Cosmology in Minkowski space

Let us begin with some general considerations on the relationship between gravitational theory, energy-momentum and geometry in the Einstein field equations, $G_{\mu\nu} = \kappa^2 T_{\mu\nu}$, where $\kappa^2 \equiv 8\pi G c^{-4}$ denotes the gravitational coupling and the metric signature $(-,+,+,+)$ will be adopted throughout the article. Suppose the gravitational theory is modified, and the energy-momentum tensor $T_{\mu\nu}$ relates to a modified tensor $M_{\mu\nu}$ rather than to the Einstein tensor $G_{\mu\nu}$ such that

$$\begin{equation} M_{\mu\nu} = \kappa^2 T_{\mu\nu} \,. \end{equation} \tag{ 1 }$$

This relation can always be recast as the usual Einstein field equation with the definition of an effective energy-momentum tensor,

$$\begin{equation} G_{\mu\nu} = \kappa^2 T_{\mu\nu} + \kappa^2 T\,{^\mathrm{eff}_{\mu\nu}} \,, \quad \quad T\,{^\mathrm{eff}_{\mu\nu}} \equiv \kappa^{-2}G_{\mu\nu} - \kappa^{-2}M_{\mu\nu} \,, \end{equation} \tag{ 2 }$$

where covariant conservation of $T_{\mu\nu}$ and the Bianchi identities imply covariant conservation of $T\,{^\mathrm{eff}_{\mu\nu}}$. This ambiguity between modifications in the metric and matter sectors has also been coined the dark degeneracy in [13, 14].

We shall now add another layer of degeneracy, stemming from the choice of geometry. Let us assume the metric $\hat{g}_{\mu\nu}$ satisfies

$$\begin{equation} \hat{G}_{\mu\nu} = \kappa^2 \hat{T}_{\mu\nu} \end{equation} \tag{ 3 }$$

for some specified matter content in $\hat{T}_{\mu\nu}$. The field equations can now be recast into a different geometry $\tilde{g}_{\mu\nu} = A_{\;\;\;\mu\nu}^{\alpha\beta} \hat{g}_{\alpha\beta} + B_{\mu\nu}$ as

$$\begin{equation} \tilde{G}_{\mu\nu} = \kappa^2 \tilde{T}^\mathrm{eff}_{\mu\nu} \,, \quad \quad \tilde{T}^\mathrm{eff}_{\mu\nu}(\tilde{g}) \equiv \left(\kappa^{-2}\tilde{G}_{\mu\nu} - \kappa^{-2}\hat{G}_{\mu\nu} + \hat{T}_{\mu\nu}\right)(\tilde{g}) \,. \end{equation} \tag{ 4 }$$

This is merely a mathematical manipulation, simply a substitution of the variable one is solving the differential equation (3) for. Just as with any other differential equation, one may always perform such a change of variable, in this case the metric. The physics remains unchanged. Note, however, that freely falling particles in $\hat{T}_{\mu\nu}(\tilde{g})$ no longer follow geodesics for $\tilde{g}_{\alpha\beta}$, which manifests as an additional interaction and breaks a degeneracy for visible matter species between the two frames set by the two different metrics. In contrast, an interacting dark sector is simply receiving additional interactions. Alternatively to equation (4), one could also define modified geometric and matter sectors such as $\tilde{M}_{\mu\nu} \equiv \hat{G}_{\mu\nu}(\tilde{g})$ and $\tilde{T}^\mathrm{eff}_{\mu\nu} \equiv \hat{T}_{\mu\nu}(\tilde{g})$ in $\tilde{M}_{\mu\nu}(\tilde{g}) = \kappa^2 \tilde{T}^\mathrm{eff}_{\mu\nu}(\tilde{g})$ or other terms that are equation (3) in disguise. A consequence of these reframed field equations is that one may cast the physics of a system into a spacetime geometry of choice. The implications of that for our physical interpretation and understanding of the observed Universe will be the main subject of this article.

Note that the transformation between the Jordan and Einstein frames of modified gravity models is related to equation (4). Hereby, a metric transformation is applied to recast the field equations $M_{\mu\nu}(g) = T_{\mu\nu}(g)$ in Jordan frame into Einstein frame with $\tilde{G}_{\mu\nu}(\tilde{g}) = \tilde{T}^\mathrm{eff}_{\mu\nu}(\tilde{g})$, where $\tilde{T}^\mathrm{eff}_{\mu\nu}(\tilde{g})$ contains $T_{\mu\nu}(\tilde{g})$ and contributions from $M_{\mu\nu}(\tilde{g})$. Thereby, $M_{\mu\nu}(g)$ and $T_{\mu\nu}(g)$ as well as $\tilde{G}_{\mu\nu}(\tilde{g})$ and $\tilde{T}^\mathrm{eff}_{\mu\nu}(\tilde{g})$ are covariantly conserved whereas $M_{\mu\nu}(\tilde{g})$ and $T_{\mu\nu}(\tilde{g})$ are not.

Finally, one may also impose a specific matter content and invert equation (1). For that, assume the following field equations hold for $g_{\mu\nu}$,

$$\begin{equation} G_{\mu\nu} = \kappa^2 T^{(1)}_{\mu\nu} \,. \end{equation} \tag{ 5 }$$

One can now enforce a different matter content $T^{(2)}_{\mu\nu}(g)$ such that

$$\begin{equation} M_{\mu\nu} = \kappa^2 T^{(2)}_{\mu\nu} \,, \quad \quad M_{\mu\nu}(g) \equiv \left(G_{\mu\nu} + \kappa^2 T^{(2)}_{\mu\nu} - \kappa^2 T^{(1)}_{\mu\nu}\right)(g) \,. \end{equation} \tag{ 6 }$$

For example, one may want to evade invoking dark matter and/or dark energy to explain our observations and rather interpret them as some effective components attributed to an underlying modification of gravity. Equation (6) implies that one can in principle construct such a modified gravity model which will be phenomenologically equivalent to ΛCDM. It can be obtained from including dark matter and/or dark energy in $T^{(1)}_{\mu\nu}$ but not in $T^{(2)}_{\mu\nu}$. The intricate part is to find $T^{(2)}_{\mu\nu}$ as a function of $g_{\mu\nu}$ so that the procedure is universal, and when adopting a different metric than $g_{\mu\nu}$ to describe some other gravitating system, the predicted phenomenology of this new system remains invariant, or at least it should remain compatible with the measurements of that. Of course, for the model to be falsifiable, one would hope that this invariance is broken in some systems.

Instead of specifying the matter content in equation (6), one could enforce a specific modified tensor $M_{\mu\nu}$ and adapt the energy-momentum tensor accordingly. One may also specify the matter content and the metric and construct the tensor $M_{\mu\nu}$ valid for that. Generally, one can specify any two components between (modified) Einstein tensor, energy-momentum tensor and metric by adaption of the third.

The form of the tensors on either side of the field equation (1) is thus not a useful discriminator for the classification between dark sector, interacting dark sector and modified gravity models. Instead, the distinction can be drawn upon whether the matter species of a theory satisfy the strong and weak equivalence principles [15]. The dark sector in Einstein's Theory of General Relativity satisfies both the weak and strong equivalence principles whereas interacting dark sector models break both. Modified gravity models satisfy the weak equivalence principle as a metric can be found, typically the Jordan frame, to which all matter species couple universally and along which they freely fall whereas they break the strong equivalence principle since this circumstance depends on a body's composition. It is not necessary, and arguably not more natural, to cast the gravitational theory in the (Jordan) frame where (some) matter species couple minimally to the metric, and physical equations and interpretations may sometimes be more practical in Einstein frame. The same applies for the geometric transformations in equation (4).

To recast the Einstein–Hilbert action and the Einstein field equations into different geometries, particularly into Minkowski spacetime, the metric decomposition will be performed employing combinations of conformal transformations and dimensional reduction. For completeness, the details of these transformations shall be provided here. Readers familiar with Riemannian geometry may want to skip this section.

Generally, for the conformal transformation of a N-dimensional metric $g_{\mu\nu}$ with the conformal factor Ω²,

$$\begin{equation} \tilde{g}_{\mu\nu}(x^{\xi}) = \Omega^2(x^{\xi}) g_{\mu\nu}(x^{\xi}) \,, \end{equation} \tag{ 7 }$$

the Ricci tensor and curvature transform as

$$\begin{eqnarray} &&R_{\mu\nu} = \tilde{R}_{\mu\nu} + \frac{N-2}{\Omega}\tilde{\nabla}_{\mu} \tilde{\nabla}_{\nu} \Omega + \left( \frac{1}{\Omega} \tilde{\nabla}_{\xi} \tilde{\nabla}^{\xi} \Omega - \frac{N-1}{\Omega^2} \tilde{\nabla}_{\xi} \Omega \tilde{\nabla}^{\xi} \Omega \right) \tilde{g}_{\mu\nu} \,, \end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray} &&R = \Omega^2 \tilde{R} + 2(N-1)\Omega\tilde{\nabla}_{\xi} \tilde{\nabla}^{\xi} \Omega - N(N-1) \tilde{\nabla}_{\xi} \Omega \tilde{\nabla}^{\xi} \Omega \,. \end{eqnarray} \tag{ 9 }$$

The Einstein tensor $G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}R g_{\mu\nu} \ (+ \; \Lambda g_{\mu\nu})$ transforms accordingly.

