able to reach this limit.

Figure 19: Sensitivities of some running and planned direct detection dark
matter experiments to the spin-independent elastic scattering cross-section.
Full curves correspond to limits from existing experiments, dashed curves to
predicted sensitivities of future experiments. The full brown, pink, blue and
yellow regions correspond respectively to the regions allowed by potential
signals observed by the DAMA, CREST, CDMS and CoGeNT experiments. The light
red disk corresponds to the region favoured by supersymmetric models. The
thick yellow line corresponds to the irreducible neutrino background.

Dark matter particles may also be searched for through their annihilation
products in massive celestal objects. This is known as indirect detection.
Indeed, because they are massive, they tend to accumulate in gravitational
potentials, such as the centre of the Sun, or the centre of our Galaxy. They
annihilate there into pairs of energetic particles, the energy of which is
directly connected with the mass $m_{X}$ of the dark matter particles. One may
therefore search for excess of energetic particles (positrons, photons,
neutrinos) in the direction of galactic centres such as in our own Milky Way.
An excess of energetic positrons (energy of a few tens to few hundred GeV) has
actually been observed by the PAMELA, Fermi and AMS-02 experiments. It remains
to be seen if this arises from the annihilation of dark matter or from
astrophysical sources, such as pulsars: we have seen at the end of Section 0.3
that a certain number of astrophysical sources produce energetic particles.
This is indeed a limitation, at least at present, of the indirect detection of
dark matter: it needs to be complemented by either direct detection or
detection at colliders.

###  0.4.3 WIMPs and physics beyond the Standard Model

We would like to stress in this section that the presence of a WIMP in a
theory is deeply connected with the naturalness of the electroweak scale.

Let us start by recalling what is the naturalness problem (see for example
[Binetruy]). As is well-known, the Higgs squared mass $m_{h}^{2}$ receives
quadratically divergent corrections. In the context of an effective theory
valid up to a cut-off scale $\Lambda$ where a more fundamental theory takes
over, $\Lambda$ is the mass of the heavy degrees of freedom of the fundamental
theory. Their contribution in loops, quadratic in their mass, destabilizes the
Higgs mass and thus the electroweak scale ($m_{h}^{2}\sim\lambda v^{2}$ where
$\lambda$ is the scalar self-coupling and $v\sim
1/(G_{{}_{F}}\sqrt{2})^{1/2}\sim 250\ \hbox{GeV}$ is the Higgs vacuum
expectation value. More precisely, we have at one loop

$\delta m_{h}^{2}={3m_{t}^{2}\over
2\pi^{2}v^{2}}\Lambda_{t}^{2}-{6M_{{}_{W}}^{2}+3M_{{}_{Z}}^{2}\over
8\pi^{2}v^{2}}\Lambda_{g}^{2}-{3m_{h}^{2}\over 8\pi^{2}v^{2}}\Lambda_{h}^{2}\
,$ (164)

where for completeness we have assumed different cut-offs for the top loops
($\Lambda_{t}$), the gauge loops ($\Lambda_{g}$) and the scalar loops
($\Lambda_{h}$) [BHR06]. The naturalness condition states that the order of
magnitude of the Higgs mass is not destabilized by the radiative corrections
i.e. $\left|\delta m_{h}^{2}\right|<m_{h}^{2}$. This translates into the
conditions:

$\displaystyle\Lambda_{t}$ $\displaystyle\sim$ $\displaystyle\sqrt{{2\over
3}}\ {\pi v\over m_{t}}m_{h}\sim 3.5m_{h}\ ,$ (165) $\displaystyle\Lambda_{g}$
$\displaystyle\sim$ $\displaystyle{2\sqrt{2}\pi
v\over\sqrt{6M_{{}_{W}}^{2}+3M_{{}_{Z}}^{2}}}m_{h}\sim 9m_{h}\ ,$ (166)
$\displaystyle\Lambda_{h}$ $\displaystyle\sim$ $\displaystyle{2\sqrt{2}\pi
v\over\sqrt{3}}\sim 1.3\ \hbox{TeV}\ .$ (167)

Thus one should introduce new physics at a scale $\Lambda_{t}\sim 3.5m_{h}$.
We will illustrate our argument with two examples: supersymmetry and extra
dimensions. In the two cases, one introduces new physics at the scale
$\Lambda_{t}$ (supersymmetric particles or Kaluza-Klein modes).

Typically, these models require the presence of a symmetry that prevents
direct coupling between the Standard Model (SM) fermions and the new fields
that one has introduced: otherwise, such couplings introduce new mixing
patterns incompatible with what is observed in flavor mixings (compatible with
the Standard Model). This symmetry is usually a parity (i.e. a discrete
symmetry) which is the low energy remnant of a continuous symmetry which
operates at the level of the underlying fundamental theory: SM fermions are
even under this parity whereas the new fields are odd. Among these new fields,
the lightest odd-parity particle (we will refer to it as the LOP) is stable:
it cannot decay into SM fermions because of the parity; it cannot decay into
the new fields because it is the lightest. It is massive and weakly
interacting. It thus provides an adequate candidate for a WIMP.

Let us take our examples in turn. In the case of supersymmetry, the parity
operation is R-parity (which usually proceeds from a continuous R-symmetry
broken by gaugino masses i.e. supersymmetry breaking). And the LOP is the
Lightest Supersymmetric Particle, the famous LSP, the lightest neutralino in
the simplest models.

In the case of extra dimensions, say a $5$-dimensional model, the local
symmetry is $5$-dimensional Lorentz invariance. It ensures conservation of the
Kaluza-Klein levels: if $A^{(n)}$ is the $n$th Kaluza-Klein mode of the
massless $5$-dimensional field $A$ (in other words, the $4$-dimensional field
with mass $m=n/R$, where $R$ is the radius of the $5$th dimension), then in
the reaction $A^{(n)}+B^{(p)}\rightarrow C^{(q)}+D^{(r)}$, we have $n+p=q+r$.
At energies smaller than $R^{-1}$, this turns into a Kaluza-Klein parity
$(-1)^{n}$. The LOP is then the lightest Kaluza-Klein mode, usually $B^{(1)}$,
the first mode of the $U(1)_{Y}$ gauge boson [ST02].

In realistic models, there is often the possibility that other odd-parity
fields are almost degenerate in mass with the LOP. This leads to the
possibility of co-annihilations, that is annihilations of the LOP against
these almost degenerate fields, and to a modification of the relic density in
the corresponding region of parameter space.

Figure 20: Upper panel: Contours in a parameter space of supersymmetry models
in the plot ($m_{0}$, $m_{1/2}$) for the discovery of the missing energy plus
jets signature of new physics by the ATLAS experiment at the LHC. The three
sets of contours correspond to levels of integrated luminosity at the LHC (in
fb-1), contours of constant squark mass, and contours of constant gluino mass
[To02].
Lower panel: Exclusion limits at $95\%$ CL for $8$ TeV, $20$ fb-1 integrated
luminosity analyses in the ($m_{0}$, $m_{1/2}$) plane for the MSUGRA/CMSSM
model with the remaining parameters set to $\tan\beta=30,A_{0}=-2m_{0},\mu>0$.
Part of the model plane accommodates a lightest neutral scalar Higgs boson
mass of 125 GeV [ATLAS_SUSY13].

Searches at LHC are based on the missing energy signal corresponding to the
LSP (see Fig. 20). Since LSP are produced in pairs, they are difficult to
reconstruct in all generality. But, in the case of a specific model, one may
be able to reconstruct the mass of the LSP as well as the relic density.

###  0.4.4 Other candidates for dark matter

Among the many other candidates proposed for dark matter, one may single out
the axion field since it is introduced to solve one puzzle of the Standard
Model known as the strong CP problem, which remains to be solved. Moreover,
from the point of view of cosmology, this is an interesting illustration of a
low-mass and wery weakly interacting particle, as we will encounter in the
next Chapter.

As we have already seen, CP symmetry is violated in weak interactions. But the
symmetries of the Standard Model allow as well a CP-violating term in the QCD
Lagrangian $(\theta g^{2}/32\pi^{2})G^{a}_{\mu\nu}\tilde{G}^{a\mu\nu}$, where
$G^{a}_{\mu\nu}$ is the gluon field strength and $g$ the QCD gauge coupling.
For non-zero quark masses, this term leads to unobserved CP-violating effects
in the strong sector191919To be more precise, it is
$\bar{\theta}\equiv\theta-\hbox{arg det}\ m_{q}$, where $m_{q}$ is the quark
mass matrix, which is observable. If $\bar{\theta}\not=0$, then strong
interactions violate P and CP. This is not compatible with the experimental
upper bound on the neutron electric dipole moment unless
$\left|\bar{\theta}\right|<10^{-10}$..

The most common way to solve the puzzle is to introduce a scalar field $a(x)$
called axion, with Lagrangian

${\cal L}_{a}={1\over 2}\partial^{\mu}a\partial_{\mu}a+{g^{2}\over
32\pi^{2}}{a(x)\over f_{a}}G^{a}_{\mu\nu}\tilde{G}^{a\mu\nu}\ ,$ (168)

where $f_{a}$ is an energy scale, called the axion decay constant (by analogy
with the pion decay constant). Strong interactions generate an effective
potential for $a(x)$ whose minimum corresponds to no violation of CP202020i.e.
a vanishing value of $\bar{\theta}=a(x)/f_{a}-\hbox{arg det}\ m_{q}$. The non-
renormalisable interaction $aG\cdot\tilde{G}$ is obtained as the low energy
effect of the breaking of a $U(1)$ global symmetry, known as the Peccei-Quinn
symmetry [PQ77a, PQ77b], spontaneously broken at the scale $f_{a}$: the axion
is the pseudo-Goldstone boson associated with this breaking [We78,
Wi78]212121The axion is not a true Goldstone boson but a “pseudo-Goldstone”
boson because the QCD vacuum which involves non-vanishing values for the quark
condensates such as $\langle\bar{u}u\rangle$ and $\langle\bar{d}d\rangle$
indices a small explicit breaking of the $U(1)$ symmetry, hence a mass for the
otherwise massless Goldstone boson associated with the spontaneous breaking..

The axion mass is

$m_{a}\sim 6.10^{-6}\ \hbox{eV}\left({10^{12}\hbox{GeV}\over f_{a}}\right)\ .$
(169)

Its coupling to ordinary matter is proportional to $1/f_{a}$ and can be
calculated in specific models. It couples to leptons and to photons, the
latter being of the form

${\cal L}_{a\gamma\gamma}=-g_{a\gamma}{\alpha\over\pi}{a(x)\over f_{a}}\ {\bf
E}.{\bf B}\ ,$ (170)

where ${\bf E}$ and ${\bf B}$ are the electric and magnetic fields, $\alpha$
is the fine structure constant and $g_{a\gamma}$ is a model-dependent
coefficient of order $1$. Moreover, since $m_{a}\ll\Lambda_{QCD}$, the axion
coupling to quarks should be described through its coupling to hadrons, which
occurs through small mixing with the $\pi_{0}$ and $\eta$ mesons. All of these
interactions can play a role in searches for the axion, and allows the axion
to be produced or detected in the laboratory and emitted by the sun or other
stars. Its non-discovery leaves us with an axion window $10^{-6}$ eV
$<m_{a}<3.10^{-3}$ eV, or correspondingly, $2.10^{9}$ GeV $<f_{a}<6.10^{12}$
GeV.

Let us describe briefly the cosmology of the axion field (see the review by P.
Sikivie [Si08] for a more thorough treatment). The breaking of the $U(1)$
symmetry corresponds to a phase transition, known as the Peccei-Quinn phase
transition, at a temperature of order $f_{a}$. This phase transition is
characterized by the formation of comic strings.

If the reheat temperature after inflation is smaller than $f_{a}$, then one
starts the evolution in the reheated universe with an homogeneous axion field.
When the temperature reaches the QCD scale, the effective potential turns on
and the axion acquires a mass. At a time $t_{*}\sim m_{a}^{-1}$ ($T_{*}\sim 1$
GeV), the axion starts to oscillate around its minimum. These oscillations do
not dissipate into other forms of energy, and thus contribute to the
cosmological energy density an amount

$\Omega_{a}\left({h_{0}\over 0.7}\right)^{2}\sim 0.15\left({f_{a}\over
10^{12}\hbox{GeV}}\right)^{7/6}\left({a(t_{*})\over f_{a}}\right)^{2}\ ,$
(171)

where $a(t_{*})$ is the axion value at $t_{*}$, which measures the
misalignment of the axion with respect to its final minimum. Such a
contribution is thus called vacuum realignment.

If the reheat temperature is larger than $f_{a}$, then the cosmic strings
produced at the Peccei-Quinn phase transition become at time $t_{*}$ the
boundaries of domain walls. This leads to a potential domain wall problem (too
much energy stored in the domain walls). There is a certain number of cases
where this can be avoided. In this case [Si08],

$\Omega_{a}\left({h_{0}\over 0.7}\right)^{2}\sim 0.7\left({f_{a}\over
10^{12}\hbox{GeV}}\right)^{7/6}\ .$ (172)

##  0.5 Light does not say it all (3): dark energy

We have seen in the introduction that the observation of the acceleration of
the expansion of the Universe in 1998-1999 provided a way out of an
increasingly uncomfortable tension between models and observation. On one
hand, the observed (luminous and dark) matter could account for only a
fraction of the critical density $\rho_{c}=10^{-26}$ kg/m3 (other forms of
energy being negligible at present time). On the other hand, the standard
theory of inflation erased any curvature and naturally led to a spatially flat
universe for which $\rho=\rho_{c}$. The discovery of the acceleration of the
expansion led to introduce a new component, named dark energy, of a type
unknown so far since all known forms of energy (non-relativistic matter,
radiation) decelerate the expansion.

