On the stability of relativistic perfect fluids with linear equations of state $p=K\rho$ where $1/3<K<1$

It can be verified by a straightforward calculation that

$$(\rho_{*},v_{*}^{i})=(t^{3(1+K)}\rho_{c},-\delta^{i}_{0}),\quad t\in(0,1],$$

(1.5)

defines a spatially homogeneous solution of the relativistic Euler equations (1.1) on the exponentially expanding FLRW spacetimes $(M,\tilde{g}{})$ for any choice of the parameter $0\leq K\leq 1$ and constant $\rho_{c}\in(0,\infty)$. From a cosmological perspective, these solutions are, in a sense, the most natural since they are also spatially isotropic and hence do not determine a preferred direction.

The future, nonlinear stability of the solutions (1.5) on the exponentially expanding FLRW spacetimes was first established in the articles⁴⁴4In these articles, stability was established in the more difficult case where the fluid is coupled to Einstein’s equations. However, the techniques used there also work in the simpler setting considered in this article where gravitational effects are neglected. articles [20, 22] for the parameter values $0<K<1/3$. Stability results for the end points $K=1/3$ and $K=0$ were established later⁵⁵5Again, stability was established in these articles in the more difficult case where the fluid is coupled to Einstein’s equations. in [13] and [8], respectively. See also [6, 10, 11, 14] for different proofs and perspectives, the articles [9, 12] for related stability results for fluids with nonlinear equations of state on the exponentially expanding FLRW spacetimes, the articles [4, 23, 26] for analogous stability results on other classes of expanding cosmological spacetimes, and [19] for related, early stability results for the Einstein-scalar field system. One of the important aspects of these works is they demonstrate that spacetime expansion can suppress shock formation in fluids, which was first discovered in the Newtonian cosmological setting [25]. This is in stark contrast to fluids on Minkowski space where arbitrary small perturbations of a class of homogeneous solutions to the relativistic Euler equations form shocks in finite time [3].

A consequence of these stability proofs is that the spatial components of the conformal four-velocity $v^{i}$ of small, nonlinear perturbations of the homogeneous solution (1.5) decay to zero at future timelike infinity, that is,

$$\lim_{t\searrow 0}v^{I}=0,$$

for the parameter values $0\leq K<1/3$. This behaviour is, of course, consistent with the isotropic homogeneous solutions (1.5). On the other hand, when $K=1/3$, the spatial components of the conformal four-velocity $v^{i}$ for perturbed solutions do not, in general, decay to zero at timelike infinity, and instead limit to a spatial vector field $\xi^{I}$ on $\mathbb{T}{}^{3}$, that is,

$$\lim_{t\searrow 0}v^{I}=\xi^{I}.$$

This behaviour is consistent with a family of non-isotropic homogeneous solutions defined by⁶⁶6More generally, we could set the spatial components of the conformal four-velocity $v_{\bullet}^{I}$ to be any non-zero vector in $\mathbb{R}{}^{3}$ and determine $v_{\bullet}^{0}$ via the conditions $g_{ij}v^{i}_{\bullet}v^{j}_{\bullet}=-1$ and $v^{0}_{\bullet}<0$. However, for simplicity, we will assume here that $v_{\bullet}^{I}$ is chosen so that it is pointing in the direction of the coordinate vector field $\partial_{1}=\frac{\partial\;}{\partial x^{1}}$.

$$(\rho_{\bullet},v_{\bullet}^{i})=(t^{3(1+K)}\rho_{c},-\sqrt{1+\nu_{c}^{2}}\delta^{i}_{0}+\nu_{c}\delta^{i}_{1}),\quad t\in(0,1],$$

(1.6)

where $(\rho_{c},\nu_{c})\in(0,\infty)\times(0,\infty)$, which satisfy the relativistic Euler equations for $K=1/3$. The known stability results for $K=1/3$ imply the future stability of nonlinear perturbations of these solutions.

