| Summary: | By employing Tsallis’ extensive but non-additive <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>δ</mi></semantics></math></inline-formula>-entropy, we formulate the first two laws of thermodynamics for gravitating systems. By invoking Carathéodory’s principle, we pay particular attention to the integrating factor for the heat one-form. We show that the latter factorizes into the product of thermal and entropic parts, where the entropic part cannot be reduced to a constant, as is the case in conventional thermodynamics, due to the non-additive nature of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mi>δ</mi></msub></semantics></math></inline-formula>. The ensuing two laws of thermodynamics imply a Tsallis cosmology, which is then applied to a radiation-dominated universe to address the Big Bang nucleosynthesis and the relic abundance of cold dark matter particles. It is demonstrated that the Tsallis cosmology with the scaling exponent <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>δ</mi></semantics></math></inline-formula>∼1.499 (or equivalently, the anomalous dimension <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Δ</mo></semantics></math></inline-formula>∼0.0013) consistently describes both the abundance of cold dark matter particles and the formation of primordial light elements, such as deuterium <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow></mrow><mn>2</mn></msup><mspace width="-0.166667em"></mspace><mi>H</mi></mrow></semantics></math></inline-formula> and helium <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow></mrow><mn>4</mn></msup><mspace width="-0.166667em"></mspace><mi>H</mi><mi>e</mi></mrow></semantics></math></inline-formula>. Salient issues, including the zeroth law of thermodynamics for the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>δ</mi></semantics></math></inline-formula>-entropy and the lithium <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow></mrow><mn>7</mn></msup><mspace width="-0.166667em"></mspace><mi>L</mi><mi>i</mi></mrow></semantics></math></inline-formula> problem, are also briefly discussed.
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