Inflation, quantum fluctuations and gravitational waves
Luuk Wagenaar
10447954
July 8, 2016
Supervisor: J.P. Van der Schaar
2
ndAssessor: J. De Boer
Course: Bachelorthesis
Pages: 45
Abstract
The Big Bang theory is a theory with several problems. Inflation theory is extremely successful in solving these problems and combining inflation theory with quantum fluctuations in the early universe shows a successful mechanism to clarify the structure in the universe. Future experiments will try to observe B mode polarisation in the CMB and perhaps directly detect inflationary gravitational waves. This paper will discuss inflation, quantum fluctuations and the different possibilities for indirect detection through a B mode pattern in the CMB caused by inflationary gravitational waves and direct detection of inflationary gravitational waves. The challenges to observe an inflationary gravitational wave signal are immense. B mode polarisation in the CMB is a possibility to learn more about the tensor power spectrum, but is subject to foreground noise. Direct detection is only possible in space and even there the signal is too small to observe in the currently planned missions.
Contents
1 Op zoek naar de vroegste momenten van het heelal 4
2 Introduction 5
3 Big Bang shortcomings 5
3.1 Homogeneity problem . . . 5 3.2 Flatness problem . . . 5 3.3 Horizon problem . . . 6 3.4 Initial conditions . . . 6 3.5 A solution . . . 7 4 Inflation 8 4.1 Conditions for inflation . . . 8
4.2 Classical Field theory . . . 8
4.3 Scalar field cosmology . . . 9
4.4 Slow roll mechanism . . . 10
4.5 Comoving horizon . . . 11
4.6 Inflation as a solution . . . 12
5 Quantum fluctuations 15 5.1 Simple one dimensional harmonic oscillator . . . 15
5.2 Quantum fluctuations in Sitter Space . . . 16
5.2.1 Qualitative description of Harmonic oscillator analogy . . . 16
5.2.2 Action . . . 17
5.2.3 Canonical quantisation . . . 18
5.2.4 Subhorizon limit of the solution . . . 19
5.2.5 Superhorizon limit of the solution . . . 19
5.2.6 Bunch-Davies mode functions . . . 19
5.3 Power Spectra . . . 20
5.3.1 Scalar power spectrum . . . 20
5.3.2 Density perturbation spectrum . . . 21
5.3.3 Tensor power spectrum . . . 22
5.4 Energy scale of inflation . . . 23
5.5 Chaotic inflation model (φ2m2) . . . . 24
6 Observations 26 6.1 The cosmic microwave background . . . 26
6.1.1 Temperature fluctuations in the CMB . . . 27
6.1.3 Polarisation power spectra . . . 33
6.1.4 Future experiments . . . 35
6.2 Direct detection of gravitational waves . . . 36
6.2.1 Frequency power spectrum . . . 36
6.2.2 Inflationary gravitational wave background (IGWB) . . . 37
6.2.3 Detector sensitivity . . . 38
6.2.4 Future experiments . . . 39
1
Op zoek naar de vroegste momenten van het heelal
Een van de grootste onopgeloste vraagstukken in de natuurkunde is het ontstaan van het heelal en daarmee het ontstaan van ruimte en tijd. De Oerknal of de Big Bang werd al vroeg voorgesteld op basis van een aantal fundamentele waarnemingen in het heelal en eist dat het heelal ooit uit een extreem heet punt, een singulariteit, met een oneindige dichtheid is ontstaan. Later echter werden observaties gedaan die niet met alleen de Oerknal theorie opgelost konden worden. Om deze problemen op te lossen is de inflatietheorie ontwikkeld door Alan Guth in 1981 en later verder ontwikkeld door Andrei Linde. Inflatie is een zeer korte periode vlak na de oerknal van ongeveer 10−35 seconde, waarbij de ruimte exponentieel is uitgedijd. In deze zeer korte periode is het heelal zo snel groter geworden, dat materie die met elkaar kon communiceren uit elkaars zicht is verdwenen. Als we inflatie theorie combineren met quantum fluctuaties in het vroege universum, verkrijgen we een beeldschoon mechanisme dat de vorming van structuur in het heelal kan verklaren.
Inflatie speelde zich af tijdens het begin van het universum, dus metingen lijken in eerste instantie onmogelijk. Er zijn echter twee mogelijkheden om inflatietheorie te testen. Beide mo-gelijkheden rusten op gravitatiegolven, die volgens Einstein’s algemene relativiteitstheorie tijdens inflatie zijn geproduceerd. Wij worden allemaal een klein beetje uitgerekt en ingedrukt door deze gravitatiegolven afkomstig van het begin van het universum. Dit effect is echter zo minimaal, dat dit vanaf aarde niet is waar te nemen. In de ruimte is er minder ruis en kunnen de experimenten op grotere schaal ingezet worden, dus daar is de kans aanwezig om gravitatiegolven direct te meten. De tweede mogelijkheid is het bestuderen van de kosmische achtergrondstraling. Gravitatiegol-ven laten kleine sporen achter in de kosmische achtergrondstraling in de vorm van B mode polar-isatie. Nauwkeurige metingen aan deze polarisatie zou een mooi patroon kunnen onthullen, dat gekoppeld kan worden aan de sterkte van de gravitatiegolven tijdens inflatie. Deze signalen zijn ontzettend klein, maar als gravitatiegolven of deze B mode polarisatie gemeten worden, zou dat een directe bevestiging zijn van inflatie theorie en kunnen natuurkundigen de vroegste momenten van het heelal bestuderen.
Het signaal van een gravitatiegolf achtergrond zou echter zo klein kunnen zijn, dat natu-urkundigen nooit in staat zullen zijn deze te meten. In dat geval blijven de vroegste momenten van het heelal wellicht voor altijd onzichtbaar.
2
Introduction
The origin of the universe is one of the big unanswered questions in physics. Big bang theory is a successful theory to clarify observed features in the universe, but it can not be the complete story and raises some fundamental questions. Inflation in combination with quantum fluctuations in the early universe obtains a mechanism that can explain the formation of structure in the universe. Inflation and quantum fluctuations together obtain both scalar and tensor perturbations. Scalar perturbations can be linked to temperature fluctuations in the CMB and it is measured accurately by Ade et al. (2015). Scalar perturbations give a lot of information and it enables scientists to constrain the energy scale of inflation to a certain maximum, but it will not fix the energy scale of inflation and the slow roll parameters. The tensor power spectrum is only dependent on the potential, V , and can therefore directly be linked to the energy scale of inflation. Measurements of a tensor power spectrum, corresponding to gravitational waves, will completely fix the energy and the slow roll parameters. In this paper both indirect detection of gravitational waves through the B mode polarisation in the CMB and direct detection of gravitational waves will be examined. The paper starts out with a general description of inflation. Subsequently quantum fluctuations in the sitter space will be derived to obtain measurable parameters. The paper is concluded with the possibilities for detecting an inflationary gravitational wave background.
3
Big Bang shortcomings
Standard Big Bang theory raises some fundamental questions about the structure of the universe. These are the flatness problem, the homogeneity problem, the horizon problem and the problem of the initial conditions.
3.1
Homogeneity problem
The universe is assumed to be homogeneous and isotropic on large scales. This is puzzling, because inhomogeneities are gravitationally unstable and will grow with time. In the early universe the inhomogeneities were therefore even smaller than today. How can we explain the extreme smoothness of the universe?
