Inflation, space-borne interferometers
and the expansion history of the Universe
Massimo Giovannini 111Electronic address<EMAIL_ADDRESS>
Department of Physics, CERN, 1211 Geneva 23, Switzerland
INFN, Section of Milan-Bicocca, 20126 Milan, Italy

Abstract

According to the common wisdom, between a fraction of the mHz and few Hz the
spectral energy density of the inflationary gravitons can be safely
disregarded even assuming the most optimistic sensitivities of the space-borne
detectors. This conclusion is evaded if, prior to nucleosynthesis, the post-
inflationary evolution includes a sequence of stages expanding either faster
or slower than radiation. As a consequence below a fraction of the Hz the
spectral energy density of the relic gravitons may exceed (even by eight
orders of magnitude) the conventional signal obtained under the hypothesis of
radiation dominance throughout the whole expansion history prior to the
formation of light nuclei. Since the slopes and the amplitudes of the spectra
specifically reflect both the inflationary dynamics and the subsequent
decelerated evolution, it is possible to disentangle the contribution of the
relic gravitons from other (late-time) bursts of gravitational radiation
associated, for instance, with a putative strongly first-order phase
transition at the TeV scale. Hence, any limit on the spectral energy density
of the relic gravitons in the mHz range simultaneously constrains the post-
inflationary expansion history and the inflationary initial data.

##  1 Introduction

A striking prediction of the early evolution of the space-time curvature is
the formation of a stochastic background of relic gravitons [2, 3, 4] whose
frequencies may extend between the aHz and the GHz regions. As originally
pointed out in Ref. [5], the spectral energy density of relic gravitons is
quasi-flat between $100$ aHz and $100$ MHz for the inflationary scenarios
relying on the conventional slow-roll evolution. Since the quasi-flat plateau
corresponds to wavelengths that left the Hubble radius during inflation and
reentered after radiation was already dominant, the spectral energy density
can only reach its maximum in the aHz region where the signal scales as
$\nu^{-2}$ with the comoving frequency $\nu$ [6]. In the aHz interval the
temperature and the polarization anisotropies of the cosmic microwave
background are customarily employed to infer the tensor to scalar ratio
$r_{T}$ here assumed in the range $r_{T}=r_{T}(\nu_{p})\leq 0.06$, as
suggested by recent determinations [7, 8, 9]. For the record
$\nu_{p}=k_{p}/(2\pi)=3.09\,\mathrm{aHz}$ and
$k_{p}=0.002\,\,\mathrm{Mpc}^{-1}$ denotes the pivot scale at which the scalar
and tensor power spectra are conventionally assigned when the relevant
wavelengths are larger than the Hubble radius prior to matter-radiation
equality.

Depending on the value of $r_{T}$, the spectral energy density in critical
units222Instead of working with the spectral energy density of the relic
gravitons in critical units (conventionally denoted by
$\Omega_{gw}(\nu,\tau_{0})$) it is practical to introduce
$h_{0}^{2}\,\Omega_{gw}(\nu,\tau_{0})$ where $h_{0}$ is the indetermination of
the Hubble rate. The spectral energy density of the relic gravitons does not
coincide with their energy density in critical units which is instead
frequency-independent. We also note that the frequencies are often mentioned
in the text by using the standard metric prefixes of the international system
of units. So, for instance, $\mathrm{aHz}=10^{-18}\,\,\mathrm{Hz}$,
$\mathrm{mHz}=10^{-3}\,\,\mathrm{Hz}$ and so on. well above $100$ aHz is
$h_{0}^{2}\Omega_{gw}(\nu,\tau_{0})<{\mathcal{O}}(10^{-17})$ and this estimate
includes the effect of the various damping source such as the late-dominance
of the dark energy, the evolution of the relativistic species and the free-
streaming of the neutrinos [10, 11]. For all these reasons the spectral energy
density of inflationary origin is too small to be detected by either ground
based or space-borne detectors even in their most advanced versions. At the
moment the only direct bounds on the relic gravitons come from the audio band
and depend upon the spectrum of the signal but we could anyway say that, for a
nearly scale-invariant spectrum,
$h_{0}^{2}\Omega_{gw}(\nu,\tau_{0})<{\mathcal{O}}(10^{-9})$ between $10$ Hz
and $80$ Hz [12, 13] (see also [14] for a recent review including earlier
bounds). For a physical comparison between the ground-based detectors and the
(futuristic) space-borne interferometers the spectral energy density can be
usefully expressed in terms of the chirp amplitude $h_{c}(\nu,\tau_{0})$ [14]
when the typical frequencies fall in the audio band:

$h_{0}^{2}\Omega_{gw}(\nu,\tau_{0})=6.26\times
10^{-9}\biggl{(}\frac{\nu}{0.1\,\,\mathrm{kHz}}\biggr{)}^{2}\biggl{[}\frac{h_{c}(\nu,\tau_{0})}{10^{-24}}\biggr{]}^{2}.$
(1.1)

If we read Eq. (1.1) from left to right we can argue that to probe
$h_{0}^{2}\Omega_{gw}(\nu,\tau_{0})={\mathcal{O}}(10^{-9})$ we would need a
sensitivity in the chirp amplitude ${\mathcal{O}}(10^{-24})$ for a typical
frequency $\nu={\mathcal{O}}(100)$ Hz. From right to left Eq. (1.1) suggests
instead that, for the same sensitivity in $h_{c}(\nu,\tau_{0})$, the minimal
detectable $h_{0}^{2}\Omega_{gw}(\nu,\tau_{0})$ gets comparatively
smaller333Besides the the absence of seismic noise this is probably one of
strongest arguments in favour of space-borne detectors for typical frequencies
ranging between a fraction of the mHz and the Hz. . This is why the minimal
detectable spectral energy density could be
$h_{0}^{2}\Omega_{gw}(\nu,\tau_{0})={\mathcal{O}}(10^{-11})$ or even
$h_{0}^{2}\Omega_{gw}(\nu,\tau_{0})={\mathcal{O}}(10^{-15})$ under the
hypothesis that the same sensitivity reached in the audio band for the chirp
amplitude can also be achieved in the mHz range. With this hope, various
space-borne detectors have been proposed so far: the Laser Interferometric
Space Antenna (LISA) [15, 16], the Deci-Hertz Interferometer Gravitational
Wave Observatory (DECIGO) [17, 18], the Ultimate-DECIGO [19] (conventionally
referred to as U-DECIGO), the Big Bang Observer (BBO) [20]. This list has been
recently enriched by the Taiji [21, 22] and by the TianQin [23, 24]
experiments. Since these instruments are not yet operational (but might come
into operation within the next twenty years) their actual sensitivities are
difficult to assess, at the moment. However, without dwelling on the specific
nature of the noise power spectra, Eq. (1.1) shows that, as long as
$h_{c}={\mathcal{O}}(10^{-23})$ the space-borne detectors might probe
$h_{0}^{2}\Omega_{gw}(\nu,\tau_{0})={\mathcal{O}}(10^{-14})$ for
$\nu_{S}={\mathcal{O}}(0.01)$ Hz and this is, roughly speaking, the daring
expectation of DECIGO [17, 18] and of U-DECIGO [19].

According to the standard lore (see e.g. [15, 16, 17, 18]) the astrophysical
sources of gravitational radiation (i.e. mostly white dwarves and solar masses
black holes) dominate the signal below $0.1$ Hz, while the bursts of gravitons
from the TeV physics are unlikely in the standard electroweak theory but
should be anyway subleading in comparison with the galactic foregrounds.
Because of the relative smallness of its spectral energy density, the
inflationary background of relic gravitons is always disregarded but this
conclusion is only based on a specific expansion history and it can be evaded
if, prior to nucleosynthesis, the evolution of the background is not
constantly dominated by radiation. Indeed, the flatness of
$h_{0}^{2}\Omega_{gw}(\nu,\tau_{0})$ for frequencies larger than $100$ aHz is
not only determined by the inflationary evolution when the relevant
wavelengths exit the Hubble radius but also by the expansion rate at reentry
[25, 26]. The high-frequency signal is maximized by a long stage expanding at
a rate that is slower than radiation [25, 26] and this possibility is realized
in various classes of quintessential inflationary scenarios [27, 28, 29, 30]
(see also [31, 32]). The signal from a long stiff phase does not imply a
reduction of $h_{0}^{2}\Omega_{gw}(\nu,\tau_{0})$ in the aHz region so that
the high-frequency measurements of wide-band detectors and the low-frequency
determinations of $r_{T}$ can be simultaneously constrained within an accurate
numerical framework [33, 34]. In this context the potential signal might be
sufficiently large both in the aHz region and in the audio band.

While in the case of a stiff post-inflationary phase the spike typically
arises for frequencies between the GHz and $100$ GHz it is also possible to
have different profiles of the spectral energy density with a number of
different peaks when the frequency is comparatively smaller or even much
smaller than the MHz. There is then a trade-off between the smallness of the
frequency and the magnitude of $h_{0}^{2}\Omega_{gw}(\nu,\tau_{0})$ [33, 34].
Since the most general post-inflationary expansion rate consists of a series
of successive stages expanding at different rates that are either faster or
smaller than radiation444 Incidentally, within the present approach the
possibility of a signal in the nHz band (recently suggested by the pulsar
timing arrays [36, 37, 37, 39]) has been specifically scrutinized by
considering a wide range of possibilities including the presence of late-time
stages of inflationary expansion [35]. In this paper we are instead concerned
with the mHz range and the potential signal from pulsar timing arrays will not
be specifically discussed. [33, 34, 35], in this paper we are going to argue
that the general approach previously explored is also applicable also to
smaller frequencies in the mHz region. In the presence of a modified post-
inflationary expansion rate the standard inflationary signal computed in Refs.
[5, 6] can be much larger below the Hz and potentially dominant against the
bursts of gravitational radiation from strongly first-order phase transitions.

The layout of this paper is the following. In section 2 the inflationary power
spectra are computed after the relevant wavelengths reentered the Hubble
radius during a post-inflationary stage that differs from radiation. In
section 3 we examine the general case where each stage of a larger sequence of
phases expands at a rate that is either faster or slower than radiation. In
this situation the spectral energy density exhibits a succession of peaks and
throughs whose frequencies are solely determined by the curvature scale. Since
the slopes of of the humps in $h_{0}^{2}\Omega_{gw}(\nu,\tau_{0})$ depend both
on the inflationary stage and on the post-inflationary evolution, in section 4
it is shown that the current limits from ground-based detectors already pin
down a well defined region of the parameter space that should be further
explored by space-borne interferometers. Section 5 contains the concluding
remarks and some comments on the future perspectives.

##  2 Spectral energy density of the inflationary gravitons

The effect of the post-inflationary evolution is not, as sometimes argued, a
purely kinematical problem that is virtually disentangled from the dynamical
evolution of the tensor modes. On the contrary the enhancement of the spectral
energy density at late times is also determined by the early expansion: it is
because of the successive occurrence of an inflationary stage and of the late
post-inflationary evolution that $\Omega_{gw}(k,\tau)$ may be enhanced at high
and intermediate frequencies [25, 26]. The flat spectrum of relic gravitons
for frequencies larger than $100$ aHz only arises if the relevant wavelengths
exit the Hubble radius during inflation and reenter in a radiation-dominated
stage of expansion, as originally assumed in Refs. [5, 6]. We may consider, in
this respect the standard form of the spectral energy density in critical
units that can be written as [14]:

$\Omega_{gw}(k,\tau_{0})=\frac{1}{24H^{2}\,a^{2}}\biggl{[}Q_{T}(k,\tau)+k^{2}\,P_{T}(k,\tau)\biggr{]},$
(2.1)

where $a(\tau)$ is the scale factor of a conformally flat background geometry,
$\tau$ is the conformal time and $H$ is the standard Hubble expansion rate. In
Eq. (2.1) $Q_{T}(k,\tau)$ and $P_{T}(k,\tau)$ are the tensor power spectra
that are defined from the evolution of the mode functions $G_{k}(\tau)$ and
$F_{k}(\tau)$:

$Q_{T}(k,\tau)=\frac{4\ell_{P}^{2}}{\pi^{2}}\,k^{3}\,\bigl{|}\,G_{k}(\tau)\bigr{|}^{2},\qquad\qquad
P_{T}(k,\tau)=\frac{4\ell_{P}^{2}}{\pi^{2}}\,k^{3}\,\bigl{|}\,F_{k}(\tau)\bigr{|}^{2},$
(2.2)

where $\ell_{P}=\sqrt{8\,\pi\,G}$; in what follows the notations for the
Planck mass are given by $\overline{M}_{P}=M_{P}/\sqrt{8\,\pi}=1/\ell_{P}$ and
$\overline{M}_{P}$ is the reduced Planck mass.