Noting that solutions of the type (1.6) can be made arbitrarily close to solutions of the type (1.5) for $K=1/3$ by choosing $\nu_{c}$ sufficiently small, from a stability point of view there seems to be verify little difference between the two classes of solutions for small $\nu_{c}$. Indeed, the future nonlinear stability of both classes of solutions, where $\nu_{c}$ is sufficiently small, can be achieved via a common proof. However, as will become clear, the essential difference between these solutions is that, from an initial data point of view, stable perturbations of solutions of the type (1.5) are generated from initial $(\rho,v^{I})|_{t=1}$ that is sufficiently close to $(\rho_{*},v_{*}^{I})|_{t=1}$ and satisfies

$$\min_{x\in\mathbb{T}{}^{3}}(g_{IJ}v^{I}v^{J})\bigl{|}_{t=1}=0,$$

(1.7)

while stable perturbations of solutions of the type (1.6) are generated from initial data $(\rho,v^{I})|_{t=1}$ that is sufficiently close to $(\rho_{\bullet},v_{\bullet}^{I})|_{t=1}$ and satisfies

$$\min_{x\in\mathbb{T}{}^{3}}(g_{IJ}v^{I}v^{J})\bigl{|}_{t=1}>0.$$

(1.8)

Until recently, it was not known if any solutions of the relativistic Euler equations were stable to the future for the parameter values $1/3<K<1$. In fact, it was widely believed that solutions to the relativistic Euler were not stable for these parameter values. This belief was due, in part, to the influential work of Rendall [18] who used formal expansion to investigate the asymptotic behaviour of relativistic fluids on exponentially expanding FLRW spacetimes with a linear equation of state. Rendall observed that the formal expansions become inconsistent for $K$ in the range $1/3<K<1$ if the leading order term in the expansion of $g_{IJ}v^{I}v^{J}$ at $t=0$ vanished somewhere. He speculated that the inconsistent behaviour is the expansions could be due inhomogeneous features developing in the fluid density that would ultimately result in the blow-up of the density contrast $\frac{\partial_{I}\rho}{\rho}$ at future timelike infinity. Speck [23, §1.2.3] added further support to Rendall’s arguments by presenting a heuristic analysis that suggested uninhibited growth would set in for solutions of the relativistic Euler equations for the parameter values $1/3<K<1$ . Combined, these considerations left the stability of solutions to the relativistic Euler equations in doubt for $K$ in the range $1/3<K<1$.

However, it was established in [16] that there exists a class of non-isotropic homogeneous solutions of the relativistic Euler equations that are stable to the future under small, nonlinear perturbations. This class of homogeneous solutions should be viewed as the natural continuation of the solutions (1.6) over the parameter range $1/3<K<1$, and they are defined by

$$\bigl{(}\rho_{\bullet},v_{\bullet}^{i})=\biggl{(}\frac{\rho_{c}t^{\frac{2(1+K)}{1-K}}}{(t^{2\mu}+e^{2u})^{\frac{1+K}{2}}},-t^{-\mu}\sqrt{e^{2u}+t^{2\mu}}\delta_{0}^{i}+t^{-\mu}e^{u}\delta_{1}^{i}\biggr{)},\quad t\in(0,1],$$

(1.9)

where

$$\mu=\frac{3K-1}{1-K}$$

(1.10)

and $u=u(t)$ solves the initial value problem (IVP)

	$\displaystyle u^{\prime}\!(t)$	$\displaystyle=\frac{K\mu t^{2\mu-1}}{t^{2\mu}+(1-K)e^{2u(t)}},\quad 0<t\leq 1,$		(1.11)
	$\displaystyle u(1)$	$\displaystyle=u_{0}.$		(1.12)

Existence of solutions to this IVP is guaranteed by Proposition 3.1. from [16], which we restate here:

The main result of [16] was a proof of the nonlinear stability to the future of the homogeneous solutions (1.9) for the parameter values $1/3<K<1/2$. It was also established in [16] that under a $\mathbb{T}{}^{2}$-symmetry assumption, future stability held for the full parameter range $1/3<K<1$. An important point that is worth emphasising is that the initial data used to generate the perturbed solutions from [16] satisfies the condition (1.8) at $t=1$, and furthermore, this positivity property propagates to the future in the sense that the perturbed solutions satisfy

$$\inf_{x\in M}(t^{2\mu}g_{IJ}v^{I}v^{J})>0.$$

It is this property of the perturbed solutions from [16] that avoids the problematic scenario identified by Rendall.

