Cosmological dynamics of Higgs potential fine tuning

The Higgs potential appears to be fine-tuned, hence very sensitive to values of other scalar fields that couple to the Higgs. We show that this feature can lead to a new epoch in the early Universe featuring violent dynamics coupling the Higgs to a scalar modulus. The oscillating modulus drives tachyonic Higgs particle production. We find a simple parametric understanding of when this process can lead to rapid modulus fragmentation, resulting in gravitational wave production. A nontrivial equation of state arising from the nonlinear dynamics also affects the time elapsed from inflation to the CMB, influencing fits of inflationary models. Supersymmetric theories automatically contain useful ingredients for this picture.

Figure 1

The shape of the Higgs-moduli potential. The global minimum of the potential is at ($\varphi ={\varphi }_{\mathrm{m}}$, $h\ne 0$), whereas ${\varphi }_0$ is the point of symmetry breaking.

Figure 2

The ratio of the spatially averaged energy density in the Higgs and modulus fields as a function of time, from our lattice simulations. This dynamics is representative of the energy transfer between the modulus and Higgs fields when $b\equiv M^4/2\lambda f^2{m}_{\varphi }^2\sim \mathcal{O}[1]$. For this plot we have chosen $\mathrm{\Delta }={10}^{-6}$, $M^2/m_{\varphi }^2={10}^2$, $M/f={10}^{-13}$ and $\lambda \sim {10}^{-24}$, which corresponds to $b=0.9$. We have confirmed that changing the parameters (for example, increasing $\lambda$ by 6 orders of magnitude), while keeping $b\sim 1$ fixed, does not qualitatively change our results.

Figure 3

The evolution of the normalized fields power spectra for the model with $\mathrm{\Delta }={10}^{-6}$, $b=0.9$, $q={10}^2$, $f=m_{\mathrm{pl}}$. The normalized power spectrum of a field $F(\mathbf{x})$ is $P_F(k)\equiv {\varphi }_{\mathrm{osc}}^{-2}(d/d\ln k)\overline{F^2(\mathbf{x})}$, where ${\varphi }_{\mathrm{osc}}$ is the amplitude of the background modulus oscillations. For this normalization, when $P_{\varphi }(k)=\mathcal{O}(1)$, the modulus becomes inhomogeneous. Initially, the tachyonic instability in the Higgs is closely followed by excitations in the modulus (due to rescattering). Comoving modes $k<m_{\varphi }{q}^{1/2}$ grow exponentially. At the third oscillation of the modulus, backreaction takes place. The spectra then settle down and power slowly propagates towards higher comoving modes.

Figure 4

Snapshots of the values of the modulus (first row) and Higgs (second row) fields on an arbitrary two-dimensional slice through the three-dimensional simulation box at four different times (the spatial coordinates are comoving). Around the time of backreaction, $t\approx 23m_{\varphi }^{-1}$ (second column), the Higgs field forms domains (“bubbles”) with $h=\pm \sqrt{2|\varphi |f}/q$. They disappear within $\mathrm{\Delta }t\sim 10m^{-1}$, due to collisions, as well as oscillations of the remnant $\varphi$ condensate. The parameters we use are $\mathrm{\Delta }={10}^{-6}$, $b=0.9$, with $q={10}^2$, $M={10}^{-13}{m}_{\mathrm{pl}}$, $f=m_{\mathrm{pl}}$.

Figure 5

Left panel: Evolution of $w$ for the Higgs-modulus system for different values of the fragmentation efficiency parameter $b\equiv M^4/2\lambda f^2{m}_{\varphi }^2$ with tuning $\mathrm{\Delta }={10}^{-6}$. For $b\sim \mathcal{O}[1]$, $1/4\lesssim w\lesssim 1/3$ is attained after fragmentation (orange curve). Smaller $b$ yields smaller late time $w$, with continued adiabatic evolution. In the untuned case ($\mathrm{\Delta }\sim \mathcal{O}[1]$, not shown above) and $b\ne 1$, we get $w\approx 0$. Right panel: For fixed $b=0.9$, varying $q=M^2/m_{\varphi }^2$ affects when $1/4\lesssim w\lesssim 1/3$ is attained. For all curves, we have averaged energy densities and pressures spatially over the simulation box and temporally over fast oscillations.

Figure 6

Dashed orange curve with $N_{\text{mod}}=0$: The gravitational waves (GWs) power spectrum today, generated by the nonlinear dynamics at $t\approx 70m_{\varphi }^{-1}$ (assuming $\mathrm{\Delta }={10}^{-6}$, $b=0.9$, $q={10}^2$, $M/f={10}^{-13}$). The GWs on intermediate frequencies are generated by the slow propagation of power towards smaller comoving scales after backreaction; see Fig. 3. Two paler dashed orange curves with $N_{\text{mod}}>0$: Rescaled versions of the top one, assuming $w_{\text{mod}}=0$. Solid black curve: Planned sensitivity of the fifth observational run of the aLIGO-AdVirgo Collaboration [17].

Figure 7

The instability chart featuring the real part of the Floquet exponent normalized by the modulus mass (left) and the Hubble rate (right), characterizing the Higgs particle production rate. When ${\varphi }_{\mathrm{in}}\sim f$, Higgs particle production is expected for $q>1$. In FRW space-time $k_{\mathrm{phys}}=k/a(t)$, implying that a given comoving mode flows towards the bottom left corner of the chart as the universe expands, as indicated with the white lines in the second chart. Note that particle production is efficient if $|\Re ({\mu }_k)|/H\sim q{m}_{\mathrm{pl}}/f\gg 1$.

Figure 8

The evolution of the equation of state, $w$, and the ratio of the mean Higgs and modulus densities, ${\rho }_h/{\rho }_{\varphi }$. After backreaction, for $q{m}_{\mathrm{pl}}/f>{10}^2$, there is a short-lived oscillatory phase. Despite this curious behavior, $w$ settles to a constant value around 0.3. We have chosen parameters such that $b=0.9$, $\mathrm{\Delta }={10}^{-6}$ in all cases. The grey and orange curves are obtained by averaging over space, with additional averaging over fast oscillations for the orange curves.

Figure 9

The growth in the amplitude of the GW power spectrum from the end of inflation to $t\approx 70m_{\varphi }^{-1}$ (with $b=0.9$, $q={10}^2$, $f=m_{\mathrm{pl}}$). The curves are output at time intervals $\mathrm{\Delta }t=6m_{\varphi }^{-1}$.

Figure 10

The lower bound on $m_{\varphi }$ as a function of $n_s$ (left) and $r$ (right) with the inflation model in Eq. (c17) and $\alpha =1$. The red solid and green dotted lines correspond to $w_{\text{mod}}=0$ and 0.1, respectively. In the left panel, the light blue shaded region corresponds to the current $1\sigma$ bounds on $n_s$ from Planck $\mathrm{TT}+\mathrm{lowP}+\mathrm{lensing}$. The narrower darker blue shaded region corresponds to the $1\sigma$ bounds of a future CMB experiment of $n_s$ with sensitivity $\pm 2\times {10}^{-3}$ [30], assuming the same central value as Planck. In the right panel, the blue shaded region corresponds to the $1\sigma$ bounds of a future CMB experiment of $r$ with sensitivity $\pm 5\times {10}^{-4}$ [30], assuming a measured central value of $r$ being 0.085.