1. Overview

• For small z values the rule is that how larger z the larger the distance.
• For large z values the rule is that how larger z value how smaller the distance. The largest z value is the CMB radiation.
• The maximum distance is roughly speaking at z = 1, but depents about the cosmological parameters.

1.1 Space expands

To say that space expands exponentially means that two inertial observers are moving farther apart with accelerating velocity.
In stationary coordinates for one observer, a patch of an inflating universe has the following polar metric:

Ds2=

This is just like an inside-out black hole metric—it has a zero in the dt ocomponent on a fixed radius sphere called the cosmological horizon. Objects are drawn away from the observer at R=0 towards the cosmological horizon, which they cross in a finite proper time. This means that any inhomogeneities are smoothed out, just as any bumps or matter on the surface of a black hole horizon are swallowed and disappear.

You can not the compare the physics of a black hole with the physics of the Big Bang. A Black hole is a stable physical object while the evolution of the Universe is not.
IMO a much better comparison is with a Super Novae which is a true explosion.

Since the space–time metric has no explicit time dependence, once an observer has crossed the cosmological horizon, observers closer in take its place. This process of falling outward and replacement points closer in are always steadily replacing points further out—an exponential expansion of space–time.

Again no physical interpretation.

This steady-state exponentially expanding spacetime is called a de Sitter space, and to sustain it there must be a cosmological constant, a vacuum energy proportional to lambda. In this case, the equation of state is p=-rho.

The physical conditions from one moment to the next are stable: the rate of expansion, called the Hubble parameter, is nearly constant, and the scale factor of the universe is proportional to E^ht.

That may be true but this is no physical explanation.

Inflation is often called a period of accelerated expansion because the distance between two fixed observers is increasing exponentially (i.e. at an accelerating rate as they move apart), while lambda can stay approximately constant (see deceleration parameter).

This is much more a mathematical explanation (using lambda) than a physical explanation, which indirectly explains nothing

1.2 Few inhomogeneities remain

1.3 Key requirement

1.4 Reheating

2 Motivations

2.1 Horizon problem

2.2 Flatness problem

Main article: Flatness problem Another problem is the flatness problem (which is sometimes called one of the Dicke coincidences, with the other being the cosmological constant problem). It had been known in the 1960s that the density of matter in the universe was comparable to the critical density necessary for a flat universe (that is, a universe whose large scale geometry is the usual Euclidean geometry, rather than a non-Euclidean hyperbolic or spherical geometry).

The problem is we are mixing here a physical problem with a mathematical problem.

• When the density at a specific moment is equal to the critical density the universe is called flat. That means the size of the universe increases at a constant rate. k=0 Lambda=0
• When the density is higher than the critical density at a certain moment the size of the universe reaches a maximum value and then decreases. k=0, Lambda < 0
• When the density is smaller than the critical density the size of the universe increases exponential k=0, Lambda > 0

In all these case the universe is mathematical flat: k=0.
The current mainstream opinion is that k=0 and Lambda>0 (At present Lamba=0,01155)
See: Friedmann Lambda=0.01155

Therefore, regardless of the shape of the universe the contribution of spatial curvature to the expansion of the universe could not be much greater than the contribution of matter. But as the universe expands, the curvature redshifts away more slowly than matter and radiation. Extrapolated into the past, this presents a fine-tuning problem because the contribution of curvature to the universe must be exponentially small (sixteen orders of magnitude less than the density of radiation at big bang nucleosynthesis, for example). This problem is exacerbated by recent observations of the cosmic microwave background that have demonstrated that the universe is flat to the accuracy of a few percent.

It is not clear what they mean. The CMB requires primary IMO a physical interpretation not a mathematical interpretation.

2.3 Magnetic-monopole problem

The magnetic monopole problem (sometimes called the exotic-relics problem) says that if the early universe were very hot, a large number of very heavy [why?], stable magnetic monopoles would be produced.

I agree with the "why?" question
How do you know that when the Universe was hot there should be stable magnetic monopoles?
Any way what is a stable magnetic monopoles and there should be two types (one positive and one negative) which makes the whole issue very problematic.

This is a problem with Grand Unified Theories, which proposes that at high temperatures (such as in the early universe) the electromagnetic force, strong, and weak nuclear forces are not actually fundamental forces but arise due to spontaneous symmetry breaking from a single gauge theory.

