Power spectrum of post-inflationary primordial magnetic fields

The origin of large scale magnetic fields is one of the most puzzling topics in cosmology and astrophysics. It is assumed that the observed magnetic fields result from the amplification of an initial field produced in the early universe. In this paper we compute the exact power spectrum of magnetic fields created after inflation best known as post-inflationary magnetic fields, using the first order cosmological perturbation theory. Our treatment differs from others’ works because we include an infrared cutoff which encodes only causal modes in the spectrum. The cross-correlation between magnetic energy density with Lorentz force and the anisotropic part of the electromagnetic field are exactly computed. We compare our results with previous works finding agreement in cases where the ratio between lower and upper cutoff is very small. However, we found that spectrum is strongly affected when this ratio is greater than 0.2. Moreover, the effect of a post-inflationary magnetic field with a lower cutoff on the angular power spectrum in the temperature distribution of cosmic microwave background was also exactly calculated. The main feature is a shift of the spectrum’s peak as a function of the infrared cutoff, therefore analyzing this effect we could infer the value of this cutoff and thus constraining the primordial magnetic fields generation models.

Figure 1

Log-log plot of magnetic energy density of PMF power spectrum $k^3{|{\rho }_B(k,\tau )|}^2$ versus $k({\mathrm{Mpc}}^{-1})$ with a smoothed magnetic field at 1 Mpc and with Batchelor spectral index ($n_B=2$) at large scale. From the top to bottom, the curves correspond to amplitude of the magnetic field of $B_{\lambda }=1,0.1,10^{-3},10^{-4},10^{-5},10^{-7},10^{-8},10^{-10}\mathrm{nG}$.

Figure 2

Plot of magnetic energy density of PMF power spectrum $k^3{|{\rho }_B(k,\tau )|}^2$ versus $k({\mathrm{Mpc}}^{-1})$ for different strength of the PMF (solid lines for $B_{\lambda }=1\mathrm{nG}$, medium dashed lines for $B_{\lambda }=5\mathrm{nG}$, and large dashed lines for $B_{\lambda }=10\mathrm{nG}$), and for different spectral indices ($n=2$ for lines with open circles, $n=7/2$ for lines with open squares, and $n=4$ for lines with filled circles).

Figure 3

Plot of Lorentz force spectra $k^3{|L^{(S)}(k,\tau )|}^2$ versus $k({\mathrm{Mpc}}^{-1})$ for different spectral indices ($n=2,3,7/2,4$ from solid to the large dashed lines).

Figure 4

Plot of scalar (solid lines), vector (short dashed lines), and tensor (large dashed lines) parts of the anisotropic trace-free part power spectrum $k^3{|\mathrm{\Pi }(k,\tau )|}^2$ versus $k({\mathrm{Mpc}}^{-1})$ for different spectral indices ($n=2$ for lines and $n=4$ for lines with filled circles).

Figure 5

Plot of cross-correlation of $k^3|{\rho }_B(k,\tau ){\mathrm{\Pi }}^{(S)}(k,\tau )|$ (just lines), $k^3|{\rho }_B(k,\tau )L^{(S)}(k,\tau )|$ (lines with filled circles), and $k^3{|{\rho }_B(k,\tau )|}^2$ (lines with open circles), for different spectral indices ($n=2$ for solid lines and $n=3$ for dashed lines).

Figure 6

Plot of magnetic energy density of PMF power spectrum $k^3{|{\rho }_B(k,\tau )|}^2$ versus $k({\mathrm{Mpc}}^{-1})$ for different values of smoothing scale ($\lambda =1\mathrm{Mpc}$ for lines with open circles, $\lambda =0.75\mathrm{Mpc}$ corresponds to lines with filled circles, and $\lambda =0.5\mathrm{Mpc}$ for lines with open squares), and for different spectral indices ($n=2$ refers to solid lines and $n=3$ for dashed lines).

Figure 7

Plot of anisotropic trace-free tensor part power spectrum $k^3{|{\mathrm{\Pi }}^{(T)}(k,\tau )|}^2$ versus $k({\mathrm{Mpc}}^{-1})$ for different values of smoothing scale ($\lambda =1\mathrm{Mpc}$ for lines with open circles, $\lambda =0.75\mathrm{Mpc}$ corresponds to lines with filled circles, and $\lambda =0.5\mathrm{Mpc}$ for lines with open squares), and for different spectral indices ($n=2$ refers to solid lines and $n=3$ for dashed lines).

