Scaling density of axion strings in terasite simulations

Abstract

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.

Topics & Concepts