Konstantinos Dimopoulos 11institutetext: Consortium for Fundamental Physics,
Physics Department, Lancaster University, Lancaster LA1 4YB, UK
11email<EMAIL_ADDRESS>
WWW home page: https://www.lancaster.ac.uk/physics/about-
us/people/konstantinos-dimopoulos

# Observable Primordial Gravitational Waves from Cosmic Inflation

Konstantinos Dimopoulos 11

###### Abstract

I will review briefly how inflation is expected to generate a stochastic
background of primordial gravitational waves (GWs). Then, I will discuss how
such GWs can be enhanced by a stiff period following inflation, enough to be
observable. I will present examples of this in the contact of hybrid inflation
with $\alpha$-attractors, or a period of hyper-kination in Palatini gravity.

###### keywords:

primordial gravitational waves, cosmic inflation, $\alpha$-attractors,
Palatini modified gravity

##  1 Introduction

The history of the Universe requires special initial conditions which are
arranged by cosmic inflation. In a nutshell, cosmic inflation can be defined
as a period of accelerated expansion in the Early Universe. Inflation results
in the Universe being large, spatially flat and uniform, in accordance to
observations [1]. Inflation also generates the primordial density
perturbations (PDPs), which are necessary for galaxies and galaxy clusters to
form [2]. The PDPs reflect themselves on the cosmic microwave background (CMB)
radiation, through the Sachs-Wolfe effect. Agreement with CMB observations is
spectacular, when inflation is quasi-de Sitter and space expands exponentially
[3].

However, there is another generic prediction of inflation beyond the acoustic
peaks observed in the CMB, namely the generation of a stochastic spectrum of
primordial gravitational waves (GWs) [4]. This prediction will soon be tested,
either indirectly, through CMB polarisation observations, or directly through
interferometers (Advanced LIGO, LISA).

How does inflation produce these GWs? Below, I attempt a brief overview. I use
natural units with $c=\hbar=1$ and $8\pi G=m_{P}^{-2}$, with $m_{P}=2.43\times
10^{18}\,$GeV being the reduced Planck mass. The signature of the metric is
positive.

##  2 Particle production of gravitational waves during cosmic inflation

Following the recipe of linearised gravity, we consider a comoving
perturbation of the metric $h_{ij}$, such that the line element of spatially
flat FRW spacetime is

${\rm d}s^{2}=a^{2}(\tau)[-{\rm d}\tau^{2}+(\delta_{ij}+h_{ij}){\rm
d}x^{i}{\rm d}x^{j}]\,,$ (1)

where $a(\tau)$ is the scale factor of the Universe as a function of conformal
time $\tau$ and $\delta_{ij}$ is equal to one when $i=j$ and to zero
otherwise, with $i,j=1,2,3$. The metric perturbation is symmetric
$h_{ij}=h_{ji}$, traceless $h_{i}^{i}=0$ and transverse $\nabla_{i}h^{ij}=0$.
This means that it is corresponds to two degrees of freedom (6-1-3=2), which
are the two polarisations, $\oplus$ and $\otimes$, of the GWs.

Thus, by Fourier transform, we can write

$h_{ij}(\tau,{\bf x})=\sqrt{16\pi G}\int\frac{{\rm
d}^{3}k}{(2\pi)^{3/2}}h_{ij}(\tau,{\bf k})\,e^{i{\bf k\cdot x}}\,,$ (2)

with

$h_{ij}(\tau,{\bf k})=\sum_{s=\oplus,\otimes}h_{k}^{s}e_{ij}^{s}({\bf k})\,,$
(3)

where $e_{ij}^{2}$ is symmetric $e_{ij}^{2}=e_{ji}^{s}$, traceless
$e_{i}^{s\;i}=0$ and transverse $k^{i}e_{ij}^{s}=0$, with $k^{i}$ being the
3-wavevector of the GWs.