Those three forces could have a common origin.

These theories predict a number of heavy, stable particles that have not yet been observed in nature.

Which impies that this part of the theory could be wrong.

The most notorious is the magnetic monopole, a kind of stable, heavy "knot" in the magnetic field. Monopoles are expected to be copiously produced in Grand Unified Theories at high temperature, and they should have persisted to the present day, to such an extent that they would become the primary constituent of the universe. Not only is that not the case, but all searches for them have failed, placing stringent limits on the density of relic magnetic monopoles in the universe.

Which again impies that this part of the theory could be wrong.

A period of inflation that occurs below the temperature where magnetic monopoles can be produced would offer a possible resolution of this problem: monopoles would be separated from each other as the universe around them expands, potentially lowering their observed density by many orders of magnitude.

Assuming that monoples never have existed is a much simpler solution.
The question can be asked if it is possible to have a piece of iron (Fe material) which has no magnetic field.

Though, as cosmologist Martin Rees has written, "Skeptics about exotic physics might not be hugely impressed by a theoretical argument to explain the absence of particles that are themselves only hypothetical. Preventive medicine can readily seem 100 percent effective against a disease that doesn't exist!"

For more about monopoles read this: Magnetic Monoples in Wikipedia.

3 History

3.1 Precursors

In the early days of General Relativity, Albert Einstein introduced the cosmological constant to allow a static solution, which was a three-dimensional sphere with a uniform density of matter. A little later, Willem de Sitter found a highly symmetric inflating universe, which described a universe with a cosmological constant that is otherwise empty. It was discovered that Einstein's solution is unstable, and if there are small fluctuations, it eventually either collapses or turns into de Sitter's.

This is correct and has nothing to do with inflation.

In the early 1970s Zeldovich noticed the serious flatness and horizon problems of big bang cosmology; before his work, cosmology was presumed to be symmetrical on purely philosophical grounds. In the Soviet Union, this and other considerations led Belinski and Khalatnikov to analyze the chaotic BKL singularity in General Relativity. Misner's Mixmaster universe attempted to use this chaotic behavior to solve the cosmological problems, with limited success.

In the late 1970s, Sidney Coleman applied the instanton techniques developed by Alexander Polyakov and collaborators to study the fate of the false vacuum in quantum field theory. Like a metastable phase in statistical mechanics—water below the freezing temperature or above the boiling point—a quantum field would need to nucleate a large enough bubble of the new vacuum, the new phase, in order to make a transition. Coleman found the most likely decay pathway for vacuum decay and calculated the inverse lifetime per unit volume. He eventually noted that gravitational effects would be significant, but he did not calculate these effects and did not apply the results to cosmology.

This may be all mathematical correct. The problem is does it have any physical relation with the evolution of the universe.

In the Soviet Union, Alexei Starobinsky noted that quantum corrections to general relativity should be important in the early universe. These generically lead to curvature-squared corrections to the Einstein–Hilbert action and a form of f(R) modified gravity. The solution to Einstein's equations in the presence of curvature squared terms, when the curvatures are large, leads to an effective cosmological constant. Therefore, he proposed that the early universe went through a de Sitter phase, an inflationary era. This resolved the problems of cosmology, and led to specific predictions for the corrections to the microwave background radiation, corrections that were calculated in detail shortly afterwards.

I think it is extremely tricky to claim that it is possible to calculate the CMB radiation pattern (power Spectrum) correctly as a function of an fast expansion burst which happened in the first second after the Big Bang

In 1978, Zeldovich noted the monopole problem, which was an unambiguous quantitative version of the horizon problem, this time in a fashionable subfield of particle physics, which led to several speculative attempts to resolve it. In 1980, working in the west, Alan Guth realized that false vacuum decay in the early universe would solve the problem, leading him to propose scalar driven inflation. Starobinsky's and Guth's scenarios both predicted an initial de Sitter phase, differing only in the details of the mechanism.

3.2 Early inflationary models

According to Andrei Linde, the earliest theory of inflation was proposed by Erast Gliner(1965) but the theory was not taken seriously except by Andrei Sakharov, 'who made an attempt to calculate density perturbations produced in this scenario." Independently, inflation was proposed in January 1980 by Alan Guth as a mechanism to explain the nonexistence of magnetic monopoles; it was Guth who coined the term "inflation".

