Time-Symmetric Gravity Today: A Closed-Time-Path Kernel for Future–Mass Projection that Preserves the ΛCDM Background and Acts Only on Structure

1. Introduction and Motivation

General Relativity (GR) with cold dark matter and a cosmological constant ($\Lambda$CDM) is extraordinarily successful on cosmological scales, yet persistent low-z growth trends (often summarized by $S_8$ or $f{\sigma }_8$) motivate carefully constrained extensions that modulate structure without spoiling the background expansion.1 Time-symmetric ideas are well established in quantum theory (CTP/Schwinger–Keldysh, influence functionals; two-time boundary views) and in modern studies of causal structure. We ask: can a controlled, time-symmetric effective response—mathematically anchored in the CTP formalism—act today like an effective dark component for fluctuations only, while leaving the homogeneous background fully $\Lambda$CDM?

Earlier FMP drafts posited a future-weighted source and showed good performance on galaxy rotation curves using a band-averaged real-space amplification $D\left(R\right)$ with explicit removal of DC offsets and shared windows [5,6]. However, the formal underpinning and background control were not yet optimal. Here we upgrade FMP to a CTP-derived kernel that (i) is a covariant bilocal bitensor obeying the Ward identity (diffeomorphism invariance ⇒ conservation), (ii) has a finite proper-time horizon $\Delta T$ and vanishing time-average (no background injection), and (iii) reduces to GR in the infrared and local limits, ensuring PPN and GW-speed safety.

2. Background: Time Symmetry, CTP, and Influence Functionals

The CTP (Schwinger–Keldysh) formalism [7] provides real-time quantum dynamics with doubled fields on a closed contour, naturally producing retarded/advanced and Keldysh (noise) kernels when environments are integrated out (Feynman–Vernon influence) [8]. In semiclassical gravity and stochastic gravity, such kernels yield causal, conserved effective equations sourced by expectation values and correlation functions of matter fields [9,10].

We adapt this to an effective response of the metric to baryons plus a short-horizon, advanced-weighted term that is strictly band-limited in time and carries zero DC offset. The time-symmetric spirit resonates with broader studies of indefinite/entangled causal order in quantum processes, while our construction remains operationally conservative (linear response for cosmological perturbations, exact conservation, and no change in tensor propagation speed at low k) [11,12].

3. CTP Action and the Divergence-Free Bitensor Kernel

Let $g_{\mu \nu }$ be the metric and $T_b^{\mu \nu }$ the baryon stress-energy. We introduce a covariant bilocal interaction in the CTP effective action

$$\begin{array}{cc}\hfill S_{\mathrm{eff}}[g;T_b]& =S_{\mathrm{GR}}\left[g\right]+S_b\left[g\right]+{\displaystyle \frac{1}{2}}\int d^{4}x{d}^{4}y\sqrt{-g_x}\sqrt{-g_y}{\mathcal{J}}_a^{\mu \nu }\left(x\right){\mathsf{K}}^{ab}\mu \nu \rho \sigma (x,y){\mathcal{J}}_{b\rho \sigma }\left(y\right),\hfill \end{array}$$

(1)

where $a,b\in \{+,-\}$ label the CTP branches, ${\mathcal{J}}_a^{\mu \nu }$ are source operators (here chosen proportional to $T_b^{\mu \nu }$ in the weak-field limit), and ${\mathsf{K}}^{ab}$ is the CTP kernel. Decomposing into retarded/advanced/Keldysh components via the usual Keldysh rotation yields a retarded part ${\mathsf{K}}^R$ that ensures causality and an advanced part ${\mathsf{K}}^A$ that we constrain to a finite future horizon$\Delta T$ with vanishing integral:

$$\begin{array}{c}\hfill {\int }_0^{\Delta T}d\tau {\mathsf{K}}^A(x;\tau )=0,\tau \equiv \mathrm{proper}\mathrm{time}\mathrm{advance}\mathrm{along}\mathrm{a}\mathrm{chosen}\mathrm{congruence}.\end{array}$$

(2)

Diffeomorphism invariance of $S_{\mathrm{eff}}$ implies the Noether identity ${\nabla }_{\mu }{T}_{\mathrm{eff}}^{\mu \nu }=0$ with $T_{\mathrm{eff}}^{\mu \nu }\equiv T_b^{\mu \nu }+T_{\mathrm{FMP}}^{\mu \nu }$, so the added nonlocal piece is conserved by construction. Parallel propagators map indices between x and y, making the bitensor structure manifest and removing the ambiguities of the earlier “projector×scalar” ansatz.

