[Paper Review] Lemaitre-Tolman-Bondi cosmological models, smoothness, and positivity of the central deceleration parameter

R. Ali Vanderveld, Éanna É. Flanagan+1 more|ArXiv.org|Apr 28, 2009

Cosmology and Gravitation Theories3 citations

TL;DR

This paper resolves a contradiction in Lemaître-Tolman-Bondi (LTB) cosmological models by demonstrating that negative central deceleration parameters ($q_0 < 0$) only occur in non-smooth models at the origin, invalidating positivity theorems that assume smoothness. The authors clarify that the deceleration parameter $q_0$, defined via luminosity distance in spherical symmetry, is constrained to be non-negative in smooth models, and that earlier claims of negative $q_0$ in LTB models stem from non-smooth solutions, not flaws in the theorems.

ABSTRACT

We argued in a previous paper [R. A. Vanderveld et al. 2006, arXiv:astro-ph/0602476] that negative deceleration parameters at the center of symmetry in Lemaitre-Tolman-Bondi cosmological models can only occur if the model is not smooth at the origin. Here we demonstrate explicitly the connection between non-smoothness and the failure of positivity theorems for deceleration. We also address some confusion that has arisen in the literature and respond to some recent criticisms of our arguments.

Motivation & Objective

• To resolve a longstanding contradiction in the literature regarding negative central deceleration parameters ($q_0 < 0$) in LTB models.
• To clarify that positivity theorems for $q_0$ rely on smoothness at the origin, and that non-smooth models violate these assumptions.
• To correct misinterpretations in recent criticism (KHCB) that falsely claim confusion between different definitions of the deceleration parameter.
• To reaffirm that only the observer-centered $q_0$ (defined via luminosity distance) is relevant for observational tests of cosmic acceleration.
• To demonstrate that not all angular diameter distance functions $D_A(z)$ can be realized in zero-energy-function LTB models, even in the linearized regime.

Proposed method

• Explicitly define the central deceleration parameter $q_0$ as derived from luminosity distance $D_L(z)$ under the assumption of a spatially flat FRW model, evaluated at $z=0$.
• Use the formalism from EF and HS to show that $q_0 \geq 0$ holds for all smooth, spherically symmetric LTB models.
• Analyze the INK models as non-smooth solutions where $q_0 < 0$ despite $q_1 > 0$, demonstrating that $q_0$ and $q_1$ diverge when smoothness fails.
• Apply linear perturbation theory to dust FRW models to study realizability of $D_A(z)$ in zero-energy-function LTB models.
• Derive the linearized differential equation for $V_1(z)$ and show that realizability requires $\int_0^{5/4} \gamma(z) \delta'(z) \, dz = 0$, proving that not all $D_A(z)$ functions are realizable.
• Clarify that all quantities, including velocity gradients, are evaluated at the observer’s location ($z=0$), as explicitly stated in VFW and EF.

Experimental results

Research questions

• RQ1 Why do some LTB models exhibit negative central deceleration parameters ($q_0 < 0$) despite positivity theorems predicting $q_0 \geq 0$?
• RQ2 What is the role of smoothness at the origin in the validity of positivity theorems for $q_0$ in spherically symmetric LTB models?
• RQ3 How do different definitions of the deceleration parameter ($q_0$, $q_1$, $q_3$, $q_4$) relate in non-smooth models, and which is physically relevant for observations?
• RQ4 Can any arbitrary angular diameter distance function $D_A(z)$ be realized in a zero-energy-function LTB model?
• RQ5 Why do some criticisms of the VFW paper (e.g., KHCB) stem from misinterpretations of definitions and evaluation points?

Key findings

• The positivity theorems for $q_0$ in LTB models require smoothness at the origin; non-smooth models can have $q_0 < 0$ without violating the theorems.
• The contradiction between $q_0 < 0$ in INK models and $q_0 \geq 0$ in smooth models is resolved by recognizing that the INK models are non-smooth at the origin.
• The deceleration parameter $q_0$ used in VFW is distinct from $q_1$ (defined via fluid element proper time), and the two do not coincide in non-smooth models.
• KHCB's criticism that VFW confused $q_0$ and $q_1$ is unfounded, as VFW used only $q_0$ throughout, and explicitly defined it before Eq. (2.16).
• The gradients of the velocity field used in the positivity theorems are evaluated at $z=0$, as explicitly stated in both EF and VFW, invalidating KHCB’s claim that this is unclear.
• Not all angular diameter distance functions $D_A(z)$ can be realized in zero-energy-function LTB models; realizability requires a specific integral condition $\int_0^{5/4} \gamma(z) \delta'(z) \, dz = 0$, which is not satisfied for arbitrary $\delta(z)$.