Unavoidable shear from quantum fluctuations in contracting cosmologies

Although inflation is the leading paradigm to describe the early universe, it is sometimes assumed to be preceded by a phase of slow contraction, followed by a bounce. This is indeed naturally expected in most theories of quantum gravity, it avoids the initial singularity that is otherwise still present in inflation when considered alone, and can also solve the trans-Planckian problem (see e.g. Refs. [1, 2, 3] and references therein). In a homogeneous cosmology, the energy density associated with the anisotropic stress however grows as $a^{-6}$, where $a$ is the scale factor. Since the background energy density grows as $a^{-3(1+w)}$, where $w$ is the equation-of-state parameter of the background fluid, any initial anisotropy grows unstable as the universe contracts, hence driving a new kind of singularity [4], unless $w>1$.

Several solutions to this problem have been proposed. One can for instance assume that the universe contracts with a stiff equation of state, $w>1$, preventing the energy density associated with the anisotropic stress to overcome the isotropic contribution. This corresponds for instance to the ekpyrotic model [5]. Another solution more rooted in quantum gravity consists in regularising the space-time curvature. A concrete example of this is Loop Quantum Cosmology of anisotropic models of the universe [6]. Here the quantisation of the gravitational degrees of freedom makes all energy densities (including the one associated with the anisotropic stress) effectively bounded from above, then guaranteeing a regular bounce for any equation of state.

Obviously, a third solution is to fine tune the initial anisotropy to be sufficiently small such that it remains negligible until the bounce occurs. The amount of required fine tuning may be large for a long phase of contraction, but, at the classical level, it is always possible to set the initial anisotropy to arbitrarily small values. However, in the presence of vacuum quantum fluctuations, this is not true anymore and there is a minimum, inevitable amount of anisotropic stress sourced by quantum fluctuations.

The goal of this work is to study this unavoidable shear production sourced by quantum fluctuations (note that vector perturbations in bouncing cosmologies were also studied at the classical level in Ref. [7]). It is organised as follows. In Sec. 2, we formulate the problem, and show why quantum fluctuations of a test scalar field provide a non-vanishing source of anisotropic stress (more precisely, we explain why the one-point function of the anisotropic stress produced by scalar-field fluctuations vanish, but not its two-point function). In Sec. 3, we derive a stochastic formalism, analogous to the stochastic-inflation formalism, to describe these quantum fluctuations in an inflating or a contracting cosmology, and to compute the expectation value of the shear they generate. In Sec. 4, we apply these tools to the case of a contracting phase with a constant equation-of-state parameter $w$. We find that, when $w>-1/9$, the shear produced out of massless scalar field fluctuations is always negligible until the energy density of the universe reaches the Planck scale, regardless of initial conditions; while for $w<-1/9$, it becomes sizeable much before then, at a point that depends on the time at which the contraction is initiated. In Sec. 5, we compare the shear backreaction with the direct contribution of the test scalar field fluctuations to the energy density of the universe, and conclude that the later is always more important than the former. We also discuss the case of inflating backgrounds, where it is found that the shear backreaction is always negligible. We present our conclusions in Sec. 6.

We consider the case where deviations from the isotropic Friedmann-Lemaître-Robertson-Walker (FLRW) space-time can be parametrised by a Bianchi I metric [8, 9],

with $a(t)$ the isotropised scale factor and $\gamma_{ij}(t)=\exp[2\beta_{i}(t)]\delta_{ij}$. The $\beta_{i}$ functions are constrained to satisfy $\sum_{i}\beta_{i}=0$, such that $\det\gamma_{ij}=1$. The shear tensor, $\sigma_{ij}$, is defined as

where a dot denotes a derivative with respect to cosmic time $t$.

We further assume that the universe is filled with a homogeneous but anisotropic matter fluid, described by its energy density $\rho$, its pressure $P$ and its anisotropic stress $\Pi_{\mu\nu}$, such that its stress-energy tensor reads

