Holographic signatures of resolved cosmological singularities II: numerical investigations

In a recent series of papers [25, 26], the AdS/CFT correspondence was used to study holographic signatures of cosmological bulk singularities in the classical gravity approximation. The setup of [25, 26] is the following.

We consider Kasner-AdS bulk spacetime geometries described by the metric

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{clmetric} {\rm d}s_5^2=\frac{1}{z^2}\left({\rm d}z^2+{\rm d}s_4^2(t)\right),\qquad\qquad {\rm d}s_4^2(t)=-{\rm d}t^2+\sum_{i=1}^3 t^{2p_i}{\rm d}x_i^2 \nonumber \end{align} \tag{ 2.1 }$$

where we have set the AdS radius to 1. As long as the exponents p _i satisfy the vacuum Kasner conditions $\sum_ip_i=1=\sum_ip_i^2$, Kasner metric ${\rm d}s_4^2$ is a solution of the 4d vacuum Einstein equations without cosmological constant, while the full metric ${\rm d}s_5^2$ is a solution of 5d vacuum Einstein equations with negative cosmological constant. It has a curvature singularity at t  =  0. In addition to the translational symmetries in the $x^1, x^2, x^3$ directions, the metric (2.1) is also invariant under the scaling transformation

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{scaling} z\longmapsto\Lambda z,\quad\quad t\longmapsto\Lambda t,\quad\quad x_i\longmapsto\Lambda^{1-p_i}x_i. \nonumber \end{align} \tag{ 2.2 }$$

Following the AdS/CFT dictionary, the dual description of this bulk system involves ${\mathcal N}=4$ Super Yang–Mills theory on a Kasner background. Alternatively, by picking a different conformal factor, the metric (2.1) can be rewritten as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{clmetric2} {\rm d}s_5^2 = \frac{t^2}{z^2} \left({\rm e}^{2\tau} {\rm d}z^2 -{\rm d}\tau^2 + \sum_{i=1}^3 {\rm e}^{-2(1-p_i) \tau} {\rm d}x_i^2 \right), \qquad t = {\rm e}^{-\tau} \nonumber \end{align} \tag{ 2.3 }$$

such that the boundary metric describes an anisotropic deformation of de Sitter space with (2.2) acting as an isometry on the boundary metric and leaving the conformal factor invariant.

In the large N semiclassical bulk limit, the leading contribution to the equal time two-point correlator of a high conformal dimension (heavy) scalar operator ${\mathcal O}$ is determined by the length of spacelike bulk geodesics connecting two points on some boundary time slice at t  =  t₀. Indeed, in the so-called geodesic approximation¹ [28], the two-point correlator of two heavy scalar operators is dominated by

$$\begin{align} \renewcommand{\braket}[1]{\langle#1\rangle} \renewcommand{\bra}[1]{\langle#1|} \newcommand{\e}{{\rm e}} \displaystyle \label{corr} \braket{{\mathcal O}(x){\mathcal O}(-x)} \sim \exp{(-\Delta\,L_{{\rm ren}})}, \nonumber \end{align} \tag{ 2.4 }$$

where $\Delta$ is the conformal dimension of ${\mathcal O}$, which for a d-dimensional boundary spacetime is related to the mass m of the bulk field corresponding to the boundary operator ${\mathcal O}$ via

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Delta=\frac{d}{2}+\sqrt{\frac{d^2}{4}+m^2}\underset{m\gg1}{\simeq}m, \nonumber \end{align} \tag{ 2.5 }$$

and $L_{{\rm ren}}$ is the renormalised (see below) length of a spacelike geodesic connecting the boundary points (t₀, −x) and (t₀, x). Notice that, as sketched in figure 1, we consider geodesics anchored on the same boundary time slice whose endpoints are separated in only one spatial direction, say x₁, hereafter denoted simply by x (and we shall henceforth refer to the corresponding Kasner exponent p ₁ as p ). This is due to the fact that geodesics can be thought of as traveling in a (2+1)-dimensional effective spacetime with coordinates $(t,x,z)$ due to the translation symmetry in the xⁱ-directions.

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In case of multiple geodesics satisfying given boundary data, a sum over the individual contributions must be included in evaluating the two-point correlator (2.4). Complex solutions have also to be taken into account as they contribute to the long distance fall-off behaviour  ∼$(L_{{\rm bdy}}){}^{-\frac{2\Delta}{1-p}}$ of the two-point correlator for geodesics with proper boundary separation $L_{{\rm bdy}}$ and not crossing the singularity. Geodesics that propagate in a direction with positive Kasner exponent (p   >  0) are curved away from the singularity, while if we consider the x-separation in a negative p  direction, geodesics are bent towards the singularity and they are thus characterised by a turning time $t_*<t_0$ for real solutions (see equation (3.4) in [26]) and t₀  >  0. In the limit $t_*\rightarrow0$, spacelike bulk geodesics approach a null boundary geodesic for p   <  0 and their tip approaches the bulk singularity. Correspondingly the two-point correlator exhibits a pole at the cosmological horizon scale which is interpreted as a dual signature of the classical bulk singularity. The presence of such a pole indicates that the state in the dual field theory description of the Kasner-AdS metric is not normalisable [26].

The possibility that quantum gravity effects might smoothen out this pole and render the two-point correlator finite at non-vanishing spatial separation was already discussed in [26]. However, no explicit mechanism for this was given. Motivated by the discussion in [26], an example of an improved CFT correlator from quantum gravity effects was provided in [24]. The strategy was to consider effective spacetimes emerging from loop quantum gravity and to repeat the computation of [26] in this context. Specifically, since the metric (2.1) is singular only in its 4d part and the 5d Einstein equations with negative cosmological constant imply that ${\rm d}s_4^2$ is Ricci-flat, the idea was to keep the z-direction classical and to consider a 1-parameter family of quantum corrected metrics for the 4d part, labelled by a parameter $\lambda$ controlling the onset of quantum effects:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{qmetric1} {\rm d}s_5^2=\frac{1}{z^2}\left({\rm d}z^2+{\rm d}s_4^2(t)\right),\quad\quad {\rm d}s_4^2(t)=-{\rm d}t^2+\frac{a_{\rm ext}^2}{\lambda^{2p}}\left(t^2+\lambda^2\right)^{\,p}{\rm d}x^2+\dots \nonumber \end{align} \tag{ 2.6 }$$

where dots refer to the other spatial directions which may have different Kasner exponents, $a_{\rm ext}$ is the extremal value of the scale factor, i.e. the value at the bounce (t  =  0). Indeed, for $\lambda>0$ the classical singularity is resolved, while the classical Kasner solution with a(t)  =  t^p is recovered in the double scaling limit $\lambda\rightarrow0$ and $a_{\rm ext}/\lambda^{\,p}\rightarrow1$. We will discuss two important shortcomings of this metric as opposed to a proper motivation from the results of [27] below.