The rescaled mode functions $f_{k}(\tau)=a(\tau)F_{k}(\tau)$ and
$g_{k}(\tau)=a(\tau)G_{k}(\tau)$ obey, in the present context, the standard
evolution equations:

$f_{k}^{\prime\prime}+\biggl{[}k^{2}-\frac{a^{\prime\prime}}{a}\biggr{]}f_{k}=0,\qquad\qquad
g_{k}=f_{k}^{\prime}-{\mathcal{H}}f_{k}.$ (2.3)

In Eq. (2.3) the prime denotes a derivation with respect to the conformal time
coordinate $\tau$; we also use the standard notation ${\mathcal{H}}=aH$ where
${\mathcal{H}}=a^{\prime}/a$ and $H$ is the conventional Hubble rate. Within
the WKB approximation Eq. (2.3) is approximately solved in the two
complementary regimes where $k^{2}$ is either larger or smaller than
$|\,a^{\prime\prime}/a|$. In particular when
$k^{2}\gg|\,a^{\prime\prime}/a\,|$ the mode functions ($f_{k}$, $g_{k}$)
oscillate while ($F_{k}$, $G_{k}$) are also suppressed as $1/a$. In the
opposite regime (i.e. $k^{2}\ll|\,a^{\prime\prime}/a|$) $f_{k}(\tau)$ is said
to be superadiabatically amplified, according to the terminology originally
introduced in Refs. [2, 3]. The oscillating and the superadiabatic regimes are
separated by a region where the solutions change their analytic behaviour and
these turning points are defined as solutions of the approximate equation
$k^{2}\simeq|\,a^{\prime\prime}/a|$ that can also be rewritten as:

$k^{2}=a^{2}\,H^{2}\biggl{[}2-\epsilon(a)\biggr{]},\qquad\qquad\epsilon(a)=-\frac{\dot{H}}{H^{2}}.$
(2.4)

During the inflationary stage of expansion $\epsilon\ll 1$ denotes the
standard slow-roll parameter; conversely in the post-inflationary phase the
background decelerates (but still expands) and $\epsilon(a)={\mathcal{O}}(1)$.
If $\epsilon\neq 2$ both turning points are regular and this means that Eq.
(2.4) can be approximately solved by $k\simeq aH$. For instance when a given
wavelength crosses the Hubble radius during inflation we have that
$\epsilon\ll 1$ and $k\simeq a_{ex}\,H_{ex}$ that also means, by definition,
$k\tau_{ex}\simeq 1$. Similarly if the given wavelength reenters in a
decelerated stage of expansion different from radiation we also have that
$k\simeq a_{re}\,H_{re}$. Finally if the reentry occurs in the radiation stage
we have that $\epsilon_{re}\to 2$ and the condition (2.4) implies that
$k\tau_{re}\ll 1$.

These considerations suggest that the spectral energy density of the relic
gravitons depends both on the exit and on the reentry of the given wavelength
and for this purpose it is appropriate to express the mode functions in the
Wentzel-Kramers-Brillouin (WKB) approximation under the further assumption
that $a_{re}\gg a_{ex}$: this requirement is verified as long as the Universe
expands as it is aways the case throughout the present discussion. The initial
conditions for the evolution of the mode functions are then assigned during
the inflationary stage and before the corresponding wavelengths exit the
Hubble radius; in this regime $f_{k}(\tau)$ and $g_{k}(\tau)$ are simply plane
waves obeying the Wronskian normalization condition:

$f_{k}(\tau)\,g_{k}^{\ast}(\tau)-f_{k}^{\ast}(\tau)g_{k}(\tau)=i,$ (2.5)

as required by the canonical commutation relations of the corresponding field
operators [14]. From the continuity of the mode functions across the turning
points of the problem, the expression of $F_{k}(\tau)$ becomes:

$F_{k}(\tau)=\frac{e^{-ik\,\tau_{ex}}}{a\sqrt{2\,k}}\,{\mathcal{Q}}_{k}(\tau_{ex},\tau_{re})\,\biggl{(}\frac{a_{re}}{a_{ex}}\biggr{)}\biggl{\\{}\frac{{\mathcal{H}}_{re}}{k}\sin{[k(\tau-\tau_{re})]}+\cos[k(\tau-\tau_{re})]\biggr{\\}},$
(2.6)

and it is valid for $k\tau\gg 1$ when all the corresponding wavelengths are
shorter than the Hubble radius. In the same approximation $G_{k}(\tau)$ is:

$G_{k}(\tau)=\frac{e^{-ik\,\tau_{ex}}}{a}\,\,\sqrt{\frac{k}{2}}\,\,{\mathcal{Q}}_{k}(\tau_{ex},\tau_{re})\,\biggl{(}\frac{a_{re}}{a_{ex}}\biggr{)}\biggl{\\{}\frac{{\mathcal{H}}_{re}}{k}\cos{[k(\tau-\tau_{re})]}-\sin[k(\tau-\tau_{re})]\biggr{\\}}.$
(2.7)

Equations (2.6)–(2.7) are valid in the two concurrent limits $k\tau\gg 1$ and
$a_{re}\gg a_{ex}$ but they are otherwise general since the expansion rates at
$\tau_{ex}$ and $\tau_{re}$ have not been specified. In Eqs. (2.6)–(2.7)
${\mathcal{Q}}_{k}(\tau_{ex},\tau_{re})$ denotes a complex amplitude defined
as:

${\mathcal{Q}}_{k}(\tau_{re},\tau_{ex})=1-(i\,k+{\mathcal{H}}_{ex})\int_{\tau_{ex}}^{\tau_{re}}\frac{a_{ex}^{2}}{a^{2}(\tau)}\,\,d\,\tau.$
(2.8)

The integral at the right hand side of Eq. (2.8) depends on the evolution
between $\tau_{ex}$ and $\tau_{re}$ but its value is always subleading so that
it is generally true555Only if $a^{2}(\tau)\simeq 1/{\mathcal{H}}$ the
contribution of the integral of Eq. (2.8) is relevant and it corresponds to
the possibility of extended stiff phases where, for instance, the energy
density is dominated by the kinetic energy of a scalar field [25, 26]; in this
case the spectral energy density and the other observables inherit a
logarithmic correction. that
$\bigl{|}{\mathcal{Q}}_{k}(\tau_{re},\tau_{ex})\bigr{|}^{2}\to 1$.

In Eqs. (2.6)–(2.7) we may note the appearance of standing waves that are
characteristic both in the case of relic gravitons and in the case of scalar
metric fluctuations and they are often referred to as Sakharov oscillations
because they arose, for the first time, in the pioneering contribution of Ref.
[40] (see also [41]). When Eqs. (2.6)–(2.7) are inserted into Eq. (2.2) we can
obtain the corresponding power spectra that determine the final expression of
the spectral energy density in critical units through Eq. (2.1):

$\Omega_{gw}(k,\tau)=\frac{k^{4}}{12\,\pi^{2}\,a^{4}\,H^{2}\,\overline{M}_{P}^{2}}\bigl{|}{\mathcal{Q}}(\tau_{ex},\tau_{re})\bigr{|}^{2}\biggl{(}\frac{a_{re}}{a_{ex}}\biggr{)}^{2}\biggl{(}1+\frac{{\mathcal{H}}_{re}^{2}}{k^{2}}\biggr{)}\biggl{[}1+{\mathcal{O}}\biggl{(}\frac{{\mathcal{H}}}{k}\biggr{)}\biggr{]}.$
(2.9)

Equation (2.9) is valid in the limit ${\mathcal{H}}/k\ll 1$ and this condition
is equivalent to $k\tau\gg 1$ since ${\mathcal{H}}={\mathcal{O}}(\tau^{-1})$.
If a given wavelength exits the Hubble radius during inflation we have:

$k\simeq
a_{ex}\,H_{ex}\simeq-\frac{1}{(1-\epsilon)\tau_{ex}}=-\frac{\beta}{\tau_{ex}},$
(2.10)

where we denoted, for the sake of convenience, $\beta=1/(1-\epsilon)$ and
$\epsilon_{ex}=\epsilon\ll 1$. When the same wavelength reenters during a
stage that is not dominated by radiation, $\epsilon_{re}\neq 2$ in Eq. (2.4)
so that, at reentry, $k\simeq{\mathcal{H}}_{re}=a_{re}\,H_{re}$.

If a given wavelength $2\pi/k$ reenters across two different regimes
characterized by a different expansion rate, the scale factor during the
$i$-th stage of expansion can be parametrized, for instance, as:

$a_{i}(\tau)=\biggl{(}\frac{\tau}{\tau_{i}}\biggr{)}^{\delta_{i}},\qquad\delta_{i}>0,\qquad\delta_{i}\neq
1,\qquad\tau\leq\tau_{i}.$ (2.11)

For $\tau>\tau_{i}$ the scale factor during the $(i+1)$-th stage of expansion
is: modified as

$a_{i+1}(\tau)\simeq\biggl{[}\frac{\delta_{i}}{\delta_{i+1}}\biggl{(}\frac{\tau}{\tau_{i}}-1\biggr{)}+1\biggr{]}^{\delta_{i+1}},\qquad\delta_{i+1}>0,\qquad\delta_{i+1}\neq
1,\qquad\tau\geq\tau_{i}.$ (2.12)

Even if $\delta_{i+1}$ and $\delta_{i}$ can be equal, the situation we want to
discuss now is the one where $\delta_{i+1}\neq\delta_{i}$. Let us then go back
to Eq. (2.9) and evaluate the spectral energy density for the modes reentering
for $\tau<\tau_{i}$

$\Omega^{(i)}_{gw}(k,\tau)=\frac{4}{3\pi}\biggl{(}\frac{H_{1}}{M_{P}}\biggr{)}^{2}\,\biggl{(}\frac{H_{1}\,a_{1}^{2}}{H\,a^{2}}\biggr{)}^{2}\biggl{|}\frac{a_{i}H_{i}}{a_{1}\,H_{1}}\biggr{|}^{4-\beta}\,\biggl{|}\frac{k}{a_{i}\,H_{i}}\biggr{|}^{n_{T}^{(i)}},\qquad\qquad
k>a_{i}\,H_{i},$ (2.13)

where we took into account that $k\simeq a_{re}H_{re}={\mathcal{H}}_{re}$. In
Eq. (2.13) $H_{1}$ is coincides with the maximal value of the Hubble rate
(e.g. at the end of inflation). We stress that, in the case $\epsilon_{re}\neq
2$, the turning points of Eq. (2.4) are determined from $k^{2}\simeq
a^{2}\,H^{2}$; the contribution of the numerical factor $(2-\epsilon_{ex})$
and $(2-\epsilon_{re})$ has been consistently neglected. The spectral index
appearing in Eq. (2.13) is given by

$n_{T}^{(i)}=2(1-\beta)+2(1-\delta_{i}),\qquad\delta_{i}>0,\qquad\beta=\frac{1}{1-\epsilon}.$
(2.14)

If we now assume the validity of the consistency relations the value of
$n_{T}^{(i)}$ depends on $\delta_{i}$ and $r_{T}$

$n_{T}^{(i)}(r_{T},\delta_{i})=\frac{32-4\,r_{T}}{16-r_{T}}-2\delta_{i}=2(1-\delta_{i})+{\mathcal{O}}(r_{T}).$
(2.15)

From Eq. (2.15) in the limit $\delta_{i}\to 1$ we have

$\lim_{\delta_{i}\to
1}\,\,n_{T}^{(i)}(r_{T},\delta_{i})=-r_{T}/8+{\mathcal{O}}(r_{T}^{2}).$ (2.16)