In this last chapter, after briefly reviewing the observational case, we will
illustrate dark energy models with the example of quintessence, identify some
of the problems posed by dark energy models, and the fundamental questions
associated with the acceleration of the expansion of the Universe.

###  0.5.1 Acceleration of the expansion of the Universe: supernovae of type
Ia as standard candle

The approach that has made the first case for the acceleration of the
expansion of the Universe uses supernovae of type Ia as standard candles (see
Section 0.3.3). Two groups, the Supernova Cosmology Project [SCP99] and the
High-$z$ Supernova Search [HZS98a] 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
expansion is accelerating.

More precisely, one uses the relation (253) between the flux $\phi$ received
on Earth and the luminosity $L$ of the supernova. Traditionally, flux and
luminosity are expressed on a log scale as apparent magnitude $m_{B}$ and
absolute magnitude $M$ (magnitude is $-2.5\ \log_{10}$ luminosity + constant).
The relation then reads

$m_{B}=5\log(H_{0}d_{L})+M-5\log H_{0}+25.$ (173)

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 $d_{L}$, varies
logarithmically with $z$ up to corrections which depend on the geometry, more
precisely on $q_{0}$ for small $z$ as can be seen from (256). This allows to
compare with data cosmological models with different components participating
to the energy budget, as can be seen from Fig. 21.

Figure 21: 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.

In the case of a model with matter and cosmological constant as dominant
components, $q_{0}=\Omega_{{}_{M}}/2-\Omega_{\Lambda}$ and the measurement can
be turned into a limit in the $\Omega_{M}-\Omega_{\Lambda}$ plane see Fig.
22).

Figure 22:  Best-fit coincidence regions in the $\Omega_{M}-\Omega_{\Lambda}$
plane, based on the analysis of 42 type Ia supernovae discovered by the
Supernova Cosmology Project [SCP99].

Let us note that this combination $\Omega_{{}_{M}}/2-\Omega_{\Lambda}$ is
‘orthogonal’ to the combination $1-\Omega_{k}=\Omega_{M}+\Omega_{\Lambda}$
measured in CMB experiments. The two measurements are therefore complementary:
this is sometimes referred to as ‘cosmic complementarity’.

An important question raised by the analysis above is whether supernovae are
truly standard candles. Otherwise, the observation could be interpreted as an
history effect: for some reasons, older supernovae would be dimmer. Indeed,
strictly speaking, supernovae of type Ia are not standard candles: dimmer
supernovae are quicker (see Section 0.3.3). In practice, one thus has to
correct the light curves using a phenomenological stretch factor. It is thus
more precise to state that supernovae of type Ia are standardizable candles.
Moreover, the type of measurement discussed above is sensitive to many
possible systematic effects (evolution besides the light-curve timescale
correction, presence of dust, etc.), and this has fuelled a healthy debate on
the significance of supernova data as well as a thorough study of possible
systematic effects by the observational groups concerned.

###  0.5.2 Cosmological constant and vacuum energy

An obvious solution to the acceleration of the expansion is the introduction
of a cosmological constant (see (45) remembering that $q_{0}$ is a
deceleration parameter). But there is a severe conceptual problem associated
with the cosmological constant, which we now describe.

Considering (25) in flat space at present time implies the general following
constraint on $\lambda$:

$|\lambda|\leq H_{0}^{2}\ .$ (174)

In other words, the length scale $\ell_{\Lambda}\equiv|\lambda|^{-1/2}$
associated with the cosmological constant must be larger than the Hubble
length $\ell_{H_{0}}\equiv cH_{0}^{-1}=h_{0}^{-1}.10^{26}$ m, and thus be a
cosmological distance.

This is not a problem as long as one remains classical: $\ell_{H_{0}}$
provides a natural cosmological scale for our present Universe. The problem
arises when one tries to combine gravity with the quantum theory. Indeed, from
Newton’s constant and the Planck constant $\hbar$, we can construct the
(reduced) Planck mass scale

$m_{{}_{P}}=\sqrt{\hbar c/(8\pi G_{{}_{N}})}=2.4\times 10^{18}\;{\rm
GeV/c}^{2}\ .$ (175)

The corresponding length scale is the Planck length

$\ell_{{}_{P}}={\hbar\over m_{{}_{P}}c}=8.1\times 10^{-35}\;{\rm m}\ .$ (176)

The above constraint now reads:

$\ell_{\Lambda}\equiv|\lambda|^{-1/2}\geq\ell_{H_{0}}={c\over H_{0}}\sim
10^{60}\ \ell_{{}_{P}}\ .$ (177)

In other words, there are more than sixty orders of magnitude between the
scale associated with the cosmological constant and the scale of quantum
gravity.

A rather obvious solution is to take $\lambda=0$. This is as valid a choice as
any other in a pure gravity theory. Unfortunately, it is an unnatural one when
one introduces any kind of matter. Indeed, set $\lambda$ to zero but assume
that there is a nonvanishing vacuum (i.e. ground state) energy:
$<T_{\mu\nu}>=\rho_{{\rm vac}}g_{\mu\nu}$; then the Einstein equations (3)
read

$R_{\mu\nu}-{1\over 2}g_{\mu\nu}R=8\pi G_{{}_{N}}T_{\mu\nu}+8\pi
G_{{}_{N}}\rho_{{\rm vac}}g_{\mu\nu}\ .$ (178)

As first noted by Zel’dovich [Ze68], the last term is interpreted as an
effective cosmological constant (from now on, we set $\hbar=c=1$):

$\lambda_{{\rm eff}}=8\pi G_{{}_{N}}\rho_{{\rm vac}}\equiv{\Lambda^{4}\over
m_{{}_{P}}^{2}}\ .$ (179)

Generically, $\rho_{{\rm vac}}$ receives a non-zero contribution from symmetry
breaking: for instance, the scale $\Lambda$ would be typically of the order of
$100$ GeV in the case of the electroweak gauge symmetry breaking or $1$ TeV in
the case of supersymmetry breaking. Moreover, it is divergent in the context
of the (non-renormalizable) theory of gravity, which would thus favour a value
as large as the Planck scale. But the constraint (177) now reads:

$\Lambda\leq 10^{-30}\;m_{{}_{P}}\sim 10^{-3}\ {\rm eV}.$ (180)

It is this very unnatural fine-tuning of parameters (in explicit cases
$\rho_{{\rm vac}}$ and thus $\Lambda$ are functions of the parameters of the
theory) that is referred to as the cosmological constant problem, or more
accurately the vacuum energy problem.

If the acceleration observed is indeed due to the cosmological constant, its
value is as large as the upper bounds obtained in the previous subsection
allow:

$\lambda\sim H_{0}^{2}\ ,\quad\ell_{\Lambda}\sim\ell_{H_{0}}\
,\quad\Lambda\sim 10^{-3}\ \hbox{eV}\ .$ (181)

Regarding the latter scale $\Lambda$, which characterizes the vacuum energy
($\rho_{{\rm vac}}\equiv\Lambda^{4}$), one may note the interesting numerical
coincidence:

${1\over\Lambda}\sim\sqrt{\ell_{H_{0}}\ell_{{}_{P}}}\sim 10^{-4}\ {\rm m}\ .$
(182)

This relation underlines the fact that the vacuum energy problem involves some
deep connection between the infrared regime (the infrared cut-off being
$\ell_{H_{0}}$) and the ultraviolet regime (the ultraviolet cut-off being
$\ell_{{}_{P}}$), between the infinitely large and the infinitely small.

As an illustration of what such a relation could tell us about some very
fundamental aspects of physics (if it is not a mere numerical coicidence), we
will follow the approach developed by T. Padmanabhan [Pa08]. As we will
discuss later, the value of the cosmological constant might be related to the
way that the short distance theory reacts to long distance fluctuations. Let
us consider a 3-dimensional domain of size $L$ (e.g. the horizon
$\ell_{H_{0}}$). From the point of view of the quantum theory, it consists of
$N=(L/\ell_{{}_{P}})^{3}$ elementary cells. For each individual cell, a
“natural value” for the energy stored is provided by the scale $m_{{}_{P}}$
characteristic of quantum gravity. This yields

$\rho\sim{m_{{}_{P}}\over\ell_{{}_{P}}^{3}}\sim{1\over\ell_{{}_{P}}^{4}}\ ,$
(183)

which is some 120 orders of magnitude larger than observed.

Alternatively, if a mechanism, yet to be determined, cancels this bulk energy,
vacuum energy may be produced by the energy fluctuations. The Poissonian
fluctuation in energy is $\Delta\epsilon\sim 1/\ell_{{}_{P}}$, which
corresponds to an energy for overall fluctuations $\Delta E^{2}\sim
N/\ell_{{}_{P}}^{2}$, or an energy density

$\rho\sim{\sqrt{N}\over\ell_{{}_{P}}L^{3}}\sim{1\over\ell_{{}_{P}}^{5/2}L^{3/2}}\
,$ (184)

which again does not reproduce (182) ($\rho_{{\rm vac}}=\Lambda^{4}$).

If we make the further assumption that the relevant degrees of freedom lie on
the surface (as the degrees of freedom of a black hole lie on the horizon),
then $N=(L/\ell_{{}_{P}})^{2}$ and

$\rho\sim{\sqrt{N}\over\ell_{{}_{P}}L^{3}}\sim{1\over\ell_{{}_{P}}^{2}L^{2}}\
,$ (185)

which is fully consistent with (182). Is this telling us something on the
quantization of spacetime, hence of gravity? We will return to this question
below.

###  0.5.3 Supersymmetry

The most natural reason why vacuum energy would be vanishing is a symmetry
argument. It turns out that, among the various spacetime symmetries available,
global supersymmetry is the symmetry intimately connected with the vanishing
of the vacuum energy.

We recall that supersymmetry is a symmetry between bosons and fermions which
plays an important rôle in high energy physics, mainly because, through a
cancellation between the boson and the fermion fields, it controls severely
the quantum fluctuations. Many believe that the Standard Model is the
effective theory of a more fundamental theory valid at higher energies, with
more boson and fermion fields. These fields should, to some level, make
themselves known through the quantum energy fluctuations to which they
participe. The fact that we find no trace of them seems to suggest that these
fluctuations are tightly constrained, i.e. that the underlying theory is
supersymmetric.

Among the various spacetime symmetries, supersymmetry is rather unique.
Indeed, in the same way as the generator of time and space translations is the
4-momentum operator $P^{\mu}\equiv(P^{0}=H,P^{i})$, and the generator of
spacetime rotations is the tensor $M^{\mu\nu}$, the only other generators
carrying a Lorentz index are the generators of supersymmetry $Q_{r}$, where
$r$ is a spinor index. In fact, the combination of two supersymmetry
transformations is merely a translation in spacetime. This is expressed by the
following algebra:

$\\{Q_{r},\bar{Q}_{s}\\}=2\gamma^{\mu}_{rs}\ P_{\mu}\ ,$ (186)

where $\bar{Q}\equiv Q\gamma^{0}$ and $\gamma^{\mu}$ are the gamma matrices
introduced by Dirac to write a relativistic fermion (electron)
equation222222The anticommutator in (186) arises from the fact that the
supersymmetry transformation parameter is an anticommuting spinor.. Since the
generator of time translations $P_{0}$ is the Hamiltonian $H$, we may easily
infer from (186) an expression for the Hamiltonian of the system232323Indeed,
(186) reads explicitly  $\\{Q_{r},Q_{t}\\}\gamma^{0}_{ts}=2\gamma^{\mu}_{rs}\
P_{\mu}\ .$ (187) Contracting with $\gamma^{0}_{sr}$, one obtains
$\sum_{r,t}\ \\{Q_{r},Q_{t}\\}\left[(\gamma^{0})^{2}\right]_{tr}=2\ {\rm
Tr}\left(\gamma^{0}\gamma^{\mu}\right)P_{\mu}\quad.$ (188) Using
$(\gamma^{0})2={\bf 1}$ and ${\rm Tr}(\gamma^{0}\gamma^{\mu})=4g^{0\mu}$, one
obtains  $\sum_{r}Q_{r}^{2}=4P^{0}=4H\quad.$ (189) :

$H={1\over 4}\sum_{r}Q_{r}^{2}\quad.$ (190)

It follows that the energy of the vacuum $|0\rangle$ can be expressed as:

$\langle 0|H|0\rangle={1\over 4}\sum_{r}\parallel Q_{r}|0\rangle\parallel^{2}\
.$ (191)

Thus, the vacuum energy vanishes if and only if supersymmetry is a symmetry of
the vacuum: $Q_{r}|0\rangle=0$ for all $r$.242424Remember that a supersymmetry
transformation $U$ is obtained by exponentiating the generators:
$U|0\rangle=|0\rangle$.

The problem however is that, at the same time, supersymmetry predicts equal
boson and fermion masses and therefore needs to be broken since this is not
observed in Nature. The amount of breaking necessary to push the
supersymmetric partners high enough not to have been observed yet, typically
$\Lambda\sim$ TeV, is incompatible with the limit (180).

Moreover, in the context of cosmology, we should consider supersymmetry in a
gravity context and thus work with its local version, supergravity (following
(186), local supersymmetry transformations are associated with local spacetime
translations which are nothing but the reparameterizations which play a
central role in general relativity). In this context, the criterion of
vanishing vacuum energy is traded for one of vanishing mass for the gravitino,
the supersymmetric partner of the graviton (which allows to cancel the
constant vacuum energy at the expense of generating a mass $m_{{}_{3/2}}$ for
the gravitino field; see e.g. Ref. [Binetruy], section 6.3 for a more complete
treatment). Local supersymmetry is then absolutely compatible with a
nonvanishing vacuum energy, preferably a negative one (although possibly also
a positive one). This is both a blessing and a problem: supersymmetry may be
broken while the cosmological constant remains small, but we have lost our
rationale for a vanishing, or very small, cosmological constant and fine-
tuning raises again its ugly head.

In some supergravity theories, however, one may recover the vanishing vacuum
energy criterion.