3.2
Flatness problem
The universe is best described by flat Euclidian space, but spacetime curves in presence of matter. How is it possible that the universe is so flat? The relevance of the problem can best be described in more detail using the Friedmann equation:
H2= 1 3ρ(a) −
κ
Now we use
Ω(a) = ρ(a) ρcrit(a)
and ρcrit= 3H(a)2 (2)
To obtain the following expression for the Friedmann equation 1 − Ω(a) = −κ
(aH)2 (3)
Ω is clearly time dependent. The comoving Hubble radius (aH)−1 grows with time, so we see that (3) goes to zero. Ω had to be extremely close to one in the early universe, to make sure that the observed flatness in our present universe, Ω(a0) ∼ 0, can be explained. It is like making a
pencil balance on its tip. If we put in the numbers we obtain (Baumann (2007)): | Ω(aBBN) − 1 |≤ O(10−16)
| Ω(aGU T) − 1 |≤ O(10−55)
| Ω(apl) − 1 |≤ O(10−61)
(4)
3.3
Horizon problem
The CMB is observed to be very homogeneous throughout the universe. This is remarkable, because the greatest part of the observable universe is not in causal contact and could have never exchanged any information. To express this fact mathematically we first take the FRW spacetime radial null geodesic, which is simply the trajectory of a photon through spacetime:
dr = ± dt
(a(t) ≡ dτ (5)
Where τ is the conformal time or the comoving horizon and describes the maximum distance a light ray can travel between t = 0 and t. This can be expressed as:
τ = Z a 0 d[ln(a)] 1 aH (6) An increasing comoving Hubble radius corresponds to an increasing comoving horizon. Radiation dominated and matter dominated universes obtain a and a1/2 respectively. This means that the
comoving horizon grows with time and that scales that enter the universe today were outside the horizon at CMB decoupling and therefore not in causal contact. It is puzzling that causally disconnected regions in the universe look so similar.
3.4
Initial conditions
These problems are not inconsistencies in the big bang model, but the initial conditions are extremely fortuitous to obtain the current universe. Physicists would like to explain this extreme fine tuning.
3.5
A solution
All these problems can be solved by a very simple idea: to invert the behaviour of the comoving Hubble radius. This means that the comoving Hubble radius decreases during inflation. Regions that are in causal contact in the early universe lose causal contact during inflation. They are stretched outside each others horizon.
4
Inflation
Inflation is a short period of time in the evolution of the universe, where the universe expanded exponentially. Inflation theory succesfully solves the big bang shortcomings, that are described in previous sections.
4.1
Conditions for inflation
Inflation theory will change the behaviour of the comoving Hubble radius and make it decrease in the early stages of the universe. The universe had to expand faster than the speed of light, so causally connected regions were stretched out of each others horizon and could not communicate anymore. d dt H−1 a < 0 ⇒ d 2a dt2 > 0 ⇒ ρ + 3P < 0 (7)
These expressions show that the comoving Hubble radius decreased, the universe had an ac-celerated expansion and must have had a driving force behind the expansion: a negative pressure. Matter gravitationally attracts, so there had to be another force that drove the expansion.
This can be solved by introducing a scalar field with special conditions. First classical field theory is shortly dealt with and subsequently the inflaton field will be introduced.
4.2
Classical Field theory
For one particle in 1D with a certain coordinate q(t) the equation of motion can be derived from the principle of least action. The action looks like S = R dtL(q, ˙q). Field theory is basically the same story, but instead of having one particle with a coordinate q(t), we have a range of spacetime dependent fields Φi(xµ). In field theory the Lagrangian can be expressed as an integral
over space of a Langrange density L(Φi, ∂
µΦi). The Langrange density is obviously a function of
the fields and the spacetime derivatives of the fields. This leads to the following expressions for the Lagrangian and subsequently the action of the fields:
L = Z d3xL(Φi, ∂µΦi) S = Z dtL = Z d4xL(Φi, ∂µΦi) (8)
The Euler-Langrange equations are found after the requirement that the action is unchanged under small variations:
Φi→ Φi+ δΦi
∂µΦi→ ∂µΦi+ δ(∂µΦi) = ∂µΦi+ ∂(δµΦi)
(9) δΦiis assumed to be small and therefore we can Taylor expand the Langrangian to first order:
L(Φi, ∂ µΦi) → L(Φi+ δΦi, ∂µΦi+ δµ(∂Φi)) = L(Φi, ∂µΦi) + ∂L ∂Φi∂Φ i+ ∂L ∂(∂µΦi ∂(∂µΦi) (10) In correspondence with this, the lagrangian goes from S → δS, where:
δS = Z d4x ∂L ∂Φi∂Φ i+ ∂L ∂(∂µΦi ∂(∂µΦi) (11) After using Stoke’s theorem (Carroll (2014)) it is possible to write the action as:
δS = Z d4x ∂L ∂Φi − ∂µ ∂L ∂(∂µΦi) ∂Φi (12) By definition δS =R d4xδS
δΦδΦ and therefore when S is at critical points:
δS δΦi = ∂L ∂Φi − ∂µ ∂L ∂(∂µΦi) = 0 (13)
These are the Euler Langrange equations for classical field theory in flat spacetime. After this derivation it is possible to look at the behaviour of the inflaton field, the proposed field for inflation.
4.3
Scalar field cosmology
The action of a gravitational field looks like: S =
Z
d4x√−g[1
2R + Lφ] (14)
R stands for the Ricci scalar, which is a measure for the curvature. The lagrangian looks like: Lφ= 1 2g µν∂µφ∂νφ − V (φ) (15) 1 2g
µν∂µφ∂νφ is the kinetic part of the equation. Now (15) is filled in (13) to obtain the
equations of motion for a scalar field with the corresponding action in (14). This leads to the following equations: ∂L ∂φ = − V (φ) ∂φ ∂L ∂(∂µφ) = gµν∂νφ ⇒ ∂S ∂φ = √ −gV (φ) ∂φ + ∂µ( √ −ggµν∂νφ) = 0 (16)
The term√−g is added in the last equation to match the action in (14). This equation has to be solved to obtain the equation of motion for the scalar field1:
V0(φ) + 1 a3∂µ(a 3gµν∂ νφ) = V0(φ) + 3H ˙φ + ¨φ = 0 (17) 1
Using√−g =pdet(gµν) =
√
a6= a3.
From the energy-momentum tensor definition it is possible to find both the pressure and density in terms of the homogeneous field. From there, the equation of state can be found:
Tµν ≡ − 2 −g δSφ δgµν = ∂µφ∂νφ − gµν 1 2∂ σφ∂ σφ + V (φ) Tνµ= diag(ρ, −P, −P, −P ) (18)
The following result is obtained after filling in the equations: ρφ= 1 2 ˙ φ2+ V (φ) Pφ= 1 2 ˙ φ2− V (φ) (19)
Therefore the equation of state is: w = Pφ
ρφ = 1 2φ˙ 2−V (φ) 1 2φ˙2+V (φ)
, which can have a negative pressure and an accelerated expansion according to (7).
4.4
Slow roll mechanism
To make inflation possible, slow-roll conditions are imposed. The potential energy has to dominate the kinetic energy for a sufficient amount of time. This implies that the second derivative ¨φ has to be small, so the conditions for inflation become:
˙ φ V (φ) ¨ φ 3H ˙φ , Vφ0 (20) The derivation of the slow roll parameters and η works as follows. The conditions for inflation and the equations for the density and pressure imply that H2= 1
3ρφ≈ V 3 and 3H ˙φ ≈ −V 0 φ. From there: 1 2φ˙ 2 V = Vφ2 6V2 ≡ 3 ˙ H H2 = −V02 φ 2V2 = − ¨ φ = −Vφφ ˙ φ 3H + Vφ 3 ⇒ ¨ φ 3H ˙φ = 1 3 −Vφφ V − (21)
Clearly the definition for the dimensionless parameter ≡ V
02 φ
2V2 and in the final equation
η ≡ Vφφ
V is defined. Both η and are much smaller than 1 in the slow roll regime. In the
derivation it is used that the background evolution is: H2 ≈ 1
3V (φ) ≈ constant and ˙φ ≈ − Vφ0 3H.
From there it is possible to show that: H = ˙a
a → da 1
a = Hdt → ln a ∼ Ht → a ∼ e
To further relate the slow roll conditions to the inflation theory: H = ˙a a → ˙a = Ha → ¨a = a ˙H + aH2 → ¨a a= (1 − ) H 2 (23)
Inflation is guaranteed when is much smaller than one. Inflation ends when slow roll condi-tions are no longer met, (φend) ≈ 1.
