Low scale inflation at high energy colliders and meson factories

Inflation occurring at energy densities less than $({10}^{14}\mathrm{GeV}{)}^4$ produces tensor perturbations too small to be measured by cosmological surveys. However, we show that it is possible to probe low scale inflation by measuring the mass of the inflaton in low energy experiments. Detection prospects and cosmological constraints are determined for low scale quartic hilltop models of inflation paired with a curvaton field, which imprints the spectrum of scalar perturbations observed in large scale structure and on the cosmic microwave background. With cosmological constraints applied, low scale quartic inflation at energies GeV–PeV can be mapped to a MeV–TeV mass inflaton resonance, discoverable through a Higgs portal coupling at upcoming collider and meson decay experiments. It is demonstrated that low scale inflatons can have detectably large couplings to Standard Model particles through a Higgs portal, permitting prompt reheating after inflation, without spoiling, through radiative corrections to the inflaton’s self-coupling, the necessary flatness of a low scale inflationary potential. A characteristic particle spectrum for a quartic inflaton-curvaton pair is identified: to within an order of magnitude, the mass of the curvaton can be predicted from the mass of the inflaton, and vice versa. Low scale inflation Higgs portal sensitivity targets are found for experiments like the LHC, SHiP, BEPC, and KEKB.

Figure 1

Schematic showing inflation driven by a hilltop inflaton, with primordial perturbations provided by a hilltop curvaton. (Ia and Ib): While the inflaton field $\varphi$ slowly rolls to its minimum, the curvaton field is perturbed by de Sitter vacuum fluctuations $\delta \sigma \sim H/2\pi$, where $H$ is the Hubble constant during inflation. (II): The inflaton field rolls to its minimum and decays. (III): Sometime later, when the curvaton energy density is the predominant energy density in the Universe, the curvaton decays. $V_0$ and $V_{0\sigma }$ are the change in potential energy of the inflaton and curvaton, respectively. In viable parameter space studied here, $V_{0\sigma }\ll V_0$.

Figure 2

Parameter space for the simplified quartic hilltop model given in Eq. (3). The requirement that the Universe inflate by 40 $e$-folds and not produce scalar primordial perturbations larger than those observed in the CMB ($P_{\zeta \varphi }\lesssim 2.2\times {10}^{-9}$) excludes the region shaded red. The inflaton’s quartic self-coupling (${\lambda }_{\varphi }$) is fixed in terms of $\mathrm{\Lambda }$ and $V_0$ by requiring that the inflaton’s potential, Eq. (3), be zero at its minimum, yielding the relation given in Eq. (5). The inflaton’s mass ($m_{\varphi }$) and vacuum expectation value ($v_{\varphi }$) at its minimum are indicated by dotted and dotted-dashed lines.

Figure 3

Constraints for inflationary energy density ($V_0$) and initial reheating temperature ($T_{\mathrm{re}}$) for a cutoff $\mathrm{\Lambda }=10^{19}\mathrm{GeV}$, assuming quartic hilltop inflation, defined in Eq. (3). The solid pink, dashed green, dotted blue, and dotted-dashed black lines indicate a postinflation average equation of state of $w_{\mathrm{re}}=1$, $\frac{1}{3}$, 0, $-\frac{1}{3}$, respectively, as described in the text. As in Fig. 2, the region shaded red is excluded because the inflaton produces primordial perturbations that are too large. The bottom horizontal line marks a big bang nucleosynthesis (BBN) reheat temperature; space below this line is excluded. The upper horizontal marks the approximate electroweak symmetry breaking temperature (100 GeV). As $N_{*}$ decreases in each panel, so does the maximum allowed reheating temperature and $V_0^{1/4}$ values, contained within a wedge of sensible equation of state values, $w_{\mathrm{re}}=[0,\frac{1}{3}]$. Note that this leads to substantially different y-axis ($T_{\mathrm{re}}$) and x-axis ($V_0^{1/4}$) ranges, as $N_{*}$ is varied.

Figure 4

The same as in Fig. 3 but for $\mathrm{\Lambda }=10^{15}\mathrm{GeV}$.

Figure 5

The same as in Fig. 3 but for $\mathrm{\Lambda }=10^{12}\mathrm{GeV}$.

Figure 6

The initial curvaton field value is plotted against the energy density during inflation, $V_0^{1/4}$. The upper blue line shows what values of the initial curvaton field value (${\sigma }_{*}$) are consistent with Planck observations of $n_s^{*}$ and $A_s^{*}$, for the quartic hilltop curvaton potential, Eq. (18). As explained in the text, the required value of ${\sigma }_{*}$ is independent of $\mathrm{\Lambda }$ and $N_{*}$. Allowing variations corresponding to Planck’s $1\sigma$ bounds on $n_s^{*}$ and $A_s^{*}$ has little effect in this plot, generating a thickness less than that of the blue line. The orange line is a lower bound for ${\sigma }_{*}$, such that $\sigma$ is slowly rolling during inflation rather than dominated by quantum fluctuations—the quartic curvaton satisfies this requirement.

