2.\hfill \end{array}$$

(77)

Moreover, Equations (71)–(73) become

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{4\pi \mathcal{G}}}\left(3a^2{H}^2\mathsf{\Phi }+3aH{\partial }_{\eta }\mathsf{\Phi }-{\nabla }^2\mathsf{\Phi }\right)-a^2{m}_a{\left|\overline{\psi }\right|}^2\mathsf{\Phi }\hfill \\ \hfill & +a^2\left[m_a+{\displaystyle \frac{|\overline{\psi }{|}^2}{m_a^2}}\left({\displaystyle \frac{{\overline{\lambda }}_4}{8}}+{\displaystyle \frac{6\xi }{a^2}}\left(3a^2{H}^2-{\nabla }^2\right)\right)\right]{\mathcal{R}}_{\delta \psi }=0.\hfill \end{array}$$

(78)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{4\pi \mathcal{G}}}\nabla \left(aH\mathsf{\Phi }+{\partial }_{\eta }\mathsf{\Phi }\right)={\displaystyle \frac{6\xi aH|\overline{\psi }{|}^2}{m_a^2}}\nabla {\mathcal{R}}_{\delta \psi }-{\displaystyle \frac{a}{2}}\nabla {\mathcal{I}}_{\delta \psi }.\hfill \end{array}$$

(79)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{8\pi \mathcal{G}}}\left({\partial }_{\eta }^2\mathsf{\Phi }+3aH{\partial }_{\eta }\mathsf{\Phi }-{\rm Y}\mathsf{\Phi }\right)+{\displaystyle \frac{1}{2}}{a}^2{m}_a{\left|\overline{\psi }\right|}^2\mathsf{\Phi }-12\xi a^2{\left|\overline{\psi }\right|}^4\mathsf{\Phi }\hfill \\ \hfill & +{\displaystyle \frac{a^2{\left|\overline{\psi }\right|}^2}{m_a^2}}\left({\displaystyle \frac{{\overline{\lambda }}_4}{16}}-{\displaystyle \frac{\xi }{a^2}}\left(3{\rm Y}-{\nabla }^2\right)\right){\mathcal{R}}_{\delta \psi }=0.\hfill \end{array}$$

(80)

Finally, the covariant continuity equations are often relevant in the dynamics of gravitational systems. The condition ${\nabla }_{\mu }{T^{\mu }}_{\nu }=0$ is a direct consequence of Einstein field equations due to contracted Bianchi identity ${\nabla }_{\mu }{G}^{\mu \nu }=0$. However, the non-minimal coupling present in this work modifies the Einstein equations, yielding

$${\nabla }_{\mu }{T^{\mu }}_{\nu }+2\xi {\partial }_{\mu }{\varphi }^4{G^{\mu }}_{\nu }=0.$$

(81)

In principle, this relation can be expanded and expressed in terms of $\psi$ to obtain the non-relativistic limit. However, these equations are automatically satisfied by any solution to Equation (11) and serve as consistency conditions rather than new constraints. For this reason, they were not explicitly evaluated in this work.

5. From Gross–Pitaevskii to Schödinger

Even though $\psi \left(x\right)=\overline{\psi }\left(\eta \right)+\delta \psi \left(x\right)$ is a convenient way to separate the background from the perturbation, the full field is still $\psi \left(x\right)$ and Equation (56) continues to describe its dynamics. This equation can be recast as

$$i{\partial }_{\eta }\psi =-{\displaystyle \frac{1}{2ma}}{\nabla }^2\psi +\mathcal{U}\psi +{\mathcal{U}}_0\psi {\left|\psi \right|}^2,$$

(82)

where

$$\begin{array}{cc}\hfill \mathcal{U}\left(x\right)=& a{m}_a\mathsf{\Phi }-{\displaystyle \frac{3ia}{2}}H+{\displaystyle \frac{i}{2}}{\partial }_{\eta }\left(\mathsf{\Phi }+3\mathsf{\Psi }\right),\hfill \end{array}$$

(83)

$$\begin{array}{cc}\hfill {\mathcal{U}}_0\left(x\right)=& {\displaystyle \frac{a{\overline{\lambda }}_4}{8m_a^2}}+{\displaystyle \frac{3\xi g\left(x\right)}{a{m}_a^2}},\hfill \end{array}$$

(84)

$$\begin{array}{cc}\hfill g\left(x\right)=& 3a^2{H}^2-3{\rm Y}-6aH{\partial }_{\eta }\left(\mathsf{\Phi }+3\mathsf{\Psi }\right)-6{\partial }_{\eta }^2\mathsf{\Psi }-2{\nabla }^2\mathsf{\Phi }+4{\nabla }^2\mathsf{\Psi }.\hfill \end{array}$$

(85)

This expression closely resembles the time-dependent Gross–Pitaevskii equation,

$$i{\partial }_t\psi =-{\displaystyle \frac{1}{2m}}{\nabla }^2\psi +\mathrm{U}\psi +{\mathrm{U}}_0\psi {\left|\psi \right|}^2,$$

(86)

where U$\left(x\right)$ is an external potential and ${\mathrm{U}}_0$ is a constant that characterizes the s-wave scattering between two bosons. Equation (86) emerges from the Hartree-Fock treatment of a system of N bosons near the ground state. Hence, it is insightful to apply the inverse procedure to Equation (82) to recover the N-body Schrödinger equation.