###  0.5.4 Why now?

In the case where the acceleration of the expansion is explained by a
cosmological constant, one has to explain why this constant contribution
appears to start to dominate precisely now. This is the “Why now?” or cosmic
coincidence problem summarized in Fig. 23, the coincidence being between the
onset of acceleration and the present time (on the scale of the age of the
Universe). In order to avoid any reference to us (and hence any anthropic
interpretation, see below), we may rephrase the problem as follows. Why does
the dark energy starts to dominate at a time $t_{\Lambda}$ (redshift
$z_{\Lambda}\sim 1$) which almost coincides with the epoch $t_{G}$ (redshift
$z_{G}\sim 3$ to $5$) of galaxy formation?

Figure 23: The cosmic coincidence problem illustrated in the case of a
cosmological constant

Exercise 5-1 : In this exercise, we will study the evolution of a flat ($k=0$)
universe with a non-relativistic matter component (energy density
$\rho_{{}_{M}}$) and a cosmological term (or equivalently vacuum energy
density $\rho_{\Lambda}=\lambda/(8\pi G_{{}_{N}})$). The relevant equations of
evolution are obtained from (23), (29) and (31):

$\displaystyle H={\dot{a}^{2}\over a^{2}}$ $\displaystyle=$
$\displaystyle{8\pi G_{{}_{N}}\over 3}(\rho_{{}_{M}}+\rho_{\Lambda})\ ,$ (192)
$\displaystyle{\ddot{a}\over a}$ $\displaystyle=$ $\displaystyle-{4\pi
G_{{}_{N}}\over 3}(\rho_{{}_{M}}-2\rho_{\Lambda})\ ,$ (193)
$\displaystyle\dot{\rho}_{{}_{M}}$ $\displaystyle=$ $\displaystyle-3H\rho_{M}\
.$ (194)

a) Defining $H_{\Lambda}\equiv\left(8\pi
G_{{}_{N}}\rho_{\Lambda}/3\right)^{1/2}$, show that the solution to the system
of equations is given by

$a(t)=C\left(\sinh{3\over 2}H_{\Lambda}t\right)^{2/3}\ ,$ (195)

where $C$ is a constant. Compute the Hubble parameter $H$ as well as the ratio
$\rho_{{}_{M}}/\rho_{\Lambda}$ as a function of time $t$ and $H_{\Lambda}$.

b) Compute the present Hubble constant $H_{0}$ in terms of $H_{\Lambda}$ and
$\Omega_{{}_{M}}$. What is the age of the Universe $t_{0}$ in terms of $H_{0}$
and $\Omega_{{}_{M}}$?

c) Plot $a(t)/C$ and $\rho_{{}_{M}}/\rho_{\Lambda}$ in terms of time. When is
the equality between matter and vacuum energy reached? Hints: a)
$H=H_{\Lambda}/\tanh(3H_{\Lambda}t/2)$ and
$\rho_{{}_{M}}/\rho_{\Lambda}=\left[\sinh(3H_{\Lambda}t/2)\right]^{-2}$.

b) $H_{0}=H_{\Lambda}/\sqrt{1-\Omega_{{}_{M}}}$,

$t_{0}={2\over
3H_{0}}{1\over\sqrt{1-\Omega_{{}_{M}}}}\ln{1+\sqrt{1-\Omega_{{}_{M}}}\over\sqrt{\Omega_{{}_{M}}}}\
.$ (196)

###  0.5.5 More dynamics: dark energy vs modification of gravity

An alternate possibility is that the cosmological constant is much smaller or
even vanishing and that the acceleration is due to some new form of energy
–known as dark energy– or some modifications of gravity. These two
possibilities correspond to modifications of either sides of Einstein’s
equations (3). Let us envisage briefly these two cases.

First, we may try to identify a new component $\rho_{{}_{X}}$ of the energy
density with negative pressure:

$p_{{}_{X}}=w_{{}_{X}}\rho_{{}_{X}},\;\;w_{{}_{X}}<0\ .$ (197)

Note that the equation of state parameter $w_{{}_{X}}$ may not be constant and
may thus evolve with time.

Observational data constrains such a dynamical component, referred to in the
literature as dark energy, just as it did with the cosmological constant. For
example, in a spatially flat Universe with only matter and this unknown
component $X$, one obtains from (31) with $\rho=\rho_{M}+\rho_{{}_{X}}$,
$p=w_{{}_{X}}\rho_{{}_{X}}$ the following form for the Hubble parameter and
the deceleration parameter (compare with (43) and (46))

$\displaystyle H^{2}(z)$ $\displaystyle=$ $\displaystyle
H_{0}^{2}\left[\Omega_{{}_{M}}(1+z)^{3}+\Omega_{{}_{X}}(1+z)^{3(1+w_{{}_{X}})}\right]\
,$ (198) $\displaystyle q(z)$ $\displaystyle=$ $\displaystyle{H_{0}^{2}\over
2H(z)^{2}}\left[\Omega_{{}_{M}}(1+z)^{3}+\Omega_{{}_{X}}(1+3w_{{}_{X}})(1+z)^{3(1+w_{{}_{X}})}\right]\
,$ (199)

where $\Omega_{X}=\rho_{{}_{X}}/\rho_{c}$. At present time (compare with
(45)),

$q_{0}={\Omega_{M}\over 2}+(1+3w_{{}_{X}}){\Omega_{{}_{X}}\over 2}\ .$ (200)

The acceleration of the expansion observed requires that $\Omega_{{}_{X}}$
dominates252525One may easily obtain from (199) the time of the onset of the
acceleration phase:  $1+z_{{\rm
acc}}=\left[-(1+3w_{{}_{X}}){\Omega_{{}_{X}}\over\Omega_{{}_{M}}}\right]^{-1/(3w_{{}_{X}})}\
.$ (201) with $w_{{}_{X}}<-1/3$.

In Fig. 24, we present constraints in the $(\Omega_{{}_{M}},w_{{}_{X}})$
plane, obtained recently [SNLS11] from combining observations using supernovae
and BAO as well as CMB results from WMAP.

Figure 24: Confidence contours in the $(\Omega_{{}_{M}},w_{{}_{X}})$ plane
arising from supernova results from SuperNova Legacy Survey (SNLS) 3 year
results (in blue) and combined BAO/WMAP7 constraints (in green) [see Ref.
[SNLS11]].

An important property of dark energy is that it does not appear to be
clustered (just as a cosmological constant). Otherwise, its effects would have
been detected locally, as is the case for dark matter. This points towards
scalar fields, which generically have this property. Indeed, an attractive
property of scalar fields is that they easily provide a diffuse background by
resisting gravitational attraction. The key quantity when discussing
gravitational clustering is the speed of sound defined as

$c_{s}^{2}\equiv{\delta p\over\delta\rho}\ .$ (202)

It is a measure of how the pressure of the field resists gravitation
clustering. In most models of dark energy, we have $c_{s}^{2}\sim 1$, which
explains why such scalar dark energy does not cluster: its own pressure
resists gravitational collapse.

It should be stressed that, until recently, no fundamental scalar field had
been observed in Nature. The discovery of a Higgs particle at the LHC high
energy collider has obviously promoted the status of fundamental scalar
particles.

In the next section, we will illustrate the dynamics of dark energy on the
example of a scalar field evolving with time along its potential. This is
often referred to as quintessence. Such scalar fields turn out to be extremely
light: the only dimensionful parameter in the problem being the Hubble
constant $H_{0}$, their mass is $\hbar H_{0}\sim 10^{-33}\ {\rm eV}$ (see e.g.
(233) below). The exchange of such fields leads to a long range force: the
range is the inverse of the mass (times $\hbar c$), typically $\ell_{H_{0}}$
i.e. the size of the observable universe. This force is therefore similar to
gravity and can hardly be disentangled from it. Gravitational tests such as
the test of the equivalence principle thus apply not to the gravitational
force alone but to the combination of gravity and this new force. This compels
this force associated with dark energy to share many properties with gravity.
One may thus talk of a gravitational type force.

The second possibility is to modify the left-hand side of Einstein’s
equations, i.e. to look for a modification of gravity (we already mentionned
that possibility in Section 0.4.1 for explaining observational data
traditionally accounted fo by dark matter). This is a notoriously difficult
task because the current theory of gravity, general relativity, has passed
many stringent experimental and observational tests: equivalence principle,
Lorentz invariance,…. Any alternative theory should first pass these tests
equally successfully before being further considered.

Again, the distinction with the previous case where one introduces a new
dynamical component is not as clear-cut as it would first seem. Indeed,
through field redefinitions, alternate theories of gravity may be rewritten as
Einstein gravity plus a dynamical scalar field. Or if one goes to more spatial
dimensions, the graviton of the higher-dimensional theory may be regarded as a
standard 4-dimensional graviton plus a collection of scalar (spin zero) or
vector (spin one) fields.

Hence, the distinction between what is often presented as the two ways to
account for the observed acceleration of the expansion is not so clear.

###  0.5.6 The example of quintessence

A scalar field $\phi$ which has reached the minimum $\phi_{0}$ of its
potential energy $V(\phi$) amounts to a cosmological constant in the form of
vacuum energy: its kinetic energy is vanishing and its potential energy is the
constant $V(\phi_{0})$. A more dynamical candidate for dark energy is a scalar
field which is still slowly evolving in its potential [We88, PR88, RP88,
CDS98]. One often refers to this field as a quintessence field.

To be more explicit, let us consider the general action which describes a real
scalar field $\phi$ minimally coupled with Einstein gravity.

${\cal S}=\int d^{4}x\sqrt{-g}\left[-{m_{{}_{P}}^{2}\over 2}R+{1\over
2}\partial^{\mu}\phi\partial_{\mu}\phi-V(\phi)\right]\ .$ (203)

Computing the corresponding energy-momentum tensor, we obtain the pressure and
energy density (note the parallel with Section 0.2.3 where we discussed
inflation models)

$\displaystyle p_{\phi}$ $\displaystyle=$ $\displaystyle{1\over
2}\dot{\phi}^{2}-V(\phi)\quad,$ (204) $\displaystyle\rho_{\phi}$
$\displaystyle=$ $\displaystyle{1\over 2}\dot{\phi}^{2}+V(\phi)\quad,$ (205)

where, in the latter, we identify the field kinetic energy $\dot{\phi}^{2}/2$
and the potential energy $V(\phi)$. The corresponding equation of motion is,
if one neglects the spatial curvature ($k\sim 0$),

$\ddot{\phi}+3H\dot{\phi}=-{dV\over d\phi}\ ,$ (206)

where, besides the standard terms, one recognizes the friction term
$3H\dot{\phi}$ due to expansion. We deduce, as expected,

$\dot{\rho}_{\phi}=-3H(p_{\phi}+\rho_{\phi})\ .$ (207)

We have for the equation of state parameter

$w_{\phi}\equiv{p_{\phi}\over\rho_{\phi}}={{1\over
2}\dot{\phi}^{2}-V(\phi)\over{1\over 2}\dot{\phi}^{2}+V(\phi)}\geq-1\quad.$
(208)

If the kinetic energy is subdominant ($\dot{\phi}^{2}/2\ll V(\phi)$), we
clearly obtain $-1\leq w_{\phi}\leq 0$. In any case $-1\leq w_{\phi}\leq+1$.

Let us look in more details at the dynamics of such a quintessence scalar
field. For this purpose, it is useful to identify scaling solutions which we
define as solutions where the $\phi$ energy density scales as a power of the
cosmic scale factor:

$\rho_{\phi}\propto a^{-n_{\phi}}\ ,\quad n_{\phi}\ \hbox{constant}\ .$ (209)

This will allow us to identify two of the main examples of dynamical potential
for quintessence:

  * •

the exponential potential:

$V(\phi)=V_{0}e^{-\lambda\phi}\ ;$ (210)

  * •

the Ratra-Peebles potential [RP88, PR88],

$V(\phi)={M^{4+\alpha}\over\phi^{\alpha}}\ ,\quad\alpha>0\ .$ (211)

As we will see, the interest of such solutions is that they correspond to
attractors in the cosmological evolution of the scalar field.

Since it follows from (209) that $\dot{\rho}_{\phi}/\rho_{\phi}=-n_{\phi}H$,
we deduce from (207) a relation between $n_{\phi}$ and the equation of state
parameter $w_{\phi}$:

$w_{\phi}={n_{\phi}\over 3}-1\ .$ (212)

Hence the scaling solutions that we look for, exist only in epochs of the
cosmological evolution where the equation of state parameter may be considered
as constant (it could still be constant piecewise).

Since the dark energy (quintessence) density is expected to emerge from the
background (radiation or matter) energy density, we consider the evolution of
a scalar field $\phi$ with constant parameter $w_{\phi}$, during a phase
dominated by a background fluid with equation of state parameter

$w_{{}_{B}}={n_{{}_{B}}\over 3}-1$ (213)

Following (30), we have $a(t)\sim t^{2/n_{{}_{B}}}$ ($n_{{}_{B}}=4$ for
radiation, $3$ for non-relativistic matter,…).

