Back-reaction of a cosmological
background of gravitational waves
Thesis advisors: Dario GRASSO
Dipartimento di Fisica ‘Enrico Fermi’ Corso di Laurea Magistrale in Fisica
UNIVERSITÀ DI PISA
Dipartimento di Fisica ‘Enrico Fermi’
Back-reaction of a cosmological background of gravitational waves
by Marta MONELLI
In this master thesis project we investigate the back-reaction effects of a background of Gravitational Waves (GWs) produced during an early phase in the Universe evo-lution. Since GWs carry energy and momentum, they contribute to the total energy density at any time, hence affecting the space-time metric itself. Requiring that the early Universe is not dominated by back-reaction effects, i.e. requiring that the en-ergy of GWs is smaller than the ‘ordinary’ one, we obtain some constraints on the tensor spectral parameters, in particular on the spectral tilt nT and the
tensor-to-scalar ratio r. The bounds are extracted for each of the main phases of Universe evolution (slow-roll inflation, radiation domination epoch and matter domination epoch, neglecting the presentΛ-dominated phase). Moreover, we make a distinction between the energy density contributed by super- and sub-Hubble modes. In fact, in these two regimes GWs show different behaviours: super-Hubble modes lead to a negative contribution to the effective energy density, and to an equation of state of curvature, while the sub-Hubble ones contributes to the effective degrees of free-dom of radiation and are therefore constrained by Big Bang nucleosynthesis. This work follows the footprints of a recent paper from Brandenberger and Takahashi , with some improvements - mostly in the consideration of a more realistic model of inflation - which however translates in small differences on the final bounds.
Firstly, I would like to express my sincere gratitude to my supervisors, Professor Dario Grasso and Professor Giovanni Marozzi, for their guidance and constant mo-tivation over these last months. Both have always proved they willingness whenever I ran into a trouble spot or had a question about my work.
I would also like to thank Professor Fabio Finelli for taking some time and giving me the opportunity to discuss with him about my thesis. His insightful comments and suggestions have been precious.
With a special mention to Luca, for his constant support throughout these tough months. Must have been a real challenge.
I am deeply grateful to my parents and family, who have always provided me through moral support in my life. I would also like to thank my friends who have supported and stand me along the way. Thanks for all your encouragement and the precious beers!
NotationsIndices Greek indices µ, ν,· · · =0, 1, 2, 3
are 4d space-time indices, while Latin indices i, j,· · · = 1, 2, 3 are 3d spatial indices.
Coordinates Space-time coordinates on a 4d manifold will be usually denoted as x, y, . . . , while coordinates on a 3d manifold (for example spatial coor-dinates) will be denoted as x, y, . . . . Thus the components of a space-time coordinate x are xµ= (t, x).
Metric The metric tensor will be always denoted as gµν, while the Minkowski
metric will be denoted as ηµν. The
signature of the metric tensor will be considered ‘mostly plus’ (− + ++), in order to make the results more eas-ily linkable to the literature.
Geometrical quantities In the following we will make use of the definitions
listed below for the Christoffel sym-bols and the Ricci tensor:
Γµ νρ ≡ 1 2g µλ ∂νgρλ+∂ρgνλ−∂λgνρ , Rµν ≡∂λΓ λ µν−∂νΓ λ µλ +Γλ λρΓ ρ µν−Γ ρ µλΓ λ νρ.
Provided with the expression for Rµν,
we can use it and its contraction, the Ricci scalar R ≡ Rµµ, to define the
Einstein tensor Gµν =Rµν−
Units Throughout all the thesis natural units, where both the speed of light c and the reduced Planck constant} ≡ h/2π are set equal to one: c= } =1. Sometimes, however}and c will ap-pear, essentially where it is important to stress the physical dimensions of a quantity.
ContentsAbstract iii Acknowledgements vii Notations ix Contents xi
Introduction and motivations 1
1 Painting the broad picture 5
1.1 Basics of FRW cosmology . . . 5
1.1.1 The FRW metric . . . 5
1.1.2 Kinematics and geometry in the FRW Universe . . . 8
1.1.3 Flatness assumption . . . 10
1.1.4 Dynamics of the expanding Universe . . . 10
1.1.5 Some remarks . . . 13
1.2 Conformal time and horizons . . . 15
1.3 Shortcomings of standard cosmology . . . 18
1.4 The inflationary paradigm . . . 20
1.4.1 Slow-roll inflation . . . 21
2 Basics of GWs theory 25 2.1 Linearized gravity. . . 25
2.1.1 The transverse-traceless gauge . . . 27
2.1.2 Linearized theory in matter . . . 29
2.2 GWs in a curved background . . . 31
2.2.1 Separation of scales/frequencies . . . 31
2.3 GWs in a FRW background . . . 34
2.3.1 Effective energy-momentum tensor of GWs. . . 34
2.3.2 Propagation of GWs . . . 36
2.3.3 Decomposition in Fourier space . . . 37
2.3.4 Propagation of GWs in the expanding Universe . . . 38
3 Irreducible GWs background 41 3.1 Irreducible GWs background from vacuum fluctuations. . . 41
3.1.1 Quantization of tensor perturbations. . . 41
3.1.2 Solution in slow-roll inflation . . . 43
3.1.3 Power spectrum . . . 46
3.2 GWs background during standard expansion . . . 47
3.2.1 A first matching attempt . . . 49
3.2.2 A second matching attempt . . . 50
3.3 Energy density and pressure of a single mode. . . 52
3.3.2 According to the second matching scheme . . . 54 4 Back-reaction effects 57 4.1 Super-Hubble modes . . . 57 4.1.1 Slow-roll . . . 58 4.1.2 Radiation domination . . . 60 4.1.3 Matter domination . . . 62 4.2 Sub-Hubble modes . . . 63 4.2.1 Radiation domination . . . 64 4.2.2 Matter domination . . . 64 Conclusions 67 A Mathematica code 69 A.1 Perturbed Einstein tensor . . . 69
A.2 Vacuum fluctuations in slow-roll inflation . . . 71
A.3 GWs background during standard expansion . . . 72
A.3.1 A first matching attempt . . . 73
A.3.2 A second matching attempt . . . 74
A.4 Energy density. . . 75
A.4.1 According to the first matching attempt . . . 77
A.4.2 A second matching attempt . . . 78
Introduction and motivations
In the framework of early Universe cosmology there are a number of mechanisms that can generate stochastic backgrounds of GWs . Because of the weakness of the gravitational interaction, as soon as these GWs are produced they decouple from matter and radiation and propagate freely until today, hence directly carrying infor-mation from the early epoch when they have been generated. Eventual detections of such GWs backgrounds, either direct - with ground-based or orbital interferometers - or indirect by observing their effect on the Cosmic Microwave Background (CMB) fluctuations, therefore, could represent powerful tests of fundamental physics and cosmology.
Given their stochastic nature, each background can be conveniently described by its power spectrumPh(k), which is usually parametrized in terms of its amplitude at a pivot scale AT(k∗) and its spectral tilt nT. For example, the irreducible GWs
background arising from quantum vacuum fluctuation during slow-roll inflation, which is the one we main focused on in this master thesis project, is characterized by a spectral tilt nT = −2e, where e is the usual slow-roll parameter.