In his book "The inflationary Universe" at the end, pages 286 and 287, Alan Guth does not mention the magnetic monople issue.

At the same time, Starobinsky argued that quantum corrections to gravity would replace the initial singularity of the universe with an exponentially expanding de Sitter phase. In October 1980, Demosthenes Kazanas suggested that exponential expansion could eliminate the particle horizon and perhaps solve the horizon problem, while Sato suggested that an exponential expansion could eliminate domain walls (another kind of exotic relic). In 1981 Einhorn and Sato published a model similar to Guth's and showed that it would resolve the puzzle of the magnetic monopole abundance in Grand Unified Theories. Like Guth, they concluded that such a model not only required fine tuning of the cosmological constant, but also would very likely lead to a much too granular universe, i.e., to large density variations resulting from bubble wall collisions

The whole issue of galaxy formation depents on inhomogeneities.

Guth proposed that as the early universe cooled, it was trapped in a false vacuum with a high energy density, which is much like a cosmological constant.

What is physical a cosmological constant?

As the very early universe cooled it was trapped in a metastablestate (it was supercooled), which it could only decay out of through the process of bubble nucleation via quantum tunneling. Bubbles of true vacuum spontaneously form in the sea of false vacuum and rapidly begin expanding at the speed of light. Guth recognized that this model was problematic because the model did not reheat properly: when the bubbles nucleated, they did not generate any radiation.

Radiation in the sense of electromagnetic radiation ?

Radiation could only be generated in collisions between bubble walls. But if inflation lasted long enough to solve the initial conditions problems, collisions between bubbles became exceedingly rare. In any one causal patch it is likely that only one bubble will nucleate

3.3 Slow-roll inflation

The bubble collision problem was solved by Andrei Linde and independently by Andreas Albrecht and Paul Steinhardt in a model named new inflation or slow-roll inflation (Guth's model then became known as old inflation). In this model, instead of tunneling out of a false vacuum state, inflation occurred by a scalar field rolling down a potential energy hill. When the field rolls very slowly compared to the expansion of the universe, inflation occurs. However, when the hill becomes steeper, inflation ends and reheating can occur.

The issue is that you have to know the physical processes that took place.
To claim that something is caused by an inflation field with a special shape is no solution because you have to answer the question what caused this special inflation field.

3.4 Effects of asymmetries

Eventually, it was shown that new inflation does not produce a perfectly symmetric universe, but that tiny quantum fluctuations in the inflaton are created.

What is a symmetric universe in this context? To assume that any process evolves harmoneous and synchroneous throughout space is a misconception. There are always inhomogeneous areas.

These tiny fluctuations form the primordial seeds for all structure created in the later universe. These fluctuations were first calculated by Viatcheslav Mukhanov and G. V. Chibisov in the Soviet Union in analyzing Starobinsky's similar model. In the context of inflation, they were worked out independently of the work of Mukhanov and Chibisov at the three-week 1982 Nuffield Workshop on the Very Early Universe at Cambridge University. The fluctuations were calculated by four groups working separately over the course of the workshop: Stephen Hawking; Starobinsky; Guth and So-Young Pi; and James M. Bardeen, Paul Steinhardt and Michael Turner.

How do you know that these calculations are correct?

4 Observational status

Inflation is a mechanism for realizing the cosmological principle, which is the basis of the standard model of physical cosmology: it accounts for the homogeneity and isotropy of the observable universe.

It is very important to point out that is written here observable universe. That means not entire universe,

In addition, it accounts for the observed flatness and absence of magnetic monopoles.

How is this flatness observed?

Since Guth's early work, each of these observations has received further confirmation, most impressively by the detailed observations of the cosmic microwave background made by the Wilkinson Microwave Anisotropy Probe (WMAP) spacecraft. This analysis shows that the universe is flat to an accuracy of at least a few percent, and that it is homogeneous and isotropic to a part in 100,000.

What is here the definition of flat ? Mathematical k=0 ? or physical Lambda=0 ?. See 2.2 Flatness Problem

In addition, inflation predicts that the structures visible in the universe today formed through the gravitational collapse of perturbations that were formed as quantum mechanical fluctuations in the inflationary epoch.

