We saw that standard cosmological models suffer from 'flatness problem' and the 'horizon problem'. To these problem one might also add 'expansion problem' which asks simply why the universe expanding at all. Therefore we augment our discussion of some cosmological models with the inflationary scenario, which seek to solve those problems (and others) and has, over the past two decades, become a fundamental part of modern cosmology. (see

which makes many complicate equations, e.g. perturbation analysis, far less cluttered and can be removed at the end if desired.

**16.1 Definition of inflation** _{.}

A possible solution of the horizon problem discussed in (15.12) is to postulate an *accelerating* phase of expansion, prior to any decelerating phase. In an accelerating phase, casual contact is better at earlier times and so remotely separated parts of our present universe could have 'coordinated' their physical characteristics in the early universe. Such an accelerating phase is called a period of *inflation*. Hence the basic definition of inflation is that

(16.1) __Ȑ > 0.__

We make recast this condition in an alternative manner considering comoving Hubble distance defined in (15.51), namely 𝜒_{𝐻}(𝑡) = 𝐻^{-1}(𝑡)/𝑅(𝑡). The derivative with respect to 𝑡 is given by

𝑑/𝑑𝑡 (𝐻^{-1}/𝑅) = 𝑑/𝑑𝑡 (1/Ṙ) = -Ȑ/Ṙ^{2},

and so the condition (16.1) can be written as

__𝑑/𝑑𝑡 (𝐻 ^{-1}/𝑅) < 0.__

Thus, an equivalent condition for inflation is that the comoving Hubble distance

Let us suppose that, at some period in the early universe, the energy density is dominated by some form of matter with density 𝜌 and pressure 𝑝. The first cosmological field equation (14.36) (with 𝛬 = 0 and 8π𝐺 = 𝑐 = 1) then reads

(16.2) Ȑ = -1/6 (𝜌 + 3𝑝)𝑅.

Thus, we see that in order for inflation to occur we require that

(16.3)

In other words, we need the 'matter' to have an equation of state with negative pressure. I fact, the above criterion can also solve the flatness problem. The second cosmological field equation (with the same condition above) reads

(16.4) Ṙ

During a period of acceleration (Ȑ > 0), the scale factor must increase faster than 𝑅(𝑡) ∝𝑡. Provided that 𝑝 < -1/3 𝜌, the quantity 𝜌𝑅

We have omitted the cosmological-constant terms since if 'matter' in the form of a

**16.2 Scalar fields and phase transitions in the very early universe** _{.}

The basic physical mechanism for producing a period of inflation in the very early universe relies on the existence of matter in terms of scalar field. Upon quantization a scalar field describes a collection of spinless particles.

The existence of such scalar fields is suggested by our best theories for the fundamental interaction in Nature, which predict that the universe experienced a succession of *phase transitions* in its early stages. Let us model this expansion by assuming that the universe followed a standard radiation-dominated Friedmann model from 𝑎(𝑡) = (2𝐻_{0}𝑡)^{1/2} in (15.29b), in which

(16.5) 𝑅(𝑡) ∝ 𝑡^{1/2} ∝1/𝛵(𝑡)

where the 'temperature' 𝛵 is related to the typical particle energy by 𝛵 ∼𝐸/𝑘_{𝐵}, (without any derivation of 𝑅(𝑡) ∝1/𝛵(𝑡) in our textbook^{※}). The basic scenario is as follows.

• 𝐸_{𝑃𝑙} ∼10^{19} Ge𝑉 > 𝐸 > 𝐸_{𝐺𝑈𝑇} ~10^{15} Ge𝑉 The earliest point at which the universe can be modeled (even approximately) as a classical system is the *Planck era*, corresponding to particle energy 𝐸_{𝑃𝑙} ∼10^{19} Ge𝑉 (or 𝛵_{𝑃𝑙} ∼10^{32} 𝐾) and time scale 𝑡_{𝑃𝑙} ∼10^{-43} s (prior to this epoch, it can be described only in terms of some quantum theory of gravity). At this high energy, grand unified theories (𝐺𝑈𝑇s) predict that the electroweak and strong forces are in fact *unified* into a single force and these interactions bring the particles present into thermal equilibrium. Once the universe has cooled to 𝐸_{𝐺𝑈𝑇} ~10^{15} Ge𝑉 (corresponding to 𝛵_{𝐺𝑈𝑇} ~10^{27} 𝐾), there is a spontaneous breaking of the larger symmetry group characterizing the 𝐺𝑈𝑇 into a product of smaller symmetry groups, and the electroweak and strong forces separate. From (16.5), this 𝐺𝑈𝑇 phase transition occurs at 𝑡_{𝐺𝑈𝑇} ~10^{-36} s.