4.5
Comoving horizon
Inverting the behaviour of the comoving Hubble radius, makes sure that all regions that are within our horizon (and even outside our horizon) were in causal contact in the early stages of the universe. In the previous section is showed that the universe expanded exponentially during inflation. The comoving Hubble radius is defined as aH1 . During inflation the Hubble parameter is approximately constant and the scale factor a grows exponentially. Therefore the comoving Hubble radius decreases during inflation. Figure 1 shows the evolution of the Hubble radius during inflation until now.
Figure 1: The comoving Hubble radius decreases during inflation and expands after inflation ends. This means that during inflation certain regions lost causal contact and moved out of each others horizon. After inflation ended, the comoving Hubble radius increased again. A large part of the universe however is not observable today, that was in causal contact before inflation started. The figure is from Baumann (2009).
4.6
Inflation as a solution
The important features of inflation have been derived in the past sections. How can this theory solve the problems stated in sections 3.3, 3.2 and 3.1?
Flatness Problem
The density parameter is the ratio of the observed density to the critical density Ω = ρρcrit. The density parameter is equal to one, when ρ = ρcrit. The flatness problem
states that the universe is extremely flat and therefore Ω = 1 should be obtained. The Friedmann equation |1 − Ω(a)| = (aH)1 2 and during inflation H ∼ constant and a = e
Ht. The universe is
driven to flatness thanks to inflation: |1 − Ω(a)| ∝ 1
a2 = e
−2Ht→ 0 as t → ∞ (24)
Horizon Problem and Homogeneity Problem
Figure 2 shows that the whole universe was in causal contact before inflation. All scales were smaller than the Hubble radius and therefore in causal contact. The universe was in thermal equilibrium and spatially homogeneous. This explains the uniformity of the CMB. The comoving Hubble radius decreased during inflation and increased after inflation. The decreasing comoving Hubble radius led to causally disconnected regions.Figure 2: The comoving Hubble radius decreases during inflation and expands after inflation ends. The figure is from Baumann (2009).
Conformal Diagram of Inflationary Cosmology
The problems and their solutions can be easily showed with the help of a conformal diagram. Conformal time is introduced in section 3.3 and describes the maximum distance a light ray can travel between t = 0 and t. See figure 3 for the horizon problem in the Big Bang model. There are many causally disconnected regions, which could have never communicated with each other. This problem is solved by inflation shown in Figure 4.Figure 3: Many causally disconnected regions at the time of last scattering in Big Bang theory puzzeled scien-tists. What made these causally disconnected regions so similar, although they could have never communicated with each other? The figure is from Baumann (2009).
Figure 4: Regions were in causal contact before inflation started, when inflation theory is included in the conformal spacetime diagram. This figure is simply an extension of figure 4 by adding the inflationary period. The figure is from Baumann (2009).
5
Quantum fluctuations
In previous sections we assumed the universe to be homogeneous and isotropic. If this were the case, no galaxies or other structures in universe could have been formed and this raises the question how structures could have formed and how they evolve in time. The combination of the inflation model with quantum mechanics obtains a beautiful mechanism, that explains the formation of structure in the universe.
Quantum fluctuations will locally delay or advance the end of inflation. This means that the evolution of the universe is slightly different on small scales and will induce density fluctuations. In the following sections the derivation of the above principle will be written down mathemat-ically. The 1 dimensional harmonic oscillator with a time dependent frequency and a constant frequency will be solved. After this basic quantum mechanical introduction, the knowledge is used to obtain the quantum fluctuations in Sitter space.
5.1
Simple one dimensional harmonic oscillator
Only a few quantum mechanical systems can be solved exactly and the 1D harmonic oscillator with a time dependent frequency is one of them. The field can be described as a combination of many harmonic oscillators and therefore the harmonic oscillator is the starting point of the derivation.
The classical action of a harmonic oscillator is: S = Z dt 1 2 ˙ x2−1 2ω 2(t)x2 ≡ Z dtL (25)
The mass of the particle is set to one. Using the Euler-Langrange equations δLδx − d dt
δL δ ˙x = 0
the following equation of motion is found: ¨
x + ω2(t)x = 0 (26)
Now standard canonical quantisation changes classical parameters into quantum operators: x → ˆx and p → ˆp and subsequently impose the following standard commutation relation: [ˆx, ˆp] = i~. Now a certain mode expansion is defined in which the mode functions still satisfy the classical equation of motion (26) with v instead of x:
ˆ
x = v(t)ˆa + v∗(t)ˆa† (27) After some algebra one finds that the commutator now looks like:
[x(t), ˙x(t)] = i~ ⇒ [(v(t)ˆa + v∗(t)ˆa†), (∂tv(t)ˆa + ∂tv∗(t)ˆa†)]
⇒ hv, vi[ˆa, ˆa†] = 1
The real number hv, vi is chosen to be positive and therefore the quantity can be rescaled to be one: hv, vi ≡ 1. This means that the commutation relation [ˆa, ˆa†] = 1. This is the standard commutation relation for raising and lowering operators of a harmonic oscillator. The operators ˆ
a and ˆa† are annihilation and creation operators and look like: ˆ
a = hv, ˆxi ˆ
a†= −hv∗, ˆxi
(29) The lowest energy state is the vacuum state and must satisfy ˆa|0i = 0. We can find the n’th energy eigenstate by working n times successively with the creation operator: |ni ≡ √1
n!(ˆa †)n|0i.
Various mode functions v(t) with a time dependent frequency can be chosen to obtain different solutions for different vacuum states. This problem can be solved by first choosing a constant frequency ω(t) = ω. The requirement that the vacuum state is the ground state of the Hamiltonian and ω(t) = ω selects a particular choice of v(t). Now v(t) can be obtained by using the Hamiltonian
ˆ
H = 12pˆ2 + 1 2ω
2xˆ2 where ˆp = ˙x and ˆx = v(t)ˆa + v∗(t)ˆa†. Subsequently using ˆa|0i and the
commutation relation [ˆa, ˆa†] = 1 the following equation is found for the Hamiltonian working on the vacuum state:
ˆ
H = ( ˙v2+ w2v2)∗ˆa†ˆa†|0i + 1 2(| ˙v|
2+ w2
|v|2)|0i (30) The first term will disappear, because |0i is an eigenstate of the Hamiltonian and therefore naturally ˆH − ( ˙v2+ w2v2)∗aˆ†ˆa†|0i = 0 . This means 1
2(| ˙v|
2+ w2|v|2) = 0 and therefore:
˙v = ±iωv (31)
After normalisation and correct choice of sign, the following solution is found: v(t) =
r ~ 2ωe
−iωt (32)
To obtain the zero point fluctuation in vacuum, the mean square expectation value of the position operator can be calculated in the ground state |0i. After some basic algebra and plugging (32) into the obtained expression leads to:
h|ˆx|2i = h0|ˆx†x|0i = h0|(vˆ ∗(t)ˆa†+ v(t)ˆa)(v(t)ˆa + v∗(t)ˆa†)|0i ⇒ |v(ω, t)|2= ~
2ω
(33)
5.2
Quantum fluctuations in Sitter Space
5.2.1
Qualitative description of Harmonic oscillator analogy
Instead of a 1D harmonic oscillator, the inflaton field has to be studied. We divide the inflaton field into a homogeneous background φ(t) and a perturbation δφ(t, x). See figure 5 for an understanding of the division of φ(t) and perturbation δφ(t, x).
Figure 5: This figure shows the quantum fluctuation δφ around the classical background evolution for a certain model, the small field model. The figure is from Baumann (2009).
The perturbation is analogous to the position coordinate in the harmonic oscillator. In fol-lowing sections will be derived that the perturbation will satisfy:
δ ¨φk+ 3Hδ ˙φk+
k2
a2δφk= 0 (34)
This looks very similar to the equation of motion of a classical harmonic oscillator. In fact (34) is identical to the equation of motion of a classical harmonic oscillator with a spring constant of ka and a damping coefficient 3H. Some interesting features can be seen immediately. If a/k is much smaller than the Hubble radius, H−1, the mode will behave as a normal oscillator, with a damping that can be neglected. When however the wavelength is greater than the Hubble radius, the mode will freeze.
