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We study brane inflation scenarios in a warped throat geometry and show that there exists a consistency condition between the non-Gaussianity of the curvature perturbation and the amplitude and scale-dependence of the primordial gravitational waves. This condition is independent of the warping of the throat and the form of the inflaton potential. We find that such a relation could be tested by a future CMB polarization experiment if the Planck satellite is able to detect both a gravitational wave background and a non-Gaussian statistic. In models where the observable stage of inflation occurs when the brane is in the tip region of the throat, we derive a further consistency condition involving the scalar spectral index, the tensor-scalar ratio and the curvature perturbation bispectrum. We show that when such a relation is combined with the WMAP3 results, it leads to a model-independent bound on the gravitational wave amplitude given by 0.001 < r < 0.01. This corresponds to the range of sensitivity of the next generation of CMB polarization experiments.

In a simple reanalysis of the KKLMMT scenario, we argue that the slow roll condition in the D3-anti-D3-brane inflationary scenario in superstring theory requires no more than a moderate tuning. The cosmic string tension is very sensitive to the conformal coupling: with less fine-tuning, the cosmic string tension (as well as the ratio of tensor to scalar perturbation mode) increases rapidly and can easily saturate the present observational bound. In a multi-throat brane inflationary scenario, this feature substantially improves the chance of detecting and measuring the properties of the cosmic strings as a window to the superstring theory and our pre-inflationary universe.

We study the consistency of large-field inflation in low-energy effective field theories of string theory. In particular, we focus on the stability of K\"ahler moduli in the particularly interesting case where the non-perturbative superpotential of the K\"ahler sector explicitly depends on the inflaton field. This situation arises generically due to one-loop corrections to the instanton action. The field dependence of the modulus potential feeds back into the inflationary dynamics, potentially impairing slow roll. We distinguish between world-sheet instantons from Euclidean D-branes, which typically yield polynomial one-loop Pfaffians, and gaugino condensates, which can yield exponential or periodic corrections. In all scenarios successful slow-roll inflation imposes bounds on the magnitude of the one-loop correction, corresponding to constraints on possible compactifications. While we put a certain emphasis on Type IIB constructions with mobile D7-branes, our results seem to apply more generally.

We propose that inflation is driven by a (complex) neutral Higgs of the MSSM extension of the SM, in a chaotic-like inflation setting. The SUSY breaking soft term masses are of order $10^{12}-10^{13}$ GeV, which is identified with the inflaton mass scale and is just enough to stabilise the SM Higgs potential. The fine-tuned SM Higgs has then a mass around 126 GeV, in agreement with LHC results. We point out that the required large field excursions of chaotic inflation may be realised in string theory with the (complex) inflaton/Higgs identified with a continuous Wilson line or D-brane position. We show specific examples and study in detail a IIB orientifold with D7-branes at singularities, with SM gauge group and MSSM Higgs sector. In this case the inflaton/Higgs fields correspond to D7-brane positions along a two-torus transverse to them. Masses and monodromy are induced by closed string $G_3$ fluxes, and the inflaton potential can be computed directly from the DBI+CS action. We show how this action sums over Planck suppressed corrections, which amount to a field dependent rescaling of the inflaton fields, leading to a linear potential in the large field regime. We study the evolution of the two components of the Higgs/inflaton and compute the slow-roll parameters for purely adiabatic perturbations. For large regions of initial conditions slow roll inflation occurs and 50-60 efolds are obtained with r>0.07, testable in forthcoming experiments. Our scheme is economical in the sense that both EWSB and inflation originate in the same sector of the theory, all inflaton couplings are known and reheating occurs efficiently.

We consider models of inflection point inflation. The main drawback of such models is that they suffer from the overshoot problem. Namely the initial condition should be fine tuned to be near the inflection point for the universe to inflate. We show that stringy realizations of inflection point inflation are common and offer a natural resolution to the overshoot problem.

String theory abounds with light scalar fields (the dilaton and various moduli) which create a host of observational problems, and notably some serious cosmological difficulties similar to the ones associated with the Polonyi field in the earliest versions of spontaneously broken supergravity. We show that all these problems are naturally avoided if a recently introduced mechanism for fixing the vacuum expectation values of the dilaton and/or moduli is at work. We study both the classical evolution and the quantum fluctuations of such scalar fields during a primordial inflationary era and find that the results are naturally compatible with observational facts. In this model, dilatons or moduli within a very wide range of masses (which includes the SUSY-breaking favored 1 TeV value and extends up to the Planck scale) qualify to define a novel type of essentially stable ultra-weakly interacting massive particles able to provide enough mass density to close the universe

We consider a possible scenario for the generation of Dirac neutrino masses motivated by Type I string theory. The smallness of the neutrino Yukawa couplings is explained by an anisotropic compactification with one compactification radius larger than the others. In addition to this we utilise small Yukawa couplings to develop strong links between the origin of neutrino masses and the physics driving inflation. We construct a minimal model which simultaneously accommodates small Dirac neutrino masses leading to bi-large lepton mixing as well as an inflationary solution to the strong CP and to the $\mu$ problem.

