Primordial magnetic field limits from cosmological data

We study limits on a primordial magnetic field arising from cosmological data, including that from big bang nucleosynthesis, cosmic microwave background polarization plane Faraday rotation limits, and large-scale structure formation. We show that the physically relevant quantity is the value of the effective magnetic field, and limits on it are independent of how the magnetic field was generated.

Figure 1

rms rotation angle ${\alpha }_{\mathrm{rms}}$ as a function of spectral index $n_B$ for the case when $B_{\mathrm{eff}}=1\mathrm{nG}$ and ${\nu }_0=100\mathrm{GHz}$. Circles correspond to the computed values.

Figure 2

Effective magnetic field limits set by the measurement of the rotation angle ${\alpha }_{\mathrm{rms}}$ for different spectral indices ($n_B=-3$, $-2$, $-1$, 0, 1, 2, from bottom to top). The horizontal solid line shows the upper limit set by BBN. Vertical dashed lines correspond to the angles ${\alpha }_{\mathrm{rms}}=3.16°$ that are set by the BBN limit on the effective magnetic field with spectral index $n_B=2$ and ${\alpha }_{\mathrm{rms}}=4.4°$ set by the WMAP 7-year data. The numerical values of the effective magnetic field constraints (in nG at 100 GHz) from the ${\alpha }_{\mathrm{rms}}=4.4°$ limit are shown on the graph for each spectral index value.

Figure 3

The mass dispersion at $z=10$ for $B_{\mathrm{eff}}=6\mathrm{nG}$ as a function of magnetic field power spectral index $n_B$. From top to bottom (at the left-hand side of the plot), the curves correspond to $n_B=2$, 1, 0, $-1$, $-2$, $-2.8$.

Figure 4

Constraint on the magnetic field strength $B_{\mathrm{eff}}$ as a function of the power spectral index $n_B$.

Figure 5

The squared spherical Bessel functions $j_l^2(x)$ for $l=5000$ (top) and $l=10000$ (bottom). Clearly $j_l^2(x)$ peaks at $x\approx l$.

Figure 6

The sum of the squared spherical Bessel function $\sum _{l=0}^{l_C}(2l+1)j_l^2(x)$ for $l_C=x_S\simeq 16000$. The sum converges to 1 (horizontal solid line).