The tensor to scalar ratio in primordial inflation with thermal dissipation

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(1)The tensor to scalar ratio in primordial inflation with thermal dissipation. Master of Science in Physics. Adviser: Wo-Lung Lee Student: Hans Brynner Lao Department of Physics National Taiwan Normal University August, 2017.

(2) Abstract. Inflationary theory is the prime candidate for solving the flatness, horizon, monopole problem that are encountered in the standard Big Bang, it also provides a way for seeding the large scale structure that we see today. In this thesis we study the behavior of the tensor to scalar ratio in the context of warm inflation where there is a dissipation present due to the interaction between the inflaton and other fields which are neglected in the standard cold inflation. We will also provide another approach on determining the observables during inflation by solving the evolution of the quantities in question. The tensor to scalar ratio can be approximated using the slow-roll parameters, this in turn can be used to check the authenticity of the approach taken. We fix the number of e-folding to be 60 before the end of inflation.. Keywords: cosmology, big bang, inflation, warm inflation. i.

(3) Contents Abstract. i. 1 Introduction. 1. 2 Standard Big Bang Cosmology. 2. 2.1. Expansion of the Universe . . . . . . . . . . . . . . . . . . . . .. 3. 2.2. Einstein’s equations . . . . . . . . . . . . . . . . . . . . . . . . .. 4. 2.3. Friedmann-Robertson-Walker Cosmology . . . . . . . . . . . . .. 4. 3 Cosmic Inflation Paradigm. 9. 3.1. Conformal Space and Horizons . . . . . . . . . . . . . . . . . . .. 3.2. Big Bang Problems . . . . . . . . . . . . . . . . . . . . . . . . . 12. 3.3. Shrinking Hubble Sphere . . . . . . . . . . . . . . . . . . . . . . 14. 3.4. Dynamics of the Inflaton . . . . . . . . . . . . . . . . . . . . . . 15. 3.5. Slow-Roll Inflation . . . . . . . . . . . . . . . . . . . . . . . . . 17. 3.6. Perturbations from Inflation . . . . . . . . . . . . . . . . . . . . 19. 3.7. Reheating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20. 4 Warm Inflation in General. 9. 22. 4.1. Warm Inflation Dynamics . . . . . . . . . . . . . . . . . . . . . 22. 4.2. Slow-Roll Warm Inflation . . . . . . . . . . . . . . . . . . . . . . 24. 4.3. Primordial Power Spectrum . . . . . . . . . . . . . . . . . . . . 27. 4.4. Analytic Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 29. ii.

(4) 5 Warm Inflation Models. 32. 5.1. Chaotic Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . 33. 5.2. Higgs-like Inflation . . . . . . . . . . . . . . . . . . . . . . . . . 43. 5.3. Axion Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . 54. 6 Conclusion. 66. A Scalar Field Theory. 67. B Perturbations during Inflation. 69. B.1 Scalar, vector, tensor decomposition . . . . . . . . . . . . . . . . 69 B.2 Quantum to Classical Scale . . . . . . . . . . . . . . . . . . . . 71 B.3 Statistics of Cosmological Perturbations . . . . . . . . . . . . . 71 B.4 Scalar Perturbations . . . . . . . . . . . . . . . . . . . . . . . . 73 B.5 Tensor Perturbations . . . . . . . . . . . . . . . . . . . . . . . . 73 C Notations and Conventions. 74. Bibliography. 75. iii.

(5) Chapter 1 Introduction Cosmology is the study of the universe as a whole, specifically its origin, evolution, and structure. As a field, cosmology has a strong restriction, we have only one universe. Whether there are other universes or not, it will be a purely semantic dispute until we are able to conduct observations that will give us some hints on the true nature of the word universe. The most important observation that had been done in the early twentieth century was that the galaxies are receding away from us thus producing a redshift in their spectrum [1]. The are some popular misconceptions about the Big Bang, to state the correct way of thinking about the Big Bang; it did not start from a single spatial point but the bang happened everywhere. We should also recognize the term observable universe where it is the farthest distance that we can see, beyond that is unknown to us but surely our universe does not stop there. As our universe expands, it does not expand into some space existing somewhere external to it but our universe contains all space and it is this space that is expanding.. 1.

(6) Chapter 2 Standard Big Bang Cosmology One of the major assumptions of cosmology is the cosmological principle, which states that the spatial distribution of matter in the universe is both homogeneous and isotropic when viewed on very large scales. Now, ”very large scales” can be very confusing since it can mean at the scale of our solar system or at the scale of our own galaxy. Based on galaxy surveys such as the Sloan Digital Sky Survey (SDSS), the Universe begins to look smooth at the scale of hundreds of megaparsecs, it is at this scale we mean that the cosmological principle makes sense. In 1965, while working with the Holmdel Horn Antenna and gathering data, Penzias and Wilson noticed a mysterious background noise that they cannot seem to remove, they even went to suspect a pigeon nesting on the antenna to be responsible for this noise. Later, with the help of some colleagues they confirmed that this is the imprint left by the Big Bang, called the Cosmic Microwave Background Radiation (CMBR) with black-body spectrum at a temperature around 3 Kelvin. This information greatly supports the Big Bang theory in which cosmology is heavily based. The Cosmic Background Explorer (COBE) confirmed that the background radiation is very close to the black-body spectrum with temperature around 2.725 Kelvin and that the temperature from different patches of the sky is approximately uniform, this 2.

(7) result greatly supports the cosmological principle [2].. 2.1. Expansion of the Universe. In 1922, Vesto Slipher had measured the redshifts of 41 galaxies. In 1927, Georges Lematre derived a solution from Einstein’s general relativity equations implying the expansion of the Universe, where he proposed a tentative relation between the recessional velocity of the galaxies and its distance from the observer; he determined the proportionality constant H0 to be 625 km Mpc−1 s−1 (later to be known as the Hubble constant). Two years later Edwin Hubble published a paper with better determination of the distance which was more convincing and became the widely cited paper for the discovery of the expansion of the Universe. He determined the Hubble constant to be 500 km Mpc s−1 [1]. From Hubble’s observations, faraway galaxies seem to be receding away from us and the farther away it is the faster it recedes from us. The recessional velocities are determined by redshifts since the emission and absorption spectra of the galaxies are well known. The relationship between the recessional velocity of a galaxy and its distance from the observer is given by Hubble’s law. ~v = H0~r.. (2.1.1). Note that Hubble’s law is an approximate relation since for galaxies which are near to us, peculiar velocities cannot be neglected and this should be taken into account with the measurement.. 3.

(8) 2.2. Einstein’s equations. In this subsection, we will show some of the results of General Theory of Relativity in order to obtain the useful solutions that we need to describe the dynamics of our Universe. Some very useful references for General Theory of Relativity are [3] and [4]. Matter and geometry are related by Einstein’s equation, 1 Rµν − Rgµν = 8πGTµν , 2. (2.2.1). where Rµν is the Ricci tensor, R is the Ricci scalar, G is Newton’s gravitational constant, and Tµν is the energy-momentum tensor. By contracting both sides of (2.2.1) with g µν then plugging back into it, we have the alternate form of (2.2.1), 1 Rµν = 8πG Tµν − gµν T . 2. (2.2.2). The Ricci tensor and scalar are given by, Rµν = ∂ρ Γρµν − ∂ν Γρµρ + Γρλρ Γλµν − Γρλν Γλµρ ,. (2.2.3). R = g µν Rµν ,. (2.2.4). where Γρµν is the Christoffel symbol and is given by, Γρµν. 2.3. 1 ρλ = g ∂µ gνλ + ∂ν gλµ − ∂λ gµν . 2. (2.2.5). Friedmann-Robertson-Walker Cosmology. A universe that is both homogeneous and isotropic on very large scales can be described by the Friedmann-Robertson-Walker (FRW) metric. 4.

(9) ds2 = −dt2 + a(t)2. ! dr2 + r2 (dθ2 + sin2 θdφ2 ) . 1 − κr2. (2.3.1). where we have used spherical coordinates (r, θ, φ) to describe the spatial part of the line element ds2 , and a(t) is the scale factor that characterizes the relative size of the hypersurface for different times t. We have explicitly indicated the curvature parameter κ such that for κ = 1 we have a hypersurface with positive curvature, for κ = −1 we have a hypersurface with negative curvature, and for κ = 0 we have a hypersurface with zero curvature (flat). Since our Universe is homogeneous and isotropic on very large scales, the energy-momentum tensor is of the form corresponding to a perfect fluid,. Tµν = (ρ + p)Uµ Uν + pgµν ,. (2.3.2). where U µ = (1, 0, 0, 0) is the comoving four-velocity, ρ is the energy density of the fluid, and p is its pressure. Tµν in matrix form is given by,  ρ 0  = 0 0. 0 p 0 0. 0 0 p 0.  −ρ 0  Tνµ =  0 0. 0 p 0 0. 0 0 p 0. Tµν.  0 0   0. (2.3.3). p. Raising one index gives,  0 0   0. (2.3.4). p. The trace of (2.3.4) is,. T = Tµµ = −ρ + 3p,. (2.3.5). by plugging this into (2.2.2) and taking note the FRW metric gµν , the µν = 00 component of (2.2.2) is the acceleration equation, 5.