For the dimensional reduction, consider a warped spacetime metric in $N = n+m$ dimensions,

$$\begin{equation} g_{\mu\nu}(x^{\xi}) dx^{\mu}dx^{\nu} = \tilde{g}_{ij}(x^k) dx^i dx^j + F^2(x^k)\hat{g}_{AB}(x^C) dx^A dx^B \,, \end{equation} \tag{ 10 }$$

where Greek indices run over n + m dimensions, lowercase Roman letters over $m = \dim(\tilde{M})$ dimensions and capital Roman letters over $n = \dim(\hat{M})$ co-dimensions. The Ricci tensor and curvature can then be decomposed as

$$\begin{eqnarray} &&R_{ij} = \tilde{R}_{ij} - \frac{n}{F} \tilde{\nabla}_i \tilde{\nabla}_j F \,, \quad R_{AB} = \hat{R}_{AB} - \hat{g}_{AB} \left[ (n - 1) \tilde{\nabla}_k F + F \tilde{\nabla}_k \right]\tilde{\nabla}^k F \,, \end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray} &&R_{Ai} = 0 \,, \quad R = \tilde{R} + \frac{1}{F^2}\left[ \hat{R} - n(n-1) \tilde{\nabla}_k F - 2 n F \tilde{\nabla}_k \right] \tilde{\nabla}^k F \,. \end{eqnarray} \tag{ 12 }$$

The conformal transformations (7) and (9) will be applied to the Einstein–Hilbert action in section 2.3. The application of the dimensional reduction of $g_{\mu\nu}$ and R to the action can be conducted analogously to [16] and shall not be performed explicitly here (see, however, sections 3.2–3.4 for an implementation in the matter sector).

For a concrete example of how the Einstein–Hilbert action and the Einstein field equations transform under the metric modifications of section 2.2, consider the general conformal transformation $\tilde{g}_{\mu\nu} = \Omega^2(\tau,\mathbf{x})g_{\mu\nu}$ of the $(3+1)$-dimensional metric $g_{\mu\nu}(\tau,\mathbf{x})$. The Einstein–Hilbert action for $g_{\mu\nu}$ reads

$$\begin{equation} S = \frac{1}{2\kappa^2} \int d^4x \sqrt{-g} \left( R - 2\Lambda \right) + \int d^4x \sqrt{-g} \mathcal{L}_m(g_{\mu\nu},\Psi_m) \,, \end{equation} \tag{ 13 }$$

where $\mathcal{L}_m$ is the Lagrangian density of the matter fields $\Psi_m$. Under the conformal transformation this becomes

$$\begin{equation} S = \frac{1}{2\kappa^2} \int d^4x \sqrt{-\tilde{g}} \left( \Omega^{-2} \tilde{R} + 6 \Omega^{-4} \tilde{\nabla}_{\mu}\Omega \tilde{\nabla}^{\mu}\Omega - 2\Omega^{-4}\Lambda \right) + S_m[\Omega^{-2}\tilde{g}_{\mu\nu},\Psi_m] \,, \end{equation} \tag{ 14 }$$

where the matter sector is given by

$$\begin{equation} S_m[\Omega^{-2}\tilde{g}_{\mu\nu},\Psi_m] = \int d^4x \sqrt{-\tilde{g}} \Omega^{-4} \mathcal{L}_m(\Omega^{-2}\tilde{g}_{\mu\nu},\Psi_m) \,. \end{equation} \tag{ 15 }$$

Note that the matter fields are no longer minimally coupled to the metric (see section 2.1).

For simplification, one may define $\psi \equiv \Omega^{-1}$ to arrive at the action

$$\begin{equation} \hat{S} \equiv -\frac{\kappa^2 S}{6} = \int d\tilde{V} \left[ -\frac{1}{2}\tilde{\nabla}_{\mu}\psi\tilde{\nabla}^{\mu}\psi -V(\psi) \right] - \frac{\kappa^2}{6} S_m[\psi^2\tilde{g}_{\mu\nu},\Psi_m] \end{equation} \tag{ 16 }$$

with volume element $d\tilde{V} = d^4x\sqrt{-\tilde{g}}$ and potential

$$\begin{equation} V(\psi) = \frac{1}{12} \left( \tilde{R} \psi^2 - 2\Lambda \psi^4 \right) \,. \end{equation} \tag{ 17 }$$

Variation of equation (16) with respect to the metric $\tilde{g}_{\mu\nu}$ yields the Einstein field equations

$$\begin{align} &\psi^2 \left(\tilde{R}_{\mu\nu} - \frac{1}{2}\tilde{g}_{\mu\nu}\tilde{R}\right) + 4 \left( \partial_{\mu}\psi \partial_{\nu}\psi - \frac{1}{4} \tilde{g}_{\mu\nu} \partial_{\alpha}\psi \partial^{\alpha}\psi \right) - 2\psi \left( \tilde{\nabla}_{\mu}\partial_{\nu}\psi - \tilde{g}_{\mu\nu} \tilde{\Box} \psi \right)\nonumber\\ & \quad + \tilde{g}_{\mu\nu}\Lambda\psi^4 = \kappa^2 \tilde{T}_{\mu\nu} \,, \end{align} \tag{ 18 }$$

where $\tilde{T}^{\mu\nu} \equiv (2/\sqrt{-\tilde{g}}) \delta L_m/\delta\tilde{g}_{\mu\nu}$ is the energy-momentum tensor in the transformed frame with $S_m = \int d^4x L_m$ and $\tilde{\Box} = (-\tilde{g})^{-1/2} \partial_{\mu} \left( \sqrt{-\tilde{g}} \tilde{g}^{\mu\nu} \partial_{\nu} \right)$. $\tilde{T}_{\mu\nu}$ relates to its analogous expression $T_{\mu\nu}$ from equation (13) as $\tilde{T}_{\mu\nu} = \psi^2 T_{\mu\nu}$ and its trace transforms as $\tilde{T} = \psi^4 T$.

The action (16) and field equations (18) resemble those of a scalar-tensor modification of gravity, where conformal transformations are used to convert between the Einstein and Jordan frames. Note, however, that for scalar-tensor theories, these frames interchange minimal and non-minimal couplings to the tensor field between the metric and matter sectors whereas the conformal transformation from equations (13) to (16) introduces a non-minimal coupling in both sectors, initially not coupled to ψ. We shall thus refrain here from using the terminology of Einstein and Jordan frames to avoid confusion.

Varying the action (16) with respect to ψ yields the field equation

$$\begin{equation} \tilde{\Box}\psi - V^{^{\prime}}(\psi) = \frac{\kappa^2}{6} \psi^{-1} \tilde{T} \,. \end{equation} \tag{ 19 }$$

Finally, note that the energy-momentum tensor is not conserved in the transformed frame,

$$\begin{equation} \tilde{\nabla}_{\mu} \tilde{T}^{\mu}_{\,\,\nu} = \psi^{-1} \tilde{T} \partial_{\nu} \psi \,. \end{equation} \tag{ 20 }$$

The $(3+1)$-dimensional spatially homogeneous and isotropic expanding cosmological background is described by the Friedmann–Lemaître–Robertson–Walker (FLRW) line element

$$\begin{equation} ds^2 = -c^2 dt^2 + a(t)^2 d\mathbf{x}^2 \,, \end{equation} \tag{ 21 }$$

where t denotes cosmic time, a(t) is the scale factor and $d\mathbf{x}^2 = (1-kr^2)^{-1}dr^2 + r^2(d\theta^2 + \sin^2\theta \: d\phi^2)$ with spatial curvature k. Cast in conformal time τ, equation (21) becomes

$$\begin{equation} ds^2 = a^2(\tau)\left[-c^2 d\tau^2 + d\mathbf{x}^2 \right] \,, \end{equation} \tag{ 22 }$$

where $d\tau = a^{-1}dt$. Hence, $a^2(\tau)$ acts as a conformal factor in equation (22).