• 𝐸_{𝐺𝑈𝑇} > 𝐸 > 𝐸_{𝐸𝑤} ~100 Ge𝑉 During this period (which is extremely long in logarithmic terms), the electroweak and strong forces are separate and these interaction sustain thermal equilibrium.This continues until the universe has cooled to 𝐸_{𝐸𝑤} ~100 Ge𝑉 (corresponding to 𝛵_{𝐸𝑤} ~10^{15} 𝐾), when the unified electroweak theory predicts that a second phase transition should occur in which eletromagnetic and weak forces separate. From (16.5), this electroweak phase transition occurs at 𝑡_{𝐸𝑤} ~10^{-11} s.

• 𝐸_{𝐸𝑤} ~100 Ge𝑉 > 𝐸 > 𝐸_{𝑄𝐻} ~100 Me𝑉 During this period the electromagnetic, weak and strong forces are separate, as they are today. When the universe has cooled to 𝐸_{𝑄𝐻} ~100 Me𝑉 (corresponding to 𝛵_{𝑄𝐻} ~10^{12} 𝐾) there is a final phase transition, according to the theory of quantum chromodynamics, in which the strong forces increases in strength and leads to the confinement of quarks into hardrons. From (16.5), this quark-hadron phase transition occurs at 𝑡_{𝑄𝐻} ~10^{-5} s.

In general, phase transitions occurs via a process called *spontaneous symmetry breaking*, which characterized by the acquisition of certain non-zero values by *scalar* parameters known as *Higgs fields*. The symmetry is manifest when the Higgs fields have the value zero; it is spontaneously broken whenever at least one of the Higgs fields becomes non-zero. Thus, the occurrence of phase transitions in the very early universe suggests the existence of scalar fields and hence provides the motivation for considering effect on the expansion of the universe. We will confine our attention to scalar fields present at, or before, the 𝐺𝑈𝑇 phase transition (the most speculative of these phase transitions).

**16.3 A scalar fields as a cosmological fluid** _{.}

Let us consider a single scalar field ε present in the very early universe. The field ⱷ^{**} is traditionally called the 'inflation' field. (see Section **19.6** at Forum #90) The Lagrangian for 𝜀 has the usual form of a kinetic term minus a potential term:

𝐿 = 1/2 𝑔^{𝜇𝜈}(𝜕_{𝜇}ⱷ)(𝜕_{𝜈}ⱷ) - 𝑉(ⱷ).

The corresponding field equation for 𝜔 is obtained from the Euler-Lagrange equations and reads

(16.6) □^{2}ⱷ + 𝑑𝑉/𝑑ⱷ = 0,

where □^{2} ≡ 𝛻^{𝜇}𝛻_{𝜈} = 𝑔^{𝜇𝜈}𝛻_{𝜇}𝛻_{𝜈} is the covariant d'Alembertian operator. A simple example is a *free* relativistic scalar field of mass 𝑚, for which the potential would be 𝑑𝑉(ⱷ) = 1/2 𝑚^{2}𝜀^{2} and the field equation becomes the covariant Klein-Gordon equation,

□^{2}𝜔 + 𝑚^{2}ⱷ = 0.

For the moment it is best to keep the potential function 𝑉(ⱷ) general.

The energy-momentum tensor 𝑇^{𝜇𝜈} for a scalar field can be derive from variational approach (see Section **19.12** at Forum #89) as follows

(16.7) 𝑇^{𝜇𝜈} = (𝜕_{𝜇}ⱷ)(𝜕_{𝜈}ⱷ) - 𝑔^{𝜇𝜈}[1/2 (𝜕_{𝜎}ⱷ)(𝜕^{𝜎}ⱷ) - 𝑉(ⱷ)].