This article has two main aims: the first is to establish the nonlinear stability to the future of the homogeneous solutions (1.9) for the full parameter range $1/3<K<1$ without the $\mathbb{T}{}^{2}$-symmetry that was required in [16]. The second aim is to provide convincing numerical evidence that shows the density contrast blow-up scenario of Rendall is realized if the condition (1.8) on the initial data is violated.

Before stating a precise version of our stability result for the homogeneous solutions (1.9), we first recall two formulations of the relativistic Euler equations from [16]. The first formulation, which was introduced in [15] and subsequently employed in [14] to establish stability for the parameter range $0<K\leq 1/3$, involves representing the fluid in terms of the modified fluid density $\zeta$ defined via

$$\rho=t^{3(1+K)}\rho_{c}e^{(1+K)\zeta}$$

(1.14)

and the spatial components $v_{I}$ of the conformal fluid four-covelocity⁷⁷7Here and in the following, all spacetime indices will be raised and lowered with the conformal metric $g_{ij}$. $v_{i}=g_{ij}v^{j}$. In terms of these variables, the relativistic Euler equations (1.1) can be formulated as the following symmetric hyperbolic system:

$$B^{k}\partial_{k}V=\frac{1}{t}\mathcal{B}{}\pi V$$

(1.15)

where

	$\displaystyle V$	$\displaystyle=(\zeta,v_{J})^{\operatorname{tr}},$		(1.16)
	$\displaystyle v_{0}$	$\displaystyle=\sqrt{|v|^{2}+1},\qquad|v|^{2}=\delta^{IJ}v_{I}v_{J},$		(1.17)
	$\displaystyle v^{i}$	$\displaystyle=\delta^{iJ}v_{J}-\delta^{i}_{0}v_{0},$		(1.18)
	$\displaystyle\mathcal{B}{}$	$\displaystyle=\frac{-1}{v^{0}}\begin{pmatrix}1&0\\ 0&\frac{1-3K}{v_{0}}\delta^{JI}\end{pmatrix},$		(1.19)
	$\displaystyle\pi$	$\displaystyle=\begin{pmatrix}0&0\\ 0&\delta_{I}^{J}\end{pmatrix},$		(1.20)
	$\displaystyle L^{k}_{I}$	$\displaystyle=\delta^{k}_{J}-\frac{v_{J}}{v_{0}}\delta^{k}_{0},$		(1.21)
	$\displaystyle M_{IJ}$	$\displaystyle=\delta_{IJ}-\frac{1}{(v_{0})^{2}}v_{I}v_{J},$		(1.22)
	$\displaystyle B^{0}$	$\displaystyle=\begin{pmatrix}K&\frac{K}{v^{0}}L^{0}_{M}\delta^{MJ}\\ \frac{K}{v^{0}}\delta^{LI}L^{0}_{L}&\delta^{LI}M_{LM}\delta^{MJ}\end{pmatrix}$		(1.23)
and
	$\displaystyle B^{K}$	$\displaystyle=\frac{1}{v^{0}}\begin{pmatrix}Kv^{K}&KL^{K}_{M}\delta^{MJ}\\ K\delta^{LI}L^{K}_{L}&\delta^{LI}M_{LM}\delta^{MJ}v^{K}\end{pmatrix}.$		(1.24)

The second formulation of the relativistic Euler equations is obtained by introducing a new density variable $\tilde{\zeta}{}$ via

$$\tilde{\zeta}{}=\zeta+\ln(v_{0})$$

(1.25)

and decomposing the spatial components of the conformal fluid four-velocity as

	$\displaystyle v_{1}$	$\displaystyle=\frac{t^{-\mu}e^{u(t)+w_{1}}}{\sqrt{t^{2\mu}\left((w_{2}-w_{3})^{2}+(w_{2}+w_{3})^{2}\right)+1}},$		(1.26)
	$\displaystyle v_{2}$	$\displaystyle=\frac{(w_{2}+w_{3})e^{u(t)+w_{1}}}{\sqrt{t^{2\mu}\left((w_{2}-w_{3})^{2}+(w_{2}+w_{3})^{2}\right)+1}}$		(1.27)
and
	$\displaystyle v_{3}$	$\displaystyle=\frac{(w_{2}-w_{3})e^{u(t)+w_{1}}}{\sqrt{t^{2\mu}\left((w_{2}-w_{3})^{2}+(w_{2}+w_{3})^{2}\right)+1}},$		(1.28)

