As shown in the timeline of Figure 8, CMBR signatures can be generally classified into two main components: primary and secondary anisotropies, separated by a Surface of Last Scattering (SoLS). Both of these components include contributions from two distinctive phases: a surface marking the threshold of decoupling of ions and electrons from hydrogen atoms in primary signals, and a surface of reionization marking the start of multiphase secondary contributions through nonlinear structure evolution, star formation, and radiative feedback from the small scales to the large.

The black body spectrum of the isotropic background is essentially due to thermal equilibrium prior to the decoupling of ions and electrons, and few photon-matter interactions after that. At sufficiently high temperatures, prior to the decoupling epoch, matter was completely ionized into free protons, neutrons, and electrons. The CMB photons easily scatter off electrons, and frequent scattering produces a blackbody spectrum of photons through three main processes that occur faster than the Universe expands:

- Compton scattering in which photons transfer their momentum and energy to electrons if they have significant energy in the electron’s rest frame. This is approximated by Thomson scattering if the photon’s energy is much less than the rest mass. Inverse Compton scattering is also possible in which sufficiently energetic (relativistic) electrons can blueshift photons.
- Double Compton scattering can both produce and absorb photons, and thus is able to thermalize photons to a Planck spectrum (unlike Compton scattering which conserves photon number, and, although it preserves a Planck spectrum, relaxes to a Bose–Einstein distribution).
- Bremsstrahlung emission of electromagnetic radiation due to the acceleration of electrons in the vicinity of ions. This also occurs in reverse (free-free absorption) since charged particles can absorb photons. In contrast to Coulomb scattering, which maintains thermal equilibrium among baryons without affecting photons, Bremsstrahlung tends to relax photons to a Planck distribution.

Although the CMBR is a unique and deep probe of both the thermal history of the early Universe and primordial perturbations in the matter distribution, the associated anisotropies are not exclusively primordial in nature. Important modifications to the CMBR spectrum, from both primary and secondary components, can arise from large scale coherent structures, even well after the photons decouple from the matter at redshift , due to gravitational redshifting, lensing, and scattering effects.

The most important contributions to primary anisotropies between the start of decoupling and the surface of last scattering include the following effects:

- Sachs–Wolfe (SW) effect: Gravitational redshift of photons between potentials at the SoLS and the present. It is the dominant effect at large angular scales comparable to the horizon size at decoupling ().
- Doppler effect: Dipolar patterns are imprinted in the energy distribution from the peculiar velocities of the matter structures acting as the last scatterers of the photons.
- Acoustic peaks: Anisotropies at intermediate angular scales () are atttributed to small scale processes occurring until decoupling, including acoustic oscillations of the baryon-photon fluid from primordial density inhomogeneities. This gives rise to acoustic peaks in the thermal spectrum representing the sound horizon scale at decoupling.
- SoLS damping: The electron capture rate is only slightly faster than photodissociation at the start of decoupling, causing the SoLS to have a finite thickness (). Scattering over this interval damps fluctuations on scales smaller than the SoLS depth ().
- Silk damping: Photons can diffuse out of overdense regions, dragging baryons with them over small angular scales. This tends to suppress both density and radiation fluctuations.

All of these mechanisms perturb the black body background radiation since thermalization processes are not
efficient at redshifts smaller than 10^{7}.

Secondary anisotropies consist of two principal effects, gravitational and scattering. Some of the more important gravitational contributions to the CMB include:

- Early ISW effect: Photon contributions to the energy density of the Universe may be non-negligible compared to ordinary matter (dark or baryonic) at the last scattering. The decreasing contribution of photons in time results in a decay of the potential, producing the early Integrated Sachs–Wolfe (ISW) effect.
- Late ISW effect: In open cosmological models or models with a cosmological constant, the gravitational potential decays at late times due to a greater rate of expansion compared to flat spacetimes, producing the late ISW effect on large angular scales.
- Rees–Sciama effect: Evolving nonlinear strucutures (e.g., galaxies and clusters) generate time-varying potentials which can seed asymmetric energy shifts in photons crossing potential wells from the SoLS to the present.
- Lensing: In contrast to ISW effects which change the energy but not directions of the photons, gravitational lensing deflects the paths without changing the energy. This effectively smears out the imaging of the SoLS.
- Proper motion: Compact objects such as galaxy clusters can imprint a dipolar pattern in the CMB as they move across the sky.
- Gravitational waves: Perturbations in the spacetime fabric affect photon paths, energies, and polarizations, predominantly at scales larger than the horizon at decoupling.

