these operators satisfy the commutation relation with an unusual $-$ sign:

$[b_{0}(\vec{k}),b_{0}(\vec{q})^{\dagger}]=-\delta(\vec{k}-\vec{q})$ (6.39)

and all the other commutators equal to zero. The mode functions
$g_{B}(\tau,q)$ can be computed explicitly at the leading order in the slow-
roll expansion and one finds the de Sitter modes [151, 13]

$g_{B}(\tau,q)\equiv\frac{H}{\bar{M}_{P}\sqrt{12q}}\left(\frac{3}{q^{2}}+\frac{3i\tau}{q}-\tau^{2}\right)e^{-iq\tau}+{\cal
R}\mbox{-terms}$ (6.40)

where “${\cal R}\mbox{-terms}$” represent a contribution due to ${\cal R}$,
which, however, vanishes at superhorizon scales and, therefore, whose form is
not needed to compute the observable predictions.

Before moving to the predictions, let us note that the recent paper [159]
projects out $B$ as well as the other components of the massive spin-2 field
as an extra first principle. The consistency of this procedure, as discussed
in [159], requires $M_{2}/H>1/2$. Since $N_{e}\gg 1$, this condition implies
$M_{2}^{2}/H^{2}\gtrsim 1/N_{e}$, which corresponds to the absence of $B$ at
superhorizon scales. As a result, the analysis of [159] gives predictions
identical to those of [151] (summarized below) in its range of validity, that
is when $M_{2}/H>1/2$ holds. This is the case for all perturbations (tensor,
vector and scalar ones) modulo the fact that in [159] only Starobinsky
inflation was studied (namely no scalars $\phi_{a}$ were considered), while
[151] analyzed a general scalar sector.

####  6.3.4 Predictions

The observable predictions can now be extracted by computing the power
spectra. These are given by the vacuum quantum averages of products of fields
computed at equal times.

As we have seen in Sec. 4.2.3, quantum averages of Dirac-Pauli variables
involve an unusual $-i$ (see Eq. (4.33)) and quantum averages of the product
of two Dirac-Pauli variables involve an unusual $-1$ (see Eq. (4.36)). We now
illustrate how this leads to positive power spectra by compensating the
unusual $-1$ appearing in the commutator of the corresponding annihilation and
creation operators, Eq. (4.10).

For this purpose we consider the vacuum quantum average of
$B(\tau,\vec{x})B(\tau,\vec{y})$, which we call $\langle
B(\tau,\vec{x})B(\tau,\vec{y})\rangle_{\rm vac}$, whose Fourier transform with
respect to $\vec{x}-\vec{y}$ gives the power spectrum of $B$. Recalling now
that $B$ should be treated as a Dirac-Pauli variable and using Eq. (4.36), we
have

$\langle B(\tau,\vec{x})B(\tau,\vec{y})\rangle_{\rm vac}=-\frac{\langle 0|\eta
B(\tau,\vec{x})B(\tau,\vec{y})|0\rangle}{\langle 0|\eta|0\rangle},$ (6.41)

where $|0\rangle$ is the vacuum state (annihilated by all annihilation
operators) and we used the fact that the norm operator of a Dirac-Pauli
canonical variable is $\eta$ (see Eq. (4.39)). Now, from the discussion of
Secs. 4.2.2 and 4.2.3 we know that $P_{H}|0\rangle=|0\rangle$ and $P_{H}=\eta$
so

$\langle B(\tau,\vec{x})B(\tau,\vec{y})\rangle_{\rm vac}=-\langle
0|B(\tau,\vec{x})B(\tau,\vec{y})|0\rangle,$ (6.42)

where we have normalized the vacuum state in a way that $\langle
0|0\rangle=1$. By using now (6.37), (6.38) and (6.40) we find

$\lim_{\tau\to 0^{-}}\langle B(\tau,\vec{x})B(\tau,\vec{y})\rangle_{\rm
vac}=-\int\frac{d^{3}q^{\prime}d^{3}q}{(2\pi)^{3}}\frac{3H^{2}}{4\bar{M}_{P}^{2}}\frac{\langle
0|[b_{0}(-\vec{q^{\prime}}),b_{0}(-\vec{q})^{\dagger}]|0\rangle}{\sqrt{q^{\prime}q}(q^{\prime}q)^{2}}e^{i\vec{q}\cdot\vec{y}-i\vec{q^{\prime}}\cdot\vec{x}}$
(6.43)

and recalling the unusual minus sign in (6.39) we find

$\lim_{\tau\to 0^{-}}\langle B(\tau,\vec{x})B(\tau,\vec{y})\rangle_{\rm
vac}=\int\frac{d^{3}q}{(2\pi)^{3}}\frac{3H^{2}}{4\bar{M}_{P}^{2}}\frac{e^{i\vec{q}\cdot(\vec{y}-\vec{x})}}{q^{5}}$
(6.44)

On the other hand, the power spectrum of (the gradient of) $B$, which we call
$P_{B}$, satisfies

$\lim_{\tau\to 0^{-}}\langle B(\tau,\vec{x})B(\tau,\vec{y})\rangle_{\rm
vac}=\int\frac{d^{3}q}{4\pi
q^{3}}e^{i\vec{q}\cdot(\vec{y}-\vec{x})}\frac{P_{B}}{q^{2}}$ (6.45)

and, comparing this expression with (6.44),

$P_{B}=\frac{3}{2\bar{M}_{P}^{2}}\left(\frac{H}{2\pi}\right)^{2}>0.$ (6.46)