5.2.2
Action
The action for the scalar field is given by S = R d4x√−gL, where the Langrangian density is given by L =12R +1
2(∂φ)
2− V (φ). This leads to an action that looks like:
S = Z d4x√−g 1 2R + 1 2(∂φ) 2− V (φ) (35) In flat space √−g can be ignored, but not in curved spacetime. From the background FRW metric ds2 = −dt2+ a2(t)dx2 follows that √−g =p−detg
µν = a3(t). The action is varied to
obtain the Klein-Gordon equation for a scalar field in the FRW metric according to Baumann (2007):
φ(t, x) = ¨φ − 1 a2∇
2φ + 3H ˙φ + V0(φ) = 0 (36)
As described in the latter subsection the inflaton field will be decomposed in a constant background φ(t) and a small perturbation δφ(t, x). The equation of motion can be found by
expanding S to the second order, but for now the perturbation is expanded in Fourier compo-nents. Fourier expansion of the perturbation effectively means a sum over all modes: δφ(t, x) = R d3k
(2π)3/2(δφk)e
−ik·x. The equation of motion for the perturbation becomes:
¨
δφk+ 3H ˙δφk+
k2
a2δφk− Vφφδφk= 0 (37)
Slow roll condition η = V,φφ
V 1 → V,φφ H
2have to be inserted to obtain:
¨
δφk+ 3H ˙δφk+
k2
a2δφk= 0 (38)
Now the Hubble friction term can be removed after defining the Mukhanov variable v = aδφ and changing to conformal time τ = − 1
Ha. In this derivation is used that ∂t = a −1∂η. ˙ φk= v0 a2 − a0v a3 ¨ φk= v00 a3 − 2v0a0 a4 − a00v a4 − a0v0 a4 + (3a0)2v a5 H = ˙a a = a0 a2 (39)
After some algebra, the following equation is obtained. vk00+ k2−a 00 a vk= 0 (40)
This looks remarkably similar to the equation of motion of a simple harmonic oscillator with a time dependent frequency (26).
5.2.3
Canonical quantisation
In order to quantise the classical description without losing the formal structure canonical quan-tisation is used. The fluctuations in the inflaton field will be quantised in the following way. First the action, which can be determined by taking the second order expansion of (36) to δφ, but for now is taken from Baumann (2007):
δ2S = 1 2 Z dτ d3x (v0)2− (∂iv)2+ a00 a v 2 (41) Canonical momentum is defined and classical fields are promoted to quantum operators:
π = ∂(δ2L) v0 = v
0 (42)
v, π → ˆv, ˆπ (43)
Canonical commutation relations should now be imposed:
The mode decomposition is now defined: ˆ v(τ, x) = Z d3k (2π)3/2 h vk(τ )ˆakeik·x+ v∗k(τ )ˆa † ke −ik·xi (45) ˆ
ak and ˆa†k are the creation and annihilation operators and satisfy the standard commutation
relation: [ˆak, ˆa†k] = δ
(3)(k − k0).
The mode functions comply with (40) and some interesting solutions can be studied.
5.2.4
Subhorizon limit of the solution
On scales inside the causal horizon, k aH,: we can neglect the aa00 term and find the following equation of motion:
vk00+ k2vk= 0 (46)
This is the same equation of motion as for the harmonic oscillator with time independence. The solution is given by:
vk=
e−ikt √
2k (47)
Which is already properly normalized. This means that modes oscillate within the horizon.
5.2.5
Superhorizon limit of the solution
The second situation is naturally that of superhorizon scales, k aH. This means that the a00 a
term dominates over the k2term and the following equation of motion is found:
vk00−a
00
a vk= 0 (48)
This leads to a growing mode solution: vk ∝ a and therefore the scalar field perturbation is
constant at superhorizon scales. This means that δφ = va = constant. For an exact calculation see Postma (2014).
5.2.6
Bunch-Davies mode functions
The equation that has to be solved in Sitter space is: vk00+ k2− 2 τ2 vk= 0 (49)
Where it is used that a00
a = 2H
2a2 = 2
τ2. The exact solution looks like this, where α and β
are free parameters that are dependent on the initial conditions: vk= α e−ikτ √ 2k 1 − i kτ + βe ikτ √ 2k 1 + i kτ (50) If the subhorizon limit is considered, one can fix the parameters α and β and obtain a familiar expression v00k+ k2vk= 0. This is simply the equation of motion for a simple harmonic oscillator.
The initial condition limkτ →∞vk = e
−ikτ
√
2k is used to find that α = 1 and β = 0 and this leads to
the Bunch-Davies mode functions.
vk= e−ikτ √ 2k 1 − i kτ (51)
5.3
Power Spectra
In cosmology the variance of the perturbations is quantified by the power spectrum as a function of the scale, k. The quantum fluctuations oscillate on subhorizon scales and freeze on superhori-zon scales and small fluctuations in the density cause the formation of structure in the universe. The metric of the perturbations can be divided into three types: scalar, vector and tensor per-turbations. Vector perturbations are associated with the rotational motion and will decay with the expansion of the universe (Lesgourgues (2006)). Scalar perturbations however are tied to the energy-momentum distribution and therefore with the density and temperature fluctuations. The tensor perturbations correspond to the perturbation of the gravitational field and will induce gravitational waves. Power spectra for the scalar perturbations and tensor perturbations will be calculated in the following sections and will be expressed in measurable or already measured parameters.
5.3.1
Scalar power spectrum
The zero point fluctuations of δφ at superhorizon scales can be calculated by taking the mean square of the mode function. Superhorizon scales mean that the mean square of the mode has to be evaluated for k aH or kτ → 0. The mean square value of the zero point field fluctuations are analogous to the 1D H.O. zero-point fluctuations. Equation (33) gave the result h|ˆx|2i = |v(ω, t)|2.
In the super horizon limit δφ =va = constant holds (See section 5.2.5). In Fourier space this results in the following value:
lim kτ →0h|δφk| 2 i ≡ lim kτ →0h0||δ ˆφk| 20i = limkτ →0|vk|2 a2 lim kτ →0|vk| 2= − 1 2k3τ2 (52) These two expressions can be combined to obtain:
lim
kτ →0h|δφk|
2i = H2
2k3 (53)
In the last step is used that: τ = −(aH)−1. Now it is possible to calculate the zero point fluctuations in real space. The following steps are easy to follow when the Fourier transform δφ(x) =R 1 (2π)3/2d 3kδφ keik·x is recalled. hδφ2i ≡ h0||δ ˆφ|2|0i = hδ ˆφ(x)δ ˆφ(x)i = 1 (2π)3 Z d3kh|δφk|2i = 1 (2π)2H 2lnkf ki (54)
The integral is cut off at a cetrain low k and high k. The cut off for low k means that these modes left the Hubble radius just after inflation started and the cut off for high k is for modes that left the Hubble radius just before the end of inflation.
From (54) and the definition that the power spectrum is defined as hδφ2i ≡ R ∆2 δφ dk k can be seen that: ∆2δφ= H 2π 2 (55)
5.3.2
Density perturbation spectrum
The Time-Delay Formalism (Wang et al. (1997)) will couple density fluctuations to the field fluctuations and the time at which inflation ends.
The time at which inflation ends, is controlled by the field φ. Inflation is possible for certain constraints on the field potential (this is what the slow roll conditions are all about). When the slope of the potential becomes too steep, it will end the inflationary period, which means that it is possible to connect the time at which inflation ends, to the field potential. The fluctuations in the field potential will locally delay or advance the end of inflation and the time difference between the end of inflation in different regions can be written as follows: δt = δφ˙
φ. In previous section is
obtained that: hδφ2i = H 2π
2R dk
k. This simply leads to:
(δt)2= δφ ˙ φ 2 = H 2π ˙φ 2Z dk k (56)
It is also possible to write: δρρ ∼ δa
a and it is known that during inflation a ∼ e
Htand therefore
that δa δt ∼ He
Ht. This leads to the simple expression:
δρ ρ ∼
δa
a ∼ Hδt (57)
Now include the slow roll result 3H ˙φ = −V,φ to obtain:
δρ ρ 2 = 9H 6 (2πV0)2 Z dk k = V 24π2 Z dk k (58)
H and therefore V (φ) is not exactly constant. Because perturbations freeze at superhorizon scales, the evaluation is done at k = aH, the horizon exit.