For such kind of bulk metric, the geodesic equations can be solved completely in the t-parametrisation. However, the affine parametrisation turns out to be more convenient for computing the renormalised geodesic length. The explicit solution $z(s)$ parametrised with respect to the geodesic length s reads as [24]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{zaff1} z(s)=\frac{z(t_*)}{\cosh{(s-s_0)}}, \nonumber \end{align} \tag{ 2.7 }$$

where we can set s₀  =  0 to start counting the proper distance from the turning point of the geodesic. Equation (2.7) shows that the length of the geodesics diverges. Therefore, by truncating the geodesics at some boundary regulator $\newcommand{\e}{{\rm e}} z=\epsilon$, which corresponds to implementing a UV cutoff at energy scale $\newcommand{\e}{{\rm e}} 1/\epsilon$ in the dual field theory, we have

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \pm s(z=\epsilon) \stackrel{\epsilon\rightarrow0}{=}\log(2z(t_*))-\log(\epsilon). \label{eq:DivergentPiece} \nonumber \end{align} \tag{ 2.8 }$$

The divergent contribution $\newcommand{\e}{{\rm e}} -\log(\epsilon)$ has to be subtracted, which can be understood as a conformal transformation that brings the boundary metric to the form ${\rm d}s_4^2(t)=-{\rm d}t^2+\sum_{i=1}^3 t^{2p_i}{\rm d}x_i^2$, i.e. it absorbs the divergent prefactor 1/z² [28]. We thus obtain the renormalised geodesic length

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{Lren1} L_{{\rm ren}}=2\log(2z(t_*)). \nonumber \end{align} \tag{ 2.9 }$$

Due to (see equation (2.4))

$$\begin{align} \renewcommand{\braket}[1]{\langle#1\rangle} \renewcommand{\bra}[1]{\langle#1|} \newcommand{\e}{{\rm e}} \displaystyle \label{corr2} \braket{{\mathcal O}(x){\mathcal O}(-x)} \sim (2z(t_*))^{-2\Delta}, \nonumber \end{align} \tag{ 2.10 }$$

z(t_*)  =  0 at finite boundary separation corresponds to a finite-distance pole in the two-point correlator.

Finally, as shown in figure 2, z(t_*) does not vanish at finite boundary separation for $\lambda=1$ corresponding to the quantum theory (blue curve), unlike the case $\lambda=0$ corresponding to the classical theory (red curve). The finite-distance pole in the two-point correlator occurring in the classical theory is thus resolved by quantum effects. There is still a pole at $(0,0)$, but this corresponds to the standard divergence occurring in the coincidence limit. Moreover, there are multiple z(t_*) values corresponding to real geodesics with the same boundary separation (same x(t₀)-value), which have to be added in the correlator. The dominant contribution comes from the local minimum of z(t_*) for $\lambda=1$, which represents a clear signature of the resolved classical pole. When t₀ gets close enough to $\lambda$, the characteristic behaviour changes (figure 2(b)), but a change of slope in the quantum corrected curve persists and z(t_*) does not vanish at finite distance, i.e. the classical pole is still resolved. Resolution of the finite distance pole along these lines persists for any $\lambda > 0$.

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The most direct way of investigating holographic signatures of quantum corrected metrics from loop quantum gravity would be to set up 5d quantum Einstein equations based on [29–31] and to extract an effective metric. Unfortunately, this is currently out of technical reach. In order to arrive at a sensible bulk metric, we therefore apply the following key simplification:

As mentioned above, the singularity in the classical bulk metric (2.1) originates in the 4d part of the metric. Resolving the singularity in ${\rm d}s^2_4$ automatically resolves it in ${\rm d}s^2_5$. The problem can therefore be approximated as one in 4d quantum cosmology, to which much of the effort in loop quantum gravity has gone. In order to ensure that this approximation remains consistent, a possible z-dependence of ${\rm d}s_4^2$ has to be kept small (at the order of $\lambda$) and we will come back to this point. For now, we remark that the techniques used to construct loop quantum gravity have been employed in a mini-superspace context, leading to the field of loop quantum cosmology [32]. Here, a quantum corrected Kasner universe has been studied in [27] using effective equations, which were later shown to be in accordance with the results directly obtained from the quantum theory [33]. Moreover, proposals for how to embed the quantised Bianchi I mini-superspace into full loop quantum gravity have been put forward [34, 35]. The resulting picture is a non-singular bounce that connects two Kasner universes at late times, however with different Kasner exponents before and after the bounce. Following the literature, we will call this feature a Kasner transition.

The onset of quantum effects is again controlled by a parameter $\lambda$ which is related to $\hbar$ and the Barbero–Immirzi parameter [36]. A crucial observation for embedding of ${\rm d}s^2_4$ into ${\rm d}s^2_5$ as in (2.6) is now that the onset of quantum effects should happen at the 5d Planck scale, and not the 4d scale which $\lambda$ sets. Since the 4d curvature is bounded by ${{\rm const}} / \lambda^2$ in loop quantum cosmology, the relation

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\del}{\partial} \displaystyle \label{curvature} R^{(5)}_{{\rm Kretschmann}}=R^{(5)}_{\alpha\beta\gamma\delta}R^{(5)\,\alpha\beta\gamma\delta}=z^4R^{(4)}_{{\rm Kretschmann}}+\dots\; \nonumber \end{align} \tag{ 2.11 }$$

motivates us to substitute $\lambda \mapsto \lambda z$ in order to obtain an onset of quantum effects in the bulk at the 5d Planck scale.

