## Author(s)

Bhaumik, Nilanjandev, Haque, Md Riajul, Jain, Rajeev Kumar, Lewicki, Marek## Abstract

Ultra-low mass primordial black holes (PBH), briefly dominating the expansion of the universe, would leave detectable imprints in the secondary stochastic gravitational wave background (SGWB). Such a scenario leads to a characteristic doubly peaked spectrum of SGWB and strongly depends on the Hawking evaporation of such light PBHs. However, these observable signatures are significantly altered if the memory burden effect during the evaporation of PBHs is taken into account. We show that for the SGWB induced by PBH density fluctuations, the memory burden effects on the Hawking evaporation of ultra-low mass PBHs can mimic the signal arising due to the non-standard reheating epoch before PBH domination. Finally, we point out that this degeneracy can be broken by the simultaneous detection of the first peak in the SGWB, which is typically induced by the inflationary adiabatic perturbations.

## Figures

The comparison of our adopted interpolation function (Eq.~\eqref{Spl}) and numerical results for $S_{\rm plateau}$ for different values of $w$.

\textbf{Left panel:} The dimensionless spectral energy density of the induced stochastic GW background (ISGWB) is plotted as a function of frequency for the initial PBH mass $\MPBH =5 \times 10^7 {\rm g}$ and $\beta_f= 6 \times 10^{-9}$ with different reheating histories. Note that we use a very tiny deviation in the EOS of the universe before PBH domination, which leads to significant changes in the ISGWB spectra. \textbf{Right panel:} We have shown the effect of memory burden on the ISGWB spectrum for PBH parameters $\MPBH =7 \times 10^5 {\rm g}$ and $\beta_f=5 \times 10^{-8}$ and compared it with the standard case. To do that, we assume standard RD before PBH domination.

\textbf{Left panel:} The dimensionless spectral energy density of the induced stochastic GW background (ISGWB) is plotted as a function of frequency for the initial PBH mass $\MPBH =5 \times 10^7 {\rm g}$ and $\beta_f= 6 \times 10^{-9}$ with different reheating histories. Note that we use a very tiny deviation in the EOS of the universe before PBH domination, which leads to significant changes in the ISGWB spectra. \textbf{Right panel:} We have shown the effect of memory burden on the ISGWB spectrum for PBH parameters $\MPBH =7 \times 10^5 {\rm g}$ and $\beta_f=5 \times 10^{-8}$ and compared it with the standard case. To do that, we assume standard RD before PBH domination.

We plot the ISGWB spectral energy density with (solid red: $PS1$) and without (dashed red) memory burden effect and also plot SGWB for the degenerate set of parameters (solid blue: $PS2$) with different background equations of state before PBH domination. Evidently, in these cases of the high-frequency peak or the PBH density fluctuation peak, PBH formation during reheating and evaporation due to standard Hawking evaporation can mimic the effect of memory burden in the case of PBH formation during RD. However, this degeneracy is broken if we can simultaneously also detect the amplitude of the first peak or the inflationary adiabatic SGWB peak.

The rainbow-coloured contours show the values of the EOS $w_2$ of $PS2$, required to mimic the memory burden effect as a function of memory burden parameter $n_1$ and PBH parameter $M_{\rm PBH1}$ for a few fixed values of the other memory burden parameter $q_1$ of $PS1$, which directly follows from Eq.~\eqref{con1} and \eqref{con2}. Colours in the contours refer to different values of $w_2$ as listed in the colour bar on the right. As we set the range of this plot $0<w_2<1$, we can see that for a fixed value of $q_1$, it is not possible to cover the full range of $n_1$ with the variation of reheating history parameter $w_2$ and PBH mass $M_{\rm PBH1}$. It is clear, however, that variations of $n_1$, $q_1$, and $M_{\rm PBH1}$ of $PS1$ can mimic the effects of any values of $w_2$ of $PS2$. We also plot the light yellow shaded region where the required PBH mass of $PS2$, $M_{\rm PBH2} > 5\times 10^8 {\rm g}$, and thus this part of the parameter space is excluded from BBN bound.

The rainbow-coloured contours show the values of the EOS $w_2$ of $PS2$, required to mimic the memory burden effect as a function of memory burden parameter $n_1$ and PBH parameter $M_{\rm PBH1}$ for a few fixed values of the other memory burden parameter $q_1$ of $PS1$, which directly follows from Eq.~\eqref{con1} and \eqref{con2}. Colours in the contours refer to different values of $w_2$ as listed in the colour bar on the right. As we set the range of this plot $0<w_2<1$, we can see that for a fixed value of $q_1$, it is not possible to cover the full range of $n_1$ with the variation of reheating history parameter $w_2$ and PBH mass $M_{\rm PBH1}$. It is clear, however, that variations of $n_1$, $q_1$, and $M_{\rm PBH1}$ of $PS1$ can mimic the effects of any values of $w_2$ of $PS2$. We also plot the light yellow shaded region where the required PBH mass of $PS2$, $M_{\rm PBH2} > 5\times 10^8 {\rm g}$, and thus this part of the parameter space is excluded from BBN bound.

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