How?
IMO it is impossible to claim a physical connection between the galaxies we see to day and what happened during the first seconds after the BIg Bang. Specific of what we call the period of inflation.

The detailed form of the spectrum of perturbations called a nearly-scale-invariant Gaussian random field (or Harrison–Zel'dovich spectrum) is very specific and has only two free parameters, the amplitude of the spectrum and the spectral index, which measures the slight deviation from scale invariance predicted by inflation (perfect scale invariance corresponds to the idealized de Sitter universe). Inflation predicts that the observed perturbations should be in thermal equilibrium with each other (these are called adiabatic or isentropic perturbations).

What is the physical link between inflation and thermal equilibrium ?

This structure for the perturbations has been confirmed by the WMAP spacecraft and other cosmic microwave background experiments, and galaxy surveys, especially the ongoing Sloan Digital Sky Survey.These experiments have shown that the one part in 100,000 inhomogeneities observed have exactly the form predicted by theory. Moreover, there is evidence for a slight deviation from scale invariance. The spectral index, ns is equal to one for a scale-invariant spectrum. The simplest models of inflation predict that this quantity is between 0.92 and 0.98.

Suppose there was never a period of rapid inflation what then should the spectral index value ns be?

From the data taken by the WMAP spacecraft it can be inferred that ns = 0.963 ± 0.012, implying that it differs from one at the level of two standard deviations (2s). This is considered an important confirmation of the theory of inflation.
A number of theories of inflation have been proposed that make radically different predictions, but they generally have much more fine tuning than is necessary.As a physical model, however, inflation is most valuable in that it robustly predicts the initial conditions of the universe based on only two adjustable parameters: the spectral index (that can only change in a small range) and the amplitude of the perturbations. Except in contrived models, this is true regardless of how inflation is realized in particle physics.
Occasionally, effects are observed that appear to contradict the simplest models of inflation. The first-year WMAP data suggested that the spectrum might not be nearly scale-invariant, but might instead have a slight curvature. However, the third-year data revealed that the effect was a statistical anomaly. Another effect has been remarked upon since the first cosmic microwave background satellite, the Cosmic Background Explorer: the amplitude of the quadrupole moment of the cosmic microwave background is unexpectedly low and the other low multipoles appear to be preferentially aligned with the ecliptic plane. Some have claimed that this is a signature of non-Gaussianity and thus contradicts the simplest models of inflation. Others have suggested that the effect may be due to other new physics, foreground contamination, or even publication bias

An experimental program is underway to further test inflation with more precise measurements of the cosmic microwave background. In particular, high precision measurements of the so-called "B-modes" of the polarization of the background radiation could provide evidence of the gravitational radiation produced by inflation, and could also show whether the energy scale of inflation predicted by the simplest models (1015–1016 GeV) is correct. In March 2014, it was announced that B-mode polarization of the background radiation consistent with that predicted from inflation had been demonstrated by a South Pole experiment, a collaboration led by four principal investigators from the California Institute of Technology, Harvard University, Stanford University, and the University of Minnesota BICEP2. Other potentially corroborating measurements are expected to be performed by the Planck spacecraft, although it is unclear if the signal will be visible, or if contamination from foreground sources will interfere with these measurements. Other forthcoming measurements, such as those of 21 centimeter radiation (radiation emitted and absorbed from neutral hydrogen before the first stars turned on), may measure the power spectrum with even greater resolution than the cosmic microwave background and galaxy surveys, although it is not known if these measurements will be possible or if interference with radio sources on earth and in the galaxy will be too great.

Dark energy is broadly similar to inflation, and is thought to be causing the expansion of the present-day universe to accelerate. However, the energy scale of dark energy is much lower, 10-12 GeV, roughly 27 orders of magnitude less than the scale of inflation.

5 Theoretical status

List of unsolved problems in physics Is the theory of cosmological inflation correct, and if so, what are the details of this epoch?

What are the physical details!

What is the hypothetical inflaton field giving rise to inflation?

That is an important unanswered question.

In the early proposal of Guth, it was thought that the inflaton was the Higgs field, the field that explains the mass of the elementary particles.

Is supposed to explain

It is now believed by some that the inflaton cannot be the Higgs field, although the recent discovery of the Higgs boson has increased the number of works considering the Higgs field as inflaton.

