Cosmic microwave background bispectrum on small angular scales

This article investigates the nonlinear evolution of cosmological perturbations on sub-Hubble scales in order to evaluate the unavoidable deviations from Gaussianity that arise from the nonlinear dynamics. It shows that the dominant contribution to modes coupling in the cosmic microwave background temperature anisotropies on small angular scales is driven by the sub-Hubble nonlinear evolution of the dark matter component. The perturbation equations, involving, in particular, the first moments of the Boltzmann equation for photons, are integrated up to second order in perturbations. An analytical analysis of the solutions gives a physical understanding of the result as well as an estimation of its order of magnitude. This allows one to quantify the expected deviation from Gaussianity of the cosmic microwave background temperature anisotropy and, in particular, to compute its bispectrum on small angular scales. Restricting to equilateral configurations, we show that the nonlinear evolution accounts for a contribution that would be equivalent to a constant primordial non-Gaussianity of order $f_{\mathrm{NL}}\sim 25$ on scales ranging approximately from $\ell \sim 1000$ to $\ell \sim 3000$.

Figure 1

Evolution of the two second-order gravitational potentials, ${\varphi }^{(2)}({\mathbf{k}}_1,{\mathbf{k}}_2)$ and ${\psi }^{(2)}({\mathbf{k}}_1,{\mathbf{k}}_2)$ for $k_1=k_2=10k_{\mathrm{eq}}$ (top panel) or $k_1=k_2=20k_{\mathrm{eq}}$ (bottom panel) and ${\mathbf{k}}_1·{\mathbf{k}}_2=0$.

Figure 2

Top panel: Comparison of the baryon and photon velocity perturbation at order 2 for $k_1=k_2=10k_{\mathrm{eq}}$ and ${\mathbf{k}}_1·{\mathbf{k}}_2=0$. It shows that $v_{\mathrm{r}}^{(2)}=v_{\mathrm{b}}^{(2)}$ with a good approximation until decoupling. Bottom panel: left-hand side and right-hand side of Eq. (45) for the adiabaticity condition at order 2. It can be seen that this adiabaticity condition holds until recombination, hence justifying the approximation of Sec. 3.

Figure 3

The time at which the contribution of cold dark matter and of radiation are comparable in the Poisson equation as a function of $k/k_{\mathrm{eq}}$. $k_{\mathrm{eq}}$ is the wave number of the mode that becomes sub-Hubble at the time of quality. For most of the scales of interest, i.e., for $k\gg k_{\mathrm{eq}}$, the contribution of CDM in the Poisson equation is dominant before equality, i.e., $y_{\star }<1$).

Figure 4

(Solid line) The second-order potential computed in the tight-coupling limit as a function of time $y$ and of $\theta$, angle between the wave vectors. It is compared to its expected late time behavior (72). Note that the convergence toward this solution is extremely rapid and takes place as soon as equality is reached, e.g., $y=1$. The results correspond to $k_1=6k_{\mathrm{eq}}$ and $k_2=12k_{\mathrm{eq}}$. The difference in the amplitude of the function is due to the fact that the baryon component has been neglected in the derivation of Eq. (72).

Figure 5

Behavior of ${\Theta }_{\mathrm{SW}}^{(2)}$ (black) and $k{v}_{\mathrm{r}}^{(2)}/\sqrt{3}$ (gray). The numerical integration is depicted with solid lines while the analytical estimates are plotted with dashed lines. From top to bottom, we have $k_1=k_2=30k_{\mathrm{eq}}$ and $k_1=k_2=40k_{\mathrm{eq}}$. The vertical gray zone represents the “surface” of last scattering.

Figure 6

Thick, black line: the bispectrum for the equilateral configuration computed in the flat sky limit. The thin gray lines represent the bispectrum that would be obtained by assuming a constant initial $f_{\mathrm{NL}}^{\Phi }$ and a linear transfer function that is neglecting the nonlinear dynamics. From bottom to top we have plotted $f_{\mathrm{NL}}^{\Phi }=1$, 10, 100, 1000, 10 000.

Figure 7

From top to bottom, left to right: evolution of the Bardeen potential, $\Phi$, and the density contrast $\delta$ for, respectively, radiation, baryons, and cold dark matter. The solid line corresponds to $k=10k_{\mathrm{eq}}$ and the dashed line corresponds to $k=20k_{\mathrm{eq}}$.

Figure 8

Left panel: Comparison of the baryon and photon velocity perturbation at first order for $k=10k_{\mathrm{eq}}$. It shows that $v_{\mathrm{r}}^{(1)}=v_{\mathrm{b}}^{(1)}$ with a good approximation until decoupling. Right panel: Comparison of $\frac{1}{4}{\delta }_{\mathrm{r}}^{(1)}$ and $\frac{1}{3}{\delta }_{\mathrm{r}}^{(1)}$. It can be seen that the adiabaticity condition holds until recombination, hence justifying the approximation of Sec. 3.

Figure 9

Left panel: thick gray line, the angular power spectrum using our code and the flat sky approximation (which does not include the late ISW). The thin black line represents the spectrum obtained using CAMB, which also takes into account the late ISW, but no reionization. Middle panel: The precision on small scale depends highly on the computation of the visibility function. On small scales, our transfer function is approximately 30% smaller than the one predicted by CAMB and is thus a good approximation. Right panel: the bispectrum obtained from a primordial constant $f_{\mathrm{NL}}=100$ evolved linearly.