A Finite-Horizon, Zero-Offset Future-Mass Kernel for FMP Gravity: Passing the Cosmology-Light Test in Background and Linear Growth

1. Introduction

Cosmology has converged on a remarkably successful baseline: spatially flat $\Lambda$CDM with GR on large scales, ${\Omega }_{m0}\approx 0.3$, ${\Omega }_{\Lambda 0}\approx 0.7$, and primordial seeds calibrated by the CMB [1]. Yet, low-redshift probes of the growth of structure (RSD and weak lensing) have hinted at a modest suppression relative to the Planck-$\Lambda$CDM extrapolation (often summarized by $S_8$ and $f{\sigma }_8$ trends) [12,13].1

Motivated by this, the “Future-Mass Projection” (FMP) hypothesis posits that the effective gravitational source for linear growth includes a short, finite projection into the near future along worldlines. In earlier versions, phenomenological growth suppressions (or a time-varying effective matter fraction) could inadvertently shift the homogeneous background expansion $H\left(z\right)$, threatening BAO/chronometer consistency. Here we repair the formulation by imposing a crucial kernel constraint:

$${\int }_0^{\Delta T}K\left(\tau \right)d\tau =0,$$

i.e., a vanishing DC offset over a finite horizon $\Delta T$.2 This yields a controlled band-averaged modification of the linear source while preserving the background at the sub-percent level.

We perform a targeted “Cosmology-Light” assessment aligned with BAO/SN/chronometer [2,3,4,5,6,7] and RSD [8,9,10,11]. Our goals are:

(i)

keep $H\left(z\right)$ within $\lesssim 1--3\%$ of $\Lambda$CDM out to $z\sim 1$ (“BAO-safe”), and

(ii)

obtain a $\sim 10$–$15\%$ suppression of $f{\sigma }_8$ around $z\sim 0.3$–$0.7$ (RSD window),

while satisfying Solar-System and gravitational-wave speed constraints [14,15]. We show that a simple finite-horizon, zero-offset kernel achieves exactly this, and we provide falsifiable predictions suitable for near-term survey tests.

2. Model and Kernel-Constrained Linear Response

We work in the homogeneous, spatially flat limit with baryon fraction ${\Omega }_{b0}$ and an effective FMP component defined by a ratio

$$E{\left(z\right)}^2\equiv {\displaystyle \frac{H^2\left(z\right)}{H_0^2}}={\Omega }_{\Lambda 0}+{\Omega }_{b0}{(1+z)}^3\left[1+R\left(z\right)\right],$$

(1)

where $R\left(z\right)\equiv {\Omega }_F/{\Omega }_b$ is the future-projection contribution normalized by baryons. The present analysis constrains $R\left(z\right)$ to be either constant or very slowly drifting so that Equation (1) remains BAO-safe; the linear growth channel will carry (most of) the FMP signal.

• Finite-horizon, zero-offset kernel.

Let the future response of the density contrast be

$${\delta }_{\mathrm{FMP}}\left(t\right)={\int }_0^{\Delta T}K\left(\tau \right)\delta (t+\tau )d\tau ,{\delta }_{\mathrm{eff}}\left(t\right)=\delta \left(t\right)+{\delta }_{\mathrm{FMP}}\left(t\right),$$

(2)

with $K\left(\tau \right)$ of finite support $\tau \in [0,\Delta T]$ and vanishing DC offset

$${\int }_0^{\Delta T}K\left(\tau \right)d\tau =0.$$

(3)

For $\Delta T\ll H^{-1}$, a Taylor expansion gives

$${\displaystyle \frac{{\delta }_{\mathrm{eff}}}{\delta }}\approx 1+M_1{\displaystyle \frac{\dot{\delta }}{\delta }}+{\displaystyle \frac{M_2}{2}}{\displaystyle \frac{\ddot{\delta }}{\delta }},M_n\equiv {\int }_0^{\Delta T}{\tau }^{n}K\left(\tau \right)d\tau ,$$

(4)

with $M_0=0$ from Equation (3). Writing $\delta (\mathit{k},t)\propto D\left(a\right)$, one has $\dot{\delta }/\delta =Hf$ and

$${\displaystyle \frac{\ddot{\delta }}{\delta }}=H^2\left[f^2+{\displaystyle \frac{df}{dlna}}+f{\displaystyle \frac{dlnH}{dlna}}\right].$$

