Potential Fluctuations in the Early Universe

In the standard hot Big Bang model, the Universe is highly ionized until a redshift $z_R \sim 1000$, the so-called recombination epoch, when protons and electrons combine to make hydrogen atoms (e.g. Peebles 1990). Prior to this epoch, photons are tightly coupled to the radiation by Thomson scattering, but once recombination is complete the Universe becomes transparent to radiation. Provided there is no subsequent energy input in the intergalactic medium that can reionize the Universe at high redshifts, photons propagate towards us along geodesics, unimpeded by the matter. Maps of the microwave background radiation therefore provide us with a picture of irregularities at the `last scattering surface' at $z_R \sim 1000$,when the Universe was about 300,000 years old.

At redshifts $z \gg 1$ an irregularity with a comoving scale $\lambda$ subtends an angle

$$\theta_\lambda \approx (\lambda/{\rm Mpc}) (\Omega_0 h) \;{\rm arcminutes}.$$ (3)

Thus, an irregularity of scale $\lambda \sim 100 h^{-1}{\rm Mpc}$,comparable in scale to the largest structures detected in galaxy surveys, subtends an angle of $\sim 100^\prime$ at the last scattering surface. The size of a causally connected region, ct (the `Hubble radius'), at the time of recombination subtends an angle

$$\theta_H \approx (\Omega_0/z_R)^{1/2} \;{\rm radians} \approx 2 (\Omega_0)^{1/2} (z_R/1000)^{-1/2}\;{\rm degrees}.$$ (4)

Temperature fluctuations in the microwave background radiation on angular scales $\gt \theta_H$ therefore cannot have been in causal contact at the time of recombination. How could such causally unconnected fluctuations have been formed? The only plausible solution is to appeal to exotic physical processes in the very early Universe that extend well beyond the Standard Model of particle physics. The properties of the microwave background anisotropies, e.g. their amplitude, power-spectrum, high-order correlations, thus provide quantitative tests of physical theories applicable at energies $\lower.5ex\hbox{$\; \buildrel \gt \over \sim \;$}10^{15}$ GeV when the Universe was less than 10^-35 seconds old.

For example, the temperature power spectra in Figure 1.3 show predictions for an inflationary universe dominated by cold dark matter. Quantum fluctuations during inflation generate macroscopic fluctuations in the gravitational potential with mean square value that scales with the scale of the perturbation as a power law

$$\left \langle \Delta \phi^2 \right \rangle \propto \lambda^{(1-n)/2},$$ (5)

(see Section Testing Theories of Inflation for further details) where n defines the spectral index of the fluctuations. Most models of inflation predict a spectral index that is close to unity $n \approx 1$ ; if n=1, the potential fluctuations are independent of scale and so this is often referred to as a `scale-invariant' spectrum.

Variations in the gravitational potential along different lines-of-sight to the last scattering surface cause temperature fluctuations in the CMB (called the Sachs-Wolfe effect) with an rms amplitude

$${\Delta T \over T} \sim {(\Delta \phi)_\lambda \over c^2} \propto \theta^{(1-n)/2}.$$ (6)

This effect dominates the anisotropy pattern at large angular scale ($\gt \theta_H$) and, since according to General Relativity the potential fluctuations $\Delta \phi$are time-independent, a map of the CMB anisotropies provides us with a picture of the potential fluctuations as they were at the time that they were generated in the early Universe.

In the more technical language of the previous Section, the Sachs-Wolfe effect leads to a temperature power spectrum of the form

C ^n-3, for >> 1
(Bond and Efstathiou 1987) and so for a scale-invariant spectrum, $\ell (\ell+1) C_\ell$ is approximately independent of multipole at $\ell \lower.5ex\hbox{$\; \buildrel < \over \sim \;$}90$ (corresponding to angular scales $\theta \gt \theta_H \sim 2^\circ$) as shown in Figure 1.3. A scale-invariant spectrum of potential fluctuations generated in the early Universe thus leads to a scale-invariant temperature fluctuation spectrum on the sky. An accurate determination of the spectral index from the CMB anisotropies provides an extremely powerful test of theories of inflation, as described in Section Testing Theories of Inflation.

[last update: 1 August 1999 by P. Fosalba]