The energy-momentum tensor for a perfect fluid is

𝑇^{𝜇𝜈} = (𝜌 + 𝑝)𝑢^{𝜇}𝑢^{𝜈} - 𝑝𝑔^{𝜇𝜈},

and by comparing the the two forms in a Cartesian inertial coordinate system (𝑔^{𝜇𝜈} = 𝜂^{𝜇𝜈}) in which the fluid at rest, we see that the scalar field acts like a perfect fluid, with energy density and pressure given by, (though there is no derivations of them in our textbook^{※},)

(16.8) __𝜌 _{ⱷ} = 1/2 ᾠ^{2} + 𝑉(ⱷ) + 1/2 (𝛻̄^{2}ⱷ)^{2}, 𝑝_{ⱷ} = 1/2 ᾠ^{2} - 𝑉(ⱷ) - 1/6 (𝛻̄^{2}ⱷ)^{2}.__

In particular, if the field ⱷ were both temporally and spatially constant, the equation would be 𝑝

**16.4 An inflationary epoch** _{.}

Let us suppose the scalar field does not interact (except gravitationally) with any other matter or radiation. In this case the scalar field will independently obey an equation of (14.39)

ῤ + 3(𝜌 + 𝑝)Ṙ/𝑅 = 0.

Substituting the (16.8) and assuming no spatial variations, i.e. with the third term at right-hand side we find the equation of motion of the scalar fields is

(16.9) __ἕ + 3𝐻ᾠ + 𝑑𝑉/𝑑ⱷ = 0.__

Let us assume further that there is some period when the scalar field dominates the energy density of the universe. Moreover we will demand the energy density is sufficiently large that we may neglect the curvature term in (16.4). Thus, we may write (16.4) using (16.8) as

(16.10) __𝐻 ^{2} = 1/3 [1/2 ᾠ^{2} + 𝑉(ⱷ)].__

These two equations thus provide a set of coupled differential equations in 𝐻 and ⱷ that determine the evolution of the scalar field and the scale factor of the universe during the epoch of scalar-field domination. From our criterion (16.3) and (16.8), we see

(16.11)

**16.5 The slow-roll approximation** _{.}

The inflation equations (16.9) and (16.10) can easily solved numerically, and even analytically for special choice of 𝑉(ⱷ). In general, however, an analytical solution is only possible in the *slow-roll approximation*, in which it is assumed that ᾠ^{2} ≪ 𝑉(ⱷ). on differentiating, this in turn implies that ᾥ ≪ 𝑑𝑉/𝑑ⱷ and so ἕ-term can be neglected in (16.9) and (16.10)

(16.12) __3𝐻ᾠ = -𝑑𝑉/𝑑ⱷ.__

(16.13) __𝐻 ^{2} = 1/3 𝑉(ⱷ).__

In this approximation the rate of change of the Hubble parameter and the scalar field can be related. Differentiating (16.13) with respect to 𝑡 and combining the result with (16.12), one obtains

(16.14) Ḣ = -1/2 ᾠ

The condition for in the slow-roll approximation can be put into a useful dimensionless form, using the two equation above and the condition ἐ

(16.15) 𝜖 ≡ 1/2 (𝑉'/𝑉)

where 𝑉' ≡ 𝑑𝑉/𝑑ⱷ. Differentiation (16.15) on also finds

(16.16) 𝜂 ≡ 𝑉"/𝑉 ≪ 1

These two equation require the potential 𝑉(ⱷ) to be sufficiently 'flat' that the field ⱷ 'rolls' slowly enough for inflation to occur.