Since GWs carry energy and momentum, they affect the background space-time they propagate in, hence inducing a gravitational back-reaction effect [3,4]. To get a grasp of how the back-reaction effects emerge, we can think about what happens if we take a background cosmological solution to the Einstein equations (metric and mat-ter) and add to it small fluctuations of relative amplitude1ewhich satisfy the linear perturbation equations. Since the Einstein equations are non-linear, the perturba-tion equaperturba-tions are not satisfied to quadratic order, and second order fluctuaperturba-tions in matter and metric build up. In particular, corrections to the background can be in-duced. Back-reaction effects have been studied extensively in literature and have been the subject of controversial analysis as well [5, 6,7,8, 9,10,11, 12, and refer-ences therein].
In a recent paper from Brandenberger and Takahashi , they have shown that, re-quiring that the effective energy density of GWs does not constitute a large part of the total energy density of the Universe, it is possible to extract some bounds on the tensor spectral parameters, in particular on the spectral tilt nT and the
tensor-to-scalar ratio r. Because of the present lack of direct experimental evidences of primordial GWs, this method could represent a smart solution to constrain the pa-rameter space. In this master thesis, we performed an analysis in a similar fashion of the one in , with a few improvements that we will outline in a minute.
Our discussion starts from two fundamental ingredients: the equation of motion (e.o.m.) of GWs in a FRW Universe and their effective energy-momentum tensor, both of which we discuss in Chapter2. In fact, by solving the e.o.m. in the vari-ous epochs of the Universe evolution and appropriately matching the solutions, we
are provided with the amplitude of GWs at any time in cosmic history, expressed in terms of their primordial amplitude hinitk . Once the amplitude of GWs is given, it is possible to derive their energy-momentum tensor. In fact, the spatially aver-aged second order Einstein tensor hG(µν2)i, which can be expressed in terms of the
GWs amplitude, enters in the Einstein equations and can hence be interpreted as a energy-momentum tensor (see, for istance ). We have first perfomed a sym-bolic calculation of the second order Einstein tensor Gµν(2) by means of the software
Mathematica (making use, in particular, of the package xPand ), and then, following the approach introduced in  and exploiting the decomposition of GWs in Fourier space, we extracted the general expressions for the effective energy den-sity and pressure contributed by a single GWs mode in terms of the amplitude hk, denoted ˜ρGWand ˜pGW respectively. Inserting then the solutions of the e.o.m. in the
expressions for ˜ρGW and ˜pGW, we are left with the contribution from a single GW
mode to the energy density and pressure.
In particular, we evaluated ˜ρGW and ˜pGW for super- and sub-Hubble modes
sepa-rately, in all the slow-roll (SR), radiation domination (RD) and matter domination (MD) phases. This choice highlights the deep differences between this two regimes. In fact,
· long-wavelength modes, which are almost constant in time, are characterized by a negative energy density and the equation of state of curvature p= −ρ/3; · short-wavelength modes, which instead oscillate (and decay) rapidly, have a
pos-itive energy density and contribute to the effective number of degrees of freedom of radiation, since their equation of state appears to be the usual p=ρ/3.
These different characteristics can be exploited to impose appropriate constraints on the energy density of GWs. In fact, since short-wavelength modes contributes to the effective number of radiative degrees of freedom, their total energy density, i.e. the quantity ˜ρGWintegrated on sub-Hubble k, can not violate the constraint on effective
radiation degrees of freedom imposed by Big Bang Nucleosynthesis . In the case of super-Hubble modes, instead, we can only assume the total energy density, i.e. the quantity ˜ρGWintegrated on super-Hubble k, to be smaller than the total energy
These bounds on the energy density of GWs translate in constraints on the param-eter space of the theory. In particular, while discussing the back-reaction during slow-roll inflation, we were able to exclude some parameter region in the(nT, Ntot)
plane, where Ntotrepresents the total number of e-folds the scale factor grew during
inflation. The bounds on the energy density during RD and MD epochs, instead, provide bounds in the(r, nT)plane.
Lastly, we explored the possibility of observing a GWs background characterized by r and nT in the constrained region through future space-based GWs detectors.
As already mentioned, the scheme outlined above takes its inspiration from . However, there are a few improvements in our work which should be mentioned
· we assumed a slow-roll inflationary scenario, instead of a less realistic de-Sitter model;
Contents 3 · we compared two different matching schemes, both different from the one
em-ployed in , and their implications on the energy density estimations.
The differences we have just outlined do not translate, however, in big discrepancies on the final bounds.
Painting the broad picture
One of the main characters of this thesis work is the background of GWs produced during inflation. Given its importance, it is fundamental to characterize the infla-tionary epoch, get a grasp of GWs theory, and finally understand why and how such primordial background emerges. The first of these topics will be matter of the present chapter, while we will discuss the second and the third ones in Chapters2 and3, respectively.
Before we get to the heart of this Chapter and study the main features of inflation, it is useful to review some aspects of Friedmann-Robertson-Walker (FRW) cosmology.
Basics of FRW cosmology
It is worth to point out that this section does not aim to provide an exhaustive dis-cussion about the FRW model, which is a fundamental topic covered in many Cos-mology textbooks (see for istance [17,18,19,20,21]), but we only want to overview some basics notions we will make use of in the rest of this thesis work. This said, let us start introducing the FRW metric.
1.1.1 The FRW metric
The FRW metric describes a homogeneous, isotropic, expanding (or otherwise con-tracting) Universe. Before writing down the expression of the metric, let us focus for a moment on the three features we have just listed and try to explain and justify them:
Isotropy: Essentially, assuming the Universe to be isotropic means that there exists no preferred direction, implying that we should get identical observational evidences regardless of the direction we are looking at. The strongest evidence support-ing this hypothesis is given by the temperature distribution of the CMB, a relic electromagnetic radiation which has be travelling to us from an early stage of the Universe history, the photon decoupling. Extremely precise measurements of CMB’s temperature show that it is characterized by a black body spectrum with a temperature of 2.72548±0.00057 K , whose fluctuations are about one part in 10−5, hence appearing extremely uniform in any direction.
Homogeneity: Homogeneity basically means that there is no preferred location in the Universe and hence observers located at different positions should have access the same observational evidences. Since we are not in a position to observe the Universe from locations other than our own, we can not directly prove homo-geneity. However, the measured abundances of light elements from Big Bang
Nucleosynthesis (BBN) provide an indirect evidence of homogeneity. In fact the FRW model, which does assume homogeneity, leads to theoretical predictions for the light elements abundances which are consistent with the present measured values.
Time-dependent scale factor: To explain and justify this last hypothesis, let us start from the Hubble’s law, a linear relation between the red-shift z of nearby galaxies and their distance d from us observed in the early 1930s by Edwin Hubble:
z 'H0d , (1.1)
where H0is said Hubble’s constant and is usually written in terms of the
adimen-sional quantity h0,
H0= h0100 km s−1Mpc−1, (1.2)
whose measured value is about 0.7 . If we try to interpret z as a Doppler shift1, we have that v'z, from which we deduce that the nearby galaxies are receding from us with velocity proportional to distance. Because we just assumed the Universe to be homogeneous, all galaxies will see other galaxies moving away from them (unless the other galaxies are part of the same gravitationally bound group or cluster of galaxies, of course). This lead us to conclude that the whole Universe is ‘intrinsically’ expanding, meaning that the scale of space itself changes with time.