Considering only variations with respect to $\psi$ and ${\psi }^{\ast }$, the action that corresponds to Equation (82) is

$$\begin{array}{cc}\hfill S=\int d\eta {\int }_{V}d\overrightarrow{r}[& i{\psi }^{\ast }(\eta ,\overrightarrow{r}){\partial }_{\eta }\psi (\eta ,\overrightarrow{r})+{\displaystyle \frac{1}{2ma}}{\psi }^{\ast }(\eta ,\overrightarrow{r}){\nabla }^2\psi (\eta ,\overrightarrow{r})\hfill \\ \hfill & -\mathcal{U}(\eta ,\overrightarrow{r})|\psi (\eta ,\overrightarrow{r}){|}^2-{\displaystyle \frac{1}{2}}{\mathcal{U}}_0(\eta ,\overrightarrow{r}){\left|\psi (\eta ,\overrightarrow{r})\right|}^4].\hfill \end{array}$$

(87)

To avoid infinities, the spatial integration is done over a finite volume V. The system is subject to

$${\displaystyle \frac{1}{V}}{\int }_{V}d\overrightarrow{r}|\psi (\eta ,\overrightarrow{r}){|}^2={\left|\overline{\psi }\left(\eta \right)\right|}^2,$$

(88)

which can be computed explicitly by solving Equation (57). The wavefunction $\chi$ of the whole system is modeled as

$$\begin{array}{c}\hfill \chi (\eta ,{\overrightarrow{r}}_1,\dots ,{\overrightarrow{r}}_N)={\displaystyle \frac{1}{|\overline{\psi }{|}^{N-1}}}\prod _{k=1}^N\psi (\eta ,{\overrightarrow{r}}_k),\end{array}$$

(89)

since it ensures that

$$\left(\prod _{j=1}^N{\displaystyle \frac{1}{V}}{\int }_{V}d{\overrightarrow{r}}_j\right){\left|\chi \right|}^2={\left|\overline{\psi }\left(\eta \right)\right|}^2.$$

(90)

A straightforward calculation shows that

$$\begin{array}{c}\hfill S={\displaystyle \frac{1}{N{V}^{N-1}}}\int d\eta \left(\prod _{j=1}^N{\int }_{V}d{\overrightarrow{r}}_j\right){\chi }^{\ast }\left(i{\partial }_{\eta }+\sum _{k=1}^N{\displaystyle \frac{{\nabla }_k^2}{2ma}}-{\mathrm{U}}_{\mathrm{ext}}-{\mathrm{U}}_{\mathrm{int}}\right)\chi ,\end{array}$$

(91)

where

$$\begin{array}{cc}\hfill {\mathrm{U}}_{\mathrm{ext}}=& \sum _{k=1}^N\left(\mathcal{U}(\eta ,{\overrightarrow{r}}_k)-{\displaystyle \frac{i(N-1)}{N^2}}{\partial }_{\eta }ln\left|\overline{\psi }\right|\right),\hfill \end{array}$$

(92)

$$\begin{array}{cc}\hfill {\mathrm{U}}_{\mathrm{int}}=& {\displaystyle \frac{V|\overline{\psi }{|}^2}{2(N-1)}}\sum _{k=1}^N\sum _{l\ne k}^N{\mathcal{U}}_0(\eta ,{\overrightarrow{r}}_k)\delta ({\overrightarrow{r}}_k-{\overrightarrow{r}}_l).\hfill \end{array}$$

(93)

Equation (91) directly leads to

$$\begin{array}{c}\hfill i{\partial }_{\eta }\chi =\left(-\sum _{k=1}^N{\displaystyle \frac{{\nabla }_k^2}{2ma}}+{\mathrm{U}}_{\mathrm{ext}}+{\mathrm{U}}_{\mathrm{int}}\right)\chi .\end{array}$$

(94)

Therefore, the Hamiltonian operator that encapsulates the system’s dynamics is

$$\begin{array}{cc}\hfill \widehat{H}=& -\sum _{k=1}^N{\displaystyle \frac{{\nabla }_k^2}{2ma}}+\sum _{k=1}^N\left(\mathcal{U}(\eta ,{\overrightarrow{r}}_k)-{\displaystyle \frac{i(N-1)}{N^2}}{\partial }_{\eta }ln\left|\overline{\psi }\right|\right)\hfill \\ \hfill & +{\displaystyle \frac{V|\overline{\psi }{|}^2}{2(N-1)}}\sum _{k=1}^N\sum _{l\ne k}^N{\mathcal{U}}_0(\eta ,{\overrightarrow{r}}_k)\delta ({\overrightarrow{r}}_k-{\overrightarrow{r}}_l).\hfill \end{array}$$

(95)

This approach is particularly useful, as it provides a convenient starting point for a variety of effective descriptions of axion systems with gravitationally mediated pairwise interactions, including statistical and dynamical treatments.

6. Summary

In this work, we have developed a systematic extension of the standard non-relativistic description of self-gravitating axion fields by incorporating a non-minimal scalar–curvature coupling of the form $\xi R{\varphi }^4$ at the relativistic level. Starting from a generally covariant action, we derived the modified Einstein and Klein–Gordon equations and consistently propagated the effects of the non-minimal coupling through cosmological perturbation theory and the non-relativistic limit. The resulting framework yields a modified Gross–Pitaevskii–Poisson system that captures gravitationally mediated pairwise interactions beyond minimal coupling.

Our analysis shows that the inclusion of the $\xi R{\varphi }^4$ term leads to additional, geometry-dependent contributions to the effective self-interaction of the axion condensate. These contributions depend explicitly on the local curvature and metric perturbations, and therefore encode genuinely relativistic effects that are absent in the standard Gross–Pitaevskii–Poisson or Schrödinger–Poisson equations. In the appropriate limits–namely vanishing non-minimal coupling, weak metric perturbations, and negligible self-interactions–the formalism smoothly reduces to the conventional equations widely used to model axion structure formation. This consistency confirms that the present framework constitutes a controlled and systematic refinement of existing approaches.

Beyond the field-theoretic formulation, we have shown that the modified Gross–Pitaevskii equation can be mapped onto an effective many-body Schrödinger description. The resulting Hamiltonian explicitly exhibits gravitationally mediated pairwise interactions, providing a convenient starting point for both dynamical and statistical analyses of axion systems. This representation opens the door to applications ranging from stability analyses and collective dynamics to thermodynamic treatments of dense axion configurations.

It is worth distinguishing the present framework from generalized curvature–matter theories of the type $f\left(R\right)L_m$, where the gravitational action itself is modified and additional scalar gravitational degrees of freedom are introduced [29,30]. In such models, stability requires specific conditions on the functional form of $f\left(R\right)$ to avoid Dolgov–Kawasaki–type instabilities or ghost modes associated with higher-derivative curvature terms. By contrast, our construction preserves the standard Einstein–Hilbert gravitational sector and therefore does not introduce additional propagating gravitational degrees of freedom or higher-derivative curvature dynamics. The non-minimal interaction $\xi R{\varphi }^4$ instead modifies the scalar sector. This change does lead to ${\nabla }_{\mu }{T}^{\mu \nu }\ne 0$ which reflects an exchange between curvature and matter. Dynamically, this induces curvature-dependent corrections to the effective axion self-interaction, which may alter stability thresholds for dense configurations such as axion stars depending on the sign and magnitude of $\xi$. A quantitative determination of the corresponding stability bounds requires dedicated numerical analysis and it is left for future work.