We see that the unusual minus sign appearing on the right-hand side of (6.41)
cancels exactly the unusual minus sign in (6.39) to give a positive power
spectrum. This is a particular case of the result in (4.36), which tells us
that that the quantum average of the square of any observable in any state is
not negative. The spectral index of $B$ defined by $n_{B}\equiv 1+\frac{d\ln
P_{B}}{d\ln q}$, where $P_{B}$ is computed at horizon exit,
$a(\tau)H(\tau)=q$, turns out to be [13] (at the leading order in the slow-
roll expansion)

$n_{B}=1-2\epsilon,$ (6.47)

where $\epsilon$ is the first slow-roll parameter in (6.15). This result holds
both in single-field and multi-field inflation. As shown in [13], although $B$
is not present in Einstein gravity $P_{B}$ is consistent with the most recent
CMB bounds [1] in some well-motivated inflationary scenarios as we will review
in this section. Future observations of the CMB background, such as CMB-S4
[162], will be able to test further these scenarios.

Analogously one can compute all the other power spectra, which turn out to be
positive as they should. As we have mentioned, $\mathcal{R}$ is the same as in
Einstein gravity (without the $W^{2}$ term) in the slow-roll approximation and
so the same is true for its power spectrum $P_{\mathcal{R}}$ (see e.g. [151,
163] for an explicit formula in the general multi-field case) and the
corresponding spectral index $n_{s}$ [151, 137, 163]. The correlation between
the power spectrum of this isocurvature mode and $\mathcal{R}$ is suppressed
at superhorizon scales [151], i.e. $P_{\mathcal{R}B}=0$. It is also convenient
to express $P_{B}$ in terms of $P_{\mathcal{R}}$ by introducing

$r^{\prime}\equiv\frac{P_{B}}{P_{\mathcal{R}}}$ (6.48)

to perform better the comparison with the observations. The power spectrum of
tensor perturbations, on the other hand, is given by [151]

$P_{t}=\frac{P_{tE}}{1+\frac{2H^{2}}{M_{2}^{2}}},$ (6.49)

where

$P_{tE}\equiv\frac{8}{\bar{M}_{\rm Pl}^{2}}\left(\frac{H}{2\pi}\right)^{2}$
(6.50)

is the tensor power spectrum in Einstein gravity. The denominator in (6.49)
comes from the quantity $\sqrt{1+2H^{2}/M_{2}^{2}}$ in the denominator of the
mode functions $y_{2}(\tau,q)$ in (6.32). When $M_{2}\gg H$ that denominator
is close to 1 as it should because in that limit the theory should approach
Einstein gravity; on the other hand, for $M_{2}\lesssim H$ it suppresses the
tensor power spectrum. By taking the ratio between (6.49) and
$P_{\mathcal{R}}$ we obtain the tensor-to-scalar ratio [151]

$r\equiv\frac{P_{t}}{P_{\mathcal{R}}}=\frac{r_{E}}{1+\frac{2H^{2}}{M_{2}^{2}}},$
(6.51)

where $r_{E}$ is the tensor-to-scalar ratio in Einstein gravity. So any model
that is in trouble because predicting a tensor-to-scalar ratio above the
observational limits [1] can be saved in agravity by taking $M_{2}/H$ small
enough (see below for some examples). Also, we can compute the spectral index
$n_{t}$ of tensor perturbations, defined by $n_{t}=\frac{d\ln P_{t}}{d\ln q}$,
where $P_{t}$ is computed at horizon exit, $a(\tau)H(\tau)=q$, to find

$n_{t}=-\frac{2\epsilon}{1+\frac{2H^{2}}{M_{2}^{2}}}$ (6.52)

independently of the number of inflatons. This result reduces to

$n_{t}=-\frac{r}{8}$ (6.53)

in single-field inflation (where $r_{E}=16\epsilon$).

In conclusion the predictions of agravity nicely reduce to those of Einstein
gravity for $M_{2}\gg H$. Using the observational bound on $H$ [1]

$H<2.7\times 10^{-5}\bar{M}_{P}\quad(2\sigma\leavevmode\nobreak\
\mbox{level})$ (6.54)

and Eq. (3.33) we see that all experimentally observable predictions of
agravity reduce to those of Einstein gravity (in any inflationary scenario)
for $f_{2}\gg\sqrt{2}\,2.7\times 10^{-5}\sim 10^{-5}$.

We now consider as examples some single-field inflationary setups and present
the corresponding specific predictions.

  * •

Planckion inflation. This is one of the choices of the inflaton that can be
realized far from the conformal regime (see Sec. 6.2.1). In the approximation
where the inflaton potential is quadratic, Eq. (6.20), the inflationary
predictions are [2, 151]

$n_{s}\approx 1-\frac{2}{N_{e}}\stackrel{{\scriptstyle N_{e}\approx
60}}{{\approx}}0.967,\qquad
r^{\prime}\approx\frac{3}{2N_{e}}\stackrel{{\scriptstyle N_{e}\approx
60}}{{\approx}}0.025\qquad
r\approx\frac{1}{1+\frac{2H^{2}}{M_{2}^{2}}}\frac{8}{N_{e}}\stackrel{{\scriptstyle
N_{e}\approx 60}}{{\approx}}\frac{0.13}{1+\frac{2H^{2}}{M_{2}^{2}}}.$

Taking into account the data in [1], the observed scalar amplitude
$P_{R}=M_{\varphi}^{2}N_{e}^{2}/(6\pi^{2}\bar{M}_{P}^{2})$ is reproduced
(taking $N_{e}\approx 60$) for

$M_{\varphi}\approx 1.4\times 10^{13}\leavevmode\nobreak\ {\rm GeV}.$ (6.55)

Quadratic inflation in Einstein gravity is ruled out by the most recent CMB
observations because it predicts a value of $r_{E}$ above the limits. As we
have seen, this problem can be cured in agravity for $H/M_{2}$ large enough,
see Eq. (6.51). The power spectrum $P_{B}$, parameterized by $r^{\prime}$, is
within the observational bounds [1] at $2\sigma$ level (see Fig. 7 of [13]).