Similarly, for the wavelengths reentering during the $(i+1)$-th stage the
spectral energy density is instead given by:

$\Omega^{(i+1)}_{gw}(k,\tau)=\frac{4}{3\pi}\biggl{(}\frac{H_{1}}{M_{P}}\biggr{)}^{2}\,\biggl{(}\frac{H_{1}\,a_{1}^{2}}{H\,a^{2}}\biggr{)}^{2}\biggl{|}\frac{a_{i}H_{i}}{a_{1}\,H_{1}}\biggr{|}^{4-\beta}\,\biggl{|}\frac{k}{a_{i}\,H_{i}}\biggr{|}^{n_{T}^{(i+1)}},\qquad\qquad
k<a_{i}\,H_{i},$ (2.17)

where $n_{T}^{(i+1)}$ is now given by

$n_{T}^{(i+1)}(r_{T},\delta_{i+1})=2(1-\beta)+2(1-\delta_{i+1})=\frac{32-4\,r_{T}}{16-r_{T}}-2\delta_{i+1}=2(1-\delta_{i+1})+{\mathcal{O}}(r_{T}).$
(2.18)

From Eqs. (2.15)–(2.18) we have in fact three complementary possibilities. If
the expansion rate is initially slower than radiation (i.e. $\delta_{i}<1$)
the spectral index for $k>a_{i}\,H_{i}$ is either blue or violet (i.e.
$n_{T}^{(i)}>0$). Then, provided $\delta_{i+1}>1$ the spectral energy density
of Eqs. (2.13) and (2.17) is characterized by a local minimum for $k\simeq
a_{i}\,H_{i}$. We may also have the opposite situation where the expansion
rate is initially faster than radiation (i.e. $\delta_{i}>1$) then it gets
smaller (i.e. $\delta_{i+1}<1$) and the spectral energy density has a local
maximum always for $k\simeq a_{i}\,H_{i}$. The third possibility suggests that
either $\delta_{i}\to 1$ or $\delta_{i+1}\to 1$: in this case the spectral
energy density exhibits a quasi-flat branch either for
$\nu<\nu_{i}=a_{i}\,H_{i}$ or for $\nu>\nu_{i}$. While this result is formally
correct, for the sake of completeness it is useful to verify it directly from
Eq. (2.9). In the case $\delta_{i}\to 1$ at reentry we have that
$k\tau_{re}\ll 1$; this means that the term ${\mathcal{H}}_{re}/k\gg 1$
dominates in Eq. (2.9) and $\Omega_{gw}(k,\tau)$ is then given by:

$\Omega_{gw}(k,\tau)=\frac{2}{3\pi}\biggl{(}\frac{k^{2}}{a_{ex}^{2}M_{P}^{2}}\biggr{)}\biggl{(}\frac{H_{re}^{2}\,a_{re}^{4}}{H^{2}\,a^{4}}\biggr{)}=\frac{2}{3\pi}\biggl{(}\frac{H_{1}^{2}\,a_{1}^{4}}{H^{2}\,a^{4}}\biggr{)}\biggl{(}\frac{H_{1}}{M_{P}}\biggr{)}^{2}\,\biggl{(}\frac{k}{a_{1}\,H_{1}}\biggr{)}^{\overline{n}_{T}},$
(2.19)

where $\overline{n}_{T}=-2\epsilon\simeq-r_{T}/8$. Thus Eqs. (2.16) and (2.19)
show that $n_{T}^{(i)}(r_{T},\delta_{i})$ evaluated in the limit
$\delta_{i}\to 1$ indeed corresponds to $\overline{n}_{T}$ up to corrections
${\mathcal{O}}(r_{T}^{2})$.

##  3 Peaks and throughs of the spectral energy density

The previous results demonstrate that ${\mathcal{H}}_{re}$ and
${\mathcal{H}}_{ex}$ are equally essential for the late-time form of the
spectral energy density. In this sense the slopes of the humps appearing in
$\Omega_{gw}(\nu,\tau_{0})$ are a simultaneous test of the expansion rate
during inflation and in the post-inflationary stage. This means that if
$n_{T}^{(i)}(r_{T},\delta_{i})$ and $n_{T}^{(i+1)}(r_{T},\delta_{i+1})$ are
observationally assessed around a given peak, their measurement ultimately
reflects the expansion history during and after inflation.

###  3.1 The profile of the effective expansion rate

It is now interesting to consider the general case illustrated in Fig. 1
where, prior to $a_{1}$, an inflationary stage dominates the evolution of the
background so that the effective expansion rate $a\,H$ increases linearly with
the scale factor. As suggested in the previous section, during this stage the
initial inhomogeneities of the tensor modes are normalized to their quantum
mechanical values. While in the conventional case we would have that, after
inflation, the effective expansion rate is immediately dominated by radiation,
in the situation illustrated in Fig. 1 we rather consider a sequence of
different stages expanding either faster or slower than radiation.

Figure 1: On the vertical axis the common logarithm of $a\,H$ is illustrated
as a function of the common logarithm of the scale factor. We consider here
the general situation where there are $N$ different stages of expansion not
necessarily coinciding with radiation. The $N$-th stage conventionally
coincides with the standard radiation-dominated evolution (i.e. $a_{r}=a_{N}$)
while the first stage starts at the end of inflation (i.e. $H_{1}=H_{max}$).
Even if, in general, $H_{r}\ll H_{1}$ we shall always require
$H_{r}>10^{-44}\,M_{P}$ implying that the dominance of radiation takes place
well before big-bang nucleosynthesis. During the $i$-th stage of the sequence
the scale factor expands as $a(\tau)\simeq\tau^{\delta_{i}}$. The dashed lines
appearing in this cartoon correspond to the pivotal frequencies of the
spectrum.

More specifically, according to Fig. 1, the post-inflationary expansion
history consists of $N$ successive stages where, by definition, $a_{1}$
coincides with the end of inflation. Moreover, since $a_{r}$ denotes the value
of the scale factor at the onset of the radiation-dominated stage of
expansion, we conventionally posit that $a_{N}=a_{r}$. During each of the
successive stages the expansion rate is characterized, in the conformal time
coordinate, by $a(\tau)\sim\tau^{\delta_{i}}$ so that the spectral index of
$h_{0}^{2}\Omega_{gw}(\nu,\tau_{0})$ is in fact the one already determined in
Eqs. (2.15) and (2.18). During the $i$-th stage of expansion the spectral
energy density in critical units scales approximately as
$(\nu/\nu_{i})^{n_{T}^{(i)}}$. The dashed lines of Fig. 1 illustrate the
values of the comoving frequencies at the transition points. Since the current
value of the scale factor is conventionally normalized to $1$ (i.e. $a_{0}=1$)
comoving and physical frequencies coincide at the present time but not earlier
on. Furthermore the largest frequency coincides with $a_{1}\,H_{1}$ while for
$\nu<\nu_{r}=a_{r}H_{r}$ the spectral energy density has the standard quasi-
flat form since the corresponding wavelengths exit the Hubble radius during
inflation and reenter when the Universe is already dominated by radiation. It
is important to appreciate that while $\nu_{1}=a_{1}\,H_{1}=\nu_{max}$ depends
on all the post-inflationary expansion rates (i.e. the different
$\delta_{i}$), $\nu_{r}=a_{r}\,H_{r}$ only depends on the hierarchy between
$H_{1}$ and $H_{r}$. To prove this statement it is practical to introduce the
ratios of the curvature scales during two successive stages of expansion,
namely:

$\xi_{i}=\frac{H_{i+1}}{H_{i}}<1,\qquad\xi=\prod_{i}^{N-1}\xi_{i}=\frac{H_{r}}{H_{1}}<1,$
(3.1)

where $\xi$ denotes the ratio between the Hubble rates at the onset of the
radiation-dominated stage666A curvature scale
$H_{r}={\mathcal{O}}(10^{-44})\,M_{P}$ correspond to a temperature of the
plasma $T={\mathcal{O}}(\mathrm{MeV})$. For the present ends it is more
practical to work directly with the curvature scales. (i.e. $H_{r}$) and at
the end of the inflationary phase (i.e. $H_{1}$); $\xi_{i}$ gives instead the
ratio of the expansion rates between two successive stages. Note that, by
definition, both $\xi_{i}$ and $\xi$ are smaller than $1$ since the largest
value of the Hubble rate always appears in the denominator. Since we
conventionally choose that $H_{1}$ concides with the expansion rate at the end
of inflation (i.e. $H_{1}\equiv H_{max}$) while $H_{N}\equiv H_{r}$, in the
simplest non-trivial situation we have that $N=3$ and Eq. (3.1) implies:

$\xi=\xi_{1}\,\xi_{2}=\frac{H_{r}}{H_{1}},\qquad\xi_{1}=\frac{H_{2}}{H_{1}},\qquad\xi_{2}=\frac{H_{r}}{H_{2}},$
(3.2)

where, following the conventions established established above and illustrated
in Fig. 1, $a_{3}=a_{r}$ and $H_{3}=H_{r}$.

###  3.2 The typical frequencies of the spectrum

As anticipated the maximal frequency of the spectrum indeed depends upon all
the successive stages of expansion and its the general expression is:

$\nu_{1}=\nu_{max}=\prod_{i=1}^{N-1}\,\,\xi_{i}^{\frac{\delta_{i}-1}{2(\delta_{i}+1)}}\,\,\overline{\nu}_{max}.$
(3.3)

In Eq. (3.3) $\overline{\nu}_{max}$ denotes the maximal frequency of the
spectrum when all the different stages of expansion appearing in Fig. 1
collapse to a single phase expanding exactly like radiation. Indeed, if
$\delta_{i}\to 1$ in Eq. (3.3) for all the $i=1,\,.\,.,\,.\,N$ we have that
$\nu_{1}=\nu_{max}\to\overline{\nu}_{max}$:

$\overline{\nu}_{max}=269.33\,\biggl{(}\frac{r_{T}}{0.06}\biggr{)}^{1/4}\,\biggl{(}\frac{{\mathcal{A}}_{{\mathcal{R}}}}{2.41\times
10^{-9}}\biggr{)}^{1/4}\,\biggl{(}\frac{h_{0}^{2}\,\Omega_{R0}}{4.15\times
10^{-5}}\biggr{)}^{1/4}\,\,\,\mathrm{MHz},$ (3.4)

where $\Omega_{R0}$ is the present fraction of relativistic species of the
concordance scenario and ${\mathcal{A}}_{{\mathcal{R}}}$ is the amplitude of
the scalar power spectrum that determines777We recall that
$(H_{1}/M_{P})=\sqrt{\pi\,\epsilon\,{\mathcal{A}}_{{\mathcal{R}}}}$. If the
consistency relations are enforced we also have that
$(H_{1}/M_{P})=\sqrt{\pi\,r_{T}\,{\mathcal{A}}_{{\mathcal{R}}}}/4$, where
$r_{T}\simeq 16\,\epsilon$ is, as usual, the tensor to scalar ratio. $H_{1}$.
In other words $\overline{\nu}_{max}$ coincides with the maximal frequency of
the spectrum in the case considered in Ref. [5] where $H_{1}\to H_{r}$ and
$\nu_{r}\to\nu_{max}={\mathcal{O}}(200)$ MHz. In the general case illustrated
in Fig. 1 we have that:

$\nu_{N}=\nu_{r}=\prod_{j=1}^{N-1}\,\sqrt{\xi_{j}}\,\,\,\overline{\nu}_{max}=\sqrt{\xi}\,\overline{\nu}_{max},$
(3.5)

where the second equality follows since, by definition,

$\prod_{j=1}^{N-1}\,\xi_{j}=\xi_{1}\,\xi_{2}\,.\,.\,.\,\xi_{N-2}\,\xi_{N-1}=\xi.$
(3.6)

Equations (3.3) and (3.5) demonstrate, as anticipated above, that while
$\nu_{max}$ is sensitive to the whole expansion history, $\nu_{r}$ only
depends upon $\sqrt{\xi}$ (where $\xi=H_{r}/H_{1}$).

