Kinetic relaxation and nucleation of Bose stars in self-interacting wave dark matter
Mudit Jain, Wisha Wanichwecharungruang, Jonathan P. Thomas
Abstract
We revisit kinetic relaxation and soliton/boson star nucleation in fuzzy scalar dark matter featuring short-ranged self-interactions <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:msub><a:mrow><a:mi mathvariant="script">H</a:mi></a:mrow><a:mrow><a:mi>int</a:mi></a:mrow></a:msub><a:mo>=</a:mo><a:mo>−</a:mo><a:mi>λ</a:mi><a:mo stretchy="false">|</a:mo><a:mi>ψ</a:mi><a:msup><a:mrow><a:mo stretchy="false">|</a:mo></a:mrow><a:mrow><a:mn>4</a:mn></a:mrow></a:msup><a:mo>/</a:mo><a:mn>2</a:mn><a:msup><a:mrow><a:mi>m</a:mi></a:mrow><a:mrow><a:mn>2</a:mn></a:mrow></a:msup></a:mrow></a:math>, alongside gravitational self-interactions. We map out the full curve of nucleation timescale for both repulsive (<f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:mi>λ</f:mi><f:mo><</f:mo><f:mn>0</f:mn></f:math>) and attractive (<h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:mi>λ</h:mi><h:mo>></h:mo><h:mn>0</h:mn></h:math>) short-ranged self-interaction strength and in doing so reveal two new points. Firstly, besides the two usual terms, <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:mo>∝</j:mo><j:msup><j:mi>G</j:mi><j:mn>2</j:mn></j:msup></j:math> and <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:mo>∝</l:mo><l:msup><l:mi>λ</l:mi><l:mn>2</l:mn></l:msup></l:math>, in the total relaxation rate <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:msub><n:mi mathvariant="normal">Γ</n:mi><n:mtext>relax</n:mtext></n:msub></n:math>, there is an additional cross term <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mo>∝</q:mo><q:mi>G</q:mi><q:mi>λ</q:mi></q:math> arising due to interference between gravitational and short-ranged self-interaction scattering amplitudes. This yields a critical repulsive interaction strength <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:msub><s:mi>λ</s:mi><s:mrow><s:mi>cr</s:mi></s:mrow></s:msub><s:mo>≃</s:mo><s:mo>−</s:mo><s:mn>2</s:mn><s:mi>π</s:mi><s:mi>G</s:mi><s:msup><s:mi>m</s:mi><s:mn>2</s:mn></s:msup><s:mo>/</s:mo><s:msubsup><s:mi>v</s:mi><s:mn>0</s:mn><s:mn>2</s:mn></s:msubsup></s:math>, at which the relaxation rate is smallest and serves as the transition point between typical net attractive self-interaction (<u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi>λ</u:mi><u:mo>≳</u:mo><u:msub><u:mi>λ</u:mi><u:mrow><u:mi>cr</u:mi></u:mrow></u:msub></u:math>) and net repulsive self-interaction (<w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mo>−</w:mo><w:mi>λ</w:mi><w:mo>≳</w:mo><w:mo>−</w:mo><w:msub><w:mi>λ</w:mi><w:mrow><w:mi>cr</w:mi></w:mrow></w:msub></w:math>). Secondly, while in the net attractive regime, nucleation timescale is similar to inverse relaxation timescale <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:msub><y:mi>τ</y:mi><y:mrow><y:mi>nuc</y:mi></y:mrow></y:msub><y:mo>∼</y:mo><y:msubsup><y:mi mathvariant="normal">Γ</y:mi><y:mtext>relax</y:mtext><y:mrow><y:mo>−</y:mo><y:mn>1</y:mn></y:mrow></y:msubsup></y:math>; in the net repulsive regime, nucleation occurs at a delayed time <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" display="inline"><bb:msub><bb:mi>τ</bb:mi><bb:mrow><bb:mi>nuc</bb:mi></bb:mrow></bb:msub><bb:mo>∼</bb:mo><bb:mo stretchy="false">(</bb:mo><bb:mi>λ</bb:mi><bb:mo>/</bb:mo><bb:msub><bb:mi>λ</bb:mi><bb:mrow><bb:mi>cr</bb:mi></bb:mrow></bb:msub><bb:mo stretchy="false">)</bb:mo><bb:msubsup><bb:mi mathvariant="normal">Γ</bb:mi><bb:mtext>relax</bb:mtext><bb:mrow><bb:mo>−</bb:mo><bb:mn>1</bb:mn></bb:mrow></bb:msubsup></bb:math>. We confirm our analytical understanding by performing 3D field simulations with varying average mass density <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:mover accent="true"><gb:mi>ρ</gb:mi><gb:mo stretchy="false">¯</gb:mo></gb:mover></gb:math>, box size <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"><kb:mi>L</kb:mi></kb:math> and grid size <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" display="inline"><mb:mi>N</mb:mi></mb:math>. Published by the American Physical Society 2024