Figure 8

Plot of comparison between our results of the anisotropic trace-free part power spectrum $k^3{|{\mathrm{\Pi }}^{(V,T)}(k,\tau )|}^2$ versus [23] (dashed line) and [11] (solid line) for values of $n=2$, $B_{\lambda }=1\mathrm{nG}$ and $\lambda =1\mathrm{Mpc}$.

Figure 9

Plot of the CMB temperature anisotropy angular power spectrum induced by scalar magnetic perturbations, where the lines with filled circles are for $n=2$ and the other ones for $n=5/2$. Here, the solid lines refer to $B_{\lambda }=10\mathrm{nG}$, large dashed lines for $B_{\lambda }=8\mathrm{nG}$, small dashed lines refer to $B_{\lambda }=5\mathrm{nG}$, and dotted lines for $B_{\lambda }=1\mathrm{nG}$.

Figure 10

Plot of the CMB temperature anisotropy angular power spectrum induced by tensor magnetic perturbations, where the lines with filled circles are for $n=2$ and the other ones for $n=4$. Here, the solid lines refer to $B_{\lambda }=1\mathrm{nG}$, large dashed lines for $B_{\lambda }=5\mathrm{nG}$, small dashed lines refer to $B_{\lambda }=8\mathrm{nG}$, and dotted lines for $B_{\lambda }=10\mathrm{nG}$.

Figure 11

Plot of magnetic energy density of PMF power spectrum $k^3{|{\rho }_B(k,\tau )|}^2$ versus $k({\mathrm{Mpc}}^{-1})$ with $n_B=2$ and for different values of infrared cutoff, lines with open squares refer to $k_m=0.9k_D$. Lines with filled squares correspond to $k_m=0.7k_D$, lines with filled circles for $k_m=0.5k_D$; dashed line refer to $k_m=0.3k_D$, and finally, lines with open circles and diamonds correspond to $k_m=0.1k_D$ and $k_m=0.001k_D$, respectively.

Figure 12

Comparison between the CMB temperature anisotropy angular power spectrum induced by scalar PMF at $k_m=0.001k_D$ lower cutoff, respect to the other ones with different values of infrared cutoff. Here, the solid horizontal line is for $k_m=0.1k_D$; small and large dashed lines refer to $k_m=0.3k_D$ and $k_m=0.4k_D$ respectively.

Figure 13

This plot shows again a comparison between the CMB temperature anisotropy angular power spectrum induced by scalar PMF at $k_m=0.001k_D$ lower cutoff, respect to the other ones with different values of infrared cutoff. Here, the solid horizontal line is for $k_m=0.1k_D$. The dashed lines describe $k_m=0.3k_D$, $k_m=0.4k_D$, $k_m=0.7k_D$, and $k_m=0.9k_D$ from the small to the longest dashed lines respectively.

Figure 14

Plot of anisotropic trace-free tensor part power spectrum $k^3{|{\mathrm{\Pi }}^{(T)}(k,\tau )|}^2$ versus $k({\mathrm{Mpc}}^{-1})$ with $n_B=2$ for different values of infrared-cutoff, lines with open squares refer to $k_m=0.9k_D$. Lines with filled squares correspond to $k_m=0.7k_D$, lines with filled circles for $k_m=0.5k_D$; dashed line refer to $k_m=0.3k_D$, and finally, lines with open circles and diamonds correspond to $k_m=0.1k_D$ and $k_m=0.001k_D$, respectively.

Figure 15

Comparison between the CMB temperature anisotropy angular power spectrum induced by tensor magnetic perturbation at $k_m=0.001k_D$ lower cutoff, respect to the other ones with different values of infrared cutoff. Here, the solid horizon line is for $k_m=0.1k_D$; small and large dashed lines refer to $k_m=0.3k_D$ and $k_m=0.4k_D$, respectively.

Figure 16

This plot shows again a comparison between the CMB temperature anisotropy angular power spectrum induced by tensor magnetic fluctuations at $k_m=0.001k_D$ lower cutoff, respect to the other ones with different values of infrared cutoff. Here, the solid horizon line is for $k_m=0.1k_D$. The dashed lines describe $k_m=0.3k_D$, $k_m=0.4k_D$, $k_m=0.7k_D$, and $k_m=0.9k_D$ from the small to the longest dashed lines, respectively.