For simplicity briefly assuming the absence of spatial curvature (k = 0), the metric $g_{\mu\nu}$ for $ds^2 = g_{\mu\nu}dx^{\mu}dx^{\nu}$ in equation (22) becomes

$$\begin{equation} g_{\mu\nu} = a^2(\tau) \eta_{\mu\nu} \,, \end{equation} \tag{ 23 }$$

where $\eta_{\mu\nu} = \textrm{diag}(-1, 1, 1, 1)$ is the Minkowski metric. Thus, $g_{\mu\nu}$ is a conformal metric transformation of Minkowski spacetime with conformal factor $a^2(\tau)$. The inverse conformal transformation $\eta_{\mu\nu} = a^{-2}(\tau) g_{\mu\nu}$ can then be used to cast the Einstein–Hilbert action into Minkowski space. One may set $\tilde{g}_{\mu\nu} = \eta_{\mu\nu}$, $\psi = a$, and $\tilde{R} = 0$ in equation (16) to arrive at

$$\begin{equation} \hat{S} = \int d^4x \left[ -\frac{1}{2}\partial_{\mu}\psi\partial^{\mu}\psi - V(\psi) \right] - \frac{\kappa^2}{6} S_m[\psi^2\eta_{\mu\nu},\Psi_m] \end{equation} \tag{ 24 }$$

with $V(\psi) = -\Lambda\psi^4/6$.

It is worthwhile noting that action (24) also holds for k ≠ 0, in which case, however, ψ becomes both time and space dependent. This is due to the FLRW metric generally being conformally flat with vanishing Weyl tensor, and so $g_{\mu\nu} = a^2(T,R) \eta_{\mu\nu}$ with k ≠ 0 for suitable coordinate choices T and R [17–20]. Section 3.2 will present a different approach to recovering the gravitational action and field equations in Minkowski space for scenarios with k ≠ 0.

Consider now the conformal transformation (7) of the metric $g_{\mu\nu}$ defined by the line element (21), allowing k ≠ 0, with $\psi = \Omega^{-1} = a(\tau)$ such that $\tilde{g}_{\mu\nu}$ is static. Importantly, as highlighted in section 2.1, given that this is merely a mathematical transformation, observed physical relations should remain unaffected by the transformation. Indeed, as shown in the following, despite the static cosmological spacetime, the Friedmann equations and the evolution of energy densities (section 3.1.1) as well as redshift of distant emissions (section 3.1.2) are recovered.

3.1.1. Friedmann equations and energy density.

Using equations (7) and (21) with $\psi = \Omega^{-1} = a(\tau)$, one finds $\tilde{R}_{00} = 0$, $\tilde{R}_{ij} = 2k\tilde{g}_{ij}$ and $\tilde{R} = 6k$ from equations (11) and (12). Hence, for the 00 component in equation (18), one obtains

$$\begin{eqnarray} H^2 + \frac{k c^2}{a^2} - \frac{\Lambda c^2}{3} & = & -\frac{\kappa^2c^2}{3} a^{-4} \tilde{T}\,{^0_{\,\,0}} = \frac{8\pi G}{3} a^{-4} \tilde{\rho} \end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray} &&= - \frac{\kappa^2c^2}{3} T^0_{\,\,0} = \frac{8\pi G}{3} \rho \,, \end{eqnarray} \tag{ 26 }$$

where $H \equiv \dot{a}/a = (da/dt)/a = (da/d\tau)/a^2$ denotes the Hubble function and dots indicate derivatives with respect to cosmic time t throughout the article. The ii component and equation (26) imply

$$\begin{eqnarray} &&\kern-9.7pt \frac{\ddot{a}}{a} = -\frac{\kappa^2c^2}{6} a^{-4} \left( -\tilde{T}\,{^0_{\,\,0}} + 3\tilde{T}^i_{\,\,i} \right) + \frac{\Lambda c^2}{3} = -\frac{4\pi G}{3} a^{-4} \left( \tilde{\rho} + \frac{3\tilde{p}}{c^2} \right) + \frac{\Lambda c^2}{3} \end{eqnarray} \tag{ 27 }$$

$$\begin{eqnarray} &&= -\frac{\kappa^2c^2}{6} \left( -T^0_{\,\,0} + 3T^i_{\,\,i} \right) + \frac{\Lambda c^2}{3} = -\frac{4\pi G}{3} \left( \rho + \frac{3p}{c^2} \right) + \frac{\Lambda c^2}{3} \,. \end{eqnarray} \tag{ 28 }$$

Hence, one recovers the first and second Friedmann equations.

Furthermore, equation (19) yields

$$\begin{eqnarray} &&\frac{\ddot{a}}{a} + H^2 + \frac{k c^2}{a^2} - \frac{2}{3}\Lambda c^2 = -\frac{\kappa^2 c^2}{6} a^{-4} \tilde{T} = \frac{8\pi G}{6} a^{-4} \left( \tilde{\rho} - \frac{3\tilde{p}}{c^2} \right) \end{eqnarray} \tag{ 29 }$$

$$\begin{eqnarray} &&= -\frac{\kappa^2 c^2}{6} T = \frac{8\pi G}{6} \left( \rho - \frac{3p}{c^2} \right) \,, \end{eqnarray} \tag{ 30 }$$

which may also be obtained from combining equations (26) and (28). Finally, equation (20) yields

$$\begin{eqnarray} \dot{\tilde{\rho}} & = & H (1-3w) \tilde{\rho} \,, \end{eqnarray} \tag{ 31 }$$

$$\begin{eqnarray} \dot{\rho} & = &\! - 3 H (1+w) \rho \end{eqnarray} \tag{ 32 }$$

with equation of state $w = \rho c^2/p$. Relations (31) and (32) provide evolution equations for the energy densities that are consistent between both frames such that the Friedmann equations and the evolution of the scale factor with cosmic time t remain invariant under the transformation.

Note that in the frame of $\tilde{g}_{\mu\nu}$, the radiation energy density $\tilde{\rho}_r$ is constant whereas the energy density associated with the cosmological constant $\tilde{\rho}_{\Lambda}$ scales as a⁴ and that of matter $\tilde{\rho}_m$ as a. Importantly, the relation of particle masses to the CMB radiation temperature remains invariant between the two frames due to the evolving masses in the Minkowski frame (see section 3.1.2), and particles become non-relativistic at the same time. Likewise, the relation of emitted photon energies to temperature remains invariant. Finally, it is also worthwhile noting that the evolution of $\psi = a$ and the evolution of the energy densities can be recovered without the Einstein field equations and thus the metric variation of action (16), only using equations (19) and (20).

3.1.2. Redshift.

Importantly, as shown in the following, despite the static geometry of the Universe in section 3.1.1, cosmological observations are still redshifted. However, rather than due to the expanding space, redshift is caused here by evolving particle masses [21, 22] (also see [23]). Thus, a photon emitted in the past as the result of some atomic line transition will be redshifted with respect to the measured emission of the same transition in our local, present laboratories. Photons still follow null geodesics in the conformally transformed frame [22] and our detectors measure them at their emission frequencies.

To illustrate the evolution of masses, consider a matter field with the simple Lagrangian

$$\begin{equation} L_m = \sqrt{-g}\left[ -\frac{1}{2}g^{\mu\nu}\partial_{\mu}\Psi_m \partial_{\nu}\Psi_m - \frac{1}{2} \left(\frac{mc}{\hbar}\right)^2 \Psi_m^2 \right] \,, \end{equation} \tag{ 33 }$$

which yields the field equation $\Box\Psi_m - (mc/\hbar)^2 \Psi_m = 0$. After conformal transformation, this becomes

$$\begin{equation} L_m = \sqrt{-\tilde{g}} \psi^2 \left[ -\frac{1}{2}\tilde{g}^{\mu\nu}\partial_{\mu}\Psi_m \partial_{\nu}\Psi_m - \frac{1}{2} \left(\frac{mc\psi}{\hbar}\right)^2 \Psi_m^2 \right] \end{equation} \tag{ 34 }$$

and

$$\begin{equation} \tilde{\Box}\Psi_m - (mc/\hbar)^2\psi^2 \Psi_m = 0 \,, \end{equation} \tag{ 35 }$$

where one may define the evolving mass $\tilde{m} \equiv m \psi$ in this frame. This evolution of $\tilde{m}$ is also more generally found for spinor fields interacting with a vector field in curved spacetime [21, 22] and has important implications for our observations.