For all the other intermediate frequencies between $\nu_{max}$ and $\nu_{r}$,
the following expression holds:

$\nu_{m}=\prod_{j=1}^{m-1}\sqrt{\xi_{j}}\,\,\prod_{i=m}^{N-1}\,\,\xi_{i}^{\frac{\delta_{i}-1}{2(\delta_{i}+1)}}\,\,\overline{\nu}_{max},\qquad\qquad
m=2,\,3,\,.\,.\,.\,N-2,\,N-1.$ (3.7)

The different frequencies are illustrated in Fig. 1 with the dashed lines are
therefore in the following hierarchy:

$\nu_{max}=\nu_{1}>\nu_{2}>\nu_{3}>\,.\,.\,.\,.>\nu_{N-2}>\nu_{N-1}>\nu_{N}=\nu_{r}.$
(3.8)

Th result of Eq. (3.8) is a direct consequence of the monotonic shape of
$a\,H$ for $a>a_{1}$. If the profile of $a\,H$ is not monotonic for $a>a_{1}$,
the hierarchy between the different frequencies of the spectrum is different
as it happens when there is a second inflationary stage of expansion between
$a_{1}$ and $a_{r}$ [35].

###  3.3 Local maxima of the spectral energy density

Since the typical frequencies probed by space-borne interferometers are below
the Hz the most interesting situation, from the practical viewpoint, involves
a maximum of $h_{0}^{2}\Omega_{gw}(\nu,\tau_{0})$ just before the final
dominance of radiation when $a_{N}=a_{r}$. In this case the extremum of the
spectral energy density occurs for $\nu_{N-1}$. Further maxima can also arise
for higher frequencies and are more constrained by the measurements of the
pulsar timing arrays and by the limits coming from wide-band detectors.
Recalling the notations of Fig. 1 together with the explicit expressions of
the slopes given in Eqs. (2.15) and (2.18), the spectral energy density has a
maximum for $\nu=\nu_{N-1}$ provided

$\delta_{N-2}\geq 1,\qquad\qquad\delta_{N-1}<1\qquad\Rightarrow\qquad
n_{T}^{(N-2)}\leq 0,\qquad\qquad n_{T}^{(N-1)}>0,$ (3.9)

where $\nu_{N-2}>\nu_{N-1}>\nu_{N}=\nu_{r}$. The spectral energy density in
critical units reaches therefore a maximum for $\nu=\nu_{N-1}$ and its value
is:

$h_{0}^{2}\,\Omega_{gw}=\overline{{\mathcal{N}}}_{\rho}(\nu_{N-1},r_{T})\biggl{(}\frac{\nu_{N-1}}{\nu_{N}}\biggr{)}^{n_{T}^{(N-1)}}.$
(3.10)

where $\overline{{\mathcal{N}}}_{\rho}(\nu,r_{T})$ is a function that weakly
depends on the frequency and it is typically smaller than $10^{-16}$ for
$r_{T}\leq 0.06$ [7, 8, 9]. This function is explicitly determined in section
5 and it contains the dependence upon the transfer functions of the problem.
According to the results deduced so far the explicit form of $\nu_{N-1}$ and
$\nu_{N}$ is given by:

$\displaystyle\nu_{N-1}$ $\displaystyle=$
$\displaystyle\sqrt{\xi_{1}}\,.\,\,.\,\,.\,\,\sqrt{\xi_{N-2}}\,\,\xi_{N-1}^{\frac{\delta_{N-1}-1}{2(\delta_{N-1}+1)}}\,\,\overline{\nu}_{max},$
(3.11) $\displaystyle\nu_{N}$ $\displaystyle=$
$\displaystyle\nu_{r}=\sqrt{\xi_{1}}\,.\,\,.\,\,.\,\,\sqrt{\xi_{N-1}}\,\,\overline{\nu}_{max}.$
(3.12)

The ratio of Eqs. (3.11) and (3.12) gives exactly the term appearing in Eq.
(3.10) so that the value of the spectral energy density at the maximum can
also be written as:

$h_{0}^{2}\,\Omega_{gw}(\nu_{N-1},\tau_{0})\simeq\overline{{\mathcal{N}}}_{\rho}(\nu_{N-1},r_{T})\,\,\xi_{N-1}^{-\frac{2(1-\delta_{N-1})}{\delta_{N-1}+1}}.$
(3.13)

The function $\overline{{\mathcal{N}}}_{\rho}(\nu_{N-1},r_{T})$ is weakly
dependent on the frequency and its explicit form is discussed in the following
section. Since the local maximum for $\nu=\nu_{N-1}$ does not depend on
different maxima possibly arising for $\nu<\nu_{N-1}$ the simplest situation,
for the present purposes, is the one where $N=3$. In this case there are only
two successive stages characterized by $\delta_{1}$ and $\delta_{2}$. The
maximal frequency of the spectrum is given by:

$\nu_{1}=\nu_{max}=\xi_{1}^{\frac{\delta_{1}-1}{2(\delta_{1}+1)}}\,\,\xi_{2}^{\frac{\delta_{2}-1}{2(\delta_{2}+1)}}\,\,\overline{\nu}_{max}.$
(3.14)

The frequencies $\nu_{2}$ and $\nu_{3}=\nu_{r}$ are instead given by:

$\nu_{2}=\sqrt{\xi_{1}}\,\,\xi_{2}^{\frac{\delta_{2}-1}{2(\delta_{2}+1)}}\,\,\overline{\nu}_{max},\qquad\qquad\nu_{r}=\nu_{3}=\sqrt{\xi_{1}}\,\,\sqrt{\xi_{2}}\,\,\overline{\nu}_{max}.$
(3.15)

We have just have one peak for $\nu=\nu_{2}$ and Eq. (3.13) gives

$h_{0}^{2}\,\Omega_{gw}(\nu_{2},\tau_{0})\simeq\,\overline{{\mathcal{N}}}_{\rho}(\nu_{2},r_{T})\,\,\xi_{2}^{-\frac{2(1-\delta_{2})}{\delta_{2}+1}}.$
(3.16)

All in all, while the existence of an early stage of accelerated expansion is
motivated by general requirements directly related to causality, the post-
inflationary expansion history is not constrained prior to big-bang
nucleosynthesis. The results obtained in this section are therefore applicable
to any post-inflationary expansion rate and do not assume the dominance of
radiation between $H_{1}$ and $H_{r}$.

###  3.4 General requirements on the total number of $e$-folds

As repeatedly stressed we always considered hereunder the possibility that
$H_{r}>10^{-44}\,M_{P}$ suggesting that the plasma is already dominated by
radiation for temperatures that are well above the MeV as it happens, for
instance, when the reheating stage is triggered by the decay of a
gravitationally coupled massive scalar field. There are however some
possibilities where the MeV-scale reheating temperature could be induced by
long-lived massive species with masses close to the weak scale, as suggested
in Refs. [42, 43]. In spite of this interesting option we simply regard the
condition $H_{r}\geq 10^{-44}\,M_{P}$ as an absolute lower limit on $H_{r}$.
Indeed the gravitational waves only couple to the expansion rate and our
purpose here is just to propose a framework where the early thermal history of
the plasma could be tested via the spectra of the inflationary gravitons.

Along this perspective it is useful to remark that the maximal number of
inflationary $e$-folds accessible to large-scale observations can be different
[25] (see also [33, 35, 44]) depending on the post-inflationary expansion
history. The maximal number of $e$-folds presently accessible to large-scale
observation (${\mathcal{N}}_{max}$ in what follows) is computed by fitting the
(redshifted) inflationary event horizon inside the current Hubble patch; in
other words we are led to require, in terms of Fig. 1, that
$H_{1}^{-1}(a_{0}/a_{1})\simeq H_{0}^{-1}$. It is clear that
${\mathcal{N}}_{max}$ does not coincide with the total number of $e$-folds
that can easily be larger (or even much larger) than ${\mathcal{N}}_{max}$.
Depending on the various $\delta_{i}$ and $\xi_{i}$ the same gap in $a\,H$ is
covered by a different amount of redshift. In the general situation of Fig. 1
the expression of ${\mathcal{N}}_{max}$ is given by:

$\displaystyle{\mathcal{N}}_{max}$ $\displaystyle=$ $\displaystyle
61.88-\ln{\biggl{(}\frac{h_{0}}{0.7}\biggr{)}}+\frac{1}{4}\ln{\biggl{(}\frac{r_{T}}{0.06}\biggr{)}}+\frac{1}{4}\ln{\biggl{(}\frac{{\mathcal{A}}_{{\mathcal{R}}}}{0.06}\biggr{)}}$
(3.17) $\displaystyle+$
$\displaystyle\frac{1}{4}\ln{\biggl{(}\frac{h_{0}^{2}\,\Omega_{R0}}{0.06}\biggr{)}}+\frac{1}{2}\sum_{i}^{N-1}\,\biggl{(}\frac{\delta_{i}-1}{\delta_{i}+1}\biggr{)}\,\ln{\xi_{i}}.$

In connection with Eq. (3.17) we have three complementary possibilities. If we
conventionally set $\delta_{i}=1$ into Eq. (3.17) we obtain the standard
result implying that ${\mathcal{N}}_{max}={\mathcal{O}}(60)$. Recalling that
all the $\xi_{i}$ are, by definition, all smaller than $1$ we have that
${\mathcal{N}}_{max}>60$ if $\delta_{i}<1$. For the same reason
${\mathcal{N}}_{max}<60$ iff $\delta_{i}>1$. Let us consider, for instance,
the case of a single phase expanding slower than radiation; in this case
${\mathcal{N}}_{max}$ can be as large as $75$ [33, 34]. In the intermediate
situations where there are different phases expanding either faster or slower
than radiation ${\mathcal{N}}_{max}$ depends on the relative duration of the
various phases and on their expansion rates.

##  4 The frequency range of space-borne interferometers

###  4.1 Approximate frequencies of the various instruments

While various space-borne interferometers have been proposed so far the
presumed sensitivity of these instruments is still under debate. For this
reason we adopt here a pragmatic viewpoint based on the considerations
developed after Eq. (1.1). In short the strategy is the following:

  * •

the fiducial frequency interval of space-borne interferometers ranges from a
fraction of the mHz to the Hz and, within this interval, the minimal
detectable spectral energy density (denoted hereunder by
$h_{0}^{2}\Omega_{gw}^{(min)}(\nu,\tau_{0})$) defines the potential
sensitivity of the hypothetical instrument;

  * •

the LISA interferometers [15, 16] might hopefully probe the following region
of the parameter space:

$h_{0}^{2}\Omega_{gw}^{(min)}(\nu,\tau_{0})={\mathcal{O}}(10^{-11.2}),\qquad\qquad
10^{-4}\mathrm{Hz}<\nu\leq 0.1\,\mathrm{Hz};$ (4.1)

  * •

in the case of the Deci-Hertz Interferometer Gravitational Wave Observatory
(DECIGO) [17, 18] the minimal detectable spectral energy density could be
smaller

$10^{-17.5}\leq
h_{0}^{2}\Omega_{gw}^{(min)}(\nu,\tau_{0})\leq{\mathcal{O}}(10^{-13.1}),\qquad\qquad
10^{-3}\mathrm{Hz}<\nu\leq 0.1\,\mathrm{Hz}.$ (4.2)

The values of Eq. (4.2) are still quite hypothetical so that it is prudent to
choose $h_{0}^{2}\Omega_{gw}^{(min)}(\nu,\tau_{0})$ between the standard
values of the hoped sensitivity of the DECIGO project [17, 18] (suggesting
$h_{0}^{2}\Omega_{gw}^{(min)}(\nu,\tau_{0})={\mathcal{O}}(10^{-13.1})$) and
the optimistic figure reachable by the Ultimate-DECIGO [19] (conventionally
referred to as U-DECIGO) where
$h_{0}^{2}\Omega_{gw}^{(min)}(\nu,\tau_{0})={\mathcal{O}}(10^{-17.5})$. For
the record, the Big Bang Observer (BBO) [20] might reach sensitivities

$h_{0}^{2}\Omega_{gw}^{(min)}(\nu,\tau_{0})={\mathcal{O}}(10^{-14.2}),\qquad
10^{-3}\mathrm{Hz}<\nu\leq 0.1\,\mathrm{Hz}.$ (4.3)

There finally exist also recent proposals such as Taiji [21, 22] and TianQin
[23, 24] leading to figures that are roughly comparable with the LISA values.
In summary for the typical frequency of the space-borne detectors we consider
the following broad range:

$0.1\,\mathrm{mHz}<\nu_{S}<0.1\,\mathrm{Hz}$ (4.4)

and suppose that in the range (4.4)
$h_{0}^{2}\Omega_{gw}^{(min)}(\nu,\tau_{0})$ may take the following two
extreme values.