The definition of the density fluctuation spectrum is given by: * δρ ρ 2+ = Z ∆2S(k)d ln k (59)
Equation (58) and (59) imply that the scalar power spectrum and scalar amplitude look like: AS(k) ≡ ∆2S(k)|k=aH= 1 24π2 V |k=aH ∝ k nS−1 (60)
A useful parameter to define is the tilt of the power spectrum. The tilt gives the derivative of the natural logarithm of the power spectrum with respect to the logarithm of the wavenumber:
nS− 1 ≡ d ln ∆2 S d ln k = 2η − 6 (61)
This definition makes it possible to write the power spectrum as a function of the tilt nS− 1:
nS− 1 = d ln ∆2 S dk k → (nS− 1) dk k = d ln ∆ 2 S ln ∆2S= (nS− 1) ln k → ∆2S = k nS−1 (62)
In the derivation of the latter equation is used that: dln k = −VV0dφ. nS = 1 corresponds to a
scale invariant power spectrum.
5.3.3
Tensor power spectrum
Gravitational waves are predicted by the General relativity theory of Einstein and are produced by accelerating masses.
The graviton corresponds to the tensor perturbation in the metric: gµν = g (0)
µν + hµν, where
hµν corresponds to gravitational waves. The second order action for the tensor perturbation looks
like (Guzzetti et al. (2016)): S(2)=
Mpl2 8
Z
dτ dx3a2[(h0ij)2− (∂thij)2] (63)
h is one of the two polarisation components of the gravitational wave. These polarisations are h+ and h× and stand for plus polarised and cross polarised gravitational waves. See figure 6.
Figure 6: The two polarisations of gravitational waves from Centrella et al. (2010) With the following Fourier transform definition and taking the two polarisations into account:
hij = Z d3k (2π)3 X λ=+,× λij(k)hλk(τ )eikx (64)
Using the definition vkλ≡a 2Mplh
λ
kthe following expression can be derived:
S(2)= X λ=+,× 1 2 Z dτ dk3[(vλk)2− k −a 00 a (vλk)2] (65) This equation for the action of the tensor perturbation looks very similar to the action for the perturbed field in (41) up to a factor 2. Now it is straitforward to find the tensor power spectrum:
δh = 2δφ ⇒ ∆2δh= 4∆2δφ= 4 H 2π
2
(66) The tensor fluctuation spectrum is then found to be:
AT(k) = ∆2T(k)|k=aH= 8 H 2π 2 = 2 3π2V k=aH ∝ knT (67)
The expression for the spectral tensor index looks like: nT ≡
d ln ∆2T
d ln k ' −2 (68)
Where is used that dln k = −VV0dφ and that (φ) =
1 2 V0 V 2
. Important to note is that detection of the tensor spectrum will directly give the energy scale at inflation, V1/4. Now the
scalar-tensor ratio is defined as:
r ≡ ∆ 2 T ∆2 S = 16 (69)
Also note that the tensor spectrum will be of crucial importance for future experiments, because of the V dependence only. Determination of ∆2
T will therefore fix the energy scale of
inflation and the slow roll parameters.
5.4
Energy scale of inflation
The energy scale of inflation can be determined by using the measured amplitude of scalar per-turbations and the relations derived in the above sections. Recall the tensor-to-scalar ratio and the expression for the tensor energy spectrum2:
r ≡ ∆ 2 T ∆2 S = 16 ∆2T(k)|k=aH= 2 3π2M4 pl V k=aH (70)
The amplitude of the scalar perturbation ∆S(k0) = (4.697 ± 0.102) × 10−5, k0 ≡ 0.05Mpc−1
is measured by the PLANCK satellite (Ade et al. (2015)). Using (70) this will lead to:
2V 3M4 plπ2 = (4.697 · 10−5)2r → V = 3rM 4 plπ 2 2 (4.697 · 10 −5)2 → E = V1/4= M pl 3π2 2 1/4 (4.697 · 10−5)1/2r1/4 = 3.273 · 1016r1/4GeV (71)
Recent constraints on r (see (83)) show that the tensor-to-scalar ratio has to be r < 0.149. This leads to an upper bound for the energy scale of inflation:
V1/4< 2.03 · 1016GeV (72) To be able to detect a gravitational wave signal, the value of r has to be larger than r ∼ 10−6 (Smith et al. (2006)), which is the ultimate goal for the DECIGO experiment. The energy scale of inflation that is possible to observe can be found by adding the lower bound3.:
1.04 · 1015GeV < V1/4< 2.03 · 1016GeV (73) It is clear that observable gravitational waves correspond to inflation that occurred at very high energies. The constraints on the detectability are of crucial importance for future experiments.
The field can be linked to the tensor-to-scalar ratio to obtain: r = 16 = 8 V 0 V 2 = 8 1 Mpl dφ dNe 2 → ∆φ Mpl ≈√60 8 √ 0.149 (74)
Where the right planck mass power is added and is integrated from the end of inflation to 60 e-folds earlier. This implies that:
∆φ ∼ Mpl (75)
This is called the Lyth bound and shows that the field range of inflation is around the Planck scale and probably even larger than the Planck scale.
5.5
Chaotic inflation model (φ
2m
2)
The potential determines all the parameters in inflation theory. In order to find the right inflation model, many potentials have been proposed and this section will deal with the φ2m2model. This model will have a potential that looks like:
V (φ) = 1 2m
2φ2 (76)
3
More conservative constraints say r = 10−3. Note that this will make the range of obervable energies at which inflation occurred even smaller.
The latest PLANCK satellite (Ade et al. (2015)) data will be used. The slow roll parameters are derived in the section 4.4 and in this model will look like:
≡ V 02 φ 2V2 = 1 2 m2φ 1 2m 2φ2 2 = 2 φ2 η ≡ Vφφ V = m2 1 2m2φ2 = 2 φ2 N = Z V Vφ0 = Z 1 2m 2φ2 m2φ dφ = Z 1 2φdφ = 1 4φ 2 ⇒ = η = 1 2N (77)
So obviously η is the same as and is related to the number of e-folds. The power spectrum can be written as follows:
∆2S(k)|k=aH = 1 24π2 V = 4 24πm 2φ2N2=m2N2 6π2 (78)
To normalize the power spectrum to the PLANCK data (See (83)), the following results are obtained:
∆2S(k0) =
m2N2
6π2 = (4.697 ± 0.102 · 10 −5)2
mN =50≈ 7.3 · 10−6in planck units = 8.9 · 1013GeV
mN =60≈ 6.0 · 10−6in planck units = 7.4 · 1013GeV
(79)
Where N = 50 and N = 60 are the amount of e folds of inflation. The values for nS and r are
easy to calculate: nS− 1 = 2η − 6 → nS = 1 − 2 N = 0.96(N =50) and 0.97(N =60) (80) r = 16 = 8 N = 0.16(N =50) and 0.13(N =60) (81) The values for nS are close to the observed value by the PLANCK satellite: nS = 0.9645 ±
0.0049. The values for r = 0.16(N =50) in the φ2m2 model exceed the experimentally obtained
upper bound r < 0.149. r = 0.13(N =60) is within the experimentally obtained upper bound. In
6
Observations
Quantum fluctuations of the field can be divided into scalar, vector and tensor perturbations. Scalar perturbations are associated with temperature fluctuations in the CMB and with the E mode polarisation of the CMB. The scalar power spectrum depends on two parameters V and , so measurements of the amplitude ∆2S gives useful information, but it won’t fix the parameters V and . The tensor power spectrum, associated with gravitational waves, has only one parameter V and measurement of the tensor power spectrum will therefore fix V and from there the potential and slow roll parameters can be obtained.