I think it is impossible to clearly establish a physical link between the higgs particle and the inflation theory ie exponential expansion.

One problem of this identification is the current tension with experimental data at the electroweak scale, which is currently under study at the Large Hadron Collider (LHC). Other models of inflation relied on the properties of grand unified theories. Since the simplest models of grand unification have failed, it is now thought by many physicists that inflation will be included in a supersymmetric theory like string theory or a supersymmetric grand unified theory. At present, while inflation is understood principally by its detailed predictions of the initial conditiën for the hot early universe, the particle physics is largely ad hoc modelling. As such, though predictions of inflation have been consistent with the results of observational tests, there are many open questions about the theory.

5.1 Fine-tuning problem

One of the most severe challenges for inflation arises from the need for fine tuning in inflationary theories.

One of the most challenges for inflation is what are the physical processes that occured throughout the universe that caused a period of rapid expansion and why did this (almost) stop all of a sudden.

In new inflation, the slow-roll conditions must be satisfied for inflation to occur. The slow-roll conditions say that the inflaton potential must be flat (compared to the large vacuum energy) and that the inflaton particles must have a small mass. In order for the new inflation theory of Linde, Albrecht and Steinhardt to be successful, therefore, it seemed that the universe must have a scalar field with an especially flat potential and special initial conditions.

Understanding the universe implies physics. To claim that you need a scalar field only moves the issues, but does not solve the issues.
Concepts like flat potential and special initial conditions also explain nothing.
How further back in time you go the more special the universe was. Special in the sense that all the processes involved were unique

However, there are ways to explain these fine-tunings. For example, classically scale invariant field theories, where scale invariance is broken by quantum effects, provide an explanation of the flatness of inflationary potentials, as long as the theory can be studied through perturbation theory.

All of this text explains nothing.

5.1.1 Andrei Linde

Andrei Linde proposed a theory known as chaotic inflation in which he suggested that the conditions for inflation are actually satisfied quite generically and inflation will occur in virtually any universe that begins in a chaotic, high energy state and has a scalar field with unbounded potential energy.

What Andrei Linde should explain why he uses a concept like unbounded potential energy.
To call a theory chaotic is very unlucky.
If you want to explain something the thermynology used should be clear.

However, in his model the inflaton field necessarily takes values larger than one Planck unit: for this reason, these are often called large field models and the competing new inflation models are called small field models. In this situation, the predictions of effective field theory are thought to be invalid, as renormalization should cause large corrections that could prevent inflation.

When you want to explain something it should be clear and unambigous....

This problem has not yet been resolved and some cosmologists argue that the small field models, in which inflation can occur at a much lower energy scale, are better models of inflation. While inflation depends on quantum field theory (and the semiclassical approximation to quantum gravity) in an important way, it has not been completely reconciled with these theories. The BICEP2 experiment detected evidence for primordial gravitational waves consistent with Linde's model.

Robert Brandenberger has commented on fine-tuning in another situation. The amplitude of the primordial inhomogeneities produced in inflation is directly tied to the energy scale of inflation. There are strong suggestions that this scale is around 1016 GeV or 10-3 times the Planck energy. The natural scale is naïvely the Planck scale so this small value could be seen as another form of fine-tuning (called a hierarchy problem): the energy density given by the scalar potential is down by 10-12 compared to the Planck density. This is not usually considered to be a critical problem, however, because the scale of inflation corresponds naturally to the scale of gauge unification.

5.2 Eternal inflation

Andry linde is a tipical proponent of eternal inflation In many models of inflation, the inflationary phase of the universe's expansion lasts forever in at least some regions of the universe.

This raises the question what is the inflation theory.

This occurs because inflating regions expand very rapidly, reproducing themselves.

This is easy to write but very difficult physical to explain

Unless the rate of decay to the non-inflating phase is sufficiently fast, new inflating regions are produced more rapidly than non-inflating regions. In such models most of the volume of the universe at any given time is inflating. All models of eternal inflation produce an infinite multiverse, typically a fractal.

Fractals are pure mathematical concepts. They do not exist in reality

Although new inflation is classically rolling down the potential, quantum fluctuations can sometimes bring it back up to previous levels. These regions in which the inflaton fluctuates upwards expand much faster than regions in which the inflaton has a lower potential energy, and tend to dominate in terms of physical volume.

