Cosmological Extension of the Electrostatic Mass Emergence (EME) Theory

Revision note (v13.0). The scope paragraph has been rewritten. It described the theory as a "local, terrestrial model" based on "the Earth's toroidal field" with a "strict no space/universe mechanisms" rule, which contradicted the rest of the document. The cross-reference to a Lagrangian "derived in Section 5 of the main report" is replaced by the Level 1 action in the main document. The modified Friedmann equations are kept; $G$ in them is the metric-sector constant. The dark-matter and dark-energy analogues are marked speculative and not computed. The earlier "unique signature" and "falsifiable CMB damping tail" are withdrawn: no CMB, matter power spectrum, growth or expansion-history calculation has been done. The linear growth formula $G_{\rm eff}(k,a)$ is stated. See section 1 of the main document. "EME" is the historical name of MCE.

1. Scope and Assumptions

MCE (historically EME) is stated in the main document as three levels. Level 0 is the metric sector, the Einstein–Hilbert action with Newton's constant $G$ taken from experiment. Level 1 is a screened scalar $\phi$ with a species-dependent conformal coupling to matter. Level 2, an emergent origin of the couplings, is an open programme. Nothing in this structure restricts the theory to a terrestrial setting. The toroidal Earth (Toroidal Field Framework) is one optional global boundary condition for the same local equations, and it plays no role in cosmology.

This document applies the Level 1 action on a Friedmann–Lemaître–Robertson–Walker (FLRW) background. It sets up the equations that a cosmological calculation would need, and it states what has and has not been computed. Everything below the level of the equations is Open: no cosmological observable has been calculated for MCE parameters.

To compare with large-scale data one averages the scalar field over cosmological volumes, which gives an effective stress-energy tensor $T_{\mu\nu}^{\text{EME(cosmo)}}$ that can be coupled to the FLRW metric. This coarse-graining is a deliberate, auditable step for comparison with $\Lambda$ CDM.

2. Effective Cosmological Stress-Energy Tensor

2.1. Justification for coarse-graining

The scalar has a range $\lambda(\rho)=\hbar/(m_{\rm eff}(\rho)c)$ that depends on the local density (see the screened scalar document). For the illustrative benchmark in that document ( $n=1$ , $\Lambda=2.4$ meV, $\beta=0.05$ , not fitted to data) the range at the cosmic mean matter density is about $3.7\times10^{19}$ m (about 1 kpc). Averaging over a volume $V$ with $\lambda^3\ll V\ll H^{-3}$ then smooths the local non-linear structure and gives a homogeneous and isotropic effective fluid. The Hubble radius is $H^{-1}\sim10^{26}$ m. The earlier text used the fixed length $\lambda_c\sim10^{-6}$ m for this step, and that length is retired.

2.2. The action and the effective fluid

We begin with the Level 1 action given in the main document:

S = \int d^4x\sqrt{-g}\left[\frac{M_{\rm Pl}^2}{2}R - \frac12(\partial\phi)^2 - V(\phi)\right] + \sum_i S_i\!\left[\psi_i,\ A_i^2(\phi)\,g_{\mu\nu}\right],\qquad A_i(\phi)=e^{\beta_i\phi/M_{\rm Pl}},

with $M_{\rm Pl}^2=1/(8\pi G)$ . The total energy-momentum tensor is $T_{\mu\nu}^{\text{Total}}=T_{\mu\nu}^{M}+T_{\mu\nu}^{\text{EME}}$ , and the scalar part is averaged over a large comoving volume:

\langle T_{\mu\nu}^{\text{EME}} \rangle = T_{\mu\nu}^{\text{EME(cosmo)}} = \text{diag}(-\rho_{\text{EME}}, p_{\text{EME}}, p_{\text{EME}}, p_{\text{EME}}).