Consider for example the transition lines from an n_i state to an n_f state of a hydrogen atom, for which one finds

$$\begin{equation} \tilde{E}_{n_i n_f} = \frac{\alpha^2 \tilde{m}_e}{4\pi\hbar}\left( \frac{1}{n_i^2} - \frac{1}{n_f^2} \right) = \psi E_{n_i n_f} \,. \end{equation} \tag{ 36 }$$

The redshift $\tilde{z}$ is given by the ratio between the energy emitted from this transition as observed in the laboratory $\tilde{E}_{n_i n_f}$ and the frequency observed from this transition occurring at a distant galaxy $\tilde{\omega}_o$. One finds

$$\begin{equation} 1+\tilde{z} = \frac{\tilde{E}_{n_i n_f}(O)}{\hbar\tilde{\omega}_o} = \frac{\psi_o}{\psi_{s}} \frac{\tilde{E}_{n_i n_f}(S)}{\hbar\tilde{\omega}_o} = \frac{\psi_o}{\psi_{s}} \frac{\tilde{E}_{n_i n_f}(S)}{\hbar\tilde{\omega}_s} = \frac{\psi_o}{\psi_{s}} \,, \end{equation} \tag{ 37 }$$

using that the photon frequency does not change with its propagation through Minkowski space, $\tilde{\omega}_o = \tilde{\omega}_s$. Inversely, one has

$$\begin{equation} \frac{\psi_o}{\psi_{s}} = \frac{\psi_o}{\psi_{s}} \frac{\tilde{E}_{n_i n_f}(S)}{\hbar\tilde{\omega}_s} = \frac{\psi_o}{\psi_{s}} \frac{\tilde{E}_{n_i n_f}(S)}{\hbar\tilde{\omega}_o} = \frac{E_{n_i n_f}(S)}{\hbar\omega_o} = \frac{\omega_s}{\omega_o} = (1+z) \,. \end{equation} \tag{ 38 }$$

This implies that redshift is invariant between the two frames, as expected for an observable quantity, and is determined by the ratio between the generally spacetime dependent $\psi(\tau,\mathbf{x})$ evaluated at source and observer.

Specifically, for the spatially homogeneous and isotropic case, $\psi_s = a(t_{em}) = a$ and $\psi_o = a(t_0) = a_0$, and thus

$$\begin{equation} \frac{\psi_o}{\psi_s} = \frac{a_0}{a} = 1+z \,. \end{equation} \tag{ 39 }$$

Note that redshift is also recovered for gravitational wave emissions, which is also attributed to the evolution of masses [22], where massless gravitons, like photons, still follow null geodesics in the conformally transformed frame.

As pointed out in section 3.1, spatially curved FLRW metrics have vanishing Weyl tensor and are thus conformally flat. They can therefore be written as $g_{\mu\nu} = a^2(T,R) \eta_{\mu\nu}$ for suitable coordinate choices T and R [17–20]. In principle, one could thus directly apply the conformally reframed Einstein field equations of section 2.3. Another procedure to transform the gravitational action and field equations into Minkowski space is to perform a combination of dimensional reduction and conformal transformations.

Specifically, the static line element obtained after the conformal transformation in section 3.1.1 is given by

$$\begin{eqnarray} &&d\tilde{s}^2 = \tilde{g}_{\mu\nu} dx^{\mu}dx^{\nu}=-c^2 d\tau^2 + \frac{dr^2}{1-kr^2} + r^2(d\theta^2 + \sin^2\theta \: d\phi^2) \end{eqnarray} \tag{ 40 }$$

$$\begin{eqnarray} & = & -c^2 d\tau^2 + d\chi^2 + r^2(\chi) (d\theta^2 + \sin^2\theta \: d\phi^2) \end{eqnarray} \tag{ 41 }$$

$$\begin{eqnarray} & = & -c^2 d\tau^2 + \left[\frac{r(\zeta)}{\zeta}\right]^2 \left[d\zeta^2 + \zeta^2(d\theta^2 + \sin^2\theta) \: d\phi^2 \right] \,, \end{eqnarray} \tag{ 42 }$$

where hyperspherical coordinates are adopted in the second line and the coordinate change $d\chi = \left[r(\zeta)/\zeta\right] d\zeta$ is performed in the third. Note that for k = 0, $r(\chi) = \chi$ and hence $d\zeta = d\chi$. To now cast the Einstein–Hilbert action (16) for $\tilde{g}_{\mu\nu}$ and $\psi = a$ with k ≠ 0 into Minkowski space, one can apply equation (10) to perform the dimensional reduction of $\tilde{R}$ to separate out the spatial Ricci scalar $\tilde{R}^{(3)}$ in the action and subsequently perform a conformal transformation of $\tilde{R}^{(3)}$ with $\hat{g}^{(3)}_{ij} = [r(\zeta)/\zeta]^{-2} \tilde{g}^{(3)}_{ij} = \eta^{(3)}_{ij}$.

The procedure shall not be carried out for the metric sector here. Of interest is rather the transformation of the matter sector. Revisiting the toy matter Lagrangian (34), one finds in Minkowski space with Cartesian coordinates the scalar field equation

$$\begin{eqnarray} &&-\frac{1}{c^2}\partial_{\tau}^2\Psi_m + \frac{1}{\xi^2} \sum_i \left( 1 + \frac{(\ln \xi)_{,i}}{(\ln \partial_i \Psi_m)_{,i}} \right) \partial_i^2 \Psi_m - \frac{m^2c^2\psi^2}{\hbar}\Psi_m =0 \,, \end{eqnarray} \tag{ 43 }$$

$$\begin{eqnarray} &&-\frac{1}{c^2}\partial_{\tau}^2\Psi_m + \frac{\ell(x^{\mu})^2}{\ell_p^2} \sum_i \partial_i^2 \Psi_m - \frac{m^2c^2\psi^2}{\hbar}\Psi_m =0 \,, \end{eqnarray} \tag{ 44 }$$

where $\xi \equiv r(\zeta)/\zeta$. Spatial curvature may thus be interpreted as a varying length scale $\ell(x^{\mu}) = \ell(\xi, \xi_{,i}, \Psi_{m,i}, \Psi_{m,ii})$ in Minkowski space. Note that for flat space k = 0, $r = \zeta$ and ξ = 1, so that $\ell = \ell_p$.

Our Universe is of course not perfectly spatially homogeneous and isotropic, and cosmological metric perturbations are typically not conformal (see sections 4.3 and 5.2 for conformal perturbations). But performing an analogous procedure to that of section 3.2, one may also recast the Einstein–Hilbert action for the perturbed Cosmos into Minkowski space.

For that, consider the metric $g_{\mu\nu}$ defined by the line element

$$\begin{equation} ds^2 = g_{\mu\nu} dx^{\mu}dx^{\nu} = a^2(\tau)\left[-c^2 e^{2\alpha(\tau,\mathbf{x})} d\tau^2 + e^{2\beta(\tau,\mathbf{x})} d\mathbf{x}^2 \right] \,, \end{equation} \tag{ 45 }$$

which for $\alpha \equiv \Psi$, $|\alpha| \ll 1$ and $\beta \equiv \Phi$, $|\beta| \ll 1$ becomes the usual linearly perturbed FLRW line element in conformal Newtonian (or longitudinal) gauge and conformal time,

$$\begin{equation} ds^2 = a^2(\tau) \left[ -c^2 (1+2\Psi(\tau,\mathbf{x})) d\tau^2 + (1+2\Phi(\tau,\mathbf{x})) d\mathbf{x}^2 \right] \,. \end{equation} \tag{ 46 }$$

For the following discussion, the perturbations may not need to be linear, however. Note that $d\mathbf{x}^2$ may not be flat (k ≠ 0), in which case a factor $r(\zeta)/\zeta$ from equation (44) can first be absorbed into e^β . Next, one can apply the conformal transformation $\tilde{g}_{\mu\nu} = a^{-2}(\tau)e^{-2\alpha(\tau,\mathbf{x})}g_{\mu\nu}$, then perform the dimensional reduction to $\tilde{R}^{(3)}$ and the conformal transformation of that with $\hat{g}^{(3)}_{ij} = e^{2(\alpha-\beta)} \tilde{g}^{(3)}_{ij} = \eta^{(3)}_{ij}$ such that the Einstein–Hilbert action is recast into Minkowski space.

We shall again refrain from computing the resulting metric sector for this frame and instead focus on the scalar field equation for $\Psi_m$. The transformations yield

$$\begin{align} &-\frac{1}{c^2}\left(1+\frac{3\gamma_{,\tau}}{(\ln \Psi_{m,\tau})_{,\tau}}\right) \partial_{\tau}^2\Psi_m + e^{-2\gamma} \sum_i \left( 1 + \frac{\gamma_{,i}}{(\ln \Psi_{m,i})_{,i}} \right) \partial_i^2 \Psi_m \nonumber\\ &\quad - \left(\frac{ae^{\alpha}mc}{\hbar} \right)^2 \Psi_m = 0 \,, \end{align} \tag{ 47 }$$

$$\begin{eqnarray} -\frac{1}{c^2}\frac{\iota^2(x^{\mu})}{\iota_0^2}\partial_{\tau}^2\Psi_m + \frac{\ell^2(x^{\mu})}{\ell_0^2} \sum_i \partial_i^2 \Psi_m - \frac{m^2(x^{\mu})c^2}{\hbar^2} \Psi_m & = & 0 \end{eqnarray} \tag{ 48 }$$

in Cartesian coordinates, where $\gamma \equiv \beta-\alpha$. Hence, one encounters a variation of time ι, length $\ell$, and mass scales m for the field $\Psi_m$ in Minkowski space. Note that one could also recast equation (48) as a varying speed of light $c(x^{\mu})$ and/or a varying Planck constant $\hbar(x^{\mu})$. Similarly, one would interpret the variation of $\ell^2(x^{\mu})c^3(x^{\mu})/\hbar(x^{\mu}) = G(x^{\mu})$ as a variation of Newton's gravitational constant.