  * •

Natural inflation. This is one of the choices of the inflaton that can be
realized in the quasi-conformal regime (see Sec. 6.2.2). In Einstein gravity
natural inflation is on the verge of being excluded because it predicts a
large value of $r_{E}$: the most recent observations of $n_{s}$ and $r$ have
retricted the parameter $f_{n}$ to lie in a very narrow band around
$f_{n}\approx 6.6\bar{M}_{P}$ [27]. Taking $f_{n}\approx 6.6\bar{M}_{P}$ we
have

$n_{s}\stackrel{{\scriptstyle N_{e}\approx 60}}{{\approx}}0.962,\qquad
r^{\prime}\stackrel{{\scriptstyle N_{e}\approx 60}}{{\approx}}0.011\qquad
r\approx\frac{r_{E}}{1+\frac{2H^{2}}{M_{2}^{2}}}\stackrel{{\scriptstyle
N_{e}\approx 60}}{{\approx}}\frac{0.061}{1+\frac{2H^{2}}{M_{2}^{2}}},$ (6.56)

Figure 3: The Einsteinian tensor-to-scalar ratio $r_{E}$ in natural inflation
as a function of $f_{n}$. In Einstein gravity $r_{E}$ is severely constrained
by the most recent observations published by the Planck collaboration in 2018
[1] because $f_{n}\approx 6.6\bar{M}_{P}$ [27]. The data also constrain
agravity for $M_{2}\ll H$ as $r_{E}$ is related to the power spectrum of the
extra isocurvature mode, however, in this case the observational bound is much
weaker and as a result the model does not suffer from any tension with data.

where we have set $\Lambda_{n}\approx 6\times 10^{-3}\bar{M}_{P}$ to fit the
observed value of the curvature power spectrum $P_{R}$. By taking $H/M_{2}$
large enough we can have $r$ well below the observational bounds. Moreover,
since $r_{E}\lesssim 0.1$ (see Fig. 3) the isocurvature power spectrum
parameterized by $r^{\prime}$ (see Eq. (6.48)) also satisfies the most recent
bounds at the 1$\sigma$ level, as illustrated by Fig. 7 of [13]. This setup
is, therefore, favored by data [1] compared to Planckion inflation. A more
detailed study of natural inflation in agravity is given by Ref. [27].

Other examples can be found in the literature, such as Starobinsky inflation,
hilltop inflation (see [151] and [27], respectively, for their study in
agravity) and Higgs inflation (whose predictions can be easily extracted by
combining the results in Einstein gravity [139, 164, 148, 149, 150] with the
general formulæ we have presented before). It is interesting to note that a
too large value of $r_{E}$ in “critical” Higgs inflation within the SM [148,
149, 150, 165, 166] can also be cured in agravity by taking $H/M_{2}$ large
enough (but other solutions are possible if critical Higgs inflation is
realized in an extended model [167]). Many more inflationary setups are
possible.

####  6.3.5 Causality

As mentioned in Sec. 4.2.4, the fact that the decay rate $\Gamma_{2}$ appears
with a negative sign in the resumed propagator (4.49) signals a violation of
causality at microscopic scales.

However, the possible acausal effects (if they really occur) are extremely
diluted by the expansion of the universe [27]. Indeed, whenever $M_{2}\lesssim
H$, the width of the massive spin-2 field, $\Gamma_{2}$, (the only source of
acausal behaviors) is strongly suppressed compared to $H$ (see Eq. (4.48)).
This is because $H$ must satisfy the observational bound in (6.54) and so
$\Gamma_{2}$ is suppressed compared to $M_{2}$ by the small factor
$f_{2}^{2}/(16\pi)\lesssim 10^{-11}$. Therefore, any acausal process is
quickly diluted by the expansion of the universe. In the opposite case,
$M_{2}\gg H$, the Weyl-squared term does not contribute to the experimentally
observable quantities as it essentially decouples (see Secs. 6.3.1, 6.3.2 and
6.3.3). Therefore, the Weyl-squared term respects all observational causality
bounds.

###  6.4 Classical metastability and chaotic inflation

As we have seen in Sec. 4.1, the classical runaways suggested by the
Ostrogradsky theorem are avoided if the typical energies are below the bounds
in (4.4) and (4.5). In this case the Ostrogradsky theorem does not imply an
instability, but at most a metastability.

In the context of inflation the typical energies we should look at are those
associated with inhomogeneities and anisotropies. This is because the
homogeneous and isotropic FRW metric is conformally flat, Eq. (6.12), and thus
has vanishing Weyl tensor. So only departures from homogeneity and isotropy
can activate the $W^{2}$ term. The homogeneous and isotropic ansatz used in
Sec. 6.2 (and regularly done in the literature on inflation) to describe the
classical part of slow-roll inflation is justified in Einstein gravity when
the energy scales of inhomogeneities ($E_{i}$) and anisotropies ($E_{a}$) are
small enough [168]:

$E_{i}\ll\left|\frac{1}{\Phi}\frac{dU}{d\Phi}\right|^{1/2},\qquad
E_{i},E_{a}\ll H,$ (6.57)

where $\Phi$ is a generic canonically normalized inflaton field. In agravity
we have the same conditions for $M_{2}\gtrsim H$ and even stronger conditions
for $M_{2}\lesssim H$ as the extra scale $M_{2}$ appears in the equations.