(10) a ¨ 4πG =− (ρ + 3p), a 3. (2.3.6). the µν = ij component of (2.2.2) is, a ¨ a˙ +2 a a. !2 +2. κ = 4πG(ρ − p), a2. (2.3.7). by substituting (2.3.6) into (2.3.7), we obtain the Friedmann equation, a˙ a. !2 =. 8πG κ ρ − 2. 3 a. (2.3.8). The rate of expansion of our universe is characterized by the Hubble parameter, a˙ H= , a. (2.3.9). which has units of inverse time. The characteristic time scale of our universe is given by the Hubble time tH ∼ H −1 , while the characteristic length scale of our universe is given by the Hubble length rH ∼ H −1 (if we did not set c = 1 then rH = cH −1 ). It is very important to take note that the Hubble constant is not really a constant since it varies through time therefore a better term would be the Hubble parameter but since the time that we are talking about for it to vary involves long periods of time that are huge compared to what we are dealing with our everyday situations, in that sense we can call the value of the Hubble parameter in our present time the Hubble constant H0 . Recent measurement of the Hubble constant is given by,. H0 = 100h km Mpc−1 s−1. (2.3.10). where h is a parameter with range 0 ≤ h ≤ 1, and we take h ∼ 0.7 today. A very useful quantity is the density parameter to which we can specify the density of our universe. From the Friedmann equation (2.3.8),. 6.

(11) H2 =. 8πG κ ρ − 2, 3 a. (2.3.11). we can divide both sides by H 2 and get, κ ρ −1= , ρc (aH)2. (2.3.12). where ρc is the critical density and is the value needed in order to make our universe flat (k = 0), it is given by,. ρc =. 3H 2 . 8πG. (2.3.13). The density parameter Ω is defined to be,. Ω=. ρ , ρc. (2.3.14). where Ω0 is the density parameter today. We can rewrite (2.3.12) as,. Ω−1=. κ , (aH)2. (2.3.15). Based on observations today, Ω0 ≈ 1 which implies that our universe is very nearly flat, this means that the Euclidean geometry applies. Based on the above discussion, it is safe to assume from now on that κ = 0. We could also solve for the zeroth component of the conservation of energy equation using the the results above,. ∇µ T0µ = 0 ∂µ T0µ + Γµµλ T0λ − Γλµ0 Tλµ = 0. (2.3.16). a˙ −∂0 ρ − 3 (ρ + p) = 0. a so that, a˙ ρ˙ + 3 (ρ + p) = 0. a 7. (2.3.17).

(12) Perfect fluids relevant to cosmology obey the equation of state,. p = wρ,. (2.3.18). such that when we substitute it to (2.3.17), we get ρ˙ a˙ = −3(1 + w) . ρ a. (2.3.19). When w is a constant, we can integrate (2.3.19) to get,. ρ = c1 a−3(1+w) . where c1 is some constant.. 8. (2.3.20).

(13) Chapter 3 Cosmic Inflation Paradigm In this section we summarize why we need the Inflationary theory in order to make sense of the problems that cannot be answered by the standard Big Bang theory. We follow very closely the treatment of [5]. One of the problems with the standard Big Bang theory is that it does not explain anything about the bang, why it banged, what banged, how it banged, and what happened before the bang. Now, inflationary theory is not an attempt to solve all the fundamental questions about the bang but it tries to solve some of the major problems that the Big Bang could not answer. Inflation is a period of exponential expansion that took place during the very early universe and is speculated to have taken place around 10−34 seconds after the bang.. 3.1. Conformal Space and Horizons. The propagation of light in the FRW spacetime determines the casual structure of our universe. Since photons travel on null geodesics and suppose that the photons travel in the radial direction (dθ = dφ = 0), then from the FRW metric (we put back c in order to avoid confusion between the definitions to follow), 9.

(14) −c2 dt2 + a(t)2 dχ2 = 0,. (3.1.1). then we have, t0. Z. dt0 , a(t0 ). χ0 − χ = −c t. (3.1.2). where we assumed that the photon is moving towards us so that we took the negative solution. χ is what we call the conformal distance, so χ0 = 0 is where we are located, then Z. t0. χ=c t. dt0 . a(t0 ). (3.1.3). We now define the conformal time τ , Z. t0. τ0 − τ = dτ = t. dt0 , a(t0 ). (3.1.4). so that (setting c = 1),. χ = (τ0 − τ ),. (3.1.5). we see that the conformal distance is just the speed of light times the conformal time but in natural units the conformal time and the conformal distance are the same so that light travels in straight lines at 45◦ in the τ − χ plane. Now, the conformal time is not a physical time but it can be interpreted as a ”clock” that slows down as the universe expands; similar to the time that slows down as observed by an observer not moving with an object that travels at a speed comparable to the speed of light. We now define two horizons that will be useful in our study of inflation. Let us define the particle horizon χph which is the largest comoving distance that light could have traveled from an initial time ti to some later time t, Z. t. χph (τ ) = τ − τi = ti. 10. dt0 . a(t0 ). (3.1.6).

(15) The initial time ti is usually considered to be at the origin of the universe (ti = 0) which corresponds to τi = 0, but this is not always the case since as we will see later in our discussion of inflation, ti = 0 does not correspond to τi = 0. Note that the Big Bang is a moment in time and not a point in space. Events separated by a distance greater than the particle horizon could not have had a chance to communicate. The physical size of the particle horizon is,. rph (t) = a(t)χph .. (3.1.7). We define another horizon called the event horizon χeh , it is the largest comoving distance that light can travel from a given moment of conformal time τ to some conformal time τmax in the future, tmax. Z χeh = t. dt0 . a(t0 ). (3.1.8). Events separated by a distance greater than the event horizon will not be able to communicate. The physical size of the event horizon is,. reh = a(t)χeh .. (3.1.9). From the previous section, we defined the Hubble radius/length to be r = H −1 , and the Hubble radius in conformal space is χ = r/a, so we can write the comoving Hubble radius χ in terms of the scale factor and Hubble parameter,. χ = (aH)−1 .. 11. (3.1.10).

(16) 3.2. Big Bang Problems. Flatness Problem In the standard Big Bang regime, the comoving Hubble radius grows with time so that the term |Ω − 1| in (2.3.15) increases with time; if κ = 0 then Ω = 1 precisely and stays at that value forever, if it deviates then our universe would evolve very quickly to become more curved. From this we can say that Ω = 1 is an unstable point. Thus, Big Bang theory without inflation would require the early universe to have a fine tuned value of Ω very close to 1. The question is, why is this so? Some observational values of different density parameters are listed,. ΩB ≈ 0.04. (3.2.1). ΩD ≈ 0.23. (3.2.2). ΩΛ ≈ 0.73. (3.2.3). where ”B” stands for baryonic matter, ”D” for dark matter, and ”Λ” for dark energy. If we add all of those density parameters, we get. Ωtotal = ΩB + ΩD + ΩΛ ≈ 1.. (3.2.4). It is from this that theoretical predictions of the density parameter of our universe being extremely close to unity seems very accurate. Horizon Problem The particle horizon is a very important quantity in the analysis of the horizon problem, we write the particle horizon in terms of the comoving Hubble radius, Z. t. χph (τ ) = ti. dt0 = a(t0 ). Z 0. a. da = Ha2. 12. Z 0. ln a. (aH)−1 d ln a.. (3.2.5).

(17) For a universe dominated by a fluid with an equation of state that is constant, we have (TASI Lectures on Inflation by Baumann p. 26) 1. (aH)−1 = H0−1 a 2 (1+3w) .. (3.2.6). As we know, matter sources satisfy the strong energy condition (1 + 3w > 0) so that it is always assumed as seen from (3.2.6) that the comoving Hubble radius always increases as our universe expands. This implies that at CMB decoupling, the comoving scales that are entering our horizon today are far outside it, but the nearly homogeneous CMB result tells us a different story, that our universe was very homogeneous at the time of last scattering on scales covering many regions that are not in causal contact with each other. The increasing comoving Hubble radius implies that the integral in (3.2.5) is dominated by the upper limit of integration, and during early times its contribution is negligible,. χph (a) =. 1 2H0−1 1 (1+3w) (1+3w) a2 = τ − τi , − ai2 (1 + 3w). (3.2.7). so that as ai → 0 with w > − 13 , we have τi =. 2H0−1 21 (1+3w) a → 0. (1 + 3w) i. (3.2.8). The comoving horizon during late times is,. χph = τ =. 2H0−1 1 (1+3w) 2 a2 = (aH)−1 . (1 + 3w) (1 + 3w). (3.2.9). This show that the comoving Hubble radius and the particle horizon are approximately the same but the two should not be confused with each other.. 13.