It is possible to discriminate between direct measurements and indirect measurements of a tensor power spectrum. Direct measurements try to find local spacetime distortions due to grav-itational waves. This is how LIGO detected gravgrav-itational waves (Abbott et al. (2016)). Indirect measurements of gravitational waves involve the polarisation of the CMB. Gravitational waves will leave signatures in the polarisation of the CMB.
Ade et al. (2015) have already measured the scalar power spectrum, ∆2S, and the tilt, nS, and
figure 7 shows the current constraints on the different inflationary models based on their data.
Figure 7: Constraints on the inflationary models in the nS - r plane by the Planck Satellite (Ade et al. (2015)).
Note that k = 0.002M pc is used as a pivot scale, instead of k = 0.05 as used before.
6.1
The cosmic microwave background
During recombination photons decoupled from matter and could stream freely through the uni-verse, constituting the CMB, the Cosmic Microwave Background. Exact measurements of the CMB show that small temperature fluctuations exist. In figure 8 these temperature fluctuations
are clearly shown. The different colours stand for small temperature fluctuations of the observed photons from different regions of the universe.
Figure 8: ESA & the Planck Collaboration (2013)
The photons not only have an energy at last scattering, but also a polarisation. Thomson scattering is responsible for the polarisation of the CMB (Baumann (2009)). The polarisation contains information about the primordial fluctuations and can therefore be linked to inflation.
To completely specify the statistics of the CMB, four spectra are needed: the TT, EE, BB and TE correlations4.
6.1.1
Temperature fluctuations in the CMB
The quantum fluctuations described and derived in the previous sections, induced small den-sity fluctuations. Because of gravitational attraction, denser regions attracted more matter and became denser. Photons carry information about the density of the universe at the time of de-coupling. In denser regions, the photons had to use more energy to move away from that region and in the less dense regions on the other hand, photons had to use less energy to move away from that region. This means that the cooler spots on the CMB temperature map (See figure 8) are slightly denser regions and the hotter spots on the map correspond to slightly less dense regions. (Postma (2014)). These temperature fluctuations are very accurately measured by the PLANCK satellite. The scalar power spectrum is linked to the density fluctuations and therefore with temperature fluctuations in the CMB:
4E stands for E mode polarisation and B for B-mode polarisation. This will be derived and explained in section
* δρ ρ 2+ = Z ∆2S(k)d ln k (82)
Ade et al. (2015) published the Planck Satellite results in 2015 and measured at a pivot scale of k = 0.05M pc the following values:
∆S(k0) = (4.697 ± 0.102) × 10−5, k0≡ 0.05Mpc−1
nS = 0.9645 ± 0.0049
r < 0.149
(83)
Temperature map of the CMB
To analyse the CMB temperature fluctuations, the de-viation from the mean temperature is defined:Θ(ˆn) =T (ˆn) − hT i
hT i (84)
In this equation, ˆn is the direction in the sky, ˆn(φ, θ). The universe is a 2D sphere around us and therefore it is useful to expand the temperature field in Spherical Harmonics:
Ylm(θ, φ) = s 2l + 1 4π (l − m)! (l + m)!P m
l (cos θ)eimφ (85)
l represents the multipole moment5and is defined as: l = 0, 1, 2, ..., ∞, −l < m < l and Plmis the Legendre Polynomial. The temperature fluctuations can be expanded:
Θ(ˆn) = l=∞ X l=0 l X m=l almYlm(ˆn) (86) In this equation alm = Rπ θ=−π R2π φ=0Θ(ˆn)Y ∗
lm(ˆn)dΩ. Analogous to what is done in Fourier space,
a power spectrum can be defined:
halma∗l0m0i = δll0δmm0Cl (87)
The maximum amount of m-modes per multipole is 2l + 1. For the power spectrum6 therefore
the following expression can be written: Cl= 1 2l + 1 l X m=−l |alm|2 (88)
The expectation value of the correlation of the temperature between two points in the universe is the real space power spectrum (Tojeiro (2006)):
5
l represents the scale on the sky α =πl. The larger l, the smaller the angular scale in the sky.
6The error in this estimation of C
l is ∆Cl = p2/(2l + 1). This is called the cosmic variance and simply is a
statistical uncertainty at larger scales, so smaller l. It means that it is hard to unravel statistical meaning from large scales in the universe.
ξΘΘ(θ) = hΘ(ˆn)Θ(ˆn0)i = 1 4π ∞ X l=0 (2l + 1)ClPlcos (θ) (89)
Figure 9: The graph starts at l = 2, because l = 0 corresponds to the average temperature fluctuation over the whole sky. For l = 1, the motion of the earth across space should be taken into account. This means that redshifted photons from the CMB and blue shifted photons from the CMB create an anisotropy that will bury relevant cosmological signals. Therefore the signals from the monopole and dipole term are discarded. The maximum multipole moment is determined by the resolution of the data. In the case of figure 9, the maximum multipole moment, lmax= 2500. This corresponds to an angle of α = 2500π ' 0.00126 rad ' 0.07◦ (Ade et al.
(2015))
The temperature fluctuations of the CMB are plotted in figure 9. Both the angular scale and multipole moment are on the x-axis and the temperature fluctuations are on the y-axis. The uncertainty (large error bars) at large scales is the cosmic variance. The smaller the scales, the smaller the error on the temperature fluctuations.
The tilt and scalar power spectrum are measured by PLANCK (Ade et al. (2015)) using the above analysis. The values for those two parameters are given in (83).
6.1.2
Polarisation of the CMB
Thomson scattering is responsible for the polarisation of the CMB (Baumann (2009)). The polarisation contains information about the primordial fluctuations and can therefore be linked to inflation.
Quadrupole Anisotropy
The CMB becomes polarised due to Thomson scattering be-tween photons and electrons at recombination. The incident radiation on the electron has to be quadrupole anisotropic in order to cause polarisation (Kaplan et al. (2003)). Radiation from hotter regions have a slightly different polarisation than radiation from colder regions and are quadrupole anisotropic (Riotto (2010)). See figure 10.Figure 10: Radiation from hotter regions and radiation from colder regions have a slightly different polarisation and will lead to a net linear polarisation (Riotto (2010))
Figure 11: If the incident radiation is isotropic (1), it can not produce a net polarisation. Dipole anisotropies (2) can not produce a net polarisation, because after adding the vertical bars and horizontal bars, no net polarisation is found. The quadrupole anisotropy (3) leads to a polarisation. After adding the vertical bars and the horizontal bars separately, a small net polarisation is found (Mansouri & Brandenberger (2012))
Mansouri & Brandenberger (2012) explain that only the quadrupole anisotropy can cause po-larisation. Figure 11 shows why isotropies and dipole anisotropies can not create a net polarisation and that in the quadrupole anisotropy case a small net polarisation can be found.
Stokes parameters
In order to explore the polarisation of the CMB, a few mathematical tools are needed and can be found in Zaldarriaga (2003). First of all the anisotropy in the CMB is characterised by the intensity tensor, Iij = δII0, which is a function of the direction
in the sky, ˆn and two directions perpendicular to the direction in the sky, ˆe1 and ˆe2. The
polarisation of electromagnetic waves can be described by Stokes Parameters, which are defined by Q = (I11− I22)/4 and U = I12/2. The stokes parameter, V , is not taken into account, because
it can’t be generated by Thomson scattering. The representation of the polarisation is done by ‘bars’ on a 2D map and therefore the last stokes parameter, the magnitude P =pQ2+ U2, is
important. The orientation of the bars on the 2D map is given by α =12arctanUQ. In the ˆe1and
ˆe2 plane, rotational transformations are given by:
(Q ± iU )0(ˆn) = e∓2iψ(Q ± iU )(ˆn) (90) (Q ± iU ) are two spin variables and therefore standard spherical harmonics cannot be used. A generalisation is used7: (Q ± iU )(ˆn) = Σ
lma±2,lm±2Ylm. The polarisation field can now be
written more compactly:
aE,lm= −(a2,lm+ a−2,lm)/2 and aB,lm= i(a2,lm− a−2,lm)/2 (91)
Two real space coordinate independent quantities can be defined: E(ˆn) = Σl,maE,lmYlm(ˆn)
B(ˆn) = Σl,maB,lmYlm(ˆn)
(92) These are the so called E mode and B mode polarisations and completely specify the polarisa-tion field. In figure 12 the E mode and B mode patterns are illustrated. The E modes are curl-free and the B modes have a curl component. Note that the symmetries in E and B are different. Both are symmetric under rotations, but only E modes are invariant under parity transformations. B modes change sign under parity transformation.