The second-order GW action is

$S_{\rm GW}=\frac{1}{64\pi G}\int{\rm
d}^{4}x\sqrt{-g}\,g^{\mu\nu}\partial_{\mu}h_{ij}\,\partial_{\nu}h^{ij}\,,$ (4)

where $g$ is the determinant of the metric $g_{\mu\nu}$ and $\mu,\nu=0,1,2,3$.
From the above action one obtains the equation of motion (EoM), which reads

$h_{ij}^{\prime\prime}+2\frac{a^{\prime}}{a}h_{ij}^{\prime}-\nabla^{2}h_{ij}=0\;\Rightarrow\;{h_{k}^{s}}^{\prime\prime}+2\frac{a^{\prime}}{a}{h_{k}^{s}}^{\prime}+k^{2}h_{k}^{s}=0\,,$
(5)

where the prime denotes derivative with respect to conformal time.

The above equations show that the metric perturbation polarisations behave as
free massless scalar fields $\psi_{k}^{s}(\tau)$ which can be written as
$h_{k}^{s}(\tau)=\sqrt{16\pi G}\psi_{k}^{s}(\tau)$. To study particle
production, we introduce the Muchanov-Sasaki variable $v_{k}^{s}(\tau)\equiv
a(\tau)\psi_{k}^{s}(\tau)$. In terms of this variable, the EoM becomes the
well-known Muchanov-Sasaki equation

${v_{k}^{s}}^{\prime\prime}+\left(k^{2}-\frac{a^{\prime\prime}}{a}\right)v_{k}^{s}=0\,,$
(6)

where $v_{k}^{2}=ah_{k}^{s}m_{P}/\sqrt{2}$.

To proceed we quantize the metric perturbations (second quantization) be
expanding them in terms of creation and annihilation operators as

$v^{s}(\tau,{\bf x})=\int\frac{{\rm
d}^{3}k}{(2\pi)^{3/2}}\left[v_{k}^{s}\hat{a}_{\bf k}^{s}\,e^{i{\bf k\cdot
x}}+(v_{k}^{s})^{*}\hat{a}_{\bf k}^{s\;\dagger}\,e^{-i{\bf k\cdot
x}}\right]\,,$ (7)

where $\hat{a}_{\bf k}^{s}$ and $\hat{a}_{\bf k}^{s\;\dagger}$ are creation
and annihilation operators respectively, which satisfy the algebra

$[\hat{a}_{\bf k}^{s},\hat{a}_{\bf
q}^{r\;\dagger}]=\delta^{sr}\delta^{(3)}({\bf q}-{\bf k})\quad{\rm
and}\quad[\hat{a}_{\bf k}^{s},\hat{a}_{\bf q}^{r}]=[\hat{a}_{\bf
k}^{s\;\dagger},\hat{a}_{\bf q}^{r\;\dagger}]=0\,,$ (8)

where the value of $\delta^{sr}$ is unity when $s=r$ or zero otherwise.

Inserting the above in Eq. (6), the solution for the mode functions is

$v_{k}^{s}(\tau)=\frac{1}{\sqrt{2k}}\left(1-\frac{i}{k\tau}\right)e^{-ik\tau}\,.$
(9)

The above solution, in the subhorizon limit $-k\tau\rightarrow+\infty$ becomes
$v_{k}^{s}\rightarrow e^{-ik\tau}/\sqrt{2k}$, which is the well-known Bunch-
Davis vacuum [5]. In the superhorizon limit $-k\tau=\frac{k}{aH}\rightarrow
0$, the above solution becomes
$v_{k}^{s}\rightarrow\frac{i}{\sqrt{2k}}\frac{aH}{k}$, where $H$ is the Hubble
parameter. Thus, in the superhorizon limit we find

$h_{k}^{s}=\frac{\sqrt{2}\,v_{k}^{s}}{am_{P}}=\frac{iH}{m_{P}k^{3/2}}\;\Rightarrow\;|h_{k}^{s}|^{2}=\frac{H^{2}}{m_{P}^{2}k^{3}}\approx{\rm
constant}\,,$ (10)

where we considered that $H\approx\,$constant in quasi-de Sitter inflation.

