Renormalization group flow and symmetry restoration in de Sitter space

We compute the renormalization group flow of <math altimg="si1.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></math> scalar field theories in de Sitter space using nonperturbative renormalization group techniques in the local potential approximation. We obtain the flow of the effective potential on superhorizon scales for arbitrary space–time dimension <math altimg="si2.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><mi>d</mi><mo>+</mo><mn>1</mn></math> . We show that, due to strong infrared fluctuations, the latter is qualitatively similar to the corresponding one in Euclidean space <math altimg="si3.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mi>D</mi></mrow></msup></math> with <math altimg="si4.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><mn>0</mn></math> . It follows that spontaneously broken symmetries are radiatively restored in any space–time dimension and for any value of N