From (204) and (205), we obtain

$\dot{\phi}^{2}={n_{\phi}\over 3}\rho_{\phi}\ ,\quad
V(\phi)=\left(1-{n_{\phi}\over 6}\right)\rho_{\phi}\ .$ (214)

Hence $\dot{\phi}^{2}\sim a^{-n_{\phi}}$ and $\dot{\phi}\sim
t^{-n_{\phi}/n_{{}_{B}}}$. We thus distinguish two cases: i)
$n_{\phi}=n_{{}_{B}}$ Clearly this implies $w_{\phi}=w_{{}_{B}}>0$ and the
quintessence field $\phi$ cannot be interpreted as the dark energy component.
We have

$\phi=\phi_{0}+{2\over\lambda}\ \ln(t/t_{0})\ ,$ (215)

with $\lambda$ constant. Then

$V(\phi)\sim\rho_{\phi}\sim a^{-n_{\phi}}\sim t^{-2}\sim e^{-\lambda\phi}\ .$
(216)

Hence, we find a scaling behavior for the exponential potential (210) in a
background such that $n_{{}_{B}}=n_{\phi}$ ($w_{{}_{B}}=w_{\phi}$). The
solution of the equation of motion (206) then reads

$\phi={1\over\lambda}\ln\left({V_{0}\lambda^{2}\over 2}{n_{{}_{B}}\over
6-n_{{}_{B}}}t^{2}\right)\ ,$ (217)

and the energy density (205)

$\rho_{\phi}={12\over\lambda^{2}n_{{}_{B}}t^{2}}\ .$ (218)

Since $H^{2}=(\rho_{{}_{B}}+\rho_{\phi})/3\sim[2/(n_{{}_{B}}t)]^{2}$,

${\rho_{\phi}\over\rho_{{}_{B}}+\rho_{\phi}}\sim{n_{{}_{B}}\over\lambda^{2}}\
.$ (219)

Hence $\rho_{\phi}/\rho_{{}_{B}}$ tends to be constant in this scenario. One
calls this property “tracking”. This is obviously compatible with our initial
assumptions only if $\lambda^{2}>n_{{}_{B}}$.

What happens if $\lambda^{2}\leq n_{{}_{B}}$?

It turns out that the scaling solution corresponds to a totally different
regime: the scalar field is the dominant contribution to the energy density.
We do not have to redo the calculation: it is identical to the previous one
with the only changes $w_{{}_{B}}\rightarrow w_{\phi}$ or
$n_{{}_{B}}\rightarrow n_{\phi}$ (for example,
$H^{2}=(\rho_{{}_{B}}+\rho_{\phi})/3\sim[2/(n_{\phi}t)]^{2}$: the scalar
energy density determines the evolution of the Universe). But then (219) reads
$1\sim n_{\phi}/\lambda^{2}$, i.e.

$w_{\phi}=-1+{\lambda^{2}\over 3}\ .$ (220)

Thus, if $\lambda^{2}<2$, the scalar field $\phi$ may provide the dark energy
component.

To summarize the two regimes that we have obtained for the exponential
potential (210):

  * •

if $\lambda^{2}\leq n_{{}_{B}}$, the scaling solution has
$w_{\phi}=-1+\lambda^{2}/3$ and $\rho_{\phi}/(\rho_{\phi}+\rho_{{}_{B}})\sim
1$ ($\phi$ is the dominant species),

  * •

if $\lambda^{2}>n_{{}_{B}}$, the scaling solution has $w_{\phi}=w_{{}_{B}}$
and $\rho_{\phi}/(\rho_{{}_{B}}+\rho_{\phi})\sim n_{{}_{B}}/\lambda^{2}$ (the
background energy density dominates; the scalar field energy density tracks
it).

ii) $n_{\phi}\not=n_{{}_{B}}$ Then $\phi\sim t^{-{n_{\phi}\over
n_{{}_{B}}}+1}$ and we now have

$V(\phi)\sim\rho_{\phi}\sim a^{-n_{\phi}}\sim
t^{-2n_{\phi}/n_{{}_{B}}}\sim\phi^{-2{n_{\phi}\over n_{{}_{B}}-n_{\phi}}}\ .$
(221)

Hence, we find a scaling behaviour for the Ratra-Peebles potential (211) in a
background characterized by $n_{{}_{B}}\not=n_{\phi}$ (or
$w_{{}_{B}}\not=w_{\phi}$). We have

$n_{\phi}={\alpha n_{{}_{B}}\over\alpha+2}\quad\hbox{or}\quad w_{\phi}={\alpha
w_{{}_{B}}-2\over\alpha+2}\ .$ (222)

The complete solution of the equation of motion (206) is

$\phi=\left({\alpha(\alpha+2)^{2}n_{{}_{B}}\ M^{4+\alpha}t^{2}\over
2\left[6(\alpha+2)-n_{{}_{B}}\alpha\right]}\right)^{{1\over\alpha+2}}\ .$
(223)

As we advertised in the beginning, these scaling solutions correspond to
attractors in the cosmological evolution of the scalar field. We show in Fig.
25 the full phase diagram for the exponential potential with $\lambda=2$
during matter domination ($n_{{}_{B}}=3$). The coordinates are
$x\equiv\dot{\phi}/(\sqrt{6}Hm_{{}_{P}}^{2})$ and
$y\equiv\sqrt{V}/(\sqrt{3}Hm_{{}_{P}}^{2})$. The stable (spiral) attractor is
found at $x=y=\sqrt{3/8}$.

Figure 25: Phase space diagram for the exponential potential (210) with
$\lambda=2$ and $n_{{}_{B}}=3$ (matter domination) [CLW98]

Exercise 5-2 : We show in this exercise that the solution (223) of the Ratra-
Peebles potential (211) is also an attractor.

a) Show that a small perturbation $\delta\phi$ satisfies the equation

$\delta\ddot{\phi}+{6\over n_{{}_{B}}t}\delta\dot{\phi}+{2(\alpha+1)\over
n_{{}_{B}}(\alpha+2)^{2}t^{2}}\left[6(\alpha+2)-n_{B}\alpha\right]\delta\phi=0\
.$ (224)

b) Look for a solution of the form $\delta\phi\sim t^{\gamma}$ and express
$\gamma$ in terms of $n_{{}_{B}}$ and $\alpha$.

c) Assume $\alpha>0$ and standard values of $n_{{}_{B}}$. Show that the two
solutions obtained in b) decay as $\delta\phi\sim
t^{-(6-n_{{}_{B}})/2n_{{}_{B}}}$: the solution (223) is an attractor.

Hints: b)

$\gamma=-{6-n_{{}_{B}}\over
2n_{{}_{B}}}\pm{\sqrt{-\left[3\alpha^{2}(3n_{{}_{B}}-2)(6-n_{{}_{B}})-12\alpha(n_{{}_{B}}^{2}-16n_{{}_{B}}+12)-4(n_{{}_{B}}^{2}-36n_{{}_{B}}+36)\right]}\over
2n_{{}_{B}}(\alpha+2)}\ .$ (225)

c) The reduced discriminant of the second order polynomial in $\alpha$ which
is under the square root is simply $288n_{{}_{B}}^{3}>0$. The corresponding
roots are then negative for $8-\sqrt{52}\sim 0.79<n_{{}_{B}}<6$ and the term
is thus negative for $\alpha>0$ and $n_{{}_{B}}$ in this range. The square
root contributes as an oscillating term and the two solutions corresponding to
(225) decay as $\delta\phi\sim t^{-(6-n_{{}_{B}})/2n_{{}_{B}}}$.

The reader may have noticed that the type of field solutions that we have
found is very similar to those encountered at the onset of inflation. This is
not surprising since we are introducing dynamics through the slow roll of a
field down its potential. Let us look a little closer at the field evolution
in the case of the Ratra-Peebles potential (211).

We have found the attractor scaling solution [RP88, PR88] $\phi\propto
a^{n_{B}/(2+\alpha)}$, $\rho_{\phi}\propto a^{-\alpha n_{B}/(2+\alpha)}$ in
the case where the background density dominates. Thus $\rho_{\phi}$ decreases
at a slower rate than the background density ($\rho_{B}\propto a^{-n_{B}}$)
and tracks it until it becomes of the same order, at a given value
$a_{{}_{Q}}$. We thus have:

$\displaystyle{\phi\over m_{{}_{P}}}$ $\displaystyle\sim$
$\displaystyle\left({a\over a_{{}_{Q}}}\right)^{n_{B}/(2+\alpha)},$ (226)
$\displaystyle{\rho_{\phi}\over\rho_{B}}$ $\displaystyle\sim$
$\displaystyle\left({a\over a_{{}_{Q}}}\right)^{2n_{B}/(2+\alpha)}.$ (227)

Exercise 5-3 : Compute the time $t_{{}_{Q}}$ at which
$\rho_{\phi}\sim\rho_{{}_{M}}$ in terms of $M$ and $m_{{}_{P}}$. Check that
$\phi$ at $t_{{}_{Q}}$ does not depend on $M$. Hints: $\rho_{M}\sim
m_{{}_{P}}^{2}/t^{2}$ and $\rho_{\phi}\sim
M^{{2(\alpha+4)\over\alpha+2}}t^{-{2\alpha\over\alpha+2}}$ give
$t_{{}_{Q}}\sim m_{{}_{P}}^{{\alpha+2\over 2}}M^{-{\alpha+4\over 2}}$. The
corresponding value for the equation of state parameter is given by (222):

$w_{\phi}=-1+{\alpha(1+w_{B})\over 2+\alpha}.$ (228)

Shortly after $\phi$ has reached for $a=a_{{}_{Q}}$ a value of order
$m_{{}_{P}}$, it satisfies the usual slow roll conditions (using the notations
(88 and (89) introduced in the context of inflation)

$\epsilon\equiv{1\over 2}\left({m_{{}_{P}}V^{\prime}\over
V}\right)^{2}=(\alpha/2)(m_{{}_{P}}/\phi)^{2}\ll 1\
,\quad\eta\equiv{m_{{}_{P}}^{2}V^{\prime\prime}\over
V}=\alpha(\alpha+1)(m_{{}_{P}}/\phi)^{2}\ll 1\ .$ (229)

Therefore (228) provides a good approximation to the present value of
$w_{\phi}$. Thus, at the end of the matter-dominated era, this field may
provide the quintessence component that we are looking for.

Two features are interesting in this respect. One is that this scaling
solution is reached for rather general initial conditions, i.e. whether
$\rho_{\phi}$ starts of the same order or much smaller than the background
energy density [ZWS99].

The second is the present value of $\rho$. Typically, since in this scenario
$\phi$ is of order $m_{{}_{P}}$ when the quintessence component emerges, we
must choose the scale $M$ in such a way that $V(m_{{}_{P}})\sim\rho_{c}$. The
constraint reads:

$M\sim\left(H^{2}_{0}m_{{}_{P}}^{2+\alpha}\right)^{1/(4+\alpha)}.$ (230)

We may note that this gives for $\alpha=2$, $M\sim 10$ MeV, not such an
atypical scale for high energy physics.

Exercise 5-4 : In the case of slow roll, the equation of motion (206) simply
reads $3H\dot{\phi}=-V^{\prime}(\phi)$.

a) Under this assumption, show that

$\ddot{\phi}=-{4\pi G_{{}_{N}}\over 3}{V^{\prime}\over
H^{2}}\sum_{i}(p_{i}+\rho_{i})$ (231)

where the summation is over all components of the Universe.

b) Deduce that, in the case where only matter and dark energy are
nonnegligible at present time $t_{0}$ ($\Omega_{M}+\Omega_{\phi}=1$),

$\left.{\ddot{\phi}\over V^{\prime}}\right|_{t_{0}}\sim-{1\over
2}(1-\Omega_{\phi})\ .$ (232)

Hence slow roll requires that $\Omega_{\phi}\sim 1$. Hints: a) Use
$\dot{H}=-4\pi G_{{}_{N}}\sum_{i}(p_{i}+\rho_{i})$.

However appealing, the quintessence idea is difficult to implement in the
context of realistic models [Ca98, KL99]. The main problem lies in the fact
that the quintessence field must be extremely weakly coupled to ordinary
matter. This problem can take several forms:

$\bullet$ The quintessence field must be very light. If we return to our
example of the Ratra-Peebles potential (211), $V^{\prime\prime}(m_{{}_{P}})$
provides an order of magnitude for the mass-squared of the quintessence
component:

$m_{\phi}\sim M\left({M\over m_{{}_{P}}}\right)^{1+\alpha/2}\sim H_{0}\sim
10^{-33}\ {\rm eV}.$ (233)

using (230). The exchange of such a field leads to a long-range force: the
range is typically $\ell_{H_{0}}$, the size of the presently observable
Universe. Since this force has not been observed yet, this means that the
field $\phi$ must be very weakly coupled to matter in order to comply with the
constraints imposed on gravitational-type forces by the very stringent tests
of the equivalence principle.

$\bullet$ The quintessence field is presently evolving with time. This may
generate a time dependence of what we call the constants of Nature. Indeed, it
turns out that, in modern particle theories, many of the constants of nature
have a dynamical origin: they are expressed in terms of quantum fields which
have settled down to their vacuum values in our present Universe. For example,
in the models discussed above, it is difficult to find a symmetry that would
prevent any coupling of the form

$\beta{\phi\over m_{{}_{P}}}F^{\mu\nu}F_{\mu\nu}$ (234)

to the gauge field kinetic term (or any given power of $\phi$). Since the
quintessence behaviour is associated with time-dependent values of the field
of order $m_{{}_{P}}$, this would generate, in the absence of fine tuning,
corrections of order one to the gauge coupling262626For example, the quantum
electrodynamics Lagrangian reads, with our conventions ${\cal
L}=-(1/4e^{2})F^{\mu\nu}F_{\mu\nu}$. The extra term would lead to an effective
electromagnetic coupling coupling $(1/4e_{{\rm
eff}}^{2})=(1/4e^{2})+\beta(\phi/m_{{}_{P}})$, hence a time-dependent fine
structure constant $\alpha=e_{{\rm eff}}^{2}/(\hbar c)$. . But the time
dependence of the fine structure constant for example is very strongly
constrained [Uz03]: $|\dot{\alpha}/\alpha|<5\times 10^{-17}{\rm yr}^{-1}$.
This yields a limit [Ca98]:

$|\beta|\leq 10^{-8}{m_{{}_{P}}H_{0}\over\langle\dot{\phi}\rangle},$ (235)

where $\langle\dot{\phi}\rangle$ is the average over the last $2\times 10^{9}$
years. Let us recall [Wi06] that the non-constancy of constants is not
compatible with the principle of local position invariance (i.e. independence
on the the location in time and space where a non-gravitational experiment is
performed) which forms part of the Einstein’s equivalence principle. This in
turn leads to violations of the universality of the weak equivalence principle
[No02].

