Future–Mass Projection with a Constant Dark-to-Baryon Background: A Renormalized FMP Variant Preserving Ω_DM/Ω_b = 5.4

1. Introduction

The $\Lambda$CDM model successfully accounts for background expansion and large-scale structure across cosmic time. The Planck 2018 legacy analysis constrains the baryon and cold-dark-matter densities to ${\Omega }_b{h}^2\simeq 0.0224$ and ${\Omega }_c{h}^2\simeq 0.120$ [1], implying a background ratio

$$R_0\equiv {\displaystyle \frac{{\Omega }_c}{{\Omega }_b}}\simeq {\displaystyle \frac{0.120}{0.0224}}\approx 5.4.$$

(1)

Distance probes (BAO, SNe Ia) and growth information (RSD, cosmic shear) broadly support this background when systematics are controlled [2,3,4,5].

Non-particle alternatives must therefore preserve the background while allowing modified inhomogeneous dynamics. We present such a construction for Future–Mass Projection (FMP), recasting it as an effective source that leaves $R\left(z\right)=R_0$ invariant while modifying structure through a scale- and time-dependent response tightly coupled to baryons.

2. Original FMP and Constant-R Renormalization

2.1. Schematic FMP

In FMP, beyond baryonic density ${\rho }_b(\mathbf{x},t)$ one introduces a gravitationally active field ${\rho }_F$ obtained by a temporal projection of baryon trajectories:

$${\rho }_F^{\mathrm{raw}}(\mathbf{x},t)={\int }_0^{\infty }K(\Delta t)\Pi \left(\mathbf{x},t+\Delta t|I_t\right)\mathrm{d}\Delta t,$$

(2)

where K is a causal kernel and $\Pi$ symbolically denotes a coarse-grained forecast of baryon mass based on information $I_t$. The modified Poisson equation reads

$${\nabla }^2\Phi (\mathbf{x},t)=4\pi G\left[{\rho }_b(\mathbf{x},t)+{\rho }_F(\mathbf{x},t)+{\rho }_{\Lambda }\right].$$

(3)

2.2. Background–Fluctuation Renormalization

We decompose ${\rho }_b={\overline{\rho }}_b+{\delta }_b$ and define a spatial average $〈{\rho }_F^{\mathrm{raw}}〉\left(t\right)$. Our renormalized model sets

$$\begin{array}{cc}\hfill {\delta }_F(\mathbf{x},t)& \equiv {\rho }_F^{\mathrm{raw}}(\mathbf{x},t)-〈{\rho }_F^{\mathrm{raw}}〉\left(t\right),\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill {\rho }_F(\mathbf{x},t)& \equiv R_0{\overline{\rho }}_b\left(t\right)+{\delta }_F(\mathbf{x},t).\hfill \end{array}$$

(5)

Equivalently, in Fourier space we impose a zero-DC condition on the kernel,

$$\tilde{K}(\mathbf{k}=\mathbf{0},\omega )=0,$$

(6)

and add the fixed homogeneous term $R_0{\overline{\rho }}_b$. Then

$${\overline{\rho }}_F=R_0{\overline{\rho }}_b\Rightarrow R\left(z\right)={\displaystyle \frac{{\Omega }_F}{{\Omega }_b}}=R_0\forall z,$$

(7)

so $H\left(a\right)$ and background distances remain those of $\Lambda$CDM for the same $({\Omega }_b,{\Omega }_c,h)$ [1,2,3].

Proposition 1

(Background invariance). With Eqs. (4)–(5) the Friedmann equation retains the standard form $H^2\left(a\right)=H_0^2\left[{\Omega }_r{a}^{-4}+{\Omega }_b(1+R_0)a^{-3}+{\Omega }_{\Lambda }\right],$ and all background distance indicators (CMB peak positions, BAO scales, SNe Hubble diagram) are unchanged by construction.

Proof. 

Spatial averaging of (3) with (5) gives ${\overline{\rho }}_{\mathrm{tot}}={\overline{\rho }}_b+R_0{\overline{\rho }}_b+{\rho }_{\Lambda }$, since $〈{\delta }_F〉=0$ by definition. Hence the homogeneous energy density equals the $\Lambda$CDM one with ${\Omega }_m={\Omega }_b(1+R_0)$. The background solution and comoving sound horizon are unaffected. □

2.3. Fluctuation Response and Scale Localization

Only ${\delta }_F$ appears in the inhomogeneous sector:

$${\nabla }^2\Phi =4\pi G\left[(1+R_0){\overline{\rho }}_b+{\delta }_b+{\delta }_F+{\rho }_{\Lambda }\right].$$

(8)

At linear order we parameterize

$${\tilde{\delta }}_F(\mathbf{k},a)=T_F(k,a)F\left(k\right){\tilde{\delta }}_b(\mathbf{k},a),$$

(9)

with a smooth high-pass $F\left(0\right)=0$; e.g.

$$F\left(k\right)={\displaystyle \frac{k^2}{k^2+k_c^2}},$$

(10)

or a band-limited generalization. The comoving scale $k_c^{-1}$ confines cFMP to galaxy–cluster regimes, mitigating conflict with large-scale growth [2,4,5].

