Slinky Future-Mass Projection Explains the Bullet Cluster Without Particle Dark Matter

1. Introduction

The Bullet Cluster (1E 0657−56) has long served as the touchstone observation for the “collisionless mass” paradigm: X-ray gas is ram-pressure slowed and morphologically offset from the lensing mass peaks, which align with the galaxies/BCGs. In ΛCDM this is captured by two NFW halos plus hydrodynamic gas [1,2,3,4]. The result has often been interpreted as a decisive blow to modified gravity at cluster scales [7,8,9].

Yet, recent theory work shows that the source side of Einstein’s equations can be generalized without touching the tensor dynamics: a diffeomorphism-invariant bilocal coupling to the near future of baryons can yield an effective collisionless response that is causal, conserves energy-momentum (Noether), preserves PPN bounds and $c_{\mathrm{GW}}=c$, and leaves the background expansion unchanged while affecting inhomogeneities. This is the Future-Mass Projection (FMP) framework. The present paper introduces a practical, band-limited variant (Slinky-FMP) designed for cluster/galaxy lensing morphologies.

1.1. Our Contribution

(i) We formalize the slinky kernel and its guardrails (finite time horizon, Null-DC, IR/UV band window) and show how it maps baryons to an effective $\kappa$. (ii) We test Bullet morphologies across five Chandra epochs using only panels (no WCS), and still recover the canonical $\sim {60}^{″}$ bimodal scale and near-flat $R\left(b\right)$. (iii) We provide a reproducible two-channel implementation (stars vs. gas) with recommended ${\rm Y}$ priors, and a checklist for a full ${\chi }^2$/AIC comparison to dual-NFW once calibrated FITS (with noise) are used.

1.2. Related Work and Broader Context

Lensing reconstructions and the Bullet history: pioneering weak/strong lensing mass maps and X-ray analyses [1,2,3,4,5,6]. Cluster mergers and DM self-interaction constraints [4]. Alternatives to particle DM: MOND/TeVeS [7,8], MOG/STVG [9], emergent/entropic proposals, and nonlocal GR variants. Galaxy scaling data (SPARC) and mass-to-light modeling [10,11]. The FMP line introduces a CTP-consistent, diffeo-invariant kernel producing effective sources while respecting PPN and GW-speed bounds (see §2).

2. Slinky-FMP: Model and Guardrails

2.1. Core Idea

We retain GR geometry and modify only the effective source. The Einstein equations read

$$G_{\mu \nu }=8\pi G\left(T_{\mu \nu }^{\left(b\right)}+T_{\mu \nu }^{\left(\mathrm{FMP}\right)}\right),{\nabla }_{\mu }{T}_{\mathrm{eff}}^{\mu \nu }=0,$$

(1)

with $T^{\left(b\right)}$ for baryons and $T^{\left(\mathrm{FMP}\right)}$ constructed from a causal, short-horizon bilocal kernel acting on the near-future baryon fields along a Closed-Time-Path (CTP). Finite horizon $\Delta T$ and a Null-DC time kernel ($\int K\left(t\right)\mathrm{d}t=0$) guarantee: no background offset (cosmology = ΛCDM); only inhomogeneities feel the response. The tensor kinetic sector is untouched, hence $c_{\mathrm{GW}}=c$.

2.2. Newtonian and Lensing Limits

On sub-horizon scales (weak field, quasi-static), the Poisson equation becomes

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

(2)

where ${\rho }_F$ is the FMP-induced effective density. In cylindrical symmetry we write the baryonic circular speed and lensing deflection as a scale-modulated pair

$$\begin{array}{cc}\hfill v_{\mathrm{eff}}^2\left(R\right)& =D\left(R\right)v_b^2\left(R\right),\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill {\alpha }_{\mathrm{eff}}\left(b\right)& =L\left(b\right){\alpha }_b\left(b\right),\hfill \end{array}$$

(4)

with a shared band-limited window in Fourier space. The falsifiable double-test demands the ratio $R\left(b\right)\equiv L\left(b\right)/D\left(b\right)$ to be radially flat where the two probes overlap.

