%0 Journal Article
%T The Evolution of the Universe Based on Principal Bundle Geometry
%A Bi Qiao
%J Journal of Modern Physics
%P 536-554
%@ 2153-120X
%D 2025
%I Scientific Research Publishing
%R 10.4236/jmp.2025.164028
%X This paper proposes a novel geometric framework for unifying dark matter, dark energy, and cosmic evolution through a modified gauge-theoretic approach. By introducing a scalar function <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>&#x03B3;</mi><mrow><mo>(</mo>
   <mi>x</mi>
  <mo>)</mo></mrow><mo>=</mo><mi>tanh</mi><mrow><mo>(</mo>
   <mrow>
    <mi>k</mi><mi>x</mi></mrow>
  <mo>)</mo></mrow></mrow>
</math>

, where <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>x</mi><mo>=</mo><mfrac>
   <mi>a</mi>
   <mrow>
    <msup>
     <mi>a</mi>
     <mn>0</mn>
    </msup>
    </mrow>
  </mfrac>
  </mrow>
</math>

 (with <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mi>a</mi>
</math>

 as the local &#8220;absolute acceleration&#8221; and <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msup>
   <mi>a</mi>
   <mn>0</mn>
  </msup>
  <mo>&#x2248;</mo><msup>
   <mrow>
    <mn>10</mn></mrow>
   <mrow>
    <mo>&#x2212;</mo><mn>8</mn></mrow>
  </msup>
  </mrow>
</math>

 cm/s&#178; a reference scale), we construct an extended principal bundle <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mover accent='true'>
   <mi>P</mi>
   <mo>&#x02DC;</mo>
  </mover>
  <mo>=</mo><mi>P</mi><mo>&#x00D7;</mo><mi>&#x211D;</mi></mrow>
</math>

 and redefine the connection form as <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mover accent='true'>
   <mi>&#x03C9;</mi>
   <mo>&#x02DC;</mo>
  </mover>
  <mo>=</mo><msub>
   <mi>&#x03C9;</mi>
   <mi>G</mi>
  </msub>
  <mo>+</mo><mi>d</mi><mi>&#x03B3;</mi></mrow>
</math>

, ensuring curvature invariance while modulating gravitational sources. In the Newtonian limit, the field equation simplifies to <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>&#x03B3;</mi><mtext>&#x200A;</mtext><msup>
   <mo>&#x2207;</mo>
   <mn>2</mn>
  </msup>
  <mi>&#x03D5;</mi><mo>=</mo><mo>&#x2212;</mo><mn>4</mn><mi>&#x03C0;</mi><mi>&#x03C1;</mi></mrow>
</math>

, revealing <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mi>&#x03B3;</mi>
</math>

&#8217;s role in scaling gravitational effects. Distinctively, dark matter and dark energy emerge as geometric consequences of spacetime curvature rather than exotic components: <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>&#x03B3;</mi><mo>&#x2192;</mo><mn>1</mn></mrow>
</math>

 restores standard gravity at small scales, <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mn>0</mn><mo>&#x003C;</mo><mi>&#x03B3;</mi><mo>&#x003C;</mo><mn>1</mn></mrow>
</math>

 enhances gravity (mimicking dark matter)
%K Generalized Gauge Transformation
%K Dark Matter
%K Dark Energy
%K Einstein Equation
%K Universal Evolution
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=141907