Inflation is now a basic ingredient of the modern cosmology. After reviewing the basic characteristics of inflation, we briefly discuss inflation in string and brane theories, focusing on the problem of the exit from inflation in these theories. We present a model, based on type-0 string theory, in which our universe after the inflationary phase, passes smoothly to a flat Robertson-Walker universe.

Natural inflation relies on the existence of an axion decay constant which is super-Planckian. In string theory only sub-Planckian axion decay constants have been found in any controlled regime. However in field theory it is possible to generate an enhanced super-Planckian decay constant by an appropriate aligned mixing between axions with individual sub-Planckian decay constants. We study the possibility of such a mechanism in string theory. In particular we construct a new realisation of an alignment scenario in type IIA string theory compactifications on a Calabi-Yau where the alignment is induced through fluxes. Within field theory the original decay constants are taken to be independent of the parameters which induce the alignment. In string theory however they are moduli dependent quantities and so interact gravitationally with the physics responsible for the mixing. We show that this gravitational effect of the fluxes on the moduli can precisely cancel any enhancement of the effective decay constant. This censorship of an effective super-Planckian decay constant depends on detailed properties of Calabi-Yau moduli spaces and occurs for all the examples and classes that we study. We expand these results to a general superpotential assuming only that the axion superpartners are fixed supersymmetrically and are able to show for a large class of Calabi-Yau manifolds, but not all, that the cancellation effect occurs and is independent of the superpotential. We also study simple models where the moduli are fixed non-supersymmetrically and find that similar cancellation behaviour can emerge. Finally we make some comments on a possible generalisation to axion monodromy inflation models.

Cosmologists have embraced a particular ad hoc formula for the primordial power spectrum from inflation for universes with $\Omega_0 < 1$. However, the so-called ``Open Inflation'' models, which are attracting renewed interest in the context of the ``string theory landscape'' give a different result, and offer a more fully developed picture of the cosmology and fundamental physics basis for inflation with $\Omega_0 < 1$. The Open Inflation power spectrum depends not only on $\Omega_0$, but on the parameters of the effective fields that drive the universe {\em before} the Big Bang (in ``another part of the landscape''). This paper considers the search for features in CMB temperature anisotropy data that might reflect a primordial spectrum of the Open Inflation form. We ask whether this search could teach us about high energy physics that described the universe before the onset of the Big Bang, and perhaps even account for the low CMB quadrupole. Unfortunately our conclusion is that the specific features we consider are unobservable even with future experiments although we note a possible loophole connected with our use of the thin wall approximation.

Inflationary models within string theory exhibit unusual scalar field dynamics involving non-minimal kinetic terms and generically referred to as k-inflation. In this situation, the standard slow-roll approach used to determine the behavior of the primordial cosmological perturbations cannot longer be used. We present a generic method, based on the uniform approximation, to analytically derive the primordial power spectra of scalar and tensor perturbations. At leading order, the scalar spectral index, its running and the tensor-to-scalar ratio are modified by the new dynamics. We provide their new expression, correct previous results at next-to-leading order and clarify the definition of what is the tensor-to-scalar ratio when the sound horizon and Hubble radius are not the same. Finally, we discuss the constraints the parameters encoding the non-minimal kinetic terms have to satisfy, such as the sound speed and the energy scale of k-inflation, in view of the fifth year Wilkinson Microwave Anisotropy Probe (WMAP5) data.

We perform a general algebraic analysis on the possibility of realising slow-roll inflation in the moduli sector of string models. This problem turns out to be very closely related to the characterisation of models admitting metastable vacua with non-negative cosmological constant. In fact, we show that the condition for the existence of viable inflationary trajectories is a deformation of the condition for the existence of metastable de Sitter vacua. This condition depends on the ratio between the scale of inflation and the gravitino mass and becomes stronger as this parameter grows. After performing a general study within arbitrary supergravity models, we analyse the implications of our results in several examples. More concretely, in the case of heterotic and orientifold string compactifications on a Calabi-Yau in the large volume limit we show that there may exist fully viable models, allowing both for inflation and stabilisation. Additionally, we show that subleading corrections breaking the no-scale property shared by these models always allow for slow-roll inflation but with an inflationary scale suppressed with respect to the gravitino scale. A scale of inflation larger than the gravitino scale can also be achieved under more restrictive circumstances and only for certain types of compactifications.

