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In this study, we extend the standard cosmological model within the quadraticenergy-momentum squared gravity (qEMSG) framework, introducing a nonminimalinteraction between the usual material field ($T_{\mu\nu}$) and itsaccompanying quadratic energy-momentum squared field (qEMSF, $T_{\mu\nu}^{\rmqEMSF}$), defined by $f(\mathbf{T}^2) = \alpha \mathbf{T}^2$ with$\mathbf{T^2}=T_{\mu\nu}T^{\mu\nu}$. Focusing on high energy scales relevant tobig bang nucleosynthesis (BBN), we employ $^4$He abundance to constrain theparameter $\alpha$. Our analysis selects the radiation-dominated universesolution compatible with the standard cosmological model limit as $\alpha\rightarrow 0$ and reveals that qEMSF interaction model can modify theradiation energy density's evolution, potentially altering neutron-protoninterconversion rates and consequently affecting $^4$He abundance in variousways. We establish the most stringent cosmological bounds on $\alpha$: $(-8.81\leq \alpha \leq 8.14) \times 10^{-27} \, \mathrm{eV}^{-4}$ (68\% CL) from Aver\textit{et al.}'s primordial $^4$He abundance measurements, aligning with$\alpha=0$. Additionally, $(3.48 \leq\alpha \leq 4.43)\,\times 10^{-27}\rm{eV}^{-4}$ (68\% CL) from Fields \textit{et al.}'s estimates, utilizing thePlanck-CMB estimated baryon density within the standard cosmological modelframework, diverges from $\alpha=0$, thereby lending support to the qEMSFinteraction model. The study also highlights the bidirectional nature ofenergy-momentum/entropy transfer in qEMSF interaction model, depending on thesign of $\alpha$. The implications of qEMSF in the presence of additionalrelativistic relics are also explored, showcasing the model's potential toaccommodate deviations from standard cosmology and the Standard Model ofparticle physics.

Quadratic energy-momentum squared gravity: constraints from big bang nucleosynthesis