7
Figure 12: The E mode and B mode representations (Krauss et al. (2010)). Different representations for certain values of the Stokes parameters. The V polarisation is not produced by Thomson scattering and will not be taken into account. The U and Q polarisation however will be important.
Implications for inflation
Scalar perturbations can only produce positive parity transfor-mations, so can only produce E mode polarisation. B mode polarisation is produced by tensor perturbations only (Kaplan et al. (2003)) and detection of a B mode polarisation pattern in the CMB can therefore directly be linked to the tensor perturbation spectrum and to the energy scale of inflation. See figure 13 for a intuitive and simplified scheme of inflation and observable parameters in the CMB that can be linked to inflation.Figure 13: This figure clearly shows the relation between inflation and different observable parameters. B-mode polarisation can only be created by tensor perturbations.
6.1.3
Polarisation power spectra
The mathematical power spectra descriptions for EE, TE and BB are: C(l)ET =aE∗ lma T lm C(l)EE=aE∗ lma E lm C(l)BB=aB∗lmaBlm (93)
TE and EE cross correlations
The TE and EE cross correlations are measured by Ade et al. (2015) and can be found in figure 14. These cross correlations can be used to reduce the degeneracy. The plots contain evidence for inflation and in particular in the TE cross correlation between 100 < l < 200. The anti-correlation is at scales that are outside the horizon at recom-bination, so a theory like inflation, where the comoving horizon shrinks, has to be considered. Further information about this map can be found in Baumann (2009)Figure 14: The dip between l=100 and l=200 is evidence for inflation. More information on this map can be found in Baumann (2009). The figure is from Ade et al. (2015). The EE cross correlation is used to reduce the degeneracy. The figure is from Ade et al. (2015)
BB cross correlation
Subject to future experiments is to measure the BB correlation. As described in the latter section, the BB correlation can only be produced by the tensor power spectrum as a result of gravitational waves. The signal of the BB power spectrum is expected to be very small and signals are distorted by foregrounds. BICEP claimed in 2013 (Ade et al. (2014a)) to have observed a B mode polarisation pattern caused by inflation in the CMB favouring the chaotic inflation model, but that claim was withdrawn after it appeared that the signal was distorted by astrophysical foregrounds. In 2014 a note was added to the original article, saying that the PLANCK results published new information on polarised dust emission and said: “more data are required to resolve the problem”.Most cosmological significant disturbances are because of reionisation and gravitational lens-ing. Reionisation distorts the power spectrum of the CMB and gravitational lensing changes the spatial pattern of the CMB polarisation, because of matter inhomogeneities in the universe. Fi-nallty galactic dust and synchrotron radiation can also cause foreground disturbances8. Figure
15 shows the BB mode polarisation measurement of Ade et al. (2014a) :
Figure 15: The expected BB mode signal for r=0.2 is shown with the dashed red line. The gravitational lensing is represented by the red curve. Different experiments and their ranges are given by the different colours. This plot is from Ade et al. (2014a) and shows that a value of r=0.2 is measured. This possible detection of B mode polarisation was later withdrawn. Ade et al. (2015) wrote in their paper that interstellar dust from the Milky Way galaxy was not taken into account. The upper bound calculated in this section is added for r = 0.149. For smaller r, the line will be lower.
From section 5.4 it is possible to predict the power spectrum for BB mode polarisation: ∆B = l(l + 1)C
BB (l)
2π (94)
Also the tensor-to-scalar ratio is important to recall: r ≡ ∆2T
∆2 S
. The constraint on the tensor-to-scalar ratio calculated in 5.4 is used and the fact that in slow roll inflation the gravitational wave spectrum is nearly invariant (Hu et al. (2003)). The B mode power spectrum is expected to peak at l ≈ 90◦. The middle value of the energy scale of inflation for a detectable tensor power spectrum, E ∼ 1016, is used for the calculation.
8
The quadrupole power in the CMB anisotropies is given by: ∆2T ∆2 S = 3.273·10E16GeV 4 . This leads to9: (∆Bmax)2≡ l(l + 1)CBB 2π ≈ (0.024)2 E 1016GeV 4 µK2 (95)
this means in particular that for r = 0.2 a value of hl(l+1)CBB
2π
i
≈ 0.013 is obtained, which matches with the curve in figure 15. For the bounds imposed in section 5.4 r = 0.149 an upper bound peak value ofhl(l+1)CBB
2π
i
≈ 0.0098 is found and for r = 10−6 a peak value ofhl(l+1)CBB
2π
i ≈ 6.6 · 10−8 is obtained. For smaller tensor-to-scalar ratios, the expected signal is even smaller.
6.1.4
Future experiments
Three types of experiments are planned for the future: ground based experiments, balloons and satellites. In the near future a limit of r = 0.1 might be reached and that value corresponds to monomial potentials for inflation according to Creminelli et al. (2015). The threshold of the trans-Planckian inflation,M∆φ
pl, corresponds to a value of r ∼ 2·10
−3as can be seen in section 5.4. Ground
based experiments that are planned are AdvACT (Advanced Atacama Cosmology Telescope), CLASS (Cosmology Large Angular Scale Surveyor), KeckBICEP (Background Imaging of Cosmic Extragalactic polarisation), Simons Array and SPT 3G (South Pole Telescope). These experiments will be able to measure r ∼ 10−2.
Balloons that are planned are EBEX 10K (E and B Experiment) and Spider. These will be able to measure r ∼ few · 10−2. Balloon-borne experiments are much cheaper than satellite experiments and have the advantage that they fly at an altitude that atmospheric absorption of radiation is avoided. Balloons can measure a smaller fraction of the sky for a shorter period of time though.
At last satellite missions that are planned or funded are COrE, CMBPol (EPIC-2m) and LiteBIRD. A detection of r ∼ 2 · 10−3 is achievable for these missions.
The great challenge is to discriminate between the signal and the noise. If the noise is accu-rately described, it can be removed from the data. A gravitational wave signal is homogeneous throughout the universe and it has Gaussian statistics, so that will help to characterise the noise. The knowledge of foregrounds however is very qualitative and therefore it is hard to discriminate. A tensor-to-scalar ratio of r ∼ 2 · 10−3 can only be measured if the noise can be reduced10.
If a value of r ∼ 2 · 10−3 can be reached, the upper limit of the gravitational wave signal will be within reach:
9The change to temperature units has to be made and the general expression from Riotto (2010) and Hu et al.
(2003) is used:hl(l+1)CBB 2π i1/2 ≈ 0.024 E 1016GeV 2
µK2, where E = 3.273 · 1016r1/4GeV which is derived in section 5.4 10
More details on future experiments and their challenges can be found in Creminelli et al. (2015) and Baumann et al. (2009)
l(l + 1)CBB
2π
≈ 1.3 · 10−4µK2 (96)
6.2
Direct detection of gravitational waves
Gravitational waves are local space time distortions. First of all the important detectable char-acteristics will be discussed, followed by how these signals can be found by a detector and in particular in the planned detectors (e)LISA, BBO, DECIGO and ALIA.
In order to explore the detection possibilities it is important to characterise the signal that we are able to measure. Inflationary gravitational waves are isotropic, unpolarised and stationary. The best way to characterise gravitational waves is through their frequency spectrum and the frequency spectrum can be used in different ways. First of all an amplitude of the GW background is a useful characterisation, hc(f ). The second way is to express the frequency power spectrum
in the spectral density of the ensemble average of the Fourier components, Sh(f ). The third
possibility is to express the power spectrum in the energy density per logarithmic interval of the frequency, h20ΩGW(f ).
