Mathematical Cosmology (5)

M. P. Hobson, G. Efstathiou and A. N. Lasenby  General Relativity  An Introduction for Physicicists  (Cambridge University Press  2006)

16. Inflationary cosmology  .
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 Fig. 3.1. and 3.2.) In particular, we will discuss the effect of inflation on the evolution of the universe as a whole and also consider how inflation gives rise to perturbation in the early universe that subsequently collapse under gravity to form all the structure we observe today. Because these topics are very complicate, we will adopt the convention throughout this chapter that
            8π𝐺 = 𝑐 = 1,
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 𝜒𝐻(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/Ṙ2,
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 decrease with time. Hence, when viewed in comoving coordinates, the characteristic length scale of the universe becomes smaller as inflation proceeds. (see Fig. 3.2. and 3.3.)
   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)    𝑝 < -1/3 𝜌.
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)    Ṙ2 = 1/3 𝜌𝑅2 - 𝑘.
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 𝜌𝑅2 will increase during such a period and can make the curvature term negligible for sufficiently long phase.
   We have omitted the cosmological-constant terms since if 'matter' in the form of a scalar field exists in the early universe then this can as an effective cosmological constant, as we will see. In order to show that the existence of such fields is likely, we must consider briefly the topic of phase transitions in the very early universe.

          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) = (2𝐻0𝑡)1/2 in (15.29b), in which
    (16.5)    𝑅(𝑡) ∝ 𝑡1/2 ∝1/𝛵(𝑡)
where the 'temperature' 𝛵 is related to the typical particle energy by 𝛵 ∼𝐸/𝑘𝐵, (without any derivation of 𝑅(𝑡) ∝1/𝛵(𝑡) in our textbook). The basic scenario is as follows.

𝐸𝑃𝑙 ∼1019 Ge𝑉 > 𝐸 > 𝐸𝐺𝑈𝑇 ~1015 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 𝐸𝑃𝑙 ∼1019 Ge𝑉 (or 𝛵𝑃𝑙 ∼1032 𝐾) and time scale 𝑡𝑃𝑙 ∼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 𝐸𝐺𝑈𝑇 ~1015 Ge𝑉 (corresponding to 𝛵𝐺𝑈𝑇 ~1027 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 𝑡𝐺𝑈𝑇 ~10-36 s.
𝐸𝐺𝑈𝑇 > 𝐸 > 𝐸𝐸𝑤 ~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 𝐸𝐸𝑤 ~100 Ge𝑉 (corresponding to 𝛵𝐸𝑤 ~1015 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 𝑡𝐸𝑤 ~10-11 s.
𝐸𝐸𝑤 ~100 Ge𝑉 > 𝐸 > 𝐸𝑄𝐻 ~100 Me𝑉    During this period the electromagnetic, weak and strong forces are separate, as they are today. When the universe has cooled to 𝐸𝑄𝐻 ~100 Me𝑉 (corresponding to 𝛵𝑄𝐻 ~1012 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 𝑡𝑄𝐻 ~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 𝑔𝜇𝜈(𝜕𝜇ⱷ)(𝜕𝜈ⱷ) - 𝑉(ⱷ). 
The corresponding field equation for 𝜔 is obtained from the Euler-Lagrange equations and reads
    (16.6)    □2ⱷ + 𝑑𝑉/𝑑ⱷ = 0,
where □2 ≡ 𝛻𝜇𝛻𝜈 = 𝑔𝜇𝜈𝛻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 𝑚2𝜀2 and the field equation becomes the covariant Klein-Gordon equation,
          □2𝜔 + 𝑚2ⱷ = 0.
For the moment  it is best to keep the potential function 𝑉(ⱷ) general.
   The energy-momentum tensor 𝑇𝜇𝜈 for a scalar field can be derive from variational approach (see Section 19.12 at Forum #89) as follows
    (16.7)     𝑇𝜇𝜈 = (𝜕𝜇ⱷ)(𝜕𝜈ⱷ) - 𝑔𝜇𝜈[1/2 (𝜕𝜎ⱷ)(𝜕𝜎ⱷ) - 𝑉(ⱷ)].
The energy-momentum tensor for a perfect fluid is
          𝑇𝜇𝜈 = (𝜌 + 𝑝)𝑢𝜇𝑢𝜈 - 𝑝𝑔𝜇𝜈,
and by comparing the the two forms in a Cartesian inertial coordinate system (𝑔𝜇𝜈 = 𝜂𝜇𝜈) 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 = 1/2 ᾠ2 + 𝑉(ⱷ) + 1/2 (𝛻̄2ⱷ)2,     𝑝 = 1/2 ᾠ2 - 𝑉(ⱷ) - 1/6 (𝛻̄2ⱷ)2.
In particular, if the field ⱷ were both temporally and spatially constant, the equation would be 𝑝 = - 𝜌, does it mean 𝜌 = 𝑉(𝜀) and 𝑝 = - 𝑉(ⱷ)?, and the so the scalar field would act as a cosmological constant with 𝛬 = 𝑉(ⱷ) (with 8π𝐺 = 𝑐 = 1). Also we assume that the spatial derivatives can be neglected, i.e. ⱷ is a function only of 𝑡 without spatial variation. i.e. 𝛻̄2ⱷ vanishes.