(18) 3.3. Shrinking Hubble Sphere. A simple conjecture might allow us to avoid the horizon problem. Suppose we assume that the comoving Hubble sphere is shrinking for a long enough period during the early universe, although this would mean breaking the strong energy condition (1 + 3w < 0), we have d(aH)−1 < 0. dt. (3.3.1). From (3.2.5), the integral is dominated by the lower limit of integration so that as ai → 0 with w < − 31 , τi =. 2H0−1 12 (1+3w) a → −∞. (1 + 3w) i. (3.3.2). This means that there was more conformal time between the singularity and decoupling than we had expected. The comoving Hubble sphere expands during the standard Big Bang evolution but shrinks during inflation and the conformal time is negative during this time. Also, instead of the spacelike singularity in the Big Bang, it is replaced by the reheating period which just means that it is just one part of the process of the evolution of the universe. The conditions for inflation to take place can be given by three equivalent equations [5], • Accelerated expansion d d a ¨ (aH)−1 = (a) ˙ −1 = − 2 < 0 , dt dt (a) ˙. (3.3.3). this immediately show that a ¨ > 0 for a shrinking Hubble sphere. • Slowly varying Hubble parameter First we define a quantity called the Hubble slow roll parameter H which characterizes accelerated expansion during inflation given by,. 14.

(19) H = −. H˙ , H2. (3.3.4). so that we write the shrinking Hubble sphere as, aH ˙ + aH˙ d 1 (aH)−1 = − = − (1 − H ) < 1, 2 dt (aH) a. (3.3.5). this implies,. H = −. H˙ < 1, H2. (3.3.6). which means that H should vary very slowly. • Negative pressure Suppose we consider a perfect fluid as a source for the accelerated expansion. (2.3.11) and (2.3.17) implies H2 3p 1 (ρ + 3p) = − (1 + ), H˙ + H 2 = − 2 6MP 2 ρ. (3.3.7). so that,. H = − which implies that w =. 3.4. p ρ. H˙ 3 p = (1 + ) < 1, H2 2 ρ. (3.3.8). < − 13 .. Dynamics of the Inflaton. In the following discussion, we will set κ = 0 (flat universe) and write q 1 equations in terms of the Planck mass MP = . Since we are going to 8πG discuss the dynamics of the inflaton, it is important to take note of how much exponential expansion has already happened, we define N to be the number 15.

(20) of e-folding which is the duration in which an exponentially evolving quantity increases by a factor of e, Z. ln(aend ). N=. 0. Z. tend. d ln(a ) =. Hdt0. (3.4.1). t. ln(a). where the subscript ”end” signifies the end point of the exponential evolution. From (3.4.1) we could write (3.3.4) in the form [5],. H = −. H˙ d ln H . =− 2 H dN. (3.4.2). This implies that the variation of the Hubble parameter per e-folding is very tiny during inflation since the the condition H < 1 should hold. The energy density and pressure of the inflaton is given by, 1 ρφ = φ˙ 2 + V (φ), 2. (3.4.3). 1 pφ = φ˙ 2 − V (φ), 2. (3.4.4). from the negative pressure condition for inflation, pφ < − 13 ρφ , we have φ˙ 2 < V (φ).. (3.4.5). This implies that for inflation to take place, the potential energy of the inflaton should dominate over its kinetic energy. Now, from the Friedmann equation (2.3.11) we can substitute (3.4.3) into it so that we get,. H2 =. 1 1 ˙2 ( φ + V ). 3MP2 2. (3.4.6). The equation of motion of the inflaton is given by the Klein-Gordon equation, φ¨ + 3H φ˙ + V,φ = 0. 16. (3.4.7).

(21) 3.5. Slow-Roll Inflation. By substituting (3.4.3) and (3.4.4) into the acceleration equation, (2.3.6), we get 1 φ˙ 2 , H˙ = − 2 MP2. (3.5.1). and substitute this to the Hubble slow-roll parameter H (3.3.4) we get,. H =. φ˙ 2 . 2MP2 H 2. (3.5.2). Slow-roll inflation is a situation where the kinetic energy of the inflaton is small compared to its total energy so that (3.5.2) satisfies,. H < 1.. (3.5.3). Now, for slow-roll inflation to last, the acceleration of the inflaton should be negligible compared to the inflaton velocity term in the Klein-Gordon equation, ¨ |3H φ|. ˙ |φ|. (3.5.4). We could quantify this by defining another Hubble slow-roll parameter ηH which characterizes how much inflation should occur,. ηH = −. φ¨ . H φ˙. (3.5.5). So, in order for inflation to last we require,. ηH < 1.. (3.5.6). The Hubble slow-roll parameters H and ηH may also be expressed in terms of an approximate expression depending on the shape of the inflationary potential given by,. 17.

(22) M2 H ≈ P 2. Vφ V. !2. Vφφ MP2 − ηH ≈ MP2 V 2. ,. Vφ V. (3.5.7) !2 ,. (3.5.8). where we define the potential slow-roll parameters , η [5], M2 = P 2 η = MP2. Vφ V. !2. Vφφ , V. ,. (3.5.9). (3.5.10). so that the relationship between the Hubble and potential slow-roll parameters are given by,. H ≈ ,. (3.5.11). ηH ≈ η − .. (3.5.12). The slow-roll conditions in terms of the potential slow-roll parameters then become,. < 1,. (3.5.13). η < 1.. (3.5.14). By looking at (3.5.9), (3.5.13) implies that the potential needs to be flat in order for slow-roll inflation to take place. By applying the conditions for slow-roll inflation to the dynamical equations, we establish the slow-roll approximation. From (3.4.5) and (3.5.3), the Friedmann equation (3.4.6) becomes, 18.

(23) H2 ≈. V . 3MP2. (3.5.15). From (3.5.4) and (3.5.6), the Klein-Gordon equation can be written as, 3H φ˙ ≈ −V,φ .. 3.6. (3.5.16). Perturbations from Inflation. At the horizon crossing (k = aH) we could compute the power spectrum of the conserved curvature perturbations ∆2R which is also equal to the power spectrum of scalar fluctuations ∆2s . The relation between the power spectrum of the curvature perturbations ∆2R and the inflaton fluctuations ∆2δφ is given by, H φ˙. ∆2s = ∆2R =. !2 ∆2δφ ,. (3.6.1). where the power spectrum of the inflaton fluctuations ∆2δφ is given by, H 2π. ∆2δφ =. !2 ,. (3.6.2). so that,. ∆2s =. H 2π. !2. H φ˙. !2 .. (3.6.3). From (3.5.2) and (3.5.11), we have

(24)

(25) 2 H

(26) ∆2s ≈ 2 2

(27) 8π MP

(28). (3.6.4) k=aH. The relation between the power spectrum of the tensor perturbations ∆2t and the inflaton fluctuations ∆2δφ is given by,. 19.

(29) ∆2t =. 8 2 ∆ , MP2 δφ. (3.6.5). so that,

(30) 2

(31) 2H

(32) ∆2t = 2 2

(33) π MP

(34). (3.6.6) k=aH. The tensor fluctuations are often normalized with respect to the amplitude of the scalar fluctuations which we call the tensor-to-scalar ratio r,. r=. ∆2t ≈ 16∗ ∆2s. (3.6.7). where we sometimes denote that we are evaluating at the horizon crossing by a subscript ”∗ ”.. 3.7. Reheating. Inflation ends when the kinetic energy of the inflaton is comparable to its potential energy for single-field models or instability due to the inflaton reaching a critical value in multi-field hybrid models[6][7]. During inflation the energy of the inflaton is dominantly potential energy, but as it rolls towards the minimum it has to transfer its energy to the standard model particles. This mechanism is called reheating and this epoch is where the standard Big Bang follows [5][7]. Suppose we assume a potential of the form V = 21 m2 φ2 . As the inflaton begins to oscillate at the bottom of the potential, it is still homogeneous such that it still satisfies the Klein Gordon equation. From the (3.5.15), mφ . H≈√ 6MP. (3.7.1). For |φ| < MP , the oscillation period becomes much shorter than the expansion time scale such that m−1 H −1 . 20.

(35) The universe cannot end up empty so the inflaton must couple to the standard model fields. If the inflaton decays into fermions then the decay is slow but if the inflaton can decay into bosons then the decay may be rapid which involves a process called parametric resonance. If the decay is rapid then there is not enough time to reach thermal equilibrium, this process is called preheating .. 21.

(36) Chapter 4 Warm Inflation in General Warm inflation is a scenario introduced by Fang [8][9] where interaction between the inflaton and other fields can dissipate the inflaton energy to some lighter fields. In the case of cold inflation, the radiation energy density quickly becomes negligible since it is inversely proportional to the fourth power of the scale factor. From General Relativity, the only requirement for inflation is that the vacuum energy density is greater than the radiation energy density [10]. By utilizing this condition, one of the key features of warm inflation is that there is nonnegligible radiation present during inflation therefore leading to a smooth exit to a radiation dominated epoch hence providing an alternative solution to the ”graceful exit” problem and does not require a separate reheating period as in the cold inflation case. Another problem that can be remedied by warm inflation is the invalidation of the simple monomial potential models due to having a prediction of the tensor to scalar ratio that exceeds the boundary given by the Planck satellite [11].. 4.1. Warm Inflation Dynamics. During warm inflation, the inflaton energy density ρφ and the radiation energy density ρr are important therefore the Friedmann equation (2.3.11) has 22.