The above three assumptions constrain the metric to take the peculiar form
ds2= −dt2+a2(t)dΣ2, (1.3) where t is called cosmic time, a(t) denotes the scale factor and Σ ranges over a 3-dimensional space of uniform curvature k and does not depend on time. The con-stant k can take different values, corresponding to different geometrical features of the space-time: k = +1 closed Universe, −1 open Universe, 0 flat Universe. (1.4)
Quasi-Cartesian coordinates In quasi-Cartesian coordinates, dΣ2takes the form
where the coordinates x are the comoving coordinates. Hence, in this case, the metric (1.3) becomes ds2= −dt2+a2(t) dx2+k(x·dx) 2 1−kx2 . (1.5)
Spherical coordinates Alternatively, one can choose to work in spherical coordi-nates (r, θ, φ) instead of the quasi-Cartesian ones. In this case, the FRW metric takes
1Warning! The analogy between cosmological redshift and Doppler shift is quite misleading, since
these two effects have different origins and meanings. However, this correspondence is not plain wrong in the limit of small distances, where the differences between the two shifts are not so important.
1.1. Basics of FRW cosmology 7 the form ds2 = −dt2+a2(t) dr2 1−kr2 +r 2dΩ2 , (1.6) where dΩ2 ≡ dθ2+sin2
θdϕ2. This form results particularly convenient while studying the propagation of light signals. In fact, photons travel along null geodesics characterized by ds=dθ =dϕ=0 , along which the metric reduces to
dt2= a2(t) dr
1−kr2 . (1.7)
Conformal form Equation (1.6) can be further simplified introducing a rescaled radial coordinate χ such that dχ≡dr/√1−kr2. In fact, this substitution leads us to
write the FRW metric in the form
k(χ)dΩ2 , (1.8)
where we have implicitly defined the quantity Sk(χ)as
Sk(χ) ≡ sinh χ k= −1 , χ k=0 , sin χ k= +1 . (1.9)
This expression for the metric becomes even more compact if we use, instead of the cosmic time t, the conformal time η, defined by
which can be integrated obtaining η=
Z t dt0
where the lower limit of integration can in principle be chosen arbitrarily. In terms of η, then, equation (1.8) takes an extremely pleasant form
ds2= a2(η)−dη2+dχ2+S2k(χ)dΩ2 . (1.12) This is usually referred as the conformal form of the FRW metric, since it is basically a static Minkowkski-like metric multiplied by a time-dependent scale factor a2(η). Besides making the FRW metric extremely compact, the use of χ and η results partic-ularly convenient in the study of light signals. In fact, in terms of these two variables, equation (1.7) takes now the form
∆χ= ±∆η , (1.13)
where the plus sign corresponds to outgoing photons while the minus sign to in-coming photons. The above equation then teaches us that the increment in con-formal time is equal to the increment in comoving distance along a photon’s path. Depending on the situation, we will work either with cosmic or conformal time.
1.1.2 Kinematics and geometry in the FRW Universe
In the FRW metric is encoded much information about the kinematical and geomet-rical properties of our Universe. A first fundamental feature is the existence of the so-called cosmological redshift.
Cosmological redshift To see how such effect emerges it is useful to consider a light signal emitted from a galaxy at comoving distance r1from us, who are located
at the origin, and recall that it propagates according to equation (1.7), so that dt
a(t) = − dr √
Assuming now that the light emitted by the galaxy has wavelength λe, we can
de-note with tethe time at which a wave crest is emitted by the galaxy and with to the
time at which we observe the very same wave crest. The above equation can then be integrated between teand to
Z to te dt a(t) = Z r1 0 dr √ 1−kr2,
and the same can be done for the subsequent wave crest
Z to+λo te+λe dt a(t) = Z r1 0 dr √ 1−kr2,
where λo can differ from λe. But we can combine these last two equations and find
λe λo = νo νe = a(te) a(to) ,
which translates in a relation between the scale radius and the usual redshift param-eter z
1+z= a(to) a(te)
It is interesting to check whether our result depends or not on the choice to write the light propagation equation (1.7) starting from the metric in spherical coordinates (1.6). There are any difference if we use the conformal form of the FRW metric (1.8) to derive the cosmological redshift? To answer this question, the simplest thing one can do is to retrace the last few equations substituting the cosmic time with its conformal counterpart. In particular, we first rewrite (1.7) in terms of η
dη= √ dr 1−kr2 ,
and integrate it for two subsequent wave crests
Z ηo ηe dη= Z r1 0 dr √ 1−kr2 , Z ηo+λ(c)o ηe+λ(c)e dη = Z r1 0 dr √ 1−kr2 ,
where the quantity λ(c)denotes the ‘conformal wavelength’. Again, the right-hand-side integrals are equals, since we are using comoving coordinates and hence r1 is
1.1. Basics of FRW cosmology 9 constant. We can then write
Z ηo ηe dη= Z ηo+λ(c)o ηe+λ(c)e dη ,
and hence λ(c)o =λ(c)e . It seems that the cosmological redshift emerges no more.
How-ever, we are talking about conformal wavelengths, which are not the physical ones we can measure. Assuming the signal to oscillate much more rapidly than the char-acteristic time-scale of the Universe evolution, we can interpret λ(c)e and λ(c)o as two
small conformal time intervals and express them as λ(c)e = λe/a(te), λ(c)o =λo/a(to),
which immediately bring us to the expression we wrote in equation (1.14).
However, there are less cumbersome approaches one could choose to see how the cosmological redshift emerges (see, for example, ) and all of them returns the result of equation (1.14).
Hubble parameter The Hubble’s constant we have defined above as the propor-tionality constant characterizing the relation between redshifts and distances we measure today can be generalized to an arbitrary time introducing another quantity: the Hubble parameter H(t). It is defined by
H(t) ≡ ˙a(t)
where the dot denotes the derivative with respect to the cosmic time. In fact, can be easily shown (see, for example, ) that the quantity H(t0)defined as ˙a(t0)/a(t0)
indeed fulfils the Hubble law z'H(t0)d.
Geometrical quantities Once we are provided with the metric of a give space-time, we are able to evaluate its Christoffel symbols, Ricci tensor and scalar, and Einstein tensor. In our case of a FRW metric
ds2= −dt2+a2(t)γijdxidxj, (1.16)
the only non vanishing Christoffel symbols are Γ0 ij = a˙aγij, Γi0j= ˙a aδ i j, Γijk = 1 2γ il( ∂jγkl+∂kγjl−∂lγjk). (1.17)
As regards the Ricci tensor components, instead, we have R00= −3 ¨a a, Rij = − " ¨a a +2 ˙a a 2 +2k a2 # gij, (1.18)
which can be combined in order to finally obtain the non-vanishing components of the Einstein tensor Gµν ≡gµλGλν:
G00 =3 " ˙a a 2 + k a2 # , Gij = " 2¨a a + ˙a a 2 + k a2 # δik. (1.19) The above expressions simplify a lot once we assume the space-time to be flat.
1.1.3 Flatness assumption
The standardΛCDM model assumes the Universe to be spatially flat, hypothesis which is supported by different observations . From here on, few exceptions excluded, we will then limit our discussion to this particular case rewriting the FRW metric for k=0. For example, equation (1.5) now reads
ds2= −dt2+a2(t)dx2. (1.20) Of course, the same can be done for the expressions in (1.6) and (1.12).