Both of these features, Kasner transitions and a z-dependent scale for the onset of quantum effects, were neglected in (2.6) to allow for an analytic computation in [24]. In principle, both of them can have important qualitative effects on the results for the two-point correlator:

First, it was shown in [26] that the divergence in the two-point correlator is due to the bulk geodesic approaching a null geodesic on the boundary. This geodesic is still present in the quantum corrected bulk spacetime since $\lambda z$ goes to zero at the boundary and the metric reduces to classical Kasner. However, our numerical computations suggest that this geodesic is isolated and not the limit of a family of bulk geodesics as in the classical case [26].

Second, the long distance behaviour of the correlator in [24] is due to geodesics passing arbitrarily close to t  =  0, where $(t^2+\lambda^2){}^{\,p}$ in (2.6) has a local maximum (for p   <  0). A Kasner transition as in [27] where a negative exponent would transition into a positive one would alter the form of the metric around t  =  0 such that no extremum could be found there.

In the following, we will successively lift these two simplifications and investigate their effect by numerical computations.

Since the geodesic starts and ends at the same time slice, there should be a turning point in t- as well as in z-direction. Furthermore, this solution should be symmetric around the turning point, i.e. it has two branches (towards the turning point and back again). A coordinate parametrisation, in particular time parametrisation, would not be a convenient choice since it does not allow to parametrise both branches at the same time (see [24, 26]). This can be achieved by using the affine parametrisation. In such a parametrisation, the geodesic equations is easily derived from the action

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle S = \int {\rm d}s \sqrt{g_{\mu \nu} \frac{{\rm d}x^{\mu}}{{\rm d}s} \frac{{\rm d}x^{\nu}}{{\rm d}s}} = \int {\rm d}s \sqrt{- \frac{\dot{t}^2}{z^2} + \frac{a(t,z)^2}{z^2} \dot{x}^2 + \frac{\dot{z}^2}{z^2} } , \nonumber \end{align} \tag{ 3.2 }$$

where dot denotes derivatives with respect to s and we chose the sign appropriate for spacelike geodesics. Using $- \frac{\dot{t}^2}{z^2} + \frac{a(t,z){}^2}{z^2} \dot{x}^2 + \frac{\dot{z}^2}{z^2} = 1$, the geodesic equations read

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\partdif}[2]{\frac{\partial #1}{\partial #2}} \displaystyle \ddot{t} - \frac{2 \dot{t} \dot{z}}{z} + \frac{\dot{x}^2}{2} \partdif{(a^2)}{t} = 0 , \nonumber \end{align} \tag{ 3.3a }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{{\rm d}}{{\rm d}s} \left(\frac{a(t,z)^2}{z^2} \dot{x} \right) = 0 \label{eq:I1} , \nonumber \end{align} \tag{ 3.3b }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\partdif}[2]{\frac{\partial #1}{\partial #2}} \displaystyle \ddot{z} - \frac{2 \dot{z}^2}{z} + z - \frac{\dot{x}^2}{2} \partdif{(a^2)}{z} = 0 . \label{geo-eq3} \nonumber \end{align} \tag{ 3.3c }$$

Note that equation (3.3b) reflects the constant of motion corresponding to translation symmetry in x-direction. Moreover, since the scale factor $a(t,z)$ depends also on z, equation (3.3c) has an additional term and we cannot explicitly solve for z only as in equation (2.7). Therefore, unlike in [24, 26], equations (3.3) do not decouple and as such they do not appear to be analytically solvable anymore.

Thus, we have to reformulate the problem in such a way that it is numerically tractable. Since we want to integrate up to the boundary, which is at infinite proper distance s form the turning point, it is convenient to compactify the parameter s as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \sigma = \tanh(s) , \quad \sigma \in (-1,1) . \nonumber \end{align} \tag{ 3.4 }$$

In this parametrisation, the boundary value problem reads

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\partdif}[2]{\frac{\partial #1}{\partial #2}} \displaystyle \label{eq:geo-bvp} \left\{\begin{array}{@{}llllll@{}} t' = p_t & \nonumber \\ x' = p_x & \nonumber \\ z' = p_z & \nonumber \\ p_t' = \frac{2 \sigma}{1-\sigma^2} p_t + \frac{2 p_t p_z}{z} - \frac{p_x^2}{2} \partdif{(a^2)}{t}& \nonumber \\ p_x' = \frac{2 \sigma}{1-\sigma^2} p_x + \frac{2 p_x p_z}{z} - \frac{p_x}{a(t,z)^2} \left(\partdif{(a^2)}{t} p_t + \partdif{(a^2)}{z} p_z \right) & \nonumber \\ p_z' = \frac{2 \sigma}{1-\sigma^2} p_z + \frac{2 p_z^2}{z} - \frac{z}{(1-\sigma^2)^2} + \frac{p_x^2}{2} \partdif{(a^2)}{z} & \nonumber \\ & \nonumber \\ t(-1) = t(1) = t_0 & \nonumber \\ x(-1) = -x(1) = -l& \nonumber \\ z(-1) = z(1) = 0 & \end{array}\right. \; , \nonumber \end{align} \tag{ 3.5 }$$

where prime denotes derivatives with respect to $\sigma$, l  =  x(t₀), and we introduced p _t, p _x, p _z to rewrite the equations as first order ODEs. The additional terms in $\sigma$ are due to the reparametrisation properties of derivatives, $\frac{{\rm d}^2 \sigma}{{\rm d}s} / \left(\frac{{\rm d} s}{{\rm d}\sigma}\right){}^2 = -2\sigma/(1-\sigma^2)$ and $\frac{{\rm d} \sigma}{{\rm d}s} = 1-\sigma^2$.