          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)    𝐻2 = 1/3 [1/2 ᾠ2 + 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 that inflation will occur, provide that
    (16.11)    2 < 𝑉(ⱷ).

          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 ᾠ2 ≪ 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)    𝐻2 = 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 ᾠ2.
   The condition for in the slow-roll approximation can be put into a useful dimensionless form, using the two equation above and the condition ἐ2 ≪ 𝑉(𝜀) as follows
    (16.15)     𝜖 ≡ 1/2 (𝑉'/𝑉)2 ≪ 1
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. However one must also assume that (16.11) holds for sufficient condition.
   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 exponentially, reminding 𝐻 ≡ Ṙ/𝑅 and using WolframAlpha quickly,
            𝑅(𝑡) ∝ exp[√{1/3 𝑉(ⱷ)}*𝑡].

          16.6 Ending inflation      .
At the field value ⱷ 'rolls' down the potential 𝑉(ⱷ), the condition ᾠ2 < 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, 𝑉min > 0 then clearly the condition ᾠ2 < 𝑉 in (16.11) is again satisfied and the universe continues to inflate indefinitely. Moreover, in this case 𝑝 = - 𝜌 and so the scalar field acts as an effective cosmological constant 𝛬 = 𝑉min. If 𝑉min = 0, 𝑉 has zero energy density, i.e. 𝜌 = 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 ∝ 𝑡1/2 ∝1/𝛵 in (16.5) we have
    (16.17)     𝑅*/𝑅0 ~ (𝑡*/𝑡0)1/2 ~ 𝛵0/𝛵*,
where 𝛵0 ~ 3K is the present-day temperature of cosmic microwave background(CMB) radiation and 𝑡0 ~ 1/𝐻0 ~ 1018s, which is a value of order, is the present age of the universe.
   Let us first consider the flatness problem. From 𝛺𝑘 = -𝑘/𝐻2𝑅2 (15.47) the ratio of the spacial curvature density 𝛺𝑘 at the inflationary epoch to that at present epoch is given by
    (16.18)     𝛺𝑘,*/𝛺𝑘,0 = (𝐻0/𝐻*)2(𝑅0/𝑅*)2 ~ 𝑡*/𝑡0,
where we have used the fact that 𝐻0/𝐻* ~ 𝑡*/𝑡0. Assuming inflation to occur between the Plank era and the GUT phase transition, so that 𝑡𝑃𝑙 < 𝑡* < 𝑡𝐺𝑈𝑇, from Section 16.2, the ratio (16.18) lies in the range ~ 10-60-10-54. Thus, if 𝛺𝑘,0 is of order unity then the required degree of fine-tuning of 𝛺𝑘,* is extreme To solve the flatness problem we require the scale factor 𝑅 to grow during inflation by a factor ~ 1027-1030. In terms of the required number 𝑁 of e-foldings (𝑒𝑁) of 𝑅, we thus have
            𝑁 ≳ 60-70     (flatness problem)
   In case of the horizon problem, the particle horizon at the inflationary epoch is
          𝑑𝑝,* = 2𝑐𝑡*,
which, taking 𝑡𝑃𝑙 < 𝑡* < 𝑡𝐺𝑈𝑇, gives the size of a causally connected region at this time as ~ 10-34-10-27m. 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,
          𝑑𝐻,0 = 𝑐𝐻0-1 ~ 1026m ~ 1010ly ~ 3000Mpc.
To solve the horizon problem, we thus require the scale factor 𝑅 to grow by a factor of ~ 1026-29 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 ⱷ1 to ⱷ2 is given by Phase one   𝑁 = t1t2 𝐻 𝑑𝑡 = 12 𝐻/ᾠ  𝑑ⱷ = - 12 𝑉/𝑉' 𝑑ⱷ ~ 12 ⱷ 𝑑ⱷ, where 𝑉' ~ 𝑉/ⱷ since the potential is reasonably smooth. Thus if 𝛥ⱷ = ∣ⱷstart - ⱷend∣ 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 = 𝑇𝑐 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 𝜌𝑟 and the scalar field ⱷ. 𝑇