$h_{0}^{2}\Omega_{gw}^{(min)}(\nu_{S},\tau_{0})={\mathcal{O}}(10^{-11}),\qquad\qquad
h_{0}^{2}\Omega_{gw}^{(min)}(\nu_{S},\tau_{0})={\mathcal{O}}(10^{-14}).$ (4.5)

While the two values of Eq. (4.5) are both quite optimistic, they are
customarily assumed by the observational proposals and, for this reason, they
are used here only for illustration.

###  4.2 The profile of the spectral energy density

The exclusion plots characterizing the parameter space of the model are
separately considered for the two illustrative values of
$h_{0}^{2}\Omega_{gw}^{(min)}(\nu_{S},\tau_{0})$ given in Eq. (4.5). For
instance in Fig. 2 we require that

$\nu_{N-1}={\mathcal{O}}(\nu_{S}),\qquad\qquad
h_{0}^{2}\Omega_{gw}(\nu_{N-1},\tau_{0})\geq 10^{-11}.$ (4.6)

The first requirement of Eq. (4.6) implies that the frequency range of the
maximum is comparable with $\nu_{S}$ while the second condition just comes
from Eq. (4.5) and it also demands, incidentally, that the inflationary signal
is larger than the spectral energy density produced by the gravitational waves
associated with a putative strongly first-order phase transition, as we shall
briefly discuss later on. The condition (4.6) can also be relaxed by assuming
the second value of $h_{0}^{2}\Omega_{gw}(\nu_{S},\tau_{0})$:

$\nu_{N-1}={\mathcal{O}}(\nu_{S}),\qquad\qquad
h_{0}^{2}\Omega_{gw}(\nu_{N-1},\tau_{0})\geq 10^{-14}.$ (4.7)

Equation (4.7) is justified by the nominal sensitivity of other space-borne
interferometers such as DECIGO [17, 18] or U-DECIGO [19]. To investigate the
phenomenological implications the simplest choice is to posit $N=3$. In this
case we just have one maximum for $\nu_{r}<\nu<\nu_{max}$ and the discussion
of the parameters is therefore simpler even if, as already mentioned, the
essential features remain the same also in more complicated situations. For
$N=3$ the spectral energy density of the model is:

$h_{0}^{2}\,\Omega_{gw}(\nu,\tau_{0})={\mathcal{N}}_{\rho}\,r_{T}(\nu_{p})\biggl{(}\frac{\nu}{\nu_{p}}\biggr{)}^{\overline{n}_{T}(r_{T})}\,\,{\mathcal{T}}^{2}_{low}(\nu/\nu_{eq})\,\,{\mathcal{T}}^{2}_{high}(\nu,\nu_{2},\nu_{r},n_{T}^{(1)},n_{T}^{(2)}),$
(4.8)

where $\overline{n}_{T}(r_{T})$ has been computed in Eqs. (2.16) and (2.19);
$\overline{n}_{T}$ is the spectral index associated with the wavelengths
leaving the Hubble radius during the inflationary phase and reentering during
the radiation stage. In Eq. (4.8) $\nu_{p}$ and $\nu_{eq}$ define the lowest
frequency range of the spectral energy density:

$\displaystyle\nu_{p}$ $\displaystyle=$
$\displaystyle\frac{k_{p}}{2\pi}=3.092\biggl{(}\frac{k_{p}}{0.002\,\,\mathrm{Mpc}^{-1}}\biggr{)}\,\mathrm{aHz},$
$\displaystyle\nu_{eq}$ $\displaystyle=$
$\displaystyle\frac{k_{\mathrm{eq}}}{2\pi}=15.97\biggl{(}\frac{h_{0}^{2}\,\Omega_{M0}}{0.1411}\biggr{)}\biggl{(}\frac{h_{0}^{2}\,\Omega_{R0}}{4.15\times
10^{-5}}\biggr{)}^{-1/2}\,\,\mathrm{aHz},$ (4.9)

where we used that $k_{eq}=0.0732\,h_{0}^{2}\,\Omega_{M0}\,\mathrm{Mpc}^{-1}$
(as usual, $\Omega_{M0}$ is the present fraction in dusty matter). The
spectral slopes $n_{T}^{(1)}$ and $n_{T}^{(2)}$ are instead determined by Eqs.
(2.15) and (2.18); up to corrections ${\mathcal{O}}(r_{T})$ we have

$n_{T}^{(1)}=2(1-\delta_{1})+{\mathcal{O}}(r_{T}),\qquad\qquad
n_{T}^{(2)}=2(1-\delta_{2})+{\mathcal{O}}(r_{T}),$ (4.10)

where $n_{T}^{(1)}<0$ and $n_{T}^{(2)}>0$ since during the first stage the
Universe expands faster than radiation (i.e. $\delta_{1}>1$) while in the
second stage it is slower than radiation (i.e. $\delta_{2}<1$). In the
simplest case where the consistency relations are enforced we have that

$\overline{n}_{T}(r_{T})=-\frac{r_{T}}{8}+{\mathcal{O}}(r_{T}^{2}),\qquad\qquad{\mathcal{N}}_{\rho}=4.165\times
10^{-15}\biggl{(}\frac{h_{0}^{2}\,\Omega_{R0}}{4.15\times 10^{-5}}\biggr{)}.$
(4.11)

In Eq. (4.9) ${\mathcal{T}}_{low}(\nu/\nu_{eq})$ is the low-frequency transfer
function of the spectral energy density [14]:

${\mathcal{T}}_{low}(\nu,\nu_{eq})=\sqrt{1+c_{1}\biggl{(}\frac{\nu_{eq}}{\nu}\biggr{)}+c_{2}\biggl{(}\frac{\nu_{eq}}{\nu}\biggr{)}^{2}},\qquad
c_{1}=0.5238,\qquad c_{2}=0.3537.$ (4.12)

The high-frequency transfer function
${\mathcal{T}}_{high}(\nu,\nu_{2},\nu_{r},\delta_{1},\delta_{2})$ appearing in
Eq. (4.8) depends on $\nu_{2}$ and $\nu_{r}$ and it is given by:

${\mathcal{T}}_{high}^{2}(\nu,\nu_{r},\nu_{2},n_{T}^{(1)},n_{T}^{(2)}))=\frac{\sqrt{1+b_{1}(\nu/\nu_{r})^{n_{T}^{(2)}}+b_{2}(\nu/\nu_{r})^{2n_{T}^{(2)}}}}{\sqrt{1+d_{1}(\nu/\nu_{2})^{n_{T}^{(2)}+|n_{T}^{(1)}|}+d_{2}(\nu/\nu_{2})^{2(n_{T}^{(2)}+|n_{T}^{(1)}|)}}},$
(4.13)

where $b_{i}$ and $d_{i}$ (with $i=1,\,2$) are numerical coefficients of order
$1$ that depend on the specific choice of $\delta_{1}$ and $\delta_{2}$ and
cannot be written in general terms. We recall that the explicit expressions of
$\nu_{2}$ and $\nu_{r}$ have been given in Eq. (3.15) and they depend
explicitly upon $\xi_{1}$ and $\xi_{2}$. Since Eq. (4.13) depends on two
different scales, there are three relevant limits of
${\mathcal{T}}_{high}^{2}(\nu,\nu_{r},\nu_{2})$ that must be considered. The
first limit stipulates that:

${\mathcal{T}}_{high}^{2}(\nu,\nu_{r},\nu_{2},n_{T}^{(1)},n_{T}^{(2)})\to\sqrt{\frac{b_{2}}{d_{2}}}\,\,\biggl{(}\frac{\nu_{2}}{\nu_{r}}\biggr{)}^{n_{T}^{(2)}}\,\,\biggl{(}\frac{\nu}{\nu_{2}}\biggr{)}^{-|n_{T}^{(1)}|},\qquad\qquad\nu\gg\nu_{2},$
(4.14)

and it corresponds to the high-frequency branch where the spectral energy
density is suppressed as $\nu^{-|n_{T}^{(1)}|}$:

$h_{0}^{2}\,\Omega(\nu,\tau_{0})=\overline{{\mathcal{N}}}_{\rho}(r_{T},\nu)\biggl{(}\frac{\nu_{2}}{\nu_{r}}\biggr{)}^{n_{T}^{(2)}}\biggl{(}\frac{\nu}{\nu_{2}}\biggr{)}^{-|n_{T}^{(1)}|},\qquad\qquad\nu_{2}<\nu<\nu_{max}.$
(4.15)

Note that since $n_{T}^{(1)}<0$, in the spectral energy density we introduced
the absolute value just to avoid potential confusions. This is not necessary
in the case of $n_{T}^{(2)}$ which is instead positive semidefinite. From Eq.
(4.13) the second relevant limit corresponds to the region where
$\nu<\nu_{2}$:

${\mathcal{T}}_{high}^{2}(\nu,\nu_{r},\nu_{2},n_{T}^{(1)},n_{T}^{(2)})\to\sqrt{b_{2}}\biggl{(}\frac{\nu}{\nu_{r}}\biggr{)}^{n_{T}^{(2)}},\qquad\qquad\nu_{r}<\nu<\nu_{2}.$
(4.16)

In this case the spectral energy density increases as $\nu^{n_{T}^{(2)}}$ and
its approximate expression is given by:

$h_{0}^{2}\,\Omega(\nu,\tau_{0})=\overline{{\mathcal{N}}}_{\rho}(r_{T},\nu)\biggl{(}\frac{\nu}{\nu_{r}}\biggr{)}^{n_{T}^{(2)}},\qquad\qquad\nu_{r}<\nu<\nu_{2}.$
(4.17)

The third relevant limit of the transfer function is finally for $\nu<\nu_{r}$
and, in this limit, Eq. (4.13) simply goes to $1$:

${\mathcal{T}}_{high}^{2}(\nu,\nu_{r},\nu_{2},n_{T}^{(1)},n_{T}^{(2)})\to
1,\qquad\qquad\nu<\nu_{r}.$ (4.18)

The function $\overline{N}_{\rho}(\nu,r_{T})$ has been already introduced in
Eq. (3.10) and, as anticipated, its explicit expression depends in fact upon
the low-frequency transfer function:

$\overline{{\mathcal{N}}}_{\rho}(r_{T},\nu)={\mathcal{N}}_{\rho}\,r_{T}\biggl{(}\frac{\nu}{\nu_{p}}\biggr{)}^{\overline{n}_{T}}\,\,{\mathcal{T}}^{2}_{low}(\nu_{r}/\nu_{eq}),\qquad\qquad\frac{d\ln{\overline{{\mathcal{N}}}_{\rho}}}{d\ln{\nu}}=-\frac{r_{T}}{8}\ll
1.$ (4.19)

Even though the prefactor $\overline{{\mathcal{N}}}_{\rho}(r_{T},\nu)$ has a
mild frequency dependence coming from neutrino free-streaming, for simplified
analytic estimates this dependence can be ignored, at least approximately;
this in fact the meaning of the second relation in Eq. (4.19). Along this
perspective we can estimate
$\overline{{\mathcal{N}}}_{\rho}={\mathcal{O}}(10^{-16.5})$ for $r_{T}=0.06$.

