Inflationary perturbations and precision cosmology

Inflationary cosmology provides a natural mechanism for the generation of primordial perturbations which seed the formation of observed cosmic structure and lead to specific signals of anisotropy in the cosmic microwave background radiation. In order to test the broad inflationary paradigm as well as particular models against precision observations, it is crucial to be able to make accurate predictions for the power spectrum of both scalar and tensor fluctuations. We present detailed calculations of these quantities utilizing direct numerical approaches as well as error-controlled uniform approximations, comparing with the (uncontrolled) traditional slow-roll approach. A simple extension of the leading-order uniform approximation yields results for the power spectra amplitudes, the spectral indices, and the running of spectral indices, with accuracy of the order of $0.1\%$—approximately the same level at which the transfer functions are known. Several representative examples are used to demonstrate these results.

Figure 1

Compilation of observations of the matter power spectrum $P_{\mathrm{o}\mathrm{b}\mathrm{s}}(k)$ taken from Tegmark et al. [54] (with permission). This particular figure assumes a flat scalar scale-invariant model with ${\Omega }_m=0.28$, $h=0.72$ and ${\Omega }_b/{\Omega }_m=0.16$, $\tau =0.17$, and a bias, $b_{*}=0.92$ for the SDSS galaxies. The solid line is the theoretical curve.

Figure 2

Imaginary (growing) part of the scalar mode function for a power-law model with $p=11$ for the mode $k=0.0214h{\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$. The solid red line is the exact analytical solution, the dashed-dotted blue line the result from the numerical solution of the exact equations, and the dashed green line the uniform approximation to leading order; the analytical and uniform approximation results are almost on top of each other. The arrow shows the turning point and the green band is the estimated error bound for the leading-order uniform approximation.

Figure 3

The absolute value of the scalar mode function for a power-law model following Fig. 2.

Figure 4

Relative error of $|u_k|$ shown in Fig. 3 for the numerical calculation (blue line). The arrow shows the turning point for the mode.

Figure 5

Relative errors of $\nu$, $ϵ$, and ${\delta }_1$; model parameters are as specified in Fig. 2.

Figure 6

(a) Scalar power spectrum $P_S(k)$ and (b) tensor power spectrum $P_T(k)$ for a chaotic $\frac{1}{2}{m}^2{\varphi }^2$ model, parameters as specified in Table . Solid red line: exact numerical results, dashed green line: uniform approximation, dashed-dotted blue line: slow-roll; dotted purple line: second-order improved uniform approximation [cf. Eq. (46)]; the green band is the estimate for the error bound for the uniform approximation, Eq. (40). The exact results and the results from the improved uniform and slow-roll approximation are visually on top of each other. The error estimate for the improved uniform approximation, Eq. (47), is so small as to be indistinguishable from the result itself.

Figure 7

(a) Scalar spectral index $n_S(k)$ and (b) tensor spectral index $n_T(k)$ for a chaotic $\frac{1}{2}{m}^2{\varphi }^2$ model, parameters as specified in Table . Solid red line: exact numerical results, dashed green line: uniform approximation, dashed-dotted blue line: slow-roll; the green band is the error estimate for the uniform approximation. Unlike the case for the power spectrum, accurate results for the uniform approximation are obtained without recourse to the improvement procedure specified by Eq. (46).

Figure 8

Tensor to scalar ratio $R(k)$ for a chaotic $\frac{1}{2}{m}^2{\varphi }^2$ model; parameters as specified in Table . All three lines lie practically on top of each other; see Fig. 9 for the relative errors.

Figure 9

Relative error of the tensor to scalar ratio $R(k)$ for a chaotic $\frac{1}{2}{m}^2{\varphi }^2$ model; parameters as specified in Table . Here the dotted purple line denotes the (second-order) improved leading order of the uniform approximation; the green error band is the error estimate for the first-order uniform approximation, the darker green band is the error estimate for the improved leading-order result.

Figure 10

(a) Scalar spectral index $n_S(k)$ and (b) tensor spectral index $n_T(k)$ for the quartic potential $\lambda {\varphi }^4$, parameters as specified in Table . Solid red line: exact numerical results, dashed green line: uniform approximation, dashed-dotted blue line: slow-roll; the green band is the error estimate for the uniform approximation.