A key open issue concerns the magnitude of the non-minimal coupling parameter $\xi$, which is not fixed a priori. While naturalness and consistency with the axion’s cosmological history impose constraints on its allowed range, a precise determination would require confronting the effective interaction induced by the non-minimal coupling with astrophysical observations or with scattering-based estimates of dark matter self-interactions. Addressing this question lies beyond the scope of the present work but represents an important direction for future research.

Several extensions of the present framework are worth pursuing. In particular, a hydrodynamic reformulation via the Madelung transformation would allow a more transparent interpretation of the modified dynamics and facilitate comparisons with previous studies of axion stars and miniclusters. Moreover, numerical simulations based on the derived equations could shed light on the impact of curvature-induced interactions on the formation, stability, and mass spectrum of compact axion structures. Finally, exploring the role of non-minimal couplings in more general cosmological or astrophysical backgrounds may provide further insight into the interplay between gravity and axion dark matter at high densities.

Finally, it is also worth emphasizing that it may be theoretically interesting to explore a natural extension of the present framework concerning the inclusion of other fundamental fields beyond scalars. For fermionic degrees of freedom, the construction would proceed by employing the vierbein formalism and introducing the spin connection to preserve local Lorentz invariance in curved spacetime. In this context, non-minimal curvature couplings such as $R\overline{\psi }\psi$ or higher-dimensional operators of the form $R{\left(\overline{\psi }\psi \right)}^2$ may arise within an effective field theory description. For massive vector (Proca) fields, the minimal coupling prescription can likewise be supplemented by curvature-dependent terms such as $R{A}_{\mu }{A}^{\mu }$ or $R_{\mu \nu }{A}^{\mu }{A}^{\nu }$, which are compatible with diffeomorphism invariance and may influence the effective mass and self-interaction structure in regions of high curvature. More generally, higher-spin fields in curved backgrounds typically require additional non-minimal couplings to preserve the correct number of propagating degrees of freedom, and their consistent formulation may impose restrictions on the background geometry. Although a detailed treatment lies beyond the scope of the present work, the effective and symmetry-based philosophy adopted here can be systematically extended to such cases, providing a pathway toward a more unified description of fundamental interactions in curved spacetime.

Appendix A.1. Christoffel Symbol

$$\begin{array}{c}{{\mathsf{\Gamma }}^{\eta }}_{\eta \eta }={\displaystyle \frac{1}{a}}{\displaystyle \frac{da}{d\eta }}+{\displaystyle \frac{{\partial }_{\eta }\mathsf{\Phi }}{1+2\mathsf{\Phi }}},{{\mathsf{\Gamma }}^r}_{rr}=-{\displaystyle \frac{{\partial }_r\mathsf{\Psi }}{1-2\mathsf{\Psi }}},{{\mathsf{\Gamma }}^{\eta }}_{\eta r}={\displaystyle \frac{{\partial }_r\mathsf{\Phi }}{1+2\mathsf{\Phi }}},{{\mathsf{\Gamma }}^{\eta }}_{\eta \theta }={\displaystyle \frac{{\partial }_{\theta }\mathsf{\Phi }}{1+2\mathsf{\Phi }}},\\ {{\mathsf{\Gamma }}^{\eta }}_{\eta \phi }={\displaystyle \frac{{\partial }_{\phi }\mathsf{\Phi }}{1+2\mathsf{\Phi }}},{{\mathsf{\Gamma }}^r}_{\eta \eta }={\displaystyle \frac{{\partial }_r\mathsf{\Phi }}{1-2\mathsf{\Psi }}},{{\mathsf{\Gamma }}^{\theta }}_{\eta \eta }={\displaystyle \frac{{\partial }_{\theta }\mathsf{\Phi }}{r^2\left(1-2\mathsf{\Psi }\right)}},{{\mathsf{\Gamma }}^{\phi }}_{\eta \eta }={\displaystyle \frac{{\partial }_{\phi }\mathsf{\Phi }}{r^2{sin}^2\left(\theta \right)\left(1-2\mathsf{\Psi }\right)}},\\ {{\mathsf{\Gamma }}^{\eta }}_{rr}={\displaystyle \frac{{{\mathsf{\Gamma }}^{\eta }}_{\theta \theta }}{r^2}}={\displaystyle \frac{{{\mathsf{\Gamma }}^{\eta }}_{\phi \phi }}{r^2{sin}^2\left(\theta \right)}}={\displaystyle \frac{\frac{1}{a}\frac{da}{d\eta }\left(1-2\mathsf{\Psi }\right)-{\partial }_{\eta }\mathsf{\Psi }}{1+2\mathsf{\Phi }}},{{\mathsf{\Gamma }}^r}_{\eta r}={{\mathsf{\Gamma }}^{\theta }}_{\eta \theta }={{\mathsf{\Gamma }}^{\phi }}_{\eta \phi }={\displaystyle \frac{1}{a}}{\displaystyle \frac{da}{d\eta }}-{\displaystyle \frac{{\partial }_{\eta }\mathsf{\Psi }}{1-2\mathsf{\Psi }}},\\ {{\mathsf{\Gamma }}^{\theta }}_{r\theta }={{\mathsf{\Gamma }}^{\phi }}_{r\phi }=-{\displaystyle \frac{{{\mathsf{\Gamma }}^r}_{\theta \theta }}{r^2}}=-{\displaystyle \frac{{{\mathsf{\Gamma }}^r}_{\phi \phi }}{r^2{sin}^2\left(\theta \right)}}={\displaystyle \frac{1}{r}}-{\displaystyle \frac{{\partial }_r\mathsf{\Psi }}{1-2\mathsf{\Psi }}},{{\mathsf{\Gamma }}^{\theta }}_{\theta \theta }={{\mathsf{\Gamma }}^r}_{r\theta }=-r^2{{\mathsf{\Gamma }}^{\theta }}_{rr}=-{\displaystyle \frac{{\partial }_{\theta }\mathsf{\Psi }}{1-2\mathsf{\Psi }}},\\ {{\mathsf{\Gamma }}^{\phi }}_{\phi \phi }={{\mathsf{\Gamma }}^r}_{r\phi }={{\mathsf{\Gamma }}^{\theta }}_{\theta \phi }=-{sin}^2\left(\theta \right){{\mathsf{\Gamma }}^{\phi }}_{\theta \theta }=-r^2{sin}^2\left(\theta \right){{\mathsf{\Gamma }}^{\phi }}_{rr}=-{\displaystyle \frac{{\partial }_{\phi }\mathsf{\Psi }}{1-2\mathsf{\Psi }}},\\ {{\mathsf{\Gamma }}^{\theta }}_{\phi \phi }=-{sin}^2\left(\theta \right){{\mathsf{\Gamma }}^{\phi }}_{\theta \phi }=-sin\left(\theta \right)cos\left(\theta \right)+{\displaystyle \frac{{sin}^2\left(\theta \right){\partial }_{\theta }\mathsf{\Psi }}{1-2\mathsf{\Psi }}},\\ {{\mathsf{\Gamma }}^{\eta }}_{r\theta }={{\mathsf{\Gamma }}^{\eta }}_{r\phi }={{\mathsf{\Gamma }}^{\eta }}_{\theta \phi }={{\mathsf{\Gamma }}^r}_{\eta \theta }={{\mathsf{\Gamma }}^r}_{\eta \phi }=0,\end{array}$$