The recent progress in embedding inflation in string theory has made it clear that the problem of moduli stabilization cannot be ignored in this context. In many models a special role is played by the volume modulus, which is modified in the presence of mobile branes. The challenge is to stabilize this modified volume while keeping the inflaton mass small compared to the Hubble parameter. It is then crucial to know not only how the volume modulus is modified, but also to have control over the dependence of the potential on the inflaton field. We address these questions within a simple setting: toroidal N=1 type IIB orientifolds. We calculate corrections to the superpotential and show how the holomorphic dependence on the properly modified volume modulus arises. The potential then explicitly involves the inflaton, leaving room for lowering the inflaton mass through moderate fine-tuning of flux quantum numbers.

The encoding of an inflating patch of space-time in terms of a dual theory is discussed. Following Bousso's interpretation of the holographic principle, we find that those are generically described not by states in the dual theory but by density matrices. We try to implement this idea on simple deformations of the AdS/CFT examples, and an argument is given as to why inflation is so elusive to string theory.

The majority of models of inflation in string theory predict an absence of measurable gravitational waves, r < < 10^{-3}. The most promising proposals for making string theoretic models that yield measurable tensor fluctuations involve axion fields with slightly broken shift symmetry. We consider such models in detail, with a particular focus on the N-flation scenario and on axion valley/natural inflation models. We find that in Calabi-Yau threefold compactifications with logarithmic Kahler potentials K it appears to be difficult to meet the conditions required for axion inflation in the supergravity regime. However, in supergravities with an (approximately) quadratic shift-symmetric K, axion inflation may be viable. Such Kahler potentials do arise in some string models, in specific limits of the moduli space. We describe the most promising classes of models; more detailed study will be required before one can conclude that working models exist.

We propose a mechanism for the natural inflation with and without modulation in the framework of type IIB string theory on toroidal orientifold or orbifold. We explicitly construct the stabilization potential of complex structure, dilaton and K\"ahler moduli, where one of the imaginary component of complex structure moduli becomes light which is identified as the inflaton. The inflaton potential is generated by the gaugino-condensation term which receives the one-loop threshold corrections determined by the field value of complex structure moduli and the axion decay constant of inflaton is enhanced by the inverse of one-loop factor. We also find the threshold corrections can also induce the modulations to the original scalar potential for the natural inflation. Depending on these modulations, we can predict several sizes of tensor-to-scalar ratio as well as the other cosmological observables reported by WMAP, Planck and/or BICEP2 collaborations.

We find a description of hybrid inflation in (3+1)-dimensions using brane dynamics of Hanany-Witten type. P-term inflation/acceleration of the universe with the hybrid potential has a slow-roll de Sitter stage and a waterfall stage which leads towards an N=2 supersymmetric ground state. We identify the slow-roll stage of inflation with a non-supersymmetric `Coulomb phase' with Fayet-Iliopoulos term. This stage ends when the mass squared of one of the scalars in the hypermultiplet becomes negative. At that moment the brane system starts undergoing a phase transition via tachyon condensation to a fully Higgsed supersymmetric vacuum which is the absolute ground state of P-term inflation. A string theory/cosmology dictionary is provided, which leads to constraints on parameters of the brane construction from cosmological experiments. We display a splitting of mass levels reminiscent of the Zeeman effect due to spontaneous supersymmetry breaking.

We review cosmological inflation and its realization in quantum field theory and in string theory. This material is a portion of a book, also entitled "Inflation and String Theory", to be published by Cambridge University Press.

At its very beginning, the universe is believed to have grown exponentially in size via the mechanism of inflation. The almost scale-invariant density perturbation spectrum predicted by inflation is strongly supported by cosmological observations, in particular the cosmic microwave background radiation. However, the universe's precise inflationary scenario remains a profound problem for cosmology and for fundamental physics. String theory, the most-studied theory as the final physical theory of nature, should provide an answer to this question. Some of the proposals on how inflation is realized in string theory are reviewed. Since everything is made of strings, some string loops of cosmological sizes are likely to survive in the hot big bang that followed inflation. They appear as cosmic strings, which can have intricate properties. Because of the warped geometry in flux compactification of the extra spatial dimensions in string theory, some of the cosmic strings may have tensions substantially below the Planck or string scale. Such strings cluster in a manner similar to dark matter leading to hugely enhanced densities. As a result, numerous fossil remnants of the low tension cosmic strings may exist within the galaxy. They can be revealed through the optical lensing of background stars in the near future and studied in detail through gravitational wave emission. We anticipate that these cosmic strings will permit us to address central questions about the properties of string theory as well as the birth of our universe.

The recent BICEP2 measurements of B-modes indicate a large tensor-to-scalar ratio in inflationary cosmology, which points towards trans-Planckian evolution of the inflaton. We propose possible string-theory realizations thereof. Schemes for natural and axion monodromy inflation are presented in the framework of the type IIB large volume scenario. The inflaton in both cases is given by the universal axion and its potential is generated at subleading order by F-terms. Our models are shown to feature a natural mechanism for inflaton decay into predominantly Standard Model particles.