###  4.3 The constrained parameter space

The shaded region n Fig. 2 illustrates the area of the parameter space where
the following pair of conditions are simultaneously verified:

$h_{0}^{2}\Omega_{gw}(\nu_{2},\tau_{0})\geq
h_{0}^{2}\Omega_{gw}^{(min)},\qquad\qquad 0.1\,\mathrm{mHz}<\nu_{2}\leq
0.1\mathrm{Hz}.$ (4.20)

Since the product of the various $\xi_{i}$ from $1$ to $(N-1)$ must equal
$\xi$, in the case of $N=3$ we have that $\xi_{1}\xi_{2}=\xi$. Then the
analysis can be simplified by using three related observations:

  * •

$\xi_{1}$ can be traded for $\xi/\xi_{2}$ by recalling that, to avoid problems
with nucleosynthesis, the lower limit $\xi>10^{-38}$ must always be separately
imposed;

  * •

as a consequence we end-up with three parameters $\xi_{2}$, $\xi$,
$\delta_{1}$ and $\delta_{2}$;

  * •

to discuss the parameter space we can fix $\delta_{1}\to 1$: this is the most
constraining value since for $\delta_{1}>1$ the high-frequency part of
$h_{0}^{2}\,\Omega_{gw}(\nu,\tau_{0})$ decreases and it is therefore less
constrained by the high-frequency limits.

If the spectral energy density decreases for $\nu>\nu_{2}$ (i.e.
$n_{T}^{(1)}<0$) all the high-frequency bounds (and in particular the LIGO-
Virgo-KAGRA limit [12, 13, 14]) are automatically satisfied provided they are
satisfied in the case $\delta_{1}\to 1$. The value of $\delta_{1}$ affects the
value of the high-frequency slope of the spectral energy density since, in
this case, $n_{T}^{(1)}(r_{T},\delta_{1})={\mathcal{O}}(r_{T})$.

In the four different plots of Fig. 2 the value of $\xi$ increases from
$10^{-36}$ to $10^{-30}$ so that the radiation dominates when the expansion
rate gets progressively larger; recall, in fact, that $\xi=H_{r}/H_{1}\geq
10^{-38}$. As $\xi$ increases the duration of the radiation phase increases,
the shaded are gets smaller and allowed region is reduced. The same logic of
Fig. 2 has been followed in the case of Fig. 3 with the difference that
$h_{0}^{2}\Omega_{gw}^{(min)}$ is now relaxed from $10^{-11}$ to $10^{-14}$.
In each of the plots appearing in Figs. 2 and 3 there are two shaded regions.
The wider area is obtained by enforcing the big-bang nucleosynthesis (BBN)
constraint [45, 46, 47]. The narrower (and darker) region in each plot of
Figs. 2 and 3 is instead obtained by imposing the limits obtained from the
operating interferometers on the backgrounds of relic gravitons, i.e. the
LIGO-Virgo-KAGRA bound [12] (see also [13, 14]).

Figure 2: The parameter space is illustrated in the plane
$(\delta_{2},\,\xi_{2})$ by fixing $\delta_{1}$ to its most constraining value
realized in the case $\delta_{1}\to 1$. In the four different plots of this
figure the value of $\xi$ progressively increases between $10^{-38}$ and
$10^{-30}$. The shaded area corresponds to the allow region. The wider area is
obtained by enforcing the big-bang nucleosynthesis limit of Eqs. (4.21)–(4.22)
while the smaller (and darker) region follows by imposing the limits coming
from the audio band (see Eqs. (4.23)) and discussion therein.

From the technical viewpoint the BBN constraint requires888The limit of Eq.
(4.22) sets a constraint on the extra-relativistic species possibly present at
the BBN time. The limit is often expressed for practical reasons in terms of
$\Delta N_{\nu}$ representing the contribution of supplementary neutrino
species. The actual bounds on $\Delta N_{\nu}$ range from $\Delta N_{\nu}\leq
0.2$ to $\Delta N_{\nu}\leq 1$; the integrated spectral density in Eq. (4.22)
is thus between $10^{-6}$ and $10^{-5}$.:

$h_{0}^{2}\int_{\nu_{bbn}}^{\nu_{max}}\Omega_{gw}(\nu,\tau_{0})d\ln{\nu}=5.61\times
10^{-6}\Delta N_{\nu}\biggl{(}\frac{h_{0}^{2}\,\Omega_{\gamma 0}}{2.47\times
10^{-5}}\biggr{)},$ (4.21)

where $\Omega_{\gamma 0}$ is the (present) critical fraction of CMB photons
and $\nu_{bbn}$ denotes the BBN frequency:

$\nu_{bbn}=2.252\times
10^{-11}\biggl{(}\frac{N_{eff}}{10.75}\biggr{)}^{1/4}\biggl{(}\frac{T_{bbn}}{\,\,\mathrm{MeV}}\biggr{)}\biggl{(}\frac{h_{0}^{2}\,\Omega_{R0}}{4.15\times
10^{-5}}\biggr{)}^{1/4}\,\,\mathrm{Hz},$ (4.22)

where $N_{eff}$ denotes the effective number of relativistic degrees of
freedom entering the total energy density of the plasma and $T_{bbn}$ is the
temperature of big-bang nucleosynthesis. The bound (4.22) can be relaxed if
the nucleosynthesis takes place in the presence of matter-antimatter domains
[46] and $\nu_{max}$ appearing in Eq. (4.21) denotes, as previously discussed,
the maximal frequency of the spectrum. However, since
$h_{0}^{2}\Omega_{gw}(\nu,\tau_{0})$ decreases for $\nu>\nu_{2}$ (and more
generally for $\nu>\nu_{N-1}$) the region between $\nu_{2}$ and $\nu_{max}$
gives a subleading contribution to the integral appearing at the left-hand
side of Eq. (4.21).

Figure 3: As in Fig. 2 the parameter space of the model is illustrated in the
plane $(\delta_{2},\,\xi_{2})$. The difference between these plots and the
ones of Fig. 2 is related to $h_{0}^{2}\Omega_{gw}^{(min)}$ which is now given
by $10^{-14}$ while it was $10^{-11}$ in Fig. 2.

According to Figs. 2 and 3 the limits imposed by Eqs. (4.21)– (4.22) are less
constraining than the ones following from the LIGO-Virgo-KAGRA bound[12].
Indeed the LIGO-Virgo-KAGRA collaboration, in its attempt to constrain the
stochastic backgrounds of relic gravitons, reported a constraint [12]
implying, in the case of a quasi-flat spectral energy density in the audio-
band

$\Omega_{gw}(\nu,\tau_{0})<5.8\times 10^{-9},\qquad\qquad
20\,\,\mathrm{Hz}<\nu_{KLV}<76.6\,\,\mathrm{Hz},$ (4.23)

where $\nu_{LVK}$ denotes the LIGO-Virgo-KAGRA frequency. The exclusion plots
of Figs. 2 are then confronted with the current phenomenological bounds in all
the available ranges of frequency with the aim of constraining the rate and
the duration post-inflationary expansion Universe. In particular, in the nHz
region, the pulsar timing arrays (PTA) recently reported a potential signal
that could be attributed to the relic gravitons [36, 37, 38, 39]. The PTA
recently reported evidence of a potential signal in the nHz band. Using the
spectral energy density in critical units as a pivotal variable the features
of this purported signal would imply, in the present notations, that:

$q_{0}^{2}\times
10^{-8.86}<\,h_{0}^{2}\,\Omega_{gw}(\nu,\tau_{0})<\,q_{0}^{2}\times
10^{-9.88},\qquad\qquad 3\,\mathrm{nHz}<\nu<100\,\mathrm{nHz}.$ (4.24)

In Eq. (4.24) we introduced the numerical factor $q_{0}$ that depends on the
specific experimental determination. The Parkes Pulsar Timing Array
collaboration [36] suggests $q_{0}=2.2$. Similarly the International Pulsar
Timing Array collaboration (IPTA in what follows) estimates $q_{0}=2.8$ [37]
while the European Pulsar Timing Array collaboration (EPTA in what follows)
[38] gives $q_{0}=2.95$ (see also [48, 49]). The results of PPTA, IPTA and
EPTA seem, at the moment, to be broadly compatble with the NANOgrav 12.5 yrs
data [39] (see also [50, 51]) implying $q_{0}=1.92$.

It is relevant to point out that neither the observations of Refs. [36, 37,
38] nor the ones of Ref. [39] can be interpreted yet as an evidence of relic
gravitons. The property of a PTA is that the signal from relic gravitons will
be correlated across the baselines, while that from the other noise will not.
Since these correlation have not been observed so far, the interpretation
suggested in Eq. (4.24) is still preliminary, to say the least. To be fair the
pragmatic strategy followed here will be to interpret Eq. (4.24) as an upper
limit whenever the corresponding theoretical signal is too low in the nHz
region. Conversely if $h_{0}^{2}\,\Omega_{gw}(\nu,\tau_{0})$ happens to be
grossly compatible with the range of Eq. (4.24) it will be interesting to see
if the associated spectral energy density fits within the PTA window. The
average of the $q_{0}$ of the different experiments is given by
$\overline{q}_{0}=2.46$. If we use $\overline{q}_{0}$ into Eq. (4.24) we get
to the requirement:

$10^{-8.07}<\,h_{0}^{2}\,\Omega_{gw}(\nu,\tau_{0})<\,\times
10^{-9.09},\qquad\qquad 3\,\mathrm{nHz}<\nu<100\,\mathrm{nHz}.$ (4.25)

The condition (4.25) is never verified for the parameter space illustrated in
Figs. 2 and 3. Thus, for the selection of parameters analyzed here,
$h_{0}^{2}\Omega_{gw}(\nu,\tau_{0})$ may lead to a relevant signal below the
Hz but not in the nHz band.

###  4.4 The spectral energy density and its signatures

In Figs. 4 and 5, the spectral energy density has been explicitly illustrated
for a selection of the parameters.

Figure 4: We illustrate the common logarithm of the spectral energy density of
the relic gravitons as a function of the common logarithm of the frequency
expressed in Hz. In this plot the dashed and the dot-dashed curves illustrate
two models that are excluded by the big-bang nucleosynthesis constraint of Eq.
(4.21) and by the LIGO-Virgo-KAGRA limit of Eq. (4.23). The parameters of the
curve at the bottom (full line) are instead drawn from the allowed region of
the parameter space. In all the examples of this plot $\delta_{1}>1$.

In Fig. 4 we selected $\xi=10^{-36}$ and $\xi_{2}=10^{-10}$ for different
values of $\delta_{1}>1$ and $\delta_{2}<1$. As expected the value of
$\nu_{r}$ is always larger than $10^{-10}$. For this reason
$h_{0}^{2}\Omega_{gw}(\nu,\tau_{0})$ does not exceed ${\mathcal{O}}(10^{-16})$
in the frequency range of the PTA and the profiles of Figs. 4 and 5 are unable
to account for the putative signal of Eqs. (4.24)–(4.25). Note, in this
respect, that the parameters of the dot-dashed and of the dashed curves of
Fig. 4 have been selected in order to get an artificially large signal that is
in fact excluded both by the constraint of Eq. (4.21) and by the limit of
ground-based detectors (see Eq. (4.23) and discussion therein). Even in this
case the PTA values are too large and must be accounted by different
mechanisms.

Figure 5: The conventions are exactly the same already explained in Fig. 4 but
the spectral energy is illustrated for a different choice of the parameters.
The dashed curves at the top of the figure is barely compatible with the
limits set by Eq. (4.23). If $\delta_{1}>1$ (see e.g. Fig. 4) the spectral
energy density decreases for $\nu>\nu_{2}$ and the most relevant constraint on
the height of the maximum comes from Eqs. (4.21)–(4.22). If $\delta_{1}\to 1$
the limits from the audio band (see, in particular, Eq. (4.23)) are the most
constraining ones and this is why we illustrated here this case where the
spectral energy density remains quasi-flat at high-frequencies.

The results of Fig. 5 correspond instead to a slightly different choice of the
parameters, namely $\xi=10^{-34}$ and $\xi_{2}=10^{-8}$. For illustration we
have chosen $\delta_{1}\to 1$ implying that between $\nu_{max}$ and $\nu_{2}$
the spectral energy density is quasi-flat. This is the most constraining case
from the viewpoint of the limits coming from wide-band detectors [12, 13, 14].