   • Phase 1   When the temperature is very high, i.e. far above the 𝐺𝑈𝑇 phase transition scale of 𝑇𝑐 ~ 1027K, from Figure 16.1 shows that it will remain ⱷ = 0. Since 𝜌𝑟 ∝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 ∝ 𝑡1/2,     𝜌𝑟 ∝ 𝑡1/2,     ⱷ = 0. ≫ 𝑇𝑐.
   • Phase 2   It is clear from the above equation that there will come a time when the scalar-field energy dominates over that of radiation. Provided that this occurs for 𝑇 > 𝑇𝑐 at ⱷ = 0, in which case it acts as effective cosmological constant of value 𝛬 = 𝑉(0). In this phase, the scale factor 𝑅(𝑡) undergo an exponential expansion:
            𝑅(𝑡) ∝ exp[√{1/3 𝑉(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 ~ 𝑇𝑐 is reached very quickly, and so this phase is extremely short-lived, and very little expansion actually achieved. Indeed, if 𝑇 ~ 𝑇𝑐 is reached before the scalar-field energy density 𝜌 dominates over that of the radiation then phase 2 does not occur at all.
   • Phase 3   Once 𝑇 ~ 𝑇𝑐, we see from Figure 16.1 the scalar field is now able to roll downhill away from 𝜀 = 0 and so the 𝐺𝑈𝑇 phase transition occurs. provided that the potential is sufficiently flat, the slow-roll approximation holds and the universe inflates, the evolution of the scalar field being determined by (16.12) and Hubble parameter by (16.13). If the potential is roughly constant then the exponential expansion continues. The rapid growth of the scale factor once again causes the evolution of the potential function as the temperature drops. The duration of inflation depends critically on the flatness and length of the plateau of the 𝑉(ⱷ) function for 𝑇 < 𝑇𝑐. For certain 'reasonable' potentials the universe can inflate at the number od 𝑒-folding 𝑁 ≳ 60 or even larger. So this is the main inflationary phase. According to detailed calculations, phase 3 occurrs between 𝑡1 ~ 10-36s and 𝑡1 ~ 10-34s and the scale factor 𝑅 increases by a factor of around 1050.
   • Phase 4   Eventually, the slow-roll approximation fails and inflation ends. The scalar field then rapidly down towards the true vacuum state at 𝜔 = 𝜎, oscillating about the minimum points, and follows the behavior outlined in Section (16.6). In particular, if 𝑉(𝜎) = 0 then the universe will revert to the standard radiation-dominated Friedmann model with
            𝑅(𝑡) ∝ 𝑡1/2.
Hence, at 𝑡 ~ 10-34s the universe starts a standard Friedmann expansion, albeit with the 'initial' conditions. Thus, the inflationary model incorporates all the observationally verified predictions of the standard cosmological models.

   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 ~ 10-43s the initial value of the scalar field ⱷstart is set chaotically, i.e. it acquires different random values in different regions of the universe. In some region ⱷstart 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 ⱷstart 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) ~ ⱷ𝑛. The potential function does not depends on the temperature 𝑇. A simple example is a free scalar field 𝑉(ⱷ) = 1/2 𝑚22 (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 ⱷstart 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: 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).         

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