In this expression, $u^{\mu}$ is the four velocity of a comoving observer, normalised such that $u_{\mu}u^{\mu}=-1$. From the metric (2.1), a comoving observer has $u^{i}=0$, and the normalisation condition imposes that $u^{\mu}=-\delta^{\mu}_{0}$. The anisotropic stress is a symmetric tensor (since the stress-energy tensor is symmetric) that parametrises deviations from a perfect fluid. Given that its “$00$” component can always be reabsorbed in the first term of Eq. (2.3), hence in the definition of $\rho$, it is set to zero without loss of generality. The “$0i$” component of the Einstein equation then imposes that $\Pi_{0i}=0$, which implies that the anisotropic stress is orthogonal to the comoving velocity, $\Pi_{\mu\nu}u^{\nu}=0$. If one decomposes $\Pi_{ij}$ as $\Pi_{ij}=\Pi_{ij}-g^{mn}\Pi_{mn}g_{ij}/3+g^{mn}\Pi_{mn}g_{ij}/3$, the component $g^{mn}\Pi_{mn}g_{ij}/3$ can always be reabsorbed in the second term of Eq. (2.3), hence in the definition of $P$, and the remaining part $\Pi_{ij}-g^{mn}\Pi_{mn}g_{ij}/3$ is traceless, so we assume that $\Pi_{ij}$ is traceless without loss of generality, $\Pi_{ij}g^{ij}=0$.

Under these conditions, the Einstein equations can be written as

which correspond respectively to the “$00$” component, the trace and the traceless part of the “$ij$” component, and where $H\equiv\dot{a}/a$ is the Hubble parameter. By combining these three equations, one obtains a conservation relation,

At the background level, one can see that the only effect of the shear is to add a contribution $\rho_{\sigma}=M_{\scriptscriptstyle{\mathrm{Pl}}}^{2}\sigma_{ij}\sigma^{ij}/2$ to the energy density, and $p_{\sigma}=\rho_{\sigma}$ to the pressure, in the Friedmann equation (2.4) and in the Raychaudhuri equation (2.5). This additional contribution is sourced by the anisotropic stress through Eq. (2.6), which can be formally solved as

From this expression, one can see that, in the absence of anisotropic stress, any initial amount of shear tensor grows as $\sigma^{i}_{j}\propto a^{-3}$, which gives rise to $\rho_{\sigma}\propto a^{-6}$. Denoting by $w$ the equation-of-state parameter of the background fluid, such that $P=w\rho$, Eq. (2.7) gives rise to $\rho\propto a^{-3(1+w)}$ in the absence of anisotropic stress, so the shear contribution eventually dominates unless $w>1$, as mentioned above.

The time at which $\rho_{\sigma}$ takes over $\rho$ obviously depends on its initial value, hence on $\sigma^{i}_{j}(t_{\mathrm{in}})$. Since the contracting phase ends at the time when the universe experiences a bounce, and given that the bounce is followed by an expanding phase, during which $\rho_{\sigma}$ decreases faster than $\rho$ (unless $w>1$), it is always possible to make the shear irrelevant by requiring its initial value to be sufficiently small. The shear problem thus becomes an initial fine-tuning problem. There is, however, no natural choice for neither the values of the shear tensor nor the probabilities associated to them, and this could have important consequences for the state of the universe after the bounce (see Refs. [10, 11] for a concrete example in Loop Quantum Cosmology). Alternatively one could imagine the existence of mechanisms in the early stage of, or prior to, the contracting phase, that would set the shear to the required small value, as could be the case in cyclic models of the universe.

These considerations are however valid only under the assumption that the anisotropic stress vanishes. While this is true in principle for a perfect fluid, at the high energy scales that operate in the early universe, field theory is the relevant framework to describe matter, and as we shall see, the unavoidable vacuum fluctuations present in quantum fields are enough to generate an anisotropic stress. For explicitness, let us consider the case of a test scalar field $\phi$, minimally coupled to gravity, described by the action

where $V(\phi)$ is the potential associated with $\phi$. The stress-energy tensor can be obtained from $T_{\mu\nu}^{(\phi)}=-(2/\sqrt{-g})\delta S^{(\phi)}/\delta g^{\mu\nu}$, which leads to