Therefore, we see that on superhorizon scales inflation generates a spectrum
of primordial GWs. The value of their spectrum is

${\cal P}_{h}(k)=\frac{k^{3}}{2\pi^{2}}\langle
h_{ij}(k)\,h^{ij}(k)\rangle=\frac{k^{3}}{\pi^{2}}\sum_{s=\oplus,\otimes}|h_{k}^{s}|^{2}=\frac{2H^{2}}{\pi^{2}m_{P}^{2}}=64\pi
G\left(\frac{H}{2\pi}\right)^{2}\,.$ (11)

The scalar perturbations corresponding to the PDPs have spectrum

${\cal P}_{\zeta}(k)=\frac{H^{2}}{8\pi^{2}\epsilon m_{P}^{2}}\,,$ (12)

where $\epsilon\equiv-\dot{H}/H^{2}$, with the dot denoting derivative with
respect to the cosmic time. From the above, we find the consistency equation
$r\equiv{\cal P}_{h}/{\cal P}_{\zeta}=16\epsilon$, which can be tested in the
near future. At the moment, the CMB observations impose only an upper bound on
$r$: $0\leq r<0.036$ [6].

##  3 Density of the gravitational waves

In view of Eq. (4), the energy-momentum tensor of the GWs is

$T_{\mu\nu}^{\rm GW}=-\frac{2}{\sqrt{-g}}\frac{\delta S_{\rm GW}}{\delta
g^{\mu\nu}}=\frac{\langle\nabla_{\mu}h_{ij}\nabla_{\nu}h^{ij}\rangle}{32\pi
G}\,,$ (13)

from which we can read the density $\rho_{\rm GW}=a^{-2}T_{00}$. Using Eq.
(5), we can write

$\langle\rho_{\rm
GW}\rangle=\int_{0}^{+\infty}\frac{k^{3}}{2\pi^{2}}\sum_{s=\oplus,\otimes}\frac{|{h_{k}^{s}}^{\prime}|^{2}+k^{2}|h_{k}^{s}|^{2}}{a^{2}}\,{\rm
d}(\ln k)\,.$ (14)

Similarly, we find the isotropic pressure $p_{\rm GW}=a^{-2}T_{i}^{i}$, for
which

$\langle p_{\rm
GW}\rangle=\int_{0}^{+\infty}\frac{k^{3}}{2\pi^{2}}\sum_{s=\oplus,\otimes}\frac{|{h_{k}^{s}}^{\prime}|^{2}-\frac{1}{3}k^{2}|h_{k}^{s}|^{2}}{a^{2}}\,{\rm
d}(\ln k)\,.$ (15)

The above integrals are dominated by the subhorizon limit $k\gg aH$. In this
limit we have $h_{k}^{s}\propto e^{-ik\tau}$, which implies
${h_{k}^{s}}^{\prime}=-ik\,h_{k}^{s}\Rightarrow|{h_{k}^{s}}^{\prime}|^{2}=k^{2}|h_{k}^{s}|^{2}$.
Therefore, for the barotropic parameter we find

$w_{{}_{\rm GW}}=\frac{p_{\rm GW}}{\rho_{\rm
GW}}=\frac{\frac{2}{3}k^{2}|h_{k}^{s}|^{2}}{2k^{2}|h_{k}^{s}|^{2}}=\frac{1}{3}\,.$
(16)

Thus, we find that the density of the gravitational waves redshifts as
radiation with the Universe expansion $\rho_{\rm GW}\propto a^{-3(1+w_{{}_{\rm
GW}})}=a^{-4}$.

In view of the above, the density parameter of the GWs per logarithmic
momentum interval is

$\Omega_{\rm GW}(k)\equiv\frac{1}{\rho_{c}}\frac{{\rm d}\rho_{\rm GW}}{{\rm
d}\ln k}=\frac{k^{3}}{2\pi^{2}}\frac{8\pi
G}{3H^{2}}\sum_{s=\oplus,\otimes}\frac{|{h_{k}^{s}}^{\prime}|^{2}+k^{2}|h_{k}^{s}|^{2}}{a^{2}}\,.$
(17)

where $\rho_{c}=3H^{2}/8\pi G$ is the critical density. The time evolution of
$\Omega_{\rm GW}(\tau,k)$ is given by