The class of signals computed here are distinguishable, at least in principle,
from the other astrophysical and cosmological foregrounds. In the region
between a fraction of the mHz and the Hz the predominant astrophysical
foregrounds of gravitational radiation are probably associated with the
galactic distribution of the white dwarves. While the signal of white dwarves
could also be used for calibration, other astrophysical foregrounds are also
expected (e.g. stellar origin black holes and even supermassive black holes
from galaxy mergers). Another cosmological foreground is given by TeV scale
early Universe and this happens since the typical frequency corresponding to
the Hubble radius at the electroweak stage is
${\mathcal{O}}(10)\,\mu\mathrm{Hz}$. To have drastic deviations from
homogeneity and the consequent production of burst of gravitational radiation
the electroweak phase transition must be strongly first-order. In this respect
two classes of related observations are in order. The first remark is that the
electroweak phase transition does not have to be strongly first-order. Since
the first perturbative analyses of the problem we actually know that that for
the measured range of Higgs masses the electroweak phase transition is not of
first-order [52, 53]. Due to the inherently non-perturbative nature of the
problem, the original perturbative estimates have been subsequently
corroborated by lattice calculations both in three [54] and four dimensions
[55]. For the measured values of the Higgs and gauge boson masses the
transition between the symmetric and the broken phase follows a cross-over
evolution that should not lead to an appreciable production of gravitational
radiation. In this case the only cosmological signal may be the one associated
with the inflationary gravitons.

The second remark is that, at the moment the hopes of observing a burst of
gravitational radiation from the electroweak scale must rely on some extension
of the standard electroweak theory. In this situation the amount of
gravitational radiation produced by the phase transition depends on the
particular model and also on the difference between the energy density in the
broken and in the symmetric phase. This energy density may be comparable with
the energy density of the ambient plasma (and in this case the phase
transition experiences a strong supercooling) or smaller than the energy
density of the surrounding radiation (and in this case the phase transition is
mildly supercooled). If the gravitational is produced from the collisions of
the bubbles of the new phase [56, 57, 58] the equivalent
$h_{0}^{2}\Omega_{gw}(\nu,\tau_{0})$ scales like $\nu^{3}$, reaches a maximum
and then decreases with a power that may be faster than $\nu^{-1}$. The
spectral energy density inherits also contribution from the sound waves of the
plasma [59] and this second component may be even larger than the one due to
bubble collisions. The key point for the present ends is that the powers
associated with a strongly first-order phase transition are typically much
steeper than the ones discussed here. In our case the rise of
$h_{0}^{2}\Omega_{gw}(\nu,\tau_{0})$ appearing in Figs. 4 and 5 always scales
with $0<n_{T}^{(2)}\leq 1$ while the corresponding slope in the case of phase
transitions is typically ${\mathcal{O}}(3)$. Another possibility not requiring
a strongly first-order phase transition is the presence of a stochastic
background of hypermagnetic fields at the electroweak phase. In this case
bursts of gravitational radiation may also be produced and the spectral energy
density is different from the one discussed here [60] (see also [61, 62]).
Overall, because of causality, the spectra associated with the TeV physics are
much steeper around their putative maximum. For this reason it seems plausible
to disentangle the inflationary contributions from other possible cosmological
foregrounds. There is of course a deeper problem that has to do with our
ability to separate the cosmological signals from the other astrophysical
foregrounds (e.g. white dwarves, massive and supermassive black-holes). The
potential difficulties associated with the astrophysical foregrounds suggested
already many years ago [25, 26] that the potential signals of post-
inflationary stages should be probably observed over much higher frequencies
${\mathcal{O}}(\mathrm{MHz})$ where electromagnetic detectors might be
operating in the future [63, 64, 65, 66, 67, 68].

##  5 Concluding remarks and future perspectives

If the wavelengths that left the Hubble radius during inflation reentered in
the radiation-dominated stage of expansion the spectral energy density of the
inflationary gravitons is today quasi-flat for typical frequencies larger than
$100$ aHz. Prior to nucleosynthesis the timeline of the expansion rate is
however unknown and we considered here a post-inflationary evolution
consisting of a sequence of stages expanding at rates that are alternatively
faster and slower than radiation. As a consequence, the spectral energy
density can even be eight orders of magnitude larger than the conventional
inflationary signal for frequencies between the $\mu$Hz and a fraction of the
Hz.

Below the Hz various space-borne detectors will probably be operational in the
next twenty years and the signals expected in the mHz region are dominated by
astrophysical sources (e.g. galactic white dwarves, solar-mass black holes,
supermassive black holes coming from galaxy mergers). The only cosmological
sources considered in this context are associated with the phase transitions
at the TeV scale even if it is well established, both perturbatively and non-
perturbatively, that the standard electroweak theory leads to a cross-over
regime where drastic deviations from homogeneity (and the consequent bursts of
gravitational radiation) should not be expected. The inflationary signal is
customarily regarded as irrelevant since its spectral energy density could be
at most $h_{0}^{2}\Omega_{gw}(\nu_{S},\tau_{0})={\mathcal{O}}(10^{-17})$ for
$r_{T}\leq 0.06$ and for $\nu_{S}={\mathcal{O}}(\mathrm{mHz})$. The relic
gravitons of inflationary origin may instead lead to
$h_{0}^{2}\Omega_{gw}(\nu_{S},\tau_{0})={\mathcal{O}}(10^{-9})$ provided the
expansion history prior to nucleosynthesis is not constantly dominated by
radiation. The slopes of the spectral energy density obtained in the case of a
putative strongly first-order phase transition are much steeper than the ones
associated with a modified expansion history. When confronted with the most
relevant phenomenological bounds the class of signals discussed here is
predominantly constrained by the limits on the massless species at the
nucleosynthesis scale and by the direct observations of ground-based detectors
(i.e. LIGO, Virgo and KAGRA). From the profiles of the spectral energy density
and from the slopes of the hump in the mHz range it is possible to infer the
post-inflationary expansion history for typical curvature scales that are
between $10^{-44}\,\,M_{P}$ and $10^{-34}\,\,M_{P}$ (i.e. roughly $10$ orders
of magnitude larger than the nucleosynthesis scale). The analysis of the
spectral energy density in different frequency ranges (e.g. nHz, mHz and MHz)
might even allow to reconstruct the expansion history of the Universe at
earlier and later times. It is finally interesting that some regions of the
parameter space that are relevant for space-borne detectors also lead to a
potentially large signal in the audio band and will probably be directly
probed or excluded in the near future.

All in all the perspective conveyed in this analysis suggests that the
frequency range below the Hz should be carefully investigated in the light of
a possible signal coming from inflationary gravitons. While a strongly first
order phase transition may be realized beyond the standard electroweak theory,
the present discussion only assumes a conventional inflationary stage
supplemented by a post-inflationary evolution that deviates from the
conventional radiation dominance prior to nucleosynthesis. The observations in
the mHz region could then simultaneously test the occurrence of an early
inflationary stage and of a post-inflationary expansion history whose details
are still unknown and might only be discovered by looking at the spectra of
relic gravitons.

## Aknowledgements

It is a pleasure to thank T. Basaglia, A. Gentil-Beccot, S. Rohr, J. Vigen and
the whole CERN Scientific Information Service for their kind help during the
preparation of this manuscript.

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# What is YOLOv5: A deep look into the internal features of the popular object
detector

Rahima Khanam* and Muhammad Hussain
Department of Computer Science, Huddersfield University, Queensgate,
Huddersfield HD1 3DH, UK; *Correspondence<EMAIL_ADDRESS>
###### Abstract

This study presents a comprehensive analysis of the YOLOv5 object detection
model, examining its architecture, training methodologies, and performance.
Key components, including the Cross Stage Partial backbone and Path
Aggregation-Network, are explored in detail. The paper reviews the model’s
performance across various metrics and hardware platforms. Additionally, the
study discusses the transition from Darknet to PyTorch and its impact on model
development. Overall, this research provides insights into YOLOv5’s
capabilities and its position within the broader landscape of object detection
and why it is a popular choice for constrained edge deployment scenarios.

_K_ eywords Automation; Computer Vision; YOLO; YOLOV5; Object Detection; Real-
Time Image processing; Convolutional Neural Networks(CNN); YOLO version
comparison

##  1 Introduction

Computer vision is a rapidly evolving field that empowers machines [1] to
interpret and comprehend visual information [2]. A critical component of this
discipline is object detection [3], which involves accurately identifying and
locating objects within images or video sequences [4]. To address this
challenge, various algorithmic approaches have been developed, with
significant advancements achieved in recent years[5].

One such groundbreaking technique is the You Only Look Once (YOLO) algorithm,
introduced by Redmon et al. in 2015 [6]. This name reflects its distinctive
approach, examining the entire image just once to identify objects and their
positions. In contrast to conventional methods employing two-stage detection
processes, YOLO treats object detection as a regression problem [6]. In the
YOLO paradigm, a single convolutional neural network is employed to predict
bounding boxes and class probabilities for an entire image [7]. This
streamlined approach differs from traditional methods with more intricate
pipelines.

Building upon the foundation laid by YOLO, this paper delves into the YOLOv5
architecture, a state-of-the-art object detection model that has garnered
significant attention due to its exceptional performance and efficiency.

###  1.1 Survey Objective

This study aims to evaluate the performance of the YOLOv5 object detection
model in comparison to state-of-the-art alternatives. The research focuses on
assessing the trade-off between model accuracy and inference speed across the
various YOLOv5 variants (n, s, m, l, x). By examining these different model
sizes, this study seeks to identify the optimal balance between performance
metrics for diverse application requirements.

The investigation will explore the factors contributing to YOLOv5’s
performance gains, with particular emphasis on:

  1. 1.

The role of PyTorch training methodologies

  2. 2.

The impact of architectural innovations such as the CSP backbone and PA-Net
neck

  3. 3.

The effectiveness of data augmentation techniques, including mosaic
augmentation

  4. 4.

The influence of loss calculation methods and bounding box anchor generation

Additionally, the study will analyze the implications of YOLOv5’s transition
from the Darknet framework to PyTorch, considering aspects such as model
development, deployment, and accessibility for custom object detection tasks.

By providing a comprehensive examination of YOLOv5’s capabilities and
innovations, this research aims to contribute to the broader understanding of
advanced object detection algorithms and their practical applications in
computer vision.

##  2 Evolution of Yolov5

The YOLOv5 [8] repository emerged as an evolution of the YOLOv3 [9] PyTorch
implementation developed by Glenn Jocher in 2020. It was released following
the release of YOLOv4 [10]. Yolov5 gained popularity as a platform for
transitioning YOLOv3 models from Darknet to PyTorch for production deployment.
Recognizing the utility of this PyTorch-based approach, Ultralytics initiated
a development trajectory focused on enhancing the YOLOv3 architecture and
training methodologies within the PyTorch framework. This endeavor aimed to
empower a broader community of developers to create and deploy custom object
detectors.

This is YOLOv5 development timeline:

  * •

April 1, 2020: Initiation of development on a series of compound-scaled
PyTorch models based on YOLOv3/YOLOv4 architectures.

  * •

May 27, 2020: Public release of the YOLOv5 repository, demonstrating state-of-
the-art performance among existing YOLO implementations.

  * •

June 9, 2020: Integration of CSP (Cross-Stage Partial) modules, contributing
to improved model speed, size, and accuracy.

  * •

June 19, 2020: Adoption of FP16 precision as default, resulting in smaller
checkpoints and faster inference.

  * •

June 22, 2020: Implementation of PANet updates, including new detection heads,
reduced parameters, and enhanced mAP (mean Average Precision).

Initially, the model incorporating these advancements was designated YOLOv4,
aligning with the contemporaneous release of the YOLOv4 architecture within
the Darknet framework. However, to avoid potential confusion and
inconsistencies, it was subsequently revised to YOLOv5. While this change
sparked some debate within the research community, a comparative analysis of
YOLOv4 and YOLOv5 was conducted to facilitate objective evaluation [11].

This study primarily focuses on the novel techniques and performance metrics
introduced within the YOLOv5 framework, as detailed within the GitHub
repository. It is essential to acknowledge the rapid evolution of the YOLOv5
model since its inception, which suggests ongoing research and development in
this domain. As such, YOLOv5 should be considered a dynamic and evolving
system rather than a static model.

##  3 Architectural footprint of Yolov5

Object detection, a primary application of YOLOv5, entails the extraction of
salient features from input images. These features are subsequently processed
by a predictive model to localize and classify objects within the image.