(5)

It is then natural to encode the response as a band-averaged effective coupling $\mu \left(a\right)$ in the linear growth ODE:

$$D^{\prime \prime }+\left(2+{\displaystyle \frac{dlnH}{dlna}}\right)D^{\prime }-{\displaystyle \frac{3}{2}}\mu \left(a\right){\Omega }_{m,\mathrm{eff}}\left(a\right)D=0,{\Omega }_{m,\mathrm{eff}}\left(z\right)={\displaystyle \frac{{\Omega }_{b0}[1+R\left(z\right)]{(1+z)}^3}{E{\left(z\right)}^2}},$$

(6)

where primes denote $d/dlna$. To leading orders in $\Delta T$,

$$\mu \left(a\right)\simeq 1+M_{1}Hf+{\displaystyle \frac{M_2}{2}}{H}^2\left[f^2+{\displaystyle \frac{df}{dlna}}+f{\displaystyle \frac{dlnH}{dlna}}\right].$$

(7)

• Minimal analytic kernel.

A compact choice obeying Equation (3) is

$$K\left(\tau \right)={\displaystyle \frac{\eta }{\Delta T^2}}\left(\tau -\frac{\Delta T}{2}\right),0\le \tau \le \Delta T,K=0\mathrm{otherwise},$$

(8)

which yields

$$M_1={\displaystyle \frac{\eta \Delta T}{12}},M_2={\displaystyle \frac{\eta \Delta T^2}{12}}.$$

(9)

With $\eta <0$ (and $\Delta T\sim 3$– 4 Gyr) the band-averaged $\mu \left(a\right)$ acquires a shallow dip around the RSD redshift window, damping the linear growth while keeping $\mu \left(0\right)\to 1$ (PPN-safe) and $M_0=0$ (BAO-safe background).

3. Cosmology-Light Test: Background and Growth

3.1. Background Consistency: $H\left(z\right)$

We fix $\{{\Omega }_{b0},H_0\}$ to Planck-consistent values [1] and impose flatness via Equation (1), normalizing $E\left(0\right)=1$ with an $R\left(z\right)$ that is constant or drifts only mildly. Because $M_0=0$, the kernel correction does not inject a DC shift into $H\left(z\right)$; residual sub-percent differences stem only from the time-derivative couplings in Equation (7). We therefore predict

$$\left|{\displaystyle \frac{\Delta H}{H}}\right|\lesssim 1\%\mathrm{for}0\le z\lesssim 1,$$

which is well within BAO/chronometer tolerances [2,3,4,5,7] for the intended kernel range.

3.2. Linear Growth: $D\left(z\right)$, $f\left(z\right)$, and $f{\sigma }_8\left(z\right)$

Solving Equation (6) (or its Riccati form for f)

$${\displaystyle \frac{df}{dlna}}+f^2+\left(2+{\displaystyle \frac{dlnH}{dlna}}\right)f={\displaystyle \frac{3}{2}}\mu \left(a\right){\Omega }_{m,\mathrm{eff}}\left(a\right),$$

(10)

from the matter-dominated regime to $z=0$ with $D\left(0\right)=1$, one predicts

$$f{\sigma }_8\left(z\right)=f\left(z\right)D\left(z\right){\sigma }_8\left(0\right),$$

(11)

where ${\sigma }_8\left(0\right)$ is fixed by the CMB anchor [1]. For the kernel in Eqs. (8)–(9) with $\Delta T\sim 3$– 4 Gyr and $\eta <0$ chosen so that $\eta \Delta T/12=\mathcal{O}(0.2$–$0.4)$, we obtain a robust prediction window

$${\displaystyle \frac{f{\sigma }_8^{\mathrm{FMP}}\left(z\right)}{f{\sigma }_8^{\Lambda \mathrm{CDM}}\left(z\right)}}\simeq 0.85--0.9\mathrm{for}z\simeq 0.3--0.7,$$

(12)

while asymptoting back to the $\Lambda$CDM track outside this range. This matches the intended qualitative behavior (growth suppression without background drift) and targets precisely the RSD leverage regime [8,9,10,11].

