1.3 The Cosmological Standard Model and Extensions
1.3.4 Inflation
We present two problems of the standard model namely the horizon and flatness problem and show that a very short period of accelerated expansion in the early Universe, a so-called inflationary phase, can solve both problems.
Horizon Problem
The comoving distance of a photon emitted at time t1 and received at t2 is rh(t1, t2) =
Z t2
t1
cdt
a(t). (1.55)
As information cannot be exchanged faster than the speed of light, the comoving distance determines the maximal distance between a region at time t1 and a region at time t2 to have been in causal contact. Therefore, if the integral is finite photons can only propagate a finite distance and if the integral diverges everything is in causal contact.
We define the particle horizon as the limit t1 →0 and t2 ≡ t which is the comoving distance a photon could have traveled since the Big Bang. On the other hand, the event horizon is the largest distance a photon can travel starting at time tuntil infinity, where t1 ≡t and t2 → ∞. For our purposes only the particle horizon is interesting. Rewriting the particle horizon in terms of the scale factor yields
rh(a) = Z a
0
c
H(a0)a02 da0. (1.56)
We find a simple solution for this integral considering aeq a 1, which is the matter-dominated era well before dark energy and curvature domination:
rh(a) = 2 c H0
r a
Ωm ' 6000
√Ωmzh−1Mpc. (1.57)
We can apply this approximation to calculate the size of the horizon at the time of the origin of the CMB radiation. We find for z ≡zrec '1100 a comoving distance of the order 100Mpc, which corresponds to an angle of 1◦ on the sky. However, today we measure the temperature of the CMB to be nearly completely isotropic although huge parts of the CMB photons were never in causal contact. This is the well known horizon problem which is a generic feature of the hot Big Bang model.
Flatness Problem
We can rewrite the Friedmann equation (1.22) in the following form:
|Ωtot(a)−1|= Kc2
a2H2(a), (1.58)
where Ωtot(a) is the total density parameter, i.e., the sum of the individual scale-dependent density parameters. We note that if the Universe is flat at some time, where Ωtot = 1, it will remain flat for all subsequent time. In the matter-dominated era the time dependence of the scale factor is a(t)∝t2/3 and thus we find
|Ωtot(t)−1| ∝t2/3. (1.59)
The deviation of Ωtot from unity is a growing function of cosmic time. Today, we measure Ωtot = 1.011(12) (Amsler et al. 2008) which is close to spatial flatness. Hence, for example at nucleosynthesis when the Universe was 1sold the deviation from flatness is
|Ωtot(tnuc)−1|.10−16. (1.60) Going to earlier times, the deviation from flatness is even smaller. This is the flatness problem, where the initial conditions set up after the Big Bang need to be extremely fine-tuned to reach the present day flatness.
The Inflationary Mechanism
The inflationary mechanism provides a solution to the horizon and flatness problem as shown in the following. At first, we rewrite the comoving horizon in the form:
rh(a)∝ Z a
0
da0 a0
1
pρa02 . (1.61)
To solve the horizon problem described above, the integral needs to diverge. This is the case when the product ρa2 is an increasing or constant function of the scale factor at early times. However, the two standard energy density forms of radiation ρ∝a−4 and matter ρ ∝a−3 do not fulfill this condition. Hence, we need to propose another unknown form of energy density dominating at early times to solve the horizon problem.
Using Eq. (1.18), we see that such an energy density needs to have an equation of state of w <−1/3 implying a negative pressure. Then we obtain from the second Friedmann equation(1.13) that¨a >0, i.e., accelerated expansion. Thus, we say at very early times the Universe underwent a phase of inflationary expansion.
Another equivalent way to define inflation is that the comoving Hubble radius is a decreasing function of cosmic time:
d dt
c H(a)a
<0. (1.62)
This is exactly the condition to solve the flatness problem as the right-hand side of Eq. (1.58) gets smaller with increasing cosmic time.
A scalar fieldφ naturally fulfills the condition of negative pressure needed to start inflation, where the scalar field is sometimes known as the inflaton. Because inflation starts at very early times, the energy at that time is well beyond the energy range
accessible to particle physics colliders. Hence, we can only try to match the predictions of inflation with cosmological observations to verify the theory. The Lagrangian of a scalar field φ with potential V(φ) is given by
L= 1
2∂µφ∂µφ−V(φ), (1.63)
where the first term is the kinetic energy and the second term the potential energy.