It is worth considering the special case in which the potential 𝑉(ⱷ) is sufficiently flat during the period of inflation, its value remains roughly constant. From (16.13) we see that in this case the 𝐻 is constant and the scale factor 𝑅(𝑡) grows

**16.6 Ending inflation** _{.}

At the field value ⱷ 'rolls' down the potential 𝑉(ⱷ), the condition ᾠ^{2} < 𝑉(ⱷ) will eventually no longer hold and inflation will cease. Equivalently, in the slow-roll approximation, the condition 1/2 (𝑉'/𝑉)^{2} ≪ 1 and 𝑉"/𝑉 ≪ 1 will eventually no longer be satisfied. If the potential 𝑉 possesses a local minimum, the field will no longer roll slowly downhill but will oscillate about the minimum, the oscillation being gradually damped by the 3𝐻ἐ friction term in (16.9). Eventually, 𝑉 is left stationary at the bottom of the potential. If the minimum value of 𝑉, 𝑉_{min} > 0 then clearly the condition ᾠ^{2} < 𝑉 in (16.11) is again satisfied and the universe continues to inflate indefinitely. Moreover, in this case 𝑝_{ⱷ} = - 𝜌_{ⱷ} and so the scalar field acts as an effective cosmological constant 𝛬 = 𝑉_{min}. If 𝑉_{min} = 0, 𝑉 has zero energy density, i.e. 𝜌_{ⱷ} = 0 and the dynamics of the universe is dominated by any other fields present.

The scenario outlined above would occur only if the scalar field ⱷ were not coupled to any other fields, which is almost certainly not the case. In practice, such couplings will cause ⱷ to decay during the oscillatory phase into pairs of elementary particles, into which the energy of ⱷ is thus converted. The universe will therefore contain roughly the same energy density as it did at the start of inflation. The process of decay of the scalar field into other particles is therefore termed *reheating.* These particles will interact with each other and subsequently decay themselves, leaving the universe filled with normal matter and radiation in thermal equilibrium and thereby providing the initial conditions for a standard cosmological model.

**16.7 The amount of inflation** _{.}

For the motivation for the introduction of inflation scenario was (in part) to solve the flatness and horizon problems, we now try to calculates the amount of inflation. From our present understanding of particle physics, it is though that inflation occurs at around the 𝐺𝑈𝑇 phase transition or earlier. We assume that the universe has followed a standard radiation-dominated Friedmann model for the history of the epoch of inflation at 𝑡 ~ 𝑡_{*}. From the relation 𝑅 ∝ 𝑡^{1/2} ∝1/𝛵 in (16.5) we have

(16.17) 𝑅_{*}/𝑅_{0} ~ (𝑡_{*}/𝑡_{0})^{1/2} ~ 𝛵_{0}/𝛵_{*},

where 𝛵_{0} ~ 3K is the present-day temperature of cosmic microwave background(CMB) radiation and 𝑡_{0} ~ 1/𝐻_{0} ~ 10^{18}s, which is a value of order, is the present age of the universe.

Let us first consider the flatness problem. From 𝛺_{𝑘} = -𝑘/𝐻^{2}𝑅^{2} (15.47) the ratio of the spacial curvature density 𝛺_{𝑘} at the inflationary epoch to that at present epoch is given by

(16.18) 𝛺_{𝑘,*}/𝛺_{𝑘,0} = (𝐻_{0}/𝐻_{*})^{2}(𝑅_{0}/𝑅_{*})^{2} ~ 𝑡_{*}/𝑡_{0},

where we have used the fact that 𝐻_{0}/𝐻_{*} ~ 𝑡_{*}/𝑡_{0}. Assuming inflation to occur between the Plank era and the GUT phase transition, so that 𝑡_{𝑃𝑙} < 𝑡_{*} < 𝑡_{𝐺𝑈𝑇}, from Section 16.2, the ratio (16.18) lies in the range ~ 10^{-60}-10^{-54}. Thus, if 𝛺_{𝑘,0} is of order unity then the required degree of fine-tuning of 𝛺_{𝑘,*} is extreme To solve the flatness problem we require the scale factor 𝑅 to grow during inflation by a factor ~ 10^{27}-10^{30}. In terms of the required number 𝑁 of e-foldings (𝑒^{𝑁}) of 𝑅, we thus have

__𝑁 ≳ 60-70 (flatness problem)__

In case of the horizon problem, the particle horizon at the inflationary epoch is

𝑑_{𝑝,*} = 2𝑐𝑡_{*},

which, taking 𝑡_{𝑃𝑙} < 𝑡_{*} < 𝑡_{𝐺𝑈𝑇}, gives the size of a causally connected region at this time as ~ 10^{-34}-10^{-27}m. From (16.17), we see that the size of such a region today would only ~10^{-3}-1m! The current size of the observable universe by the present-day Hubble distance,

𝑑_{𝐻,0} = 𝑐𝐻_{0}^{-1} ~ 10^{26}m ~ 10^{10}ly ~ 3000Mpc.