where $u(t)$ solves the IVP (1.11)-(1.12). Then setting

	$\displaystyle\breve{w}{}_{1}$	$\displaystyle=u+w_{1},$		(1.29)
	$\displaystyle\psi=$	$\displaystyle t^{2\mu}+e^{2\breve{w}{}_{1}},$		(1.30)
	$\displaystyle\chi=$	$\displaystyle t^{2\mu}-(K-1)e^{2\breve{w}{}_{1}},$		(1.31)
	$\displaystyle\phi$	$\displaystyle=2t^{2\mu}\left(w_{2}^{2}+w_{3}^{2}\right)+1,$		(1.32)
	$\displaystyle\eta_{\Lambda}$	$\displaystyle=\left(2w_{\Lambda}t^{2\mu}(w_{2}-w_{3})+(-1)^{\Lambda}1\right),\quad\Lambda=2,3,$		(1.33)
and
	$\displaystyle\xi_{\Lambda}$	$\displaystyle=\left(2w_{\Lambda}t^{2\mu}(w_{2}+w_{3})+1\right),\quad\Lambda=2,3,$		(1.34)

it was shown in [16, §3.2] that in terms of the variables

$$W=(\tilde{\zeta}{},w_{1},w_{2},w_{3})^{\operatorname{tr}}$$

(1.35)

the relativistic Euler equations become

$$\partial_{t}W+\mathcal{A}{}^{I}\partial_{I}W=-\frac{\mu}{t}\Pi W+t^{\mu-1}\mathcal{G}{}$$

(1.36)

where

$$\mathcal{A}{}^{1}=\frac{1}{\sqrt{\frac{t^{2\mu}}{e^{2\tilde{w}{}_{1}}}+1}}\begin{pmatrix}-\frac{1}{\sqrt{\phi}}&-\frac{t^{2\mu}}{\psi\sqrt{\phi}}&\frac{2t^{2\mu}w_{2}}{\phi^{3/2}}&\frac{t^{2\mu}w_{3}}{\phi^{3/2}}\\ -\frac{Kt^{2\mu}e^{-2\breve{w}{}_{1}}\psi}{\sqrt{\phi}\chi}&\frac{(2K-1)t^{2\mu}+(K-1)e^{2\breve{w}{}_{1}}}{\sqrt{\phi}\chi}&-\frac{2Kt^{2\mu}\psi w_{2}}{\phi^{3/2}\chi}&-\frac{2Kt^{2\mu}\psi w_{3}}{\phi^{3/2}\chi}\\ Kt^{2\mu}w_{2}e^{-2\breve{w}{}_{1}}\sqrt{\phi}&-\frac{Kt^{2\mu}w_{2}\sqrt{\phi}}{\psi}&-\frac{1}{\sqrt{\phi}}&0\\ Kt^{2\mu}w_{3}e^{-2\breve{w}{}_{1}}\sqrt{\phi}&-\frac{Kt^{2\mu}w3\sqrt{\phi}}{\psi}&0&-\frac{1}{\sqrt{\phi}}\\ \end{pmatrix},$$

