Cosmological zoom-in perturbation theory as a consistent beyond point-particle approximation framework

Abstract

Modelling structure formation across the full dynamical range of the Universe remains a major challenge in cosmology. This difficulty originates from a fundamental limitation of geodesics in general relativity: a one-parameter family of geodesics can cease to be geodesic at a finite time. This implies that the conventional point-particle approximation is not the primary issue; rather, the breakdown of geodesic flow restricts a consistent description across scales. We develop a covariant multi-scale framework that resolves this problem by decomposing spacetime into hierarchical regions separated by matter horizons. We show how to match shared boundary consistently at the level of the action, leading to a covariant backreaction contribution. The resulting construction provides a first-principles theoretical foundation for cosmological zoom-in simulations and yields an effective energy-momentum tensor capturing the impact of the geometric backreaction effect. As an application, we demonstrate that this backreaction naturally produces flat galaxy rotation curves without invoking an additional dark matter component. Our results establish a new perspective on nonlinear structure formation, in which long dynamical range is resolved through a hierarchy of discrete geodesic domains.

Figures (7)

• Figure 1: The left panel shows the divergence of the initial relative velocity vector field, it has both negative and positive values. We evaluated the initial data at a redshift of $z =10$. Right panel shows two one-parameter families of geodesics emanating from a deformed initial hypersurface $h^{ab}_{\rm{ini}}(x)$. The blue curves denote the geodesics that started out from an initial over-density (initial negative local expansion scalar $\Theta_{L}<0$.) tend to converge/focus in the future. The red curves denote the geodesics that start out from an under-dense region tend to de-focus $\Theta_{L} >0$. For the red curves, there exists a finite time in the future where $\Theta_{H} + \Theta_{L} = 0$.
• Figure 2: The figure shows the plot of the expansion scalar as a function of proper time for various values of the density contrast. The decoupling timescale from the Hubble flow, $\Theta_{H}$, is very sensitive to the initial density contrast.
• Figure 3: We show a typical local geodesic coordinate system. The vertical line at the centre is the central geodesic $\gamma(\tau)$ parametrised by the proper time $\tau$. At a given hypersurface, $\Sigma$, we have a spatial triad. The time coordinate is synchronised with the observer's proper time along the worldline. The circle in the middle around the central geodesics denotes the astrophysical matter horizon.
• Figure 4: We show the convergence of the spacelike geodesics as a function of the physical distance from the centre of a gravitational system. Here $\sigma_{v}^{2 } = 200$[ km/s]. We made use of the NFW halo density profile. The left panel was estimated at $z =0.5$, while the right panel was estimated at $z=10.5$.
• Figure 5: Left panel: This is a schematic illustration of the timeline of structure formation in the universe, starting from light elements(particles) coming together through gravitational pull to form more massive particles. The left vertical line has crucial time scales when a collection of these particles decoupled from the Hubble flow: $\tau_{ \rm{H}}$, $\tau_{ \rm{star}}$, $\tau_{ \rm{gal}}$ and $\tau_{ \rm{clus}}$. Right panel: The right panel is a zoom-in on one section of the world-tube shown on the left panel in order to illustrate the glueing together of two manifolds along timelike and spacelike hypersurfaces. We indicated the initial and final spatial hypersurface surfaces $\Sigma_{\tau_+}^{\text{ini}}$ and $\Sigma_{\tau_-}^{\text{final}}$, and the common boundary hypersurface at the matter horizon (dashed oval) at $\tau = t_{\star}$. The timelike boundaries $B_\pm$ enclose the spatial region.