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Turnaround physics beyond spherical symmetry

The concept of turnaround surface in an accelerating universe is generalized to arbitrarily large deviations from spherical symmetry, to close the gap between the idealized theoretical literature and the real world observed by astronomers. As an analytical application, the characterization of turnaround surface is applied to small deviations from spherical symmetry, recovering a previous result while extending it to scalar-tensor gravity.


Introduction
Thanks to the study of Type Ia supernovae [1,2], we have known since 1998 that the present expansion of the universe is accelerated. In the context of general relativity, this acceleration is attributed to the presence of a mysterious form of dark energy permeating the universe and responsible for approximately seventy percent of its energy content [3]. Since this dark energy is, however, introduced completely ad hoc, there has been much activity in explaining the cosmic acceleration by modifying gravity at large scales and dispensing with dark energy ( [4,5], see Refs. [6][7][8][9][10] for reviews). Whatever the explanation for the cosmic acceleration, there are physical phenomena peculiar to an accelerated universe. One of them is the turnaround radius of cosmic structures [11][12][13][14], which has attracted much attention recently [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29] because of its potential to test the Λ-Cold Dark Matter (ΛCDM) model or modified gravity.
Consider an accelerating Friedmann-Lemaître-Robertson-Walker (FLRW) universe and superpose a spherical matter condensation acting as a perturbation of the FLRW metric. The local gravitational attraction due to this overdensity tends to make it collapse, while the cosmic expansion tends to disperse it (if this structure is sufficiently large to feel the effect of the cosmic expansion appreciably). The turnaround radius is the minimum scale at which a spherical shell of test particles can not collapse because of the accelerated cosmic expansion (or, vice-versa, the upper limit to the radius of spherical bound structures in an accelerated universe). At the turnaround radius the local attraction balances the cosmic expansion.
Thus far, with the exception of [29,30], the literature on turnaround physics has been restricted to spherically symmetric situations. Ref. [30] studies analytically small deviations from spherical symmetry (Ref. [29], instead, examines larger deviations numerically). However, this idealized situation is still far from being realistic and can easily induce large errors (cf. Refs. [25][26][27][28]). In the presence of spherical symmetry, the turnaround radius trivially defines the "turnaround surface", i.e., the sphere of radius equal to the turnaround radius but the generalization of this turnaround surface to non-spherical situations of astrophysical interest has not been discussed in the theoretical literature. As a consequence, astronomers attempting to determine the turnaround surface and deduce information on the cosmology have to grapple with ill-defined theory and basic concepts that are unclear, in addition to major observational challenges.
Here we identify the salient features of the turnaround surface in spherical symmetry and characterize it with a definition suitable for geometries with arbitrarily large deviations from spherical symmetry. The key idea is to identify the turnaround surface with an equipotential surface of the (local) metric perturbation potential with the special property that, if test particles initially sit on this surface with zero velocity with respect to it, they remain on this surface at later times while it evolves. This characterization captures the essence of turnaround sphere in spherical symmetry and generalizes this concept, while shifting the emphasis from the size of this surface (the turnaround radius) to the surface itself.
The study of the relevant equations for specific cosmic structures (observed or hypothetical) requires, in general, a numerical implementation. We can, however, apply our definition to an analytical discussion of small non-sphericities and test our characterization in this situation, which has already been studied in Ref. [30] with a completely different method based on the splitting of the Hawking-Hayward quasilocal energy contained in the turnaround surface into local and cosmological parts. We recover the results of [30] in our new, general description.
In Sec. 2 we calculate the timelike geodesics needed in the rest of this paper, while Sec. 3 provides the general definition of turnaround surface. The application to small non-sphericities is detailed in Sec. 4, while Sec. 5 extends this result to scalar-tensor gravity. Sec. 6 contains a discussion and the conclusions. We follow the notation of Ref. [31].