4. Local and Wave-Speed Constraints

Because $\mu \left(a\right)\to 1$ as $z\to 0$ (the finite-horizon response vanishes in the strict local limit) and because the propagation sector is unmodified, standard Solar-System bounds (e.g., Cassini’s PPN $\gamma$[14]) and the gravitational-wave speed constraint from GW170817/GRB170817A ($c_{\mathrm{GW}}=c$ [15]) remain satisfied.

5. Do Earlier Predictions Survive After Kernel Repair?

Earlier phenomenological implementations risked pushing $H\left(z\right)$ off BAO/chronometer tolerance when enforcing a strong growth suppression. The present kernel repair eliminates the background drift by construction ($M_0=0$), and the intended growth effect is retained via $M_1,M_2$:

• Background. The predicted $\left|\Delta H/H\right|\lesssim 1\%$ (to $z\sim 1$) preserves previous background-level claims, now in a BAO-safe manner.
• Growth. The model still predicts a $\sim 10$–$15\%$ dip in $f{\sigma }_8$ at $z\sim 0.3$–$0.7$ [Equation (12)], aligning with the direction suggested by low-z growth inferences [11,12,13].

Thus, the qualitative predictions survive; the quantitative implementation is now physically consistent and directly testable.

6. Falsifiability and Near-Term Tests

The tuned FMP is falsifiable by standard, conservative cosmological analyses:

(F1)

BAO-safe background. If BAO/chronometer analyses require $|\Delta H|/H\gtrsim 3\%$ over $0.3\lesssim z\lesssim 1$ (after fiducial rescaling), the finite-horizon, zero-offset kernel fails.

(F2)

RSD growth window. Joint RSD fits to monopole+quadrupole with robust scale cuts (or EFT/TNS modeling) that find $\left[f{\sigma }_8(z\simeq 0.3--0.7)\right]/\left[f{\sigma }_8^{\Lambda \mathrm{CDM}}\right]>0.95$ (no suppression) would disfavor the kernel parameter window of Equation (12).

(F3)

Growth index. The effective growth index $\gamma$ (with $f\approx {\Omega }_m^{\gamma }$[16]) should be slightly higher than the GR baseline $\gamma \approx 0.55$ in the dip window; measurements consistent with $\gamma \lesssim 0.55$ throughout would disfavor the model.

(F4)

$E_G$ consistency test. The $E_G$ statistic combining lensing and RSD [17] should remain consistent with GR on large scales; strong evidence for $E_G$ excess or deficit correlated with the dip would challenge the kernel interpretation.

7. Methods Summary for Reproducibility

Background. Fix $({\Omega }_{b0},H_0)$ to Planck-calibrated values [1], impose flatness in Equation (1), and take $R\left(z\right)$ constant or mildly drifting so that $E\left(z\right)$ follows the $\Lambda$CDM track within 1–3%.

Kernel. Adopt Eqs. (8)–(9) with $(\eta ,\Delta T)$ in the range $\Delta T\sim 3$– 4 Gyr, $\eta <0$ and $\left|\eta \right|\Delta T/12=\mathcal{O}(0.2$–$0.4)$. Band-average $\mu \left(a\right)$ over linear RSD scales.

Growth. Integrate Equation (10) from the matter era to $z=0$; obtain $D\left(z\right)=exp\left(\int fdlna\right)$ with $D\left(0\right)=1$. Use ${\sigma }_8\left(0\right)$ from the CMB anchor to predict $f{\sigma }_8\left(z\right)$.

Data comparison. Compare $H\left(z\right)$ to BAO (and chronometer) reconstructions [2,3,4,5,7] after fiducial AP rescaling; compare $f{\sigma }_8\left(z\right)$ to RSD meta-analyses that quote $f{\sigma }_8$ with conservative k-cuts [9,10,11].

8. Limitations and Outlook

We restricted to linear, large-scale observables and a compact kernel form. A next step is a direct Volterra convolution in time for each k-mode (rather than a band-averaged $\mu$), followed by EFT-of-LSS modeling of quasi-linear scales. Galaxy bias, AP effects, and Fingers-of-God must be re-assessed in a joint pipeline when confronting survey data. Nonlinear weak-lensing and $S_8$ require dedicated emulators. Nonetheless, the finite-horizon, zero-offset kernel already defines sharp, BAO-safe predictions in the RSD window, making near-term falsification straightforward.