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Phys. J. Plus"],"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We investigate how the cosmological equation of state can be used for scrutinizing extended theories of gravity, in particular, the Palatini <jats:inline-formula><jats:alternatives><jats:tex-math>$$f({{\\mathcal {R}}})$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>f<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>R<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> gravity. Specifically, the approach consists, at first, in investigating the effective equation of state produced by a given model. Then, the inverse problem can also be considered in view of determining which models are compatible with a given effective equation of state. We consider and solve some cases and show that, for example, power-law models are (the only models) capable of transforming barotropic Equations of State into effective barotropic ones. Moreover, the form of equation of state is preserved (only) for <jats:inline-formula><jats:alternatives><jats:tex-math>$$f({{\\mathcal {R}}})={{\\mathcal {R}}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>f<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>R<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mi>R<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, as expected. In this perspective, modified Equations of State are a feature capable of distinguishing extended gravity with respect to general relativity. We investigate also quadratic and non-homogeneous effective Equations of State showing, in particular, that they contain the Starobinsky model and other ones.<\/jats:p>","DOI":"10.1140\/epjp\/s13360-023-03676-0","type":"journal-article","created":{"date-parts":[[2023,2,27]],"date-time":"2023-02-27T09:02:39Z","timestamp":1677488559000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["The effective equation of state in Palatini $$f({{\\mathcal {R}}})$$ cosmology"],"prefix":"10.1140","volume":"138","author":[{"given":"Stefano","family":"Camera","sequence":"first","affiliation":[]},{"given":"Salvatore","family":"Capozziello","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3159-1272","authenticated-orcid":false,"given":"Lorenzo","family":"Fatibene","sequence":"additional","affiliation":[]},{"given":"Andrea","family":"Orizzonte","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,2,27]]},"reference":[{"key":"3676_CR1","first-page":"110","volume":"6","author":"F Zwicky","year":"1933","unstructured":"F. 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