Geometrical quantities Here we list the flat counterparts of all the geometrical ob-jects we have just discussed. In particular, we are interested in the explicit form of the Einstein tensor which will be of fundamental importance in the study of the dy-namics in FRW Universe. The non-vanishing components of the Christoffel symbols in a flat FRW space-time are
ij = a2Hδij, Γi0j= Hδij. (1.21)
The non-vanishing components of the Ricci tensor is then given by
R00 = −3(H˙ +H2), Rij =a2(H˙ +3H2)δij, (1.22)
from which we can deduce the Ricci scalar
R=6(H˙ +2H2). (1.23) Finally, the Einstein tensor takes the form
G00 = −3H2, Gij = −(2 ˙H+3H2)δij, (1.24) where we have omitted all the vanishing terms. Once we know the components of Gµν, we are ready to make a step forward and discuss the dynamics of our FRW
1.1.4 Dynamics of the expanding Universe
The dynamics of the scale factor a(t)is governed by the Einstein equations Gµν =8πG T
where the energy-momentum tensor, at the level of background evolution, is taken to be that of a perfect fluid:
Tµν = (ρ+p)u µu
ν , (1.26)
where ρ represents the energy-density and p the pressure. This choice is motivated by the peculiar features of the FRW Universe, i.e. its homogeneity and isotropy . Let us now focus on the case of a single fluid and try to extrapolate more significant information from the equations above.
1.1. Basics of FRW cosmology 11
Conservation equation First, because of the Bianchi identity∇µGµν, the
energy-momentum tensor is conserved, meaning that ∇µT µ ν =δ ρ µ∇ρT µ ν =δ ρ µ ∂ρT µ ν +Γ µ ρσT σ ν −Γ σ ρνT µ σ =0 .
Recalling the expressions of the Christoffel symbols (1.21) and of the energy-momentum tensor (1.26), we can write the ν=0 component of the conservation equation as
˙ρ+3H(ρ+p) =0 . (1.27)
Friedmann equations Now, let us consider the µ = ν = 0 component of the Ein-stein equations (1.25) and write explicitly both the Einstein and the energy-momentum tensor, G0
0 = −3H2and T00 = −ρ, respectively. We then find
3 ρ, (1.28)
commonly referred to as the first Friedmann equation. Similarly, we find that the (i, 0)and(0, i)components are two identities 0=0, while the(i, j)component reads
2 ˙H+3H2= −8πGp .
Anyway, it can be easily shown that the last equation follows from (1.27) and (1.28). This means that the two latter can be taken as the only independent equations gov-erning the Universe dynamics.
Equation of state The last equation we need to close the system is the equation of state of the cosmological fluid, which is supposed to take the form
p(t) =w(t)ρ(t), (1.29) where we assume w(t)to be constant during a given cosmological epoch:
. if the Universe is filled by relativistic particles, w=1/3;
. for non-relativistic matter we can set, in first approximation, w=0, meaning the pressure of such fluid is approximatively zero;
. finally, for a cosmological constant, we have w = −1, as can be shown directly from Einstein equation adding a termΛδij.
Solutions of Friedmann equations We are finally ready to solve the Friedmann equations in (1.28) and then find the time evolution of the scale factor a(t). First, we assume p(t) =w(t)ρ(t), so that the conservation equation (1.27) becomes
˙ρ+3(1+w)Hρ=0 . (1.30) The above equation leads us to the formula
where ρ0is the present value of the energy density. We can then specify the
expres-sion of ρ(a)in the various cases above ρr(a) =
a4 , ρm(a) =
a3 , ρΛ =const. , (1.32)
where the subscripts r, m andΛ obviously indicate radiation, matter and cosmolog-ical constant, while the subscript 0 denotes the present values. Substituting these expressions in (1.28), we get a differential equation which can be integrated finding
a(t)∝ t1/2 (RD) , t2/3 (MD) , eHt (ΛD) . (1.33)
Once we have the time-dependence of a(t), it is immediate to find
H(t) = 1/(2t) (RD) , 2/(3t) (MD) , √ Λ/3 (ΛD) . (1.34)
These are essentially the equations we need to move ahead in our discussion. Any-way, it is interesting to relax the hypotheses of single fluid and spatial flatness for a moment and recall some fundamental results which apply to that more generic framework.
Relaxing some assumptions We now consider the case of a FRW Universe with curvature k which is fulfilled with radiation, matter and a cosmological constant. Assuming an arbitrary curvature k implies that we can no longer work with equation (1.20), but we need the general form of the FRW metric, equation (1.5). This means that the Christoffel symbols, and hence the Ricci and Einstein tensors, change. In particular, the Einstein equations become 
−2k a2 − 2 ˙a2 a2 − ¨a a = −4πG(ρ−p), 3 ¨a a = −4πG(3p+ρ).
The above equations are equivalent to the system composed by the conservation equation (1.27) and the first Friedmann equation, which now reads
H2+ k a2 =
3 ρ. (1.35)
Assuming now that in the Universe are present radiation, matter and a cosmological constant with energy density ρr, ρm and ρΛ, respectively, we can rewrite the total
1.1. Basics of FRW cosmology 13 energy density ρ at a given time t as2
ρ(t) =ρr(t) +ρm(t) +ρΛ(t) =ρr,0a−4(t) +ρm,0a−3(t) +ρΛ,0 = 3H 2 0 8πGΩra −4(t) +Ω ma−3(t) +ΩΛ , (1.36)
where we have introduced the critical energy density ρc≡
corresponding to the value of ρ such that k=0, and the cosmological density parame-tersΩr,Ωm andΩΛ defined as
Extending this notation also to the term k/a2 in equation (1.35), i.e. definingΩk ≡
0, the Friedmann equation can be rewritten as
H(a) = H0ΩΛ+Ωka−2+Ωma−3+Ωra−4
Hence, by measuring the values of the Hubble constant H0 and of the density
pa-rametersΩΛ,Ωk,ΩmandΩrat the present time, we are able to evaluate the Hubble
parameter at any scale factor. Substituting the temporal dependence of a(t), then, one could (in principle) determine completely H(t). Notice that from (1.35) follows that
ΩΛ+Ωk+Ωm+Ωr =1 . (1.40)
1.1.5 Some remarks
The main results we need to go on with our work are essentially the ones we ob-tained above. However, there are some less fundamental notions which are worth to mention and briefly discuss.
Big Bang nucleosynthesis
In the first stages of the Universe evolution, the temperature was so high and there was so much radiation that there were no neutral atoms or even bound nuclei. In fact, at that time, any atom or nucleus produced would be immediately destroyed by a high energy photon. As the Universe cooled well below the binding energies of typical nuclei (T ∼ 0.1 MeV), light elements began to form. This production of nuclei is referred to as Big Bang nucleosynthesis. As we have already mentioned at the beginning on this chapter, the capability of BBN theory to provide predictions on light elements abundances (essentially, deuterium,3He,4He and7Li) which are in good agreement with the measured values represents one pillar of Big Bang theory. However, the measurements do even more. In fact, in the theoretical predictions enter also parameters of the underlying particle theory, which are then constrained in order not to spoil the agreement. In particular, the prediction is sensitive to the
2Notice that this equation is exact only in the case of non-interacting fluids. However, this
effective number of species at time of nucleosynthesis, g∗,BBN = g∗(T ∼ 0.1 MeV),
which we are going to define in a minute.
The equilibrium between radiation and matter is defined by the condition ρr(teq) =
ρm(teq), or equivalently ρm(teq) ρr(teq) = Ωm Ωr aeq =1 .