$\Omega_{\rm GW}(\tau,k)=\frac{k^{2}\Delta_{h}^{2}(\tau,k)}{12a^{2}H^{2}}\,,$
(18)

where $\Delta_{h}^{2}(\tau,k)=T_{h}(\tau,k){\cal P}_{h}(k)$. The transfer
function is given by $T_{h}=\frac{1}{2}(a_{k}/a)^{2}$, where $a_{k}$ is the
value of $a(\tau)$ at the moment of horizon re-entry of the scale with
momentum $k$. Today we have
$T_{h}^{0}=\frac{1}{2}(a_{k}/a_{0})^{2}=\Omega_{R}\left(\frac{a_{0}H_{0}}{a_{k}H_{k}}\right)^{2}$,
where $\Omega_{R}\simeq 10^{-4}$ is the density parameter of radiation at
present and ‘0’ denotes today.

Switching to frequency $f$, we employ the relation
$f=\frac{k}{2\pi}\frac{a_{k}}{a_{0}}$. We end up with the expression [7]

$\Omega_{\rm GW}(f)\propto f^{-2(\frac{1-3w}{1+3w})}\,,$ (19)

where $w$ is the barotropic parameter of the Universe at the time of horizon
reentry. Thus, we see that, for modes which re-enter the horizon during the
radiation era, because $w=\frac{1}{3}$, we have $\Omega_{\rm
GW}(f)=\,$constant. That is, the spectrum is flat and unfortunately, its value
is unobservable in the near future (see Fig. 1).

##  4 Kination

The inflationary paradigm suggests that the Universe inflates when dominated
by the potential density of a scalar field, called the inflaton field. Non-
oscillatory inflation considers a runaway inflaton scalar potential, with its
minimum displaced at infinity [8]. Is such models, after the inflaton field
$\phi$ rolls away from the inflationary plateau, which is a relatively flat
part of its scalar potential $V(\phi)$, it becomes dominated by its kinetic
energy density $\frac{1}{2}\dot{\phi}^{2}$, because the potential reduces
drastically and becomes negligible. The Universe is still dominated by the
inflaton field, but the latter is kinetically dominated, with barotropic
parameter
$w=\frac{\frac{1}{2}\dot{\phi}^{2}-V}{\frac{1}{2}\dot{\phi}^{2}+V}\approx 1$.
This phase is called kination [9]. Eq. (19) suggests that, for the modes that
re-enter the horizon during kination we have $\Omega_{\rm GW}(f)\propto f$.

Therefore, the GW spectrum features a peak. The corresponding frequencies,
however, are unobservable at the moment because kination cannot be extended
arbitrarily to later times, and therefore, lower frequencies. The reason is
that, if kination is prolonged, the GW peak becomes too large and threatens to
destabilise the delicate process of Big Bang Nucleosynthesis (BBN). The upper
bound to $\Omega_{\rm GW}$ is obtained as follows.

At the time of BBN we require that $\Omega^{\rm BBN}_{\rm GW}<10^{-2}$ so that
BBN is not harmed. Using that the density of GWs redshifts as radiation we can
estimate the corresponding bound at present. We find

$\Omega_{\rm GW}^{0}=\frac{\rho_{\rm
GW}^{0}}{\rho_{c}^{0}}=\left.\frac{\rho_{\rm
GW}}{\rho_{r}}\right|_{0}\Omega_{R}=\left.\frac{\rho_{\rm
GW}}{\rho_{r}}\right|_{\rm BBN}\Omega_{R}=\Omega_{\rm GW}^{\rm
BBN}\Omega_{R}<10^{-6}\,,$ (20)

where $\rho_{r}$ is the density of radiation and we used that $\rho_{\rm
GW}/\rho_{r}=\,$constant. Thus, the sharp peak in $\Omega_{\rm GW}(f)$ of
kination cannot be extended to observable frequencies (see Fig. 1).