The second condition in (6.57), together with the observational bound on $H$
in Eq. (6.54), ensures that both (4.4) and (4.5) are satisfied for
$f_{2}\gtrsim 10^{-9}$. The first condition in (6.57) is instead model
dependent as the potential $U$ appears explicitly. In Planckion inflation in
the quadratic approximation of Eq. (6.20), using (6.55), one has

$\left|\frac{1}{\Phi}\frac{dU}{d\Phi}\right|^{1/2}\sim 10^{-5}\bar{M}_{P}$
(6.58)

and we have a similar result in natural inflation for the observationally
allowed values of the parameters given in Sec. 6.3.4 (as well as in
Starobinsky, Higgs and hilltop inflation [13]). So we see that both (4.4) and
(4.5) are satisfied for $f_{2}\gtrsim 10^{-9}$ also assuming the first
condition in (6.57). This bound on $f_{2}$ is very close but compatible with
the upper bound required to have a naturally small ratio $M_{h}/\bar{M}_{P}$
(see Sec. 5.2).

The proximity of these two bounds is actually a good thing because it allows
us to understand why we live in a nearly homogeneous and isotropic universe
[13]. Indeed, for $f_{2}$ close to the maximal value required by Higgs mass
naturalness, those patches where (6.57) is violated typically have scales of
inhomogeneities and/or anisotropies exceeding the bound in (4.4) because of
(6.58) and the fact that $H$ is typically191919This is the case, for example,
for all inflationary scenarios mentioned above: Planckion, Starobinsky,
hilltop, Higgs and natural inflation. around the maximal value allowed in
(6.54). Life is rendered impossible there by the Ostrogradsky instabilities
[13]. What we have found provides a mechanism to implement the original
chaotic inflation idea by Linde [20], which identified the patches compatible
with life with those satisfying (6.57). In Einstein gravity it is not clear if
the patches that were largely inhomogeneous and anisotropic (where Conditions
(6.57) were violated) are incompatible with life: the fact that we cannot use
an isotropic and homogeneous classical metric in that case is not sufficient
to reach this conclusion. On the other hand, the classical runaways that can
be triggered when those conditions are violated in the presence of the $W^{2}$
term certainly render the universe inhospitable. So we see that taking $f_{2}$
close to $10^{-8}$ is not only motivated by Higgs mass naturalness, but also
allows us to understand the homogeneity and isotropy of the universe.

###  6.5 Reheating

After inflation has taken place, the universe has to be heated. Several
articles have shown that it is possible to obtain successful reheating, even
keeping the Higgs mass naturally small compared to the Planck mass. Let us
give a couple of examples that are relevant for agravity.

In the case of Planckion, Starobinsky and Higgs inflation, which are viable
options only when we are far from the conformal regime, reheating can occur
because the inflaton(s) couple to the particles of the SM. For Higgs inflation
the couplings are simply the Yukawa and gauge couplings of the Higgs field
[169]. However, Higgs inflation does not occur when $f_{0}$ is small enough to
ensure a small $M_{h}/\bar{M}_{P}$ without fine tuning [7]. In the Planckion
and Starobinsky case the inflaton couplings to the SM particles appear because
the divergence of the dilatation current $\mathscr{D}^{\mu}$ and the trace of
the energy-momentum tensor are not zero [7]. Indeed, both
$\partial_{\mu}\mathscr{D}^{\mu}$ and $T^{\mu}_{\leavevmode\nobreak\ \mu}$
contain (among others) the fields of the SM. The inflaton(s) can, therefore,
decay into the SM particles because the typical value of the inflaton mass is
the one in Eq. (6.55) [7], which exceeds by several orders of magnitude the
masses of all SM particles.

In natural inflation, on the other hand, what allows us to heat the universe
is the presence of many weakly coupled scalars, which have sizable couplings
to the observed particles and undergo large quantum fluctuations of order
$H/(2\pi)$ [27]. Many weakly coupled scalars are, for example, present in
asymptotically-free extensions of the SM, which allow us to extend agravity up
to infinite energies (as seen in Sec. 2.2).

##  7 Phase transitions and gravitational waves

It is well known that the phase transitions in models featuring a scale
invariant or almost scale invariant behavior are generically of first order
and can, therefore, lead to observable GWs [170]. This behavior is typical of
theories where all scales are generated through dimensional transmutation, as
in this case the breaking of scale invariance only occurs radiatively.
Therefore, another important motivation for considering theories of this type
is the possibility to test them through GW experiments. These include ground-
based interferometers (such as advanced LIGO in Hanford and Livingston [171,
172], Cosmic Explorer [173, 174] and Einstein Telescope [175, 176, 177]) as
well as space-based interferometers (BBO [178, 179, 180], DECIGO [181, 182],
and LISA [183]), etc.

To illustrate this very interesting feature we consider the case of CW
symmetry breaking described in Sec. 3.1. This case has been amply studied in
the literature (see e.g. Refs. [184, 185, 186, 187] for some studies of GWs
and phase transitions in models with this type of breaking).

