Large blue isocurvature spectral index signals time-dependent mass

We show that if a spectator linear isocurvature dark matter field degree of freedom has a constant mass through its entire evolution history, the maximum measurable isocurvature spectral index that is consistent with the current tensor-to-scalar ratio bound of about $r\lesssim 0.1$ is about $n_I\lesssim 2.4$, even if experiments can be sensitive to a ${10}^{-6}$ contamination of the predominantly adiabatic power spectrum with an isocurvature power spectrum at the shortest observable length scales. Hence, any foreseeable future measurement of a blue isocurvature spectral index larger than $\sim 2.4$ may provide nontrivial evidence for dynamical degrees of freedom with time-dependent masses during inflation. The bound is not sensitive to the details of the reheating scenario and can be made mildly smaller if $r$ is better constrained in the future.

Figure 1

The maximum measurable spectral index $n_I-1$ assuming $\{E_{k_{\max }}=1,k_{\max }/k_{\min }=10^5;r_b={10}^{-1},{10}^{-3};n_{\mathcal{O}}=5,,6;S=(8\pi {)}^{-1},{10}^{-2}(8\pi {)}^{-1}\}$ is plotted as a function of $T_{\mathrm{RH}}$. Left: The bound on the tensor to scalar ratio $r_b$ has been taken to be ${10}^{-1}$. The maximum spectral index with the amplitude parametrized by $\{E_{k_{\max }}=1,k_{\max }/k_{\min }=10^5\}$ is around $n_I=2.11$. For $T\lesssim 10^9\mathrm{GeV}$, the lower solid curve corresponds to the reheating nonrenormalizable operator dimension of $n_{\mathcal{O}}=5$ while the upper solid curve corresponds to the case of the nonrenormalizable decay operator dimension of $n_{\mathcal{O}}=6$. For $T\gtrsim 10^9\mathrm{GeV}$, the two curves merge. The dashed curve corresponds to weakening the coefficient of the nonrenormalizable operator by a factor of 10. The vertical curve on the left portion of the boundary curves occurs because the expansion rate there is too small in that parametric regime to produce measurable isocurvature perturbations (i.e. constraint 8). The dotted curve corresponds to the evaluation of the fiducial spectral index at $k_{\max }$ instead of $k_{\min }$. The correction is small (except at the highest $T_{\mathrm{RH}}$ where the dip occurs) because constraint 7 is not saturated for $r_b=0.1$. Right: Similar to the left plot except with a probably possible future bound of $r_b={10}^{-3}$.

Figure 2

The maximum measurable spectral index $n_I-1$ as a function of $T_{\mathrm{RH}}\in [100,5\times 10^{15}]\mathrm{GeV}$ assuming an experimental sensitivity of $E_{k_{\max }}={10}^{-3}$ corresponding to resolving ${\mathrm{\Delta }}_s^2/{\mathrm{\Delta }}_{\zeta }^2$ to $O({10}^{-4}\%)$ at about 1 Mpc length scale. The rest of the parameters are set at $\{k_{\max }/k_{\min }=10^5;r_b={10}^{-1},{10}^{-3};n_{\mathcal{O}}=5,6;S=(8\pi {)}^{-1},{10}^{-2}(8\pi {)}^{-1}\}$. Left: The bound on the tensor-to-scalar ratio $r$ has been taken to be ${10}^{-1}$. The lower solid curve for $T_{\mathrm{RH}}\lesssim 10^9\mathrm{GeV}$ corresponds to the reheating nonrenormalizable operator dimension of $n_{\mathcal{O}}=5$ just as in Fig. 1. The dotted curve corresponds to evaluation of the fiducial spectral index at $k_{\max }$ instead of $k_{\min }$. As in Fig. 1, the correction is small because constraint 7 is not saturated even for $r_b=0.1$. As expected from increasing the experimental resolution by $10^3$, the maximum measurable spectral index has only gone up mildly to $n_I=2.25$ (from $n_I=2.12$). Note that unlike in Fig. 1, the bounds for $n_{\mathcal{O}}=6$ end at $T_{\mathrm{RH}}=10^2\mathrm{GeV}$ because we simply truncated the plot there (and not because ${\omega }_{\chi }>1$ there). The dashed curve corresponds to weakening the coefficient of the nonrenormalizable operator by a factor of 10 just as in Fig. 1. Right: Similar to the left plot except that $r_b$ has been set to ${10}^{-3}$.

Figure 3

Similar to the left panel in Fig. 2 except with $k_{\max }/k_{\min }=10^7$ (i.e. $k_{\max }$ is at the scale of 10 kpc). The bound on the maximum spectral index $n_I$ is only logarithmically sensitive to $k_{\max }/k_{\min }$ as it is now about 2.35 instead of 2.25. The features of the plots are explained as in previous figures. As before, the dotted curve corresponds to the evaluation of the spectral index with the fiducial $k$ value of $k_{\max }$ instead of $k_{\min }$.

Figure 4

$m/H(t_{k_{\max }})$ is plotted as a function of $n_I-1$ defined according to Eq. (8) for the severest parametric choices of $r_b=0.1$ and $k_{\max }/k_{\min }=10^7$. This shows that constraint 13 is satisfied for $n_I-1\lesssim 1.6$.