To solve boundary value problems numerically, there are well established methods like the relaxation method (see e.g. [37, 38]) or the shooting method [39]. In our case, the problem is more tractable once reformulated as a initial value problem. This makes the numerical solution much simpler and has other advantages, which we will discuss later on. The basic idea is the following: we are only interested in geodesics which have a turning point in t and z. Geodesics which do not come back to the boundary or come back but on a different time slice are not solutions of equations (3.5). Denoting by t_* and z_* the values of t and z at the turning point, the exact behaviour of the geodesic at this point can be characterised in terms of t_* and z_* only. The turning point itself is then given by the coordinates $(t_*, 0 , z_*)$, where, as already stressed, we can use the translation invariance to fix the value x_* of x at the turning point to be 0. Furthermore, we can always shift the parametrisation such that the geodesic turns around at s  =  0 $\Leftrightarrow$ $\sigma = \tanh(0) = 0$. At the turning point, the velocities of t and z should vanish, i.e.:

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle &0 = \dot{t}(s=0) = t'(\sigma=0) \cdot \frac{{\rm d}\sigma}{{\rm d}s}(\sigma=0) = t'(\sigma=0) , \nonumber \\ &0 = \dot{z}(s=0) = z'(\sigma=0) \cdot \frac{{\rm d}\sigma}{{\rm d}s}(\sigma=0) = z'(\sigma=0) . \nonumber \end{align*}$$

Using again $- \frac{\dot{t}^2}{z^2} + \frac{a(t,z){}^2}{z^2} \dot{x}^2 + \frac{\dot{z}^2}{z^2} = 1$, we find:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \dot{x}(s = 0) = \frac{z_*}{a(t_*,z_*)} = \frac{z_* \lambda^{\,p}}{a_{\rm ext} \left(t_*^2 + \lambda^2 z_*^2 \right)^{\frac{p}{2}}} = x'(\sigma=0) \cdot \frac{{\rm d}\sigma}{{\rm d}s}(\sigma=0) = x'(\sigma=0) . \nonumber \end{align} \tag{ 3.6 }$$

The boundary value problem (3.5) can be then rephrased as the following initial value problem

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\partdif}[2]{\frac{\partial #1}{\partial #2}} \displaystyle \label{eq:geo-ivp} \left\{\begin{array}{@{}llllll@{}} t' = p_t & \nonumber \\ x' = p_x & \nonumber \\ z' = p_z & \nonumber \\ p_t' = \frac{2 \sigma}{1-\sigma^2} p_t + \frac{2 p_t p_z}{z} - \frac{p_x^2}{2} \partdif{(a^2)}{t}& \nonumber \\ p_x' = \frac{2 \sigma}{1-\sigma^2} p_x + \frac{2 p_x p_z}{z} - \frac{p_x}{a(t,z)^2} \left(\partdif{(a^2)}{t} p_t + \partdif{(a^2)}{z} p_z \right) & \nonumber \\ p_z' = \frac{2 \sigma}{1-\sigma^2} p_z + \frac{2 p_z^2}{z} - \frac{z}{(1-\sigma^2)^2} + \frac{p_x^2}{2} \partdif{(a^2)}{z} & \nonumber \\ & \nonumber \\ t(0) = t_*,\; x(0) = 0,\; z(0) = z_* & \nonumber \\ p_t(0) = 0,\; p_x(0) = z_*/a(t_*,z_*), \; p_z(0) = 0 & \end{array}\right. , \nonumber \end{align} \tag{ 3.7 }$$

where the initial data $(t_*, 0, z_*)$ and the velocities $(0, z_*/a(t_*,z_*), 0)$ are expressed only in terms of the turning point values $t_*, z_*$, which we give as input. The parameter $\sigma$ runs from 0 to 1, that means we only cover one of the two branches. This is not a problem, since around $\sigma = 0$ the solution is symmetric in t and z and anti-symmetric in x.

The crucial point is to relate t_* and z_* with the boundary data t₀ and l. For this, let us define the map

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle f\!:\; &\mathbb{R}^2 \longrightarrow \mathbb{R}^2 \nonumber \\ \quad\quad &\;(t_*, z_*) \longmapsto f(t_*, z_*) = \left(\,f^1(t_*, z_*), f^2(t_*,z_*) \right) \nonumber \end{align} \tag{ 3.8 }$$

with

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:bdymapping} & f^1(t_*, z_*) = t(\sigma=1) = t_0(t_*,z_*) \nonumber \\ &f^2(t_*, z_*) = x(\sigma=1) = l(t_*,z_*) \nonumber \end{align} \tag{ 3.9 }$$

where $t(\sigma)$, $x(\sigma)$ are solutions of equation (3.7). This map is well-defined due to the uniqueness of solutions to initial value problems. For solving the boundary value problem, it would be necessary to 'invert' this map³. Indeed, we are interested in those $(t_*, z_*)$, whose corresponding solutions end at fixed t₀. Such points give the level lines of $t_0(t_*,z_*)$ and relate z_* with t_* (see equation (5.2) in [26] for such a relation in the classical case). This can be formalised by considering a curve $c_{t_0}^{\mu}(\tau) = (t_*(\tau), z_*(\tau)){}^{\mu}$, parametrised by $\tau$, which satisfies:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\partdif}[2]{\frac{\partial #1}{\partial #2}} \displaystyle \frac{{\rm d} c_{t_0}^{\mu}}{{\rm d}\tau} \; \partdif{}{x^{\mu}} t_0(t_*(\tau), z_*(\tau)) = 0, \quad \quad x^{\mu} = (t_*, z_*)^{\mu} . \nonumber \end{align} \tag{ 3.10 }$$

Similarly, the level lines $c_{l}^{\mu}(\tau)$ of l satisfy

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\partdif}[2]{\frac{\partial #1}{\partial #2}} \displaystyle \frac{{\rm d} c_{l}^{\mu}}{{\rm d}\tau} \; \partdif{}{x^{\mu}} l(t_*(\tau), z_*(\tau)) = 0, \quad \quad x^{\mu} = (t_*, z_*)^{\mu} . \nonumber \end{align} \tag{ 3.11 }$$

The boundary value problem equation (3.5) is then solved by all intersection points $(t_*^i,z_*^i)$, $i = 1,2,\dots$ of the two level lines.

The boundary value problem can therefore be reformulated in terms of an initial value problem, the calculation of two level lines, and the calculation of their intersections. For doing so, we have to solve the initial value problem for numerous values of t_* and z_*. For our problem this is convenient since we are not interested in specific values of l. Instead, we want to vary l. Furthermore, we are in principle interested in different values of t₀. This means it is possible to solve the initial value problem for a given grid of t_* and z_* and use this data to calculate different t₀-level lines, which decreases the total numerical effort⁴.