###  7.1 Thermal effective potential

In order to investigate the nature of a given phase transition, one should
take into account both quantum and thermal corrections. The effective
potential reads

$V_{\rm eff}(\chi,T)\equiv V_{\rm CW}(\chi)+V_{T}(\chi)+\Lambda_{0},$ (7.1)

where $V_{\rm CW}$ is given around the minimum $\chi_{0}$ in Eq. (3.7), the
term $V_{T}$ is the thermal correction to the effective potential at finite
temperature $T$, which at one-loop level is [188] (see also [189])

$V_{T}(\chi)=\frac{T^{4}}{2\pi^{2}}\left(\sum_{b}n_{b}J_{B}(M^{2}_{b}(\chi)/T^{2})-\sum_{f}n_{f}J_{F}(M^{2}_{f}(\chi)/T^{2})\right),$
(7.2)

with

$J_{B,F}(x)\equiv\int_{0}^{\infty}dy\,y^{2}\ln\left[1\mp\exp\left(-\sqrt{y^{2}+x}\right)\right],$
(7.3)

and we have included in $V_{\rm eff}(\chi,T)$ a constant term $\Lambda_{0}$ to
account for the observed value of the cosmological constant $\Lambda$ when
$\chi=\chi_{0}$: setting $T=0$, from (3.7) we obtain

$\Lambda=\Lambda_{0}-\frac{\bar{\beta}_{\lambda_{\chi}}}{16}\chi_{0}^{4}.$
(7.4)

In the expression in (7.2) the sum over $b$ runs over all bosons (with number
of degrees of freedom $n_{b}$), that over $f$ runs over all fermions (with
number of degrees of freedom $n_{f}$) and $M_{b,f}(\chi)$ are the
corresponding background-dependent masses.

In any QFT all background-dependent squared masses of spinor and vector fields
are positive, but in some cases the squared masses of the scalar fields (which
are the eigenvalues of the Hessian matrix of the classical potential) are
negative. This always happens in models where symmetry breaking is triggered
through the Higgs mechanism, via a tachyonic mass term in the classical
potential, because in this case the potential is concave for some values of
the scalar fields. When this happens $V_{\rm eff}$ has a non-vanishing
imaginary part, which signals the breaking of the perturbative expansion. This
cannot happen when symmetry breaking occurs through the CW mechanism: since
the classical potential is purely quartic in this case, Eq. (2.8), the Hessian
matrix

$M_{ab}^{2}\equiv\frac{\partial^{2}V_{\rm
ns}}{\partial\phi_{a}\partial\phi_{b}}$ (7.5)

evaluated at the CW flat direction must be proportional to $\chi^{2}$ via some
quartic couplings, namely

$M_{ab}^{2}(\chi)=\frac{1}{2}\lambda_{abcd}\nu_{c}\nu_{d}\chi^{2}.$ (7.6)

Clearly all eigenvalues of $M_{ab}^{2}$ must be positive to have a potential
bounded from below. Therefore, the CW symmetry breaking supports the validity
of perturbation theory.

###  7.2 Phase transitions

The quantum part of the effective potential is very shallow because its
departure from flatness is only due to perturbatively small loop effects.
Therefore, the thermal correction can dominate the effective potential. By
expanding the thermal functions for small values of their argument one finds

$\displaystyle J_{B}(x)$ $\displaystyle=$
$\displaystyle-\frac{\pi^{4}}{45}+\frac{\pi^{2}}{12}x-\frac{\pi}{6}x^{3/2}-\frac{x^{2}}{32}\ln\left(\frac{x}{a_{B}}\right)+...,$
(7.7) $\displaystyle J_{F}(x)$ $\displaystyle=$
$\displaystyle\frac{7\pi^{4}}{360}-\frac{\pi^{2}}{24}x-\frac{x^{2}}{32}\ln\left(\frac{x}{a_{F}}\right)+...,$
(7.8)

where $a_{B}=16\pi^{2}\exp(3/2-2\gamma_{E})$,
$a_{F}=\pi^{2}\exp(3/2-2\gamma_{E})$ and $\gamma_{E}$ is the Euler-Mascheroni
constant (the complete series expansion of $J_{B}$ and $J_{F}$ can be found in
e.g. [189]). So the thermal effects induce a positive quadratic term in $\chi$
near $\chi=0$ and generically lead to two minima, separated by a potential
barrier. If one lowers $T$ from very high values, where the symmetry is
unbroken, down to some critical value of the temperature, $T_{c}$, the two
minima are degenerate and a first-order phase transition starts. This typical
situation is illustrated in Fig. 4, where we show $V_{\rm eff}$ as a function
of $\chi$ for two values of the temperature, the critical temperature $T_{c}$
and $T=0$.

Figure 4: The effective potential adding a constant such that it vanishes at
$\chi=0$.

The absolute minimum of the thermal effective potential is at
$\langle\chi\rangle=0$ for $T>T_{c}$, while, for $T<T_{c}$, is at a non-
vanishing temperature-dependent value. In the latter case the decay rate per
unit volume $\Gamma$ of the false vacuum $\chi=0$ into the true vacuum
$\chi=\langle\chi\rangle\neq 0$ can be computed with the formalism of [190,
191, 192, 193]:

$\Gamma\approx\max\left(T^{4}\left(\frac{S_{3}}{2\pi
T}\right)^{3/2}\exp(-S_{3}/T)\,,\,\,\frac{1}{R_{4}^{4}}\left(\frac{S_{4}}{2\pi}\right)^{2}\exp(-S_{4})\right).$
(7.9)

Here $S_{d}$ is the Euclidean $d$-dimensional thermal effective action of
$\chi$ evaluated at the regular solution of the corresponding Euclidean EOM
with O($d$) symmetry (which thus only depends on some radial coordinate
$\rho$) satisfying the boundary condition
$\chi(\rho)\stackrel{{\scriptstyle\rho\to\infty}}{{\to}}0$. This solution is
known as the O($d$) bounce. Also, $R_{4}$ is the size of the $O(4)$ bounce.
The first entry in the $\max$ appearing in (7.9) is typically bigger than the
second one when the potential is dominated by thermal effects.