(37) the form (κ = 0),. H2 =. 1 (ρφ + ρr ), 3MP2. (4.1.1). or using (3.4.3) we can write (4.1.1) as,. H2 =. 1 1 ˙2 ( φ + V + ρr ). 3MP2 2. (4.1.2). The continuity equations for the inflaton and radiation take the form respectively [8], ρ˙ φ + 3H(ρφ + pφ ) = −Γφ˙ 2 ,. (4.1.3). ρ˙ r + 4Hρr = Γφ˙ 2 ,. (4.1.4). where Γ is the dissipation term. Inflation in the context of particle physics implies that the inflaton interacts with other fields but during the cold inflationary scenario it is assumed that the inflationary dynamics is not affected by the interaction between those fields and the inflaton. Despite of this, the interaction between them may lead to dissipation of the inflaton energy into other degrees of freedom. The dissipation in (4.1.3) will appear as an extra friction term in the inflaton equation of motion [11]. By substituting (3.4.3) and (3.4.4) in (4.1.3), we get the inflaton equation of motion in warm inflation, φ¨ + (3H + Γ)φ˙ + V,φ = 0, or in terms of the quantity Q =. Γ 3H. (4.1.5). which represents the effectiveness at which. inflaton energy is being transformed into radiation, we have φ¨ + 3H(1 + Q)φ˙ + V,φ = 0. 23. (4.1.6).

(38) If Q > 1 , there is a strong dissipative regime, and if Q < 1 there is a weak dissipative regime. Also, we will consider a constant Q throughout the discussion though generally Q is a function of φ and T . The temperature of the background radiation can be calculated by, π2 ρr = geff T 4 , 30. (4.1.7). where geff is the effective number of degrees of freedom at temperature T [9]. During warm inflation, radiation is continuously produced due to the dissipation of the inflaton energy, so it is not necessarily the case during inflation that radiation is redshifted away. One of the conditions for inflation to take place is that the energy density of the inflaton should dominate over the radiation energy density but it can be larger than the expansion rate such that 1/4. ρr. > H [11], from (4.1.7) we can write it as,. T > H,. (4.1.8). this is one of the most important conditions for warm inflation. A scenario where even if this condition holds and warm inflationary period is not guaranteed is studied by Ramos & da Silva [12].. 4.2. Slow-Roll Warm Inflation. During warm inflation, additional conditions are added aside from the con¨ |3H φ|); ˙ ditions imposed in cold inflation (φ˙ 2 V and |φ| dark energy predominates over radiation (ρr ρφ ), the production of radiation is quasi¨ |3H(1 + Q)φ|) ˙ [13]. static (ρ˙ r 4Hρr ), and the similar condition (|φ| Equations (4.1.2), (4.1.4), and (4.1.6) can be approximated by,. H2 ≈. V , 3MP2. 24. (4.2.1).

(39) φ˙ ≈. ρr ≈. −V,φ , 3H(1 + Q). (4.2.2). Γφ˙ 2 3 = Qφ˙ 2 . 4H 4. (4.2.3). The Hubble slow-roll parameters H and ηH that characterizes the accelerated expansion and its duration in warm inflation are modified to, H˙ H2. (4.2.4). φ¨ . H(1 + Q)φ˙. (4.2.5). H = −. ηH = −. The Hubble slow-roll parameters H and ηH in warm inflation may also be expressed in terms of the potential; by taking the first derivative of (4.2.1), we have. V,φ2 φ˙ ¨ (φ + V,φ ) ⇒ 2H H˙ ≈ − 3 9H(1 + Q) 2 V,φ H˙ ≈ − , 2 18H (1 + Q). 2H H˙ = ⇒. (4.2.6). so that V,φ2 H˙ MP2 V,φ H = − 2 ≈ ≈ H 18H 4 (1 + Q) 2(1 + Q) V. !2 .. (4.2.7). Taking the first derivative of (4.2.2), we have φ¨ ≈ −. V,φφ φ˙ V,φ H˙ + , 3H(1 + Q) 3H 2 (1 + Q). so that,. 25. (4.2.8).

(40) φ¨ V,φφ V,φ H˙ ≈ − 3H 2 (1 + Q)2 3H 3 (1 + Q)3 φ˙ H(1 + Q)φ˙ MP2 V,φφ H˙ ≈ + (1 + Q)2 V (1 + Q)H 2 !2 MP2 V,φ MP2 V,φφ − . ≈ (1 + Q)2 V 2(1 + Q)2 V. ηH = −. (4.2.9). We now establish the relationship between the Hubble and potential slow-roll parameters in warm inflation, , (1 + Q). (4.2.10). 1 (η − ) . (1 + Q)2. (4.2.11). H ≈. ηH ≈. The slow-roll conditions H < 1 and ηH < 1 imply that the flatness of the potential is no longer necessary and the potential slow-roll parameters can be greater than unity which puts much lesser constraint on the shape of the potentials studied in different models,. < 1 + Q,. (4.2.12). η < + (1 + Q).. (4.2.13). We rewrite (4.2.2) in terms of the potential slow roll parameter ,. φ˙ 2 ≈. V,φ2 H 2 V,φ2 H 2 MP4 V,φ2 2MP2 H 2 ≈ ≈ = . 9H 2 (1 + Q)2 9H 4 (1 + Q)2 (1 + Q)2 V 2 (1 + Q)2. (4.2.14). Inflationary expansion ends when the slow-roll conditions are violated such that,. H (φend ) = 1 ,. ηH (φend ) = 1 . 26. (4.2.15).

(41) with, !2

(42) V,φ

(43)

(44) =

(45)

(46) (1 + Q) 2(1 + Q) V MP2. ≈1 φend. 1 MP2 MP2 V,φφ − (η − ) = (1 + Q)2 (1 + Q)2 V 2(1 + Q)2. V,φ V. !2

(47)

(48)

(49)

(50)

(51). (4.2.16) ≈1 φend. The number of e-folding in warm inflation is modified using (4.2.1) and (4.2.2), Z. tend. Z. φ φend. t. −(1 + Q) ≈ MP2. 4.3. φend. Hdt =. N=. Z φ. H dφ φ˙. V (1 + Q) dφ = √ V,φ 2MP. Z. φ. φend. dφ √ . . (4.2.17). Primordial Power Spectrum. In this section we will consider the warm inflation version of the scalar power spectrum, tensor power spectrum, and scalar spectral index. Scalar power spectrum The evolution of the inflaton perturbations is modified by the fluctuationdissipation dynamics in which the source is a gaussian white noise ξk . √ k2 δ φ¨k + 3H(1 + Q)δ φ˙ k + 2 δφk ≈ 2ΓT a−3/2 ξk , a The scalar power spectrum is modified as [12][17],. ∆2R =. H∗ 2π. !2. H∗ φ˙ ∗. !2. . 1 + 2n∗ + ω∗ ,. (4.3.1). (4.3.2). where ”∗ ” indicates that the quantity is evaluated at the horizon crossing k0 = 0.05Mpc−1 (N = 60 e-folds before the end of inflation), n∗ is the inflaton occupation number and ω∗ is,. ω∗ =. T∗ H∗. !. √ 2 3πQ∗ √ , 3 + 4πQ∗. 27. (4.3.3).

(52) As n∗ , Q∗ , T∗ → 0 we recover the scalar power spectrum in the cold inflation case. The Planck data [14] constrained the scalar power spectrum with 68% confidence level at the fixed wave number k0 = 0.05Mpc−1 to be, −9 ∆2R = (2.215+0.032 −0.079 ) × 10 .. (4.3.4). Tensor power spectrum The tensor perturbations do not incorporate strongly with the thermal background such that gravitational waves are generated only by the quantum fluctuations as in the standard cold inflation [16]. The tensor power spectrum is thus the same as in the standard cold inflation,. ∆2t =. 2 H∗2 . π 2 MP2. (4.3.5). Tensor to scalar ratio In the limit where the inflaton particle production is negligible n∗ 1, we compute the tensor to scalar ratio r by [12][17], " #−1 √ ∆2t 8φ˙ 2∗ T∗ 2 3πQ∗ √ r = 2 = 2 2 1+ . ∆R MP H∗ H∗ 3 + 4πQ∗. (4.3.6). By applying the slow-roll approximation, we obtain ". r≈. 16 1+ (1 + Q)2. s 4. 18MP6 Q 61π 2 (1 + Q)2. #−1 √ V,φ2 2 3πQ √ . V 3 3 + 4πQ. (4.3.7). As Q → 0, we obtain the tensor to scalar ratio r → 16 as in the cold inflation case. The Planck data [14] constrained the tensor to scalar ratio at k0 = 0.05Mpc−1 to be. r < 0.12 95%C.L.. 28. (4.3.8).