$\bullet$ We have seen that, in the simplest models, the regime of interest is
reached when the quintessence field $\phi$ value becomes larger than
$m_{{}_{P}}$. In this instance, as well as in the context of (single field)
chaotic inflation, there has been discussions whether this remains in the sub-
Planck domain: strictly speaking, the answer is yes since what characterizes
the Planck domain are energy densities of order $m_{{}_{P}}^{4}$, whereas here
$\rho_{\phi}$ remains much smaller because of the specific form of the
potential (see e.g. (210) or (211)). It remains that, in such a context, one
must take into account all non-renormalisable interactions of order
$(\phi/m_{{}_{P}})^{n}$ compatible with the symmetries.

All the preceding shows that there is extreme fine tuning in the couplings of
the quintessence field to ordinary matter, unless they are forbidden by some
symmetry. This is somewhat reminiscent of the fine tuning associated with the
cosmological constant. Let us stress however that the quintessence solution
does not claim to solve the cosmological constant (vacuum energy) problem
described above. Quintessence may explain the acceleration of the expansion of
the Universe but has nothing to say about the cancellation of the bulk of the
vacuum energy arising from quantum fluctuations.

###  0.5.7 Back to the cosmological constant

Let us conclude this Section by reviewing some of the attempts to address the
problem of the cosmological constant.

#### Relaxation mechanisms

In the days where it was believed that vacuum energy was vanishing, one had to
look for a mechanism to fully cancel the contribution of order
$m_{{}_{P}}^{4}$. One naturally advocated mechanisms that relaxed the
cosmological constant to zero through the equations of motions of some
dynamical fields.

For example, in the context of string models, any dimensionful parameter is
expressed in terms of the fundamental string scale $M_{{}_{S}}$ and of vacuum
expectation values of scalar fields. The physics of the cosmological constant
and of its relaxation to a vanishing value would then be associated with the
dynamics of the corresponding scalar fields.

However, Steven Weinberg [We89] has constrained the possible mechanisms for
the relaxation of the cosmological constant by proving the following “no-go”
theorem: it is not possible to obtain a vanishing cosmological constant as a
consequence of the equations of motion of a finite number of fields.
Weinberg’s no-go theorem relies on a series of assumptions: Lorentz
invariance, finite number of constant fields, possibility of globally
redefining these fields… All attempts to propose a relaxation mechanism have
tried to avoid the conclusions of the theorem by relaxing one of these
assumptions.

#### Anthropic considerations

The anthropic principle approach can be sketched as follows. We consider
regions of spacetime with different values of $t_{G}$ (time of galaxy
formation) and $t_{\Lambda}$, the time when the cosmological constant starts
to dominate i.e. when the Universe enters a de Sitter phase of exponential
expansion. Clearly galaxy formation must precede this phase otherwise no
observer (similar to us) would be able to witness it. Thus $t_{G}\leq
t_{\Lambda}$. On the other hand, regions with $t_{\Lambda}\gg t_{G}$ have not
yet undergone any de Sitter phase of re-acceleration and are thus “phase-space
suppressed” compared with regions with $t_{\Lambda}\sim t_{G}$. Hence the
regions favoured have $t_{\Lambda}\stackrel{{\scriptstyle>}}{{\sim}}t_{G}$ and
thus $\rho_{\Lambda}\sim\rho_{M}$.

This was quantified by S. Weinberg [We87, MSW97], who obtained the following
bound:

$\rho_{\Lambda}<{\pi^{2}\over 3}\rho_{0}(1+z_{G})^{3}\ ,$ (236)

where $\rho_{0}$ is the present energy density and $z_{G}$ the redshift
corresponding to galaxy formation. Using $z_{G}=4.5$ as originally chosen by
Weinberg [We87], one finds $0<\rho_{\Lambda}/\rho_{{}_{M}}<550$. More recent
observations of a galaxy at $z=8.6$ [Le10] or the existence of dwarf galaxies
at $z\sim 10$ [Lo06] give a larger anthropic range:

$0<\rho_{\Lambda}/\rho_{{}_{M}}<4000\ .$ (237)

#### Emergent gravity

The alternative approach is to return to the origin of the vacuum energy
problem. We stressed in Section 0.5.2 that this problem arises in the context
of a quantum treatment of gravity (both $\hbar$ and $G_{{}_{N}}$ are
involved). At present we do not have a fully valid theory of quantum gravity.
Presumably, it involves as well a quantum version of spacetime. It is probable
that, just as our notion of continuous and elastic matter is only valid in a
large distance approximation, our notion of continuous space and time is also
only valid at large distance. Of course, it remains to be seen by which
"objects" one should replace continuous space and time, for distances smaller
than the Planck length, and what is the corresponding theory. In any case,
space and time would be emergent notions, and probably also gravity. For what
concerns us here, it could be that the solution to the vacuum problem should
be searched in this deeper context. And maybe dark energy is telling us
something about this underlying theory. Let us also note that, if spacetime is
an emergent notion, then its symmetries are also emergent: one may expect at
some level violations of Lorentz invariance for example, which lead to
violations of Einstein’s equivalence principle.

#### Holography

Until now we have considered gravity as a fundamental force which is on the
same footing as the other three. However, one aspect of gravity is strikingly
different from what we encounter with other interactions: it is the phenomenon
of gravitational collapse. As we have seen in Section 0.3.2, if a quantity $E$
of (gravitating) energy is localized in a region of spacetime of size $R$
smaller than the Schwarzschild radius defined as:

$R_{S}\equiv 2{G_{{}_{N}}E\over c^{2}}\,$ (238)

it undergoes gravitational collapse. This has been used by some (e.g. [DG10])
to consider that the high-energy (ultraviolet) regime of gravity is classical:
before reaching Planckian energies, regions of spacetime undergo gravitational
collapse and turn into black holes, which are classical objects. This may have
some far reaching consequences for the issues we are dealing with here,
especially vacuum energy.

Indeed, let us return to the considerations that led to the estimate $\rho\sim
m_{{}_{P}}/\ell_{{}_{P}}^{3}$ (see (183)) for the vacuum energy density in the
context of quantum field theory. Consider a spherical region of radius $R$ and
energy density given by (183): $\rho=m_{{}_{P}}^{4}$. Then the total energy
reads:

$E={4\pi\over 3}R^{3}\rho={4\pi\over 3}m_{{}_{P}}\left(Rm_{{}_{P}}\right)^{3}\
.$ (239)

But the system will undergo gravitational collapse when $R<R_{S}$ that is,
using (238) $R<\left(Rm_{{}_{P}}\right)^{3}/3m_{{}_{P}}$ i.e.
$R>1/m_{{}_{P}}=\ell_{{}_{P}}$. Hence, for any volume larger than the
elementary cell, on cannot concentrate a vacuum energy density
$\rho=m_{{}_{P}}^{4}$, at least in the case (that we consider here) that
vacuum energy is gravitating: the system is unstable and undergoes
gravitational collapse. The maximal energy density for a macroscopic region of
size $R$ is ($E<R/(2G_{{}_{N}})$)

$\rho_{max}=E/(4\pi R^{3}/3)={3\over 8\pi G_{{}_{N}}R^{2}}$ (240)

Let us extend these considerations to the whole observable Universe of size
$R\sim H_{0}^{-1}$: we can only store a vacuum energy density

$\rho<{3H_{0}^{2}\over 8\pi G_{N}}=\rho_{c}\ .$ (241)

Taken at face value, this would mean that the vacuum energy density has the
value it has because our observable Universe is very large. This cannot be
true at all times: otherwise, one can easily check that the presence of dark
energy can be absorbed in a redefinition of Newton’s constant; up to this
redefinition, the Universe would behave as if there is no dark energy and thus
the recent phase of acceleration of the expansion would remain unexplained. If
pushed to its full consequences, this leads to a new way of considering the
quantum evolution of the Universe [Bi12].

###  0.5.8 Concluding remarks

The most fascinating aspect of the dark energy problem is the number of
fundamental questions it connects with: how did the Universe emerge from a
quantum state to become so large and so old? why does dark energy emerge so
late in the evolution of the Universe? does it relate to the nature of space
and time as we know them? has spacetime emerged from something else? is
general relativity the ultimate theory of gravity? if not, are its basic
principles violated at some scale? what is a quantum state of the Universe?
what is the status of an observer in such a Universe? are there multiple
universes? are there more than four dimensions?…

One of the reasons is that dark energy appears to be connected with vacuum
energy, which is the most fundamental issue faced by theorists in fundamental
physics, an issue that illustrates the difficulties encountered at the
interface between general relativity, the present theory of gravity, and the
quantum theory. In some sense, the situation is reminiscent of the one
encountered at the end of the XIXth century, where one had two very successful
theories, Newtonian gravity and electromagnetism (summarized into the Maxwell
equations). The Michelson-Morley experiment in 1887 was the experimental
observation that led Einstein and others to reconsider the foundations.
Similarly, both general relativity and the quantum theory, the latter
described at the level of (non-gravitational) fundamental interactions by the
Standard Model, are extremely successful theories. Moreover, the recent
successes of cosmology have shown that our picture of the early Universe based
on these two pillars is not simply qualitative but is supported quantitatively
by increasingly precise observations. Is dark energy the signal of a new era?
It remains to be seen its exact connection with the issue of vacuum energy.
But more importantly, it is at present a conceptual difficulty, rather than a
clear experimental sign of the inconsistency of the overall picture. We are
still lacking our Michelson-Morley experiment.

From this perspective, it is reassuring that we have in front of us in the
next decade or so a very substantial experimental programme, which includes
not only increasingly precise and complete observational data, but also
experiments of many types, that might help us identify the road to follow in
order to reconsider the foundations of physics.

##  .6 Astrophysical constants and scales

Constants The tradition in astrophysics is to use the CGS system. Whereas
there are in some specific cases useful quantities to be defined (such as the
parsec), centimeter and gram seem hardly relevant. We thus use here the
international system. Note that $1$ kg.m${}^{-3}=10^{-3}$ g.cm-3 , $1$ J
$=10^{7}$ erg, $1$ W $=10^{7}$ erg.s-1. Speed of light: $c=299\ 792\ 458$ m
s-1

Newtonian gravitational constant: $G_{{}_{N}}=6.6742\times 10^{-11}$ m3 kg-1
s-2

$\alpha_{G}\equiv G_{{}_{N}}m_{p}^{2}/(\hbar c)=5.906\times 10^{-39}$

Fine structure constant: $\alpha\equiv e^{2}/(4\pi\epsilon_{0}\hbar
c)=7.297\times 10^{-3}=1/137$

Thomson cross section: $\sigma_{T}=8\pi r_{e}^{2}/3=0.665$ barn $=0.665\times
10^{-28}$ m2

Boltzmann constant: $k_{B}=1.380\times 10^{-23}$ J.K${}^{-1}=8.617\times
10^{-5}$ eV.K-1

Planck constant: $\hbar=1.054\times 10^{-34}$ J.s

Typical length scales

Planck length: $\ell_{{}_{P}}=\sqrt{8\pi G_{{}_{N}}\hbar/c^{3}}=8.1\times
10^{-35}$ m

Classical electron radius:
$r_{e}=e^{2}/(4\pi\epsilon_{0}m_{e}c^{2})=2.817\times 10^{-15}$ m

Solar radius: $R_{\odot}=6.9598\times 10^{8}$ m

Astronomical unit (au) = Sun-Earth distance = $1.4960\times 10^{11}$ m

Parsec (au/arc sec): $1$ pc $=3.262$ light-year $=3.086\times 10^{16}$ m

Sun-galactic center distance: $10$ kpc

Milky way galaxy disk radius (luminous matter): $15$ kpc

Presently visible universe: $6$ Gpc

Typical mass scales

Reduced Planck mass: $m_{{}_{P}}=\sqrt{\hbar c/(8\pi G_{{}_{N}})}=2.14\times
10^{18}$ GeV$/c^{2}=3.81\times 10^{-9}$ kg

Solar mass: $M_{\odot}=1.989\times 10^{30}$ kg

Milky Way galaxy mass: $4$ to $10\times 10^{11}\ M_{\odot}$

Typical luminosities

Solar luminosity: $L_{\odot}=3.85\times 10^{33}$ erg/s $=3.85\times 10^{26}$ W

Typical densities

Present mean density of the universe: $\rho_{0}\sim\rho_{c}=10^{-26}$ kg.m-3

Interstellar medium: $10^{-22}$ kg.m-3

Sun: $\rho_{\odot}=1408$ kg/m-3

Neutron star: $10^{18}$ kg.m-3

##  .7 General relativity

In the context of general relativity, one defines the Christoffel symbol or
affine connection $\Gamma^{\rho}{}_{\mu\nu}$ which is the analogue of the
gauge field (it appears in covariant derivatives). It is defined in terms of
the metric as:

$\Gamma^{\rho}{}_{\mu\nu}={1\over
2}g^{\rho\sigma}\left[\partial_{\mu}g_{\nu\sigma}+\partial_{\nu}g_{\mu\sigma}-\partial_{\sigma}g_{\mu\nu}\right]\quad,$
(242)

where $g^{\rho\sigma}$ is the inverse metric tensor:
$g^{\rho\sigma}g_{\sigma\tau}=\delta^{\rho}_{\tau}$.