(1.37)

$$\mathcal{A}{}^{2}=\frac{1}{\sqrt{\frac{t^{2\mu}}{e^{2\tilde{w}{}_{1}}}+1}}\begin{pmatrix}-\frac{t^{\mu}(w_{3}+w_{2})}{\sqrt{\phi}}&-\frac{t^{3\mu}(w_{3}+w_{2})}{\psi\sqrt{\phi}}&\frac{t^{\mu}\eta_{3}}{\phi^{3/2}}&-\frac{t^{\mu}\eta_{2}}{\phi^{3/2}}\\ -\frac{Kt^{3\mu}(w_{2}+w_{3})e^{-2\breve{w}{}_{1}}\psi}{\sqrt{\phi}\chi}&\frac{t^{\mu}(w_{2}+w_{3})\left((2K-1)t^{2\mu}+(K-1)e^{2\breve{w}{}_{1}}\right)}{\sqrt{\phi}\chi}&-\frac{Kt^{\mu}\psi\eta_{3}}{\phi^{3/2}\chi}&\frac{Kt^{\mu}\psi\eta_{2}}{\phi^{3/2}\chi}\\ -\frac{1}{2}Kt^{\mu}e^{-2\breve{w}{}_{1}}\sqrt{\phi}&\frac{Kt^{\mu}\sqrt{\phi}}{2\psi}&-\frac{t^{\mu}(w_{3}+w_{2})}{\sqrt{\phi}}&0\\ -\frac{1}{2}Kt^{\mu}e^{-2\breve{w}{}_{1}}\sqrt{\phi}&\frac{Kt^{\mu}\sqrt{\phi}}{2\psi}&0&-\frac{t^{\mu}(w_{3}+w_{2})}{\sqrt{\phi}}\\ \end{pmatrix},$$

(1.38)

$$\mathcal{A}{}^{3}=\frac{1}{\sqrt{\frac{t^{2\mu}}{e^{2\tilde{w}{}_{1}}}+1}}\begin{pmatrix}\frac{t^{\mu}(w_{3}-w_{2})}{\sqrt{\phi}}&\frac{t^{3\mu}(w_{3}-w_{2})}{\psi\sqrt{\phi}}&-\frac{t^{\mu}\xi_{3}}{\phi^{3/2}}&\frac{t^{\mu}\xi_{2}}{\phi^{3/2}}\\ -\frac{Kt^{3\mu}(w_{2}-w_{3})e^{-2\breve{w}{}_{1}}\psi}{\sqrt{\phi}\chi}&\frac{t^{\mu}(w_{2}-w_{3})\left((2K-1)t^{2\mu}+(K-1)e^{2\breve{w}{}_{1}}\right)}{\sqrt{\phi}\chi}&\frac{Kt^{\mu}\psi\xi_{3}}{\phi^{3/2}\chi}&-\frac{Kt^{\mu}\psi\xi_{2}}{\phi^{3/2}\chi}\\ -\frac{1}{2}Kt^{\mu}e^{-2\breve{w}{}_{1}}\sqrt{\phi}&\frac{Kt^{\mu}\sqrt{\phi}}{2\psi}&\frac{t^{\mu}(w_{3}-w_{2})}{\sqrt{\phi}}&0\\ \frac{1}{2}Kt^{\mu}e^{-2\breve{w}{}_{1}}\sqrt{\phi}&-\frac{Kt^{\mu}\sqrt{\phi}}{2\psi}&0&\frac{t^{\mu}(w_{3}-w_{2})}{\sqrt{\phi}}\\ \end{pmatrix},$$

(1.39)

$$\mathcal{G}{}=\begin{pmatrix}0\\ -\frac{K(3K-1)\left(e^{2w_{1}}-1\right)e^{2u}}{\left((K-1)e^{2u}-t^{2\mu}\right)\left((K-1)e^{2\breve{w}{}_{1}}-t^{2\mu}\right)}\\ 0\\ 0\end{pmatrix},$$

(1.40)

and

$$\Pi=\operatorname{diag}(0,0,1,1).$$

(1.41)

For later use, we also define

$$\Pi^{\perp}=\mathord{{\mathrm{1}}\kern-2.70004pt{\mathrm{I}}}\kern 3.50006pt-\Pi,$$

(1.42)

and observe that $\Pi$ and $\Pi^{\perp}$ satisfy the relations

$$\Pi^{2}=\Pi,\quad(\Pi^{\perp})^{2}=\Pi^{\perp},\quad\Pi\Pi^{\perp}=\Pi^{\perp}\Pi=0{\quad\text{and}\quad}\Pi+\Pi^{\perp}=\mathord{{\mathrm{1}}\kern-2.70004pt{\mathrm{I}}}\kern 3.50006pt.$$