To solve the horizon problem, we thus require the scale factor 𝑅 to grow by a factor of ~ 10^{26}-^{29} during the period of inflation. Expressing this result in terms of the required number 𝑁 of 𝑒-foldings, we once again find

__𝑁 ≳ 60-70 (horizon problem).__

We have thus found that *both* the flatness and the horizon problems can be solved. In the slow-roll approximation, the number of 𝑒-foldings that occur while 𝑅 'rolls' from ⱷ_{1} to ⱷ_{2} is given by
*Phase one* 𝑁 = ∫_{t1}^{t2} 𝐻 𝑑𝑡 = ∫_{ⱷ1}^{ⱷ2} 𝐻/ᾠ 𝑑ⱷ = - ∫_{ⱷ1}^{ⱷ2} 𝑉/𝑉' 𝑑ⱷ ~ ∫_{ⱷ1}^{ⱷ2} ⱷ 𝑑ⱷ,
where 𝑉' ~ 𝑉/ⱷ since the potential is reasonably smooth. Thus if 𝛥ⱷ = ∣ⱷ_{start} - ⱷ_{end}∣ is the range of ⱷ-values over which inflation occur. One finds 𝑁 ~ (𝛥ⱷ)^{2}.

**16.8 Starting inflation** _{.}

How the universe can arrive at an appropriate starting state of inflation? There are two main classes of model of inflationary cosmology. In early models of inflation, the inflationary epoch is an 'interlude' in the evolution of a standard cosmological model. In such models, the inflation field ⱷ in the **Figure 16.1** and **16.2** of our textbook) is usually identified with a scalar Higgs field operating during the 𝐺𝑈𝑇 phase transition. It is thus assumed that this state was relatively homogeneous and large enough to survive until the beginning of inflation at the 𝐺𝑈𝑇 era; an example of this sort is the *'new' inflation* model in next Section 16.9. In more recent model of inflation, the scalar field ⱷ is some generic scalar field present in the very early universe. In this models the universe may inflate soon after exits the Plank era. An example of such a model is the *chaotic inflation* scenario in Section 16.10 and the its natural extension is *stochastic inflation (or eternal inflation)* in Section 16.11.

**16.9 'New' inflation** _{.}

In the 'new' inflation models, (which is so called to distinguish it from original 'old' inflation model of Guth in 1981 and is the variant proposed independently by Linde (1982) and Albrechit and Steinhart (1982),) the inflationary epoch occurs when the universe goes through the 𝐺𝑈𝑇 phase transition. This model requires a rather special form for the potential 𝑉(ⱷ) in order to produce an effective period of inflation. As shown in **Figure 16.1** there are several potential curves for different values of 𝑇 which in fact happen by sequence. (Refer to other version of schematic illustration of an inflationary potential [Kolb and Turner 1990 p.271]^{***} which is more understandable.) At very high temperatures the potential is parabolic with a minimum at ⱷ = 0, which is *true vacuum* state, i.e. the state of lowest energy. For lower temperature the form of 𝑉(ⱷ) changes until at the critical temperature 𝑇 = 𝑇_{𝑐} it develops a lower energy state than that at ⱷ = 0. Thus this new non-zero value of 𝜀 is now the true vacuum state, and ⱷ = 0 is now a *false vacuum* state. For even lower values the new true vacuum state becomes more pronounced until a final form is reached for 'low' temperatures.