Figure 7

The minimum number of $e$-folds during reheating, $N_{\mathrm{re}}$, as a function of $w_{\mathrm{re}}$, the average equation of state during reheating, using Planck’s constraint on isocurvature perturbations [6], ${\beta }_{\mathrm{iso}}\le 0.0013$. The region below the curve is excluded. A stiffer equation of state during reheating ($w_{\mathrm{re}}\gg 0$) implies that the period of reheating must last longer, because it will take the curvaton energy density longer to become the predominant energy density in the Universe.

Figure 8

The relationship among the energy density during inflation ($V_0$), the average equation of state during reheating ($w_{\mathrm{re}}$), and the number of $e$-folds during reheating ($N_{\mathrm{re}}$), for a low scale quartic inflaton paired with a quartic curvaton and a given value of the cutoff $\mathrm{\Lambda }$, as detailed in this section. The cosmological consistency conditions outlined in Sec. 3 uniquely determine the number of $e$-folds ($N_{*}$) for each point in parameter space. The parameter space is bounded at each edge by various factors. Top ($P_{\zeta \varphi }<10^{-9}$): The inflaton must not overproduce perturbations, Eq. (11). Bottom (No curvaton inflation bound): The curvaton field must not produce more than an $e$-fold of inflation, Eq. (34). Left (Isocurvature bound): Reheating must last long enough that the curvaton comes to dominate (constitute $\ge 99.1\%$) of the energy density of the Universe, Eqs. (38) and (41). Note that the bound in the “left” bound assumes that the curvaton equation of state after inflation is immediately matterlike, i.e. $w_{\sigma ,ei}\simeq 0$; relaxing this assumption would allow for a broader range of parameters. The viable parameter space is shown for cutoffs $\mathrm{\Lambda }=10^{19}$ and $10^{15}\mathrm{GeV}$, as indicated.

Figure 9

Similar to Fig. 2, but with the addition of constraints from requiring a consistent curvaton cosmology. The top red line is again an upper bound, from requiring that the inflaton not overproduce perturbations, Eq. (11). For this bound, the number of $e$-folds has been set with Eq. (42), which was derived using results in Sec. 3. As in Fig. 2, the dotted black lines show the implied mass of the inflaton. All the other lines are lower bounds on viable parameter space. The blue dashed line marks the constraint that $\sigma$ not generate a second wave of inflation which lasts longer than an $e$-fold, Eq. (34), using Eq. (43) to set the maximum number of $e$-folds. The region above the pink dotted-dashed line indicates parameter space where $V_0>10V_{0\sigma }$.

Figure 10

Parameter space for low scale inflation, which reheats the Universe through a Higgs portal coupling. Constraints from meson decay and collider searches are indicated with thick dashed lines. Indirect constraints from the muon’s lifetime along with $W$, $Z$-boson masses ($\mathrm{\Delta }r$) are indicated with a thin orange line, and the indirect constraint from the Higgs boson’s decay width measured at the LHC is indicated with a thin gray line. The long-dashed blue line excludes parameter space where the Higgs-inflaton coupling (${\lambda }_{\varphi \mathrm{h}}$) spoils the flatness of the inflaton’s potential during inflation. The dotted pink lines show parameter space where $\varphi$ decays promptly at the end of inflation for $\mathrm{\Lambda }=10^{19}\mathrm{GeV}$ and ${\lambda }_{\varphi }={10}^{-13}$, where $\varphi$ decays when the energy density of the Universe is ${\sim (100\mathrm{GeV})}^4$, and excludes where $\varphi$ decays after big-bang nucleosynthesis. On top of the plot, the correspondence between the energy scale during inflation and the quartic inflaton mass is indicated. The range of inflationary energy scales is derived from relations shown in Fig. 2; these ranges hold for a generic quartic inflaton, irrespective of the possible addition of a curvaton. With a curvaton model specified, the scale of inflation is more tightly predicted; see Table 1.

Figure 11

The bounds and prospects are the same as in Fig. 10, but here we show example quartic curvaton parameter points (Curv 1, Curv 2, Curv 3), alongside the corresponding predicted quartic inflaton parameter space (Inf 1, Inf 2, Inf 3), where these have been found using results in Sec. 5. Note that the curvaton and inflaton parameters roughly match those shown in Table 1.

Figure 12

The bounds and prospects are the same as in Fig. 10, but here we show example quartic inflaton parameter points (Inf 1, Inf 2, Inf 3), alongside the corresponding predicted quartic curvaton parameter space (Curv 1, Curv 2, Curv 3), where these have been found using results in Sec. 5. Note that the inflaton and curvaton parameters roughly match those shown in Table 1.