Reexamination of inflation in noncommutative space-time after Planck results

An inflationary model in the framework of noncommutative space-time may generate a nontrivial running of the scalar spectral index, but usually induces a large tensor-to-scalar ratio simultaneously. With the latest observational data from the Planck mission, we reexamine the inflationary scenarios in a noncommutative space-time. We find that either the running of the spectral index is tiny compared with the recent observational result, or the tensor-to-scalar ratio is too large to allow a sufficient number of e-folds. As examples, we show that the chaotic and power-law inflation models with the noncommutative effects are not favored by the current Planck data.

Figure 1

The expected region of the tensor-to-scalar ratio $r$ and the noncommutative parameter $\mu$ in the ${\varphi }^2$ inflation model under the consideration of the $1\sigma$ ranges of $n_{\mathrm{s}}$ and ${\alpha }_{\mathrm{s}}$ in Eq. (7). We find that the noncommutative effects lead to a too large tensor-to-scalar ratio, $0.41<r<0.75$. Here we simply treat $s$ and ${\alpha }_{\mathrm{s}}$ as free parameters, and the lower bound of $r$ will be even larger, if the correlation of $s$ and ${\alpha }_{\mathrm{s}}$ is considered.

Figure 2

The predicted relationship of the running of the spectral index ${\alpha }_{\mathrm{s}}$ and the spectral index $n_{\mathrm{s}}$ (black line) in the ${\varphi }^2$ inflation model in noncommutative space-time, with the number of e-folds chosen to be $N=40$. Contours (68% and 95% C.L.) in the ${\alpha }_{\mathrm{s}}–n_{\mathrm{s}}$ plane are from the joint $\mathrm{\text{Planck}}+\mathrm{WP}+\mathrm{BAO}$ constraints; here we use a copy of Fig. 2 in Ref. 8. Blue contours (upper) are for $\Lambda \mathrm{CDM}+{\alpha }_{\mathrm{s}}$ cosmology, and red contours (lower) are for $\Lambda \mathrm{CDM}+{\alpha }_{\mathrm{s}}+r$ cosmology. For comparison, we keep the purple strip that shows the prediction for usual chaotic inflation models in commutative space-time with $50<N<60$. Note that in the usual slow-roll chaotic inflation models, ${\alpha }_{\mathrm{s}}$ is of order $(1-n_{\mathrm{s}}{)}^2$, and thus under the constraints of current data this purple strip is so narrow that it looks like a purple line. On the contrary to this strip, the noncommutative effects lead to a more negative slope for the ${\alpha }_{\mathrm{s}}–n_{\mathrm{s}}$ relation. We find a rather small ${\alpha }_{\mathrm{s}}$ in the noncommutative model, ${\alpha }_{\mathrm{s}}=-0.0005$ at $n_{\mathrm{s}}=0.92$ and ${\alpha }_{\mathrm{s}}=-0.0025$ at $n_{\mathrm{s}}=1$. For the central value of the spectral index $n_{\mathrm{s}}=0.9607$ [8], we get ${\alpha }_{\mathrm{s}}=-0.001518$. These values are tiny compared with the central value ${\alpha }_{\mathrm{s}}=-0.021$ [8], indicating that the noncommutative ${\varphi }^2$ inflation model is not favored by the current $\mathrm{\text{Planck}}+\mathrm{WP}+\mathrm{BAO}$ observations at the 68% C.L. (red contours). Larger number of e-folds makes things worse, as the slope of the function in Eq. (15) is even smaller.

Figure 3

The predicted relationship of the tensor-to-scalar ratio $r$ and the spectral index $n_{\mathrm{s}}$ in the ${\varphi }^2$ inflation model in noncommutative space-time. Contours (68% and 95% C.L.) in the $r–n_{\mathrm{s}}$ plane are from the joint $\mathrm{\text{Planck}}+\mathrm{WP}+\mathrm{BAO}$ constraints; here we use a copy of Fig. 4 in Ref. 8. Blue contours (lower) are for $\Lambda \mathrm{CDM}+r$ cosmology, and red contours (upper) are for $\Lambda \mathrm{CDM}+r+{\alpha }_{\mathrm{s}}$ cosmology. The black line indicates the ${\varphi }^2$ inflation model in commutative space-time, and the green, red, and blue lines show the $r–n_{\mathrm{s}}$ relationships, with the noncommutative parameter $\mu$ taken to be 0.05, 0.10, and 0.15, respectively. We find that the larger $\mu$ is, the more inconsistent the noncommutative models are with the $\mathrm{\text{Planck}}+\mathrm{WP}+\mathrm{BAO}$ data. If $r$ is lower than its upper bound $r<0.25$ [8], a large enough number of e-folds $N$ is needed. Here we take$N=40$, 50, and 60, respectively. Since the noncommutative ${\varphi }^2$ inflation model predicts a negligible running, given a reasonable number of e-folds, the theoretical predictions (the green, red, and blue lines) should be compared to the blue contours in the $r–n_{\mathrm{s}}$ plane. We see that the noncommutative ${\varphi }^2$ inflation model with $\mu \gtrsim 0.05$ is ruled out at the 95% C.L.. Even if the comparison is made to the red contours, the model with $\mu \gtrsim 0.05$ is ruled out at the 68% C.L., and with $\mu \gtrsim 0.15$ is ruled out at the 95% C.L.

Figure 4

The expected region of the tensor-to-scalar ratio $r$ and the noncommutative parameter $\mu$ in the ${\varphi }^4$ inflation model under the consideration of the $1\sigma$ ranges of $n_{\mathrm{s}}$ and ${\alpha }_{\mathrm{s}}$ in Eq. (7). The noncommutative effects lead to a even larger tensor-to-scalar ratio, $0.51<r<0.91$. This means that the noncommutative ${\varphi }^4$ inflation model is more inconsistent with the current observational data.