Timelike geodesics in the perturbed FLRW universe
The definition of turnaround surface requires one to consider test particles lying on this surface. They follow timelike geodesics in spacetime, therefore we first discuss these special curves traced by test particles and clouds (or shells) of dust in the perturbed FLRW spacetime.
The spacetime metric in the conformal Newtonian gauge is where η is the conformal time of the unperturbed FLRW universe and Φ(x i ) describes the Newtonian perturbation. Since we will consider only structures of size much smaller than the Hubble radius H −1 0 , the time dependence of Φ can be safely neglected.
Timelike geodesics parametrized by the proper time τ have four-tangents u µ = dx µ /dτ that satisfy the geodesic equation Since the only non-vanishing Christoffel symbols of the perturbed FLRW universe (2.1) are the components of the timelike geodesic equation are (2.14) In an unperturbed FLRW space, the four-velocity of a comoving observer reads in coordinates (η, r, θ, ϕ). Adding a perturbation as in Eq. (2.1), the four-tangent to a timelike geodesic becomes to first order in δu µ and Φ. Substituting this expression of δu 0 in the normalization of u µ , one finds We can now use these results to study the geodesic equations to first order in the perturbation. In detail, it is easy to show that Eq. (5.12) reduces to an identity to O(Φ), whereas for the spatial components one finds

General definition of turnaround surface
Here we generalize the definition of turnaround surface to the case in which deviations from spherical symmetry can be arbitrarily large. An examination of the salient features of the turnaround sphere in spherical symmetry shows that this surface is an equipotential surface of the perturbation potential Φ with the property that, if test particles initially lay on this surface and have zero initial velocity with respect to it, they remain on this surface as it evolves. Therefore, we require the turnaround surface Σ t at (comoving) time t to be a two-dimensional, closed, simply connected surface that, at all times t, is an equipotential surface of the perturbation Φ such that: i) The time evolution of the surface is such that the three-dimensional components of the tangent to the timelike geodesics crossing Σ t are locally proportional to the gradient ∇Φ (and therefore perpendicular to Σ t in the three-dimensional sense): ii) A dust particle initially comoving with the surface remains on this surface. In other words, if a dust particle is initially comoving with Σ t0 , namely u i | Σ0 = α(t 0 ) g ij ∂ j Φ| Σt 0 , then at a time t > t 0 its 3-velocity will satisfy u i | Σt = α(t) g ij ∂ j Φ| Σt .
Clearly u and ∇Φ are parallel. Multiplying this equation by f (t), whereḟ (t) ≡ 2Hf (t), one obtains which yields Thus, we have proved that the only 3-velocity perturbation component δu 1 is proportional to the gradient of Φ on the turnaround sphere. Since this sphere evolves with time in coordinates (η, r, θ, ϕ) the proportionality constant α depends on time (and only on time), as described by Eq. (3.3). In the general non-spherical case, this relation becomes δu i = α(t)g ij ∂ j Φ.

Small non-sphericities
Let us apply now the previous considerations to small deviations from spherical symmetry. This situation is studied in [30] with a conceptually different method. Ref. [30] is based on the splitting of the Hawking-Hayward quasilocal energy enclosed by a 2-surface Σ into a local and a cosmological part: the local part due to the perturbation Φ dominates inside the turnaround surface, while the cosmological part due to the cosmic mass-energy enclosed by Σ dominates outside of the turnaround surface. This surface is defined by the equality of these two contributions (previously, this quasilocal energy method was applied to the spherical case [32,33]). The result of Ref. [30] is that, to first order in the metric perturbations and in a parameter ǫ describing the deviations from spherical symmetry, the non-sphericities do not matter and the turnaround surface is still described by the turnaround radius obtained to zero order in ǫ. Given the very different methods used in the present paper and in [30], one should check that the results obtained coincide and that the two methods are compatible. Indeed, the results of [30] are recovered in our new, general description of Sec. 2.
To wit: let us go back to the perturbed geodesic equations Small non-sphericities are introduced by perturbing the otherwise spherical potential Φ 0 (r) as The non-sphericity leads to a further correction in δu, specifically with δu 1 (0) denoting the (radial) velocity perturbation in the unperturbed spherical case. Hence, perturbing the geodesic equations and using the results of the spherical case, one finds the equation satisfied by the non-sphericities to O(ǫ), where H ≡ȧ/a is the Hubble function with respect to comoving time.. Finally, since the perturbed surface must be close to a sphere at all times, the perturbation function f (r, θ, ϕ) must be controlled so that it does not blow up. Unless some regulating condition is imposed to this regard, the function f could grow very fast and the modified surface could deviate arbitrarily from a sphere even if the expansion parameter ǫ remains small. Therefore, we impose that 1 ∇f = O(ǫ), then δ = O(ǫ). This means that, for small deviations from sphericity, approximating the non-spherical turnaround surface with the unperturbed (spherical) one still gives the correct result to first order in the parameter ǫ that quantifies the non-sphericity in Eq. (4.2).

