Testing Theories of Inflation

In 1981 Guth proposed a radical new theory of the early universe, the inflationary model, which can explain a number of fundamental problems in Cosmology: e.g. why is the Universe so nearly homogeneous and spatially flat, and why are there no magnetic monopoles or other remnants of Grand Unified phase transitions? The inflationary model has since become a paradigm in modern cosmology (see e.g. Kolb and Turner 1990, Linde 1990). The key idea, common to all models of inflation, is that at some very early time after the Big Bang, the Universe underwent a rapid near-exponential expansion increasing in size by a factor $\lower.5ex\hbox{$\; \buildrel \gt \over \sim \;$}e^{60}$. The large expansion during the inflationary phase dilutes the abundance of remnant monopoles and smooths any initial spatial curvature leading to a Universe that is very nearly spatially flat at the present epoch. Furthermore, quantum fluctuations produced during inflation are stretched in scale by many orders of magnitude producing density fluctuations that can then grow by gravity to make the galaxies and clusters of galaxies seen in the Universe today (Guth & Pi 1982, Hawking 1982, Bardeen, Steinhardt & Turner 1983).

Figure 1.6: The table gives values of the spectral indices n_s and n_t of the density and gravitational wave spectra and the ratio r of their amplitudes, for several common inflationary potentials V(). The box in the figure shows the range of r-n_s consistent with inflationary models. Most models of inflation are constrained to lie along the grey lines. (Adapted from Steinhardt 1996).

By studying the CMB anisotropies we can determine the nature of the fluctuations as they were at the time that they were generated, less than $\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}10^{-35}$ seconds after the Big Bang. The CMB anisotropies offer a direct test, not only of whether inflation took place, but also of the specific details of the inflationary mechanism. For example, in the simplest inflationary models, the energy density of the Universe is dominated by a single scalar field $\phi$ (called the inflaton field). The dynamics of the inflaton field is governed by an effective potential $V(\phi)$. In principle, the potential $V(\phi)$, the nature of the inflaton field and its couplings to other fields should be deducible from fundamental physics, but at present we have so few experimental constraints on physics at energies > 10¹⁵ GeV that the parameters of inflationary models, which determine the amplitudes and spectra of the primordial fluctuations, are almost arbitrary. Observations of the CMB anisotropies are thus one of the few ways of setting firm experimental constraints on theories at ultra-high energies of $\lower.5ex\hbox{$\; \buildrel \gt \over \sim \;$}10^{15}$ GeV. Nevertheless, successful inflationary models (i.e. models that produce sufficient expansion) lead to some predictions that depend only weakly on specific details such as the shape of the inflaton potential. These are as follows:

• Spatial flatness: Most models of inflation lead to a Universe which is spatially flat with $\Omega = 1 \pm \epsilon$, where $\epsilon$ is exponentially small. Here $\Omega$ includes all contributions to the energy density including matter, radiation and any vacuum energy associated with a non-zero cosmological constant.
• Gaussian perturbations: Quantum fluctuations during inflation generate irregularities that obey Gaussian statistics. Any deviations from Gaussian statistics in the CMB temperature anisotropies would thus be difficult to reconcile with the inflationary model.
• Gravitational wave fluctuations: Quantum fluctuations of massless gravitons can lead to a spectrum of primordial gravitational waves after inflation is complete. These produce a special signature in the CMB radiation, unique to inflationary models, as illustrated in Figure 1.3.
• Nearly scale invariant fluctuation spectra: The spectra of the density and gravitational wave fluctuations resulting from inflation can be characterized by power law indices n_s and n_t respectively (for scalar and tensor modes). Most models of inflation predict nearly scale-invariant fluctuations (cf Section Potential Fluctuations in the Early Universe.) with $n_s \approx 1$ and $n_t \approx 0$.

An exactly exponential (de Sitter) expansion during inflation would lead to precisely scale invariant density and gravitational wave spectra with indices n_s = 1 and n_t =0 respectively. In order to match on to Friedman-Robertson-Walker models, the expansion rate must slow down at the end of inflation. This leads to small deviations from a purely scale invariant spectrum in any realistic inflationary model that depend on the precise shape of the potential $V(\phi)$. However, extreme fine tuning is required to produce inflationary models with spectral indices that lie outside the bounds $0.7 \lower.5ex\hbox{$\; \buildrel < \over \sim \;$}n_s \lower.5ex\hbox{$\; \buildrel < \over \sim \;$}1.2$ and $0.3 \lower.5ex\hbox{$\; \buildrel < \over \sim \;$}n_t \lower.5ex\hbox{$\; \buildrel < \over \sim \;$}0$ (Steinhardt 1996, Bond 1996) shown by the heavy box in Figure 1.6.

Furthermore, the ratio r=C₂^t/C₂^s of the CMB quadrupoles from gravitational wave (tensor) and density perturbation (scalar) components depends on the shape of the potential and hence on the spectral indices n_s and n_t. Inflationary models can be grouped into two categories, one in which n_t = n_s - 1 and the ratio

r = 7 ( 1 - n_s),

shown by the diagonal grey line in Figure 1.6, and another in which negligible gravitational waves are produced, shown by the grey line along the abscissa. Any experimentally determined deviation of the CMB anisotropies from the grey lines in Figure 1.6 would be extremely problematic for inflationary models.

Figure 1.7 illustrates the power of PLANCK for testing inflationary models. Here we have assumed a pure power law fluctuation spectrum with scalar spectral index n_s=1 and we have estimated the error on the determination of the spectral index from an all-sky experiment as the number of multipoles is increased. The dotted lines show the limits on n_s from an experiment with the same angular resolution and sensitivity as the 2 year COBE data; the $1\sigma$ error on n approaches $\pm 0.3$ at multipoles $\ell \sim 20$ corresponding to the resolution limit of the COBE maps. The solid lines show the $1\sigma$ errors on n_s from an experiment with the sensitivity of PLANCK. Here the error on n_s is determined primarily by ``cosmic variance'' (equation 1.9) and decreases rapidly to $\delta n_s = 0.023$ at $\ell =100$. An experiment with the sensitivity and resolution of PLANCK can therefore measure the spectral index with an accuracy that is much smaller than the width of the heavy box plotted in Figure 1.8 and can therefore distinguish between different theories of inflation. The ability to measure small deviations from a precise scale invariant spectrum will provide tight constraints on the form of the inflationary potential and hence on fundamental physics at energies $\lower.5ex\hbox{$\; \buildrel \gt \over \sim \;$}10^{15}\;$ GeV.

Figure 1.7: This figure illustrates the fundamental limits on the accuracy with which the scalar spectral index n_s can be determined from observations of the CMB anisotropies at multipoles $\ell \lower.5ex\hbox{$\; \buildrel < \over \sim \;$}100$. The initial spectrum is assumed to be a scale-invariant power law with n_s=1 and gravitational waves have been ignored. The lines show the $1\sigma$ scatter in n_s from a maximum likelihood fit to the temperature power spectrum as the number of multipoles is increased. The dotted lines show the limits from the 2 year COBE data, which contain no useful information on multipoles $\ell \gt 15$. The shaded regions show the $1\sigma$ scatter in n_s expected from PLANCK.

In the next section, we extend the simple analysis of Figure 1.7 to the full range of multipoles accessible to PLANCK and to a wide range of cosmological parameters including the inflationary parameters n_s, n_t and r.

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