Let us now consider the evolution of scale factor 𝑅(𝑡), the radiation energy density 𝜌_{𝑟} and the scalar field ⱷ. 𝑇

• *Phase 1* When the temperature is very high, i.e. far above the 𝐺𝑈𝑇 phase transition scale of 𝑇_{𝑐} ~ 10^{27}K, from **Figure 16.1** shows that it will remain ⱷ = 0. Since 𝜌_{𝑟} ∝𝑅^{-4} from Section (15.6) the radiation dominates over ⱷ at very early epochs. Thus we have the standard early-time Radiation-only Friedmann model, in which we can neglect the curvature constant 𝑘. Thus, for

__𝑅 ∝ 𝑡 ^{1/2}, 𝜌_{𝑟} ∝ 𝑡^{1/2}, ⱷ = 0.__ ≫ 𝑇

•

As a result, there is a corresponding exponential decrease in the temperature 𝑇, which result in a rapid change of the potential function. Thus 𝑇 ~ 𝑇

•

•

Hence, at 𝑡 ~ 10

Although the 'new' inflation model still has advocates, it suffer from undesirable features. In particular, the scenario only provides an effective period of inflation if 𝑉(𝜀, 𝑇) has a very flat plateau near ⱷ = 0, which is somewhat artificial. Moreover, the period of thermal equilibrium prior to the inflationary phase requires many particles to interact with one another, and so already one requires the universe to be very large and contain many particles. Finally, the universe could easily recollapse before inflation starts. As a result of these difficulties, new inflation may not be a viable model, and so there are strong theoretical reasons to believe that the inflation field 𝜀 *cannot* be identified with the 𝐺𝑈𝑇 symmetry breaking Higgs field, Thus 𝐺𝑈𝑇's could provide the mechanism for the homogeneity and flatness of the universe may be abandoned.

**16.10 Chaotic inflation** _{.}

There are more recent models of inflation in that the scalar field ⱷ is not identified with the Higgs field in the 𝐺𝑈𝑇 phase transition but is regarded as a generic scalar field present in the very early universe. These models invoke the idea of *chaotic inflation*. In this scenario, as the universe exits the Plank era at 𝑡 ~ 10^{-43}s the initial value of the scalar field ⱷ_{start} is set chaotically, i.e. it acquires different random values in different regions of the universe. In some region ⱷ_{start} is somewhat displaced from the minimum of the potential and so the field field subsequently rolls downhill. If the potentials sufficiently flat, the field is more likely to displaced a greater distance from its minimum and will roll slowly enough, and for the regions to undergo an effective period of inflation. Conversely, in other regions ⱷ_{start} may not be placed sufficiently from the minimum of the potential for the region to inflate. Thus, on the largest scales the universe is highly inhomogeneous, but our observable universe lies (well) within a region that underwent a period of inflation.

According to this scenario, inflation may occur even in theories with very simple potential, such as 𝑉(ⱷ) ~ ⱷ^{𝑛}. The potential function does not depends on the temperature 𝑇. A simple example is a free scalar field 𝑉(ⱷ) = 1/2 𝑚^{2}ⱷ^{2} (see **Figure 16.2**). Moreover, in the chaotic scenario, inflation may begin even if there is no thermal equilibrium in the early universe, and i may even start just after the lank epoch.

**16.11 Stochastic inflation** _{.}

A natural extension to the chaotic inflation model is the mechanism of *stochastic* (eternal) inflation. Its main idea is to to take account of quantum fluctuations in the evolution of the scalar field, which we have thus far ignored by modelling the field entirely classically. If, in the chaotic assignment of initial values of the scalar field, some regions have a large value of ⱷ_{start} then quantum fluctuation can cause 𝜀 to move further *uphill* in the potential 𝑉(ⱷ). These region inflate as a greater rate than the surrounding ones, and the fraction of the total volume of the universe containing the growing ⱷ-field increase. Quantum fluctuations within these regions lead in tern to the production of some new inflationary regions that expand still faster.This process thus lead to eternal self-reproduction of the inflationary universe.

* **Fig. 3.1-3** from Andrew L. Liddle and David H. Lyth *Cosmological inflation and Large-Scale Structure* (Cambridge University Press 2000)

** The 𝜙 in the textbook is replaced by ⱷ for using ᾠ or ᾥ which denotes the first or second time derivative.

*** Edward W. Kolb and Michael S. Turner *The Early Universe* (Addison-Wesley Publishing Company 1990)

^{※} attention notes when some rigorous derivation might be required

p.s. The 2014 Kavli Prize in Astrophysics is awarded to "for pioneering the theory of cosmic Inflation", Alan H. Guth (MIT), Andrei D. Linde

(Stanford University) and Alexei A. Starobinsky (Landau Institute for Theoretical Physics, Russia).