Importantly, the same metric transformations performed on the metric part of the action (and on the other matter fields) will leave physics invariant. Predictions for observable quantities will remain the same, but interpretations may change such as for the redshift in section 3.1.2. For a loose analogy, consider the observed angular positions of planets in the sky, whose predicted motions are unaffected by whether we adopt a heliocentric view or an exact transformation of that into geocentric coordinates. The heliocentric system provides a much simpler and universal physical picture than the resulting epicycles, but one would not adopt heliocentric coordinates to describe a small-scale experiment in a laboratory on Earth. Similarly, the FLRW metric may not be optimal for describing small-scale physical processes in our nonlinear Universe, and a change of mathematical frame may become more suitable (see section 5).

It is worthwhile noting that if not performing the analogous transformations in the metric sector, or only performing them in the metric but not in the matter sector, one does modify physics. For instance, if not applying the conformal transformation for the matter sector in equation (16), one recovers a scalar-tensor modification of gravity. One may furthermore consider non-universal couplings of different matter species to this conformal factor, thereby breaking the weak equivalence principle [15] (see section 2.1). This multitude of possibilities motivates generally testing, across the vast spacetime, for phenomenological variations in the effective time, length and mass scales of the observed behaviour of different matter species as in equation (48), or likewise in the fundamental couplings. Note in this context that while the Minkowskian picture is physically equivalent to the standard cosmological picture both at the classical and quantum level [24] and observationally indistinguishable from it, it offers a different basis for theoretical extensions. These may appear more natural in one picture over the other and if verified, could favour one framework over the other.

Finally, consider the symmetric traceless tensor perturbation h_ij of $d\mathbf{x}^2$ in equation (45). This term will remain in $\hat{g}^{(3)}_{ij}$ after the transformations of sections 3.2 and 3.3. To then cast the Einstein–Hilbert action into Minkowski space, one may perform a metric transformation of the form $\hat{g}^{(3)}_{ij} = A_{ij}^{\:\: kl} \eta_{kl}^{(3)} + B_{ij}$. With suitable transformations and coordinates, this amounts to simply separating out the tensor field h_ij in addition to the scalar fields from the three and four-dimensional conformal transformations. One can see from equation (48) that this will lead to oscillations in time and length scales. Since gravitational waves are not studied further in this article, a more detailed calculation is left to future work.

In an Einstein-de Sitter Universe, the scale factor evolves as $\tilde{a} \propto \tilde{t}^{2/3} \propto \tau^2$. Since ΛCDM is Einstein-de Sitter at early times, in the limit of $\tau\rightarrow0$, one has $\tilde{a} = a$ and $\tilde{t} = t$. The scale factor can be normalised at a_eq as in equation (49). Given the absence of an era of Λ domination, in Einstein-de Sitter spacetime, a_eq can be thought of as a normalisation for when $c^2\kappa^2\tilde{\rho}$ falls below a given threshold Λ. It follows that at early times, $\tilde{m} = m$, and at late times,

$$\begin{equation} \tilde{m} = \left(\frac{t_{\Lambda}}{a_{eq}}\right)^3\frac{4}{\tau^2(\tau_{\infty}-\tau)} m \,, \end{equation} \tag{ 56 }$$

approaching infinity in finite conformal time $\tau_{\infty}$.

Hence, cosmic acceleration can be interpreted in this frame as increasing and diverging particle masses that start to evolve once cosmic time approaches $\tilde{t} \sim t_{\Lambda}$ set by the threshold Λ.

In de Sitter spacetime, one has a constant Hubble function $\tilde{H} = t_{\Lambda}^{-1}$, the scale factor evolves as $\tilde{a} = \tilde{a}_{dS} e^{\tilde{t}/t_{\Lambda}} = t_{\Lambda}(\tilde{\tau}_{\infty}-\tau)^{-1}$, and conformal and cosmic times relate as $\tau = \tilde{\tau}_{\infty} - \tilde{a}_{dS}^{-1} t_{\Lambda} e^{-\tilde{t}/t_{\Lambda}}$. Normalising $\tilde{\tau}_{\infty} = \tau_{\infty}$ by suitable choice of ψ, one finds that $\tilde{m} = m$ at late times and

$$\begin{equation} \tilde{m} = \frac{1}{4} \left(\frac{a_{eq}}{t_{\Lambda}}\right)^3 \tau^2(\tau_{\infty}-\tau) m \simeq \frac{\tau_{\infty}}{4} \left(\frac{a_{eq}}{t_{\Lambda}}\right)^3 \tau^2 m \end{equation} \tag{ 57 }$$

at early times.

Hence, similarly to interpreting dark energy as the late-time divergent increase of particle masses, in this frame, one may interpret dark matter as the effect of an early-time increase of particle masses with τ² that converges to a constant mass m at late times.

Rather than casting the cosmic structure encompassed in the nonlinearly perturbed FLRW metric (45) into Minkowski space, one may cast it into smooth FLRW spacetime instead. This will lead to a similar interpretation of changing mass, length and time scales as in section 3.3, which vary locally with respect to the cosmological background.

The following discussion will focus on conformal inhomogeneities, which are a special type of nonlinear inhomogeneities that have not been studied in much detail. Particularly, in [25, 26], they were proposed as a simultaneous solution of cosmological tensions in the measured Hubble constant H₀, the amplitude of matter fluctuations σ₈, and the amplitude of CMB lensing A_L , which would be found from residing in a local conformal void. The nonlinearly conformally perturbed FLRW metric has also been dubbed Conformal FLRW (CFLRW) cosmology in [27].

Consider the scenario where we occupy a local conformal inhomogeneity in a ΛCDM FLRW (or CFLRW) Universe such that $a_{\Lambda \mathrm{CDM}}(\tau) \rightarrow a_{\Lambda \mathrm{CDM}}(\tau, \mathbf{x}) \equiv C(\tau, \mathbf{x})\bar{a}_{\Lambda \mathrm{CDM}}(\tau)$ with conformal factor C². Proceeding as in sections 4.1 and 4.2, let the target cosmology be a smooth ΛCDM FLRW metric, $\tilde{a}_{\Lambda \mathrm{CDM}}(\tau) = \bar{a}_{\Lambda \mathrm{CDM}}(\tau)$. This implies the evolution of masses in this frame as

$$\begin{equation} \tilde{m} = \frac{a_{\Lambda \mathrm{CDM}}(\tau, \mathbf{x})}{\bar{a}_{\Lambda \mathrm{CDM}}(\tau)} m = C(\tau, \mathbf{x}) m \,. \end{equation} \tag{ 58 }$$

Equations (37) and (38) imply that redshifts relate as $1+z = C_0 (1 + \bar{z}) / C(\tau_s,\mathbf{x}_s) = 1+\tilde{z}$, where the conformal factor $C_0^2 \equiv C^2_{obs}$ is defined at the spacetime location of the observer and $\bar{z}$ is the redshift one would obtain if both source and observer resided in the smooth background. Defining masses at the location of the observer, one has

$$\begin{equation} \tilde{m} = \frac{C(\tau,\mathbf{x})}{C_0} \tilde{m}_0 \,. \end{equation} \tag{ 59 }$$

For a simple application, one may adopt a top-hat inhomogeneity of radius R with $C =$ $(1+\delta)^{-1/3}$, where $\rho_m = \bar{\rho}_m (1+\delta)$. At the spacetime location of the observer, $C_0 =$ $(1+\delta_0)^{-1/3}$, and for a source outside the top hat in the smooth background, $C_s = 1$. In this case, masses evolve as $\tilde{m} = m = C_0^{-1} \tilde{m}_0 = (1+\delta_0)^{1/3} \tilde{m}_0$ and redshifts relate as $1 + z = C_0 (1 + \bar{z}) = (1+\delta_0)^{-1/3} (1 + \bar{z}) = 1 + \tilde{z}$.

This implies that if assuming particles, or larger massive objects, to be located in the smooth cosmological FLRW background, one ought to correct for the local evolution of their masses and adapt redshifts accordingly to correctly interpret observations.