(53) Scalar spectral index In the limit where the inflaton particle production is negligible n∗ 1, the scalar spectral index ns is given by [10][16][18],. ns − 1 = 4. H˙ ∗ φ¨∗ ω˙ ∗ − 2 − . 2 H∗ H∗ φ˙ ∗ H∗ (1 + ω∗ ). (4.3.9). By applying the slow-roll approximation and suppose we are considering a constant Q such that Q˙ = 0, we obtain. ns − 1 ≈. 1 (5η − 15). 2(1 + Q). (4.3.10). The Planck data [14] constrained the scalar spectral index at k0 = 0.05Mpc−1 to be,. ns = 0.9655 ± 0.0062 68%C.L.. 4.4. (4.3.11). Analytic Solution. In this section we will find the warm inflation solution of (4.1.2), (4.1.4), (4.1.6) analytically without resorting to slow-roll approximation [19]; we first solve for the inflation equation of motion (4.1.6), φ¨ + 3H(1 + Q)φ˙ + V,φ = 0,. (4.4.1). the auxiliary equation of this second order differential equation is,. x2 + 3H(1 + Q)x + c = 0, where c is the coefficient of V,φ (V,φ = cφ) and the solution is,. 29. (4.4.2).

(54) p. 9H 2 (1 + Q)2 − 4c 2p 3H(1 + Q)[−1 ± 1 − 4c/9H 2 (1 + Q)2 ] = 2 3H(1 + Q)[−1 ± (1 − 2c/9H 2 (1 + Q)2 ) ] ≈ 2 c ≈− . 3H(1 + Q). x=. −3H(1 + Q) ±. (4.4.3). Thus,. φ = φi eαHt ,. (4.4.4). where φi is the initial value of the inflaton field and α is given by,. α=−. c 3H 2 (1. + Q). .. (4.4.5). Substituting (4.4.4) into (4.1.4), we get. ρr = C1 e2αHt + C2 e−4Ht , where C1 =. 3α2 φ2i QH 2 (2α+4). (4.4.6). and C2 = ρr (0) − C1 .. From (4.4.6) it is evident that there are two phases of the evolution of radiation where phase 1 is the period where the C2 term is dominant due to inflationary expansion and radiation density drops substantially; while phase 2 is where the C1 term is dominant and the radiation density begins to increase due to the friction of the inflaton. In phase 2, inflation and heating are both in progress such that this can be interpreted as the stage of inflation as well as reheating. This is one of the reasosn why warm inflation does not need to employ any reheating epoch after inflation since it is being done during inflation and it can exit smoothly into a radiation dominated epoch after the slow-roll conditions are violated. The temperature at the transition from phase 1 to phase 2 is called the rebound temperature[19]. 30.

(55) The number of e-folding during warm inflation is given by, Z. tend. Hdt ≈. N= t. 2 T ln , α Tb. (4.4.7). Isolating T , we get [19]. T ≈ Tb e. Nα 2. .. (4.4.8). Now, we can substitute this into (4.3.6) and get, " #−1 √ N α∗ Tb e 2 2 3πQ∗ 8φ˙ 2∗ √ r = 2 2 1+ . MP H∗ H∗ 3 + 4πQ∗. 31. (4.4.9).

(56) Chapter 5 Warm Inflation Models In this chapter we consider different models in the context of warm inflation, we describe the behavior of the observables by considering the models in the strong dissipation regime Q > 1 and in the weak dissipation regime Q < 1. We will present three models which are, chaotic inflation, higgs-like particle inflation, and axion-like particle inflation. Regardless of the mechanism, each model should be tested against the measurements obtained by the Planck mission, although some models considered in the cold inflation scenario are disfavored by the Planck data, warm inflation can remedy this situation by the introduction of the dissipation rate Γ. The Planck data that we are going to base our work on will be Planck 2015 results XX. Constraints on inflation [14]. We also note that this work is not meant to be exhaustive but to merely point out the possible pitfall that might lurk in dealing with warm inflation models i.e. by using the usual slow-roll approximation and setting N = 60 will ensure that the observables obtained are within the conditions for warm inflation. Without explicitly showing that the condition T > H is satisfied, we cannot be ensured that we are indeed in warm inflation. In section (4.4), the rebound temperature can be well below the Hubble constant which if it is the case then we are in the cold inflation regime. Details on the numerical calculations for this work are discussed in Appendix C. 32.

(57) 5.1. Chaotic Inflation. We consider in this section the chaotic potential 1 V = m2 φ2 , 2. (5.1.1). where m is the inflaton mass and φ is the inflaton field. The inflaton mass is constrained using the scalar power spectrum and its value is found to be m ≈ 1013 GeV [15]. The potential slow-roll parameters and η are M2 = P 2. V,φ V. !2. M2 = P 2. m2 φ 1 2 2 mφ 2 m2 1 2 2 mφ 2. V,φφ = MP2 η = MP2 V. !2 =. ! =. 2MP2 , φ2. 2MP2 φ2 .. (5.1.2). (5.1.3). From (4.2.17),. Z (1 + Q) φ dφ √ N≈ √ 2MP φend 1+Q Z φ φ dφ ≈ 2MP2 φend 1+Q ≈ (φ2 − φ2end ) . 4MP2. (5.1.4). 2MP2 = ≈ 1. (1 + Q) (1 + Q)φ2end. (5.1.5). From (4.2.16),. Thus, r φend ≈ MP. 2 . 1+Q. From (5.1.4) we can produce an expression for φ, 33. (5.1.6).

(58) φ2 ≈ φ2end +. 2MP2 4N MP2 (2 + 4N ) 2 4N MP2 ≈ + = MP . 1+Q 1+Q 1+Q 1+Q. (5.1.7). Now, we can calculate the tensor to scalar ratio from (4.3.7) for N = 60 before the end of inflation, " #−1 r √ 16 36MP2 Q 2 3πQ 4 √ r≈ 1+ . 121(1 + Q) 893101π 2 m2 3 + 4πQ. (5.1.8). Figure 5.1: The tensor to scalar ratio vs. Q for Q < 1 using the slow-roll approximation. The dashed line indicates the upper bound given by Planck with 95% confidence level.. 34.

(59) Figure 5.2: The tensor to scalar ratio vs. Q for Q > 1 using the slow-roll approximation. The dashed line indicates the upper bound given by Planck with 95% confidence level.. It can be seen from figure 5.1 that for very low Q, the tensor to scalar ratio exceeds the bound given by the Planck data as is the case in the cold inflation but as the dissipation increases the tensor to scalar ratio is suppressed to values that are within the given bound and it can be seen that this starts to happen around Q ≈ 10−2 . In figure 5.2 for high Q the tensor to scalar ratio starts off very low and continues to decrease as the dissipation gets stronger. The scalar spectral index can be calculated from (4.3.10) for N = 60 before the end of inflation,. ns ≈ 1 −. 10MP2 10 = 1 − = 0.958677. (1 + Q)φ2 242. (5.1.9). This shows that for chaotic quadratic potential, the scalar spectral index is approximately independent of the dissipation Γ, in which we plot the tensor to scalar ratio as a function of the scalar spectral index in figure 5.3 and figure 5.4.. 35.

(60) Figure 5.3: The tensor to scalar ratio vs. ns for Q < 1 using the slow-roll approximation. The light blue region shows the range for ns with 95% confidence level and the yellow region shows the range for ns with 68% confidence level as given by Planck.. 36.

(61) Figure 5.4: The tensor to scalar ratio vs. ns for Q > 1 using the slow-roll approximation. The light blue region shows the range for ns with 95% confidence level and the yellow region shows the range for ns with 68% confidence level as given by Planck.. Numerical Calculation One of the problems with the approximations and procedures we have done above is that it does not give any reference if the total duration of inflation for a given Q is enough to solve the problems of the standard Big Bang such that it should also satisfy the condition T > H. The keypoint here is that we just assumed that if we set N = 60 and calculate the observables for a given Q we would be able to get the observables at the horizon exit in warm inflation, but it could also be the case that at N = 60 the temperature is still less than the Hubble constant. So, the best procedure to take is to first numerically solve the evolution of the temperature and Hubble constant during inflation and acquire the values of the corresponding quantities for a given Q at the horizon exit assuring that it is in the region where T > H. The initial conditions used in this model were φ˜0 = 27 and φ˜00 = 10−4 (See Appendix C).. 37.

(62) Figure 5.5: The temperature evolution for Q = 10−5 and N ≈ 58.. Figure 5.6: The temperature evolution for Q = 30, N ≈ 1020.. It can be seen in figure 5.5 that the rebound temperature can go below the Hubble constant during inflation, but we only consider the region where T > H in which this is the regime of warm inflation; this is where we question if the 38.

(63) duration is enough in order to solve the standard problems of Big Bang. For Q = 10−5 , N ≈ 58 so that this can be considered as the lower limit for the model to be a legitimate warm inflation. In figure 5.6 it can be seen that for Q = 30, N ≈ 1020, the rebound temperature is above the Hubble constant.. Figure 5.7: The evolution of the inflaton and radiation energy density for Q = 10−5 .. 39.