Numerically, we implemented this method with use of Matlab and its built in library. For the solution of the ODE we used the routine ode45, which is based on fifth-order Runge–Kutta method with adaptive step size. For the calculation of the level lines we used the routine contourc. Since the coefficients of equations (3.7) diverge at $\sigma=\pm1$, we cut the integration interval such that $\newcommand{\e}{{\rm e}} \sigma \in [0, 1-\epsilon]$, where $\newcommand{\e}{{\rm e}} \epsilon = 0.00001$. We have performed extensive cross-checks on the numerics, including the reproduction of the analytical results of [24].

For calculating the two-point correlator, it is necessary to compute the renormalised geodesic length. We are interested in the length of the complete geodesic, i.e. from boundary to boundary. In affine parametrisation, with s  =  0 at the turning point, this is L  =  2s. However, since by construction the conformal boundary lies at infinity, this value diverges and a renormalisation procedure is necessary. The geodesic length is thus only evaluated up to a small value $z_{\rm UV}/t \rightarrow 0$. This ratio is preserved by the scaling symmetry (2.2) and, as discussed in [26], a constant value of the conformal factor in the metric (2.3) corresponds to a constant UV cutoff in pure AdS. Then the divergent piece is subtracted and the limit, which should be finite, is taken. Numerically, this is not a trivial task, since the analytical expression is not available and hence the singular part cannot be isolated (see equation (2.9)). Nevertheless, the strategy is the same: define $L_{{\rm ren}} = L - L_0$, where L and L₀ are evaluated at a z-cutoff $z_{\rm UV}$ and L₀ is the divergent piece from (2.8) (see also [37, 38]). The final result should be independent of $z_{\rm UV}$ and hence coincide in the limit $z_{\rm UV}/t \rightarrow 0$. Let us recall that the range of $\sigma$ is $[0,1]$⁵ and that s and $\sigma$ are related by $s={{\rm arctanh}}\left(\sigma\right)$. Hence, as expected, s diverges for $\sigma \rightarrow 1$. To renormalise the geodesic length, the solution is evaluated up to the given $z_{\rm UV}$ and the corresponding value of the curve parameter $\sigma$ is read off, say $\bar{\sigma}$ such that $z(\bar{\sigma})=z_{\rm UV}$. With this cutoff, we have

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\del}{\partial} \displaystyle \sigma\in[0,1-\delta] \quad, \quad \delta=1-\bar\sigma \ll 1 , \nonumber \end{align} \tag{ 3.12 }$$

where the condition $\newcommand{\del}{\partial} \delta \ll 1$ is satisfied as long as $z_{\rm UV}/t \ll z_*/t_*$. Subtracting the divergent part, which is $-\log(\vert z_{\rm UV}/t(\bar{\sigma}) \vert)$ for a single branch, this gives the renormalised length

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\del}{\partial} \displaystyle \label{renlen} L_{{\rm ren}}=2{{\rm arctanh}}\left(1-\delta\right)+2\log{\left(\left|\frac{z_{\rm UV}}{t(\bar{\sigma}) }\right|\right)} . \nonumber \end{align} \tag{ 3.13 }$$

To check that the limit $z_{\rm UV}/t \rightarrow 0$ (i.e. $\bar{\sigma} \rightarrow 1$, $\newcommand{\del}{\partial} \delta \rightarrow 0$) is finite, let us notice that

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\del}{\partial} \displaystyle {{\rm arctanh}}\left(1-\delta\right)+\log\left(\left|\frac{z_{\rm UV}}{t(\bar{\sigma})}\right|\right)=-\frac{1}{2}\log{\left(\frac{\delta \cdot t(\bar{\sigma})^{2}}{z_{\rm UV}^{2}} \right)}+\frac{1}{2}\log{(2)}+{\mathcal O}(\delta) , \nonumber \end{align} \tag{ 3.14 }$$

and, as can be checked numerically, for small values of $z_{\rm UV}$, the quantity (3.13) approaches a non-zero constant.

In order to conclude that the two-point correlator is non-singular, we need to check that $L_{{\rm ren}}$ remains finite. Let us recall from section 2 that both in the classical case [25, 26] and the quantum corrected case considered in [24], the relation was⁶

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:Lrenofzstar} \frac{L_{{{\rm ren}}}}{2} = \log\left(\frac{2z_*}{t_0}\right) . \nonumber \end{align} \tag{ 3.15 }$$

As discussed in section 2.2, this corresponds to the z_*-dependence (2.10) in the two-point correlator. Using the method of this section we are able to relate numerically z_* and l. We expect the same or a similar dependence also in the cases under consideration. What changes is the relation z_*(l), which differs from case to case. The results of the numerical calculation are presented in the next section.

We first applied this method to the metric

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{qmetric2} {\rm d}s_5^2=\frac{1}{z^2}\left({\rm d}z^2-{\rm d}t^2+\frac{a_{\rm ext}^2}{\lambda^{2p}}\left(t^2+\lambda^2z^2\right)^{\,p}{\rm d}x^2+\dots\right), \nonumber \end{align} \tag{ 4.1 }$$

which, as can be checked by calculating the Kretschmann scalar, features an onset of quantum gravity effects at the 5d Planck scale (see section 2.3) but neglects Kasner transitions. For $t \gg z \lambda$, where quantum corrections are negligible in (4.1), the classical Kasner-AdS solution of the 5d-Einstein equations is recovered. The z-derivative of $a(t,z)$ is ${\mathcal O}(\lambda)$ with finite coefficients and can thus be accounted for by quantum corrections in the z-direction, which we systematically neglect here. Also, the proper classical boundary limit exists. These points will be relevant and highly non-trivial also in section 4.2 where Kasner transitions are included.