The phase transition occurs through the nucleation of bubbles of the true
vacuum inside a space filled by the false vacuum $\chi=0$. In the situation we
are studying, when $T$ goes below $T_{c}$, the scalar field $\chi$ is
typically trapped in the false vacuum until $T$ is much below $T_{c}$, in
other words the universe features a phase of strong supercooling.

To understand why this happens note that the bubbles created are diluted by
the expansion of the universe and they cannot collide until $T$ reaches the
nucleation temperature $T_{n}$, which is defined as the temperature when
$\Gamma/H^{4}=1$, where $H$ the is Hubble rate. When the decay is dominated by
the O(3) bounce, $T_{n}$ can be equivalently defined as the solution to the
following equation

$\frac{S_{3}}{T}-\frac{3}{2}\ln\left(\frac{S_{3}/T}{2\pi}\right)=4\ln\left(\frac{T}{H}\right)$
(7.10)

with rrespect to $T$. The key quantity that allows us to understand how long
cooling lasts is the inverse of the duration of the phase transition which,
following Ref. [186], we define through

$\beta_{n}\equiv\left[\frac{1}{\Gamma}\frac{d\Gamma}{dt}\right]_{T_{n}}.$
(7.11)

This quantity, using $dT/T=-Hdt$ and the fact that the transition is dominated
by thermal effects, can be computed with the formula

$\frac{\beta_{n}}{H_{n}}\approx\left[T\frac{d}{dT}(S_{3}/T)-4-\frac{3}{2}T\frac{d}{dT}\ln(S_{3}/T)\right]_{T=T_{n}},$
(7.12)

where $H_{n}$ is the Hubble rate evaluated at $T=T_{n}.$ Now, in a quasi-scale
invariant regime $S_{3}/T$ is nearly temperature independent and so the phase
transition lasts a very long time (supercooling). In this situation one can
neglect the last term in (7.12). Supercooling gives often a $T_{n}$ that is
several orders of magnitude below the typical scale of the potential,
$\chi_{0}$.

During supercooling the energy density is dominated by the vacuum energy of
$\chi$ and the universe grows exponentially like during primordial inflation,
but with Hubble rate

$H_{f}=\frac{\sqrt{\bar{\beta}_{\lambda_{\chi}}}\,\chi_{0}^{2}}{4\sqrt{3}\bar{M}_{P}}$
(7.13)

associated with the false vacuum. At the end of this inflationary period the
universe has to be reheated so the energy stored in $\chi$ should be
transferred to the SM particles. The details of reheating depend on the model
and on the specific transition under study so we do not discuss it here.

Finally, the strength of the phase transition is measured by the parameter
$\alpha$ defined as the ratio between

$\rho(T_{n})\equiv\left[\frac{T}{4}\frac{d}{dT}\Delta V_{\rm
eff}(\langle\chi\rangle,T)-\Delta V_{\rm
eff}(\langle\chi\rangle,T)\right]_{T=T_{n}},$ (7.14)

where $\Delta V_{\rm eff}(\langle\chi\rangle,T)\equiv V_{\rm
eff}(\langle\chi\rangle,T)-V_{\rm eff}(0,T)$, and the energy density of the
thermal plasma (more details can be found in Refs [194, 195]). So

$\alpha\equiv\frac{30\rho(T_{n})}{\pi^{2}g_{*}(T_{n})T_{n}^{4}},$ (7.15)

where $g_{*}(T)$ is the effective number of relativistic species in thermal
equilibrium at temperature $T$. The parameter $\alpha$ is typically very large
because of supercooling and so the phase transition is very strong.

###  7.3 Gravitational waves

When the temperature drops below $T_{n}$ GWs are produced. The production is
maximised when the energy scale $\Lambda_{G}$ above which gravity is softened
(see Sec. 5) is larger than the relevant scales. These include the typical
scale of the potential, $\chi_{0}$ (which in turn is typically much larger
than $T_{n}$, as we have seen), the values of fields and their derivatives as
well as the temperature $T_{\rm RH}$ associated with the reheating after
supercooling. Therefore, we focus from now on values of $\Lambda_{G}$ such
that this condition is satisfied.

In this case the gravitational corrections to the false vacuum decay studied
in Sec. 7.2 are amply negligible for $\Lambda_{G}\ll\bar{M}_{P}$, which is
necessary true if one wants to solve the hierarchy problem through softened
gravity (see Eq. (5.5)). These corrections are, indeed, suppressed by factors
at least as small as $\Lambda_{G}^{2}/\bar{M}_{P}^{2}$ [196, 197].

The dominant source of GWs are typically bubble collisions that take place in
the vacuum: in the era when $T$ reaches $T_{n}$ the energy density is
dominated since a long time by the vacuum energy density associated with
$\chi$, which leads to an exponential growth of the cosmological scale factor,
as we have seen. This inflationary behavior as usual dilutes preexisting
matter and radiation and thus one can typically neglect the GW production due
to turbulence and sound waves in the cosmic fluid [198].

From [199] (which used, among other things, the results of [200] based on the
“envelope approximation”) one finds the following GW spectrum due to vacuum
bubble collisions (valid in the presence of supercooling and $\alpha\gg 1$)

$h^{2}\Omega_{\rm GW}(f_{\rm GW})\approx 1.29\times
10^{-6}\left(\frac{H(T_{\rm
RH})}{\beta_{n}}\right)^{2}\left(\frac{100}{g_{*}(T_{\rm
RH})}\right)^{1/3}\frac{3.8(f_{\rm GW}/f_{\rm peak})^{2.8}}{1+2.8(f_{\rm
GW}/f_{\rm peak})^{3.8}},$ (7.16)

where $f_{\rm peak}$ is the red-shifted frequency peak today and is given by
[199]

$f_{\rm peak}\approx 3.79\times 10^{2}\frac{\beta_{n}}{H(T_{\rm
RH})}\frac{T_{\rm RH}}{10^{10}{\rm GeV}}\left(\frac{g_{*}(T_{\rm
RH})}{100}\right)^{1/6}\,{\rm Hz}.$ (7.17)

Some progress has been made to compute the GW spectrum beyond the envelope
approximation [201, 202, 203], but Eq. (7.16) remains to date a reasonable and
simple approximation for the bubble collisions that take place in the vacuum
[202] (which is what we are mainly interested in for quasi scale-invariant
theories).