This is easy to write but very difficult physical to accept.

This steady state, which first developed by Vilenkin, is called "eternal inflation". It has been shown that any inflationary theory with an unbounded potential is eternal. It is a popular conclusion among physicists that this steady state cannot continue forever into the past. The inflationary spacetime, which is similar to de Sitter space, is incomplete without a contracting region. However, unlike de Sitter space, fluctuations in a contracting inflationary space will collapse to form a gravitational singularity, a point where densities become infinite. Therefore, it is necessary to have a theory for the universe's initial conditions. Linde, however, believes inflation may be past eternal.

In eternal inflation, regions with inflation have an exponentially growing volume, while regions that are not inflating don't.

In dutch we call this: "De waarheid van een koe"
Translated: "The truth of a cow". The issue is what are the physical conditions that cause this.

This suggests that the volume of the inflating part of the universe in the global picture is always unimaginably larger than the part that has stopped inflating, even though inflation eventually ends as seen by any single pre-inflationary observer.

The observer has of course nothing to do with this.
Inflation creates something that is much larger than what was before. Again: What caused this to happen all of a sudden.

Scientists disagree about how to assign a probability distribution to this hypothetical anthropic landscape.

Very strange sentence. Why use concepts like probability? when the probability is zer why mention.
I can imagine that scientist disagree.

If the probability of different regions is counted by volume, one should expect that inflation will never end, or applying boundary conditions that a local observer exists to observe it, that inflation will end as late as possible. Some physicists believe this paradox can be resolved by weighting observers by their pre-inflationary volume.

Apperently there is no agreement, making the concept of eternel inflation doubtfull. Any way what is the different with eternal inflation and eternal space expansion (without inflation).

5.3 Initial conditions

Some physicists have tried to avoid the initial conditions problem by proposing models for an eternally inflating universe with no origin. These models propose that while the universe, on the largest scales, expands exponentially it was, is and always will be, spatially infinite and has existed, and will exist, forever.

You cannot avoid the initial problem. In some way you must discuss (think about) what is and what caused the Big Bang or to make it "simpler" what started all what exists at present. We are no ostrich.

Other proposals attempt to describe the ex nihilo creation of the universe based on quantum cosmology and the following inflation. Vilenkin put forth one such scenario. Hartle and Hawking offered the no-boundary proposal for the initial creation of the universe in which inflation comes about naturally.

This range of all possible scanarios does not make a very scientific impression.

Alan Guth has described the inflationary universe as the "ultimate free lunch": new universes, similar to our own, are continually produced in a vast inflating background.

Yes

Gravitational interactions, in this case, circumvent (but do not violate) the first law of thermodynamics (energy conservation) and the second law of thermodynamics (entropy and the arrow of time problem). However, while there is consensus that this solves the initial conditions problem, some have disputed this, as it is much more likely that the universe came about by a quantum fluctuation.

This raises the "ultimate" question what is a quantum fluctuation.

Donald Page was an outspoken critic of inflation because of this anomaly.He stressed that the thermodynamic arrow of time necessitates low entropy initial conditions, which would be highly unlikely. According to them, rather than solving this problem, the inflation theory further aggravates it – the reheating at the end of the inflation era increases entropy, making it necessary for the initial state of the Universe to be even more orderly than in other Big Bang theories with no inflation phase.

I fully agree with Donald Page that the inflation theory in a sense creates more problems than it solves because all the possible solutions should be clear how they physical opperate.
At the same time if you use the concept of "arrow of time" you have to explain clearly what you mean. If you don't than don't use it. The same with "entropy".

Hawking and Page later found ambiguous results when they attempted to compute the probability of inflation in the Hartle-Hawking initial state.

You cannot calculate this probability.

Other authors have argued that, since inflation is eternal, the probability doesn't matter as long as it is not precisely zero: once it starts, inflation perpetuates itself and quickly dominates the universe.

I agree that probabilty is no issue. Why do you want to calculate this probabilty anyway?
How do you know that eternal inflation has started ?