3. Linear Growth and Effective Coupling

Defining ${\Omega }_m={\Omega }_b(1+R_0)$, the effective matter fluctuation is

$${\delta }_{\mathrm{eff}}\equiv {\displaystyle \frac{\delta {\rho }_b+{\delta }_F}{{\overline{\rho }}_m}}={\displaystyle \frac{{\Omega }_b}{{\Omega }_m}}\left[1+T_F(k,a)F\left(k\right)\right]{\delta }_b.$$

(11)

In a growth-equation form one may write

$${\delta }_{\mathrm{eff}}^{\prime \prime }+\left(2+{\displaystyle \frac{H^{\prime }}{H}}\right){\delta }_{\mathrm{eff}}^{\prime }-{\displaystyle \frac{3}{2}}{\Omega }_m\left(a\right)\mu (k,a){\delta }_{\mathrm{eff}}=0,$$

(12)

with an effective modification $\mu (k,a)$ determined by the baryon fraction and $T_F$.1 Current shear and RSD data bound any sizeable, broad-band deviation [2,4,5]. cFMP therefore adopts $T_F$ that is localized in k and (optionally) limited in redshift.

4. Nonlinear Phenomenology

4.1. Galaxies: Dynamics and Lensing

Because ${\delta }_F$ is baryon-tethered, the same kernel parameters must describe rotation curves and lensing deflection for a given galaxy. The tight radial acceleration relation (RAR) [9] suggests precisely such a baryon–gravity linkage. Joint lensing–dynamics fits thus provide a stringent, object-by-object test.

4.2. Clusters and the Baryon Fraction

Relaxed-cluster gas mass fractions $f_{\mathrm{gas}}$ constrain the background and show near-constancy with redshift after calibration [6,7]. Since cFMP fixes $R\left(z\right)=R_0$, it preserves these results at the background level; residual effects arise only through inhomogeneous physics already considered in $\Lambda$CDM analyses.

4.3. Substructure in Strong Lenses

Strong-lensing “gravitational imaging” detects low-mass subhalos [15,16]. cFMP generically predicts a lower incidence of truly baryon-free subhalos than particle CDM due to baryon tethering. A statistically significant population of dark, baryonless subhalos would therefore challenge the mechanism.

4.4. Merging Systems

In Bullet-like mergers, lensing mass peaks offset from X-ray gas are well established [17]. cFMP predicts that the lensing pattern should be reconcilable with the time-integrated baryon flow; failure to achieve such consistency with plausible baryon end-states would falsify cFMP on these scales.

5. Consistency with High-Energy and Solar-System Tests

cFMP is an effective source within the Poisson sector of GR; it does not modify the tensor kinetic term or GW propagation speed. Consequently, the GW170817/GRB170817A constraints are automatically satisfied [10,11]. Parametrized post-Newtonian bounds are likewise respected, since no new long-range force is introduced.

6. Relation to Known Small-Scale Issues

Baryon–DM small-scale tensions (cores, satellites, too-big-to-fail) have various proposed resolutions within baryonic astrophysics and/or new DM physics [12]. cFMP offers an alternative effective description: extra, baryon-tied mass that can mimic some small-scale trends while remaining invisible to non-gravitational searches.

7. Observational Program and Falsifiability

We highlight four decisive tests:

• Joint fits in galaxies: For spirals and ellipticals, fit dynamics and lensing with a common $(A_F,k_{*},\Delta lnk)$ entering $T_F$. Failure to fit both with the same parameters would disfavor cFMP.
• Subhalo demographics: Use strong-lensing samples to measure the abundance of baryon-free subhalos [15,16]. A high fraction at $M\lesssim {10}^9{M}_{\odot }$ would be problematic for cFMP.
• Growth tomography: RSD $f{\sigma }_8\left(z\right)$ and weak lensing two-point statistics constrain broad-band $\mu (k,a)$[2,4,5]. Detection of a large, wide k-range enhancement would force $T_F$ to be narrow or small; a contradictory requirement from galaxy scales would challenge viability.
• Merging clusters: Test whether lensing peaks can be matched by reasonable forward models of the baryon flow over merger timescales in systems like 1E 0657−56 [17].

8. Conclusions

We presented a constant-R renormalization of FMP that leaves the $\Lambda$CDM background intact while enabling baryon-tethered, scale-localized modifications to structure. The construction removes the kernel DC mode and adds a fixed homogeneous term $R_0{\overline{\rho }}_b$, guaranteeing $R\left(z\right)=R_0=5.4$ consistent with CMB/BAO/SNe [1,2,3]. The remaining degrees of freedom enter through $T_F(k,a)$, to be constrained by shear and RSD [4,5]. cFMP is falsifiable via joint galaxy lensing–dynamics fits, subhalo statistics, and merger chronologies, and it predicts null results in non-gravitational DM searches.