Let us briefly consider the consequences of failing to do so. The matter contribution to the Hubble function in the background is given by

$$\begin{equation} \bar{H}^2 \supset \frac{\kappa^2\bar{\rho}_{m0}}{3} (1 + \bar{z} )^3 = \frac{\kappa^2\bar{\rho}_{m0}}{3} (1 + \delta_0) (1 + z )^3 \,. \end{equation} \tag{ 60 }$$

Importantly, as discussed in section 3.1.2, one observes the frame-invariant redshift $z = \tilde{z}$ and not $\bar{z}$. Thus, wrongly assuming $\bar{z} = z$ for a source outside the top hat with background density $\bar{\rho}_m$ will give $(1+\delta_0)^{-1/2} \bar{H}_0$ rather than the Hubble parameter $\bar{H}_0$. Hence, if we reside in a conformal top-hat underdensity $\delta_0 \lt 0$ and interpret our observations by placing ourselves into the smooth cosmological background instead, we infer an enhanced effective Hubble parameter. Specifically, a ∼10% enhancement of $\bar{H}_0$ requires a ∼20% conformal underdensity [25, 26]. Note that the size of the conformal void R does not need to be large to change the inferred Hubble parameter as opposed to that of conventional voids. A diameter of a few tens of megaparsecs may be sufficient [25, 26]. This scenario will be further investigated in section 5.2.

In light of the importance of the cosmological constant problem (section 1), it is worthwhile revisiting it in the Minkowski frame of section 3 for a possible reinterpretation of the problem. Section 5.1.1 examines the naturalness of the value of the cosmological constant for the Universe cast into Minkowski space. Section 5.1.2 inspects the consistency with the gravitating ground state energy density from canonical quantisation in this frame. In this process, the scale factor of the metric will be treated as a scalar field and quantised. It should be emphasised, however, that one could instead proceed analogously with the quantisation of a matter field $\Psi_m$ as given by equation (33) to arrive at the same conclusions.

5.1.1. The scale factor as a scalar field.

For the following discussion, let us for simplicity assume $\tilde{R} = 0$ and $\tilde{T} = 0$ and redefine $\psi \rightarrow i \psi$ in equation (16). Note that the new field ψ must not need to be imaginary as one may also take $a\rightarrow ia$, representing a rotation of metric signature, such that $\psi = a$ remains real. This gives the scalar field action

$$\begin{equation} S_{\psi} = \int d^4x \left[ - \frac{1}{2} \partial_{\mu}\psi \partial^{\mu}\psi - V(\psi) \right] \,, \end{equation} \tag{ 61 }$$

where $V = \Lambda \psi^4/6$. The scalar field equation becomes

$$\begin{equation} \tilde{\Box} \psi - \frac{2}{3} \Lambda \psi^3 \,, \end{equation} \tag{ 62 }$$

where one may identify the effective mass $m^2 = V^{^{\prime\prime}}(\psi) = 2\Lambda (\psi\hbar/c)^2$. More formally, consider a small perturbation δψ around $\psi_* = a(\tau_*)$ such that $\psi = \psi_*(1+\delta \psi)$. Additionally making the field redefinition $\chi = 1 + 2\delta\psi$, the potential can be rewritten as $6V/\Lambda = \psi^4 \simeq \psi_*^4(1+2\delta\psi)^2 = \psi_*^4\chi^2$. Variation of the action with respect to χ yields the field equation $\tilde{\Box}\chi - (4/3)\Lambda\psi_*^2\chi$ with the Klein–Gordan mass given by $m^2_{\chi} = (4/3)\Lambda (\psi_*\hbar/c)^2$. The mass associated with the field ψ thus varies as

$$\begin{equation} m \sim \frac{\hbar}{c} \sqrt{\Lambda} a \,. \end{equation} \tag{ 63 }$$

This time dependence of m has important implications for the question of the naturalness of Λ as inferring its value from a mass scale, $\Lambda \sim m^2/a^2$, implies a choice of a.

Consider the Planck time t_p with $a_p \equiv a(t_p)$ and the present t₀ with $a_0 \equiv a(t_0)$. Let us furthermore normalise the scale factor at Planck time, $a_p = 1$ (one could also normalise masses as $\hat{m} = m/a_p$ instead), and assume that the volume of our currently observable Universe originated from a Planck volume at Planck time $\ell_u^3 = (a_0/a_p)^3\ell_p^3$ with Planck length $\ell_p$. Our observations imply $\ell_u\sim10^{61}\ell_p$. For a Plank mass m_p of the field ψ today, $m_0 = m_p$, one therefore finds

$$\begin{equation} \Lambda \sim a_0^{-2} \frac{c^2}{\hbar^2} m_0^2 = a_0^{-2} \frac{c^2}{\hbar^2} m_p^2 = a_0^{-2} \ell_p^{-2} \sim \left(\frac{\ell_p}{\ell_u}\right)^2 \ell_p^{-2} \sim 10^{-122} \ell_p^{-2} \,. \end{equation} \tag{ 64 }$$

Thus, in this frame, Λ takes on a natural value set by the Planck scale and the size of the present Universe.

This may not seem surprising given that we are located at an era of equality between Λ and $\kappa^2c^2\rho_m$, where the size of the Universe is set by the scale of Λ, $\ell_u \sim \ell_{\Lambda} \equiv c t_{\Lambda}$. Importantly, however, one should consider $m_* = \sqrt{\Lambda}\hbar c^{-1}$ as the fundamental mass scale whereas the field mass m_χ is subject to the time of evaluation of a(t) in equation (63) and only becomes m_p at a₀. The fact that $\ell_u \sim \ell_{\Lambda}$ is therefore attributed to the coincidence or Why now? problem of why $\Lambda \kappa^{-2} \approx \rho_m c^2$ today whereas these energy densities differ by orders of magnitude in the past and future of the Universe. To make this connection, note that $\ell_u/\ell_p \sim t_0/t_p$. Matter domination implies $H_0\sim t_0^{-1}$ and when Λ dominates, $H\sim t_{\Lambda}^{-1}$. Hence, at matter-Λ equality, $t_0 \approx t_{\Lambda}$ such that $\ell_u/\ell_p \sim t_{\Lambda}/t_p$.

A solution to the coincidence problem in turn is provided by anthropic considerations. This can simply be in the sense of an observer's bias on the observer's location in cosmic history without the need to invoke a multiverse. For that, assume a Universe configured by the fundamental mass scale given by $\sqrt{\Lambda}\hbar c^{-1}$ and the matter energy density ρ_m at their observed values. As described in [28], this Universe produces a star formation peak at the age of ${\sim} 4$ Gyr whereas terrestrial planet formation peaks at ${\sim} 8$ Gyr, making Earth a typical sample from this distribution. If the time it took on Earth to develop observers can be considered typical, the most likely epoch for observers to emerge in the Universe is around the equality between Λ and $\kappa^2c^2\rho_m$, setting $t_0 \approx t_{\Lambda}$, $\ell_u \approx \ell_{\Lambda}$ and $m_p \sim a(t_0) \sqrt{\Lambda} \hbar c^{-1}$.

Thus, neither the observed value of Λ nor the comparable size of $\kappa^2c^2\rho_m$ today seem problematic in Minkowski space.

5.1.2. Canonical quantisation of the scale factor.

For completeness, let us compute Λ from the ground state energy density attributed to canonical quantisation of the field ψ. For the Lagrangian density $\mathcal{L}$ in equation (62), the canonical momentum and classical Hamiltonian are given by

$$\begin{equation} \pi = \frac{\delta \mathcal{L}}{\delta \partial_0\psi} \,, \quad \quad H = \int d^3x \left[ \pi \partial_0\psi - \mathcal{L} \right] \,, \end{equation} \tag{ 65 }$$

where the model is quantised with the commutator

$$\begin{equation} \left[ \hat{\psi}(\textbf{x}), \hat{\pi}(\textbf{y}) \right] = i \hbar \delta(\textbf{x} - \textbf{y}) \,. \end{equation} \tag{ 66 }$$

As in section 5.1.1, we shall define the Klein–Gordan field χ with mass m_χ for small perturbations around $\psi_*$. The ground state energy density for that is

$$\begin{equation} \epsilon_0 = \frac{1}{2} \int \frac{d^3k}{(2\pi)^3} \sqrt{(k\hbar c)^2 +(m_{\chi}c^2)^2} \,. \end{equation} \tag{ 67 }$$

Taking m_χ as the ultraviolet cutoff, one obtains

$$\begin{equation} \epsilon_0 \sim \frac{m_{\chi}^4 c^5} {\hbar^3} \sim \hbar c \Lambda^2 a^4 \,. \end{equation} \tag{ 68 }$$

For the gravitating vacuum energy density in Minkowski space, one therefore has

$$\begin{equation} \frac{\hbar}{m_*^2 c^3} \epsilon_0 \sim \frac{\hbar^2}{m_*^2 c^2} \Lambda^2 a^4 \sim \Lambda a^4 = \kappa^2 c^2 \tilde{\rho}_{\Lambda} \,. \end{equation} \tag{ 69 }$$