In the same way that one defines the field strength by differentiating the
gauge field, one introduces the Riemann curvature tensor:

$R^{\mu}{}_{\nu\alpha\beta}=\partial_{\alpha}\Gamma^{\mu}{}_{\nu\beta}-\partial_{\beta}\Gamma^{\mu}{}_{\nu\alpha}+\Gamma^{\mu}{}_{\alpha\sigma}\Gamma^{\sigma}{}_{\nu\beta}-\Gamma^{\mu}{}_{\beta\sigma}\Gamma^{\sigma}{}_{\nu\alpha}\quad.$
(243)

By contracting indices, one then defines the Ricci tensor $R_{\mu\nu}$ and the
curvature scalar $R$

$R_{\mu\nu}\equiv R^{\alpha}{}_{\mu\alpha\nu},\quad,\quad R\equiv
g^{\mu\nu}R_{\mu\nu}\quad.$ (244)

One also uses the Christoffel symbols to define the covariant derivatives:

$\displaystyle V_{\mu;\nu}$ $\displaystyle=$
$\displaystyle\nabla_{\nu}V_{\mu}\equiv\partial_{\nu}V_{\mu}-\Gamma^{\rho}{}_{\mu\nu}V_{\rho}\
,$ $\displaystyle V^{\mu}{}_{;\nu}$ $\displaystyle=$
$\displaystyle\nabla_{\nu}V^{\mu}\equiv\partial_{\nu}V^{\mu}+\Gamma^{\mu}{}_{\nu\rho}V^{\rho}\
.$ (245)

Exercise B-1 : In the case of the Robertson-Walker metric (19),

a) compute the non-vanishing Christoffel symbols (242),

b) using the fact that the Ricci tensor associated with the 3-dimensional
metric $\gamma_{ij}$ is simply $R_{ij}(\gamma)=2k\gamma_{ij}$, compute the
components of the Ricci tensor and the scalar curvature (244),

c) deduce the components of the Einstein tensor $G_{\mu\nu}$ defined in (3):
The components of the Einstein tensor now read (see Exercise 2-1):

$\displaystyle G_{tt}$ $\displaystyle=$ $\displaystyle
3\left({{\dot{a}}^{2}\over a^{2}}+{k\over a^{2}}\right),$ (246) $\displaystyle
G_{ij}$ $\displaystyle=$
$\displaystyle-\gamma_{ij}\left({\dot{a}}^{2}+2a{\ddot{a}}+k\right),$ (247)

Hints: a) $\Gamma^{i}{}_{jt}=\delta^{i}_{j}\dot{a}/a$,
$\Gamma^{t}{}_{ij}=a\dot{a}\gamma_{ij}$,
$\Gamma^{i}{}_{jk}=\Gamma^{i}{}_{jk}(\gamma)$.

b) $R_{tt}=-3\ddot{a}/a$,
$R_{ij}=\left(2k+\ddot{a}a+2\dot{a}^{2}\right)\gamma_{ij}$,
$R=-6\left(k+\ddot{a}a+\dot{a}^{2}\right)/a^{2}$.

##  .8 Measure of distances

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

First consider a photon travelling in an expanding or contracting Friedmann
universe. Its equation of motion is fixed by the condition $ds^{2}=0$ (as in
Eq. (32) of Chapter 0.1). One then defines the proper distance as

$d(t)\equiv
a(t)\int_{0}^{r}{dr\over\sqrt{1-kr^{2}}}=a(t)\int_{t}^{t_{0}}{cdt^{\prime}\over
a(t^{\prime})}\ .$ (248)

Using

$\int_{t}^{t_{0}}{cdt\over a(t)}=\int_{a(t)}^{a_{0}}{cda\over
a\dot{a}}=\int_{a(t)}^{a_{0}}{cda\over a^{2}H}=\int_{0}^{z}{cdz\over H(z)}\,$

we may extract from (248) the proper distance at time $t_{0}$:

$\displaystyle d(t_{0})=a_{0}\int_{0}^{r}{dr\over\sqrt{1-kr^{2}}}$
$\displaystyle=$ $\displaystyle
a_{0}\left\\{\begin{array}[]{ll}\sin^{-1}r&k=+1\\\ r&k=0\\\
\sinh^{-1}r&k=-1\end{array}\right.$ $\displaystyle=$
$\displaystyle\ell_{H_{0}}\int_{0}^{z}{dz\over\left[\Omega_{{}_{M}}(1+z)^{3}+\Omega_{{}_{R}}(1+z)^{4}+\Omega_{k}(1+z)^{2}+\Omega_{\Lambda}\right]^{1/2}}$

where $\ell_{H_{0}}=cH_{0}^{-1}$.

If a photon source of luminosity $L$ (energy per unit time) is placed at a
distance $r$ from the observer, then the energy flux $\phi$ (energy per unit
time and unit area) received by the observer is given by

$\phi={L\over 4\pi a_{0}^{2}r^{2}(1+z)^{2}}\equiv{L\over 4\pi
d_{L}^{2}}\quad.$ (253)

The two powers of $1+z$ account for the photon energy redshift and the time
dilatation between emission and observation. The quantity $d_{L}\equiv
a_{0}r(1+z)$ is called luminosity distance.

If the source is at a redshift $z$ of order one or smaller, the effect of
spatial curvature is unimportant and we can approximate the integral
$\int_{0}^{r}dr/\sqrt{1-kr^{2}}$ in (248) by simply $r$ (i.e. the value for
$k=0$). This equation gives

$a_{0}r\sim\int_{t}^{t_{0}}{a_{0}cdt\over a(t)}=\int_{a}^{a_{0}}{a_{0}cda\over
a\dot{a}}\sim\ell_{H_{0}}\int_{a}^{a_{0}}{da\over
a\left[1-q_{0}H_{0}(t-t_{0})\right]}$ (254)

where we have used the development (47) with
$t_{H_{0}}=\ell_{H_{0}}/c=H_{0}^{-1}$. Using
$H_{0}(t-t_{0})\sim(a-a_{0})/a_{0}\ll 1$ and $a=a_{0}/(1+z)$, we obtain for
$z\ll 1$

$a_{0}r=\ell_{H_{0}}z\left(1-{1+q_{0}\over 2}z+\cdots\right)\quad.$ (255)

Thus, the luminosity distance reads, for $z\ll 1$,

$d_{L}=\ell_{H_{0}}z\left(1-{1+q_{0}\over
2}z+\cdots\right)(1+z)=\ell_{H_{0}}z\left(1+{1-q_{0}\over
2}z+\cdots\right)\quad.$ (256)

Hence measurement of deviations to the Hubble law ($d_{L}=\ell_{H_{0}}z$) at
moderate redshift allow to measure the combination
$\Omega_{{}_{M}}/2-\Omega_{\Lambda}$ (see (45)).

Another distance is defined in cases where one measures the angular diameter
$\delta$ of a source in the sky. If $D$ is the diameter of the source, then
$D/\delta$ would be the distance of the source in Euclidean geometry. In a
universe with a Robertson-Walker metric, it turns out to be
$a(t)r=a_{0}r/(1+z)$. This defines the angular diameter distance $d_{A}$

$d_{A}={d_{L}\over(1+z)^{2}}\quad.$ (257)

Several distance measurements tend to point towards an evolution of the
present universe dominated by the cosmological constant contribution272727at
least when analyzed in the framework of the model discussed in this section,
i.e. including non-relativistic matter, radiation and a cosmological constant.
and thus a late acceleration of its expansion, as we will now see.

Exercise C-1 : We compute exactly the luminosity distance $d_{L}=a_{0}r(1+z)$
or angular distance $d_{A}=a_{0}r/(1+z)$ in the case of a matter-dominated
universe. Defining

$\zeta_{k}(r)\equiv\left\\{\begin{array}[]{ll}\sin^{-1}r&k=+1\\\ r&k=0\\\
\sinh^{-1}r&k=-1\end{array}\right.\ ,$ (258)

use (.8) which reads, in the case of a matter-dominated universe,

$a_{0}\zeta_{k}(r)=\ell_{H_{0}}\int_{0}^{z}{dz\over\left[\Omega_{{}_{M}}(1+z)^{3}+(1-\Omega_{{}_{M}})(1+z)^{2}\right]^{1/2}}$
(259)

to prove Mattig’s formula [Ma58]:

$a_{0}r=2\ell_{H_{0}}{\Omega_{{}_{M}}z+\left(\Omega_{{}_{M}}-2\right)\left[\sqrt{1+\Omega_{{}_{M}}z}-1\right]\over\Omega_{{}_{M}}^{2}(1+z)}\
.$ (260)

Hints: For $k\not=0$, change to the coordinate
$u^{2}=k(\Omega-1)/\left[\Omega(1+z)\right]$ in order to compute the integral
(259). Using the last of equations (40), which reads
$\ell_{H_{0}}^{2}/a_{0}^{2}=k(\Omega-1)$, one obtains

$\zeta_{k}(r)=2\left(\zeta_{k}\left[\sqrt{{k(\Omega-1)\over\Omega}}\right]-\zeta_{k}\left[\sqrt{{k(\Omega-1)\over(1+z)\Omega}}\right]\right)\
,$

from which (260) can be inferred.

##  .9 Perturbations with scalar fields

We study in this Appendix the perturbations of a scalar field coupled to
gravity, following Ref. [GM99]. This has obvious implications for the study of
inflation or dark energy models.

We consider the most general local action for a scalar field coupled to
Einstein gravity:

${\cal S}=-{m_{{}_{P}}^{2}\over 2}\int d^{4}x\sqrt{g}R+\int d^{4}x\sqrt{g}\
p(X,\phi)\ ,$ (261)

where we have defined

$X\equiv{1\over 2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi\ .$ (262)

One may describe this system as a perfect fluid, with the standard energy-
momentum tensor (21):

$T_{\mu\nu}=-pg_{\mu\nu}+(p+\rho)U_{\mu}U_{\nu}\quad,$ (263)

Indeed, varying with respect to the metric, we find

$\displaystyle\delta{\cal S}$ $\displaystyle=$
$\displaystyle\int\sqrt{g}\left[Xp_{,X}U_{\mu}U_{\nu}-{1\over
2}pg_{\mu\nu}\right]\delta g^{\mu\nu}$ (264) $\displaystyle\equiv$
$\displaystyle{1\over 2}\int T_{\mu\nu}\delta g^{\mu\nu}\ ,$

with $U_{\mu}\equiv\partial_{\mu}\phi/(2X)^{1/2}$. Thus, the energy-momentum
tensor has the form (263) with the function $p(X,\phi)$, i.e. the scalar
Lagrangian, as the pressure (hence the notation) and the energy density:

$\rho=2Xp_{,X}-p\ .$ (265)

In the case where $p=X-V(\phi)$, one recovers (205,204).

In the following, a quantity will play an important role; it is the speed of
sound:

$c_{s}^{2}\equiv{\delta p\over\delta\rho}={p_{,X}\over\rho_{,X}}={p+\rho\over
2X\rho_{,X}}\ .$ (266)

We start with a background metric described by (19) (for simplicity, we assume
that space is flat: $k=0$; for the general case, see Ref. [GM99]) and with a
background scalar configuration $\varphi(t)$ which satisfies (29) (or
equivalently the scalar field equation of motion).

Perturbing this background, we write in the longitudinal gauge [MFB92]282828We
use the fact that the spatial part of the energy-momentum tensor is diagonal.
Otherwise, two different functions $\Phi$ wand $\Psi$ would appear
respectively as $(1+2\Phi)$ in the time component and $(1-2\Psi)$ in the space
component [MFB92].

$ds^{2}=(1+2\Phi)dt^{2}-(1-2\Phi)a^{2}(t)\gamma_{ij}dx^{i}dx^{j}\ ,$ (267)

where $\Phi$ is the Newtonian potential, and we take for the scalar field

$\phi(t,x)=\varphi(t)+\delta\varphi(t,x)\ .$ (268)

Then

$\displaystyle\delta G^{0}{}_{0}$ $\displaystyle=$ $\displaystyle
2\left[{1\over a^{2}}\Delta\Phi-3H\dot{\Phi}-3H^{2}\Phi\right]\ ,$ (269)
$\displaystyle\delta G^{0}{}_{i}$ $\displaystyle=$ $\displaystyle
2\left[\dot{\Phi}+H\Phi\right]_{,i}\ ,$ (270)

where, as usual, $H^{2}=8\pi G_{{}_{N}}\rho/3$. As for the variation of the
energy-momentum tensor, we have

$\delta T^{0}{}_{0}=\delta\rho=\rho_{,X}\delta X+\rho_{,\phi}\delta\varphi\
,\quad\delta T^{0}{}_{i}=(p+\rho)\delta U_{i}\ .$ (271)

Note for the latter that $U_{0}=1$, $U_{i}=0$ but $\delta
U_{i}=(\delta\varphi/\dot{\varphi})_{,i}\not=0$. For the former, we use
$\dot{\rho}=-3H(p+\rho)=\rho_{,X}\dot{X}+\rho_{,\phi}\dot{\phi}$. One finds

$\displaystyle\delta T^{0}{}_{0}$ $\displaystyle=$
$\displaystyle-3H(p+\rho){\delta\varphi\over\dot{\varphi}}+{p+\rho\over
c_{s}^{2}}\left[\left({\delta\varphi\over\dot{\varphi}}\right)^{\cdot}-\Phi\right]\
,$ (272) $\displaystyle\delta T^{0}{}_{i}$ $\displaystyle=$
$\displaystyle(p+\rho)\left({\delta\varphi\over\dot{\varphi}}\right)_{,i}\ .$
(273)

We thus obtain from $\delta G_{\mu\nu}=8\pi G_{{}_{N}}\delta T_{\mu\nu}$

$\displaystyle\left({\delta\varphi\over\dot{\varphi}}\right)^{\cdot}$
$\displaystyle=$ $\displaystyle\left(1+{c_{s}^{2}\over 4\pi
G_{{}_{N}}a^{2}(p+\rho)}\Delta\right)\Phi\ ,$ (274)
$\displaystyle\left(a\Phi\right)^{\cdot}$ $\displaystyle=$ $\displaystyle 4\pi
G_{{}_{N}}a(p+\rho)\left({\delta\varphi\over\dot{\varphi}}\right)\ .$ (275)

The other Einstein’s equations are redundant. We may now define the new
variables $\xi$ and $\zeta$:

$a\Phi=4\pi G_{{}_{N}}H\xi\
,\quad{\delta\varphi\over\dot{\varphi}}={\zeta\over H}-{4\pi G_{{}_{N}}\over
a}\xi\ ,$ (276)

which satisfy the equations of motion

$\displaystyle\dot{\xi}$ $\displaystyle=$ $\displaystyle{a(p+\rho)\over
H^{2}}\ \zeta\ ,$ (277) $\displaystyle\dot{\zeta}$ $\displaystyle=$
$\displaystyle{c_{s}^{2}H^{2}\over a^{3}(p+\rho)}\ \Delta\xi\ .$ (278)

Defining

$z\equiv{a(p+\rho)^{1/2}\over c_{s}H}$ (279)

and differentiating with respect to conformal time ($\xi^{\prime}\equiv\
d\xi/d\eta=a\dot{\xi}$ and so on), we may write the system of differential
equations simply as

$\xi^{\prime}=c_{s}^{2}z\zeta\ ,\quad\zeta^{\prime}={1\over z^{2}}\Delta\xi\
,$ (280)

which can be turned into a single differential equation for $\zeta$. Indeed,
defining $v\equiv z\zeta$, we find

$v^{\prime\prime}-c_{s}^{2}\Delta v-{z^{\prime\prime}\over z}v=0\ .$ (281)

This can be derived from the following action:

${\cal S}={1\over 2}\int\left[v^{\prime 2}+c_{s}^{2}v\Delta
v+{z^{\prime\prime}\over z}v^{2}\right]d\eta d^{3}x\ .$ (282)

If we look for plane wave solutions of (281) i.e. $v=v_{\bf k}e^{-i{\bf
k}.{\bf x}}$, we find two regimes:

  * •

at long wavelength (${\bf k}^{2}c_{s}^{2}\ll|z^{\prime\prime}/z|$), a non-
decaying solution $v_{\bf k}\propto z$.

