The Quantum Structure of Space and Time (293 pages)

Cosmology 247

Quite a few models of inflation were derived since 2003 within compactified string
theory with stabilized moduli. The inflaton field, whose evolution drives inflation

is the only field which is not stabilized before the exit from inflation. Each of these

models relies on particular assumptions. Some of these models have clear predictions
for observables and therefore are FALSIFIABLE by the future observations. We will
shortly comment here on few recently constructed models of inflation where, under
clearly specified assumptions, one can predict three important observables.

(1) Tilt of the primordial spectrum of fluctuations, n,
(2) The tensor to scalar ratio, r = 5
(3) Light cosmic strings produced by the end of inflation
One can approximate the spectrum of the scalar and tensor perturbations of the
metric by a power-law, writing

where n,, nt are known as the scalar spectral index and the gravitational spectral

index, respectively, and k, is a normalization point, r is the tensor/scalar ration , the

relative amplitude of the tensor to scalars modes. The observations require n, close
to one, which corresponds to the perturbations in the curvature being independent
of scale. The deviation of the spectral index from one, n, - 1, is a measure of the
violation of the scale invariance of the spectrum of primordial fluctuations.
The only known at present viable mechanism for generating the observed per-
turbations is the inflationary cosmology, which posits a period of accelerated ex-
pansion in the Universe's early stages. In the simplest class of inflationary model
the dynamics are equivalent to that of a single scalar field 4 slowly rolling on an ef-

fective potential V(4). Inflation generates perturbations through the amplification

of quantum fluctuations, which are stretched to astrophysical scales by the rapid

expansion. The simplest models generate two types of density perturbations which

come from fluctuations in the scalar field and its corresponding scalar metric per-

turbation, and gravitational waves which are tensor metric fluctuations. Defining

slow-roll parameters, with primes indicating derivatives with respect to the scalar

field, as

167r 87r v '

the spectra can be computed using the slow-roll approximation (t, 171 << 1). In each

case, the expressions on the right-hand side are to be evaluated when the scale k

is equal to the Hubble radius during inflation. The spectral indices and tensor to

scalar ratio follow

n, N 1 - 6t + (^27) ; nt N -26,
The last relation is known as the consistency equation for the single field inflation
models, it becomes an inequality for multi-field inflationary models.
r N 16~ N -8nt,