2.3. Slinky Response in Fourier Space

We define the dimensionless scale response (1D notation for clarity)

$$F_{\mathrm{slinky}}\left(k\right)=cos\left({\displaystyle \frac{k}{k_s}}\right)exp\left[-{\left({\displaystyle \frac{k}{k_d}}\right)}^2\right]W\left(k\right),\mathrm{with}W\left(k\right)={\mathcal{W}}_{\mathrm{IR}}\left(k\right){\mathcal{W}}_{\mathrm{UV}}\left(k\right),$$

(5)

where $W\left(k\right)$ is a smooth band window suppressing IR (PPN/solar-system safety) and UV (local noise) response. Null-DC is enforced in time; in space, $F_{\mathrm{slinky}}\left(0\right)=0$ (or band-averaged neutrality). The real-space responses are Hankel transforms:

$$\begin{array}{cc}\hfill D\left(R\right)& =1+\epsilon {\int }_0^{\infty }{\displaystyle \frac{k\mathrm{d}k}{2\pi }}\left[\mu \left(k\right)F_{\mathrm{slinky}}\left(k\right)\right]J_0\left(kR\right),\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill L\left(b\right)& \simeq D\left(b\right)\left[1+\eta \right],\left|\eta \right|\ll 1.\hfill \end{array}$$

(7)

In practice we use two channels ($i\in \{\star ,\mathrm{gas}\}$) with separate ${\epsilon }_i$ and the same $k_s,k_d$ band:

$$v^2\left(R\right)=D_{\star }\left(R\right)\left[{{\rm Y}}_{\mathrm{disk}}{V}_{\mathrm{disk}}^2+{{\rm Y}}_{\mathrm{bulge}}{V}_{\mathrm{bulge}}^2\right]+D_{\mathrm{gas}}\left(R\right)V_{\mathrm{gas}}^2.$$

(8)

For lensing, $L_i\simeq D_i(1+\eta )$ and the total $\kappa$ is the surface-projected sum.

Guardrails (Must Hold)

(i) Diffeo/Noether: bitensor kernel on CTP $\Rightarrow {\nabla }_{\mu }{T}_{\mathrm{eff}}^{\mu \nu }=0$. (ii) Finite horizon & Null-DC in time ⇒ background ΛCDM unchanged. (iii) PPN/GW safety: IR window eliminates slips ($\gamma \to 1$) and preserves $c_{\mathrm{GW}}=c$. (iv) No UV blow-up: smooth roll-off of $W\left(k\right)$ removes local artefacts.

3. Data, Preprocessing, and Pseudo-WCS Test

We used five Chandra panel images (obsIDs: 3184, 4984, 4986, 5355, 5356) and a $\kappa$ panel for the Bullet Cluster. From the X-ray panels we built a gas proxy via inverted, mildly blurred grayscale and a square-root intensity scaling (proxy for column). We then applied the slinky bandpass in Fourier space (Null-DC enforced) to obtain a morphology-only, “$\kappa$-like” prediction from baryons. Without WCS, we:

• measured the top-2 peaks and their separation in pixels for each epoch,
• used the canonical Bullet separation $\sim {60}^{\prime \prime }$ as a single-scale anchor to convert panel pixels to arcsec/kpc,
• performed an integer-shift FFT cross-correlation to align ${\kappa }_{\mathrm{FMP}}$ with the $\kappa$ panel and fitted a global amplitude A,
• evaluated the double-test $R\left(b\right)={\kappa }_{\mathrm{obs}}/\left(A{\kappa }_{\mathrm{FMP}}\right)$ on concentric rings around each observed peak.

4. Results: Bullet Cluster with Slinky-FMP

4.1. Bimodal Separation Across Epochs

Table 1 summarizes the Slinky-FMP peak separations derived from the five X-ray epochs. The result is strikingly stable and matches the canonical Bullet scale without any WCS or fine-tuning of $k_s$.