$$\begin{array}{c}{{\mathsf{\Gamma }}^r}_{\theta \phi }={{\mathsf{\Gamma }}^{\theta }}_{\eta r}={{\mathsf{\Gamma }}^{\theta }}_{\eta \phi }={{\mathsf{\Gamma }}^{\theta }}_{r\phi }={{\mathsf{\Gamma }}^{\phi }}_{\eta \theta }={{\mathsf{\Gamma }}^{\phi }}_{\eta r}={{\mathsf{\Gamma }}^{\phi }}_{r\theta }=0.\end{array}$$

(A1)

Appendix A.2. Einstein Tensor

$$\begin{array}{cc}\hfill G_{\eta \eta }=& {\displaystyle \frac{3}{a^2}}{\left({\displaystyle \frac{da}{d\eta }}\right)}^2-{\displaystyle \frac{6}{a}}{\displaystyle \frac{da}{d\eta }}{\displaystyle \frac{{\partial }_{\eta }\mathsf{\Psi }}{1-2\mathsf{\Psi }}}+{\displaystyle \frac{3{\left({\partial }_{\eta }\mathsf{\Psi }\right)}^2}{{\left(1-2\mathsf{\Psi }\right)}^2}}+{\displaystyle \frac{2\left(1+2\mathsf{\Phi }\right)}{{\left(1-2\mathsf{\Psi }\right)}^2}}{\nabla }^2\mathsf{\Psi }\hfill \\ \hfill & +{\displaystyle \frac{3\left(1+2\mathsf{\Phi }\right)}{{\left(1-2\mathsf{\Psi }\right)}^3}}{\left(\nabla \mathsf{\Psi }\right)}^2,\hfill \end{array}$$

(A2)

$$\begin{array}{cc}\hfill G_{\eta r}=& {\displaystyle \frac{2}{a}}{\displaystyle \frac{da}{d\eta }}{\displaystyle \frac{{\partial }_r\mathsf{\Phi }}{1+2\mathsf{\Phi }}}+{\displaystyle \frac{2{\partial }_{\eta }{\partial }_r\mathsf{\Psi }}{1-2\mathsf{\Psi }}}-{\displaystyle \frac{2{\partial }_{\eta }\mathsf{\Psi }{\partial }_r\mathsf{\Phi }}{\left(1+2\mathsf{\Phi }\right)\left(1-2\mathsf{\Psi }\right)}}+{\displaystyle \frac{4{\partial }_{\eta }\mathsf{\Psi }{\partial }_r\mathsf{\Psi }}{{\left(1-2\mathsf{\Psi }\right)}^2}},\hfill \end{array}$$

(A3)

$$\begin{array}{cc}\hfill G_{\eta \theta }=& {\displaystyle \frac{2}{a}}{\displaystyle \frac{da}{d\eta }}{\displaystyle \frac{{\partial }_{\theta }\mathsf{\Phi }}{1+2\mathsf{\Phi }}}+{\displaystyle \frac{2{\partial }_{\eta }{\partial }_{\theta }\mathsf{\Psi }}{1-2\mathsf{\Psi }}}-{\displaystyle \frac{2{\partial }_{\eta }\mathsf{\Psi }{\partial }_{\theta }\mathsf{\Phi }}{\left(1+2\mathsf{\Phi }\right)\left(1-2\mathsf{\Psi }\right)}}+{\displaystyle \frac{4{\partial }_{\eta }\mathsf{\Psi }{\partial }_{\theta }\mathsf{\Psi }}{{\left(1-2\mathsf{\Psi }\right)}^2}},\hfill \end{array}$$

(A4)

$$\begin{array}{cc}\hfill G_{\eta \phi }=& {\displaystyle \frac{2}{a}}{\displaystyle \frac{da}{d\eta }}{\displaystyle \frac{{\partial }_{\phi }\mathsf{\Phi }}{1+2\mathsf{\Phi }}}+{\displaystyle \frac{2{\partial }_{\eta }{\partial }_{\phi }\mathsf{\Psi }}{1-2\mathsf{\Psi }}}-{\displaystyle \frac{2{\partial }_{\eta }\mathsf{\Psi }{\partial }_{\phi }\mathsf{\Phi }}{\left(1+2\mathsf{\Phi }\right)\left(1-2\mathsf{\Psi }\right)}}+{\displaystyle \frac{4{\partial }_{\eta }\mathsf{\Psi }{\partial }_{\phi }\mathsf{\Psi }}{{\left(1-2\mathsf{\Psi }\right)}^2}},\hfill \end{array}$$