From the Euler-Lagrange equations, we obtain the equation of motion of the scalar field φ¨+ 3Hφ˙+ dV
dφ = 0, (1.64)
where we assumed a homogeneous field that only depends on time, i.e., φ =φ(t). The energy density and pressure are obtained from the energy-momentum tensor
Tµν =∂µφ∂νφ−gµνL. (1.65) In the end, we find
ρφ= φ˙2
2 +V(φ), pφ = φ˙2
2 −V(φ). (1.66)
Hence, we infer the important result that as long as φ˙2/2V(φ), we find ρφ =−pφ leading to the desired accelerated expansion. This condition is for example fulfilled by a scalar field that is trapped in a false vacuum which is the original interpretation of inflation developed by Guth (1981). However, in this model the field never attains the true vacuum necessary for ending the phase of inflation. To avoid this problem, subsequent models proposed that the field is slowly rolling down its potential to the true vacuum. This scenario also fulfills the condition that the kinetic energy of the field is much smaller than the potential energy. In addition, the equation of motion of a time-varying field in Eq. (1.64) has to satisfy the slow roll conditions:
= m2pl 16π
V0 V
2
1, η= m2pl 8π
V00 V
1, (1.67)
where a prime denotes the derivative of the potential with respect to φ and mpl is the Planck mass given by
mpl= r ~c
GN ≈2.43×1018GeV/c2. (1.68) Besides solving the horizon and flatness problem inflation provides a mechanism to explain the origin of tiny density fluctuations that were present in the early Universe and which built the seeds of structure formation. In fact, quantum fluctuations of the inflaton field are amplified during inflation such that at the end of inflation the perturbations are sufficiently large to grow by gravitational instability. The next chapter will describe in detail how the large-scale structure we observe today is formed out of these fluctuations.
Chapter 2
Cosmological Perturbation Theory and Correlation Functions
Cosmological linear perturbation theory is one of the pillars for predictions of the temperature and polarization anisotropies of the cosmic microwave background. This is due to the fact that the temperature fluctuations are only of the order 10−5 and are thus well modeled by a linear perturbative approach. However, probes of the large-scale structure of the Universe like cosmic shear and galaxy redshift surveys use scales that are also in the nonlinear regime. A lot of effort was put into a perturbative approach that is still accurate in the mildly nonlinear or quasilinear regime (see the comprehensive review by Bernardeau et al. 2002, and references therein). Nevertheless, the nonlinear regime cannot be accessed with a perturbative approach since the description of the fluid equations breaks down. Several approaches try to cure this problem. Dark matter N-body simulations start from a Gaussian distribution of the density field and then evolve the density and velocity fields according to the Vlasov equation (which we introduce below). The caveat is that they are computationally costly and lack a physical interpretation of the process of structure formation via gravitational instability. They are also limited by the resolution of the simulation. Due to the large computational requirements, fitting functions were developed to interpolate between simulation results (most notably the fitting formulas for the dark matter power spectrum by Peacock
& Dodds 1996; Smith et al. 2003). Another approach is the dark matter halo model which combines results from simulations for dark matter halos and theoretical models for gravitational clustering. We will discuss this model in detail in the next chapter.
Recently, we have seen a revival of perturbation theory mainly due to the feature of baryonic acoustic oscillations observed in the galaxy two-point correlation function (Eisenstein et al. 2005). The so-called baryonic acoustic peak is important at scales which essentially lie in the quasilinear regime. This regime can be modeled within 1% accuracy (Jeong & Komatsu 2006) using perturbation theory. In addition, the importance of baryonic acoustic oscillations to break parameter degeneracies has led to a remarkable progress in perturbation theory, e.g., in the development of renormalized perturbation theory and the renormalization group approach (Crocce & Scoccimarro 2006; McDonald 2007). Furthermore, these techniques may help to find a model which is also applicable in the nonlinear regime.
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The outline of this chapter is as follows: In Sect. 2.1, we derive the basic equations governing cosmological structure formation and present a general perturbative solution scheme to these nonlinear equations. We present the concept and properties of n-point correlation functions in Sect. 2.2. In particular, we show the perturbative results for the two-, three- and four-point correlation function in lowest order of the density field. As perturbation theory breaks down in the nonlinear regime, we give fitting functions for the dark matter power spectrum and the dark matter bispectrum in Sect. 2.3 . Finally, in Sect. 2.4 we show the effect of radiation on structure formation, and introduce the transfer function which describes the transition of the radiation-dominated era to the matter-dominated era.
2.1 Perturbation Theory
This section provides the basic framework of cosmological perturbation theory in the matter-dominated regime. First, we derive the fundamental equations governing the process of structure formation, namely the fluid equations, from the collisionless Boltzmann equation which describes the conservation of the number of dark matter particles in a given phase-space element. The resulting fluid equations are highly nonlinear and are in general not analytically solvable. Nevertheless, it is possible to find a closed solution in the linear regime. For the nonlinear regime we apply a perturbative ansatz around this linear solution. Most of the presented results and further issues can be found in the review paper by Bernardeau et al. (2002) which provides a thorough introduction to this field.