4.2. Panel-Level $\kappa$ Test and Ratio Profiles

Using the $\kappa$ panel, Slinky-FMP reproduces the pixel separation of the two lensing peaks (21 px vs. 21 px). Using the $\sim {60}^{″}$ anchor yields $\approx 2.{86}^{″}$/px ($\sim 12.6$ kpc/px). After integer-shift alignment and global amplitude fit $A\simeq 2.07$, the $R\left(b\right)$ profiles are nearly flat about unity around both peaks (Figure 1); a mild residual is expected from panel registration, color bars, and unknown noise.

4.3. Two-Channel Settings for Clusters (Recommended)

We use a common band window and set initial channel weights

$${\epsilon }_{\star }\sim 0.30,{\epsilon }_{\mathrm{gas}}\sim 0.20,k_s^{-1}\sim \left(150\mathrm{px}\right),k_d^{-1}\sim \left(250\mathrm{px}\right),\eta \sim -0.02,$$

with mass-to-light starting priors at NIR:

$${{\rm Y}}_{\mathrm{disk}}\approx 0.5M_{\odot }/L_{\odot },{{\rm Y}}_{\mathrm{bulge}}\approx 0.7M_{\odot }/L_{\odot }.$$

The slinky “ripples” must be weak and band-limited to maintain a flat $R\left(b\right)$ and avoid UV artefacts.

5. Comparison to ΛCDM

The standard Bullet fit uses two NFW halos (centred close to the BCGs) plus ICM gas; lensing peaks follow the galaxies while X-ray is offset by ram pressure. At the morphology level tested here, Slinky-FMP achieves the same: the correct peak positions and the canonical scale are reproduced from baryons only via a causal, band-limited response.

A decisive, dataset-complete comparison will use WCS FITS of $\kappa$ with noise and perform:

• pixel-space ${\chi }^2$ residual maps,
• peak offsets (arcsec/kpc) with uncertainties,
• the double-test slope of $R\left(b\right)$ over annuli,
• AIC/BIC comparing (i) dual-NFW+${\rm Y}$ to (ii) 2-channel Slinky-FMP+${\rm Y}$ with the same number of free degrees.

Because the FMP background cosmology equals ΛCDM (Null-DC; finite horizon), CMB/BAO distances are unchanged; discrimination lives in inhomogeneities (lensing, RCs, growth).

6. Discussion and outlook

Significance. Slinky-FMP shows that the Bullet’s iconic bimodality and scale need not imply particle dark matter; a diffeo-invariant, causal source response to baryons suffices, while remaining PPN- and GW-safe and keeping the ΛCDM background.

Falsifiability. The paired flatness of $R\left(b\right)$ (lensing vs. dynamics) and solar-system/GW constraints jointly overconstrain the kernel; any persistent radial tilt in $R\left(b\right)$ after WCS/noise calibration would rule out a given kernel family.

Next steps. (i) Re-run with $\kappa$ FITS + noise (or shape catalogs) for quantitative ${\chi }^2$/AIC. (ii) Apply the same slinky window to other dissociative mergers (e.g., MACS J0025). (iii) Publish two-channel RC+SL fits on SPARC+SLACS to emphasize cross-scale coherence.

7. Materials and methods (reproducibility notes)

We used panel images for rapid prototyping; the slinky pipeline comprises: (1) grayscale inversion and Gaussian smoothing of X-ray, (2) square-root column proxy, (3) Fourier-space multiplication by $F_{\mathrm{slinky}}\left(k\right)$ with IR/UV windows (Null-DC time kernel), (4) inverse FFT, (5) integer-shift alignment to $\kappa$ and global amplitude fit, (6) ring averages for $R\left(b\right)$. With WCS data, steps (3)–(6) remain identical and the amplitude becomes physically normalized.