(A5)

$$\begin{array}{cc}\hfill G_{rr}=& -{\displaystyle \frac{2\left(1-2\mathsf{\Psi }\right)}{1+2\mathsf{\Phi }}}{\displaystyle \frac{1}{a}}{\displaystyle \frac{d^{2}a}{d{\eta }^2}}+{\displaystyle \frac{1-2\mathsf{\Psi }}{1+2\mathsf{\Phi }}}{\displaystyle \frac{1}{a^2}}{\left({\displaystyle \frac{da}{d\eta }}\right)}^2+{\displaystyle \frac{2\left(1-2\mathsf{\Psi }\right)}{{\left(1+2\mathsf{\Phi }\right)}^2}}{\displaystyle \frac{1}{a}}{\displaystyle \frac{da}{d\eta }}{\partial }_{\eta }\mathsf{\Phi }\hfill \\ \hfill & +{\displaystyle \frac{4}{1+2\mathsf{\Phi }}}{\displaystyle \frac{1}{a}}{\displaystyle \frac{da}{d\eta }}{\partial }_{\eta }\mathsf{\Psi }+{\displaystyle \frac{2{\partial }_{\eta }^2\mathsf{\Psi }}{1+2\mathsf{\Phi }}}+{\displaystyle \frac{{\left({\partial }_{\eta }\mathsf{\Psi }\right)}^2}{\left(1+2\mathsf{\Phi }\right)\left(1-2\mathsf{\Psi }\right)}}-{\displaystyle \frac{2{\partial }_{\eta }\mathsf{\Phi }{\partial }_{\eta }\mathsf{\Psi }}{{\left(1+2\mathsf{\Phi }\right)}^2}}\hfill \\ \hfill & +{\displaystyle \frac{{\nabla }^2\mathsf{\Phi }-{\partial }_r^2\mathsf{\Phi }}{1+2\mathsf{\Phi }}}-{\displaystyle \frac{{\nabla }^2\mathsf{\Psi }-{\partial }_r^2\mathsf{\Psi }}{1-2\mathsf{\Psi }}}-{\displaystyle \frac{{\left(\nabla \mathsf{\Phi }\right)}^2-{\left({\partial }_r\mathsf{\Phi }\right)}^2}{{\left(1+2\mathsf{\Phi }\right)}^2}}-{\displaystyle \frac{2{\left(\nabla \mathsf{\Psi }\right)}^2-3{\left({\partial }_r\mathsf{\Psi }\right)}^2}{{\left(1-2\mathsf{\Psi }\right)}^2}}\hfill \\ \hfill & -{\displaystyle \frac{2{\partial }_r\mathsf{\Phi }{\partial }_r\mathsf{\Psi }}{\left(1+2\mathsf{\Phi }\right)\left(1-2\mathsf{\Psi }\right)}},\hfill \end{array}$$

(A6)

$$\begin{array}{cc}\hfill G_{r\theta }=& -{\displaystyle \frac{r}{1+2\mathsf{\Phi }}}{\partial }_r\left({\displaystyle \frac{{\partial }_{\theta }\mathsf{\Phi }}{r}}\right)+{\displaystyle \frac{r}{1-2\mathsf{\Psi }}}{\partial }_r\left({\displaystyle \frac{{\partial }_{\theta }\mathsf{\Psi }}{r}}\right)+{\displaystyle \frac{{\partial }_r\mathsf{\Phi }{\partial }_{\theta }\mathsf{\Phi }}{{\left(1+2\mathsf{\Phi }\right)}^2}}+{\displaystyle \frac{3{\partial }_r\mathsf{\Psi }{\partial }_{\theta }\mathsf{\Psi }}{{\left(1-2\mathsf{\Psi }\right)}^2}}\hfill \\ \hfill & -{\displaystyle \frac{{\partial }_{\theta }\mathsf{\Phi }{\partial }_r\mathsf{\Psi }+{\partial }_r\mathsf{\Phi }{\partial }_{\theta }\mathsf{\Psi }}{\left(1+2\mathsf{\Phi }\right)\left(1-2\mathsf{\Psi }\right)}},\hfill \end{array}$$

(A7)

$$\begin{array}{cc}\hfill G_{r\phi }=& -{\displaystyle \frac{r}{1+2\mathsf{\Phi }}}{\partial }_r\left({\displaystyle \frac{{\partial }_{\phi }\mathsf{\Phi }}{r}}\right)+{\displaystyle \frac{r}{1-2\mathsf{\Psi }}}{\partial }_r\left({\displaystyle \frac{{\partial }_{\phi }\mathsf{\Psi }}{r}}\right)+{\displaystyle \frac{{\partial }_r\mathsf{\Phi }{\partial }_{\phi }\mathsf{\Phi }}{{\left(1+2\mathsf{\Phi }\right)}^2}}+{\displaystyle \frac{3{\partial }_r\mathsf{\Psi }{\partial }_{\phi }\mathsf{\Psi }}{{\left(1-2\mathsf{\Psi }\right)}^2}}\hfill \\ \hfill & -{\displaystyle \frac{{\partial }_{\phi }\mathsf{\Phi }{\partial }_r\mathsf{\Psi }+{\partial }_r\mathsf{\Phi }{\partial }_{\phi }\mathsf{\Psi }}{\left(1+2\mathsf{\Phi }\right)\left(1-2\mathsf{\Psi }\right)}},\hfill \end{array}$$

