New inflation versus chaotic inflation, higher degree potentials, and the reconstruction program in light of WMAP 3-year data

The cosmic microwave background power spectra are studied for different families of single field new and chaotic inflation models in the effective field theory approach to inflation. We implement a systematic expansion in $1/N_e$, where $N_e\sim 50$ is the number of e-folds before the end of inflation. We study the dependence of the observables ($n_s$, $r$ and $d{n}_s/d\ln k$) on the degree of the potential ($2n$) and confront them to the WMAP3 and large scale structure data: This shows in general that fourth degree potentials ($n=2$) provide the best fit to the data; the window of consistency with the WMAP3 and LSS data narrows for growing $n$. New inflation yields a good fit to the $r$ and $n_s$ data in a wide range of field and parameter space. Small field inflation yields $r<0.16$ while large field inflation yields $r>0.16$ (for $N_e=50$). All members of the new inflation family predict a small but negative running $-4(n+1)\times {10}^{-4}\le d{n}_s/d\ln k\le -2\times {10}^{-4}$. (The values of $r$, $n_s$, $d{n}_s/d\ln k$ for arbitrary $N_e$ follow by a simple rescaling from the $N_e=50$ values.) A reconstruction program is carried out suggesting quite generally that for $n_s$ consistent with the WMAP3 and LSS data and $r<0.1$ the symmetry breaking scale for new inflation is $|{\varphi }_0|\sim 10M_{\mathrm{Pl}}$ while the field scale at Hubble crossing is $|{\varphi }_c|\sim M_{\mathrm{Pl}}$. The family of chaotic models features $r\ge 0.16$ (for $N_e=50$) and only a restricted subset of chaotic models are consistent with the combined WMAP3 bounds on $r$, $n_s$, $d{n}_s/d\ln k$ with a narrow window in field amplitude around $|{\varphi }_c|\sim 15M_{\mathrm{Pl}}$. We conclude that a measurement of $r<0.16$ (for $N_e=50$) distinctly rules out a large class of chaotic scenarios and favors small field new inflationary models. As a general consequence, new inflation emerges more favored than chaotic inflation.

Figure 1

Left panel: broken symmetry potential for $n=2$, small and large field cases. Right panel: unbroken symmetry potential for $n=2$, large field inflation.

Figure 2

The coupling $g$ as a function of $X$, for the degrees of the new inflation potential $n=2$, 3, 4. For $X\to 1$, $g$ vanishes as $[\frac{1-X}{2}{]}^{2(n-1)}$. The point $X=1$, $g=0$ corresponds to the monomial $m^2{\varphi }^2/2$; $g$ increases both for $X\to 0$ and for large $X$ as $(\log \frac{1}{X}{)}^{n-1}$ and $(\frac{X^2}{4n}{)}^{n-1}$, respectively.

Figure 3

Slow roll parameters as a function of $X$ for $N_e=50$. Left panel ${ϵ}_v$, right panel ${\eta }_v$, for new inflation with the degrees of the potential $n=2$, 3, 4. The results for arbitrary values of $N_e$ are obtained by multiplying by the factor $\frac{50}{N_e}$.

Figure 4

Scalar spectral index $n_s$ for the degrees of the potential $n=2$, 3, 4 for new inflation with $N_e=50$. The vertical line delimits the smallest value of $n_s$ (for $r=0$) [6]. The gray dot at $n_s=0.96$, $X=1$ corresponds to the value for the monomial potential $n=1$, $m^2{\varphi }^2/2$. Notice that the small field behavior is $n$ independent. For arbitrary $N_e$ the result follows directly from the $N_e=50$ value by using Eq. (16).

Figure 5

Tensor to scalar ratio $r$ vs $X$ for the degrees of the potential $n=2$, 3, 4 for new inflation with $N_e=50$. The horizontal dashed line corresponds to the upper limit $r=0.28$ (95% C.L.) from WMAP3 without running. The vertical dashed line determines the minimum value of $X$, $X\sim 0.2$, consistent with the WMAP limits for $n_s$ as in Fig. 4. The gray dot at $X=1$, $r=0.16$ corresponds to the value for the monomial potential $m^2{\varphi }^2/2$. The small field limit is nearly independent of $n$. For arbitrary $N_e$ the result follows directly from the $N_e=50$ value by using Eq. (16).

