Small-scale nonlinear dynamics of K-mouflage theories

We investigate the small-scale static configurations of K-mouflage models defined by a general function $K(\chi )$ of the kinetic terms. The fifth force is screened by the nonlinear K-mouflage mechanism if $K^{\prime }(\chi )$ grows sufficiently fast for large negative $\chi$. In the general nonspherically symmetric case, the fifth force is not aligned with the Newtonian force. For spherically symmetric static matter density profiles, we show that the results depend on the potential function $W_{-}(y)=y{K}^{\prime }(-y^2/2)$; i.e., $W_{-}(y)$ must be monotonically increasing to $+\infty$ for $y\ge 0$ to guarantee the existence of a single solution throughout space for any matter density profile. Small radial perturbations around these static profiles propagate as travelling waves with a velocity greater than the speed of light. Starting from vanishing initial conditions for the scalar field and for a time-dependent matter density corresponding to the formation of an overdensity, we numerically check that the scalar field converges to the static solution. If $W_{-}$ is bounded, for high-density objects there are no static solutions throughout space, but one can still define a static solution restricted to large radii. Our dynamical study shows that the scalar field relaxes to this static solution at large radii, whereas spatial gradients keep growing with time at smaller radii. If $W_{-}$ is not bounded but nonmonotonic, there is an infinite number of discontinuous static solutions. However, the Klein–Gordon equation is no longer a well-defined hyperbolic equation, which leads to complex characteristic speeds and exponential instabilities. Therefore, these discontinuous static solutions are not physical, and these models are not theoretically sound. Such K-mouflage scenarios provide an example of theories that can appear viable at the cosmological level, for the cosmological background and perturbative analysis, while being meaningless at a nonlinear level for small-scale configurations. This shows the importance of small-scale nonlinear analysis of screening models. All healthy K-mouflage models should satisfy $K^{\prime }>0$, and $W_{\pm }(y)=y{K}^{\prime }(\pm y^2/2)$ are monotonically increasing to $+\infty$ when $y\ge 0$.

Figure 1

Time evolution of the scalar field derivatives and of the characteristic speeds, for the polynomial kinetic function (82) and the Gaussian matter profiles (77), with $R/R_K=0.5$ (left panels) and $R/R_K=1.5$ (right panels), starting with the initial condition $u=v=0$ at $\tau =0$. We choose a formation time scale ${\tau }_{\mathrm{f}}$ in Eq. (80) with ${\alpha }_{\mathrm{f}}=0.1$. Upper panels: time derivative $u(x,\tau )=\partial \varphi /\partial \tau$ as a function of radius, at times $\tau =0.25,0.5,1$, and $2\times {\tau }_{\mathrm{f}}$. Middle panels: spatial derivative $v(x,\tau )=\partial \varphi /\partial x$. The solid lines are the quasistatic profiles defined by Eq. (81) at $\tau =0.25,0.5$ and $1\times {\tau }_{\mathrm{f}}$ [these quasistatic solutions remain equal to the final static solution defined by Eqs. (77) and (81) after ${\tau }_{\mathrm{f}}$]. Lower panels: characteristic speeds $c_{\pm }(x,\tau )$ from Eq. (70). The solid lines are the results (72) on the final static profile (77).

Figure 2

Time evolution of the scalar field derivatives and of the characteristic speeds, for the ${\mathrm{DBI}}^{+}$ kinetic function (83) and the Gaussian matter profiles (77), with $R/R_K=0.5$ (left panels) and $R/R_K=1.5$ (right panels), starting with the initial condition $u=v=0$ at $\tau =0$ and with ${\alpha }_{\mathrm{f}}=0.1$. Upper panels: time derivative $u(x,\tau )=\partial \varphi /\partial \tau$ as a function of radius, at times $\tau =0.25,0.5,1$, and $2\times {\tau }_{\mathrm{f}}$. Middle panels: spatial derivative $v(x,\tau )=\partial \varphi /\partial x$. The solid lines are the quasistatic profiles defined by Eq. (81). Lower panels: characteristic speeds $c_{\pm }(x,\tau )$ from Eq. (70). The solid lines are the results (72) on the final static profile.

Figure 3

Time evolution of the scalar field derivatives and of the characteristic speeds, for the ${\mathrm{DBI}}^{-}$ kinetic function (84) and the Gaussian matter profiles (77), with $R/R_K=0.5$ (left panels) and $R/R_K=1.5$ (right panels), starting with the initial condition $u=v=0$ at $\tau =0$ and with ${\alpha }_{\mathrm{f}}=0.1$. Upper panels: time derivative $u(x,\tau )=\partial \varphi /\partial \tau$ as a function of radius, at times $\tau =0.25,0.5,1$, and $2\times {\tau }_{\mathrm{f}}$. Middle panels: spatial derivative $v(x,\tau )=\partial \varphi /\partial x$. The solid lines are the quasistatic profiles defined by Eq. (81). Lower panels: characteristic speeds $c_{\pm }(x,\tau )$ from Eq. (70). The solid lines are the results (72) on the final static profile.