Self-interacting dark matter with observable ΔNeff

We propose a GeV-scale self-interacting dark matter (SIDM) candidate within a dark <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi>U</a:mi><a:mo stretchy="false">(</a:mo><a:mn>1</a:mn><a:msub><a:mrow><a:mo stretchy="false">)</a:mo></a:mrow><a:mrow><a:mi>D</a:mi></a:mrow></a:msub></a:mrow></a:math> gauged extension of the Standard Model (SM), addressing small-scale structure issues in <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi mathvariant="normal">Λ</e:mi><e:mi>CDM</e:mi></e:math> while predicting an observable contribution to <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:mi mathvariant="normal">Δ</h:mi><h:msub><h:mi>N</h:mi><h:mi>eff</h:mi></h:msub></h:math> in the form of dark radiation. The model introduces a fermionic DM candidate <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>χ</k:mi></k:math> and a scalar <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>ϕ</m:mi></m:math>, both charged under an unbroken <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>U</o:mi><o:mo stretchy="false">(</o:mo><o:mn>1</o:mn><o:msub><o:mo stretchy="false">)</o:mo><o:mi>D</o:mi></o:msub></o:math> gauge symmetry. The self-interactions of <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mi>χ</s:mi></s:math> are mediated by a light vector boson <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:msup><u:mi>X</u:mi><u:mi>μ</u:mi></u:msup></u:math>, whose mass is generated via the Stueckelberg mechanism. The relic abundance of <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mi>χ</w:mi></w:math> is determined by thermal freeze-out through annihilations into <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:msup><y:mi>X</y:mi><y:mi>μ</y:mi></y:msup></y:math>, supplemented by a nonthermal component from the late decay of <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:mi>ϕ</ab:mi></ab:math>. Crucially, <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:mi>ϕ</cb:mi></cb:math> decays after the big bang nucleosynthesis (BBN) but before the cosmic microwave background (CMB) epoch, producing additional <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"><eb:mi>χ</eb:mi></eb:math> and a dark radiation species (<gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:msub><gb:mi>ν</gb:mi><gb:mi>S</gb:mi></gb:msub></gb:math>). This late-time production compensates for thermal underabundance due to efficient annihilation into light mediators, while remaining consistent with structure formation constraints. The accompanying dark radiation yields a detectable <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"><ib:mi mathvariant="normal">Δ</ib:mi><ib:msub><ib:mi>N</ib:mi><ib:mi>eff</ib:mi></ib:msub></ib:math>, compatible with Planck 2018 bounds and within reach of next-generation experiments such as SPT-3G, CMB-S4, and CMB-HD.