(64) Figure 5.8: The evolution of the inflaton and radiation energy density for Q = 30.. From section 4.1, one of the conditions for warm inflation is that the energy density of the inflaton should dominate over the radiation energy density, but if this is violated or in other words the radiation density is comparable to the inflaton energy density (ρr ≈ ρφ ) then inflation is terminated. It can be seen from figure 5.7 that at the end of inflation for Q = 10−5 , the radiation energy density is barely comparable to the energy density of the inflaton, this just implies that Q is too small and its behavior is close to the cold inflation case; while in figure 5.8 for Q = 30, the radiation energy density is comparable to the energy density of the inflaton at the end of inflation. The effect of Q is that as it increases the number of e-folding also increases so that we should determine the tensor to scalar ratio (4.3.6) and scalar spectral index (4.3.9) only for each Q at the horizon crossing, and we do this for all the Q to be studied.. 40.

(65) Figure 5.9: The tensor to scalar ratio as a function of Q (Q < 1) obtained from (4.3.6). The dashed line indicates the upper bound given by Planck with 95% confidence level.. Figure 5.10: The tensor to scalar ratio as a function of Q (Q > 1) obtained from (4.3.6). The dashed line indicates the upper bound given by Planck with 95% confidence level.. 41.

(66) Figure 5.11: The tensor to scalar ratio as a function of the scalar spectral index ns (Q < 1) obtained from (4.3.9). The light blue region shows the range for ns with 95% confidence level and the yellow region shows the range for ns with 68% confidence level as given by Planck.. Figure 5.12: The tensor to scalar ratio as a function of the scalar spectral index ns (Q > 1) obtained from (4.3.9). The light blue region shows the range for ns with 95% confidence level and the yellow region shows the range for ns with 68% confidence level as given by Planck. For the numerical calculation, in figure 5.9 for Q < 1 the tensor to scalar 42.

(67) ratio starts off at values above the bound given by Planck but starts to enter the bound around Q ≈ 10−2 , while in figure 5.10 for Q > 1 it is well below the bound. In figure 5.11 and 5.12 it can be seen that the scalar spectral index is approximately independent of the dissipation so that ns ≈ 0.96 which is in the 95% confidence region.. 5.2. Higgs-like Inflation. We consider in this section the Higgs-like potential,. V = λ(φ2 − σ 2 )2 ,. (5.2.1). where σ is the energy scale of symmetry breaking and λ is the coupling constant. The value that the parameter σ can take is constrained by how much freedom we would like the initial conditions to have, but given that constrain we should also regulate its value so as to generate enough e-folding for a viable inflation, we took σ = 20MP [20]. The coupling constant λ is constrained using the scalar power spectrum from the Planck results, its value is found to be λ ≈ 10−14 , which is in the range of the accepted region [21]. The potential slow-roll parameters and η are, M2 = P 2. V,φ V. !2 = 8MP2. φ2 (φ2 − σ 2 )2. V,φφ 4 8φ2 η = MP2 = MP2 + V φ2 − σ 2 (φ2 − σ 2 )2 From (4.2.17),. 43. (5.2.2) ! (5.2.3).

(68) N≈ ≈ ≈ ≈ ≈. Z (1 + Q) φ dφ √ √ 2MP φend Z (1 + Q) φ σ 2 − φ2 dφ 4MP2 φ φend Z (1 + Q)σ 2 φ dφ 4MP2 φend φ

(69) φ 2 (1 + Q)σ

(70) ln φ

(71) 4MP2 φend ! (1 + Q)σ 2 φ ln . 4MP2 φend. (5.2.4). where we neglected the φ2 term in the second line since φ σ during the slow-roll stage. From (4.2.16), 8MP2 φ2end = ≈ 1. (1 + Q) (1 + Q) (φ2end − σ 2 )2. (5.2.5). Thus, we have. φend. 1 ≈ 2. "s. 8MP2 + 1+Q. s. # 8MP2 + 4σ 2 . 1+Q. (5.2.6). From (5.2.4) we can produce an expression for φ,. φ ≈ φend e. 2 4N MP (1+Q)σ 2. 1 = 2. "s. 8MP2 1+Q. s +. # 2 4N MP 8MP2 2 2 (1+Q)σ + 4σ e . 1+Q. (5.2.7). From (4.3.7) we can calculate the tensor to scalar ratio for N = 60 before the end of inflation,. r≈. (1. 128MP2 φ2 + Q)2 (φ2 −. " σ 2 )2. 1+. s 4. #−1 √ 2 3πQ 288MP6 Qφ2 √ 61π 2 λ(1 + Q)2 (φ2 − σ 2 )4 3 + 4πQ (5.2.8). 44.

(72) Figure 5.13: The tensor to scalar ratio vs. Q for Q < 1 using the slow-roll approximation. The dashed line indicates the upper bound given by Planck with 95% confidence level.. The value of the tensor to scalar ratio also depends on the value of σ where we chose σ = 20 since it produces enough inflation in our numerical calculation but it can be seen in figure 5.13 that for low Q the tensor to scalar ratio exceeded the bound given by the Planck data, although we could have chosen a different σ in order to suppress the tensor to scalar ratio, this is just to compare to the result of the numerical calculation, while in figure 5.14 the tensor to scalar ratio is well inside the bound. It should be taken note that there is an accepted region where we can choose the value of σ with range 15MP ≤ σ ≤ 40MP [20].. 45.

(73) Figure 5.14: The tensor to scalar ratio vs. Q for Q > 1 using the slow-roll approximation. The dashed line indicates the upper bound given by Planck with 95% confidence level.. The scalar spectral index (4.3.10) for N = 60 before the end of inflation can be calculated as,. ns ≈ 1 −. 10MP2 3φ2 + σ 2 1 + Q (φ2 − σ 2 )2. (5.2.9). From figure 5.15 it can be seen that for low dissipation the scalar spectral index is well within the bound given by the Planck data but as the dissipation gets stronger the scalar spectral index deviates tremendously from 1, while in figure 5.16 the scalar spectral index continues to deviate farther from 1 which can be seen to be extremely beyond the 95% confidence region.. 46.

(74) Figure 5.15: The tensor to scalar ratio vs. ns for Q < 1 using the slow-roll approximation. The light blue region shows the range for ns with 95% confidence level and the yellow region shows the range for ns with 68% confidence level as given by Planck.. 47.

(75) Figure 5.16: The tensor to scalar ratio vs. ns for Q > 1 using the slow-roll approximation. The light blue region shows the range for ns with 95% confidence level and the yellow region shows the range for ns with 68% confidence level as given by Planck.. 48.

(76) Numerical Calculation The initial conditions used in this model were φ˜0 = 3 and φ˜00 = 0.0029 (See Appendix C).. Figure 5.17: The temperature evolution for Q = 10−5 and N ≈ 75. In figure 5.17, for Q = 10−5 the temperature went below the Hubble parameter but dominated it later to produce a total duration of warm inflation around N ≈ 75, while in figure 5.18 for Q = 30 the dissipation is strong enough to induce a rebound temperature above the Hubble parameter with a total duration of warm inflation around N ≈ 885.. 49.

(77) Figure 5.18: The temperature evolution for Q = 30, N ≈ 885.. Figure 5.19: The evolution of the inflaton and radiation energy density for Q = 10−5 .. 50.

(78) Figure 5.20: The evolution of the inflaton and radiation energy density for Q = 30.. As with the chaotic inflation, the total duration of inflation increases as the dissipation increases. In figure 5.19 it can be seen that for Q = 10−5 the radiation energy density is barely comparable to the energy density of the inflaton at the end of inflation while in figure 5.20 for Q = 30 the radiation energy density is very comparable to the energy density of the inflaton.. 51.

(79) Figure 5.21: The tensor to scalar ratio as a function of Q (Q < 1) obtained from (4.3.6). The dashed line indicates the upper bound given by Planck with 95% confidence level.. Figure 5.22: The tensor to scalar ratio as a function of Q (Q > 1) obtained from (4.3.6). The dashed line indicates the upper bound given by Planck with 95% confidence level.. 52.

(80) Figure 5.23: The tensor to scalar ratio as a function of the scalar spectral index ns (Q < 1) obtained from (4.3.9). The light blue region shows the range for ns with 95% confidence level and the yellow region shows the range for ns with 68% confidence level as given by Planck.. 53.

(81) Figure 5.24: The tensor to scalar ratio as a function of the scalar spectral index ns (Q > 1) obtained from (4.3.9). The light blue region shows the range for ns with 95% confidence level and the yellow region shows the range for ns with 68% confidence level as given by Planck.. In figure 5.21 it can be seen that the tensor to scalar ratio starts of well below the bound given by the Planck data as is the case in the standard cold inflation. This is in contrast to the slow-roll result in figure 5.13, while in figure 5.22 the tensor to scalar ratio is well below the bound and continues to do so as Q gets stronger. In figure 5.23, the tensor to scalar ratio vs. scalar spectral index data points fall within the 95% confidence level region given by the Planck data but starts to enter the 68% confidence level region as Q increases. This is in contrast to the result in figure 5.15 where as Q increases the scalar spectral index shifts towards values 1. In figure 5.24 the scalar spectral index data points are well inside the 68% confidence region.. 5.3. Axion Inflation. We consider in this section the axion-like particle potential, 54.

