1.4 How to Detect Inflation
1.4.2 CMB Polarization
1.4.2.2 Thomson Scattering
in §1.4.1 can be used to create power spectra and cross-spectra for the temperature and polarization of the CMB sky:
haT`m∗aT`0m0i=C`T Tδ``0δmm0 haE∗`maE`0m0i=C`EEδ``0δmm0 (1.68) haB∗`maB`0m0i=C`BBδ``0δmm0 haT`m∗aE`0m0i=C`T Eδ``0δmm0 (1.69) haT`m∗aB`0m0i=C`T Bδ``0δmm0 haE∗`maB`0m0i=C`EBδ``0δmm0. (1.70) (1.71) Since scalar spherical harmonics and the E-mode tensor harmonics have parity (−1)`, while the B-mode tensor harmonics have parity (−1)`+1, symmetry under parity transitions will require that C`T B = C`EB = 0. These moments are typically used to monitor foreground emission, though a non-zero detection of the C`T B or C`EB spectra would be a remarkable finding.
A non-zero detection of the C`BB spectrum is frequently called the “smoking gun” of inflation, since tensor modes are one of the unique predictions of inflation. The C`BB spectrum can be related back to the primordial tensor power spectrum as follows [15]:
C`BB = (4π)2 Z
k2 dk AT(k) ∆B`(k), (1.72) whereAT(k) is the primordial tensor power spectrum and ∆B`(k) is the transfer function for B-modes. A similar, though more complicated, equation can be written to relate the E-mode power spectrum to the primordial scalar and tensor modes. Both the tensor (AT(k)) and scalar (AS(k)) spectra will be necessary, since gravitational waves contribute to both E- and B-mode polarization. The transfer function is how the primordial spectrum is “processed”
by the plasma physics of the early universe into the CMB polarization spectra we see today.
As mentioned in §1.3, the amplitude of the gravitational wave spectrum is typically parameterized by the tensor-to-scalar ratio,r(Eqn 1.44). A measurement of the polarization power spectra allows us to get an estimate forr, and therefore the energy scale of inflation.
Figure 1.5: Thomson scattering of a quadrupole anisotropy. Figure from [15].
These anisotropies can be caused by either density anisotropies or by gravitational waves.
Density anisotropies (scalar perturbations of the metric) result in E-mode only polarization.
Gravitational waves (tensor perturbations of the metric) result in both E and B-mode polarization.
The Thomson scattering cross-section depends on polarization. The energy flux radi- ated into polarization state by an incident plane wave with propagation vector k0 and polarization vector 0 is
dσ
dΩ = 3σT
8π |∗·0|2, (1.73)
where σT is the total Thomson cross-section. The incoming light causes the electron to vibrate in the direction of its electric field vector with the same frequency as its own.
This vibration reradiates the light with a polarization direction parallel to the direction of the shaking. If radiation is incident upon the electron equally in all directions, no net polarization will occur. However, if the intensity of the radiation varies at 90◦ (i.e., a quadrupole pattern), then the resulting radiation will have a net polarization (see Fig. 1.5).
Let us assume that radiation is incident upon the scattering electron from all directions with intensityI(φ, θ) (Fig. 1.6). We can model an unpolarized incident beam as the linear superposition of two linearly polarized beams of equal intensity:
E~incident=E~i,1+E~i,2. (1.74)
The polarization of E~i,1 will point along the directioni,1 and the polarization of E~i,2 will
Figure 1.6: Scattering diagram for an incoming wave ki scattering off an electron at the origin, producing a scattered wave ks.
point alongi,2:
i,1 = sinθ xˆ+ cosθyˆ (1.75)
i,2= cosθcosφxˆ+ sinθcosφyˆ+ sinφz.ˆ (1.76) Note thati,1 andi,2 are not unit vectors.
We are viewing the scattered radiation from the axis ˆz. The polarization we will see from the scattered waves will be perpendicular to the scattering direction. The scattered waves will be polarized along s,1 and s,2:
s,1 = sinθxˆ+ cosθ yˆ (1.77)
s,2= cosθcosφxˆ+ sinθcosφy.ˆ (1.78) From here, we can calculate the polarization fraction along the x-axis. Here I am stating the result found in [86]:
Πx= R2π
0 dθRπ
0 sinφ dφ I(φ, θ)(sin2θ−cos2θ) sin2φ R2π
0 dθRπ
0 sinφ dφ I(φ, θ)(1 + cos2φ) . (1.79) An examination of this equation will provide some useful intuition. If there is azimuthal symmetry with respect to ki (i.e., if I(φ, θ) = I(φ)), the polarization fraction will be zero.
In fact, if we assume I(φ, θ) is of the form
I(φ, θ) =I0Y`m(φ, θ), (1.80)
then the polarization fraction will be zero for all spherical harmonics except for ` = 2.
Therefore, only scalar (Y20), vector (Y2±1), and tensor quadrupoles (Y2±2) will result in a net polarization in the scattered light.
As mentioned above, the scalar quadrupoles are the result of density perturbations in the early universe and are the dominant polarization mechanism of the CMB at angular scales less than 10◦. Due to the axial symmetry of Y20 spherical harmonics, scalar perturbations result only in E-mode polarization (no B-modes).
The vector modes (Y2±1) are not typically generated by inflationary theories (although some theories based on topological defects do predict them). The vector modes arise due to vorticity in the photon-baryon fluid. However, as the universe expands, the rotation rate will decrease (due to conservation of angular momentum) and the vector modes will quickly decay. We typically ignore vector perturbations in cosmological calculations for these reasons.
The tensor quadrupoles are the result of inflationary gravitational waves, which are predicted by most inflationary models. The quadrupoles are created by the gravitational wave stretching and compressing the light around the scattering electron (see Fig. 1.7b). The Thomson scattering of this quadrupole (Y2±2) will generate both E- and B-mode polarization in equal amounts. As mentioned before, B-mode polarization is uniquely generated by gravitational waves and is considered to be the “smoking gun” of inflation.
It should be noted here that B-mode polarization at small angular scales can also be caused by the gravitational lensing of the E-mode spectrum [4]. These lensing B-modes come from gravitational lensing of the E-mode spectrum by all the matter between us and the CMB. This lensing distorts the E-mode signal and creates both E and B modes. The lensing B-mode spectrum can be used to probe things like the matter distribution, neutrino masses, and dark energy. Since these angular scales (∼ 1 arcminute) are out of reach of the SPIDER instrument due to our large-aperture telescopes, I do not discuss them further here.
(a) Quadrupole temperature anisotropies created by density waves.
(b) Quadrupole temperature anisotropies created by gravitational waves.
Figure 1.7: Temperature anisotropies created by scalar and tensor perturbations.