Supersymmetry. Theory, Experiment, and Cosmology

Friedmann–Robertson–Walker Universes 459

A reference energy density at present time t0 is obtained from the Friedmann equation for vanishing cosmological (λ = 0) and flat space (k = 0):

This corresponds to approximately one galaxy per Mpc3 or 5 protons per m3. In fundamental units where h¯ = c = 1, this is of the order of 10−3eV 4. In the case of a vanishing cosmological constant, it follows from (D.27) that, depending on whether the present energy density of the Universe ρ0 is larger, equal or smaller than ρc, the present Universe is spatially open (k > 0), flat (k = 0) or closed (k < 0). Hence the name critical density for ρc.

It has become customary to normalize the di erent forms of energy density in the present Universe in terms of this critical density. Separating the energy density ρM 0 presently stored in nonrelativistic matter (baryons, neutrinos, dark matter, etc.) from the density ρR0 presently stored in radiation (photons, relativistic neutrino, if any), one defines:

The last term comes from the spatial curvature and is not strictly speaking a contribution to the energy density.

Then the Friedmann equation taken at time t0 simply reads

Since matter dominates over radiation in the present Universe, we may neglect ΩR in the preceding equation. Using the dependence of the di erent components with the scale factor a(t), one may then rewrite the Friedmann equation at any time as:

where a0 is the present value of the cosmic scale factor and all time dependences have been written explicitly. We note that, even if ΩR is negligible in (D.33), this is not so in the early Universe because the radiation term increases faster than the matter term in (D.34) as one gets back in time (i.e. as a(t) decreases). If we add an extra component X with equation of state pX = wX ρX , it contributes an extra term

ΩX (a0/a(t))3(1+wX) where ΩX = ρX /ρc.

Important information about the evolution of the Universe at a given time is whether its expansion is accelerating or decelerating. This is obtained by studying the

460 An introduction to cosmology

second derivative of the cosmic scale factor, which is easily extracted from (D.25) and (D.26):

The acceleration of our Universe is usually measured by the deceleration parameter q which is defined as:

Using (D.35) and separating again matter and radiation, we may write it at present time t0 as:

Once again, the radiation term ΩR can be neglected in this relation. We see that in order to have an acceleration of the expansion (q0 < 0), we need the cosmological constant to dominate over the other terms. We can also write the deceleration parameter (D.36) at a given time t as

If we introduce an extra component X as above, it contributes a term ΩX (a0/ a(t))3(1+wX )(1 + 3wX )/2: only components with equation of state parameter wX <

−1/3 tend to accelerate the expansion of the Universe.

The measurement of the Hubble constant and of the deceleration parameter allows us to obtain the behavior of the cosmic scale factor in the last stages of the evolution of the Universe:

D.2.2 Measure of distances

Measuring cosmological distances allows us to study the geometry of spacetime. Depending on the type of observation, one may define several distances.

In an expanding or contracting Universe, the light emitted by a distant source undergoes a frequency shift which gives a direct information on the time dependence of the cosmic scale factor a(t). To obtain the explicit relation, we consider a photon propagating in a fixed direction (θ and φ fixed). Its equation of motion is given as in special relativity by setting ds2 = 0 in (D.18):

Friedmann–Robertson–Walker Universes 461

Thus, if a photon (an electromagnetic wave) leaves at time t a galaxy located at distance r from us, it will reach us at time t0 such that

The electromagnetic wave is emitted with the same amplitude at a time t + T where the period T is related to the wavelength of the emitted wave λ by the relation λ = cT . It is thus received with the same amplitude at the time t0 + T0 given by

One may thus replace time by redshift since time decreases monotonically as redshift increases. For example, the expression of the Hubble parameter in (D.34) can be turned into

Friedmann–Robertson–Walker Universes 463

Several distance measurements tend to point towards an evolution of the present Universe dominated by the cosmological constant contribution3 and thus a late acceleration of its expansion.

The approach that has made the first strong case for such an hypothesis uses supernovae of type Ia as standard candles4. Two groups, the Supernova Cosmology Project [88] and the High-z Supernova Search [140] have found that distant supernovae appear to be fainter than expected in a flat matter-dominated Universe. If this is to have a cosmological origin, this means that, at fixed redshift, they are at larger distances than expected in such a context and thus that the Universe is accelerating its expansion.

More precisely, one uses the relation (D.48) between the flux φ received on Earth and the luminosity L of the supernova. Traditionally, flux and luminosity are expressed on a log scale as apparent magnitude mB and absolute magnitude M (magnitude is −2.5 log10 luminosity + constant). The relation then reads

The last terms are z-independent, if one assumes that supernovae of type Ia are standard candles; they are then measured by using low z supernovae. The first term, which involves the luminosity distance dL, varies logarithmically with z up to corrections which depend on the geometry, more precisely on q0 = ΩM /2 − ΩΛ for small z as can be seen from (D.51). This allows us to compare with data cosmological models with di erent components participating to the energy budget, as can be seen from Fig. D.1.

This can be turned into a limit in the ΩM − ΩΛ plane for the model considered here (see Fig. D.2).

