Revisiting constraints on small scale perturbations from big-bang nucleosynthesis

We revisit the constraints on the small scale density perturbations ($10^4{\mathrm{Mpc}}^{-1}\lesssim k\lesssim 10^5{\mathrm{Mpc}}^{-1}$) from the modification of the freeze-out value of the neutron-proton ratio at the big-bang nucleosynthesis era. Around the freeze-out temperature $T\sim 0.5\mathrm{MeV}$, the universe can be divided into several local patches that have different temperatures since any perturbation that enters the horizon after the neutrino decoupling has not diffused yet. Taking account of this situation, we calculate the freeze-out value in detail. We find that the small scale perturbations decrease the $n$-$p$ ratio in contrast to previous works. With the use of the latest observed ${}^4\mathrm{He}$ abundance, we obtain the constraint on the power spectrum of the curvature perturbations as ${\mathrm{\Delta }}_{\mathcal{R}}^2\lesssim 0.018$ on $10^4{\mathrm{Mpc}}^{-1}\lesssim k\lesssim 10^5{\mathrm{Mpc}}^{-1}$.

Figure 1

Plot of $\mathrm{\Phi }$, ${\mathrm{\Theta }}_0$, ${\mathrm{\Theta }}_1$, and ${\mathrm{\Theta }}_0^2+3{\mathrm{\Theta }}_1^2$ ($0.1\mathrm{MeV}<T<1.5\mathrm{MeV}$), where we set ${\mathrm{\Phi }}_p=1$.

Figure 2

Evolution of $X_n$ without the perturbations ($0.1\mathrm{MeV}<T<1.5\mathrm{MeV}$). In this plot, it can be seen that the freeze-out occurs around $T\sim 0.5\mathrm{MeV}$.

Figure 3

Schematic image of the two patches in the horizon, one of which has the slightly high temperature $T(1+\mathrm{\Theta })$ and the other of which has the slightly low temperature $T(1-\mathrm{\Theta })$.

Figure 4

Evolution of $X_n^{\text{exact}}$ (orange curves) and $X_n^{\mathrm{app}}$ (blue curves) for $\zeta =0.1$ (dotted curves) and 0.2 (solid curves).

Figure 5

Ratio of $X_n^{\text{exact}}(0,\zeta )$ to $X_n^{\mathrm{app}}(T,\zeta )$ with $\zeta =0$, 0.1, and 0.2.

Figure 6

Plot of ${\overline{X}}_n^{\mathrm{ave}}(0,\sigma )$ (blue solid line) and the latest observational bounds (black solid lines) [2].

Figure 7

The ratios of the approximated reaction rates ${\lambda }^{\mathrm{app}}$ to the exact ones ${\lambda }^{\text{exact}}$.