  * •

at short wavelength (${\bf k}^{2}c_{s}^{2}\ll|z^{\prime\prime}/z|$), an
oscillating solution $v_{\bf k}\propto\exp(ikc_{s}\eta)$.

Quantization of scalar field in curved spacetime

We now turn to the quantization of the scalar degrees of freedom. The fact
that we are in a non-trivial background gravitational field brings some new
features but, since the background is only time dependent, the quantization
procedure may be broadly inspired by the flat spacetime case.

Let us consider a generic scalar field (this could be for example the
gravitational potential). The standard commutation relations

$\left[\Phi(\eta,{\bf x}),\Phi(\eta,{\bf
x^{\prime}})\right]=\left[\Pi(\eta,{\bf x}),\Pi(\eta,{\bf
x^{\prime}})\right]=0\ ,\quad\left[\Phi(\eta,{\bf x}),\Pi(\eta,{\bf
x^{\prime}})\right]=i\delta^{3}({\bf x}-{\bf x^{\prime}})$ (283)

involves the canonical momentum $\Pi=\delta{\cal L}/\delta\partial_{\eta}\Phi$
(we are using here the conformal time $\eta$).

One may decompose the operator $\Phi$ over the complete orthonormal basis of
the eigenfunctions of the Laplace operator. In the spatially flat case that we
are considering here, these are simply the plane waves: $\chi_{\bf
k}(\eta)e^{-i{\bf k}.{\bf x}}$ (we note that we are making full use of spatial
translation invariance, which remains a symmetry). We thus write

$\Phi(\eta,{\bf
x})={1\over\sqrt{2}}\int{d^{3}k\over(2\pi)^{3/2}}\left[e^{-i{\bf k}.{\bf
x}}\chi_{\bf k}(\eta)a^{\dagger}_{\bf k}+e^{i{\bf k}.{\bf x}}\chi_{\bf
k}^{*}(\eta)a_{\bf k}\right]\ ,$ (284)

where the operators $a_{\bf k}$ and $a^{\dagger}_{\bf k}$ satisfy the
commutation rules

$\left[a_{\bf k},a_{\bf k^{\prime}}\right]=\left[a^{\dagger}_{\bf
k},a^{\dagger}_{\bf k^{\prime}}\right]=0\ ,\quad\left[a_{\bf
k},a^{\dagger}_{\bf k^{\prime}}\right]=\delta^{3}({\bf k}-{\bf k^{\prime}})\
.$ (285)

This is consistent with (283) under the condition

$\chi^{\prime}_{\bf k}(\eta)\chi_{\bf k}^{*}(\eta)-\chi^{\prime*}_{\bf
k}(\eta)\chi_{\bf k}(\eta)=2i\ .$ (286)

The $\chi_{k}(\eta)$ modes satisfy an equation of the type

$\chi^{\prime\prime}_{\bf k}(\eta)+E_{\bf k}^{2}\chi_{\bf k}(\eta)=0\ ,$ (287)

where $E_{\bf k}^{2}$ includes a mass-squared term and possibly other
contributions such as the one that would arise from a non-minimal coupling of
the scalar field to gravity.

In Minkowski spacetime, on constructs a Fock space of states obtained by
applying a product of creation (negative frequency) operators on the vacuum
state $|0\rangle$, defined as the state annihilated by all positive frequency
operators $a_{\bf k}$: $a_{\bf k}|0\rangle=0$. This relies on the invariance
under the Poincaré group which gives an absolute meaning to these notions.
More precisely, in Minkowski spacetime, the operator $\partial/\partial t$ is
a Killing vector orthogonal to the spacelike hypersurfaces $t=$ constant and
the plane wave modes $e^{-ik.x}$ are eigenfunctions of this Killing vector
with eigenvalues $-ik_{0}=-i\omega$ of a given sign.

In curved spacetime (see for example the book by Birrell and Davies
[Birrell]), the Poincaré group is no longer a symmetry group of spacetime and
correspondingly there is no time-invariant notion of positive or negative
frequency. There is thus no possibility of agreeing on a specific vacuum state
for all inertial measuring devices. One may still rely, in some cases, on
specific symmetries such as translation invariance, conformal symmetry or the
de Sitter group to constrain the description of vacuum states.

[See the review by Mukhanov, Feldman and Brandenberger [MFB92]]

We thus choose a given time $\eta_{0}$ in order to define a vacuum state
$|0\rangle_{\eta_{0}}$ such that, for all ${\bf k}$, $a_{\bf
k}|0\rangle_{\eta_{0}}=0$. These annihilation operators are the operator
factors of the positive frequency modes $\chi^{+}_{\bf k}\equiv\chi_{\bf
k}^{*}(\eta)$ in the expansion (284) (similarly we define the negative
frequency modes $\chi^{-}_{\bf k}\equiv\chi_{\bf k}(\eta)$). If all $E_{\bf
k}$ are positive, it turns out that one find such modes: they are the
solutions of (287) with the following initial conditions at time $\eta_{0}$:

$\chi_{\bf k}(\eta_{0})=E_{\bf k}^{-1/2}(\eta_{0})\ ,\quad\chi^{\prime}_{\bf
k}(\eta_{0})=iE_{\bf k}^{1/2}(\eta_{0})\ ,$ (288)

consistent with the consistency condition (286). Since these solutions
obviously depend on $\eta_{0}$, we will affect them a superscript $(0)$ in
what follows.

At a later time $\eta_{1}$, we define along the same lines a new vacuum
$|0\rangle_{\eta_{1}}$, which is annihilated by all operators $b_{\bf k}$.
These operators appear in an expansion of the type (284) but with new positive
frequency modes $\chi^{+(1)}_{\bf k}$. Since equation (287) is linear, there
is a linear relation between the positive and negative frequency modes at
$\eta_{0}$ and $\eta_{1}$:

$\displaystyle\chi_{\bf k}^{(1)+}$ $\displaystyle=$ $\displaystyle\alpha_{\bf
k}\chi_{\bf k}^{(0)+}+\beta_{\bf k}\chi_{\bf k}^{(0)-}\
,\quad\left|\alpha_{\bf k}\right|^{2}-\left|\beta_{\bf k}\right|^{2}=1\ ,$
$\displaystyle\chi_{\bf k}^{(1)-}$ $\displaystyle=$
$\displaystyle\beta^{*}_{\bf k}\chi_{\bf k}^{(0)+}+\alpha^{*}_{\bf k}\chi_{\bf
k}^{(0)-}\ ,$ (289)

where we have used (286).

This defines the Bogoliubov coefficients $\alpha_{\bf k}$ and $\beta_{\bf k}$.
Obviously we have in parallel for the operators

$\displaystyle b_{\bf k}$ $\displaystyle=$ $\displaystyle+\alpha^{*}_{\bf
k}a_{\bf k}-\beta^{*}_{\bf k}a^{\dagger}_{\bf k}\ ,$ $\displaystyle
b^{\dagger}_{\bf k}$ $\displaystyle=$ $\displaystyle-\beta_{\bf k}a_{\bf
k}+\beta_{\bf k}a^{\dagger}_{\bf k}\ .$ (290)

Let us give an example to illustrate the physical meaning of the Bogoliubov
coefficients. We start with the vacuum $|0\rangle_{\eta_{0}}$ at time
$\eta_{0}$ and compute at $\eta_{1}$ the number of particles $b^{\dagger}_{\bf
k}b_{\bf k}$. It is given by

${}_{\eta_{0}}\langle 0\left|b^{\dagger}_{\bf k}b_{\bf
k}\right|0\rangle_{\eta_{0}}=\left|\beta_{\bf k}\right|^{2}\ ,$ (291)

where we have used (290). Thus, even though we have prepared the system in the
vacuum state at time $\eta_{0}$, the number of particles is non-vanishing at
time $\eta_{1}$. Fluctuations can be produced quantum mechanically from the
vacuum through the coupling of the scalar field to gravity.

If not all energies are positive, then we cannot define a set of modes through
the boundary conditions (288). This is in particular the situation encountered
in the case of inflation. There, the symmetries of de Sitter space help to
define the so-called de Sitter invariant vacuum through the conditions:

$\chi_{\bf k}(\eta_{0})={1\over k^{3/2}}\left({\cal H}_{0}+ik\right)\
,\quad\chi^{\prime}_{\bf k}(\eta_{0})={i\over k^{1/2}}\left({\cal
H}_{0}+ik-i{\cal H}^{\prime}_{0}/k\right)\ ,$ (292)

where ${\cal H}_{0}=a^{\prime}(\eta_{0})/a(\eta_{0})=a(\eta_{0})H_{0}$. We
recover (288) at small wavelength i.e. for $k\ll{\cal H}_{0}$. We note that,
whereas the small wavelength behavior is universal, the large wavelength
behaviour strongly depends on the choice of vacuum.

Perturbations

Let us now apply this formalism to the quantum generation of perturbations. We
are interested in fluctuations of the Newtonian gravitational potential. Using
(275) and (276), we have

$\zeta=\Phi\left[1+{2\over 3}{\rho\over p+\rho}\right]+{2\over 3}{\rho\over
p+\rho}{\dot{\Phi}\over H}\ .$ (293)

Since $\Phi$ is constant in any phase where $p/\rho$ is constant (say matter
or radiation domination), then in such a phase, $\zeta$ is simply proportional
to the gravitational potential. We thus consider the scalar variable $\zeta$
in what follows and write:

$\zeta(\eta,{\bf
x})={1\over\sqrt{2}}\int{d^{3}k\over(2\pi)^{3/2}}\left[e^{-i{\bf k}.{\bf
x}}\zeta_{\bf k}(\eta)a^{\dagger}_{\bf k}+e^{i{\bf k}.{\bf x}}\zeta_{\bf
k}^{*}(\eta)a_{\bf k}\right]\ ,$ (294)

We have $\zeta_{\bf k}=v_{\bf k}/z$, where $v_{\bf k}$ satisfies, according to
(281),

$v^{\prime\prime}_{\bf k}+\left(c_{s}^{2}{\bf k}^{2}-{z^{\prime\prime}\over
z}\right)v_{\bf k}=0\ .$ (295)

One often characterizes the fluctuations through the power spectrum ${\cal
P}^{\zeta}_{\bf k}(\eta)$ which is defined from the $2$-point correlation
function:

$\langle 0\left|\zeta({\bf x},\eta)\zeta({\bf x}+{\bf
r},\eta)\right|0\rangle=\int_{k=0}^{k=+\infty}{dk\over k}{\sin kr\over
kr}{\cal P}^{\zeta}_{\bf k}(\eta)\ .$ (296)

Using the decomposition (294), one easily obtains

${\cal P}^{\zeta}_{\bf k}(\eta)={k^{3}\over 2\pi^{2}}\left|\zeta_{\bf
k}\right|^{2}={k^{3}\over 2\pi^{2}}{\left|v_{\bf k}\right|^{2}\over|z|^{2}}\
.$ (297)

As we have seen above, Eq. (295) has two distinct regimes depending of the
relative magnitude of $c_{s}k$ and $z^{\prime\prime}/z$. In the case of slow
roll inflation, the main dependence with time in $z$, as given in (279), comes
from the scale factor. Hence $z^{\prime\prime}/z\sim
a^{\prime\prime}/a\sim(aH)^{2}$. Hence we have to compare $c_{s}k$ with $aH$
i.e. the comoving wavelength $a/k$ with the sound horizon length $c_{s}/H$.

In the case of short wavelength (smaller than the sound horizon), the
normalized solution is

$v_{\bf k}={1\over(2kc_{s})^{1/2}}e^{ikc_{s}\eta}\ .$ (298)

For long wavelengths (larger than the sound horizon), it is

$v_{\bf k}=C_{\bf k}z\ ,$ (299)

where the constant $C_{\bf k}$ may be obtained by continuity between the two
approximate solutions at the scale of sound horizon: $|C_{\bf
k}|^{2}=1/(2kc_{s}z_{s})$, with $z_{s}$ the value of $z$ at horizon crossing.