The formalism explained in this and the previous sections can be used to
calculate the key features of the phase transitions and the GW spectrum in a
variety of models, where the scales are generated through DT. For example, the
EW phase transition and the corresponding GW spectrum has been studied in
[204, 205, 206]. The Peccei-Quinn (PQ) [207] case has been investigated in
[185, 186], which focused on effective models, and in [187], which considered
a fundamental asymptotically free Peccei-Quinn model previously built in
[208]. The analysis of Refs. [187, 208] also provides an example of how
predictive a theory with CDT can be, especially if combined with requirement
of having UV fixed points for all couplings: several couplings of this model
are indeed predicted in the IR, where the observations are made. The
predictivity of the model leads to a rigid dependence of the phase transition
(like its duration and the nucleation temperature) and the GW spectrum on the
PQ symmetry breaking scale and the QCD gauge coupling.

##  8 Summary and conclusions

To conclude, we have provided a review (with some original results) of
theories of all interactions where the mass scales are generated quantum
mechanically through DT. We have referred to this scenario as agravity to
emphasize that gravitational interactions are included.

Let us provide here a summary of the main points of the paper, which can be
useful after reading the bulk of the review (for a more introductory summary
see Sec. 1).

  * •

The general field content and action of agravity has been provided in Sec. 2.
Quantum consistency requires that, for a given field content, all terms
compatible with the symmetries must be present, including, among others, the
$W^{2}$ and the $R^{2}$ term. The (strong) equivalence principle (as precisely
stated in e.g. [9]) has been assumed. This fixes
$\Gamma_{\mu\,\nu}^{\,\sigma}$ to be the Levi-Civita connection. Any other
connection is either physically equivalent or implies a violation of the
equivalence principle. The matter content includes all types of fields and
interactions compatible with renormalizability.

In Sec. 2 we have also reviewed how agravity can be UV complete: it can flow
to conformal gravity ($1/f_{0}\to 0$ and $\xi_{ab}\to-\delta_{ab}/6$) in the
infinite energy limit provided that all matter couplings reach UV fixed points
(which can be either free or interacting). This allows us to avoid tachyonic
instabilities in the gravity sector.

  * •

Various mechanisms for DT in theories including gravity have been described in
Sec. 3.

In the perturbative approach of Sec. 3.1 a gravitational generalization of the
CW mechanism has been described, including a novel extension to the case of
multiple scalar fields. The key conditions for the gravitational CW mechanism
to successfully occur are given in (3.23). Both the standard [57, 58] and the
gravitational [2] CW mechanisms can contribute to $M_{h}$.

The non-perturbative generation of mass scales (including $\bar{M}_{P}$,
$\Lambda$ and $M_{h}$) has been described in Sec. 3.2.

After DT has generated $\bar{M}_{P}$, the graviparticles acquire a non-trivial
mass spectrum (that of quadratic gravity), which has been described in Sec.
3.3.

  * •

One of the graviparticle has spin-2, mass $M_{2}=f_{2}\bar{M}_{P}/\sqrt{2}$
and negative classical kinetic energy. Because of the latter abnormal feature
this particle is sometimes called “a ghost”. Nevertheless, in Sec. 4 we have
provided possible solutions of the related issues both at the classical and
quantum level. It must be kept in mind, though, that our approach has been
intrinsically perturbative because based on the metric split in Eq. (4.1).

At the classical level, we have reviewed how the runaway solutions suggested
by the Ostrogradsky theorem can be avoided. This is the case if the energies
associated with the derivative of the spin-2 fields respect Condition (4.4)
and those associated with the derivatives or mass terms of matter fields
and/or matter-field values times coupling constants fulfill (4.5). Both
conditions should be imposed at the space and time boundaries.

At the quantum level the theory has been shown to be unitary even beyond the
scattering theory: all probabilities are non-negative and sum up to one. This
emerges once a correct definition of probabilities (which takes into account
how they are actually introduced in experiments) is used: the frequency
argument of Sec. 4.2.2 (which for the first time has been given in a complete
pedagogical form) has shown that the indefinite metric is not the one that
should be used to compute probability. Other metrics, which are positively
defined and observable dependent, should be used to this purpose. Therefore,
the solution proposed here makes use of a modification of quantum mechanics.
Another non-standard quantum feature is the fact that the canonical
coordinates of the abnormal graviton must be treated as Dirac-Pauli variables,
as described in Sec. 4.2.3. Using this theoretical framework the ghost decay
rate is non-vanishing and positive; it is given at the leading non-trivial
order in $f_{2}$ (and at zero order in the other couplings) in Eq. (4.48).
Causality may be violated at microscopic scales, but, as discussed in Secs.
4.2.4 and 6.3.5, this is within the observational bounds, at least for the
values of $f_{2}$ of interest.

