Analytical sensitivity curves for additional gravitational-wave polarizations of second-generation time-delay interferometry

Abstract In addition to the gravitational-wave (GW) tensor modes of general relativity, more general metric theories of gravity allow up to four additional polarization states. Sensitivity curves for these additional GW polarizations are key quantities for assessing how well a detector can constrain such theories. In this work, we derive analytical expressions and high-accuracy approximate formulas for the sensitivity curves for the vector- x , vector- y , breathing, and longitudinal modes of the second-generation time-delay interferometry (TDI). Our analysis covers the TDI Michelson, $$(\alpha ,\beta ,\gamma ),$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:mi>β</mml:mi> <mml:mo>,</mml:mo> <mml:mi>γ</mml:mi> <mml:mo>)</mml:mo> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> Monitor, Beacon, Relay, and Sagnac combinations, together with the orthogonal A , E , T channels constructed from them. The validity of analytical expressions is confirmed by Monte Carlo integration. We find that, in the high-frequency limit, the sensitivity curves for the tensor and breathing modes scale as $$f^{2},$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>f</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> whereas those for the vector and longitudinal modes approach the explicit asymptotic forms $$\frac{c^{\textrm{op}}f^{2}}{\ln f}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfrac> <mml:mrow> <mml:msup> <mml:mi>c</mml:mi> <mml:mtext>op</mml:mtext> </mml:msup> <mml:msup> <mml:mi>f</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mrow> <mml:mo>ln</mml:mo> <mml:mi>f</mml:mi> </mml:mrow> </mml:mfrac> </mml:math> and $$\frac{4c^{\textrm{op}}}{\pi ^{2} L^{2}} f,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mfrac> <mml:mrow> <mml:mn>4</mml:mn> <mml:msup> <mml:mi>c</mml:mi> <mml:mtext>op</mml:mtext> </mml:msup> </mml:mrow> <mml:mrow> <mml:msup> <mml:mi>π</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:mfrac> <mml:mi>f</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> respectively. In the low-frequency limit, for all GW modes, the sensitivity curves of the $$\zeta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ζ</mml:mi> </mml:math> combination and of the T channel scale as $$f^{-6},$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>6</mml:mn> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> whereas those of the other TDI combinations and of the A , E channels scale as $$f^{-4}.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:msup> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> In this limit, the sensitivity curves for the tensor and vector modes coincide, and likewise for the breathing and longitudinal modes. For the breathing mode, the sensitivity curves of the $$(\alpha ,\beta ,\gamma )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:mi>β</mml:mi> <mml:mo>,</mml:mo> <mml:mi>γ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and $$\zeta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ζ</mml:mi> </mml:math> combinations and of the T channel exhibit singularities at frequencies $$f = k/L$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>=</mml:mo> <mml:mi>k</mml:mi> <mml:mo>/</mml:mo> <mml:mi>L</mml: