Next: Discussion Up: Scaling Stress Seeds Previous: Metric Fluctuations

## CMB Anisotropies

Anisotropy and structure formation in causally seeded, or in fact any isocurvature model, proceeds by a qualitatively different route than the conventional adiabatic inflationary picture. As we have seen, fluctuations in the metric are only generated inside the horizon rather than at the initial conditions (see §VB). Since CMB anisotropies probe scales outside the horizon at last scattering, one would hope that this striking difference can be seen in the CMB. Unfortunately, gravitational redshifts between last scattering and today masks the signature in the temperature anisotropy. The scaling ansatz for the sources described in §VA in fact leads to near scale invariance in the large angle temperature because fluctuations are stimulated in the same way for each k-mode as it crosses the horizon between last scattering and the present. While these models generically leave a different signature in modes which cross the horizon before last scattering [22,27], models which mimic adiabatic inflationary predictions can be constructed [31].

Polarization provides a more direct test in that it can only be generated through scattering. The large angle polarization reflects fluctuations near the horizon at last scattering and so may provide a direct window on such causal, non-inflationary models of structure formation. One must be careful however to separate scalar, vector and tensor modes whose different large angle behaviors may obscure the issue. Let us now illustrate these considerations with the specific examples introduced in the last section.

The metric fluctuations produced by the seed sources generate CMB anisotropies through the Boltzmann equation (60). We display an example with B1=1 and B2=0.5 in Fig. 8. Notice that scaling in the sources does indeed lead to near scale invariance in the large angle temperature but not the large angle polarization. The small rise toward the quadrupole for the tensor temperature is due to the contribution of long-wavelength gravity waves that are currently being generated and depends on how rapidly they are generated after horizon crossing. Inside the horizon at last scattering (here ), scalar fluctuations generate acoustic waves as discussed in §IV which dominate for small characteristic times . On the other hand, these contributions are strongly damped below the thickness of the last scattering surface by dissipational processes. Note that features in the vector and tensor spectrum shown here are artifacts of our choice of source function. In a realistic model, the superposition of many sources of this type will wash out such features. The general tendencies however do not depend on the detailed form of the source. Note that vector and tensor contributions damp more slowly and hence may contribute significantly to the small-angle temperature anisotropy.

Figure 8: Temperature and polarization power spectra for a scaling anisotropic stress seeds with the minimal characteristic time B1=1 for scalars (S, solid), vectors (V, short-dashed), and tensors (T, long-dashed). Scalar temperature fluctuations at intermediate scales are dominated by acoustic contributions which then damp at small scales. B-parity polarization contributions are absent for the scalars, larger by an order of magnitude than E-parity contributions for the vectors and similar to but smaller than the E-parity for the tensors. Features in the vector and tensor spectra are artifacts of our choice of source and are unlikely to be present in a realistic model. The background cosmology is set to ,h=0.5, .

Polarization can only be generated by scattering of a quadrupole temperature anisotropy. For seeded models, scales outside the horizon at last scattering have not formed significant metric fluctuations (see e.g. Fig. 7). Hence quadrupole fluctuations, generated from the metric fluctuations through Eqns. (90), (92), and (93), are also suppressed. The power in k of the polarization thus drops sharply below . This drop of course corresponds to a lack of large angle power in the polarization. However its form at low depends on geometric aspects of the projection from k to . In these models, the large angle polarization is dominated by projection aliasing of power from small scales . The asymptotic expressions of Eqn. (77) thus determine the large angle behavior of the polarization: for scalars (EE), for vectors and for tensors (EE and BB); the cross spectrum goes as for each contribution. For comparison, the scale invariant adiabatic inflationary prediction has scalar polarization (EE) dropping off as and cross spectrum () as from Eqn. (92) because of the constant potential above the horizon (see Fig. 6). Seeded models thus predict a more rapid reduction in the scalar polarization for the same background cosmology.

Polarization can also help separate the three types of fluctuations. In accord with the general prediction (see §IIID), scalars produce no B-parity polarization, whereas vector B-parity dominates E-parity polarization by a factor of 6 and tensor B-parity is suppressed by a factor of 8/13 (see Eqn. (23)). Differences also arise in the temperature-polarization cross power spectra shown in Fig. 9. Independent of the nature of the source, above the angle the horizon subtends at last scattering, scalar and vector temperature perturbations from the last scattering surface [21] are anticorrelated with polarization, whereas they are correlated for tensor perturbations (see §IVB and [9]). Inside the horizon, the scalar polarization follows the scalar velocity which is out of phase with the effective temperature (see Eqn. 89). In the adiabatic model, scalar cross correlation reverses signs before the first acoustic peak, as compression overcomes the gravitational redshift of the Sachs-Wolfe effect, unlike the isocurvature models (see Fig. 6 and [28]). The sign test to distinguish scalars from tensors must thus be performed on scales larger than twice the first peak. Conversely, to use the cross correlation to distinguish adiabatic from isocurvature fluctuations, the scalar and tensor contributions must be separated.

Figure 9: Temperature-polarization cross power spectrum for the model of Fig. 8. Independent of the nature of the sources, the cross power at angles larger than that subtended by the horizon at last scattering is negative for the scalars and vectors and positive for the tensors. The more complex structure for the scalars at small angular scales reflects the correlation between the acoustic effective temperature and velocity at last scattering.

How do these results change with the model for the seeds? As we increase the characteristic time xc by decreasing B1=0.2 (see Fig. 10), the main effect comes from differences in the generation of metric fluctuations discussed in §VB. For the same amplitude anisotropic stress, scalars contributions dominate the vector and tensor contributions by factors of xc = B1-1 (see Eqn. 103). Note however that the scalar contributions come from the gravitational redshifts between last scattering and today rather than the acoustic oscillations (see §IVA) and hence produce no strong features. Because of the late generation of metric fluctuations in these models, the peak in the polarization spectra is also shifted with xc. Note however that the qualitative behavior of the polarization described above remains the same.

Figure 10: Same as Fig. 8 except with a larger characteristic time B1 =0.2, B2=0.1. Scalar gravitational redshift effects now dominate over scalar acoustic as well as vector and tensor contributions for the same stress source due the process by which stress perturbations generate metric fluctuations (see Fig. 7).

Although these examples do not exhaust the full range of possibilities for scaling seeded models, the general behavior is representative. Equal amplitude anisotropic stress sources tend to produce similar large angle temperature anisotropies if the source is active as soon as causally allowed . Large angle scalar polarization is reduced as compared with adiabatic inflationary models because of causal constraints on their formation. This behavior is not as marked in vectors and tensors due to the projection geometry but the relative amplitudes of the E-parity and B-parity polarization as well as the cross correlation can be used to separate them independently of assumptions for the seed sources. Of course in practice these tests at large angles will be difficult to apply due to the smallness of the expected signal.

Reionization increases the large angle polarization signal because the quadrupole anisotropies that generate it can be much larger [32]. This occurs since decoupling occurs gradually and the scattering is no longer rapid enough to suppress anisotropies. The prospects for separating the scalars, vectors and tensors based on polarization consequently also improve [33].

For angles smaller than that subtended by the horizon at last scattering, the relative contributions of these effects depends on a competition between scalar gravitational and acoustic effects and the differences in the generation and damping behavior of the scalar, vector and tensor perturbations.

Next: Discussion Up: Scaling Stress Seeds Previous: Metric Fluctuations
Wayne Hu
9/9/1997