| Summary: | It is shown that the inflationary model is the result of the symmetry of the generalized <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>R</mi><mo>,</mo><mi>T</mi><mo>,</mo><mi>X</mi><mo>,</mo><mi>φ</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-cosmological model using the Noether symmetry. It leads to a solution, a particular case of which is Starobinsky’s cosmological model. It is shown that even in the more particular case of cosmological models <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>R</mi><mo>,</mo><mi>X</mi><mo>,</mo><mi>φ</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><mi>X</mi><mo>,</mo><mi>φ</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> the Monge–Ampère equation is still obtained, one of the solutions including the Starobinsky model. For these models, it is shown that one can obtain both power-law and exponential solutions for the scale factor from the Euler–Lagrange equations. In this case, the scalar field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>φ</mi></semantics></math></inline-formula> has similar time dependences, exponential and exponential. The resulting form of the Lagrangian of the model allows us to consider it as a model with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>R</mi><mn>2</mn></msup></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>X</mi><mn>2</mn></msup></semantics></math></inline-formula>. However, it is also shown that previously less studied models with a non-minimal relationship between <i>R</i> and <i>X</i> are important, as one of the possible models. It is shown that in this case the power-law model can have a limited evolutionary period with a negative value of the kinetic term.
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