Of course, such type of measurement is sensitive to many possible systematic e ects (extra corrections besides the phenomenological stretch factor applied to the lightcurves, presence of dust, etc.), and this has fueled a healthy debate on the significance of supernova data, as well as a thorough study of possible systematic e ects by the observational groups concerned.

Let us note that the combination ΩM /2 − ΩΛ is ‘orthogonal’ to the combination ΩM + ΩΛ measured in CMB experiments (see Section D.3.4). The two measurements are therefore complementary: this is sometimes referred to as ‘cosmic complementarity’.

Other results come from gravitational lensing. The deviation of light rays by an accumulation of matter along the line of sight depends on the distance to the source (D.50), and thus on the cosmological parameters ΩM and ΩΛ. As q0 decreases (i.e. as the Universe accelerates), there is more volume and thus more lenses between the observer and the object at redshift z. Several methods are used: abundance of multiplyimaged quasar sources [255], strong lensing by massive clusters of galaxies (providing multiple images or arcs) [356], and weak lensing [286].

3At least when analyzed in the framework of the model discussed in this section, i.e. including nonrelativistic matter, radiation and a cosmological constant. As discussed in Section 12.2.2 of Chapter 12, the cosmological constant may be replaced by a dynamical component.

4By calibrating them according to the timescale of their brightening and fading.

464 An introduction to cosmology

Type Ia Supernovae

Fig. D.1 Hubble plot (magnitude versus redshift) for Type Ia supernovae observed at low redshift by the Calan–Tololo Supernova Survey and at moderate redshift by the Supernova Cosmology Project.

D.2.3 Age of the Universe

Since 1 + z = a0/a(t), we can write (D.45) as

(D.54) This is easily integrated to obtain the time–redshift relation. Sending z to infinity (or, for that matter, to the value of z corresponding to nucleosynthesis, up to which we think we understand the evolution of the Universe), we obtain the age of the Universe:

(we have set t = 0 at the Big Bang singularity; hence t0 is the age of the Universe). One may neglect the radiation dominated era since it is short on the scale of the age of the Universe (see Table D.1 below). One can check, using (D.33) to express Ωk, that t0 increases with ΩΛ at fixed ΩM . Thus a nonvanishing cosmological constant helps to solve what was once known as the “age of Universe crisis”: the Universe should be older than the oldest globular clusters in the Milky way whose age is estimated at 10 Gyr. Present data on (ΩM ,ΩΛ) favors an age of the Universe around 15 Gyr.

466 An introduction to cosmology

Table D.1 The di erent stages of the cosmological evolution in the standard scenario, given in terms of time t since the Big Bang singularity, the energy kT of the background photons, and the redshift z. The line following nucleosynthesis indicates the part of the evolution which has been tested through observation. The values (h0 = 0.7, ΩM = 0.3, ΩΛ = 0.7) are adopted to compute explicit values.

where g is the number of internal degrees of freedom (g = 1 for a neutrino or an antineutrino and g = 2 for a massless vector field such as the photon, for an electron or a positron). We will need the explicit form in the relativistic limit (kT m):

p = ρ/3,

where the parenthesis (· · · ) indicates the extra factor to be taken into account in the

F

case of the Fermi–Dirac distribution and aBB is the blackbody constant (we restore here the powers of c):

For example we have at present time (T0 = 2.725 K) nγ0 = 411 cm−3.

In the nonrelativistic limit (kT m), n and ρ = mn are exponentially small:

The hot Big Bang scenario 467

We conclude that, as long as some species remain relativistic, the matter energy density

where we have taken into account the possibility that the species i may have a thermal distribution at a temperature Ti di erent from the temperature T of the photons. If radiation dominates the energy density of the Universe, one obtains from (D.27)

It is possible to show, using the second law of thermodynamics (T dS = dE + pdV ),

and that the entropy in a covolume sa3 remains constant. The entropy density is dominated by relativistic particles and reads

where the sum extends only to the species in thermal equilibrium. Note that, when all species have the temperature of the photons T , which is true through most of the history of the Universe, gs = g .

We deduce from the constancy of sa3 that gs (aT )3 remains constant. Hence the temperature T of the Universe behaves as a−1 whenever gs remains constant. This is so except when some species drop out of equilibrium. Indeed, a given species drops out of equilibrium when its interaction rate Γ drops below the expansion rate H. We will return to this question in Section D.3.3 and see for example that neutrinos decouple at temperatures below 1 MeV. Their temperature continues to decrease as a−1 and thus remains equal to T . However when kT drops below 2me, electrons annihilate against positrons with no possibility of being regenerated and the entropy of the electron– positron pairs is transferred to the photons. Since gs|γ,e± = 2 + 4 · 7/8 = 11/2 and gs|γ = 2, the temperature of the photons becomes multiplied by a factor (11/4)1/3. Since the neutrinos have already decoupled, they are not a ected by this entropy

We can now compute the value of gs for temperatures much smaller than me: gs = 2 + (7/8)6(4/11) = 3.91. We deduce that, at the present time (T0 = 2.725 K), s0/k = 2890 cm−3.