On the viability of f(Q) gravity models

<jats:title>Abstract</jats:title>
<jats:p>In general relativity, the contracted Bianchi identity makes the field equation compatible with the energy conservation, likewise in <jats:italic>f</jats:italic>(<jats:italic>R</jats:italic>) theories of gravity. We show that this classical phenomenon is not guaranteed in the symmetric teleparallel theory, and rather generally <jats:italic>f</jats:italic>(<jats:italic>Q</jats:italic>) model specific. We further prove that the energy conservation criterion is equivalent to the affine connection’s field equation of <jats:italic>f</jats:italic>(<jats:italic>Q</jats:italic>) theory, and except the <jats:inline-formula>
<jats:tex-math><?CDATA $f(Q) = \alpha Q+\beta$?></jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mml:mi>f</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>Q</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>=</mml:mo>
<mml:mi>α</mml:mi>
<mml:mi>Q</mml:mi>
<mml:mo>+</mml:mo>
<mml:mi>β</mml:mi>
</mml:math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cqgaccef7ieqn1.gif" xlink:type="simple" />
</jats:inline-formula> model, the non-linear <jats:italic>f</jats:italic>(<jats:italic>Q</jats:italic>) models do not satisfy the energy conservation or, equivalently the second field equation in every spacetime geometry; unless <jats:italic>Q</jats:italic> itself is a constant. So the problem is deep-rooted in the theory, several physically motivated examples are provided in the support.</jats:p>