Testing the inflationary slow-roll condition with tensor modes

A major goal of modern cosmology is the detection of $B$-modes in the cosmic microwave background polarization originated from primordial gravitational waves. Their detection not only could provide substantial evidence for primordial inflation but also could shed light on its physical nature. Under the assumption of single-field slow-roll inflation, a set of conditions exist for the scalar and tensor parameters. In particular, given a constraint on the scalar spectral index $n_{\mathrm{s}}$, its running ${\alpha }_{\mathrm{s}}$, its running of running ${\beta }_{\mathrm{s}}$, and the tensor-to-scalar ratio $r$, constraints can be derived on the tensor spectral index $n_{\mathrm{t}}$, its running ${\alpha }_{\mathrm{t}}$, its running of running ${\beta }_{\mathrm{t}}$, and its running of running of running ${\gamma }_{\mathrm{t}}$. Using current bounds from the Planck 2015 and BICEP2 datasets and under the slow condition we found the following constraints at 95% C.L.: $n_{\mathrm{t}}>-0.0157$, ${\alpha }_{\mathrm{t}}=-0.00018_{-0.00024}^{+0.00019}$, ${\beta }_{\mathrm{t}}=0.00004_{-0.00013}^{+0.00022}$, and ${\gamma }_{\mathrm{t}}=0.00017_{-0.00019}^{+0.00040}$. Future measurements of the tensor spectrum could therefore be used to test these bounds and the slow-roll condition.

Figure 1

On the left side panel the consistency relation (20) for different values of $n_{\mathrm{s}}$. On the right side panel the consistency relation (21) for $n_{\mathrm{s}}=0.959$ and the different indicated values of ${\alpha }_{\mathrm{s}}$.

Figure 2

Marginalized joint confidence contours at 68% and 95% C.L. in the plane $(n_{\mathrm{t}},r)$ for the indicated datasets. On the left (right) side panel the contours obtained relaxing (imposing) the consistency relations. The black dashed line in the left panel represents the consistency relation $n_{\mathrm{t}}=-\frac{r}{8}$.

Figure 3

Marginalized joint confidence contours at 68% and 95% C.L. in the plane $({\alpha }_{\mathrm{t}},r)$ for the indicated datasets. On the left (right) side panel the contours obtained relaxing (imposing) the consistency relations. The colored dashed lines represent the consistency relations (20) for different values of $n_{\mathrm{s}}$.

Figure 4

Marginalized joint confidence contours at 68% and 95% C.L. in the plane $({\beta }_{\mathrm{t}},r)$ for the indicated datasets. On the left (right) side panel the contours obtained relaxing (imposing) the consistency relations. The colored dashed lines represent the consistency relations (21) for $n_{\mathrm{s}}=0.959$ and the different indicated values of ${\alpha }_{\mathrm{s}}$.

Figure 5

Marginalized joint confidence contours at 68% and 95% C.L. obtained imposing the consistency relations for the indicated datasets. On the left side panels the plane $(n_{\mathrm{s}},{\alpha }_{\mathrm{t}})$. On the right side panel the plane $({\alpha }_{\mathrm{s}},{\beta }_{\mathrm{t}})$.

Figure 6

Marginalized joint confidence contours at 68% and 95% C.L. in the plane $(n_{\mathrm{t}},{\alpha }_{\mathrm{t}})$ for the indicated datasets. On the left (right) panels the contours obtained relaxing (imposing) the consistency relations.

Figure 7

Marginalized joint confidence contours at 68% and 95% C.L. in the plane $(n_{\mathrm{t}},{\beta }_{\mathrm{t}})$ for the indicated datasets. On the left (right) panels the contours obtained relaxing (imposing) the consistency relations.

Figure 8

Marginalized joint confidence contours at 68% and 95% C.L. in the plane $({\alpha }_{\mathrm{t}},{\beta }_{\mathrm{t}})$ for the indicated datasets. On the left (right) panels the contours obtained relaxing (imposing) the consistency relations.

Figure 9

Marginalized joint confidence contours at 68% and 95% C.L. in the planes $(n_{\mathrm{t}},\frac{d{\beta }_{\mathrm{t}}}{d\log k})$ (left panels) and $(n_{\mathrm{s}},\frac{d{\beta }_{\mathrm{t}}}{d\log k})$ (right panels), for the indicated datasets.

Figure 10

Marginalized joint confidence contours at 68% and 95% C.L. in the planes $({\alpha }_{\mathrm{t}},\frac{d{\beta }_{\mathrm{t}}}{d\log k})$ (left panels) and $({\alpha }_{\mathrm{s}},\frac{d{\beta }_{\mathrm{t}}}{d\log k})$ (right panels), for the indicated datasets.

Figure 11

Marginalized joint confidence contours at 68% and 95% C.L. in the planes $({\beta }_{\mathrm{t}},\frac{d{\beta }_{\mathrm{t}}}{d\log k})$ (left panels) and $({\beta }_{\mathrm{s}},\frac{d{\beta }_{\mathrm{t}}}{d\log k})$ (right panels), for the indicated datasets.