Measuring the present values of the density parametersΩr andΩm, therefore, we
can evaluate aeqand all the related quantities such as the redshift 1+zeq =Ωm/Ωr,
from equation (1.14), and the temperature Teq = (1+zeq)T0 at radiation-matter
equilibrium (in fact, it follows from entropy conservation that T(t)a(t) = const, see for example ). Using the most recent estimates ofΩm andΩrwe find zeq =
3365±44, so that Teq '8000 K'0.7 eV.
Photon decoupling and CMB
Photon decoupling occurred during the epoch known as the recombination. During this time, electrons combined with protons to form hydrogen atoms, resulting in a sudden drop in free electron density. Decoupling occurred abruptly when the rateΓ of Compton scattering of photonsΓ was approximately equal to the rate of expan-sion of the Universe H, or alternatively when the mean free path of the photons λ was approximately equal to the horizon size of the universe H−1. After this pho-tons were able to stream freely, producing the Cosmic Microwave Background as we know it, and the universe became transparent. In this case we have zdec ' 1100
and Tdec '3000 K'0.3 eV.
Effective number of relativistic species
For equation (1.39) to be correct we must assume that the species which are rela-tivistic today are the same species which were relarela-tivistic at earlier times and that for each of them the temperature always scaled as T ∝ a−1. However, this is an oversimplified description, in fact
. It is natural that, depending on the temperature, the number of relativistic species changes. For example, at T1 GeV protons and neutrons that are non relativis-tic today were replaced by relativisrelativis-tic deconfined quarks. We should then take into account the actual relativistic degrees of freedom at each temperature T. . Another more subtle issue emerges if we consider a species, let us denote it s1,
which becomes non relativistic at a given temperature T1. Then, it starts
annihi-lating and its entropy is transferred to the particles to which it is coupled. The temperature of the species coupled with s1, therefore is raised, compared with
the expected T∝ a−1.
These are the two fundamental remarks that allow us to define the effective number of relativistic species g∗as g∗(T) ≡
∑i=bosons gi Ti T 4 +7 8i=fermions
∑gi Ti T 4 , (1.41)
1.2. Conformal time and horizons 15 where gi represents the number of spin or helicity states of the i-th species, Ti is
its temperature and T denotes the photon temperature. The effective number of relativistic species enters in the evaluation of the total radiation energy density as
ρr(T) =g∗(T)π 2
A for the energy density of radiation, also entropy is dominated by the contribution of relativistic species. This analogy make us introduce another parameter, g∗s, which
characterizes the entropy density s of the Universe: g∗s(T) ≡
∑i=bosons gi Ti T 3 + 7 8i=fermions
∑gi Ti T 3 . (1.43)
In fact, in terms of g∗s, we have
s(T) = 2π
The parameter g∗,s, therefore, can be considered the ‘entropic counterpart’ of the
effective number of relativistic species g∗.
Conformal time and horizons
The conformal time η, which we have defined en passant in last section, is actually an important quantity which is worth to discuss in deeper detail. Of course, it could be used to express the evolution of the scale factor in place of the cosmic time. In fact, the Friedmann equation (1.28) translates in the following
H2 = 8πG 3 ρa
2 ∝ a−1−3w ⇒ a(3w−1)/2da∝ dη ,
where we introduced the conformal Hubble parameterH ≡ a0(η)/a(η) = aH and expressed the energy density as in equation (1.31). In particular, inserting the value of w typical of the various phases, we obtain
a(η)∝ η (RD) , η2 (MD) , η−1 (ΛD) . (1.45)
From the above expressions of a(η), it is immediate to find
H(η) = 1/η (RD) , 2/η (MD) , −1/η (ΛD) . (1.46)
This done, we can take a step forward and discuss the really interesting features of conformal time, focusing first on the advantages we gain studying the propagation of light in the expanding Universe in terms of η. As we already shown in equation (1.13), the path of a photon is defined by dχ(η) = ±dη, implying that
. Light rays correspond to straight lines at 45◦angles in the χ-η coordinates, rep-resenting a fair simplification in graphs and diagrams. In fact, if instead we had used physical time t, then the light cones for curved space-times would be curve. . Conformal time essentially represents the distance travelled by light in a certain amount of time, but this, in turn, determines the size of the casually connected patch at that time.
Hence, one can exploit this last correspondence and use conformal time to define two different types of cosmological horizons: the particle and the event horizon.
Particle horizon If the Universe has a finite age, then light travels only a finite dis-tance in that time and the volume of space from which we can receive information at a given moment of time is limited. The boundary of this volume is called the particle horizon, dph. To see how this quantity is related to the conformal time, let us consider
equation (1.13). We see that the maximal comoving distance that light can travel between two times η1 and η2 > η1 is simply∆η ≡ η2−η1. Hence, assuming that
the Big Bang started with the singularity at ti ≡ 0, we can express the (comoving)
particle horizon as dph(η) =η−ηi = Z t ti dt a(t). (1.47)
The size of the particle horizon at time η may be visualised by the intersection of the past light cone of an observer p with the spacelike surface η =ηi. Causal influences have to come from within this region. Only comoving particles whose worldlines intersect the past light cone of p can send a signal to an observer at p. The boundary of the region containing such worldlines is the particle horizon at p. Notice that every observer has his or her own particle horizon.
Event horizon Just as there are past events that we cannot see now, there may be future events that we will never be able to see (and distant regions that we will never be able to influence). The event horizon encloses the set of points from which signals sent at a given moment of time η will never be received by an observer in the future. These points are limited by
deh(η) =ηf −η=
Here, ηf denotes the ‘final moment of conformal time’. Notice that this may be finite
even if physical time is infinite, tf = +∞. Whether this is the case or not depends on
the form of a(t). In particular, ηf is finite for our Universe, if dark energy is really a
Hubble sphere Let us note now that equation (1.47) can be rewritten as dph(η) = Z t ti dt a(t) = Z a ai da a˙a = Z log a log ai (aH)−1d log a , (1.49) where ai ≡ 0 corresponds to the Big Bang singularity. The causal structure of the
spacetime can hence be related to the evolution of the comoving Hubble radius
1.2. Conformal time and horizons 17 where we recalled the definition of the conformal Hubble parameterH ≡ aH. For a Universe dominated by a barotropic fluid, we get
dH= H0−1a12(1+3w). (1.51)
Notice that, for 1+3w > 0, the comoving Hubble radius increases as the Universe expands. Moreover, one can show that the comoving Hubble radius and the particle horizon are connected by the relation
dph(η) = 2 1+3wH
In standard cosmology w is between 0 and 1/3, corresponding to dph(η) = 2H−1 and dph(η) = H−1, respectively. This has lead to the confusing practice of referring to both the particle horizon and the Hubble radius as the ‘horizon’ even though there is a deep conceptual difference between the two. In fact, while the particle horizon represents the maximum distance from which particles could have travelled to the observer in the age of the Universe, the Hubble radiusH−1is the (comoving)
distance over which particles can travel in the course of one expansion time. From the equation above we see that this difference is no big deal during the standard expansion but it is however convenient to keep it in mind. The time evolution of the particle and event horizons, as well as the Hubble sphere is shown in Figure1.1.