(A8)

$$\begin{array}{cc}\hfill {\displaystyle \frac{G_{\theta \theta }}{r^2}}=& -{\displaystyle \frac{2\left(1-2\mathsf{\Psi }\right)}{1+2\mathsf{\Phi }}}{\displaystyle \frac{1}{a}}{\displaystyle \frac{d^{2}a}{d{\eta }^2}}+{\displaystyle \frac{1-2\mathsf{\Psi }}{1+2\mathsf{\Phi }}}{\displaystyle \frac{1}{a^2}}{\left({\displaystyle \frac{da}{d\eta }}\right)}^2+{\displaystyle \frac{2\left(1-2\mathsf{\Psi }\right)}{{\left(1+2\mathsf{\Phi }\right)}^2}}{\displaystyle \frac{1}{a}}{\displaystyle \frac{da}{d\eta }}{\partial }_{\eta }\mathsf{\Phi }\hfill \\ \hfill & +{\displaystyle \frac{4}{1+2\mathsf{\Phi }}}{\displaystyle \frac{1}{a}}{\displaystyle \frac{da}{d\eta }}{\partial }_{\eta }\mathsf{\Psi }+{\displaystyle \frac{2{\partial }_{\eta }^2\mathsf{\Psi }}{1+2\mathsf{\Phi }}}+{\displaystyle \frac{{\left({\partial }_{\eta }\mathsf{\Psi }\right)}^2}{\left(1+2\mathsf{\Phi }\right)\left(1-2\mathsf{\Psi }\right)}}-{\displaystyle \frac{2{\partial }_{\eta }\mathsf{\Phi }{\partial }_{\eta }\mathsf{\Psi }}{{\left(1+2\mathsf{\Phi }\right)}^2}}\hfill \\ \hfill & +{\displaystyle \frac{{\nabla }^2\mathsf{\Phi }-\frac{1}{r}{\partial }_r\mathsf{\Phi }-\frac{1}{r^2}{\partial }_{\theta }^2\mathsf{\Phi }}{1+2\mathsf{\Phi }}}-{\displaystyle \frac{{\nabla }^2\mathsf{\Psi }-\frac{1}{r}{\partial }_r\mathsf{\Psi }-\frac{1}{r^2}{\partial }_{\theta }^2\mathsf{\Psi }}{1-2\mathsf{\Psi }}}-{\displaystyle \frac{{\left(\nabla \mathsf{\Phi }\right)}^2-\frac{1}{r^2}{\left({\partial }_{\theta }\mathsf{\Phi }\right)}^2}{{\left(1+2\mathsf{\Phi }\right)}^2}}\hfill \\ \hfill & -{\displaystyle \frac{2{\left(\nabla \mathsf{\Psi }\right)}^2-\frac{3}{r^2}{\left({\partial }_{\theta }\mathsf{\Psi }\right)}^2}{{\left(1-2\mathsf{\Psi }\right)}^2}}-{\displaystyle \frac{\frac{2}{r^2}{\partial }_{\theta }\mathsf{\Phi }{\partial }_{\theta }\mathsf{\Psi }}{\left(1+2\mathsf{\Phi }\right)\left(1-2\mathsf{\Psi }\right)}},\hfill \end{array}$$

(A9)

$$\begin{array}{cc}\hfill G_{\theta \phi }=& -{\displaystyle \frac{sin\left(\theta \right)}{1+2\mathsf{\Phi }}}{\partial }_{\theta }\left({\displaystyle \frac{{\partial }_{\phi }\mathsf{\Phi }}{sin\left(\theta \right)}}\right)+{\displaystyle \frac{sin\left(\theta \right)}{1-2\mathsf{\Psi }}}{\partial }_{\theta }\left({\displaystyle \frac{{\partial }_{\phi }\mathsf{\Psi }}{sin\left(\theta \right)}}\right)+{\displaystyle \frac{{\partial }_{\theta }\mathsf{\Phi }{\partial }_{\phi }\mathsf{\Phi }}{{\left(1+2\mathsf{\Phi }\right)}^2}}+{\displaystyle \frac{3{\partial }_{\theta }\mathsf{\Psi }{\partial }_{\phi }\mathsf{\Psi }}{{\left(1-2\mathsf{\Psi }\right)}^2}}\hfill \\ \hfill & -{\displaystyle \frac{{\partial }_{\theta }\mathsf{\Phi }{\partial }_{\phi }\mathsf{\Psi }+{\partial }_{\phi }\mathsf{\Phi }{\partial }_{\theta }\mathsf{\Psi }}{\left(1+2\mathsf{\Phi }\right)\left(1-2\mathsf{\Psi }\right)}},\hfill \end{array}$$

(A10)

$$\begin{array}{cc}\hfill {\displaystyle \frac{G_{\phi \phi }}{r^2{sin}^2\left(\theta \right)}}=& -{\displaystyle \frac{2\left(1-2\mathsf{\Psi }\right)}{1+2\mathsf{\Phi }}}{\displaystyle \frac{1}{a}}{\displaystyle \frac{d^{2}a}{d{\eta }^2}}+{\displaystyle \frac{1-2\mathsf{\Psi }}{1+2\mathsf{\Phi }}}{\displaystyle \frac{1}{a^2}}{\left({\displaystyle \frac{da}{d\eta }}\right)}^2+{\displaystyle \frac{2\left(1-2\mathsf{\Psi }\right)}{{\left(1+2\mathsf{\Phi }\right)}^2}}{\displaystyle \frac{1}{a}}{\displaystyle \frac{da}{d\eta }}{\partial }_{\eta }\mathsf{\Phi }\hfill \\ \hfill & +{\displaystyle \frac{4}{1+2\mathsf{\Phi }}}{\displaystyle \frac{1}{a}}{\displaystyle \frac{da}{d\eta }}{\partial }_{\eta }\mathsf{\Psi }+{\displaystyle \frac{2{\partial }_{\eta }^2\mathsf{\Psi }}{1+2\mathsf{\Phi }}}+{\displaystyle \frac{{\left({\partial }_{\eta }\mathsf{\Psi }\right)}^2}{\left(1+2\mathsf{\Phi }\right)\left(1-2\mathsf{\Psi }\right)}}-{\displaystyle \frac{2{\partial }_{\eta }\mathsf{\Phi }{\partial }_{\eta }\mathsf{\Psi }}{{\left(1+2\mathsf{\Phi }\right)}^2}}\hfill \\ \hfill & +{\displaystyle \frac{{\partial }_r^2\mathsf{\Phi }+\frac{1}{r}{\partial }_r\mathsf{\Phi }+\frac{1}{r^2}{\partial }_{\theta }^2\mathsf{\Phi }}{1+2\mathsf{\Phi }}}-{\displaystyle \frac{{\partial }_r^2\mathsf{\Psi }+\frac{1}{r}{\partial }_r\mathsf{\Psi }+\frac{1}{r^2}{\partial }_{\theta }^2\mathsf{\Psi }}{1-2\mathsf{\Psi }}}-{\displaystyle \frac{{\left(\nabla \mathsf{\Phi }\right)}^2-\frac{1}{r^2{sin}^2\left(\theta \right)}{\left({\partial }_{\phi }\mathsf{\Phi }\right)}^2}{{\left(1+2\mathsf{\Phi }\right)}^2}}\hfill \\ \hfill & -{\displaystyle \frac{2{\left(\nabla \mathsf{\Psi }\right)}^2-\frac{3}{r^2{sin}^2\left(\theta \right)}{\left({\partial }_{\phi }\mathsf{\Psi }\right)}^2}{{\left(1-2\mathsf{\Psi }\right)}^2}}-{\displaystyle \frac{\frac{2}{r^2{sin}^2\left(\theta \right)}{\partial }_{\phi }\mathsf{\Phi }{\partial }_{\phi }\mathsf{\Psi }}{\left(1+2\mathsf{\Phi }\right)\left(1-2\mathsf{\Psi }\right)}}.\hfill \end{array}$$