We thus have the behavior indicated on Fig. 9: the quantum fluctuations are
created at small wavelength and grow until they cross the (sound) horizon.
From then on they grow mechanically until they reneter the horizon and are
observed in the matter dominated epoch. We thus obtain the power spectrum:

${\cal P}^{\zeta}_{\bf k}=\left.{k^{3}\over 2\pi^{2}}|C_{\bf
k}|^{2}\right|_{k=aH/c_{s}}=\left.{1\over c_{s}}{H^{2}\over
p+\rho}\left({H\over 2\pi}\right)^{2}\right|_{k=aH/c_{s}}\ .$ (300)

The spectral index is defined as

$n_{S}(k)-1={d\ln{\cal P}^{\zeta}_{\bf k}\over d\ln k}\ .$ (301)

Thus, inflation predicts a departure from a scale invariant spectrum
($n_{S}=1$).

Exercise D-1 : Show that, in the case of slow roll inflation (Section 0.2.3),
the spectral index is simply given by

$n_{S}=1-6\epsilon+2\eta\ .$ (302)

Hints: In this case, $c_{s}=1$. Moreover,

$n_{S}(k)-1\sim{1\over H}{d\ln{\cal P}^{\zeta}_{\bf k}\over
dt}=-6\left(1+{p\over\rho}\right)-2{\ddot{\phi}\over H\dot{\phi}}.$

Use then (90).


# Bridging the Gap: Addressing Discrepancies in Diffusion Model Training for
Classifier-Free Guidance

Niket Patel
University of California, Los Angeles
Los Angeles, CA 90095
<EMAIL_ADDRESS>
&Luis Salamanca
Swiss Data Science Center, ETH Zürich,
Zürich, 8092, Switzerland
<EMAIL_ADDRESS>
&Luis Barba
Swiss Data Science Center, Paul Scherrer Institute,
Villigen, 5232, Switzerland
<EMAIL_ADDRESS>
*Work done during internship at Swiss Data Science Center.

###### Abstract

Diffusion models have emerged as a pivotal advancement in generative models,
setting new standards to the quality of the generated instances. In the
current paper we aim to underscore a discrepancy between conventional training
methods and the desired conditional sampling behavior of these models. While
the prevalent classifier-free guidance technique works well, it’s not without
flaws. At higher values for the guidance scale parameter $w$, we often get out
of distribution samples and mode collapse, whereas at lower values for $w$ we
may not get the desired specificity. To address these challenges, we introduce
an updated loss function that better aligns training objectives with sampling
behaviors. Experimental validation with FID scores on CIFAR-10 elucidates our
method’s ability to produce higher quality samples with fewer sampling
timesteps, and be more robust to the choice of guidance scale $w$. We also
experiment with fine-tuning Stable Diffusion on the proposed loss, to provide
early evidence that large diffusion models may also benefit from this refined
loss function.

##  1 Introduction

Image generation has witnessed monumental shifts with the advent of diffusion
models, which have set new gold standards for generation quality. However, the
efficacy of conditional diffusion models can be heavily reliant on the
guidance method employed. However, we find the loss function used during
training may not directly align with the desired behavior of models during
sampling.

###  1.1 Background on Diffusion Models

Diffusion models function on the principle of score-based generative models
[10]. In practice, they are typically formulated as having some forward
direction that adds Gaussian noise, denoted $q(z_{t}|z_{t-1})$, and a
parameterized reverse process that attempts to predict the noise added to the
model, denoted $p_{\theta}(z_{t-1}|z_{t})$. In this paper we typically
distinguish between the Denoising Diffusion Probablistic Model (DDPM) and the
Denoising Diffusion Implicit Model (DDIM) sampling methods [3, 9]. DDIM uses
the same training procedure as DDPM, but enables to sample more efficiently
with fewer timesteps.

###  1.2 Guidance

We usually apply guidance to intervene in the reverse process to create
conditional samples. Classically, we can use classifier guidance on these
score-based models by training a separate classifier network [4]. For this
approach, the sample is then modified during training by integrating the
gradients from this classifier, as expressed in Eq. 1.

$\displaystyle\tilde{\epsilon}_{\theta}(z_{t},c)=\epsilon_{\theta}(z_{t},c)-w\sigma_{t}\nabla_{z_{t}}\log
p_{\phi}(c|z_{t})$ (1)

Notice here that multiplying the second term by $w$ means that we are
effectively guiding our sample towards the distribution given by
$p_{\phi}(c|z_{t})^{w}$, which for $w>1$ should decrease sample diversity and
increase sample class specificity [4].

However, due to the added complexity introduced by having to train a separate
model, capable of classifying with different levels of noise, we see a shift
towards classifier-free guidance [4]. This method, which involves taking a
linear combination of the outputs of both a conditional and an unconditional
models to predict the noise during sampling, as shown in Eq. 2, has set the
precedent for many subsequent works. This approach is facilitated by a
stochastic training regimen where conditional information is randomly dropped
out, or replaced with a null token, during training, effectively creating a
conditional and an unconditional model.

$\displaystyle\tilde{\epsilon}_{\theta}(z_{t},c)=(1+w)\epsilon_{\theta}(z_{t},c)-w\epsilon_{\theta}(z_{t},\emptyset)$
(2)

###  1.3 Shortcomings in Classifier-Free Guidance

Despite its widespread acceptance, classifier-free guidance presents some
drawbacks. Most notably, larger values of the guidance parameter $w$ may lead
to mode collapse and the generation of samples that deviate from expected
distributions. We can see an example of this phenomena in Fig. 1.
Nevertheless, in practice, high values of $w$ are still often preferred to
create samples that more accurately follow the conditioning information. The
conditional samples provided in the Stable Diffusion paper use $w=5$ and $w=3$
[8], and the primary text-to-video latent diffusion model in [1] uses guidance
values between $w=8$ and $w=13$.

Previous literature notes similar problems when high values of $w$ are used
[6]. This paper claims this occurs due to the scale of the resulting
classifier-free guided noise becoming too high. They accordingly attempt to
fix this by scaling to classifier-free guided sample to have the same variance
as the purely conditional sample, then taking a convex combination of this re-
scaled sample with a classical classifier-free guided sample.

##  2 Methods

Classically, classifier free-guidance (c.f.g) models are trained to predict
the noise added to the image with the loss function introduced by [3], where
conditioning $c$ is dropped out at random during the training process.

$\mathcal{L}_{\text{standard}}=||\epsilon-\epsilon_{\theta}(z_{t},c)||^{2}_{2}$

However, during sampling the unconditional noise estimate is corrected with an
extra term $w(\epsilon_{\theta}(z_{t},c)-\epsilon_{\theta}(z_{t},\emptyset))$,
which guides the generation. This disparity between the training loss and the
sampling process presents a disconnect: during sampling, a model that was
trained to minimize $\mathcal{L}_{\text{standard}}$ for both the conditional
and unconditional case, doesn’t accurately predict $\epsilon$ when its
prediction is modified by the classifier free guidance using Eq. 2. If we
train a model such that $\epsilon_{\theta}(z_{t},c)$ and
$\epsilon_{\theta}(z_{t},\emptyset)$ are both good estimates of $\epsilon$,
then while a convex combination of the two may also be a good estimate of
$\epsilon$, this isn’t necessarily the case for non-convex linear combinations
of the two, which is what we have when we use higher values of the guidance
scale parameter $w$.

We root our rationale in the previous observation, and propose to incorporate
the conditioning process more explicitly during the training process.
Formally, our approach is notated by the following equation:

$\mathcal{L}_{\text{updated}}=||\epsilon-(1+w)\epsilon_{\theta}(z_{t},c)+w\epsilon_{\theta}(z_{t},\emptyset)||^{2}_{2}$

In Fig. 1, row “Updated”, we observe how the generated distribution more
closely matches the training distribution, presumably due to the training loss
being better aligned with the sampling process.

Given that the proposed loss somewhat differentiates the conditional and
unconditional models, we don’t expect that the unconditional model would
produce high quality samples. It is also important to note that the proposed
method comes with the downside that the effective computation required per
batch is doubled, meaning in practice that, for some cases with limited VRAM,
we may need to half the batch size. Nevertheless, the proposed loss could also
be used in a fine-tuning step, which can potentially help alleviate the issues
mentioned for models trained with the standard classifier-free guidance
approach.

We find empirically that this method performs well for broader ranges of $w$
during sampling, as well as for values of $w$ different from those used in
training. We choose $w=1$ during training for most of the results presented in
this paper. A figure detailing an ablation study on this training guidance
parameter scale is presented in the appendix 4.

Figure 1: Comparison of standard classifier-free guidance (top row) to the
updated loss function (bottom row) to target distribution (left-most column).
The updated method more robustly represents accurate sampling guided from the
distribution $p(c|z_{0})^{w}$ for different values of $w$.

##  3 Experiments

###  3.1 CIFAR-10 Benchmarking

To showcase the potential of the proposed loss, we first train two models on
the CIFAR-10 dataset [5] from scratch, one with the updated loss and the other
with the standard c.f.g method. Accounting for the increased computational
demand of our updated loss, and to perform a fair comparison, batch sizes were
adjusted accordingly for the standard c.f.g. model. We used Frechet Inception
Distance (FID) [2] as our chosen metric for performance evaluation. The
quantitative results in Fig. 2 show a consistent performance improvement
across varying $w$ values and sampling steps, using a DDIM sampler [9].
Notably, the model trained with our updated loss retains its quality for fewer
sampling steps, leading to an improved sampling efficiency. Over all timesteps
tested in 2, we see an average 15.62% gain over the standard approach.

Figure 2: FID score with respect to number of DDIM sampling timesteps on two
individually trained models on the CIFAR-10 Dataset. Figure 3: Results of
fine-tuning Stable Diffusion on the Pokemon dataset. The top row is the model
fine-tuned with the standard loss, while the bottom row is fine-tuned with our
updated loss function.

###  3.2 Fine-Tuning Existing Models

To further evaluate our approach, we fine-tuned the Stable Diffusion v1.4
model [8] using our updated loss on the Pokemon text-to-image dataset [7].
Qualitative assessments of generation quality are showcased in Fig. 3. We find
that our model performs better at higher values of $w$, and follows more
precisely the prompt throughout the range of guidance values. We do note a
failure case here for $w=2$, which could be due to the likely aforementioned
poor performance of the unconditional model as a generator. More samples are
provided in the appendix Fig. 5.

##  4 Discussion

Our experiments highlight the potential of our updated loss function for
training and fine-tuning diffusion models. By directly addressing the
limitations of classifier-free guidance, we open the door to more reliable,
robust, and faster image generation. Significantly, the enhanced efficiency of
our model can allow for faster more accurate sampling, an improvement with
critical implications for applications demanding real-time responses.

A tangible limitation of our approach is its impact on the maximum allowable
batch size during training. Our method essentially halves the permissible
batch size, posing potential challenges for setups constrained by memory and
computational resources. We also loose the ability to generate samples
unconditionally, so we have then a trade-off between generality of the model
and specificity of conditional generation.

Further research could work towards deploying our enhanced loss function in
more scenarios, particularly with larger-scale models and bigger datasets.
Moreover, we envision an opportunity to further investigate and refine the
proposed loss for fine-tuning, and assess its potential as a default fine-
tuning step to improve established large diffusion models.

## References

  * [1] Andreas Blattmann, Robin Rombach, Huan Ling, Tim Dockhorn, Seung Wook Kim, Sanja Fidler, and Karsten Kreis.  Align your latents: High-resolution video synthesis with latent diffusion models, 2023.
  * [2] Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, and Sepp Hochreiter.  Gans trained by a two time-scale update rule converge to a local nash equilibrium, 2018.
  * [3] Jonathan Ho, Ajay Jain, and Pieter Abbeel.  Denoising diffusion probabilistic models, 2020.
  * [4] Jonathan Ho and Tim Salimans.  Classifier-free diffusion guidance, 2022.
  * [5] Alex Krizhevsky, Vinod Nair, and Geoffrey Hinton.  Cifar-10 (canadian institute for advanced research).  N/A, 2009.
  * [6] Shanchuan Lin, Bingchen Liu, Jiashi Li, and Xiao Yang.  Common diffusion noise schedules and sample steps are flawed, 2023.
  * [7] Justin N. M. Pinkney.  Pokemon blip captions.  https://huggingface.co/datasets/lambdalabs/pokemon-blip-captions/, 2022.
  * [8] Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Björn Ommer.  High-resolution image synthesis with latent diffusion models, June 2022.
  * [9] Jiaming Song, Chenlin Meng, and Stefano Ermon.  Denoising diffusion implicit models, 2022.
  * [10] Yang Song, Jascha Sohl-Dickstein, Diederik P. Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole.  Score-based generative modeling through stochastic differential equations, 2021.

## Appendix

### Ablation on guidance parameter used in training

We show in Fig. 4 the effects of using various guidance parameters in training
(using the same target distribution as in Fig. 1). We observe that we best
match the target distribution when the sampling $w$ matches that used in
training. It follows that, to some extent, samples from a model trained with
$w_{\text{train}}$, could be related to the model trained with $w=1$, by
$w_{\text{relative}}\approx\frac{w_{\text{sample}}}{w_{\text{train}}}$

To ensure that we can get high quality generation at various values for $w$
and evaluate effectively against the standard c.f.g. model, we choose $w=1$
for most of the experiments in this paper.

Figure 4: Ablation study of guidance parameter used during training (rows)
versus the guidance parameter used during sampling (columns).

### Stable Diffusion samples

Figure 5: More samples comparing the standard c.f.g. guidance and the updated
loss. These are all computed the same way as in 3 (with 100 DDIM timesteps).

### CIFAR-10 samples

Figure 6: Example of samples generated with the respective models at $w=1.8$.
Even though it is difficult to qualitatively assess these results, we can
observe a more robust generation for the updated loss across all the values of
$w$ used.