(1.43)

An important point regarding the formulation (1.36) is that it is symmetrizable. Indeed, as shown in [16], multiplying (1.36) by the positive definite, symmetric matrix

$$A^{0}=\begin{pmatrix}K&0&0&0\\ 0&\frac{t^{2\mu}e^{2\breve{w}{}_{1}}-(K-1)e^{4\breve{w}{}_{1}}}{\psi^{2}}&0&0\\ 0&0&\frac{2e^{2\breve{w}{}_{1}}\left(2w_{3}^{2}t^{2\mu}+1\right)}{\phi^{2}}&-\frac{4w_{2}w_{3}t^{2\mu}e^{2\breve{w}{}_{1}}}{\phi^{2}}\\ 0&0&-\frac{4w_{2}w_{3}t^{2\mu}e^{2\breve{w}{}_{1}}}{\phi^{2}}&\frac{2e^{2\breve{w}{}_{1}}\left(2w_{2}^{2}t^{2\mu}+1\right)}{\phi^{2}}\\ \end{pmatrix}$$

(1.44)

yields

$$A^{0}\partial_{t}W+A^{I}\partial_{I}W=-\frac{\mu}{t}A^{0}\Pi W+t^{\mu-1}A^{0}\mathcal{G}{}$$

(1.45)

where it is straightforward to verify from (1.37)-(1.39) that the matrices

$$A^{I}=A^{0}\mathcal{A}{}^{I}$$

(1.46)

are symmetric, that is,

$$(A^{I})^{\operatorname{tr}}=A^{I}.$$

(1.47)

We are now in a position to state the main stability theorem of this article. The proof is presented in Section 2. Before stating the theorem, it is important to note that, due to change of variables defined via (1.25)-(1.28) and (1.35), the homogeneous solutions (1.9) correspond to the trivial solution $W=0$ of (1.36).

It is essential for the stability result stated in Theorem 1.2 to hold that the initial data used to generated the nonlinear perturbations of homogeneous solutions of the type (1.9) satisfies the condition (1.8). This leaves the question of what happens when this condition is violated, which would be guaranteed to happen for some choice of initial data from any given open set of initial data that contains initial data corresponding to an isotropic homogeneous solution (1.5). To investigate this situation, we consider a $\mathbb{T}{}^{2}$-symmetric reduction of the system (1.15) obtained by the ansatz

	$\displaystyle\tilde{\zeta}(t,x^{1},x^{2},x^{3})$	$\displaystyle=\mathtt{z}{}(t,x^{1}),$		(1.56)
	$\displaystyle v_{I}(t,x^{1},x^{2},x^{3})$	$\displaystyle=t^{-\mu}\mathtt{w}{}(t,x^{1})\delta_{I}^{1},$		(1.57)

where $\tilde{\zeta}{}$ is as defined above by (1.25). It is not difficult to verify via a straightforward calculation that the relativistic Euler equations (1.15) will be satisfied provided that $\mathtt{z}{}$ and $\mathtt{w}{}$ solve⁹⁹9Here, we set $x=x^{1}$.

	$\displaystyle\partial_{t}\mathtt{z}{}-\frac{\mathtt{w}{}}{(t^{2\mu}+\mathtt{w}{}^{2})^{\frac{1}{2}}}\partial_{x}\mathtt{z}{}-\frac{t^{2\mu}}{(t^{2\mu}+\mathtt{w}{}^{2})^{\frac{3}{2}}}\partial_{x}\mathtt{w}{}$	$\displaystyle=0,$		(1.58)
	$\displaystyle\partial_{t}\mathtt{w}{}-\frac{Kt^{2\mu}(t^{2\mu}+\mathtt{w}{}^{2})^{\frac{1}{2}}}{(t^{2\mu}-(K-1)\mathtt{w}{}^{2})}\partial_{x}\mathtt{z}{}+\frac{\bigl{(}(2K-1)t^{2\mu}+(K-1)\mathtt{w}{}^{2}\bigr{)}\mathtt{w}{}}{(t^{2\mu}+\mathtt{w}{}^{2})^{\frac{1}{2}}(t^{2\mu}-(K-1)\mathtt{w}{}^{2})}\partial_{x}\mathtt{w}{}$	$\displaystyle=\frac{t^{2\mu-1}(-3K+\mu+1)\mathtt{w}{}}{t^{2\mu}-(K-1)\mathtt{w}{}^{2}}.$		(1.59)