The dominant contribution comes from the scalar field. After coarse-graining, the effective density and pressure are

\rho_{\text{EME}}(a) = \tfrac{1}{2} \langle \dot{\phi}^2 \rangle + \tfrac{1}{2} \langle (\nabla \phi)^2 \rangle + \langle V(\phi) \rangle,

p_{\text{EME}}(a) = \tfrac{1}{2} \langle \dot{\phi}^2 \rangle - \tfrac{1}{6} \langle (\nabla \phi)^2 \rangle - \langle V(\phi) \rangle,

where $a$ is the scale factor. The v12 expressions also contained an interaction term $\langle\kappa\phi T\rangle$ . In the Einstein-frame form used here the coupling sits in the matter sector: the energy density of non-relativistic matter scales as $a^{-3}e^{\beta\phi/M_{\rm Pl}}$ , and the exchange of energy between matter and the scalar appears in the continuity equations. The scalar equation of motion on the background is $\ddot\phi+3H\dot\phi=-V'(\phi)-(\beta/M_{\rm Pl})\,\rho_m e^{\beta\phi/M_{\rm Pl}}$ , with $\rho_m$ the conserved matter density (a standard result for conformally coupled scalars; see Khoury and Weltman, Physical Review D 69, 044026, 2004). Its static linearised limit is electrostatics in a screening medium (Section 2.1 of the screened scalar document).

3. Modified Friedmann Equations

The standard Friedmann equation, $H^2=(8\pi G/3)\rho_{\rm Total}$ , is modified by the inclusion of the scalar's effective density $\rho_{\text{EME}}$ :

H^2 = \left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3} \left( \rho_b + \rho_r + \rho_{\Lambda} + \rho_{\text{EME}}(a) \right),

where $\rho_b$ is the baryonic matter density, $\rho_r$ the radiation density and $\rho_\Lambda$ a cosmological-constant density. The acceleration equation is

\frac{\ddot{a}}{a} = -\frac{4\pi G}{3} \left( \rho_b + 3p_b + \rho_r + 3p_r + \rho_{\Lambda} + 3p_{\Lambda} + \rho_{\text{EME}} + 3p_{\text{EME}} \right).

$G$ in these equations is the Newton constant of the metric sector (Level 0). It is taken from experiment, and the equations hold for any total energy-momentum tensor. The statement in v12 that the scalar, rather than curvature, generates the force of gravity is withdrawn. If the potential $V(\phi)$ supplies the late-time acceleration, the separate term $\rho_\Lambda$ should be dropped to avoid counting the same contribution twice.

4. Dark-Matter and Dark-Energy Analogues (Speculative, Not Computed)

The coarse-grained scalar fluid is a candidate for the missing components of the cosmic inventory only in the sense that it has an energy density and pressure that can enter the equations of section 3. No calculation has been done that shows it supplies either component. Both subsections are Open and speculative.

4.1. Dark-matter analogue

If the scalar is non-relativistic and $V(\phi)$ is negligible at late times, the fluid can behave like pressureless matter. The continuity equation is

\dot{\rho}_{\text{EME}} + 3H(\rho_{\text{EME}} + p_{\text{EME}}) = 0,

and for $w_{\text{EME}}=p_{\text{EME}}/\rho_{\text{EME}}\approx0$ it gives $\rho_{\text{EME}}(a)=\rho_{\text{EME},0}\,a^{-3}$ .

Two statements from v12 are withdrawn, and one assumption is flagged.

Withdrawn: that "the effective charge field of baryonic matter itself generates the required extra gravitational pull". For the legacy value of the coupling, $2\beta_0^2=5.7\times10^{-3}$ , the unscreened scalar force is 0.57% of the Newtonian force. That is far smaller than the gap between the baryon density and the cosmic dark-matter density, which is a factor of several. Screening reduces the scalar force further.
Withdrawn: that the model has a "unique signature" in a non-zero, scale-dependent sound speed $c_s^2(k,a)$ . A non-zero sound speed is a property of many dark-sector models. No $c_s^2(k,a)$ has been derived for MCE.
Assumption: that the scalar is non-relativistic with a negligible potential at late times. The benchmark potential, $V=\Lambda^{4+n}/\phi^n$ , has no minimum in vacuum, so the field rolls instead of oscillating. A matter-like fluid is therefore not what the benchmark gives without further assumptions, and whether any Level 1 potential gives one has not been shown.