However, Albrecht and Lorenzo Sorbo have argued that the probability of an inflationary cosmos, consistent with today's observations, emerging by a random fluctuation from some pre-existent state, compared with a non-inflationary cosmos overwhelmingly favours the inflationary scenario, simply because the "seed" amount of non-gravitational energy required for the inflationary cosmos is so much less than any required for a non-inflationary alternative, which outweighs any entropic considerations.

The biggest question is how did the Big Bang started anyway. That question is not adressed.

Another problem that has occasionally been mentioned is the trans-Planckian problem or trans-Planckian effects. Since the energy scale of inflation and the Planck scale are relatively close, some of the quantum fluctuations that have made up the structure in our universe were smaller than the Planck length before inflation. Therefore, there ought to be corrections from Planck-scale physics, in particular the unknown quantum theory of gravity. There has been some disagreement about the magnitude of this effect: about whether it is just on the threshold of detectability or completely undetectable.

Physics has in principle nothing to do with the Planck-scale. When you discuss quantum fluctuations you must explain what you are discussing. Photons, gravitons, atoms, molecules etc ?

5.4 Hybrid inflation

Another kind of inflation, called hybrid inflation, is an extension of new inflation. It introduces additional scalar fields, so that while one of the scalar fields is responsible for normal slow roll inflation, another triggers the end of inflation: when inflation has continued for sufficiently long, it becomes favorable to the second field to decay into a much lower energy state.

I do not call introducing adhoc fields rock bottom science.

In hybrid inflation, one of the scalar fields is responsible for most of the energy density (thus determining the rate of expansion), while the other is responsible for the slow roll (thus determining the period of inflation and its termination). Thus fluctuations in the former inflaton would not affect inflation termination, while fluctuations in the latter would not affect the rate of expansion. Therefore hybrid inflation is not eternal. When the second (slow-rolling) inflaton reaches the bottom of its potential, it changes the location of the minimum of the first inflaton's potential, which leads to a fast roll of the inflaton down its potential, leading to termination of inflation.

You have to explain the physical processes responsible for each type of field If you don't this type of science does not make sense.

5.5 Inflation and string cosmology

The discovery of flux compactifications have opened the way for reconciling inflation and string theory. A new theory, called brane inflation suggests that inflation arises from the motion of D-branes in the compactified geometry, usually towards a stack of anti-D-branes. This theory, governed by the Dirac-Born-Infeld action, is very different from ordinary inflation. The dynamics are not completely understood. It appears that special conditions are necessary since inflation occurs in tunneling between two vacua in the string landscape. The process of tunneling between two vacua is a form of old inflation, but new inflation must then occur by some other mechanism.

You can never explain something clearly if you don't understand yourself what you want to explain.

5.6 Inflation and loop quantum gravity

When investigating the effects the theory of loop quantum gravity would have on cosmology, a loop quantum cosmology model has evolved that provides a possible mechanism for cosmological inflation. Loop quantum gravity assumes a quantized spacetime. If the energy density is larger than can be held by the quantized spacetime, it is thought to bounce back.

Same comment as above:
You can never explain something clearly if you don't understand yourself what you want to explain.

5.7 Inflation and generalized uncertainty principle (GUP)

The effects of generalized uncertainty principle (GUP) on the inflationary dynamics and the thermodynamics of the early Universe are studied.

You should first explain the difference between the uncertainty principle and the GUP. The uncertainty principle is in principle a concept which describes our human limitations to describe (measure) the state of the universe accurately. This principle can not be used as a tool to describe the evolution of the universe, the Big Bang or inflation.

Using the GUP approach, Tawfik et al. evaluated the tensorial and scalar density fluctuations in the inflation era and compared them with the standard case. They found a good agreement with the Wilkinson Microwave Anisotropy Probe data. Assuming that a quantum gas of scalar particles is confined within a thin layer near the apparent horizon of the Friedmann-Lemaitre-Robertson-Walker Universe that satisfies the boundary condition, Tawfik et al. calculated the number and entropy densities and the free energy arising from the quantum states using the GUP approach. Furthermore, a qualitative estimation for effects of the quantum gravity on all these thermodynamic quantities was introduced.

6 Alternatives to inflation

The flatness and horizon problems are naturally solved in the Einstein-Cartan-Sciama-Kibble theory of gravity, without needing an exotic form of matter and introducing free parameters.

The flatness and the horizon problem require physical solutions.