3.0.0.1. Minimal analytic kernel.

In a comoving frame for cosmology or in local Fermi normal coordinates for galaxies, a compact representation of the (advanced) time kernel that satisfies (2) is

$$\begin{array}{c}\hfill K_A\left(\tau \right)=\eta {\displaystyle \frac{\tau -\Delta T/2}{\Delta T^2}}\Theta \left(\tau \right)\Theta (\Delta T-\tau ),{\int }_0^{\Delta T}{K}_A\left(\tau \right)d\tau =0,\end{array}$$

(3)

optionally multiplied by a smooth spatial window (Bitensor generalization omitted for brevity). Band-limited forms ensure that only derivatives of fluctuations are probed, never a DC shift.

4. Cosmological Limit: Fixing $R\left(z\right)$ and Acting Only on Growth

Define $R\left(z\right)\equiv {\Omega }_F/{\Omega }_b\simeq {\rho }_F/{\rho }_b$. We fix

$$\begin{array}{c}\hfill R\left(z\right)\equiv R_0=5.4,\end{array}$$

(4)

so that the homogeneous expansion $H^2/H_0^2={\Omega }_{\Lambda 0}+{\Omega }_{b0}{(1+z)}^3(1+R_0)$ is exactly the $\Lambda$CDM one inferred from precision probes. The FMP correction then enters only the linear growth ODE via an effective coupling $\mu (a,k)$:

$$\begin{array}{c}\hfill D^{\prime \prime }+\left(2+{\displaystyle \frac{dlnH}{dlna}}\right)D^{\prime }-{\displaystyle \frac{3}{2}}\mu (a,k){\Omega }_{m,\mathrm{eff}}\left(a\right)D=0,{\Omega }_{m,\mathrm{eff}}\left(a\right)={\displaystyle \frac{{\Omega }_{b0}(1+R_0)a^{-3}}{H^2/H_0^2}},\end{array}$$

(5)

with primes $d/dlna$. For a short horizon $\Delta T\ll H^{-1}$, the advanced piece leads to a band-averaged correction (schematically)

$$\begin{array}{c}\hfill \mu (a,k)\simeq 1+M_1\left(a\right)Hf+{\displaystyle \frac{M_2\left(a\right)}{2}}{H}^2\left[f^2+{\displaystyle \frac{df}{dlna}}+f{\displaystyle \frac{dlnH}{dlna}}\right]+\dots ,M_n\equiv {\int }_0^{\Delta T}{\tau }^n{K}_A\left(\tau \right)d\tau ,\end{array}$$

(6)

with $M_0=0$ from (2). Thus $\mu \to 1$ as $k\to 0$ and in the local limit, keeping PPN parameters and $c_{\mathrm{GW}}$ within measured bounds.

5. Galaxy Scale: Mapping to Real-Space Amplification

In the Newtonian/weak-field limit, the kernel induces a Fourier response $\Phi \left(\mathbf{k}\right)=-(4\pi G/k^2){\rho }_b\left(\mathbf{k}\right)[1+{\Xi }_d\left(k\right)]$ and a real-space amplification of the circular speed

$$\begin{array}{c}\hfill v_c^2\left(R\right)=D\left(R\right)v_b^2\left(R\right),D\left(R\right)=1+{\int }_{k_{min}}^{k_{max}}{\displaystyle \frac{kdk}{2\pi }}\left[\mu (a_0,k)W\left(k\right)-1\right]J_0\left(kR\right),\end{array}$$

(7)

where $W\left(k\right)$ is a shared window and $(k_{min},k_{max})$ are harmless regulators that remove DC offsets. This band-averaged $D\left(R\right)$ mapping—already used successfully in M31/MW fits—now arises from a conserved CTP kernel rather than a phenomenological scalar [5,6].