(A11)

Appendix A.3. Curvature Scalar

$$\begin{array}{cc}\hfill R=& {\displaystyle \frac{6}{a^3\left(1+2\mathsf{\Phi }\right)}}{\displaystyle \frac{d^{2}a}{d{\eta }^2}}-{\displaystyle \frac{6{\partial }_{\eta }\mathsf{\Phi }}{a^3{\left(1+2\mathsf{\Phi }\right)}^2}}{\displaystyle \frac{da}{d\eta }}-{\displaystyle \frac{18{\partial }_{\eta }\mathsf{\Psi }}{a^3\left(1+2\mathsf{\Phi }\right)\left(1-2\mathsf{\Psi }\right)}}{\displaystyle \frac{da}{d\eta }}\hfill \\ \hfill & +{\displaystyle \frac{6{\partial }_{\eta }\mathsf{\Phi }{\partial }_{\eta }\mathsf{\Psi }}{a^2{\left(1+2\mathsf{\Phi }\right)}^2\left(1-2\mathsf{\Psi }\right)}}-{\displaystyle \frac{6{\partial }_{\eta }^2\mathsf{\Psi }}{a^2\left(1+2\mathsf{\Phi }\right)\left(1-2\mathsf{\Psi }\right)}}-{\displaystyle \frac{2{\nabla }^2\mathsf{\Phi }}{a^2\left(1+2\mathsf{\Phi }\right)\left(1-2\mathsf{\Psi }\right)}}\hfill \\ \hfill & +{\displaystyle \frac{4{\nabla }^2\mathsf{\Psi }}{a^2{\left(1-2\mathsf{\Psi }\right)}^2}}+{\displaystyle \frac{2{\left(\nabla \mathsf{\Phi }\right)}^2}{a^2{\left(1+2\mathsf{\Phi }\right)}^2\left(1-2\mathsf{\Psi }\right)}}+{\displaystyle \frac{6{\left(\nabla \mathsf{\Psi }\right)}^2}{a^2{\left(1-2\mathsf{\Psi }\right)}^3}}\hfill \\ \hfill & +{\displaystyle \frac{2\nabla \mathsf{\Phi }·\nabla \mathsf{\Psi }}{a^2\left(1+2\mathsf{\Phi }\right){\left(1-2\mathsf{\Psi }\right)}^2}}.\hfill \end{array}$$

(A12)

Appendix A.4. Double Covariant Derivative on a Scalar

$$\begin{array}{cc}\hfill {\nabla }_{\eta }{\nabla }_{\eta }\varphi =& {\partial }_{\eta }^2\varphi -{\displaystyle \frac{1}{a}}{\displaystyle \frac{da}{d\eta }}{\partial }_{\eta }\varphi -{\displaystyle \frac{{\partial }_{\eta }\mathsf{\Phi }{\partial }_{\eta }\varphi }{1+2\mathsf{\Phi }}}-{\displaystyle \frac{\nabla \mathsf{\Phi }·\nabla \varphi }{1-2\mathsf{\Psi }}},\hfill \end{array}$$

(A13)

$$\begin{array}{cc}\hfill {\nabla }_{\eta }{\nabla }_r\varphi =& {\partial }_{\eta }{\partial }_r\varphi -{\displaystyle \frac{1}{a}}{\displaystyle \frac{da}{d\eta }}{\partial }_r\varphi -{\displaystyle \frac{{\partial }_r\mathsf{\Phi }{\partial }_{\eta }\varphi }{1+2\mathsf{\Phi }}}+{\displaystyle \frac{{\partial }_{\eta }\mathsf{\Psi }{\partial }_r\varphi }{1-2\mathsf{\Psi }}},\hfill \end{array}$$

(A14)

$$\begin{array}{cc}\hfill {\nabla }_{\eta }{\nabla }_{\theta }\varphi =& {\partial }_{\eta }{\partial }_{\theta }\varphi -{\displaystyle \frac{1}{a}}{\displaystyle \frac{da}{d\eta }}{\partial }_{\theta }\varphi -{\displaystyle \frac{{\partial }_{\theta }\mathsf{\Phi }{\partial }_{\eta }\varphi }{1+2\mathsf{\Phi }}}+{\displaystyle \frac{{\partial }_{\eta }\mathsf{\Psi }{\partial }_{\theta }\varphi }{1-2\mathsf{\Psi }}},\hfill \end{array}$$

(A15)

$$\begin{array}{cc}\hfill {\nabla }_{\eta }{\nabla }_{\phi }\varphi =& {\partial }_{\eta }{\partial }_{\phi }\varphi -{\displaystyle \frac{1}{a}}{\displaystyle \frac{da}{d\eta }}{\partial }_{\phi }\varphi -{\displaystyle \frac{{\partial }_{\phi }\mathsf{\Phi }{\partial }_{\eta }\varphi }{1+2\mathsf{\Phi }}}+{\displaystyle \frac{{\partial }_{\eta }\mathsf{\Psi }{\partial }_{\phi }\varphi }{1-2\mathsf{\Psi }}},\hfill \end{array}$$