Since $\tilde{\rho}_{\Lambda}(a) = \rho_{\Lambda} a^4$ (see section 3.1), one recovers

$$\begin{equation} \kappa^2 c^2 \rho_{\Lambda} = \Lambda \,. \end{equation} \tag{ 70 }$$

A conformal transformation was used in section 4.3 to cast a Universe where we occupy a conformal top-hat inhomogeneity into one where we reside in the smooth cosmological FRLW background geometry. The implication of that was that masses of particles and larger objects as well as redshifts would rescale with the local density field. Failing to account for these modifications shifts inferred cosmological quantities such as the cosmic expansion rate. References [25, 26] showed that corrections from residing in a local conformal cosmic void of ${\sim}20\%$ underdensity of a few tens of megaparsecs in diameter simultaneously solves the current tensions between measurements of the Hubble constant H₀ as well as of the amplitude of matter fluctuations σ₈ and restores a preference for a consistent CMB lensing amplitude of $A_L = 1$ (section 1). Specifically, in the frame adopted in [26], the local CMB temperature T₀ deviates from that of the cosmological background $\bar{T}_0$ as $T_0 = \bar{T}_0(1+\delta_0)^{1/3}$, which changes parameters inferred from the CMB, particularly lowering σ₈ and A_L and increasing H₀. Allowing for spatial curvature, Planck, baryon acoustic oscillations and H₀ data then prefer an open and hotter Universe at $\gt3\sigma$ [26].

The framework established in sections 2–4 can be used to study further properties and observable signatures of conformal inhomogeneities. In the following, section 5.2.1 will focus on their nonlinear evolution whereas section 5.2.2 will explore the linear limit, where observable signatures will be highlighted for both regimes.

5.2.1. Conformal cosmic void.

Consider a local conformal top-hat inhomogeneity with scale factor $\hat{a}$ of a smooth FLRW background metric with scale factor a. Following section 4.3, this defines the conformal factor $\psi = \hat{a}/a = (1+\delta)^{-1/3}$. Using the second Friedmann equation (28) both for $\hat{a}$ and a, one recovers the spherical collapse equation,

$$\begin{equation} \ddot{\delta} + 2H\dot{\delta} - \frac{4}{3}\frac{\dot{\delta}^2}{1+\delta} = \frac{\kappa^2c^4}{2} \bar{\rho}_m (1+\delta)\delta \,, \end{equation} \tag{ 71 }$$

where dots denote derivatives with respect to the cosmic time t of the smooth FLRW background, $dt = a\,d\tau$. Note that while conformal times between the two frames agree, cosmic times do not, $t\neq\hat{t}$. The evolution of δ and thus of $\hat{a}$ can be solved for setting the initial conditions at $a_i\ll1$ in the matter-dominated regime, where $\delta \propto a$. The interior of the conformal top-hat inhomogeneity thus evolves as a usual top-hat density contrast. This is not surprising in light of Birkhoff's theorem. Density perturbations can be viewed as separate universes with different background density and curvature. For a top-hat density perturbation this becomes manifest adopting Conformal Fermi Coordinates [29]. Also recall from section 3.1 that any smooth FLRW metric describing the interior of the top hat has vanishing Weyl tensor and is thus conformally flat.

Besides the impact of a local conformal void on currently measured cosmological parameter tensions, the presence of conformal perturbations can have further observable implications. Revisiting the relation of measured redshift z to its background counterpart $\bar{z}$ in section 4.3,

$$\begin{equation} 1 + z = \left(\frac{1+\delta_s}{1+\delta_0}\right)^{1/3} (1 + \bar{z}) \,, \end{equation} \tag{ 72 }$$

one can see that redshifts are not only impacted by the environment of the observer but also by that of the sources, where both satisfy the evolution equation (71). Particularly, this implies that photons emitted from galaxy clusters ($\delta_s\gt0$) and detected in our underdense environment ($\delta_0\lt0$) may be significantly more redshifted than expected from $\bar{z}$. As a consequence, one may attribute larger distances and older ages to observed galaxies in clusters, and consequently find larger masses and population sizes for those than expected. This effect may offer an explanation for recent observations [30–32]. Of course, the local underdense environment may also lead to astronomical signatures in our immediate neighbourhood, and it is worthwhile noting that measurements of nearby galaxy groups [33], clusters [34] and peculiar velocities [35] indeed seem to indicate such an underdense local region.

In view of these effects, one may wonder about the nature of the coupling of matter fields with the local scale factor at a source $\psi = a_s$ suggested by equations (16)–(20). After all, photon-emitting sources typically lie in virialised regions, which have decoupled from the cosmic expansion, or inside compact bodies, and so $a_s = \mathrm{const}.$ at the immediate location of the emitting particle process. Likewise, $a_0 = \mathrm{const}.$ at the immediate position of our detectors. However, when observing sources from virialised regions that are detached from ours, an expansion takes place between them. In the non-expanding frame, the coupling hence corresponds to an effective coupling of the entire virialised region to the evolving scale factor of its environment and of that to the cosmological background or the space separating this region from ours. This coupling can then be viewed as acting on the individual particles composing those regions. Schematically, one has $\psi = a = a_\mathrm{vir} \cdot a_\mathrm{env} \cdot \bar{a}$, where $a_\mathrm{vir} = \mathrm{const}.$ and $a_\mathrm{env}$ may represent a composition of environments, $a_\mathrm{env} = a_\mathrm{env(1)} \cdot\cdot\cdot a_{\mathrm{env}(n)}$.

5.2.2. Linear cosmological perturbations.

To explore further observable signatures attributed to the presence of conformal inhomogeneities in the Cosmos, one may resort to linear perturbation theory. For that, consider the linearised conformal factor $\psi^2(\tau,\mathbf{x}) = 1 + 2\delta \psi(\tau,\mathbf{x})$, where $\left|\delta \psi\right|\ll1$, acting in front of the linearly perturbed FLRW line element in conformal Newtonian (or longitudinal) gauge, equation (46). Operating in the linear regime, one may simply absorb the conformal factor by defining the new potentials $\hat{\Psi} \equiv \Psi+\delta \psi$ and $\hat{\Phi} \equiv \Phi+\delta \psi$. The usual linearly perturbed Einstein field equations apply for $\hat{\Psi}$ and $\hat{\Phi}$. Importantly, however, the notion of cosmic time may change with the absorption of the conformal factor, $dt = a\,d\tau \neq a(1+\delta\psi)d\tau$. Given linearity, one may split the perturbed energy-momentum tensor as $\delta T_{\mu\nu} = \delta T_{\mu\nu}^{(\psi)} + \delta T_{\mu\nu}^{(0)}$, where $\delta T_{\mu\nu}^{(0)}$ determines Ψ and Φ. Thus, the linearised field equations for δψ become

$$\begin{eqnarray} \nabla^2 \delta\psi -\frac{3a^2H}{c^2} \left ( \dot{\delta\psi} - H \delta\psi \right) & = & -\frac{\kappa^2c^2}{2}a^2 \delta\rho^{(\psi)} \,, \end{eqnarray} \tag{ 73 }$$

$$\begin{eqnarray} \kern59.5pt \nabla^2\dot{\delta \psi} - H \nabla^2\delta\psi & = & \frac{\kappa^2c^2}{2} a\left(1+w\right)\bar{\rho}\theta^{(\psi)} \,, \end{eqnarray} \tag{ 74 }$$

$$\begin{eqnarray} \nabla^2\delta\psi - \frac{3a^2}{2c^2} \left[\ddot{\delta\psi} + 2H\dot{\delta\psi} - \left( H^2 + 2\frac{\ddot{a}}{a}\right) \delta\psi \right] & = & \frac{3\kappa^2c^2}{4} a^2 \delta P^{(\psi)} \,, \end{eqnarray} \tag{ 75 }$$

$$\begin{eqnarray} &&\nabla^2 \delta\psi = \frac{3\kappa^2c^2}{4}a^2\left(1+w \right) \bar{\rho} \sigma^{(\psi)} \,, \end{eqnarray} \tag{ 76 }$$

where the density perturbation $\delta\rho^{(\psi)} = \bar{\rho}\delta^{(\psi)}$, the divergence of the fluid velocity $\theta^{(\psi)}$, the pressure perturbation $\delta P^{(\psi)}$, and the shear (or anisotropic) stress $\sigma^{(\psi)}$ describe the fluid giving rise to the conformal inhomogeneity. Energy-momentum conservation implies

$$\begin{eqnarray} \dot{\delta}^{(\psi)} & = & -(1+w) \left( \frac{\theta^{(\psi)}}{a} + 3 \dot{\delta\psi} \right) - 3 H \left( \frac{c_s^{2(\psi)}}{c^2} - w \right) \delta^{(\psi)} \,, \end{eqnarray} \tag{ 77 }$$

$$\begin{eqnarray} \dot{\theta}^{(\psi)} & = & -H(1-3w)\theta^{(\psi)} - \frac{\dot{w}}{1+w}\theta^{(\psi)} - \frac{1}{a} \left(\! \frac{c_s^{2(\psi)}}{1+w} \nabla^2 \delta^{(\psi)} - c^2\nabla^2\sigma^{(\psi)} + c^2\nabla^2\delta^{(\psi)} \psi \!\!\right) \!, \end{eqnarray} \tag{ 78 }$$

where $c_s^{2(\psi)} = c^2\delta P^{(\psi)} / \delta \rho^{(\psi)}$ denotes the sound speed squared.