Secondary scattering effects are associated with reionization and their significance depends on when and over what scales it takes place. Early reionization leads to large optical depths and greater damping due to secondary scattering. Over large scales, reionization has little effect since these scales are not in causal contact. At small scales, primordial anisotropies can be wiped out entirely and replaced by secondary ones. Some of the more important secondary scattering effects include:

- Thomson scattering: Photons are scattered by free electrons at sufficiently large optical depths achieved when the Universe undergoes a global reionization at late times. This damps out fluctuations since energies are averaged over different directions in space.
- Vishniac effect: In a reionized Universe, high order coupling between the bulk flow of electrons and their density fluctuations generates new anisotropies at small angles.
- Thermal Sunyaev–Zel’dovich effect: Inverse Compton scattering of the CMB by hot electrons in the intracluster gas of a cluster of galaxies distorts the black body spectrum of the CMB. Low frequency photons will be shifted to high frequencies.
- Kinetic Sunyaev–Zel’dovich effect: The peculiar velocities of clusters produces anisotropies via a Doppler effect to shift the temperature without distorting the spectral form. Its effect is proportional to the product of velocity and optical depth.
- Polarization: Scattering of anisotropic radiation affects polarization due to the angular dependence of scattering. Polarization in turn affects anisotropies through a similar dependency and tends to damp anisotropies.

To make meaningful comparisons between numerical models and observed data, all of these (low and high order) effects from both the primary and secondary contributions (see for example Section 4.1.4 and [94, 101]) must be incorporated self-consistently into any numerical model, and to high accuracy in order to resolve and distinguish amongst the various weak signals. The following sections describe some work focused on incorporating many of these effects into a variety of large-scale numerical cosmological models.

Many efforts based on linear perturbation theory have been carried out to estimate temperature anisotropies in our Universe (for example see [114] and references cited in [131, 94]). Although such linearized approaches yield reasonable results, they are not well-suited to discussing the expected imaging of the developing nonlinear structures in the microwave background. Also, because photons are intrinsically coupled to the baryon and dark matter thermal and gravitational states at all spatial scales, a fully self-consistent treatment is needed to accurately resolve the more subtle features of the CMBR. This can be achieved with a ray-tracing approach based on Monte-Carlo methods to track individual photons and their interactions through the evolving matter distributions. A fairly complete simulation involves solving the geodesic equations of motion for the collisionless dark matter which dominate potential interactions, the hydrodynamic equations for baryonic matter with high Mach number shock capturing capability, the transport equations for photon trajectories, a reionization model to reheat the Universe at late times, the chemical kinetics equations for the ion and electron concentrations of the dominant hydrogen and helium gases, and the photon-matter interaction terms describing scattering, redshifting, depletion, lensing, and Doppler effects.

Such an approach has been developed by Anninos et al. [15], and applied to a Hot Dark Matter (HDM) model of structure formation. In order to match both the observed galaxy-galaxy correlation function and COBE measurements of the CMBR, they find, for that model and neglecting reionization, the cosmological parameters are severely constrained to , where and are the density and Hubble parameters respectively.

In models where the IGM does not reionize, the probability of scattering after the photon-matter decoupling epoch is low, and the Sachs–Wolfe effect dominates the anisotropies at angular scales larger than a few degrees. However, if reionization occurs, the scattering probability increases substantially and the matter structures, which develop large bulk motions relative to the comoving background, induce Doppler shifts on the scattered CMBR photons and leave an imprint of the surface of last scattering. The induced fluctuations on subhorizon scales in reionization scenarios can be a significant fraction of the primordial anisotropies, as observed by Tuluie et al. [157] also using ray-tracing methods. They considered two possible scenarios of reionization: A model that suffers early and gradual (EG) reionization of the IGM as caused by the photoionizing UV radiation emitted by decaying neutrinos, and the late and sudden (LS) scenario as might be applicable to the case of an early generation of star formation activity at high redshifts. Considering the HDM model with and , which produces CMBR anisotropies above current COBE limits when no reionization is included (see Section 4.1.4), they find that the EG scenario effectively reduces the anisotropies to the levels observed by COBE and generates smaller Doppler shift anisotropies than the LS model, as demonstrated in Figure 9. The LS scenario of reionization is not able to reduce the anisotropy levels below the COBE limits, and can even give rise to greater Doppler shifts than expected at decoupling.

Additional sources of CMBR anisotropy can arise from the interactions of photons with dynamically evolving matter structures and nonstatic gravitational potentials. Tuluie et al. [156] considered the impact of nonlinear matter condensations on the CMBR in Cold Dark Matter (CDM) models, focusing on the relative importance of secondary temperature anisotropies due to three different effects: (i) time-dependent variations in the gravitational potential of nonlinear structures as a result of collapse or expansion (the Rees–Sciama effect), (ii) proper motion of nonlinear structures such as clusters and superclusters across the sky, and (iii) the decaying gravitational potential effect from the evolution of perturbations in open models. They applied the ray-tracing procedure of [15] to explore the relative importance of these secondary anisotropies as a function of the density parameter and the scale of matter distributions. They find that secondary temperature anisotropies are dominated by the decaying potential effect at large scales, but that all three sources of anisotropy can produce signatures of order as shown in Figure 10.

In addition to the effects discussed in this section, many other sources of secondary anisotropies (as mentioned in Section 4.1, including gravitational lensing, the Vishniac effect accounting for matter velocities and flows into local potential wells, and the Sunyaev–Zel’dovich (SZ) (Section 4.5.4) distortions from the Compton scattering of CMB photons by electrons in the hot cluster medium) can also be fairly significant. See [94, 152, 28, 80, 93] for more thorough discussions of the different sources of CMBR anisotropies.

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