The YOLO architecture introduced the end-to-end, differentiable approach to
object detection by unifying the tasks of bounding box regression and object
classification into a single neural network [12]. Fundamentally, the YOLO
network comprises three core components. The backbone, a convolutional neural
network, is responsible for encoding image information into feature maps at
varying scales. These feature maps are then processed by the neck, a series of
layers designed to integrate and refine feature representations. Finally, the
head module generates predictions for object bounding boxes and class labels
based on the processed features.

Figure 1: Process of Object Detection [13]

While there exists considerable flexibility in the architectural design of
each component, YOLOv4 and YOLOv5 have significantly advanced the field by
incorporating techniques from other computer vision domains. This synergistic
approach has demonstrated substantial improvements in object detection
performance within the YOLO framework.

###  3.1 YOLOv5 Training Methods

The efficacy of an object detection system is contingent not only on its
underlying architecture but also on the employed training methodologies. While
architectural innovations often garner significant attention, the role of
training procedures in achieving optimal performance is equally critical.
There are two primary training techniques employed in YOLOv5:

  * •

Data augmentation is a pivotal component of the YOLOv5 training pipeline. By
introducing diverse transformations to the training dataset, this technique
enhances model robustness and generalization capabilities. Consequently, the
model becomes more resilient to variations in real-world image conditions.

  * •

The Loss function is a composite metric calculated from three primary
components: Generalized Intersection over Union (GIoU), objectness, and
classification loss. These loss components are carefully designed to optimize
mean average precision (mAP), a widely adopted evaluation metric for object
detection models.

###  3.2 Transition to PyTorch

YOLOv5 represents a significant advancement by transitioning the YOLO
architecture from the Darknet framework to PyTorch [14]. The Darknet
framework, primarily implemented in C, affords researchers granular control
over network operations. While this level of control is advantageous for
experimentation, it often hinders the rapid integration of novel research
findings due to the necessity of custom gradient calculations for each new
implementation.

Porting the training procedures of Darknet to PyTorch, as achieved in YOLOv3,
is a complex undertaking in itself. YOLOv5 extends this effort by further
optimizing and refining these procedures within the PyTorch ecosystem.

###  3.3 Data augumentation

YOLOv5 incorporates data augmentation techniques within its training pipeline
to enhance model robustness and generalization. During each training epoch,
images are subjected to a series of augmentations through an online data
loader, including:

  * •

Scaling: Adjustments to image size.

  * •

Color space manipulation: Modifications to color channels.

  * •

Mosaic augmentation: A novel technique that combines four images into four
randomly sized tiles.

.

Introduced in the YOLOv3 PyTorch repository and subsequently integrated into
YOLOv5, mosaic data augmentation has proven particularly effective in
addressing the challenge of small object detection, a common limitation in
datasets by the Common Objects in Context (COCO) object detection benchmark.
By combining multiple images into a single training example, this technique
exposes the model to a wider variety of object scales and spatial
arrangements, thereby enhancing its ability to accurately detect smaller
objects.

###  3.4 Bounding Box Anchors

The YOLOv3 PyTorch repository introduced a novel approach to anchor box
generation, employing K-means clustering and genetic algorithms to derive
anchor box dimensions directly from the distribution of bounding boxes within
a given dataset. This methodology is particularly critical for custom object
detection tasks, as the scale and aspect ratios of objects often diverge
significantly from those commonly found in standard datasets like COCO.

The YOLOv5 architecture predicts bounding box coordinates as offsets relative
to a predefined set of anchor box dimensions. These anchor dimensions are
essential for initializing the prediction process and can significantly
influence the model’s performance.

Figure 2: Bounding box prediction based on an anchor box [15]

###  3.5 Loss Calculation

The YOLOv5 loss function is a composite of three components: Binary Cross-
Entropy (BCE) for class prediction and objectness, and Complete Intersection
over Union (CIoU) for localization. The overall loss is computed as a weighted
sum of these individual losses:

$\text{Loss}=\lambda_{1}\cdot L_{\text{cls}}+\lambda_{2}\cdot
L_{\text{obj}}+\lambda_{3}\cdot L_{\text{loc}}$ (1)

where $L_{\text{cls}}$, $L_{\text{obj}}$, and $L_{\text{loc}}$ represent the
BCE loss for class prediction, BCE loss for objectness, and CIoU loss for
localization, respectively. The coefficients $\lambda_{1}$, $\lambda_{2}$, and
$\lambda_{3}$ are hyperparameters that balance the contributions of each loss
component to the overall optimization process.

###  3.6 16 Bit Floating Point Precision

The PyTorch framework offers the capability to reduce computational precision
from 32-bit to 16-bit floating-point numbers during both training and
inference. When applied to YOLOv5, this technique has demonstrated potential
for significant acceleration in inference speed. However, it is essential to
note that these performance gains is only contingent upon specific GPU
architectures, notably the V100 and T4 models for Yolov5. While the
limitations of hardware compatibility exist at present, ongoing developments
by NVIDIA indicate a prospective expansion of this efficiency enhancement to a
wider range of hardware platforms.

###  3.7 CSP Backbone

Both YOLOv4 and YOLOv5 incorporate the CSP (Cross-Stage Partial) bottleneck
module for feature extraction. This architectural innovation, introduced by
WongKinYiu et al., addresses the issue of redundant gradient information
prevalent in larger convolutional neural network backbones. By decoupling
feature maps into two main parts and recombining them, the CSP module
effectively reduces computational cost and model complexity without
compromising performance. This efficiency enhancement is particularly
advantageous for the YOLO family, where rapid inference and compact model size
are paramount.

CSP models draw inspiration from the DenseNet architecture as shown in Figure
3, addressing the challenges inherent in deep CNNs, including the vanishing
gradient problem. By fostering direct connections between layers, DenseNet
aimed to enhance feature propagation, promote feature reuse, and reduce the
overall number of network parameters.

Figure 3: Interconnected dense layers in DenseNet [16]

The CSPResNext50 and CSPDarknet53 architectures incorporate modifications to
the DenseNet structure as illustrated in Figure 4. Specifically, the feature
maps generated by the base layer are divided into two branches. One branch is
processed through a dense block, while the other is forwarded directly to the
subsequent stage. This architectural modification aims to mitigate
computational bottlenecks inherent to DenseNet and enhance learning by
preserving an unaltered representation of the original feature map.

Figure 4: (a) DenseNet and (b) CSPDenseNet [17]

###  3.8 PA-Net Neck

Both YOLOv4 and YOLOv5 employ the PA-Net architecture for feature aggregation
[18]. As illustrated in Figure 5, each P_i represents a distinct feature layer
extracted from the CSP backbone. This architectural choice is inspired by the
EfficientDet object detection framework, where the BiFPN module was deemed
optimal for feature integration. While the BiFPN approach serves as a
benchmark, it is plausible that alternative implementations within the YOLO
framework could yield further performance improvements.

Figure 5: Variations of FPN architectures [19]

It is evident that YOLOv5 builds upon the foundational research of YOLOv4 in
selecting the most suitable neck architecture. YOLOv4 comprehensively
evaluated a range of options including FPN, PAN, NAS-FPN, BiFPN, ASFF, and
SFAM.

##  4 YOLOv5 Models

The YOLOv5 architecture encompasses five distinct models, ranging from the
computationally efficient YOLOv5n to the high-precision YOLOv5x.

  * •

YOLOv5n: Designed for resource-constrained environments, YOLOv5n is the
smallest and fastest model in the series. Its compact size, less than 2.5 MB
in INT8 format and approximately 4 MB in FP32 format, makes it suitable for
deployment on edge devices and IoT platforms. Compatibility with OpenCV DNN
further enhances its utility for mobile applications.

  * •

YOLOv5s: Representing the baseline model, YOLOv5s incorporates approximately
7.2 million parameters. Its performance characteristics make it suitable for
CPU-based inference tasks.

  * •

YOLOv5m: Positioned as a mid-sized model with 21.2 million parameters, YOLOv5m
offers a balanced combination of speed and accuracy. This model is often
considered a versatile choice for a wide range of object detection
applications and datasets.

  * •

YOLOv5l: With 46.5 million parameters, YOLOv5l is designed for scenarios
requiring higher precision, particularly in detecting smaller objects within
images.

  * •

YOLOv5x: As the largest and most complex model in the series, YOLOv5x achieves
the highest mAP among its counterparts. However, this performance comes at the
cost of increased computational requirements due to its 86.7 million
parameters.

A comprehensive overview of the model variants, encompassing inference speed
across CPU and GPU platforms as well as the number of parameters for a
640-pixel image size, is presented in Table 1.

Table 1: YOLOv5 Model Performance Model | Params (Million) | Accuracy (mAP 0.5) | CPU Time (ms) | GPU Time (ms)
---|---|---|---|---
YOLOv5n | 1.9 | 45.7 | 45 | 6.3
YOLOv5s | 7.2 | 56.8 | 98 | 6.4
YOLOv5m | 21.2 | 64.1 | 224 | 8.2
YOLOv5l | 46.5 | 67.3 | 430 | 10.1
YOLOv5x | 86.7 | 68.9 | 766 | 12.1

##  5 YOLOv5 Annotation Format

YOLOv5 employs a PyTorch TXT annotation format that closely resembles the YOLO
Darknet TXT standard, with the addition of a YAML file specifying model
configuration and class labels. To facilitate compatibility with YOLOv5,
annotation data generated from various tools may require conversion. Platforms
such as Roboflow offer support for multiple annotation formats and can export
data directly into the YOLOv5-compatible format. Popular annotation tools
including VOTT, LabelImg, and CVAT can also be utilized, with appropriate data
conversion steps.

##  6 YOLOv5 Labelling tools

For efficient data management and annotation, Ultralytics, the developers of
YOLOv5, recommends Roboflow as a compatible labeling tool [20]. Table 2
provides an overview of third-party integration platforms compatible with the
YOLOv5. For each platform, the table outlines its primary functionalities when
integrated with YOLOv5.

Table 2: YOLOv5 Integrations Integration Platform | Functionality
---|---
Deci [21] | Automated compilation and quantization of YOLOv5 for enhanced inference performance.
ClearML [22] | Tracking, visualization, and remote training of YOLOv5 models using an open-source platform.
Roboflow [18] | Labeling and exporting custom datasets directly compatible with YOLOv5 training.
Weights and Biases [23] | Tracking and visualization of YOLOv5 training runs in the cloud.

##  7 Discussion

YOLOv5 represents a significant advancement in the field of object detection,
building upon the foundations laid by its predecessors while introducing novel
improvements are listed below:

  1. 1.

Architectural Advancements: YOLOv5 represents a significant advancement in
object detection, building upon previous YOLO iterations. The incorporation of
the CSP backbone and PA-Net neck enhances computational efficiency without
compromising accuracy. The CSP module effectively addresses redundant gradient
information, while the PA-Net architecture optimizes feature aggregation.

  2. 2.

Model Versatility: The range of YOLOv5 models (n, s, m, l, x) provides
flexibility for diverse hardware and application requirements. The smallest
model, YOLOv5n, expands object detection capabilities to edge devices and IoT
platforms.

  3. 3.

Training Methodology Innovations: YOLOv5’s emphasis on data augmentation,
particularly mosaic augmentation, improves small object detection and reduces
dataset size requirements. The adoption of 16-bit floating-point precision
demonstrates a forward-looking approach to optimization.

  4. 4.

Performance and Impact: YOLOv5’s high mAP scores and low inference times
position it as a strong contender for real-time object detection. The
transition to PyTorch has democratized access to the model, fostering broader
research and development.

##  8 Conclusion

YOLOv5 has emerged as a significant advancement in object detection,
demonstrating a compelling balance of speed, accuracy, and user-friendliness.
While the core architecture builds upon established principles, the model’s
implementation within the PyTorch framework represents a substantial leap
forward, enhancing both development efficiency and deployment capabilities.
The availability of multiple model variants tailored to diverse computational
constraints expands YOLOv5’s applicability across various domains from
renewable energy [24] to quality inspection in Manufacturing [25].

The model’s accessibility, coupled with its compatibility with existing tools
and platforms, positions YOLOv5 as a versatile solution for both research and
practical applications. As the field of computer vision continues to evolve
rapidly, YOLOv5 serves as a strong foundation for future advancements in real-
time object detection and related tasks.

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