de Sitter vacua in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi mathvariant="normal">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>d</mml:mi><mml:mo>,</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy="false">)</m

We perform a thorough analysis of de Sitter solutions in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi mathvariant="normal">O</a:mi><a:mo stretchy="false">(</a:mo><a:mi>d</a:mi><a:mo>,</a:mo><a:mi>d</a:mi><a:mo stretchy="false">)</a:mo></a:math> invariant cosmologies. Starting with a homogeneous and isotropic framework we examine conditions for the existence of such solutions to the vacuum field equations, nonperturbative in <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:msup><f:mi>α</f:mi><f:mo>′</f:mo></f:msup></f:math> in both the string frame and the Einstein frame. We elucidate the nature of the instability in the string frame vacuum. For the Einstein frame, we demonstrate that the de Sitter solutions cannot be eternal. We then extend our analysis to include Bianchi I universes where the <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:mi mathvariant="normal">O</h:mi><h:mo stretchy="false">(</h:mo><h:mi>d</h:mi><h:mo>,</h:mo><h:mi>d</h:mi><h:mo stretchy="false">)</h:mo></h:math> symmetry includes scale factor exchange as well as scale factor duality. We show how the theory can be extended to the anisotropic case so that it admits de Sitter solutions, noting the crucial role played by the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi mathvariant="normal">O</m:mi><m:mo stretchy="false">(</m:mo><m:mi>d</m:mi><m:mo>,</m:mo><m:mi>d</m:mi><m:mo stretchy="false">)</m:mo></m:math> symmetry in satisfying any additional constraints. Published by the American Physical Society 2024