-60 -40 -20 0 20 40 60
Proper Distance, D, (Glyr)
0 5 10 15 20 25 Time, t , (Gyr) 0.2 0.4 0.6 0.8 1.0 1.2 1.5 2.0 Scalefactor, a particle horizon now Hubble sphere event horizon light cone 1 1 3 3 10 10 1000 1000 0 -60 -40 -20 0 20 40 60 Comoving Distance, R 0χ, (Glyr) 0 5 10 15 20 25 Time, t , (Gyr) 1000 10 3 1 0 1 3 10 1000 0.2 0.4 0.6 0.8 1.0 1.2 1.5 2.0 Scalefactor, a Hubble now particle horizon light cone event horizon sphere -60 -40 -20 0 20 40 60 Comoving Distance, R 0χ, (Glyr) 0 10 20 30 40 50 60 Conformal Time, τ , (Gyr) 1000 10 3 1 0 1 3 10 1000 0.2 0.4 0.6 0.8 1.0 2.0 3.0 infinity 0.01 0.1 0.001 Scalefactor, a Hubble sphere
now particle horizon
light cone event horizon
Figure 1: Spacetime diagrams showing the main features of the general relativistic description of the expansion of the universe for the (ΩM, ΩΛ) = (0.3, 0.7) model with H0= 70 km s−1M pc−1. Dotted
lines show the worldlines of comoving objects. We are the central vertical worldline. The current redshifts of the comoving galaxies shown appear labeled on each comoving worldline. The normalized scalefactor, a = R/R0, is drawn as an alternate vertical axis. All events that we currently observe
are on our past light cone (with apex at t = now). All comoving objects beyond the Hubble sphere (thin solid line) are receding faster than the speed of light. Top panel (proper distance): The speed of photons relative to us (the slope of the light cone) is not constant, but is rather vrec−c. Photons we receive that were emitted by objects beyond the Hubble sphere were initially receding from us (outward sloping lightcone at t<
∼5 Gyr). Only when they passed from the region of superluminal
recession vrec > c (gray crosshatching) to the region of subluminal recession (no shading) can the
photons approach us. More detail about early times and the horizons is visible in comoving coordinates (middle panel) and conformal coordinates (lower panel). Our past light cone in comoving coordinates appears to approach the horizontal (t = 0) axis asymptotically. However it is clear in the lower panel that the past light cone at t = 0 only reaches a finite distance: about 46 Glyr, the current distance to the particle horizon. Currently observable light that has been travelling towards us since the beginning of the universe, was emitted from comoving positions that are now 46 Glyr from us. The distance to the particle horizon as a function of time is represented by the dashed line. Our event horizon is our past light cone at the end of time, t = ∞ in this case. It asymptotically approaches χ = 0 as t → ∞. The vertical axis of the lower panel shows conformal time. An infinite proper time is transformed into a finite conformal time so this diagram is complete on the vertical axis. The aspect ratio of ∼ 3/1 in the top two panels represents the ratio between the radius of the observable universe and the age of the universe, 46 Glyr/13.5 Gyr. 3
FIGURE 1.1: Space-time diagram showing the main features of the general
rela-tivistic description of the expansion of the universe for the(Ωm,ΩΛ) = (0.3, 0.7)
model with H0=70 km s−1Mpc−1. The central vertical worldline is our one. All
events that we currently observe are on our past light cone. All comoving objects beyond the Hubble sphere (thin solid line) are receding faster than the speed of light. conformal coordinates past light cone at t=0 only reaches a finite distance: about 46 Glyr, the current distance to the particle horizon. Currently observable light that has been travelling towards us since the beginning of the Universe, was emitted from comoving positions that are now 46 Glyr from us. The distance to the particle horizon as a function of time is represented by the dashed line. Our event horizon is our past light cone at the end of time, t=∞ in this case. It asymptotically
Shortcomings of standard cosmology
The FRW model we have just introduced is strongly supported by many observa-tional evidences. For example, it gives predictions about the CMB temperature and the abundances of light elements which are confirmed by measurements. Anyway, the standard cosmological model it is not free of puzzles. Here we are going to list some of the main ones.
Flatness problem: A few pages ago, we stated that there are many observational evidences favouring the vanishing of the spatial curvature parameter Ωk, and
hence supporting the spatial flatness of the Universe. The strongest experimental constraint comes from the Planck collaboration  and limitsΩkbetween
Ωk = −0.005+−0.0160.017 (95% CL).
Therefore, the possibility thatΩk 6= 0 is not certainly excluded by observations,
but we can surely conclude that its non-vanishing value should be small, i.e. |Ωk| < 1. But, since the curvature parameter scales as 1/(aH)2 = 1/ ˙a2, a small
value of Ωk today corresponds to an even smaller value at earlier times. For
example, for the curvature parameter to be less than unity today,|k|/ ˙a2should have been at most 10−16 at electron-positron annihilation. Nothing forbids the curvature parameter to be arbitrarily small, but FRW Cosmology is not able to give any ‘a priori’ explanation of such peculiar initial condition.
Horizon problem: One might point out that the FRW metric describes a homoge-neous and isotropic space, while the most generic cosmological solutions to Ein-stein equations are not isotropic nor homogeneous. How, then, can we justify the realization of this particular space-time? This question can be reworded from a more practical point of view recalling that the main evidence of isotropy is given by the almost uniform temperature distribution of the CMB. How, then, can we explain the extreme smoothness in the CMB temperature? We now try to answer this question with the tools provided us by standard cosmology, in particular, making use of the conformal Hubble radius dH(η) = H−1(η)we de-fined in equation (1.50). From equation (1.46) it is clear that dH(η) ∼ η both during radiation and matted dominance so that, given that a(η)has increased as t2/3 since the time of last scattering, the horizon scale at last scattering was of order dH(tL) ∼ H0−1a3/2(tL). Inserting the present value of the Hubble
parame-ter and the scale factor at last scatparame-tering, and using the definition of the angular diameter distance3 dA, one finds that the horizon at the time of last scattering
now subtends an angle of order dH(tL)/dA,L ∼ 1.6◦. Therefore, causally
con-nected patches of the surface of last scattering subtend small angles meaning that no physical process could have smoothed out initial inhomogeneities on larger scales. A schematic representation of the horizon problem is shown in Figure1.2.
Unwanted relics: Within the context of unified gauge theories there are a variety of stable, superheavy particle species X that should have been produced in the early Universe via the spontaneous breaking of local symmetries to the symmetry group of the Standard Model SU(3) ×SU(2) ×U(1). Since any particle species with very a large mass typically has a very small annihilation cross section and easily survive for a long time, these superheavy particles should contribute too
3The angular diameter distance d
Ais a distance measure used in astronomy. It is defined in terms
1.3. Shortcomings of standard cosmology 19 generously to the present energy density, that isΩX 1. The Standard
Cosmol-ogy has no mechanism of ridding the Universe of relics that are overproduced early in its history.
Small scale inhomogeneity: Once the Universe becomes matter-dominated, small, primeval density inhomogeneities grow via Jeans instability into the rich array of structure present today4. Since the relic CMB photons did not take part in
the gravitational collapse that gave rise to the structures, they remain as a fossil record of the primeval inhomogeneity. Measuring CMB temperature, hence, we can deduce the relative amplitude of the primaeval perturbations, which results ∼ 10−4. However, one can estimate the mass of baryons contained within the horizon at a given time and find 
Mhor'0.29M(ΩBh2)g−∗1/2 m Pl T 3 (RD), Mhor'9.4×1021M(ΩB/Ω3/20 h)(1+z)−3/2 (MD).
From the above expression it is clear that perturbations on the scale of cosmolog-ical interest today were well outside the horizon at early times. This observation leads us to the conclusion that no casual, microphysical processes acting during the earliest moments of the Universe could give rise to primaeval density pertur-bations on the scales of interest.