This theory extends general relativity by removing a constraint of the symmetry of the affine connection and regarding its antisymmetric part, the torsion tensor, as a dynamical variable. The minimal coupling between torsion and Dirac spinors generates a spin-spin interaction that is significant in fermionic matter at extremely high densities. Such an interaction averts the unphysical Big Bang singularity, replacing it with a cusp-like bounce at a finite minimum scale factor, before which the Universe was contracting. The rapid expansion immediately after the Big Bounce explains why the present Universe at largest scales appears spatially flat, homogeneous and isotropic.

Appears!

As the density of the Universe decreases, the effects of torsion weaken and the Universe smoothly enters the radiation-dominated era.

How do you know that this happens smoothly? Most probably all changes in the universe happened discontinuous.

There are models that explain some of the observations explained by inflation. However none of these "alternatives" has the same breadth of explanation as inflation, and still require inflation for a more complete fit with observation; they should therefore be regarded as adjuncts to inflation, rather than as alternatives.

I think it is even very difficult to clearly pinpoint observations to inflation (assuming it has a clear and precise definition).
The meaning of adjunct is addition.

String theory requires that, in addition to the three observable spatial dimensions, there exist additional dimensions that are curled up or compactified (see also Kaluza–Klein theory). Extra dimensions appear as a frequent component of supergravity models and other approaches to quantum gravity. This raised the contingent question of why four space-time dimensions became large and the rest became unobservably small. An attempt to address this question, called string gas cosmology, was proposed by Robert Brandenberger and Cumrun Vafa. This model focuses on the dynamics of the early universe considered as a hot gas of strings. Brandenberger and Vafa show that a dimension of spacetime can only expand if the strings that wind around it can efficiently annihilate each other. Each string is a one-dimensional object, and the largest number of dimensions in which two strings will generically intersect (and, presumably, annihilate) is three. Therefore, one argues that the most likely number of non-compact (large) spatial dimensions is three. Current work on this model centers on whether it can succeed in stabilizing the size of the compactified dimensions and produce the correct spectrum of primordial density perturbations. For a recent review, see... The authors admits that their model "does not solve the entropy and flatness problems of standard cosmology ..... and we can provide no explanation for why the current universe is so close to being spatially flat".

This document is about inflation and not about string theory.

The ekpyrotic and cyclic models are also considered adjuncts to inflation. These models solve the horizon problem through an expanding epoch well before the Big Bang, and then generate the required spectrum of primordial density perturbations during a contracting phase leading to a Big Crunch.

You first have to explain physical how a big crunch works. Mathematical it is "easy", but what we are discussing is physics.

The universe passes through the Big Crunch and emerges in a hot Big Bang phase. In this sense they are reminiscent of the oscillatory universe proposed by Richard Chace Tolman: however in Tolman's model the total age of the universe is necessarily finite, while in these models this is not necessarily so. Whether the correct spectrum of density fluctuations can be produced, and whether the universe can successfully navigate the Big Bang/Big Crunch transition, remains a topic of controversy and current research.

It will always be a topic of controversy.

Ekpyrotic models avoid the magnetic monopole problem as long as the temperature at the Big Crunch/Big Bang transition remains below the Grand Unified Scale, as this is the temperature required to produce magnetic monopoles in the first place. As things stand, there is no evidence of any 'slowing down' of the expansion, but this is not surprising as each cycle is expected to last on the order of a trillion years.

That is easy to write....

Another adjunct, the varying speed of light model has also been theorized by Jean-Pierre Petit in 1988, John Moffat in 1992 as well Andreas Albrecht and João Magueijo in 1999, instead of superluminal expansion the speed of light was 60 orders of magnitude faster than its current value solving the horizon and homogeneity problems in the early universe.

The speed of light has nothing to do with the process involved in the evolution universe. What is much more important is the speed of gravity. When the speed of gravity is higher than the speed of light it becomes easier to explain why the observed universe is more homogeneous because the underlying physical processes could faster communicate gravitational.

7 Criticisms

• The inflation debate April 2011 by Paul Steinhardt Is the theory at the heart of modern cosmology deeply flawed.
• Big Bang blunder bursts the multiverse bubble In this document of 3 June 2014 the supposed detection of gravity waves are discussed. Gravity waves are the proof that the Inflation theory is correct.