Figure 6

Running of the scalar index $d{n}_s/d\ln k$ vs $X$ for the degrees of the potential $n=2$, 3, 4, respectively, for new inflation with $N_e=50$. The small field behavior is independent of $n$. For arbitrary $N_e$ the result follows directly from the $N_e=50$ value by using Eq. (16).

Figure 7

Tensor to scalar ratio $r$ vs $n_s$ for degrees of the potential $n=2$, 3, 4, respectively, for new inflation with $N_e=50$; $r$ turns to be a double-valued function of $n_s$ exhibiting a maximum value for $n_s$. The values inside the box between the dashed lines correspond to the WMAP3 marginalized region of the $(n_s,r)$ plane with (95% C.L.): $r<0.28$, $0.942+0.12r\le n_s\le 0.974+0.12r$, see Eq. (20). The gray dot corresponds to the values for the monomial potential $m^2{\varphi }^2/2$ and the value $X=1$: $r=0.16$, $n_s=0.96$.

Figure 8

Running of the scalar index $d{n}_s/d\ln k$ vs $n_s$ for degrees of the potential $n=2$, 3, 4, respectively, for new inflation with $N_e=50$. The values for arbitrary $N_e$ follow directly from the $N_e=50$ value by using Eq. (16).

Figure 9

Reconstruction program for broken symmetry potentials with $N_e=50$, small field case $X<1$. ${\chi }_{50}\equiv {\chi }_c$ and ${\chi }_0$ vs $n_s$ for degrees of the potential $n=2$, 3, 4, respectively. These values of ${\chi }_c$, $n_s$ correspond to the region $r<0.16$.

Figure 10

Reconstruction program for broken symmetry potentials with $N_e=50$, large field case $X>1$. ${\chi }_{50}\equiv {\chi }_c$ and ${\chi }_0$ vs $n_s$ for degrees of the potential $n=2$, 3, 4, respectively. These values of $\chi$, $n_s$ correspond to the region $r>0.16$.

Figure 11

Coupling $g$ as a function of $X$ for $N_e=50$, for $n=2$, 3, 4 for chaotic inflation; $g$ turns to be a monotonically increasing function of $X$; $g$ vanishes for $X\to 0$ as $(\frac{X}{2}{)}^{2n-2}$ in sharp contrast with new inflation where $g$ strongly increases for $X\to 0$.

Figure 12

Left panel ${ϵ}_v$, right panel ${\eta }_v$ as a function of $X$ for $N_e=50$, for chaotic inflation with degrees of the potential $n=2$, 3, 4. The small $X$ behavior is $n$ independent.

Figure 13

Scalar spectral index $n_s$ for degrees of the potential $n=2$, 3, 4, respectively, for chaotic inflation with $N_e=50$. For $X\to 0$, $n_s$ reaches for all $n$ the value $n_s=0.96$ corresponding to the monomial potential $m^2{\varphi }^2/2$.

Figure 14

Tensor to scalar ratio $r$ vs $X$ for degrees of the potential $n=2$, 3, 4, respectively, for chaotic inflation with $N_e=50$. The horizontal dashed lines with the downward arrows delimit the region of 95% C.L. given by WMAP3 with no running $r<0.28$ [6].

Figure 15

Running of the scalar index $d{n}_s/d\ln k$ vs $X$ for degrees of the potential $n=2$, 3, 4, respectively, for chaotic inflation with $N_e=50$. The $X\to 0$ behavior is $n$ independent; $d{n}_s/d\ln k$ features a maximum value that gets stronger with increasing $n$. For chaotic inflation $d{n}_s/d\ln k$ takes negative as well as positive values, in contrast with new inflation where $d{n}_s/d\ln k<0$.

Figure 16

Tensor to scalar ratio $r$ vs $n_s$ for degrees of the potential $n=2$, 3, 4, respectively, for chaotic inflation with $N_e=50$. The range of 95% C.L. as determined by WMAP3 [6] is within the tilted box delimited by: $r<0.28$, $0.942+0.12r\le n_s\le 0.974+0.12r$, see Eq. (20).

Figure 17

Running of the scalar index $d{n}_s/d\ln k$ vs $n_s$ for degrees of the potential $n=2$, 3, 4, respectively, for chaotic inflation with $N_e=50$.

Figure 18

Reconstruction program for chaotic inflation with $N_e=50$, ${\chi }_c$ vs $n_s$ for $n=2$, 3, 4, respectively.