4.2. Dark-energy analogue

If the scalar is dominated by its potential energy, then $p_{\text{EME}}\approx-\rho_{\text{EME}}$ and $\rho_{\text{EME}}\approx\langle V(\phi)\rangle$ , as for any slowly rolling scalar. The statement in v12 that this "suggests that the quantum vacuum polarisation that gives rise to the effective charge density is also the source of cosmic acceleration" is withdrawn. It has no calculation behind it. In the screened scalar document the scale $\Lambda=2.4$ meV is chosen as a reference equal to the dark-energy scale, and it is not derived. The cosmological-constant problem is not solved by MCE: the vacuum-energy toy model reduces a discrepancy of about 123 orders of magnitude to about 40, which is not a solution (see the main document).

5. Observational Signatures: What Has and Has Not Been Computed

No CMB, matter power spectrum, growth or expansion-history calculation has been done for MCE. The statements in v12 that the model has a "unique equation of state and scale-dependent coupling" and "makes specific, falsifiable predictions regarding the CMB and the growth of structure" are withdrawn.

5.1. CMB anisotropies

The earlier text asserted a shift in the third and higher acoustic peaks and a "suppression of the power in the damping tail", described as "a highly falsifiable signature". Neither was calculated, and both are withdrawn. A CMB calculation would require:

The background evolution of $\phi$ with its matter coupling, for specified $(\beta_0,C,\Lambda,n)$ .
The perturbation equations for the scalar and the modified Boltzmann hierarchy for the species that couple to it.
An implementation in a Boltzmann code (CLASS or CAMB) and a comparison with data.

None of these has been done. Until it is, there is no statement about the CMB.

5.2. Growth of structure

For a scalar-tensor model with a screened scalar, the effective gravitational coupling in linear perturbation theory is scale dependent:

G_{\rm eff}(k,a)=G\left[1+\frac{2\beta^2k^2}{k^2+a^2m^2(a)}\right].

This is a standard scalar-tensor result, not a derivation specific to MCE. It has not been evaluated for MCE parameters. Two limits follow from the formula. For $k\ll a\,m(a)$ , $G_{\rm eff}\to G$ : scales larger than the scalar range feel no extra force. For $k\gg a\,m(a)$ , $G_{\rm eff}\to G(1+2\beta^2)$ , which for the legacy value $2\beta_0^2=5.7\times10^{-3}$ is an enhancement of 0.57% before any screening.

For a conformally coupled scalar with real $\beta$ , $G_{\rm eff}\ge G$ . The linear-theory effect is therefore an enhancement of growth on scales below the scalar range, not the suppression of $P(k)$ at high $k$ that the v12 text claimed. A suppression from a pressure or sound-speed effect (a Jeans length) is a separate mechanism that has not been derived. The v12 claim of a growth index $\gamma(a,k)$ that deviates from the $\Lambda$ CDM value of about 0.55 is withdrawn for the same reason.

The Euclid, DESI and DES-type forecast tables in the v12 documents were withdrawn because their inputs were wrong. One input carried a unit error of about 23 orders of magnitude in the wavenumber $k_c$ , and another used $\kappa^2C\sim10^{-21}$ , which cannot produce per cent effects. Replacing them requires the method above, with the field equation solved for the screening in each environment. The forecast is not yet computed.

5.3. What would constrain the model

Once computed, a cosmological observable excludes a region of $(\beta_0,\Lambda,n)$ in the same way as the laboratory tests: the parameters for which the calculated signal exceeds the observed bound are excluded. Local tests (MICROSCOPE, the Eöt-Wash torsion balance, the Cassini bound, atom interferometry near a source mass) already restrict that space, as set out in section 7 of the screened scalar document.

6. Conclusion

This document gives the equations for applying the Level 1 scalar sector on an FLRW background: the coarse-grained fluid, the modified Friedmann and acceleration equations, and the linear growth coupling $G_{\rm eff}(k,a)$ . It does not give a result. The dark-matter and dark-energy analogues are speculative. The linear-theory formula suggests a growth enhancement of order $2\beta^2$ below the scalar range rather than a suppression. No CMB or $P(k)$ calculation has been done, and the earlier claims of a unique signature are withdrawn. The cosmological extension remains an open programme, separate from the laboratory tests that define the testable part of MCE.

References

Khoury, J., Weltman, A. (2004). Chameleon cosmology. Physical Review D 69, 044026.
Khoury, J., Weltman, A. (2004). Chameleon fields: awaiting surprises for tests of gravity in space. Physical Review Letters 93, 171104.