(82) where Λ =. √. h φ i V = Λ4 1 + cos N . f. (5.3.1). mf , m ∼ 1013 GeV is the inflaton mass, f is the decay constant,. and N an integer. We will take N = 1 so that the potential has a unique minimum at φ = πf . The decay constant has a lower bound close to the Planck mass so we are free to choose beyond that, we chose f = 7MP such that the number of e-folding is enough to generate a viable inflation. The potential slow-roll parameters , η are,. M2 = P 2. Vφ V. !2. M2 = P 2. −m2 f sin(φ/f ) m2 f 2 (1 + cos(φ/f )). !2 (5.3.2). M2 MP2 1 − cos(φ/f ) sin2 (φ/f ) = P2 = . 2f (1 + cos(φ/f ))2 2f 2 1 + cos(φ/f ) −m2 cos(φ/f ) V,φφ = MP2 η = MP2 V m2 f 2 (1 + cos(φ/f )). ! =−. MP2 cos(φ/f ) f 2 1 + cos(φ/f ). (5.3.3). From (4.2.17),. N≈ ≈. ≈. ≈ ≈ ≈. Z Z −(1 + Q) φend V (1 + Q)f φend 1 + cos(φ/f ) dφ ≈ dφ MP2 Vφ MP2 sin(φ/f ) φ φ "Z # Z φend φend (1 + Q)f csc(φ/f )dφ + cot(φ/f )dφ MP2 φ φ " #

(83) φend

(84) (1 + Q)f 2 sin(φ/f )

(85) ln

(86) 2 MP csc(φ/f ) + cot(φ/f )

(87) φ " #

(88) φend (1 + Q)f 2 2 sin(φ/2f ) cos(φ/2f )

(89)

(90) ln

(91)

(92) MP2 cot(φ/2f ) φ h i

(93) 2 φ end (1 + Q)f

(94) ln 2 sin2 (φ/2f )

(95) 2 MP φ " # (1 + Q)f 2 sin2 (φend /2f ) ln . MP2 sin2 (φ/2f ) 55. (5.3.4).

(96) From (4.2.16),. MP2 sin2 (φend /f ) = (1 + Q) 2(1 + Q)f 2 (1 + cos(φend /f ))2 MP2 1 − cos(φend /f ) ≈1 = 2 2(1 + Q)f 1 + cos(φend /f ). (5.3.5). 1 − cos(φend /f ) ≈β, 1 + cos(φend /f ). (5.3.6). Thus, we have. where we define,. β=. 2(1 + Q)f 2 . MP2. (5.3.7). By computing for φend , we have φend ≈ f cos−1. 1−β 1+β. ! (5.3.8). From (5.3.4) we can produce an expression for φ, " φ = 2f sin−1. ". 1 cos−1 sin 2. 1−β 1+β. !# e. # −N β. (5.3.9). From (4.3.7), we can calculate the tensor to scalar ratio for N = 60 before the end of inflation,. r≈. (1. 8MP2 + Q)2 f 2. #−1 √ 6 1 − cos(φ/f ) 18M Q 1 − cos(φ/f ) 2 3πQ P √ 1+ 4 1 + cos(φ/f ) 61π 2 m2 f 4 (1 + Q)2 (1 + cos(φ/f ))2 3 + 4πQ (5.3.10) ". s. 56.

(97) Figure 5.25: The tensor to scalar ratio vs. Q for Q < 1 using the slow-roll approximation. The dashed line indicates the upper bound given by Planck with 95% confidence level.. Figure 5.26: The tensor to scalar ratio vs. Q for Q > 1 using the slow-roll approximation. The dashed line indicates the upper bound given by Planck with 95% confidence level. From figure 5.25 for low Q it can be seen that the tensor to scalar ratio 57.

(98) starts off well below the bound given by the Planck data and gets suppressed as the dissipation increases, while in figure 5.26 for high Q the tensor to scalar ratio is well below the bound and continues to do so as Q gets stronger. The scalar spectral index (4.3.10) for N = 60 before the end of inflation can be calculated as, 5MP2 ns ≈ 1 − 4(1 + Q)f 2. 3 − cos(φ/f ) 1 + cos(φ/f ). ! (5.3.11). Figure 5.27: The tensor to scalar ratio vs. ns for Q < 1 using the slow-roll approximation. The light blue region shows the range for ns with 95% confidence level and the yellow region shows the range for ns with 68% confidence level as given by Planck.. 58.

(99) Figure 5.28: The tensor to scalar ratio vs. ns for Q > 1 using the slow-roll approximation. The light blue region shows the range for ns with 95% confidence level and the yellow region shows the range for ns with 68% confidence level as given by Planck.. In figure 5.27, the tensor to scalar ratio vs. scalar spectral index data points starts from the 95% confidence region and slowly shifts towards the 68% confidence region as Q increases, while in figure 5.28 it somehow is in the border between the 68% and 95% confidence region.. 59.

(100) Numerical Calculation The initial conditions used in this model were φ˜0 = 0.104 and φ˜00 = 2.88 × 10−4 (See Appendix C).. Figure 5.29: The temperature evolution for Q = 10−5 and N ≈ 40.. 60.

(101) Figure 5.30: The temperature evolution for Q = 30, N ≈ 3028.. Figure 5.31: The evolution of the inflaton and radiation energy density for Q = 10−5 .. 61.

(102) Figure 5.32: The evolution of the inflaton and radiation energy density for Q = 30.. From figure 5.29 and 5.30 it can be seen that the effect of the dissipation on the total duration of inflation is much more drastic than with the two models previously studied. An example where warm inflation does not have enough e-folding can be seen in figure 5.29 where the temperature T is only above H for approximately N ≈ 40. In figure 5.31, it can be seen that the radiation energy density is not comparable to the inflaton energy density at the end of inflation such that it can be the case that the behavior is close to the cold inflation, while in figure 5.32 it can be seen that the radiation energy density is comparable to the inflaton energy density.. 62.

(103) Figure 5.33: The tensor to scalar ratio as a function of Q (Q < 1) obtained from (4.3.6). The dashed line indicates the upper bound given by Planck with 95% confidence level.. In figure 5.33, the tensor to scalar ratio starts off well below the bound given by the Planck data but deviated a little from the result in figure 5.25, while in figure 5.34 it can be seen that the tensor to scalar ratio is well below the bound and continues to do so as Q gets stronger.. 63.

(104) Figure 5.34: The tensor to scalar ratio as a function of Q (Q > 1) obtained from (4.3.6). The dashed line indicates the upper bound given by Planck with 95% confidence level.. Figure 5.35: The tensor to scalar ratio as a function of the scalar spectral index ns (Q < 1) obtained from (4.3.9). The light blue region shows the range for ns with 95% confidence level and the yellow region shows the range for ns with 68% confidence level as given by Planck.. 64.

(105) Figure 5.36: The tensor to scalar ratio as a function of the scalar spectral index ns (Q > 1) obtained from (4.3.9). The light blue region shows the range for ns with 95% confidence level and the yellow region shows the range for ns with 68% confidence level as given by Planck.. It can be seen in figure 5.35 that the tensor to scalar ratio vs. scalar spectral index data points started well inside the 68% confidence region and shifts towards 1 as the dissipation increases. This is also in contrast to the result in figure 5.27, while in figure 5.36 the tensor to scalar ratio vs. scalar spectral index data points are outside of the confidence regions given by Planck.. 65.

(106) Chapter 6 Conclusion We have shown that by showing explicitly the evolution of the temperature throughout inflation we could ensure that at the horizon exit the observables are within the context of warm inflation, this is in contrast to the slow-roll approach where there is no explicit process in order to determine if the warm inflation conditions are satisfied. For the numerical calculation, the result for the tensor to scalar ratio in chaotic inflation is now within the bound given by the Planck data in contrast to if it is considered in the cold inflation case, the scalar spectral index also seems to be fairly independent of the dissipation rate. The higgs-like inflation and axion inflation are both an example of small-field models such that its tensor to scalar ratio are well inside the bound and are not surprising since they do not have the same problem in the cold inflation case as in chaotic inflation. The scalar spectral index results indicate that there seems to be more consistency for weak dissipation since for the axion inflation, the scalar spectral index is out of the region for the strong dissipation case, while for the higgs inflation either case are within the region. In general, the slow-roll approach and the numerical approach gives a different result, however, for the Chaotic inflation it just so happened that the scalar spectral index derived from slow-roll is approximately independent of the dissipation which in contrast to the Higgs and Axion case where it is dependent on the dissipation. 66.

(107) Appendix A Scalar Field Theory For a single scalar field, the Lagrangian density L is given by, 1 L = − ∂ µ φ∂µ φ − V (φ). 2. (A.0.1). where φ is the scalar field, the first term in the Lagrangian density is the kinetic term and V (φ) is the scalar field potential. In curved spacetime, the action takes the form, Z S=. √ d4 x −gL,. (A.0.2). where g is the determinant of the Friedmann-Robertson-Walker metric gµν . By varying the action, we get φ¨ + 3H φ˙ − a−2 ∇2 φ + V,φ = 0.. (A.0.3). For a homogeneous scalar field, we have φ¨ + 3H φ˙ + V,φ = 0. This equation is called the Klein-Gordon equation.. 67. (A.0.4).