32 2. Inflation
coming from p and q “know” that they should be at almost exactly the same temperature? The same question applies to any two points in the CMB that are separated by more than 1 degree in the sky. The homogeneity of the CMB spans scales that are much larger than the particle horizon at the time when the CMB was formed. In fact, in the standard cosmology the CMB is made of about 104disconnected patches of space. If there wasn’t enough time for these regions
to communicate, why do they look so similar? This is the horizon problem.
1100 10 3 1 0 1 1100 0.2 0.4 0.6 0.8 1.0 0.01 0.1 0.001 Hubble sphere now light cone
comoving distance [Glyr]
conformal time [Gyr]
-40 -20 0 20 40 50 40 30 20 10 0 3 10 CMB our worldline p q
Figure 2.2: The horizon problem in the conventional Big Bang model. All events that we currently observe are on our past light cone. The intersection of our past light cone with the spacelike slice labelled CMB corresponds to two opposite points in the observed CMB. Their past light cones don’t overlap before they hit the singularity, a = 0, so the points appear never to have been in causal contact. The same applies to any two points in the CMB that are separated by more than 1 degree on the sky.
A Shrinking Hubble Sphere
Our description of the horizon problem has highlighted the fundamental role played by the growing Hubble sphere of the standard Big Bang cosmology. A simple solution to the horizon problem therefore suggests itself: let us conjecture a phase of decreasing Hubble radius in the early universe,
1 < 0 . (2.2.10)
If this lasts long enough, the horizon problem can be avoided. Physically, the shrinking Hubble sphere requires a SEC-violating fluid, 1 + 3w < 0.
2.2.1 Solution of the Horizon Problem
For a shrinking Hubble sphere, the integral in (2.1.5) is dominated by the lower limit. The Big Bang singularity is now pushed to negative conformal time,
⌧i= 2H 1 0 (1 + 3w)a 1 2(1+3w) i ai!0 , w< 13 ! 1 . (2.2.11)
This implies that there was “much more conformal time between the singularity and decoupling than we had thought”! Fig. 2.3 shows the new spacetime diagram. The past light cones of
FIGURE1.2: The horizon problem in the conventional Big Bang model. All events that we currently observe are on our past light cone. The intersection of our past light cone with the spacelike slice labelled CMB corresponds to two opposite points in the observed CMB. Their past light cones don’t overlap before they hit the sin-gularity, a=0, so the points appear never to have been in causal contact. The same applies to any two points in the CMB that are separated by more than 1 degree on
the sky. (Figure and caption from .)
4The Jeans instability causes the collapse of interstellar gas clouds and subsequent star formation.
It occurs when the internal gas pressure is not strong enough to prevent gravitational collapse of a region filled with matter.
The inflationary paradigm
Various attempts have been made to solve the shortcomings we just listed, but infla-tion has definitely been the most successful. Basically, when we speak about inflainfla-tion we refer to an primordial phase, before the RD era, when the energy density of the Universe was dominated by a slowly varying vacuum energy density, and hence a(t)grew more or less exponentially. As we will see in a moment, assuming an ini-tial inflationary phase dynamically induces an isotropic, homogeneous, flat Universe and gets rid on any topological defects, hence solving three of the four issues above.
Flatness problem: Since the scale factor grows quasi-exponentially during inflation, ˙a/a is almost constant and|Ωk| = |k|/(aH)2decreases with time more or less like
a−2. So, if the Big Bang was preceded by a sufficiently long lasting inflationary phase, the spatial curvature parameter at the beginning of RD would be suffi-ciently small. In  is shown how one can set a lower bound on the number of e-foldings needed to ensure the density parameter to be small at the Big Bang. In particular, once one assumes that a(t)increases by a factor eN during inflation, he finds different lower bounds for N depending by the value he assumes for the energy density at the end of inflation. However, a typical reference value is N∼60.
Horizon problem: Inflation could solve this problem if, during that phase, the por-tion of the Universe we can observe today would have occupied a tiny space, and there would have been plenty of time for everything in this space to be homoge-nized (see Figure1.3for an schematic explanation). Again, in  is shown that the condition to solve the large scale smoothness problem is identical to the one needed to solve the spatial flatness-oldness one. Hence, besides solving the flat-ness problem, a number of e-folds N ∼60 gets also rid of the horizon problem.
Unwanted relics: In inflationary cosmological models the production of monopoles and other topological defects happens at a given time during inflation, while the Universe is expanding exponentially. The expansion that occurs before the relics production greatly extends the horizon, while the exponential expansion that occurs after the production of topological defects (but before photons are created in a period of reheating) greatly reduced the relic to photon ratio, hence lowering their abundance.
Small scales inhomogeneities In Subsection1.3we noticed that, since ‘perturbations on the scale of cosmological interest today were well outside the horizon at early times’, Standard Cosmology does not provide an explanation for the rise of primaeval fluctuations. However, during the period of inflation that is supposed to precede the RD era, H(t)was roughly constant, while a(t)increased quasi-exponentially, so even if a perturbation was outside the horizon at the end of inflation it would inevitably have been found deep inside the horizon sufficiently early in the era of inflation. Basically, what happens is that the exponential growth of the scale factor during inflation causes quantum fluctuations of the fields to be stretched to macroscopic scales and leaving the horizon. At the later stages of RD and MD, these fluctuations re-entered the horizon, and set the initial conditions for structure formation.
We will come back to the study of primaeval perturbations in Section 3.1, with a particular attention to the case of the primordial GWs background, but first let us discuss the slow-roll inflationary model.
1.4. The inflationary paradigm 21 33 2. Inflation 1100 10 3 1 0 1 1100 0.2 0.4 0.6 0.8 1.0 0.01 0.1 0.001 Hubble sphere now light cone scale factor
conformal time [Gyr]
50 40 30 20 10 3 10 CMB reheating -10 -20 -30 -40 inflation causal contact
Figure 2.3: Inflationary solution to the horizon problem. The comoving Hubble sphere shrinks during inflation and expands during the conventional Big Bang evolution (at least until dark energy takes over at a⇡ 0.5). Conformal time during inflation is negative. The spacelike singularity of the standard Big Bang is replaced by the reheating surface, i.e. rather than marking the beginning of time it now corresponds simply to the transition from inflation to the standard Big Bang evolution. All points in the CMB have overlapping past light cones and therefore originated from a causally connected region of space.
widely separated points in the CMB now had enough time to intersect before the time ⌧i. The
uniformity of the CMB is not a mystery anymore. In inflationary cosmology, ⌧ = 0 isn’t the initial singularity, but instead becomes only a transition point between inflation and the standard Big Bang evolution. There is time both before and after ⌧ = 0.
2.2.2 Hubble Radius vs. Particle Horizon
A quick word of warning about bad (but unfortunately standard) language in the inflationary literature: Both the particle horizon ph and the Hubble radius (aH) 1 are often referred to
simply as the “horizon”. In the standard FRW evolution (with ordinary matter) the two are roughly the same—cf. eq. (2.1.9)—so giving them the same name isn’t an issue. However, the whole point of inflation is to make the particle horizon much larger than the Hubble radius.
The Hubble radius (aH) 1 is the (comoving) distance over which particles can travel in the course of one expansion time.3 It is therefore another way of measuring whether particles are causally connected with each other: comparing the comoving separation of two particles with (aH) 1determines whether the particles can communicate with each other at a given moment (i.e. within the next Hubble time). This makes it clear that ph and (aH) 1are conceptually
3The expansion time, t
H⌘ H 1= dt/d ln a, is roughly the time in which the scale factor doubles.
