The paper by Salvatore Capozziello explores the possibility of achieving quintessence, a concept often associated with dark energy, through higher-order theories of gravity. Quintessence is a time-varying, spatially inhomogeneous component of cosmic density with negative pressure, typically described by a scalar field minimally coupled to gravity. The author investigates whether this quintessential scheme can be realized geometrically by considering higher-order theories of gravity, where the gravitational Lagrangian is not restricted to a linear function of the Ricci scalar. In the first section, Capozziello derives the Friedmann-Einstein equations for generic fourth-order models, showing that these theories naturally exhibit inflationary behaviors and can be made to match observational data. The second section discusses the conditions necessary to obtain quintessence, specifically focusing on the curvature pressure and density. The third section presents exact solutions to these conditions, demonstrating that accelerated expansion can be achieved for certain forms of the function \( f(R) \) and scale factor \( a(t) \). The paper concludes that curvature quintessence can be recovered in the framework of higher-order theories of gravity, suggesting that quintessence might be related to effective theories of quantum gravity. However, further improvements and comparisons with observations are needed to constrain the form of \( f(R) \) and validate this approach.The paper by Salvatore Capozziello explores the possibility of achieving quintessence, a concept often associated with dark energy, through higher-order theories of gravity. Quintessence is a time-varying, spatially inhomogeneous component of cosmic density with negative pressure, typically described by a scalar field minimally coupled to gravity. The author investigates whether this quintessential scheme can be realized geometrically by considering higher-order theories of gravity, where the gravitational Lagrangian is not restricted to a linear function of the Ricci scalar. In the first section, Capozziello derives the Friedmann-Einstein equations for generic fourth-order models, showing that these theories naturally exhibit inflationary behaviors and can be made to match observational data. The second section discusses the conditions necessary to obtain quintessence, specifically focusing on the curvature pressure and density. The third section presents exact solutions to these conditions, demonstrating that accelerated expansion can be achieved for certain forms of the function \( f(R) \) and scale factor \( a(t) \). The paper concludes that curvature quintessence can be recovered in the framework of higher-order theories of gravity, suggesting that quintessence might be related to effective theories of quantum gravity. However, further improvements and comparisons with observations are needed to constrain the form of \( f(R) \) and validate this approach.