In cosmological inflation, within the slow-roll paradigm, the Lyth argument places a theoretical upper bound on the amount of gravitational waves produced during inflation, given the amount of departure from the homogeneity of the cosmic microwave background (CMB).
Summary During slow-roll inflation, the ratio of gravitational waves to inhomogeneities of the CMB is correlated to the inflationary potential steepness. Temperature inhomogeneities of the CMB were successfully and accurately measured. in the CMB. There are current CMB polarization experiments (see this article for instance for an overview of gravitational wave observatories) aimed at measuring the primordial gravitational wave signature in the CMB. However, to date, a significant signal of primordial gravitational waves was not detected. Thus the ratio cannot exceed a certain value. Thus the steepness of the inflationary potential is bounded.
Detail The argument was first introduced by David H. Lyth in his 1997 paper "What Would We Learn by Detecting a Gravitational Wave Signal in the Cosmic Microwave Background Anisotropy?" The detailed argument is as follows: The power spectrum for curvature perturbations Ψ {\displaystyle \Psi } is given by:
P Ψ ( k ) = 8 π 9 k 3 H 2 ϵ M p l 2 | a H = k {\displaystyle P_{\Psi }(k)={\frac {8\pi }{9k^{3}}}{\frac {H^{2}}{\epsilon M_{pl}^{2}}}{\big |}_{aH=k}} , Whereas the power spectrum for tensor perturbations is given by:
P h ( k ) = 8 π k 3 H 2 M p l 2 | a H = k {\displaystyle P_{h}(k)={\frac {8\pi }{k^{3}}}{\frac {H^{2}}{M_{pl}^{2}}}{\big |}_{aH=k}} , in which H {\displaystyle H} is the Hubble parameter, k {\displaystyle k} is the wave number, M p l {\displaystyle M_{pl}} is the Planck mass and ϵ {\displaystyle \epsilon } is the first slow-roll parameter given by − H ˙ H 2 {\displaystyle {\frac {-{\dot {H}}}{H^{2}}}} . Thus the ratio of tensor to scalar power spectra at a certain wave number k {\displaystyle k} , denoted as the so-called tensor-to-scalar ratio r {\displaystyle r} , is given by:
r ( k ) ≡ P Ψ ( k ) P h ( k ) | a H = k = 1 9 ϵ ( k ) {\displaystyle r(k)\equiv {\frac {P_{\Psi }(k)}{P_{h}(k)}}{\big |}_{aH=k}={\frac {1}{9\epsilon (k)}}} . While strictly speaking ϵ {\displaystyle \epsilon } is a function of k {\displaystyle k} , during slow-roll inflation, it is understood to change very mildly, thus it is customary to simply omit the wavenumber dependence. Additionally, the numeric pre-factor is susceptible to slight changes owing to more detailed calculations but is usually between 1 9 ∼ 1 16 {\displaystyle {\frac {1}{9}}\sim {\frac {1}{16}}} . Although the slow-roll parameter is given as above, it was shown that in the slow-roll limit, this parameter can be given by the slope of the inflationary potential such that:
… excerpt ends here. Continue reading the full article.
