Analytic approximation for the primordial spectra of single scalar potential models and its use in their reconstruction

We give final shape to a recent formalism for deriving the functional forms of the primordial power spectra of single-scalar potential models and theories which are related to them by conformal transformation. An excellent analytic approximation is derived for the nonlocal correction factors which are crucial to capture the “ringing” that can result from features in the potential. We also present the full algorithm for using our representation, including the nonlocal factors, to reconstruct the inflationary geometry from the power spectra.

Figure 1

The left hand graph shows one model’s scalar power spectrum as a function of $n$, the number of e-foldings from the beginning of inflation to first horizon crossing. The right hand graph shows the same power spectrum versus $N$, the number of e-foldings until the end of inflation. Early times correspond to small $n$ and large $N$, whereas late times correspond to large $n$ and small $N$. Recall that $n+N=n_e$, where $n_e$ is the total number of e-foldings during inflation.

Figure 2

The left-hand graph shows the local slow roll correction factor $C(\epsilon )$ (solid blue), which was defined in expression (5). Also shown is its global approximation of $1-\epsilon$ (dashed yellow) over the full inflationary range of $0\le \epsilon <1$. The right-hand graph shows $C(\epsilon )$ (solid blue) versus the better approximation of $1-0.55\epsilon$ (large dots) relevant to the range $0\le \epsilon <0.02$ favored by current data.

Figure 3

The left-hand graph shows the $\epsilon =0$ Green’s function $G(e^{\mathrm{\Delta }n})$ given in expression (50). The right-hand graph shows the coefficient of ${ϵ}^{\prime \prime }(n)$ in the small $\epsilon$ form (58) for $S_h(n,k)$. This function ${\mathcal{E}}_1(x)$ is defined by expressions (52), (54), and (59). The solid blue curve gives the exact numerical result while the large dots give the approximation resulting from the series expansion on the right-hand side of expression (54).

Figure 4

The coefficients of $[{ϵ}^{\prime }(n){]}^2$ (left) and ${ϵ}^{\prime }(n)$ (right) in the small $\epsilon$ form (58) for $S_h(n,k)$. In each case the solid blue curve gives the exact numerical result, while the large dots give the result of using the series approximations on the far right of (54)–(56) in expressions (60) and (61).

Figure 5

The left-hand figure shows the Hubble parameter and the right shows the first slow roll parameter for a model with features. This model which was proposed [28, 29] to explain the observed features in the scalar power spectrum at $\ell \approx 22$ and $\ell \approx 40$ which are visible in the data reported from both WMAP [27, 30, 31] and PLANCK [32, 33]. Note that the feature has little impact on $H(n)$ but it does lead to a distinct bump in $\epsilon (n)$. Recall that $n$ is the number of e-foldings from the start of inflation and that it relates to the number of e-foldings until the end of inflation as $N=n_e-n$, where $n_e$ is the total number of inflationary e-foldings.

Figure 6

These graphs show the scalar power spectrum for the model of Fig. 5. The left-hand figure compares the exact result (solid blue) with the local slow roll approximation ${\mathrm{\Delta }}_{\mathcal{R}}^2(k)\approx G{H}_k^2/\pi {\epsilon }_k\times C({\epsilon }_k)$ (yellow dashed). The right-hand figure compares the exact result (solid blue) with the much better approximation (yellow dashed) obtained from multiplying by $\exp [\sigma [\epsilon ](k)]$, using our analytic approximation (64) for $\sigma [\epsilon ](k)]$. Recall that $n$ is the number of e-foldings from the start of inflation and that it relates to the number of e-foldings until the end of inflation as $N=n_e-n$, where $n_e$ is the total number of inflationary e-foldings.

Figure 7

These graphs show the tensor power spectrum for the model of Figure 5. The left-hand figure compares the exact result (solid blue) with the local slow roll approximation ${\mathrm{\Delta }}_h^2(k)\approx \frac{16}{\pi }G{H}_k^{2}C({\epsilon }_k)$ (yellow dashed). The solid blue line on the right-hand graph shows the logarithm of the ratio of ${\mathrm{\Delta }}_h^2(k)$ to its local slow roll approximation. The yellow dashed line gives the nonlocal corrections of expression (63). Recall that $n$ is the number of e-foldings from the start of inflation and that it relates to the number of e-foldings until the end of inflation as $N=n_e-n$, where $n_e$ is the total number of inflationary e-foldings.

Figure 8

Numerical values of exponents 1, 2, and 4 for the model of Fig. 5. The left-hand graph gives separate results for expression (71) in dashed blue, expression (72) in dot-dashed yellow, and expression (74) in solid green. The right-hand graph shows the sum of all three exponents. Recall that $n$ is the number of e-foldings from the start of inflation and that it relates to the number of e-foldings until the end of inflation as $N=n_e-n$, where $n_e$ is the total number of inflationary e-foldings.

Figure 9

Numerical values of ${\exp }_3(n)$ and ${\exp }_5(n)$ for the model of Fig. 5. The left-hand graph gives separate results for expression (73) in dashed blue, and expression (75) in solid yellow. Note that ${\exp }_5(n)$ is responsible for correcting the small, systematic underprediction of the slow roll approximation before and after the feature. The right hand graph shows the sum. Recall that $n$ is the number of e-foldings from the start of inflation and that it relates to the number of e-foldings until the end of inflation as $N=n_e-n$, where $n_e$ is the total number of inflationary e-foldings.

Figure 10

Various choices for the left-hand side of the first pass reconstruction equation for the model of Fig. 5. The left-hand graph shows the first pass source $-\ln [\delta (n)]+2\ln [h(n)]$ in solid blue with $\ln [ϵ(n)]$ overlaid in dashed yellow. The poor agreement between the two curves is why using just $\ln [ϵ(n)]$ as the left hand side of the first pass reconstruction fails to converge when features are present. The right-hand graph shows the much better agreement between the same source (solid blue) and $\ln [ϵ(n)]-{\exp }_1(n)-{\exp }_2(n)-{\exp }_4(n)$ (dashed yellow). Recall that $n$ is the number of e-foldings from the start of inflation and that it relates to the number of e-foldings until the end of inflation as $N=n_e-n$, where $n_e$ is the total number of inflationary e-foldings.

Figure 11

These graphs show numerical reconstructions of $\ln [\epsilon (n)]$ for the power spectrum of Fig. 6. The solid blue line of the left hand graph shows the exact result while the yellow dashed line gives the result of integrating ${\mathcal{G}}_0(n-m)$—using the first six terms of the sum over $\ell$ in expression (83)—against the first pass source on the right-hand side of (78). The right-hand graph shows the result of adding the first order improvement ${\mathcal{G}}_1(n-m)$—computed using the first four terms of the sum over $m$ in expression (92). Recall that $n$ is the number of e-foldings from the start of inflation and that it relates to the number of e-foldings until the end of inflation as $N=n_e-n$, where $n_e$ is the total number of inflationary e-foldings.

Figure 12

The solid blue line of the left hand graph shows a numerical evaluation of the integral $\mathcal{I}(s)$ of expression (86). The 0th order approximation ${\mathcal{I}}_0(s)$ of expression (87) is overlaid in large dots. The solid blue line of the right-hand graph shows the deviation $\mathrm{\Delta }\mathcal{I}(s)\equiv \mathcal{I}(s)-{\mathcal{I}}_0(s)$. Our fit ${\mathcal{I}}_1(s)$ of expression (88) is overlaid in large dots.