6.2.1
Frequency power spectrum
The intensity of primordial gravitational waves can be expressed as a dimensionless quantity ΩGW(f ) =
1 ρc
dρGW
d ln f (97)
This clearly represents the energy density of gravitational waves, ρGW per logarithmic interval
of the frequency. h2
0 is included in the equation to get rid of the Hubble parameter uncertainty.
This uncertainty has nothing to do with the gravitational wave signal11: h2
0ΩGW(f )
The amplitude and spectral density of the ensemble average are important to find the effect on the detector. In section 5.3.3 the Fourier component of a gravitational wave is already discussed. The ensemble average can be found by the following (Maggiore (2000)):
D˜h+,×(f, Ω)˜h0+,×(f 0, Ω0)E= δ(f − f0) 1 4πδ 2( ˆΩ, ˆΩ0)δ +,×δ+,× 1 2Sh(f ) (98) Now a characteristic amplitude can be found with the help of (98) and the earlier expression for the gravitational wave in Fourier space:
hij = X λ=+,× Z ∞ −∞ df Z ˆ Ωλij( ˆΩ)˜hλ(f, ˆΩ)e−2πif t (99)
The difference between this expression and the one that is used in 5.3.3 is the angular depen-dence and the change to frequency instead of wavenumber. The ensemble average is then given by:
11H
hijhij = 2 Z ∞ −∞ df Sh(f ) = 4 Z ∞ 0 d(ln f )Sh(f )f (100)
Now the characteristic amplitude is defined as: hijhij ≡ 2
Z ∞
−∞
d(ln f )h2c (101)
The factor 2 is part of the definition because the plus and cross polarisations are added. It is clear that (101) is the characteristic amplitude per logarithmic scale. This leads to:
h2c= 2Sh(f )f (102)
To find the relation between the energy density of gravitational waves, h2
0ΩGW(f ) and the
characteristic amplitude h2c can be found in the following way. It is impossible to find the energy
carried within a certain wavelength according to Maggiore (2000), but only over a certain region in space within different wavelengths. The stress-energy tensor can be written:
TµνGW = 1 32πG D δµhT Tij δνhijT T E (103) The 00 component, the energy component, leaves us with the following expresion:
ρGW =
1
32πGδ0hijδ0h
ij
(104) Equation (99) and (100) can be filled into (103) to obtain:
ρGW = 1 32πG(2πf ) 2 hijhij = 4 32πG Z ∞ 0 d(ln f )Sh(f )f (2πf )2 (105) This expression can be rewritten into:
dρGW
d(ln f )=
πSh(f )f3
2G (106)
Finally the expression for the critical density and (97) can be used to write the intensity in terms of the characteristic amplitude or the spectral density:
ΩGW(f ) = 2π2 3H02f 2h2 c(f ) ΩGW(f ) = 4π2 3H2 0 f3Sh(f ) (107)
6.2.2
Inflationary gravitational wave background (IGWB)
From (107) the intensity of inflationary gravitational waves can be written in the following form: ΩGWh20= k2h20 6H0 |hk|2 (108)
Where it is used that k = 2πf . The evolution of inflationary gravitational waves after horizon entry is determined by the transfer function derived by Smith et al. (2006):
T = 2.1 · 10−20
k
6.5 · 1013Mpc−1
−1
g−1/6100 (109) Where g100∼ g∗(Tk)/10012. Equation (109) is scaled to the peak frequency of the BBO/DECIGO
experiment: f = 0.1Hz or corresponding to k = 6.47 · 1013Mpc−1. The root of the variance of the inflationary gravitational wave background today is equal to the product of the tensor power spectrum and the transfer function: |hk|2
1/2
= ∆TT (k). The parameter k is taken to be the
wavenumber at horizon crossing: k = ak
a0Hk =
ak
a0
1.66g1/2∗sTk2
Mpl and because during the radiation
dominated era a ∝ T−1 the following expression for k can be written: k = 1.66g
1/2 ∗s TkT0
Mpl
(110) According to Smith et al. (2006) the energy density of the IGW spectrum can be written:
ΩGWh20= k2h20 6H0 ∆2TT2= 2.74 · 10−6g−1/3 100 ∆ 2 T (111)
The PLANCK data (Ade et al. (2015)) can be used to obtain an upper bound for ΩGWh20.
The constraints on the tensor-to-scalar ratio r ≤ 0.149 and accurate measurements of the scalar power spectrum ∆2
S ' (4.697 · 10−5)2= 2.21 · 10−9 are used to obtain:
∆2T = r∆2S ' 3.29 · 10−10 (112) The upper bound for the energy density reads13:
ΩmaxGWh20≈ 9 · 10−16 (113)
6.2.3
Detector sensitivity
The signal that is detected contains two pieces of information. It contains the relevant gravita-tional background signal, but also background noise. This can be expressed straight forward in the following way:
S(t) = s(t) + n(t) (114)
The signal-to-noise ratio must be greater than unity in order to be able to observe a relevant background (Maggiore (2000) and Smith et al. (2006)). First the ensemble average over the Fourier components of the noise is taken:
12
g∗ is the effective number of relativistic degrees of freedom. g∗(T ) ≈ 100 (Smith et al. (2006)) and therefore g100
will be close to one
13g−1/3
100 should be added to the expression, but the value is close to one and therefore will not change the order of
n2 Z ∞
0
df Sn(f ) (115)
Sn is a measure for the square spectral noise density. The strain sensitivity, the noise level of
the detector, can now be defined as:
˜ hf =
p
Sn (116)
A background can now be detected if: Sh>
Sn
F (117)
F is a factor to correct for the fact that not the entire sky is measured, but only a small part of it. Equation (117) can be rewritten using (102) into the following form:
|hc(f )|2 1/2 > 2f Sn F 1/2 (118) The expression for the the energy density of inflationary gravitational waves can be rewritten into the strain sensitivity at f = 0.1Hz by using (107):
˜
hf≈ 5.2 · 10−25 (119)
6.2.4
Future experiments
Future experiments for detecting a gravitational wave background originating from inflation are the space interferometer (e)LISA, BBO (Big Bang Observer), DECIGO (Deci-hertz Interferometer Gravitational wave Observatory) and ALIA (Advanced Laser Interferometer Antenna) as can be seen in figure 16. It is clear that all detectors are unlikely to detect a stochastic background. The energy density of inflationary gravitational waves is smaller than the detectors can reach.
In (113) an upper bound is found for the stochastic background ΩGWh20. The current designs
for different experiments probe the following sensitivities at a frequency of f = 0.1Hz:
Detector Energy density ΩGWh20 Strain sensitivity ˜hf
LISA 10−12 10−23
BBO 10−14 10−24
DECIGO 10−13 10−24
ALIA 10−11 10−23
The values are taken from Smith et al. (2006) (LISA, BBO, ALIA and DECIGO), from Yagi & Seto (2011) (DECIGO), from Crowder & Cornish (2005) (ALIA and LISA) and from Moore
et al. (2015). A special alignment of BBO and DECIGO together can lower this limit to 10−15 or even 10−16 according to Yagi & Seto (2011). The energy density and strain are evaluated at f = 0.1Hz in currently planned mission, but future experiments might be able to explore lower frequency ranges, where the energy density is higher. If the same sensitivity can be reached at lower frequencies, the detection of an inflationary gravitational wave signal might be possible.
Figure 16: This figure shows the sensitivity of a number of experiments as a function of the frequency. The coloured lines show the sensitivity of the different experiments. The DECIGO/BBO cross correlation is not added, but will come closer to the expected upper bound for gravitational wave energy density at f = 0.1Hz: Ωmax
GWh 2
0= 9.01 · 10−16. All experiments in this figure are unlikely to observe an inflationary gravitational wave
background. Long term DECIGO/BBO cross correlations are more likely to detect an IGWB in the future for it will reach a sensitivity of of 10−15or even 10−16. In comparison the LIGO experiment is added to the figure. Both the frequency and energy spectrum sensitivity for LIGO are no where near the possible detection of an inflationary gravitational wave background. For other frequencies, the inflationary gravitational wave energy density is higher. Figure adapted from Moore et al. (2015)