6. Old vs. New: What the CTP Kernel Fixes

• Conservation & covariance: the new bitensor kernel follows from a diffeo-invariant action, so ${\nabla }_{\mu }{T}_{\mathrm{eff}}^{\mu \nu }=0$. The old scalar/projector form had to enforce this by hand.
• No background drift: zero DC offset and constant $R\left(z\right)$ guarantee a strictly $\Lambda$CDM background; earlier drafts risked injecting a homogeneous component.
• Safety gates: $\mu \to 1$ as $k\to 0$ and locally, so PPN $(\gamma ,\beta )$ and $\dot{G}/G$ remain within Solar-System bounds; tensor speed equals c as required by GW170817 ($|c_{\mathrm{GW}}/c-1|\lesssim {10}^{-15}$), since we do not modify the tensor sector [13,14].
• Targeted phenomenology: a shallow $\sim 10$–$15\%$ dip of $\mu$ around $z\sim 0.3$–$0.7$ can ease low-z growth trends without touching BAO/SN/chronometers, consistent with your finite-horizon, zero-offset preprint.

7. Minimal Parameter Set and Priors

A practical two-parameter kernel suffices for LSS:

$$K_A\left(\tau \right)=\eta {\displaystyle \frac{\tau -\Delta T/2}{\Delta T^2}}\Theta \left(\tau \right)\Theta (\Delta T-\tau ),\{\eta ,\Delta T\}\mathrm{with}\mathrm{hard}\mathrm{prior}{\int }_0^{\Delta T}{K}_{A}d\tau =0.$$

In galaxy fits one additionally specifies a smooth $W\left(k\right)$ encoding disk thickness/time filtering and maps to $D\left(R\right)$ via (7). Cosmology blocks fix $({\Omega }_{b0},H_0,R_0)$ to Planck/BAO-calibrated values and never allow a homogeneous FMP component.

8. Observational Gates and Falsifiability

Cosmology-Light gate. (i) $|H\left(z\right)-H_{\Lambda \mathrm{CDM}}\left(z\right)|/H<1$–$3\%$ for $0\le z\lesssim 1$ (BAO/chronometers); (ii) $f{\sigma }_8\left(z\right)$ suppression of ∼10–15% in $z\sim 0.3$–$0.7$ with scale cuts avoiding nonlinear systematics.

Local gate. PPN $\gamma -1$, $\beta -1$, and $|\dot{G}/G|$ within Solar-System bounds; $c_{\mathrm{GW}}=c$.

Galaxy gate. Predictive $D\left(R\right)$ with the shared window and no DC leakage; fits like those in M31/MW must survive ablations and baryon prior variations.

9. Discussion: Relation to QFT and Causal Structure

Our CTP construction is orthodox QFT: the effective action with an influence kernel contains retarded/advanced pieces; we engineer the advanced sector to have finite horizon and zero mean so it acts only on fluctuations, not on the background. This implements a controlled form of time symmetry without paradoxes (global self-consistency selection) and does not violate microcausality in observable channels we test (growth and Newtonian-scale dynamics). Connections to broader time-symmetric/indefinite-causal-order ideas are conceptual, but our phenomenology is deliberately modest and testable in near-term surveys.

10. Conclusions

We provide a complete, conserved, and cosmology-safe CTP kernel realization of FMP. It preserves the $\Lambda$CDM background by fixing $R\left(z\right)=5.4$ and confines all new effects to inhomogeneous growth and galaxy-scale responses. The model is minimal, falsifiable, compatible with GW170817, and ready for LSS and rotation-curve pipelines. A reference implementation (growth ODE and $D\left(R\right)$ Hankel mapping) can accompany the preprint as supplementary code.