For this metric the derivatives of a(t, z)² entering equations (3.7) are given by

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\partdif}[2]{\frac{\partial #1}{\partial #2}} \displaystyle \partdif{(a^2)}{t} = 2 p \frac{a_{\rm ext}^2}{\lambda^{2p}} t \left(t^2 + \lambda^2 z^2 \right)^{\,p-1} = 2 a(t,z)^2 \frac{p t}{t^2 + \lambda^2 z^2} , \nonumber \end{align} \tag{ 4.2 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\partdif}[2]{\frac{\partial #1}{\partial #2}} \displaystyle \partdif{(a^2)}{z} = 2 p \frac{a_{\rm ext}^2}{\lambda^{2p}} \lambda^2 z \left(t^2 + \lambda^2 z^2 \right)^{\,p-1} = 2 a(t,z)^2 \frac{p \lambda^2 z}{t^2 + \lambda^2 z^2} .\label{zderivative} \nonumber \end{align} \tag{ 4.3 }$$

The solutions of $t_0(t_*,z_*)$ and $l(t_*,z_*)$ describe surfaces in a 3d space spanned by $(t_*,z_*,t_0)$ and $(t_*, z_*, l)$, respectively. To visualise them we report z_* versus t_* in figure 3 where the third direction (respectively t₀ and l) is replaced by a colour scale. For this calculation we fixed the parameters to p   =  −1/4 and $\lambda = 0.06$. The range of t_* and z_* is chosen to be between $[0,10]$. Let us focus on figure 3(a) first. Among the level lines corresponding to different values of t₀, we selected for instance the one for t₀  =  4 (red curve). This level line relates z_* and t_* for that given constant value of t₀. We can compare this now with the classical case (see figure 4). The classical region is in the area where $t^2 \gg \lambda^2 z^2$, but since $\lambda = 0.06$ is chosen very small the 'dividing line' $t^2 = \lambda^2 z^2$ is close to the z_*-axis (see black dashed line in figure 3(a)). Indeed for large t_* (here $\gtrsim 1$) and small z_* (here  <10) we see exactly the classical behaviour (see figure 4). On the other hand, going to the quantum regime ($t^2\ll \lambda^2 z^2$), i.e. close to the z_*-axis, we see that the level lines exhibit turning points. Therefore, unlike the classical case where the finite-distance pole in the two-point correlator was due to bulk geodesics approaching a null geodesic lying entirely on the boundary ($z_* \rightarrow 0$ for $t_* \rightarrow 0$ on a constant t₀ level line), quantum corrections of the metric induce a turning point which leads to a growing z_* for $t_* \rightarrow 0$, thus showing, within our numerical accuracy ($t_*\gtrsim10^{-8}$), that this null-boundary solution is isolated and not the limit of a family of bulk geodesics.

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The contour plot of l is reported in figure 3(b) where we also included the t₀  =  4 level line shown in figure 3(a) (red curve). As we see in the plot, l increases as z_* becomes larger. Following the level line and reading off the corresponding values of l leads to the plot of figure 5, where we plotted both the quantum corrected metric (blue line) and the classical result (red line). The classical result agrees with the results of [25, 26] (see red line in figure 2). According to the analytical results, the red line should hit z_*  =  0 for finite l. However, since we introduced a cutoff in z_*, the classical red line does not reach the z_*  =  0 axis. Close to the cutoff, some numerical uncertainties occur and we are not able to see the classical pole. Nevertheless, the classical curve converges towards z_*  =  0 for finite l within numerical accuracy and hence, as argued in section 2, this leads to a pole in the two-point correlator of the boundary theory. Concerning the quantum case (blue line), we see a turning point in z_*(l) at finite l, and then z_* increases again as it was already visible in figure 3. Therefore, z_* never hits 0 for finite non-zero values of l. Moreover, as in the analytical case (figure 2), there are multiple solutions corresponding to the same boundary separation, whose contribution has to be added in the two-point correlator. Note that because of the cutoff in z_* both the classical and quantum curves do not start at z_*  =  l  =  0.

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The next step is to calculate $L_{{\rm ren}}$ by means of the procedure described in section 3.3. A possible dependence like equation (3.15) can be easily visualised in a log-plot, where a straight line is expected (red dashed line in figure 6). As shown in figure 6, our numerical solutions (blue line) exhibit such a dependence. Nevertheless, there are some subtleties to be discussed. First of all, the upper region provides a purely quantum contribution to the long distance behaviour of the two-point correlator. As can be checked, in agreement with the results of [24], this contribution decays faster than the lower region (short distance) contribution. Moreover, there are turning points in the line which reflects the above mentioned existence of multiple solutions for a given boundary separation. Indeed, the renormalised length $L_{{\rm ren}}$ is calculated for each point along the same t₀  =  4 level line selected before in figure 3. The turning points are also present in figure 6, where some values of z_* are passed more than once. Nevertheless, the shape of a straight line is kept. In the lower region, which contributes to the short distance behaviour of the two-point correlator, deviations from this log-dependence occur below our chosen range of evaluation, but these are just numerical artifacts. Indeed, in this region, z_* comes close to $z_{\rm UV}$, the approximation $\frac{z_{\rm UV}}{t}\ll \frac{z_*}{t_*}$ fails, and the error increases. However, our main interest concerns the behaviour $L_{{\rm ren}} \propto \log(z_*)$, or in other words, that $L_{{\rm ren}}(z_*)$ is well-behaved in the sense that it does not diverge at finite values of z_*. Together with the above results, we can finally conclude that the resolution of the singularity in the bulk also resolves the singularity of the two-point correlator of the boundary theory when the onset of quantum gravity effects happens at the 5d bulk Planck scale.

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We did not find a completely satisfactory 5d metric that incorporates 4d Kasner transitions as found in loop quantum cosmology, the reason for which will be explained below. For the purpose of this paper, which is to show that the finite-distance pole in the two-point correlator can be resolved by quantum gravity effects, we will ignore this point and simply conclude that for two possible straightforward proposals for a metric incorporating Kasner transitions, the pole remains resolved. While this does not settle the issue in that we do not have access to a completely satisfactory 5d effective metric, it again supports and strengthens our previous results.