(A16)

$$\begin{array}{cc}\hfill {\nabla }_r{\nabla }_r\varphi =& {\partial }_r^2\varphi -{\displaystyle \frac{1}{a}}{\displaystyle \frac{da}{d\eta }}{\displaystyle \frac{1-2\mathsf{\Psi }}{1+2\mathsf{\Phi }}}{\partial }_{\eta }\varphi +{\displaystyle \frac{{\partial }_{\eta }\mathsf{\Psi }{\partial }_{\eta }\varphi }{1+2\mathsf{\Phi }}}-{\displaystyle \frac{\nabla \mathsf{\Psi }·\nabla \varphi }{1-2\mathsf{\Psi }}}+{\displaystyle \frac{2{\partial }_r\mathsf{\Psi }{\partial }_r\varphi }{1-2\mathsf{\Psi }}},\hfill \end{array}$$

(A17)

$$\begin{array}{cc}\hfill {\nabla }_r{\nabla }_{\theta }\varphi =& r{\partial }_r\left({\displaystyle \frac{{\partial }_{\theta }\varphi }{r}}\right)+{\displaystyle \frac{{\partial }_{\theta }\mathsf{\Psi }{\partial }_r\varphi +{\partial }_r\mathsf{\Psi }{\partial }_{\theta }\varphi }{1-2\mathsf{\Psi }}},\hfill \end{array}$$

(A18)

$$\begin{array}{cc}\hfill {\nabla }_r{\nabla }_{\phi }\varphi =& r{\partial }_r\left({\displaystyle \frac{{\partial }_{\phi }\varphi }{r}}\right)+{\displaystyle \frac{{\partial }_{\phi }\mathsf{\Psi }{\partial }_r\varphi +{\partial }_r\mathsf{\Psi }{\partial }_{\phi }\varphi }{1-2\mathsf{\Psi }}},\hfill \end{array}$$

(A19)

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\nabla }_{\theta }{\nabla }_{\theta }\varphi }{r^2}}=& {\displaystyle \frac{{\partial }_{\theta }^2\varphi }{r^2}}-{\displaystyle \frac{1}{a}}{\displaystyle \frac{da}{d\eta }}{\displaystyle \frac{1-2\mathsf{\Psi }}{1+2\mathsf{\Phi }}}{\partial }_{\eta }\varphi +{\displaystyle \frac{{\partial }_{\eta }\mathsf{\Psi }{\partial }_{\eta }\varphi }{1+2\mathsf{\Phi }}}+{\displaystyle \frac{{\partial }_r\varphi }{r}}-{\displaystyle \frac{\nabla \mathsf{\Psi }·\nabla \varphi }{1-2\mathsf{\Psi }}}\hfill \\ \hfill & +{\displaystyle \frac{2{\partial }_{\theta }\mathsf{\Psi }{\partial }_{\theta }\varphi }{r^2\left(1-2\mathsf{\Psi }\right)}},\hfill \end{array}$$

(A20)

$$\begin{array}{cc}\hfill {\nabla }_{\theta }{\nabla }_{\phi }\varphi =& sin\left(\theta \right){\partial }_{\theta }\left({\displaystyle \frac{{\partial }_{\phi }\varphi }{sin\left(\theta \right)}}\right)+{\displaystyle \frac{{\partial }_{\phi }\mathsf{\Psi }{\partial }_{\theta }\varphi +{\partial }_{\theta }\mathsf{\Psi }{\partial }_{\phi }\varphi }{1-2\mathsf{\Psi }}},\hfill \end{array}$$

(A21)

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\nabla }_{\phi }{\nabla }_{\phi }\varphi }{r^2{sin}^2\left(\theta \right)}}=& {\displaystyle \frac{{\partial }_{\phi }^2\varphi }{r^2{sin}^2\left(\theta \right)}}-{\displaystyle \frac{1}{a}}{\displaystyle \frac{da}{d\eta }}{\displaystyle \frac{1-2\mathsf{\Psi }}{1+2\mathsf{\Phi }}}{\partial }_{\eta }\varphi +{\displaystyle \frac{{\partial }_{\eta }\mathsf{\Psi }{\partial }_{\eta }\varphi }{1+2\mathsf{\Phi }}}+{\displaystyle \frac{{\partial }_r\varphi }{r}}+{\displaystyle \frac{cos\left(\theta \right)}{r^{2}sin\left(\theta \right)}}{\partial }_{\theta }\varphi \hfill \\ \hfill & -{\displaystyle \frac{\nabla \mathsf{\Psi }·\nabla \varphi }{1-2\mathsf{\Psi }}}+{\displaystyle \frac{2{\partial }_{\phi }\mathsf{\Psi }{\partial }_{\phi }\varphi }{r^2{sin}^2\left(\theta \right)\left(1-2\mathsf{\Psi }\right)}}.\hfill \end{array}$$

(A22)

Appendix A.5. Covariant D’Alembertian on a Scalar

$$\begin{array}{cc}\hfill {\nabla }^{\alpha }{\nabla }_{\alpha }\varphi =& -{\displaystyle \frac{{\partial }_{\eta }^2\varphi }{a^2\left(1+2\mathsf{\Phi }\right)}}+{\displaystyle \frac{{\nabla }^2\varphi }{a^2\left(1-2\mathsf{\Psi }\right)}}-{\displaystyle \frac{2}{a^3}}{\displaystyle \frac{da}{d\eta }}{\displaystyle \frac{{\partial }_{\eta }\varphi }{1+2\mathsf{\Phi }}}+{\displaystyle \frac{{\partial }_{\eta }\mathsf{\Phi }{\partial }_{\eta }\varphi }{a^2{\left(1+2\mathsf{\Phi }\right)}^2}}\hfill \\ \hfill & +{\displaystyle \frac{3{\partial }_{\eta }\mathsf{\Psi }{\partial }_{\eta }\varphi }{a^2\left(1+2\mathsf{\Phi }\right)\left(1-2\mathsf{\Psi }\right)}}+{\displaystyle \frac{\nabla \mathsf{\Phi }·\nabla \varphi }{a^2\left(1+2\mathsf{\Phi }\right)\left(1-2\mathsf{\Psi }\right)}}-{\displaystyle \frac{\nabla \mathsf{\Psi }·\nabla \varphi }{a^2{\left(1-2\mathsf{\Psi }\right)}^2}}.\hfill \end{array}$$

(A23)