FIGURE 1.3: Inflationary solution to the horizon problem. The comoving Hub-ble sphere shrinks during inflation and expands during the conventional Big Bang evolution (at least until dark energy takes over at a∼0.5). Conformal time during inflation is negative. The spacelike singularity of the standard Big Bang is replaced by the reheating surface, i.e. rather than marking the beginning of time it now cor-responds simply to the transition from inflation to the standard Big Bang evolution. All points in the CMB have overlapping past light cones and therefore originated
from a causally connected region of space. (Figure and caption from .)
1.4.1 Slow-roll inflation
Slow-roll (SR) inflation is one of the simplest inflationary models. Basically, it in-volves a single scalar field φ, usually referred to as the inflaton, characterized by the following action S= Z d4xp−g " m2Pl 2 R− 1 2∂ µ φ∂µφ−V(φ) # . (1.53)
Now, we suppose that at some time φ takes a value at which its potential V(φ) is large but quite flat. This is actually the key assumption ensuring the functioning of the slow-roll model. In fact, because of the flatness of the potential, the field φ ‘rolls’ very slowly down V(φ) (hence the name slow-roll inflation), so that its ki-netic energy is negligible with respect to the potential one and the Universe is then embedded with a vacuum energy density V(φ) ∼ const. But this is essentially the situation we described at the beginning of the present section while introducing the general features of inflation! It seems that we are on the right track, but one should still require that the field slow-rolls long enough to ensure a sufficient scale radius growth, which, in turns, implies the solution of the Standard Cosmology shortcom-ings. To do that, it is convenient to first rewrite the Friedmann equation (1.28) and the energy conservation equation (1.27) in terms of the inflaton energy density and pressure. For a spatially homogeneous scalar field φ(t)with potential V(φ)in a FRW
space-time we have  ρ= 1 2 ˙φ 2+V( φ), p= 1 2 ˙φ 2−V( φ), (1.54)
from which we get
H2 = 8πG 3 1 2 ˙φ 2+V( φ) , (1.55a) ¨ φ+3H ˙φ+V0(φ) =0 . (1.55b) The above couple of equations are essential to derive conditions which ensure the so-lution on the Standard Cosmology shortcomings and hence the success of the slow-roll model. First, by taking the time derivative of (1.55a) and the using (1.55b), one finds
2H ˙H= 8πG
3 (˙φ ¨φ+V 0(
φ)˙φ) = −8πGH ˙φ2, (1.56) where the prime now denotes the derivative with respect to φ. We then have
H= −4π ˙φ2. (1.57)
Slow-roll conditions This said, in order to have the scale factor to grow quasi-exponentially during inflation, we impose the first slow-roll condition
1 2 ˙φ
In fact, if the above condition is satisfied, the Universe is dominated by the vacuum energy density V(φ)and it experiences a quasi-exponential expansion. We should also require that this condition holds for a long enough period of time, so that we get rid of all the shortcomings. This can be done imposing that, in absolute value, the time derivative of the left-hand side of equation (1.58) should be much smaller than the time derivative of the right-hand side, i.e.
|φ¨| |V0(φ)|, (1.59) which is usually referred to as second slow-roll condition. When the two slow-roll conditions are satisfied, equations (1.55a) and (1.55b) read
3 V(φ), (1.60a)
3H ˙φ' −V0(φ). (1.60b) But then the second slow-roll condition can be written as
|φ¨| 3H|˙φ|, (1.61) which means that, to ensure the slow-roll phase to be long enough, ˙φ must change slowly on the typical time-scale of the problem, which is given by the Hubble param-eter. Another formulation of the second slow-roll condition can be found combining
1.4. The inflationary paradigm 23 equations (1.57), (1.58) and (1.60a):
−H˙ 3H2, (1.62)
which implies that also H changes slowly on the typical time-scale. We can also translate the two slow roll conditions in terms of the potential only 
1 48πG V0 V 1 , (1.63a) 1 24πG V00 V 1 . (1.63b)
Slow-roll parameters We have just obtained some conditions needed for the slow-roll approximation to hold. Usually, these conditions are more compactly expressed in terms of the dimensionless slow-roll parameters
eV ≡ 1 16πG V0 V 2 , ηV ≡ 1 8πG V00 V , (1.64)
where we used the subscript V to highlight that this is the definition of slow-roll parameters made in terms of the potential. However, we can also use the Hubble parameter to define eH ≡ − ˙ H H2 , ηH ≡eH− ¨ H 2H ˙H, (1.65) so that ¨ H H ˙H =2(eH−ηH). (1.66) In a perfect analogy, one can also introduce
eφ ≡ 3 2 ˙φ2 V , ηφ ≡ − ¨ φ H ˙φ. (1.67)
Putting together the equations (1.64), (1.65) and (1.67), one finds that, to lowest order in slow-roll parameters, eV ' eH ' eφ and we then choose to simply denote all the
three parameters as e.
Quasi-exponential expansion Recalling equation (1.54), the equation of state p= wρ in the slow-roll regime is characterized by
w= p ρ ' 1 2 ˙φ2−V(φ) 1 2 ˙φ2+V(φ) ' −1+2 3e, (1.68)
hence describing a quasi-exponential expansion a(t) ∼ eHt, with H slowly decreas-ing in time. In particular, because of equation (1.65),
H 'e∆N , (1.69)
where N represents the umber of e-folds dN=d log a' H dt. We can also parametrize the scale factor evolution by means of the conformal time η. In fact, recalling the con-siderations we have made at the beginning of Section1.2, we it is straightforward to find that a(η)∝ η−1−eduring slow-roll inflation.
Basics of GWs theory
Now that we have a quite good understanding of the fundamental ideas of the in-flationary mechanism, the only ingredient we need to get on with our work and dis-cuss the irreducible GWs background produced during inflation is some grounding in GWs theory. In particular, each section in which this chapter is divided introduces GWs on a different background: we consider a vacuum Minkowski background in Subsection2.1.1, a Minkowski background with matter perturbations in Subsection 2.1.2, a general curved background in Section 2.2, and then, finally, the particular case of a FRW background in Section2.3. This will be the last topic covered in the chapter, and the results therein will be intensively used in the following discussion.
The most straightforward approach to introduce GWs and see how they emerge from General Relativity is linearized gravity. In this approximation scheme, GWs are defined as metric perturbations in asymptotically flat space-times. In practice, this means that we can split the metric tensor gµν as a sum of a flat Minkowski background
metric ηµνand a small perturbation, denoted as hµν:
gµν =ηµν+hµν, |hµν| 1 , (2.1)
where hµν represents the GWs field. In fact, as we will show in the next pages,
massaging the Einstein equations1 Gµν= Rµν−
2gµνR=8πG Tµν, (2.2) we will obtain a wave equation for hµν, hence confirming the wave nature of the
Symmetry under slowly varying infinitesimal coordinate transformations Since the numerical values of a tensor components depend on the reference frame we use to evaluate them, the requirement|hµν| 1 translates in the choice of a reference
frame where such condition holds on a sufficiently large region of space. Once this choice is made, it is easy to show that linearized theory is invariant under slowly varying infinitesimal coordinate transformations, i.e. gauge transformations
xµ →x0µ= xµ+
ξµ(x), |∂µξν| . |hµν|, (2.3) 1Notice that, since we are assuming a flat Minkowski background, the Ricci tensor R