Propagating speed of primordial gravitational waves

Primordial gravitational waves, i.e., a background of metric perturbations sourced by the quantum inflationary fluctuations, if measured, could both provide substantial evidence for primordial inflation and shed light on physics at extremely high energy scales. In this work we focus on their propagating speed. Using an effective field theory approach we introduce a time-dependent propagating speed $c_{\mathrm{T}}(t)$ showing that also small deviations from the general relativity (GR) prediction $c_{\mathrm{T}}(t)=c$ can lead to testable consequences. We derive a set of equations that relate the propagating speed and its time dependence to the inflationary parameters and that generalize the usual slow roll consistency relations. Imposing the new generalized consistency relations and combining small and large scales data, we derive model-independent constraints on inflation with nontrivial primordial tensor speed. In particular, we constrain its scale dependence to be $d\log c_{\mathrm{T}}/d\log k=0.082_{-0.11}^{+0.047}$ at 68% C.L. while we only derive the lower bound $c_{\mathrm{T}}>0.22c$ at 95% C.L. We also constrain the tensor-to-scalar ratio at the pivot scale $k_{*}=0.05{\mathrm{Mpc}}^{-1}$ to be $r<0.0599$ at 95% C.L. in agreement with the result provided by the Planck Collaboration. Thanks to a proper small scale parametrization of the tensor spectrum we derive stringent constraints on the tensor tilt $n_{\mathrm{T}}=-0.084_{-0.047}^{+0.10}$ at 68% C.L. and on its runnings ${\alpha }_{\mathrm{T}}=d{n}_{\mathrm{T}}/d\log k=0.0141_{-0.021}^{+0.0035}$ and ${\beta }_{\mathrm{T}}=d{\alpha }_{\mathrm{T}}/d\log k=-0.0061_{-0.0014}^{+0.010}$ both at 68% C.L. Our results show a remarkable agreement with the standard slow roll predictions and prove that current data can significantly constrain deviations from GR on the inflationary energy scales.

Figure 1

Small-scale constraints in the plane ($r$, $n_{\mathrm{T}}$) derived by ${\mathrm{\Omega }}_{\mathrm{GW}}(k)$ through Eq. (3) or equivalently its scale-dependent generalization (60). The red region is excluded at 95% C.L. by the LIGO/VIRGO limit (1) (red solid line). The green regions represent the sensitivity of future GWs experiment, such as LISA [42] (green dashed line) and ET [43] (green solid line). If the tensor tilt is assumed to be scale independent, all these constraints refer also to its value on the CMB scale: $n_{\mathrm{T}}(k_{*})$. Instead in the scale-dependent parametrization Eq. (59) they refer to different scales: $n_{\mathrm{T}}(k_{\mathrm{ET}})\lesssim n_{\mathrm{T}}(k_{\mathrm{LISA}})<n_{\mathrm{T}}(k_{\mathrm{LV}})$, as discussed in Sec. 4b.

Figure 2

Marginalized 2D and 1D posteriors for the combination of Planck 2018 [105, 106] and Biceps/Keck 2015 [18] data for the parameters of the tensor spectrum and their combination with the LIGO/VIRGO upper limit on the stochastic background amplitude [36, 37] ($\mathrm{P}18+\mathrm{BK}15+\mathrm{LV}$).

Figure 3

Marginalized 2D and 1D posteriors for the combination of Planck 2018 [105, 106] and Biceps/Keck 2015 [18] data ($\mathrm{P}18+\mathrm{BK}15$) for the first and second-order slow parameters and their combination with the LIGO/VIRGO upper limit on the stochastic background amplitude [36, 37] ($\mathrm{P}18+\mathrm{BK}15+\mathrm{LV}$).

Figure 4

Marginalized 2D posterior in the planes ($n_{\mathrm{T}},r$) and (${\alpha }_{\mathrm{T}},r$). The blue contours are derived from the combination of Planck 2018 [105, 106] and Biceps/Keck 2015 [18] data (see Sec. 4a) while the red contours include also the LIGO/VIRGO data on the stochastic background [36, 37] (see Sec. 4c). The yellow dashed lines represent the relations (50) and (51) we used to derive the small scale constraints in Sec. 4b.

Figure 5

Marginalized 2D posterior for the combination of Planck 2018 [105, 106] and Biceps/Keck 2015 [18] data in the plane ($r,{\epsilon }_1^{\mathrm{T}}$). The red region is excluded by the LIGO/VIRGO data on the stochastic background of GWs (see Sec. 4b).

Figure 6

Marginalized 2D posterior in the planes ($n_{\mathrm{T}},r$) and (${\alpha }_{\mathrm{T}},r$). The blue contours are derived from the combination of Planck 2018 [105, 106] and Biceps/Keck 2015 [18] data (see Sec. 4a) while the red contours take into account the LIGO/VIRGO data on the stochastic background [36, 37] (see Sec. 4c). The yellow dashed lines represent the standard slow roll relations in the GR limit, i.e., $c_{\mathrm{T}}=1$ and ${\epsilon }_1^{\mathrm{T}}=0$.

Figure 7

Marginalized 2D and 1D posteriors for the combination of Planck 2018 [105, 106] and Biceps/Keck 2015 [18] data for the parameters of the tensor spectrum. The blue contours are those obtained exploring only subluminal velocities while the green contours are obtained extending the prior to superluminal velocities $c_{\mathrm{T}}<5$. As one can see, the choice of exploring only subluminal velocities does not lead to significant bias on the inflationary parameters. Nevertheless, once the superluminal velocities are considered, since the Planck data prefer a vanishing tensor amplitude, the posterior of $c_{\mathrm{T}}$ is pushed to $c_{\mathrm{T}}\gg 1$ leading to a prior dependent upper (and lower) bound.

Figure 8

Constraints on the propagating speed $c_{\mathrm{T}}$ at the generic scale $k$ extrapolate from the constraints on the CMB scales fixing ${\epsilon }_1^{\mathrm{T}}=0.082$ and $c_{\mathrm{T}}(k_{*})>0.2$. Remarkably on the LIGO/VIRGO scales we can extrapolate the lower limit $c_{\mathrm{T}}(k_{\mathrm{LV}})\gtrsim 0.94$, in perfect agreement with the constraints on the astrophysics GWs.