Scalar-tensor gravity
The turnaround radius has been studied also in scalar-tensor gravity and can, in principle, provide information about the theory of gravity at large scales [18][19][20][21][22][23]29]. In scalar-tensor gravity there is a gravitational slip and a Newtonian perturbation describing a bound structure is described by two metric potentials Ψ and Φ. The perturbed FLRW line element is now where it is assumed that the small metric perturbations Ψ and Φ are time-independent and of the same order. The general definition of turnaround surface given above can still be applied, provided that this surface is now an equipotential surface of Ψ.
The only non-vanishing Christoffel symbols are The equations of timelike geodesics with four-tangents u µ are now The four-velocities u µ = u µ (0) + δu µ are given again by Eq. (2.18). Specifically, to first order in both Ψ and δu µ , the normalization u µ u µ = −1 gives Besides, plugging the latter again into the normalization of u µ one finds The spatial components give, to first order: Now we expand the potentials to describe small deviations from spherical symmetry as Ψ (r, θ, ϕ) = Ψ 0 (r) + ǫ f (r, θ, ϕ) , (5.21) Φ (r, θ, ϕ) = Φ 0 (r) + ǫ h (r, θ, ϕ) , (5.22) with ǫ a smallness parameter. Again, the four-velocities become u µ = u µ (0) + δu µ = δu 0 (0) + ǫ δ 0 , δu 1 (0) + ǫ δ 1 , ǫ δ 2 , ǫ δ 3 (5.23) Inserting this expansion into the spatial components of the geodesic equations yields Again, one needs to control the behaviour of the deviations from sphericity by limiting the growth of the function f . This leads to the same results derived above for general relativity. One can conclude that, also in scalar-tensor gravity, small deviations from sphericity can be neglected in the identification of the turnaround surface.

Discussion and conclusions
In spherical symmetry, the turnaround radius clearly corresponds to a sphere of instability. Test particles that start on this surface with zero initial velocity with respect to it remain on it; analogous test particles inside this critical sphere must collapse, while those outside never form a bound system and disperse. The turnaround sphere corresponds to a delicate balance between the local gravitational attraction, which tends to make a dust shell collapse, and the cosmic expansion that tends to disperse it. On either side of the turnaround surface, one of these two forces prevails and moves a test particle away from it, so the position of (dynamical) equilibrium at the turnaround surface is clearly unstable. The turnaround radius is not a fixed point in the phase space of radial timelike geodesics, unless the background universe is de Sitter, which is locally static [34][35][36]. In a general FLRW background, the (proper or areal) turnaround radius is not constant but depends on time and the turnaround sphere expands (but the dust particles sitting on it have zero accelerationR = 0, where R is the areal radius). The turnaround sphere is not comoving.
Since the turnaround sphere is a sphere of unstable equilibrium, it marks the upper bound on the radius of any (spherical) bound structure. Because of the instability, a spherical bound structure with radius equal to the turnaround radius will not occur in nature. The turnaround radius is presented correctly in the literature as marking the upper limit to the largest possible size of a bound spherical structure.
Realistic structures in the sky, however, are not spherical nor approximately spherical. This fact is a challenge for observational astronomers attempting to identify the turnaround surface from observations of bound cosmic structures. This observational challenge is, of course, more complicated if one does not know what a turnaround surface is from the theoretical point of view. This is the gap addressed in the previous sections.
In the absence of spherical symmetry, the "size" of an asymmetric bound structure, or cluster, may be defined operationally in various ways. Each one of them will have advantages and disadvantages and will be somehow questionable. However, it is more important to focus on the turnaround surface and to identify it, rather than discussing its "size". Here we have identified the turnaround surface with an equipotential surface of the metric perturbation potential Φ satisfying the special property that dust particles initially sitting on this (non-spherical) surface with zero velocity with respect to it, will remain on it as this surface evolves in time. This definition is completely general and seems to be the correct generalization of turnaround sphere. Then, the turnaround radius no longer exists 2 and the "size" of the critical turnaround surface ceases to play a primary role in the discussion of turnaround physics.
The application of our characterization of turnaround surface to small deviations from spherical symmetry reproduces the previous result of Ref. [30], which was obtained with a completely different method (the splitting of the Hawking mass contained in the turnaround surface into local and cosmological parts [32]). Further applications to realistic situations will be presented in the future.