Non-metricity with boundary terms: 𝖿(𝖰,𝖢) gravity and cosmology

<jats:title>Abstract</jats:title>
<jats:p>We formulate <jats:italic>f</jats:italic>(<jats:italic>Q,C</jats:italic>) gravity and cosmology. Such a
construction is based on the symmetric teleparallel geometry, but apart
form the non-metricity scalar <jats:italic>Q</jats:italic> we incorporate in the Lagrangian the
boundary term <jats:italic>C</jats:italic> of its difference from the standard Levi-Civita Ricci scalar
 <jats:italic>R̊</jats:italic>. We extract the general metric and affine connection field equations,
we apply them at a cosmological framework, and
adopting three different types of symmetric teleparallel affine connections
we obtain the modified Friedmann equations. As we show, we acquire an
effective dark-energy sector of geometrical origin, which can lead to
interesting cosmological phenomenology.
Additionally,
we may obtain an effective interaction between matter and dark energy.
Finally, examining a specific model, we
show that we can obtain the usual thermal
history of the universe, with the sequence of matter and dark-energy epochs,
while the effective dark-energy equation-of-state
parameter can be quintessence-like, phantom-like, or cross the phantom-divide
during evolution.</jats:p>