  * •

The presence of the extra graviparticles softens gravity when the energy
exceeds a critical value $\Lambda_{G}$ (see Fig. 2), which generically depends
on $M_{0}$ and $M_{2}$. Far from the conformal regime $\Lambda_{G}$ is given
in Eq. (5.1). Here, for the first time, we have given $\Lambda_{G}$ in the
quasi-conformal regime of Refs. [3, 27], that is $f_{0}\gg 1$ and
$\xi_{ab}\approx-\delta_{ab}/6$ (see Eq. (5.2)).

The softening of gravity has several phenomenological applications. The small
ratio $M_{h}/\bar{M}_{P}$ can be made natural thanks to the Higgs shift
symmetry that can be softly broken at the scale $M_{h}$ (see Sec. 5.2). In
order for this to happen not only gravity should be softened at scales
$\Lambda_{G}\lesssim 10^{11}$ GeV, but the matter sector should preserve this
shift symmetry as well. This leads to new physics not far from 10 TeV. On the
other hand, the softening of gravity is not sufficient to address the
cosmological constant problem, whose solution should be found somewhere else.

However, the softening of gravity does imply that microscopic BHs of Einstein
gravity (satisfying Condition (5.13)), which can be considered as the endpoint
of gravitational collapse, are replaced by horizonless ultracompact objects
that are free from any singularity, as discussed in Sec. 5.4. This has
interesting astrophysical implications, e.g. such objects can play the role of
DM. Moreover, in this way one can avoid the formation of microscopic BHs with
horizon $r_{h}$ smaller than $1/h_{\rm max}$, where $h_{\rm max}$ is the value
of the Higgs radial field for which the effective Higgs potential acquires its
maximum. This is useful because such BHs have been proved to be very dangerous
if the EW vacuum is metastable.

  * •

An important part of this work is the discussion of the physical implications
regarding the early universe in Sec. 6.

We have remarked that, since quadratic gravity can flow to conformal gravity
in the UV and the FRW metric is conformally flat, the initial-time
cosmological singularity of GR can be elegantly avoided.

We have mentioned examples of inflationary scenarios that work far from the
conformal regime (e.g. Planckion, Starobinsky, etc.) and others that work in
the quasi-conformal regime where $f_{0}\gg 1$ and
$\xi_{ab}\approx-\delta_{ab}/6$ (natural inflation and hilltop inflation).

Sec. 6.3 was devoted to the study of quantum fluctuations generated during
inflation and the corresponding predictions. By using the non-standard quantum
theory developed earlier in Sec. 4.2, the power spectra have been proved to be
positive (such proof appears here for the first time). When $M_{2}\gg H$ the
abnormal graviton does not affect the predictions, which then turn out to be
the same as in Einstein gravity. For $M_{2}\lesssim H$, on the other hand, we
find different predictions that can be tested with future CMB observations:
$r$ is suppressed as shown in Eq. (6.51) and (for $M_{2}/H\lesssim
1/\sqrt{N_{e}}$) an isocurvature mode of gravitational nature (one of the spin
components of the massive spin-2 field) appears and is compatible with the
most recent CMB data [1]. We have also briefly reviewed how reheating can
happen in Sec. 6.5.

Moreover, in Sec. 6.4, we have shown (for a larger class of inflationary
models compared to previous papers) that taking $f_{2}$ close to the maximal
value compatible with Higgs naturalness ($f_{2}\lesssim 10^{-8}$) allows us to
explain the nearly homogeneity and isotropy of our universe: patches where the
FRW ansatz in the classical part of inflation is not justified (which do not
satisfy (6.57)) are rendered inhospitable by the Ostrogradsky instabilities.
On the other hand, patches (like ours) where those conditions are satisfied
avoid the Ostrogradsky runaway solutions. So the presence of the $W^{2}$ term
gives us a concrete implementation of Linde’s chaotic inflation.

  * •

Last but not least, theories that are nearly scale invariant (like those where
DT is a small perturbative effect) typically predict very long and strong
phase transitions, which can lead to observable GWs at detectors. We have
reviewed this topic in Sec. 7, which also provided the main theoretical tools
to analyse phase transitions and GWs in this type of theories: the effective
thermal potential, the theory of vacuum decay at finite temperature and the
main source of GWs. It has also been pointed out for the first time that the
CW symmetry breaking supports the validity of perturbation theory in studying
phase transition because the imaginary part of the thermal effective potential
must vanish in that case. One should keep in mind that in order to maximise
the production of GWs, $\Lambda_{G}$ has to be chosen above the typical scale
of the potential, the values of the fields and their derivatives as well as
$T_{\rm RH}$.

Some challenges are left for future research. First, our approach to quantum
gravity has been intrinsically perturbative (see Eq. (4.1)). How to define
agravity non perturbatively is not clear. This does not only regards the
matter sector202020Currently we do not have a non-perturbative operative
definition (such as the lattice) of realistic QFTs that feature chiral fermion
representations of the gauge group, such as the SM., but also the gravity
sector. Perhaps this could be done with some sort of lattice Euclidean path
integral with respect to both the metric and the matter fields.

Another interesting topic for future developments is the cosmological constant
problem. Is it possible to have a naturally small $\Lambda/\bar{M}_{P}$? If
not, is it possible to understand the smallness of the cosmological constant
in some other way?

The list of unanswered question does not stop here. For example, can we have a
simultaneous solution of all fine-tuning problems of the SM and GR? Can we
understand the flavor and gauge-group structure of the SM? Etc.

### Acknowledgments

I thank the International Journal of Modern Physics A for inviting me to write
this review. I also thank my collaborators on the topics discussed here: A.
Strumia, H. Veermäe, G. F. Giudice, G. Isidori, K. Kannike, L. Pizza, A.
Racioppi, M. Raidal, G. M. Pelaggi, A. Ghoshal, A. D. Plascencia, F. Sannino,
J. Smirnov.

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