Scaling density of axion strings in terasite simulations

We report on a study of axion string networks using fixed-grid simulations of up to 16384 points per side. The length of string can be characterized in terms of standard dimensionless parameters <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:msub><a:mrow><a:mi>ζ</a:mi></a:mrow><a:mrow><a:mi mathvariant="normal">w</a:mi></a:mrow></a:msub></a:mrow></a:math> and <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:msub><d:mi>ζ</d:mi><d:mi mathvariant="normal">r</d:mi></d:msub></d:math>, the length density measured in the cosmic rest frame and the string rest frame, scaled with the cosmic time <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>t</g:mi></g:math>. The motion of the string can be characterized by the root-mean-square (RMS) velocity of the string. Starting from a range of initial length densities and velocities, we analyze the string network in the standard scaling framework and find evolution toward a fixed point with estimated values <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mrow><i:msub><i:mrow><i:mover accent="true"><i:mrow><i:mi>ζ</i:mi></i:mrow><i:mrow><i:mo stretchy="false">^</i:mo></i:mrow></i:mover></i:mrow><i:mrow><i:mi mathvariant="normal">w</i:mi><i:mo>,</i:mo><i:mo>*</i:mo></i:mrow></i:msub><i:mo>=</i:mo><i:mn>1.220</i:mn><i:mo stretchy="false">(</i:mo><i:mn>57</i:mn><i:mo stretchy="false">)</i:mo></i:mrow></i:math> and <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:msub><p:mover accent="true"><p:mi>ζ</p:mi><p:mo stretchy="false">^</p:mo></p:mover><p:mrow><p:mi mathvariant="normal">r</p:mi><p:mo>,</p:mo><p:mo>*</p:mo></p:mrow></p:msub><p:mo>=</p:mo><p:mn>1.491</p:mn><p:mo stretchy="false">(</p:mo><p:mn>93</p:mn><p:mo stretchy="false">)</p:mo></p:math>. The two measures are related by the RMS velocity, which we estimate to be <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:msub><w:mover accent="true"><w:mi>v</w:mi><w:mo stretchy="false">^</w:mo></w:mover><w:mo>*</w:mo></w:msub><w:mo>=</w:mo><w:mn>0.5705</w:mn><w:mo stretchy="false">(</w:mo><w:mn>93</w:mn><w:mo stretchy="false">)</w:mo></w:math>. The length density is consistent with previous measurements, while the velocity is about 5% lower. For simulations starting from low enough density, the length density parameters <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:msub><cb:mi>ζ</cb:mi><cb:mi mathvariant="normal">w</cb:mi></cb:msub></cb:math> and <fb:math xmlns:fb="http://www.w3.org/1998/Math/MathML" display="inline"><fb:msub><fb:mi>ζ</fb:mi><fb:mi mathvariant="normal">r</fb:mi></fb:msub></fb:math> remain below their fixed point values throughout, while growing slowly, giving rise to an impression of approximately logarithmic increase with time. This has been proposed as the true long-term behavior. We find that the growth tends to slow down as the values of <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"><ib:msub><ib:mi>ζ</ib:mi><ib:mi mathvariant="normal">w</ib:mi></ib:msub></ib:math> and <lb:math xmlns:lb="http://www.w3.org/1998/Math/MathML" display="inline"><lb:msub><lb:mi>ζ</lb:mi><lb:mi mathvariant="normal">r</lb:mi></lb:msub></lb:math> identified as fixed points are approached. In the case of <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"><ob:msub><ob:mi>ζ</ob:mi><ob:mi mathvariant="normal">r</ob:mi></ob:msub></ob:math>, the growth stops for simulations that started close to the fixed point length density. The difference between <rb:math xmlns:rb="http://www.w3.org/1998/Math/MathML" display="inline"><rb:msub><rb:mi>ζ</rb:mi><rb:mi mathvariant="normal">w</rb:mi></rb:msub></rb:math> and <ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML" display="inline"><ub:msub><ub:mi>ζ</ub:mi><ub:mi mathvariant="normal">r</ub:mi></ub:msub></ub:math> can be understood to result from the continuing velocity evolution. Our results indicate that the growth of <xb:math xmlns:xb="http://www.w3.org/1998/Math/MathML" display="inline"><xb:msub><xb:mi>ζ</xb:mi><xb:mi mathvariant="normal">w</xb:mi></xb:msub></xb:math> is a transient appearing at low densities and while the velocity is converging. This highlights the importance of studying the string density and the velocity together, and the preparation of initial conditions.