Cosmological scaling of precursor domain walls

Domain wall (DW) networks have a large impact on cosmology and present interesting dynamics that can be controlled by various scaling regimes. In the first stage after spontaneous breaking of the discrete symmetry, the network is seeded with “DW precursors,” the zeros of a tachyonic field. At sufficiently weak coupling, this stage can be quite long. The network is then driven to a nonrelativistic scaling regime; in flat spacetime the correlation length grows like <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>L</a:mi><a:mo>∼</a:mo><a:msup><a:mi>t</a:mi><a:mi>κ</a:mi></a:msup></a:math> with <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>κ</c:mi><c:mo>=</c:mo><c:mn>1</c:mn><c:mo>/</c:mo><c:mn>2</c:mn></c:math>. We focus on the precursor regime in cosmology, assuming a power law scale factor <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>a</e:mi><e:mo>∝</e:mo><e:msup><e:mi>t</e:mi><e:mi>α</e:mi></e:msup></e:math>. We obtain the scaling exponent as a function of the external parameter, <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>κ</g:mi><g:mo stretchy="false">(</g:mo><g:mi>α</g:mi><g:mo stretchy="false">)</g:mo></g:math>, by explicit computation in <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mn>1</k:mn><k:mo>+</k:mo><k:mn>1</k:mn></k:math> and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mn>2</m:mn><m:mo>+</m:mo><m:mn>1</m:mn></m:math> dimensions, and find a smooth transition from nonrelativistic scaling with <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>κ</o:mi><o:mo>≃</o:mo><o:mn>1</o:mn><o:mo>/</o:mo><o:mn>2</o:mn></o:math> for <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>α</q:mi><q:mo>≲</q:mo><q:mn>1</q:mn><q:mo>/</q:mo><q:mn>2</q:mn></q:math> to DW gas regime <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mi>κ</s:mi><s:mo>≃</s:mo><s:mi>α</s:mi></s:math> for <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi>α</u:mi><u:mo>≳</u:mo><u:mn>1</u:mn><u:mo>/</u:mo><u:mn>2</u:mn></u:math>, confirming previous arguments. The precise form of the transition <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mi>κ</w:mi><w:mo stretchy="false">(</w:mo><w:mi>α</w:mi><w:mo stretchy="false">)</w:mo></w:math> is surprisingly independent of dimension, suggesting that similar results should also be valid in <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:mn>3</ab:mn><ab:mo>+</ab:mo><ab:mn>1</ab:mn></ab:math> dimensions.