Jump in fluid properties of inflationary universe to reconcile scalar and tensor spectra

The recent detection of the B-mode polarization from the BICEP2 observation, if confirmed to be primordial, seems to be in tension with the upper bound on the amplitude of tensor perturbations from the PLANCK data. We consider a phenomenological model of inflation in which the microscopical properties of the inflationary fluid such as the equation of state $w$ or the sound speed $c_s$ jump in a sharp manner. We show that the amplitude of the scalar perturbations is controlled by a nontrivial combination of $w$ and $c_s$ before and after the phase transition while the tensor perturbations remains nearly intact. With an appropriate choice of the fluid parameters $w$ and $c_s$ one can suppress the scalar perturbation power spectrum on large scales to accommodate a large tensor amplitude.

Figure 1

The plot of the inverse comoving sound horizon $\mathcal{H}/c_s$ as a function of the number of e-folds $n$ for the case II in which ${c_s}_2>{c_s}_1$. There are three different distinct behaviors for the modes. The lower dashed line represents modes which are outside the sound horizon during both stages of inflation. The middle solid line represents modes which leave the sound horizon during the first stage of inflation, re-enter the sound horizon during the second stage of inflation and leave the sound horizon again during the second period of inflation. The upper dashed-dotted line represents modes which are inside the sound horizon during both periods of inflation. Here $c_{s1}=0.2$, $c_{s2}=1.0$, $w_1=-0.96$, $w_2=-0.996$.

Figure 2

Case I: $c_{s1}=1.0$, $c_{s2}=0.8$, ${\epsilon }_1=0.012$, ${\epsilon }_2=0.01$. The left plot is for an infinitely sharp phase transition while the right plot represents the case in which the phase transition takes 1.5 $e$-folds. The bottom plot if for ${\mathcal{P}}_T$. As we see, the tensor perturbations are nearly insensitive to the phase transition.

Figure 3

The same plot as in Fig. 2 but for case II: $c_{s1}=0.8$, $c_{s2}=1.0$, ${\epsilon }_1=0.016$, ${\epsilon }_2=0.007$. Note that the qualitative shape of the jump in ${\mathcal{P}}_{\mathcal{R}}$ is different than the case in Fig. 2. The bottom plot represents the tensor power spectrum. Interestingly, the tensor perturbations are more affected compared to case I in Fig. 2 since ${\epsilon }_2$ has decreased significantly in order to maintain ${\epsilon }_2{c_s}_2<{\epsilon }_1{c_s}_1$.

Figure 4

Here we present the TT power spectrum in which ${\mathcal{D}}_{\ell }$ is related to $C_{\ell }$ via ${\mathcal{D}}_{\ell }=\ell (\ell +1)C_{\ell }/2\pi$. The red dashed-dotted curve and the blue dashed curve, as in previous plots, correspond to case I ($c_s<1$) and II ($c_s>1$) respectively. The solid black curve represent the standard $\mathrm{\Lambda }\mathrm{CDM}$ with $r=0$. The upper dotted-dashed-black curve is for $\mathrm{\Lambda }\mathrm{CDM}$ with $r=0.2$.

Figure 5

The EE power spectrum with ${\mathcal{D}}_{\ell }^{EE}=\ell (\ell +1)C_{\ell }^{EE}/2\pi$. The curves’ descriptions and the parameters are the same as in Fig. 4.

Figure 6

The BB power spectrum with ${\mathcal{D}}_{\ell }^{BB}=\ell (\ell +1)C_{\ell }^{BB}/2\pi$. The dashed and dotted-dashed curves are the same as in previous plots. The solid black curve is for $\mathrm{\Lambda }\mathrm{CDM}$ with $r=0.004$.

Figure 7

Tensor power spectrum. ${\mathcal{D}}_{\ell }^T=\ell (\ell +1)C_{\ell }^T/2\pi$. All plots are similar to Fig. 6.