It should be emphasised again that dots represent here derivatives with respect to the cosmic time $dt = a \, d\tau$ rather than $dt = a(1+\delta\psi)d\tau$, which can give rise to effective fluid perturbations. To illustrate this point, consider separating off a fraction of the scale factor of the smooth FLRW metric as $a \rightarrow a(1+\delta a)$ such that $\delta\psi = \delta a(t)$ and $\nabla^2\delta a = 0$ everywhere. Equations (73) and (75) then directly follow from the linearised first and second Friedmann equations (26) and (28), respectively, under the transformations of the scale factor and of cosmic time, where $\delta\rho^{(a)}$ and $\delta P^{(a)}$ are the density and pressure perturbations attributed to $\delta a$. For instance, one may separate out a different background matter density $\bar{\rho}_m^{(a)}$ (see section 2), giving rise to an effective density contrast $\delta\rho^{(a)}_m = \bar{\rho}^{(a)}_m \delta^{(a)}$ and $\delta P^{(a)}_m = 0$. As another example, given that spatially curved FRLW metrics are conformally flat (section 3.1), one may treat a small spatial curvature as a linear conformal perturbation $\delta \psi = \delta a$ of a flat background.

From the field equations (73)–(76), one can already identify some interesting observable implications of linear conformal perturbations without the need for solving the system of differential equations. Consider the fluid comoving density in Fourier space,

$$\begin{equation} \Delta = \delta + 3(1+w) \frac{aH}{c^2k^2} \theta \,. \end{equation} \tag{ 79 }$$

Note that for $\delta \psi = \delta a (t)$, this density vanishes, $\Delta^{(a)} = 0$. Equations (73), (74) and (76) more generally imply

$$\begin{equation} k^2\delta\psi = \frac{\kappa^2c^2}{2}a^2\Delta^{(\psi)} = -\frac{3\kappa^2c^2}{4}a^2\left(1+w\right)\bar{\rho} \sigma^{(\psi)} \,. \end{equation} \tag{ 80 }$$

For a matter perturbation, one has $2k^2\Phi = \kappa^2c^2a^2\bar{\rho}_m \Delta_m$ such that the total potential satisfies

$$\begin{equation} k^2 \hat{\Phi} = \frac{\kappa^2c^2}{2} a^2 \bar{\rho}_m \left( \Delta_m + \Delta^{(\psi)} \right) \,. \end{equation} \tag{ 81 }$$

Since $\sigma_m = 0$, the potentials relate as $\Psi = -\Phi$ and

$$\begin{equation} -k^2 \hat{\Psi} = \frac{\kappa^2c^2}{2} a^2 \bar{\rho}_m \left( \Delta_m - \Delta^{(\psi)} \right) \,. \end{equation} \tag{ 82 }$$

For the lensing potential, one finds

$$\begin{equation} \frac{k^2}{2} \left( \hat{\Phi}-\hat{\Psi} \right) = \frac{\kappa^2c^2}{2} a^2 \bar{\rho}_m \Delta_m \,. \end{equation} \tag{ 83 }$$

Thus, lensing remains unaffected by the presence of the conformal inhomogeneity, as expected for a conformal factor, whereas dynamic, or kinematic, effects such as clustering are modified in equation (82). The conformal inhomogeneity acts like an effective modification of gravity with

$$\begin{eqnarray} \mu(a,k) & \equiv & -\frac{2 k^2 \hat{\Psi}}{\kappa^2c^2 a^2 \bar{\rho}_m \Delta_m} = 1 - \frac{\Delta^{(\psi)}}{\Delta_m} \,, \end{eqnarray} \tag{ 84 }$$

$$\begin{eqnarray} \Sigma(a,k) & \equiv & -\frac{k^2\left( \hat{\Phi}-\hat{\Psi} \right)}{\kappa^2c^2 a^2 \bar{\rho}_m \Delta_m} = 1 \,. \end{eqnarray} \tag{ 85 }$$

Thus, for clusters with $\Delta_m \gt 0$ and for $\Delta^{(\psi)} \lt \Delta_m$, this implies that cluster masses inferred dynamically, or kinematically, M_dyn , are underestimated compared to masses inferred from lensing, M_len . This mass bias is indeed observed but typically associated with deviations from hydrostatic equilibrium affecting the estimations of M_dyn [36]. The growth suppression µ < 1 in equation (84) may also act to lower σ₈ as preferred by galaxy surveys [9]. Note that in contrast, modified gravity models typically produce µ > 1 as well as $M_{dyn} \gt M_{len}$ [37] for theoretical stability reasons [38]. It is also worthwhile noting that for conformal inhomogeneities, $\Delta^{(\psi)}/\Delta_m$ may also be negative, for example in voids, such that µ > 1 and $M_{dyn} \gt M_{len}$ instead.

In section 4.2, an interpretation of dark matter as the effect of an early-time convergent increase of particle masses was given. Similarly, as discussed in section 4.1, dark energy could be interpreted as the late-time divergent increase of particle masses. These interpretations were motivated by the change of the cosmic geometry into de Sitter and Einstein-de Sitter space, respectively. In the following, another candidate of geometric origin for dark matter as well as for inflation shall briefly be explored, which is motivated from treating the scale factor as a scalar field as in section 5.1.

Let us revisit the action (61), which for the field $\chi \equiv 1 + 2\delta\psi$ becomes

$$\begin{equation} S_{\chi} = \int d^4x \sqrt{-g} \psi_*^2\left[ - \frac{1}{2} \partial_{\mu}\chi \partial^{\mu}\chi - \frac{1}{2} m_{\chi}^2\chi^2 \right] \,, \end{equation} \tag{ 86 }$$

where $m^2_{\chi} = 4\Lambda\psi_*^2/3$ and $S_{\chi}\simeq4S_{\psi}$. Instead of the Minkowski space in section 5.1, however, $g_{\mu\nu}$ shall describe an evolving smooth FLRW background obtained after a conformal transformation, for example, of a conformal inhomogeneity similar to section 4.3. Thus, δψ describes the perturbation of the relative scale factor around $\psi_* = a_*/\bar{a}_*$. In the interior, the field is only time dependent such that the scalar field equation implies

$$\begin{equation} \ddot{\chi} + 3H\dot{\chi} + m_{\chi}^2\chi = 0 \,. \end{equation} \tag{ 87 }$$

Equation (87) suggests that χ may act as an axion [39]. Let us assume the initial condition $\dot{\chi} = 0$. For $H \gt m_{\chi}$, the axion field is overdamped and frozen to its initial value with an equation of state of $w_{\chi} = -1$ such that it may act as dark energy or drive inflation. For $H \lt m_{\chi}$, the axion field is underdamped, oscillating around $w_{\chi} = 0$ with energy density $\rho_{\chi} \sim a^{-3}$. Hence, in this case, it may act as dark matter.

Finally, the transformed action (16) could also have implications for baryogenesis, irrespective of the axion treatment, where couplings could introduce a CP violation causing baryogenesis [39]. For this, consider quantum gravity corrections in the action such as the term R². Besides being a potential source of inflation [40], this term can also provide a coupling relevant to baryogenesis in the conformally transformed action. Specifically, with equation (9) one finds from integration by parts the term

$$\begin{equation} \sqrt{-g}R^2 \propto -\sqrt{-\tilde{g}} \tilde{J}^{\mu} \partial_{\mu}\tilde{R} + \ldots \,, \end{equation} \tag{ 88 }$$

defining the current $\tilde{J}^{\mu} \equiv \partial^{\mu}\ln \psi = \tilde{T}^{-1} \tilde{\nabla}^{\sigma} \tilde{T}^{\mu}_{\ \sigma}$ with equation (20), where $\tilde{\nabla}_{\mu} \tilde{J}^{\mu}$ can be non-vanishing with equation (19). This is reminiscent of the CP-violating coupling $\sqrt{-g}J_{B-L}^{\mu}\partial_{\mu}R$ for gravitational baryogenesis [41–43], where for a non-conserved baryon-lepton current $J_{B-L}$ net B − L charge is generated.

A more detailed analysis of equations (87) and (88) is left for future work.