(108) The Klein-Gordon equation is similar to the harmonic oscillator equation,. x¨ + β x˙ + kx = 0. (A.0.5). such that the friction term in the Klein Gordon equation in accordance to the friction β in the harmonic oscillator equation is 3H. Assuming that the scalar field φ is large initially and suppose we take a potential of the form V = 21 m2 φ2 , then from the Friedmann equation, V , 3MP2. (A.0.6). mφ H≈√ . 6MP. (A.0.7). H2 ≈ we have,. This means that if the scalar field φ is large then H is also large which implies that the friction is large so that the scalar field φ moves very slowly down the potential in such a way that it approximately does not move from its original position, therefore H is approximately constant.. H=. a˙ ≈ constant a. (A.0.8). a ≈ eHt. (A.0.9). 68.

(109) Appendix B Perturbations during Inflation In this appendix we follow closely the treatment by Baumann [5]. We consider the metric tensor of the perturbed FRW universe gµν ,. gµν = g¯µν + δgµν ,. (B.0.1). where g¯µν is the background metric and δgµν is the assumed small perturbations. The line element for the perturbed FRW universe in conformal time is given by,. ds2 = a2 (τ )[−(1 + 2A)dτ 2 − 2Bi dηdxi + ((1 − 2D)δij + 2Eij )dxi dxj ]. (B.0.2). B.1. Scalar, vector, tensor decomposition. The homogeneous, isotropic, and spatially flat background possesses symmetry such that these symmetries allow a decomposition of the metric into scalar, vector, and tensor components. In other words, the metric tensor is symmetric so that instead of having sixteen degrees of freedom, it has only ten degrees of freedom. A and D transform as scalars, Bi transforms as a 3-vector, and Eij as a 3-tensor. 69.

(110) From vector calculus, a vector field can be decomposed into two parts such that, ~ = B~S + B~V , B. (B.1.1). with ∇ × B~S = 0 (ijk ∂i Bj = 0) and ∇ · B~V = 0 (δ ij ∂j BiV = 0); so that we can write B~S = −∇B for some scalar B. In component notation, we have. Bi = ∂i B + BiV .. (B.1.2). We see that the first term of (35) is a scalar and the second term is a vector. BiV has one constraint so it has two independent components. Also, any rank-2 symmetric tensor can be decomposed such that,. Eij = EijS + EijV + EijT. (B.1.3). where EijV satisfies the constraint as in the second term of (34). EijT satisfies the four constraints EiiT = 0 (traceless), δ ij EijT = 0 (transverse/divergenceless) so that it has only two independent components. Also, EijS and EijV can be written as, 1 EijS = (∂i ∂j − δij ∇2 )E 3. (B.1.4). 1 EijV = (∂j Ei + ∂i Ej ) 2. (B.1.5). The ten degrees of freedom of the metric tensor have now been decomposed into a 4+4+2 degrees of freedom where the scalar part consists of A, B, D, and E; the vector part consists of BiV and Ei ; the tensor part consists of EijT . At first order perturbation, it can be seen that the scalar, vector, and tensor part do not couple to each other and evolve independently so that we can study them separately. We are interested in studying the scalar and tensor perturbations which are responsible for the density fluctuations and 70.

(111) gravitational waves. Vector perturbations are not important in our context since it couples to the rotational velocity perturbations in the cosmic fluid which tend to decay as the universe expands.. B.2. Quantum to Classical Scale. During early times all the modes were well inside the horizon such that a collection of harmonic oscillators describe the quantum fluctuations in the inflaton field on small scales. Inflation stretches these quantum fluctuations to scales beyond the horizon. It is convenient to switch from the inflaton fluctuations δφ to fluctuations in the conserved curvature perturbations R at the horizon crossing k = aH such that the relationship between R and δφ in spatially flat gauge is, H δφ φ˙. R=−. (B.2.1). where δφ is the inflaton fluctuations in spatially flat gauge and the variance of the curvature perturbations is,. h|Rk |2 i =. B.3. H 2 φ˙. h|δφk |2 i. (B.2.2). Statistics of Cosmological Perturbations. The power spectrum PR (k) of the primordial scalar fluctuations is given by, hRk Rk0 i = (2π)3 δ(k + k0 )PR (k).. (B.3.1). We could also write the dimensionless power spectrum ∆2s as, ∆2s = ∆2R =. k3 PR (k). 2π 2. 71. (B.3.2).

(112) P (k) and ∆2 (k) both refer to the word power spectrum. ∆2 (k) has the more obvious physical meaning since it gives the contribution of a logarithmic interval of scales. ∆2 (k) is dimensionless while P (k) has dimensions Mpc3 . The scalar spectral index ns characterizes the scale-dependence of the power spectrum for scalar perturbations and is given by,. ns − 1 =. d ln ∆2s , d ln k. (B.3.3). where scale-invariance means that ns = 1. The power spectrum Ph (k) of the two polarization modes for the gravitational wave hij is given by, hhk hk0 i = (2π)3 δ(k + k0 )Ph (k).. (B.3.4). The dimensionless power spectrum ∆2h is, ∆2h. k3 = 2 Ph (k). 2π. (B.3.5). We also define the power spectrum of the tensor perturbations as the sum of the two polarization mode power spectra.. ∆2t = 2∆2h .. (B.3.6). nt characterizes the scale-dependence of the power spectrum for tensor perturbations and is given by,. nt =. d ln ∆2t . d ln k. (B.3.7). The power spectrum of the inflaton fluctuations is given by,. hδφk δφk0 i = (2π)3 δ(k + k0 ) so that the dimensionless power spectrum is, 72. 2. 2π k3. H 2π 2. !2 ,. (B.3.8).

(113) H 2π 2. ∆2δφ =. B.4. !2 .. (B.3.9). Scalar Perturbations. The power spectrum of the inflaton fluctuations δφ and the curvature perturbations R are related by, H φ˙. hRk Rk0 i =. !2 hδφk δφk0 i,. (B.4.1). from this relation we can conclude that the inflationary quantum fluctuations produce the power spectrum for R given by,. ∆2R (k) =. B.5. H2 H2 . (2π)2 φ˙ 2. (B.4.2). Tensor Perturbations. By quantizing the tensor perturbations, each polarization of the gravitational wave is just a renormalized massless field in de Sitter space so that,. hk =. 2 δφ. MP. (B.5.1). The power spectrum of the inflaton fluctuations δφ and a single polarization of tensor perturbations hij are related by, 4 ∆2h (k) = 2 MP. H 2π. !2 .. (B.5.2). The dimensionless power spectrum for tensor fluctuations is,. ∆2t (k) = 2∆2h (k) =. 73. 2 H2 . π 2 MP2. (B.5.3).

(114) Appendix C Notations and Conventions The metric convention for this work is (−, +, +, +). The numerical calculations were done using Mathematica and the equations are rescaled using the reduced Planck mass MP such that the dimensionless quantities are listed as follows,. t˜ = MP t, ρ ρ˜ = 4 , MP. φ φ˜ = , MP p p˜ = 4 MP. ˜ = H H MP. (C.0.1). Differentiation with respect to t˜ will be denoted by 0 . All the quantities for the numerical calculation are evaluated at the horizon crossing k0 = 0.05Mpc−1 where N = 60 e-folds before the end of inflation, an example would be calculating the tensor to scalar ratio r, " #−1 √ N α∗ 8φ˙ 2∗ Tb e 2 2 3πQ∗ √ r = 2 2 1+ . MP H∗ H∗ 3 + 4πQ∗. (C.0.2). ˙ the Hubble parameter H, and the ratio Q (in this the velocity of the inflaton φ, work Q is constant for the whole period of inflation) are evaluated at N = 60 e-folds before the end of inflation then the values are plugged into r.. 74.

(115) Bibliography [1] M. Livio and A. Riess. Measuring the Hubble constant. Phys. Today 66(10) 41, 2013 [2] A. Liddle. Introduction to Modern Cosmology. John Wiley, 2015 [3] B. Schutz. A First Course in General Relativity. Cambridge University Press, 2009 [4] S. Carroll. Spacetime and Geometry: An Introduction to General Relativity. Pearson, 2003 [5] D. Baumann. TASI Lectures on Inflation. arXiv:0907.5424, 2009 [6] A. Linde. Hybrid Inflation. Phys. Rev. D 49,748, 1994 [7] B. Bassett, S. Tsujikawa, and D. Wands. Inflation Dynamics and Reheating. Reviews of Modern Physics, vol 78, no. 2, pp. 537-589, 2006 [8] L.Z. Fang and A. Berera. Thermally Induced Density Perturbations in the Inflation Era. Phys. Rev. Lett. 74, 1912, 1995 [9] A. Berera. Warm Inflation. Physical Review Letters, 75, 18, p. 3218-3221, 1995 [10] A. Berera. Warm inflation in the adiabatic regime - A model, an existence proof for inflationary dynamics in quantum field theory. Nuclear Physics B, 585, 3, p. 666-714, 2000 75.