Let us first consider the following 5d quantum corrected bulk metric

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{qmetric3a} {\rm d}s_5^2=\frac{1}{z^2}\left({\rm d}z^2-{\rm d}t^2+a(t,z)^2{\rm d}x^2+\dots\right) \nonumber \end{align} \tag{ 4.4 }$$

with

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{qmetric3b} a(t,z)^2=\frac{a_{\rm ext}^2}{\lambda^{2p}}\left(t^2+\lambda^2 z^2\right)^{\,p}\exp{\left[2\Delta p\sinh^{-1}\left(\frac{t}{z\lambda}\right)\right]},\qquad\Delta p\in{\mathbb R}. \nonumber \end{align} \tag{ 4.5 }$$

The explicit form of the metric (4.5) has not been derived from any specific quantum gravity model. Nevertheless, our choice can be motivated as follows. We first consider a 4d Planck scale quantum corrected bulk metric with

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{CM} a(t)=\frac{a_{\rm ext}}{\lambda^{\,p}}\left(t^2+\lambda^2\right)^{\,p/2}\exp{\left[\Delta p\sinh^{-1}\left(\frac{t}{\lambda}\right)\right]}, \nonumber \end{align} \tag{ 4.6 }$$

which implements a smooth transition between two Kasner universes at late and early times (more details below).

The four-dimensional part of this metric can be directly related to the metric proposed by Chamseddine and Mukhanov [40] in their modified version of General Relativity that implements the idea of a limiting curvature $\newcommand{\e}{{\rm e}} \epsilon_m$ if we identify p   =  1/3, $\lambda=1/\sqrt{3\varepsilon_m}$, and specify the values of $a_{\rm ext}$ and $\Delta p$ accordingly (the explicit expressions are not relevant for our present purposes but they can be easily derived by direct comparison with equation (40) in [40]). Here however, we do not specify the values of $p, a_{\rm ext}$ and $\Delta p$ leaving them as generic input parameters in our numerical analysis. The Chamseddine–Mukhanov model [40] has been proposed as a toy model for an effective theory of quantum gravity in [41, 42] by showing that it agrees with the effective dynamics of loop quantum cosmology (LQC) in the spatially flat, homogeneous and isotropic sector if one identifies the limiting curvature with a multiple of the Planck curvature⁷.

Now, as for the 5d Planck scale quantum corrected metric considered in equation (4.1), the bulk metric defined in equations (4.4), (4.5) is not singular at t  =  0. Hence, as argued in section 2.3, following the usual AdS/CFT logic we take a 5d bulk quantum gravity point of view and replace $\lambda$ with a 5d effective scale $z\lambda$ in (4.6), thus leading us to the metric (4.4), (4.5).

The metric (4.5) allows to include Kasner transitions in our analysis. Indeed, for $\left|\frac{t}{z\lambda}\right|\gg1$, the following approximation holds

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{asymp} \sinh^{-1}\left(\frac{t}{z\lambda}\right)\simeq\pm\log{\left| 2 \frac{t}{z\lambda}\right|}, \nonumber \end{align} \tag{ 4.7 }$$

where the plus and minus signs are to be taken for $t\gg z\lambda$ and $t\ll -z\lambda$, respectively. The scale factor (4.5) then simplifies to

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{classlim} a(t,z)^2\simeq\frac{a_{\rm ext}^2}{\lambda^{2p}} \,\left(\frac{2}{\lambda z}\right)^{\pm 2 \Delta p}\, t^{2p_\pm}, \nonumber \end{align} \tag{ 4.8 }$$

with Kasner exponents

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{kasnerexp} p_\pm=p\pm\Delta p. \nonumber \end{align} \tag{ 4.9 }$$

Therefore, as also shown in figure 7, asymptotically far in the past and in the future, the scale factor (4.5) for fixed non-zero z reduces to that of the classical 4d-Kasner metric (up to the scaling factor $(\frac{2}{\lambda z}){}^{\pm2\Delta p}$ to be discussed below) with Kasner exponents given in (4.9). Quantum effects become dominant for $\left|\frac{t}{z\lambda}\right|\ll1$. In such a regime, we have

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle a(t,z)^2\simeq a_{\rm ext}^2z^{2p}\left(1+2\Delta p\frac{t}{z\lambda}\right), \nonumber \end{align} \tag{ 4.10 }$$

and the metric describes a regular bounce (around t  =  0) during which the exponents characterising the Kasner universe are changing from p ₋ to $p_+=p_-+2\Delta p$ after the bounce. This qualitatively agrees with the behaviour expected from LQC predictions [27]⁸. Obviously for $\Delta p=0$ there are no Kasner transitions (i.e. $p_-=p_+$), and coherently the metric (4.4), (4.5) reduces to (4.1).

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Although, the metric (4.4), (4.5), seems to be a good candidate for the 5d-quantum metric, it has the following problems. First, it is not a solution of the 5d-Einstein equations up to quantum corrections of order $\mathcal{O}(\lambda)$. This a priori unexpected issue comes about as follows:

As shown in figure 7, a scale factor of the type (4.5) connects two Kasner branches, one with a negative and one with a positive exponent. They are matched such that the solution is monotonically increasing (or decreasing). In order to achieve this, one needs to scale the prefactors of $t^{2p_+}$ as well as $t^{2p_-}$ relatively to each other. In the classical limit, which is achieved either by $\lambda \rightarrow 0$ or here equivalently for the 4d-part by $z \rightarrow 0$, the relative scaling diverges as shown in equation (4.8). At finite z, this effect is responsible for (4.4) not solving the classical Einstein equations at large t, since this matching during the bounce induces a z-dependence in $a(t,z)$ which persists large t.

This indicates the second problem: since equation (4.8) diverges for $\lambda=0$, it is not possible to recover the 4d classical Kasner geometry globally. Moreover, since equation (4.8) diverges also for z  =  0, the boundary limit is not well-defined. One may try to circumvent this issue by modifying the metric (4.5), e.g. as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{qmetric4} a(t,z)^2=\frac{a_{\rm ext}^2}{\lambda^{2p}}\left(t^2+\lambda^2 z^2\right)^{\,p}\exp{\left[2\Delta p\sinh^{-1}\left(\frac{t}{z\lambda}\right)\right]}(\lambda z)^{2\Delta p\,\tanh{\left(\frac{t}{z\lambda}\right)}}, \nonumber \end{align} \tag{ 4.11 }$$

where the factor $(\lambda z){}^{2\Delta p\tanh{\left(\frac{t}{z\lambda}\right)}}$ allows to recover the proper boundary (classical) limit. Indeed for $\left|\frac{t}{z\lambda}\right|\gg1$, taking into account that $\tanh{\left(\frac{t}{z\lambda}\right)}\simeq{{\rm sign}}(t)$ ($z,\lambda\geqslant0$) together with equation (4.7), the scale factor (4.11) reduces to