As explained above, the anisotropic stress corresponds to the transverse and traceless part of the stress-energy tensor. Transverseness is obtained by projecting $T_{\mu\nu}^{(\phi)}$ in the hyperplane orthogonal to $u^{\mu}$, by using the projector $\perp_{\mu\nu}=g_{\mu\nu}+u_{\mu}u_{\nu}$.¹¹1From the normalisation condition of $u^{\mu}$, $u_{\mu}u^{\mu}=-1$, one can indeed check that (i) $\perp_{\mu\nu}u^{\nu}=0$, (ii) if $v^{\mu}$ is a vector orthogonal to $u^{\mu}$, i.e. $u_{\mu}v^{\mu}=0$, then $\perp_{\mu\nu}v^{\nu}=v_{\mu}$, and (iii) $\perp_{\mu}^{\rho}\perp_{\rho\sigma}=\perp_{\mu\nu}$. In the Bianchi I space time (2.1), its only non-zero components are $\perp_{ij}=g_{ij}$, which is no more than the induced metric. Tracelessness is obtained by removing the trace of the projected tensor, and one obtains $\Pi_{ij}^{(\phi)}=T^{(\phi)}_{ij}-\frac{1}{3}g_{ij}g^{mn}T_{mn}^{(\phi)}$, hence

Let us consider the case where the initial shear vanishes, so space time is initially isotropic. If the quantum state of the scalar field $\phi$ shares the same symmetries as the background, for instance if it is in its vacuum state $|0\rangle$, then there is no preferred direction and $\langle 0|\Pi_{ij}^{(\phi)}|0\rangle=0$ (this will be shown explicitly below). This implies that, when taking the expectation value of Eq. (2.8), the anisotropic stress disappears, so $\langle\sigma^{i}_{j}\rangle\propto a^{-3}$ and the above considerations still apply. However, the two-point function of the anisotropic stress does not necessarily vanish, and since the self contracted-shear appears in $\rho_{\sigma}$, this means that quantum fluctuations of a scalar field can source an effective shear. In other words, by self contracting Eq. (2.8), in the absence of initial shear, one has

The goal of the following is thus to compute the two-point function of the anisotropic stress for a test scalar field, in order to evaluate Eq. (2.12). Let us note that, since $\phi$ must be homogeneous at the background level, Eq. (2.11) is of second order in the field perturbations, hence $\rho_{\sigma}$ is of fourth order in those perturbations. This small contribution is however enhanced by $a^{-6}$, so the (ir)relevance of the effect is a priori not obvious. Let us also note that other types of fields, such as vector fields or gauge fields, can give a non-vanishing mean anisotropic stress, hence source the shear at second rather than fourth order in perturbations. In this work, we choose to study the case of a scalar field, in order to extract the minimum amount of shear one cannot avoid, simply by postulating the existence of light scalar degrees of freedom during the contraction.

A last word is in order regarding the strategy followed in this work. Hereafter, the question we are interested in is whether or not a homogeneous and isotropic contracting cosmology is stable under quantum fluctuations of a test scalar field. In practice, the stability analysis is performed in a specific “direction”, namely the would-be contribution to the anisotropic stress coming from a test scalar field, in a Bianchi I model. This is because, at the classical level, the shear instability signals the presence of a potential danger in this direction. However, this does not allow us to draw conclusions about stability in other “directions” (this would for instance require to go beyond the Bianchi I space-time). Moreover, our formalism is not meant to describe backreaction, but rather to provide an indicator of whether or not backreaction might become important. This is done by computing an effective energy density associated to the anisotropic stress, and by comparing this quantity to the overall energy density of the background, while still assuming that the space-time geometry is fixed, and close to isotropic. If one found situations where backreaction becomes important, then one would also have to include fluctuations in the metric sector (which are known to couple in anisotropic universes [9]) to describe the full dynamics of the problem. This however goes beyond the scope of this paper.

In order to compute the statistics of the anisotropic stress generated by a scalar field in a contracting phase, we employ a stochastic formalism reminiscent of the “stochastic inflation” formalism [12, 13, 14]. While this makes practical calculations easier, a comparison with a full quantum field theoretic treatment is performed in Appendix A, where we check that the “stochastic contraction” approach developed in this work is indeed valid (a result that is well-established in the context of inflation, see e.g. Refs. [15, 16, 17]). Another reason for using a stochastic formalism is that, although only the case of a test field is considered hereafter, it may help to generalise our results to describe the backreaction of this field on the background dynamics, see the discussion in Sec. 6.