In Section 3, we numerically solve this system for specific choices of initial data

$$(\mathtt{z}{},\mathtt{w}{})|_{t={t_{0}}}=(\mathtt{z}{}_{0},\mathtt{w}{}_{0})\quad\text{in $\mathbb{T}{}^{1}$.}$$

Importantly, these choices include initial data for which $\mathtt{w}{}_{0}$ crosses zero at two points in $\mathbb{T}{}^{1}$, and as a consequence, violates the condition (1.8). From our numerical solutions, we observe the following behaviour:

1. (1)

   For all $K\in(1/3,1)$ and all choices of initial data $(\mathtt{z}{}_{0},\mathtt{w}{}_{0})$ that are sufficiently close to homogeneous initial data of either family of solutions (1.5) and (1.9), $\mathtt{z}{}$ and $\mathtt{w}{}$ remain bounded and converge pointwise as $t\searrow 0$.

2. (2)

   For each $K\in(1/3,1)$ and each choice of initial data $(\mathtt{z}{}_{0},\mathtt{w}{}_{0})$ that violates (1.8) and is sufficiently close to homogeneous initial data of the family of solutions (1.5), there exists a $\ell=\ell(K)\in\mathbb{Z}{}_{\geq 0}$ such that

   $$\sup_{x\in\mathbb{T}{}^{1}}\bigl{(}|\partial_{x}^{\ell}\mathtt{z}{}(t,x)|+|\partial_{x}^{\ell}\mathtt{w}{}(t,x)|\bigr{)}\nearrow\infty\quad\text{as $t\searrow 0$.}$$

   This indicates an instability in the $H^{\ell}$-spaces for solutions of (1.58)-(1.59) that is not present, c.f. Theorem 1.2, in solutions generated from initial data satisfying (1.8). We also observe that the integer $\ell$ is a monotonically decreasing function of $K$ with a minimum value of $1$. For the initial data we tested, the blow-up at $t=0$ in the derivatives occurs at a finite set of spatial points.

3. (3)

   For all $K\in(1/3,1)$ and all choices of initial data $(\mathtt{z}{}_{0},\mathtt{w}{}_{0})$ that are sufficiently close to homogeneous initial data of either family of solutions (1.5) and (1.9), solutions to (1.58)-(1.59) are approximated remarkably well, for times sufficiently close to zero, by solutions to the asymptotic system¹⁰¹⁰10Note this system is obtain from (1.58)-(1.59) simply by discarding the terms involving spatial derivatives.

   	$\displaystyle\partial_{t}\tilde{\mathtt{z}{}}{}$	$\displaystyle=0,$		(1.60)
   	$\displaystyle\partial_{t}\tilde{\mathtt{w}{}}{}$	$\displaystyle=\frac{t^{2\mu-1}(-3K+\mu+1)\tilde{\mathtt{w}{}}{}}{t^{2\mu}-(K-1)\tilde{\mathtt{w}{}}{}^{2}},$		(1.61)

   everywhere except, possibly, at a finite set of points where steep gradients form in $z$, which only happens for $K$ large enough and initial data violating (1.8).

4. (4)

   For each $K\in(1/3,1)$ and each choice of initial data $(\mathtt{z}{}_{0},\mathtt{w}{}_{0})$ that violates (1.8) and is sufficiently close to homogeneous initial data of the family of solutions (1.5), the density contrast $\frac{\partial_{x}\rho}{\rho}$ develops steep gradients near a finite number of spatial points where it becomes unbounded as $t\searrow 0$. This behaviour was anticipated by Rendall in [18], and it is not consistent with either the standard picture for inflation in cosmology where the density contrast remains bounded as $t\searrow 0$, or with the behaviour of the density contrast of solutions generated from initial data satisfying (1.8), c.f. Theorem 1.2.