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle a(t,z)^2\simeq a(t)^2=\frac{a_{\rm ext}^2}{\lambda^{2p}}\,t^{2p_\pm}, \nonumber \end{align} \tag{ 4.12 }$$

which does not depend on z, and the classical Kasner case is recovered in the double scaling limit $\lambda\rightarrow0, a_{\rm ext}/\lambda^{\,p}\rightarrow1$. This modification ensures the correct classical and boundary limits. Figure 8 shows the time dependence of the scale factor for different values of z. In contrast to the previous case for all values of z the same classical solution is approached for large t. However, due to the lack of relative rescaling of the prefactors, we necessarily have a non-monotonous behaviour around t  =  0 for small z. Indeed, around t  =  0, the scale factor can be written as

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \displaystyle a(t,z)^2\simeq a^2_{\rm ext}\,z^{2p}\left[1+2\Delta p\frac{t}{\lambda z}\bigl(1+\log{(\lambda z)}\bigr)\right], \nonumber \end{align} \tag{ 4.13 }$$

and, as reported in figure 8, for small z the logarithmic term is responsible for the change in sign of the slope around t  =  0. This differs qualitatively from the LQC behaviour.

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Unlike (4.5), the metric (4.11) is a solution of the 5d-Einstein equations in zeroth order in $\lambda$. Indeed, as already remarked in section 4.1, in order to have a plausible embedding of the 4d metric into the 5d bulk space, two conditions have to be satisfied: first, the boundary limit is well defined and solves the 4d-Einstein equations; second, the z-derivative of the scale factor vanishes in zeroth order in $\lambda$. On the one hand, for the metric (4.5) the proper boundary limit is not recovered and the z-derivative does not vanish in zeroth order in $\lambda$ (see equation (4.15) below). On the other hand both conditions are satisfied for the metric (4.11). However, the Kretschmann scalar for t  =  0 is not constant in z and furthermore diverges for $z\rightarrow 0$ due to a blow-up of the ${\mathcal O}(\lambda)$ coefficient, as can be checked using computer algebra. Therefore, unlike (4.5), we cannot interpret (4.11) as an effective metric of a quantum gravity theory that resolves singularities by means of a limiting 5d bulk curvature.

Hence, finding a plausible metric that cures all problems at once turns out to be a highly non-trivial task. In principle the effective form of the quantum corrected metric should be derived from full quantum gravity, which would amount to set up 5d quantum Einstein equations and to extract an effective metric, at least in a suitable midisuperspace. We will leave this for further research. Here in this work, we decided to continue, as a case study, with the metric (4.5)⁹ since, unlike (4.11), it shows qualitatively the behaviour we expect from LQC in the 4d part of metric, the Kretschmann scalar exhibits the right behaviour, and it is also symmetric under the scaling symmetry (2.2). As we are interested here in studying the two-point correlator, we use (4.5) only as an example to check the genericity of the absence of the finite-distance pole. We do not claim that it is the correct effective metric.

4.2.1. Solution of the geodesic equations.

The same procedure as for the metric (4.1) is now repeated for the case of Kasner transitions, i.e. for the metric defined in equations (4.4) and (4.5). The derivatives of a(t,z)² entering equations(3.7) are now given by:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\partdif}[2]{\frac{\partial #1}{\partial #2}} \displaystyle \partdif{(a^2)}{t} = 2 a(t,z)^2 \left[ \frac{p t}{t^2 + \lambda^2 z^2} + \Delta p \frac{1}{\sqrt{t^2+\lambda^2 z^2}} \right] , \nonumber \end{align} \tag{ 4.14 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\partdif}[2]{\frac{\partial #1}{\partial #2}} \displaystyle \partdif{(a^2)}{z} = 2 a(t,z)^2 \left[ \frac{p \lambda^2 z}{t^2 + \lambda^2 z^2} - \frac{\Delta p}{z} \frac{t}{\sqrt{t^2+\lambda^2 z^2}} \right]. \label{zderivative2} \nonumber \end{align} \tag{ 4.15 }$$

The corresponding colour plot of t₀ is shown in figure 9. Unlike the previous case, the range of t_* includes also negative values so that we have now the possibility to study geodesics passing the resolved singularity at t  =  0. In figure 9 the level lines for t₀  =  −5 (purple) and t₀  =  3 (blue) are singled out. Again, there is a turning point in z_*, close to t_*  =  0. Moreover, there are solutions that start at a negative t₀, pass through the resolved singularity, have their turning point at positive t_* and come back to the t₀-value they started from. For example, there are points of the level line for t₀  =  −5 that reach positive t_*. Furthermore, it is possible to observe the dominant behaviour of the different values of p . In the region t_*  <  0, the plot looks similar to figure 3(a), just mirrored. For t  >  0, the plot has exactly the behaviour of positive p  solutions. As expected, the transition area is smoothened out. Coherently with the analysis of [26], geodesics with t₀  <  0 are bent towards t  =  0, while geodesics with t₀  >  0 are bent away from t  =  0. Accordingly, as reported in figure 10, the former correspond to the behaviour of z_*(l) described by the purple line, while the latter to the blue line. In both cases, z_* never hits z_*  =  0 for finite values of l. However, since we are interested in probing the resolved bulk singularity, in what follows we shall focus on solutions of the first kind (t₀  <  0).

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4.2.2. Renormalised length.

Similarly to section 4.1, figure 11 shows a plot of the renormalised length $L_{{\rm ren}}$ against z_* in a logarithmic scale. Up to the above-mentioned error occurring in the lower region of the line, we again have linear asymptotic behaviour, i.e. $L_{{\rm ren}} \propto \log(z_*)$, staying away form z_*  =  0 for finite l. For such kind of solutions (purple line in figures 10, 11) $L_{{\rm ren}}$ remains finite for $l \neq 0$, and hence the two-point correlator does not diverge. For completeness, we also include in figure 11 the plot of a positive t₀ solution (blue). This kind of solutions are bent away from the region where quantum effects dominate, hence they behave as in the classical case and there is no pole in the correlator.

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