Let us now apply the calculational program sketched above to the case of a contracting cosmology (expanding cosmologies, in particular inflating cosmologies, are discussed in Sec. 5). In an isotropic universe, defining $u=a\phi$, Eqs. (3.1)-(3.2) give rise to

where a prime denotes derivation with respect to conformal time $\eta$, related to cosmic time through $\mathrm{d}t=a\mathrm{d}\eta$, and where $\mathcal{H}=a^{\prime}/a$. In order to set initial conditions in the Bunch-Davies vacuum state [22], the pulsation $\omega_{k}^{2}$, defined as the term between the braces in Eq. (4.1), must be dominated by $k^{2}$ at early time. In a contracting cosmology, this clearly imposes $m=0$, and for simplicity, we consider that this is the case hereafter. If the equation-of-state parameter of the background is constant, one has $\omega_{k}^{2}=k^{2}-2(1-3w)/[(1+3w)^{2}\eta^{2}]$. The resulting mode equation can be solved by means of Hankel functions, and setting integration constants such that the mode functions are normalised according to the Klein-Gordon product mentioned above, one obtains

in the vacuum state, where $H_{\nu}^{(1)}$ is the first Hankel function and where, hereafter, we assume that $-1/3<w<1$. On super-Hubble scales, the Hankel function can be expanded, and one finds $\phi_{k\ll aH}\simeq e^{i\pi(2\nu-1)/4}2^{\nu-1}\Gamma(\nu)(-k\eta)^{1/2-\nu}/(\sqrt{\pi k}a)$ and $\pi_{k}=a^{2}\phi_{k}^{\prime}$. Plugging these expressions into Eq. (3.12), one obtains

If the mass vanishes, the background equation of motion (3.16) can be readily solved, and one obtains for the Green function

Recalling that $\mathrm{sinc}^{\prime\prime}(0)=-1/3$, by making use of Eqs. (3.12) and (3.1), Eq. (3.24) then gives rise to

where $a_{1}$ and $a_{2}$ denote $a(t_{1})$ and $a(t_{2})$ respectively. The contribution from the shear to the overall energy density can then be computed with Eq. (2.12). In order to specify the term $\mathrm{max}(a_{1},a_{2})$ appearing in Eq. (4.10), the rectangular integration domain of Eq. (2.12) can be split into two triangles, one where $t_{1}<t_{2}$ and one where $t_{1}>t_{2}$. Since the integrand is in fact symmetric in $t_{1}$ and $t_{2}$, both these triangular regions give the same contribution and it is enough to compute the integral over one of the triangles and multiply it by two. This leads to

One can check that, when $t=t_{\mathrm{in}}$, $\left\langle\rho_{\sigma}\right\rangle=0$. When $w>-1/9$, the terms have been ordered in decreasing power, while for $w<-1/9$, the dominant term is the third one (i.e. the first one on the second line), which increases as $\propto a^{2+18w}$. These two cases need therefore to be distinguished.

When $w>-1/9$, since the dominant term, i.e. the one proportional to $1/(5+27w)$, does not depend on the initial time (contrary to the other terms, it does not involve $a_{\mathrm{in}}$), the contribution from the shear to the energy density quickly reaches an attractor, which scales as $\langle\rho_{\sigma}\rangle\propto H^{6}/M_{\scriptscriptstyle{\mathrm{Pl}}}^{2}$, hence $\langle\rho_{\sigma}\rangle/\rho_{\mathrm{tot}}\propto(H/M_{\scriptscriptstyle{\mathrm{Pl}}})^{4}$. As a consequence, until the background energy density reaches the Planck scale, the contribution from the shear to that energy density is negligible. This can be clearly seen in the left panel of Fig. 1 (see the cases $w=0$ and $w=1/3$). In fact, because of the prefactor in Eq. (LABEL:eq:rho_shear:contraction), the ratio $\langle\rho_{\sigma}\rangle/\rho_{\mathrm{tot}}$ is still negligible when $H$ reaches the Planck scale. In bouncing cosmologies, $|H|$ is bounded from above since the bounce demands for a smooth transition from contraction, i.e. $H<0$, to expansion, i.e. $H>0$. The upper-bound of $|H|$ can be safely taken to be of the order of the Planck scale at most⁵⁵5Either the bounce is classical and we can expect the maximal value of $|H|$ to be below the Planck scale; or the bounce is of quantum-gravitational origin, hence expected to occur at the Planck energy density, as can explicitly be proven in Loop Quantum Cosmology for example. and the maximal value that $\langle\rho_{\sigma}\rangle/\rho_{\mathrm{tot}}$ can achieve is thus of order $\sim\epsilon^{4\frac{1+9w}{1+3w}}$. Hence, one concludes that there is no shear backreaction problem in this case.