Inflationary gravitational waves as a probe of the unknown postinflationary primordial Universe

One of the key predictions of the standard inflationary paradigm is the quantum mechanical generation of the transverse and traceless tensor fluctuations due to the rapid accelerated expansion of space, which later constitute a stochastic background of primordial gravitational waves (GWs). The amplitude of the (nearly) scale-invariant inflationary tensor power spectrum at large scales provides us with crucial information about the energy scale of inflation in the case of the minimal inflaton coupling to gravity. Furthermore, the spectral energy density, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msub> <a:mi mathvariant="normal">Ω</a:mi> <a:mrow> <a:mi>GW</a:mi> </a:mrow> </a:msub> <a:mo stretchy="false">(</a:mo> <a:mi>f</a:mi> <a:mo stretchy="false">)</a:mo> </a:math> , of the GWs at sufficiently small scales (or, large frequencies <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:mi>f</f:mi> </f:math> ) serves as an important observational probe of postinflationary primordial dynamics. In fact, the small-scale spectral tilt, <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"> <h:msub> <h:mi>n</h:mi> <h:mrow> <h:mi>GW</h:mi> </h:mrow> </h:msub> <h:mo>=</h:mo> <h:mfrac> <h:mrow> <h:mi mathvariant="normal">d</h:mi> <h:mi>log</h:mi> <h:msub> <h:mi mathvariant="normal">Ω</h:mi> <h:mrow> <h:mi>GW</h:mi> </h:mrow> </h:msub> </h:mrow> <h:mrow> <h:mi mathvariant="normal">d</h:mi> <h:mi>log</h:mi> <h:mi>f</h:mi> </h:mrow> </h:mfrac> </h:math> , of the spectral energy density of GWs is sensitive to the (unknown) postinflationary equation of state (EOS), <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mi>w</m:mi> </m:math> , of the Universe; with a softer EOS ( <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mi>w</o:mi> <o:mo>&lt;</o:mo> <o:mn>1</o:mn> <o:mo>/</o:mo> <o:mn>3</o:mn> </o:math> ) leading to a red tilt, <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:msub> <q:mi>n</q:mi> <q:mrow> <q:mi>GW</q:mi> </q:mrow> </q:msub> <q:mo>&lt;</q:mo> <q:mn>0</q:mn> </q:math> , while a stiffer EOS ( <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:mi>w</s:mi> <s:mo>&gt;</s:mo> <s:mn>1</s:mn> <s:mo>/</s:mo> <s:mn>3</s:mn> </s:math> ) resulting in a blue tilt, <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:msub> <u:mi>n</u:mi> <u:mrow> <u:mi>GW</u:mi> </u:mrow> </u:msub> <u:mo>&gt;</u:mo> <u:mn>0</u:mn> </u:math> . The postinflationary dynamics, however, is generically expected to be quite complex, potentially involving a number of distinct phases. Hence, in this work, we discuss the possibility of multiple sharp transitions, namely <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"> <w:mrow> <w:msub> <w:mrow> <w:mi>w</w:mi> </w:mrow> <w:mrow> <w:mn>1</w:mn> </w:mrow> </w:msub> <w:mo stretchy="false">→</w:mo> <w:msub> <w:mrow> <w:mi>w</w:mi> </w:mrow> <w:mrow> <w:mn>2</w:mn> </w:mrow> </w:msub> <w:mo stretchy="false">→</w:mo> <w:msub> <w:mrow> <w:mi>w</w:mi> </w:mrow> <w:mrow> <w:mn>3</w:mn> </w:mrow> </w:msub> <w:mo stretchy="false">→</w:mo> <w:mo>…</w:mo> <w:mo stretchy="false">→</w:mo> <w:msub> <w:mrow> <w:mi>w</w:mi> </w:mrow> <w:mrow> <w:mi>n</w:mi> </w:mrow> </w:msub> </w:mrow> </w:math> , in the EOS of the postinflationary Universe and compute the corresponding spectral energy density of the inflationary GWs. We explicitly determine the region of the parameter space <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"> <cb:mo stretchy="false">{</cb:mo> <cb:mrow> <cb:msub> <cb:mi>w</cb:mi> <cb:mn>1</cb:mn> </cb:msub> <cb:mo>,</cb:mo> <cb:msub> <cb:mi>w</cb:mi> <cb:mn>2</cb:mn> </cb:msub> <cb:mo>,</cb:mo> <cb:msub> <cb:mi>w</cb:mi> <cb:mn>3</cb:mn> </cb:msub> <cb:mo>,</cb:mo> <cb:mo>…</cb:mo> <cb:mo>,</cb:mo> <cb:msub> <cb:mi>w</cb:mi> <cb:mi>n</cb:mi> </cb:msub> <cb:mo stretchy="false">}</cb:mo> </cb:mrow> </cb:math> which leads to a potentially detectable signal in the upcoming GW detectors, without violating the current constraints.