Figure 11

$z^{\prime \prime }/(z{a}^2)$ and ${\nu }_S^2$ and $a^{\prime \prime }/a^3$ and ${\nu }_T$ for the $C^2$ potential; the point ${\varphi }_{*}$ is reached at $N\approx 12.4$. The beginning in time of the numerical calculation is at $N=0$. The inflationary attractor is reached at $N\simeq 1.8$, checked by varying $\dot{\varphi }(0)$ and determining at which e-fold the $\varphi$ behavior becomes independent of the initial velocities. The quantities $a^{\prime \prime }/a^3$ and ${\nu }_T$, relevant for tensor perturbations, are much smoother than the corresponding quantities $z^{\prime \prime }/z{a}^2$ and ${\nu }_S$ for the scalar perturbations, leading to smaller errors in the approximations as discussed in the text.

Figure 12

(a) Scalar power spectrum $P_S(k)$ and (b) tensor power spectrum $P_T(k)$ for the $C^2$ potential in Eqs. (69, 70); parameters: $\alpha =-100$, $m^2=(1.90\pm 0.21)\times {10}^{-12}/8\pi G$, ${\varphi }_{*}=15.2/\sqrt{8\pi G}$, $\varphi (0)=17.5/\sqrt{8\pi G}$, $\dot{\varphi }(0)=-0.2/\sqrt{8\pi G}\mathrm{s}$. Solid red line: exact numerical results, dashed green line: uniform approximation, dashed-dotted blue line: slow-roll; the green band is the estimate for the error bound for the (unimproved) uniform approximation. Again, the exact results and the results from the improved uniform and slow-roll approximation are on top of each other.

Figure 13

(a) Relative errors for the scalar power spectrum $P_S(k)$ and (b) tensor power spectrum $P_T(k)$ for the $C^2$ potential, Eqs. (69, 70). In both cases, the light green band denotes the estimated error bound for the first-order uniform approximation (40) and the dark green band denotes the best estimated error for the improvement procedure (46). The results are nicely consistent with these estimates showing where the second-order improvement can be enhanced by going to higher order, and where it cannot.

Figure 14

(a) Scalar spectral index $n_S(k)$ and (b) tensor spectral index $n_T(k)$ for the $C^2$ potential in Eqs. (69, 70) in the region around the kink; parameters specified as in Fig. 12. Solid red line: exact numerical results, dashed green line: uniform approximation, dashed-dotted blue line: slow-roll; the green band is the error estimate for the uniform approximation.

Figure 15

Relative error of the (a) scalar and (b) tensor spectral index [see Figs. 14a, 14b] for the $C^2$ potential in Eqs. (69, 70). The green band is the error estimate for the uniform approximation. In addition to the uniform and the slow-roll approximation we also show the slow-roll approximation including higher-order derivatives of $V(\varphi )$ [see Eqs. (71, 72)]. Light brown dotted line: first-order slow-roll, dark brown dashed-dotted line: second-order slow-roll.

Figure 16

Ratio $R(k)$ of tensor to scalar perturbations for the $C^2$ potential in Eqs. (69, 70); the green band is the error estimate for the uniform approximation to leading order; the relative difference between all three approximations is below $2\%$ (see Fig. 17), making it difficult to distinguish the curves.

Figure 17

Relative error for the ratio $R(k)$ [ratio of tensor to scalar perturbations] for the $C^2$ potential in Eqs. (69, 70); the green band is the error estimate for the uniform approximation to leading order, the darker green band is the estimate for the improved leading order.

Figure 18

Tensor to scalar ratio $R(k)$ against the scalar spectral index $n_S(k)$ for the exact numerical results of the inflation models investigated here; via the $k$-dependence (running of the spectral index and the tensor to scalar ratio) each model produces a line—the solid red line denotes the ${\varphi }^2$ model, the dashed green line the ${\varphi }^4$ model and the dashed-dotted blue line the glued $C^2$ potential; the black crosses denote the central WMAP pivot $k=0.05{\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$. The two dotted lines delimit small field, large field and hybrid inflation models according to the classification in Ref. 50. In the inset we show a zoom into the region where the $C^2$ potential goes over into the ${\varphi }^2$ potential. It is interesting to note that at this point the curve bends over sharply.

Figure 19

Tensor to scalar ratio $R(k)$ against the tensor spectral index $n_T(k)$; specifications as in Fig. 18 except that the dotted line here denotes $R=-8n_T$. By fitting the ${\varphi }^2$- and the ${\varphi }^4$-model results, they are found to lie on a straight line $R=-(7.757\pm 0.004)n_T$. Although the numerical prefactor is different from the slow-roll estimate of $-8$, the degeneracy is still obvious. As in the previous figure, the inset shows a zoom into the region where the curve of the $C^2$ potential bends over to join the line from the ${\varphi }^2$ potential.