##  5 Stiff period

If the peak in $\Omega_{\rm GW}(f)$ is not so sharp then it might be extended
to observable frequencies without disturbing BBN. This may be possible if
$\frac{1}{3}<w<1$. Indeed, In Ref. [10] it is shown that, when $0.46\leq w\leq
0.56$ and 1 MeV$\,<T_{\rm reh}<150\,$MeV, then the GW peak can be extended to
frequencies low enough to be observable in the near future by Advanced LIGO
and LISA, where $T_{\rm reh}$ is the reheating temperature, that is the
temperature of the thermal bath at the onset of the usual radiation era of the
hot Big Bang.

How can this possibility be realised? I have presented a concrete model to
this end in Ref. [11]. Consider two flat directions $\varphi$ and $\sigma$ in
field space, which meet at an Enhanced Symmetry Point (ESP) such that they are
characterised by the standard hybrid potential [12]

$V(\varphi,\sigma)=\frac{1}{2}g^{2}\sigma^{2}\varphi^{2}+\frac{1}{4}\lambda(\varphi^{2}-M^{2})^{2}+V(\sigma)\,,$
(21)

where $g<1$ is a perturbative interaction coupling, $\lambda<1$ is a
perturbative self-coupling and $V(\sigma)$ is some unknown potential for the
inflaton field $\sigma$, which forces it to vary (roll) to smaller values. In
the above $M$ is the vacuum expectation value (VEV) of the waterfall field
$\varphi$. Below we consider that $M\sim m_{P}$, which is why $\varphi$ can be
called a flat direction (only lifted by Planck-suppressed interactions).

The waterfall field is non-canonical. Indeed, the the Lagrangian density is

${\cal
L}=-\frac{1}{2}\partial_{\mu}\sigma\partial^{\mu}\sigma-\frac{\frac{1}{2}\partial_{\mu}\varphi\partial^{\mu}\varphi}{(1-\varphi^{2}/M^{2})^{2}}-V(\varphi,\sigma)\,,$
(22)

i.e. there are poles at the VEVs of $\varphi$ due to non-trivial geometry in
field space. This is the standard setup in $\alpha$-attractors, where the
poles could be due to a non-trivial Kähler metric in supergravity [13]. We can
define a canonically normalised waterfall field $\phi$ using the
transformation $\varphi=M\tanh(\phi/M)$. In terms of $\phi$, the scalar
potential in Eq. (21) assumes the form

$V(\phi,\sigma)=\frac{1}{2}g^{2}M^{2}\sigma^{2}\tanh^{2}(\phi/M)+\frac{\frac{1}{4}\lambda
M^{4}}{\cosh^{4}(\phi/M)}+V(\sigma)\,,$ (23)

where the minima along the waterfall direction have been displaced at
infinity.

When the inflaton expectation value is large, the waterfall field is heavy and
is pushed towards the origin. At the origin, $\varphi$ is canonical, so the
hybrid mechanism operates normally. The waterfall transition occurs when the
inflaton reaches the critical value $\sigma_{c}=(\sqrt{\lambda}/g)M$ [12].
Afterwards, the waterfall field finds itself on top of a potential hill and is
released along its runaway direction towards large (absolute) values.

Near the origin, when $\phi\ll M$ (without loss of generality we assume
$\phi>0$), the runaway waterfall potential is approximated as

$V\simeq\frac{\frac{1}{4}\lambda
M^{4}}{\left[1+\frac{1}{2}(\phi/M)^{2}\right]^{4}}\simeq\frac{1}{4}\lambda
M^{4}\left[1-2\left(\frac{\phi}{M}\right)^{2}\right]\,.$ (24)

Because $M\sim m_{P}$ (see below), the waterfall field undergoes a period of
quadratic hilltop inflation, while $\phi$ dominates the Universe [14].

Eventually, $\phi\gg M$ and the waterfall potential is approximated as

$V\simeq\frac{\frac{1}{4}\lambda
M^{4}}{\left[\frac{1}{2}\exp(\phi/M)\right]^{4}}\simeq 4\lambda
M^{4}e^{-4\phi/M}\,.$ (25)

During this roll, there is an attractor solution (power-law inflation [15]) in
which the barotropic parameter of the rolling scalar field is

$w=-1+\frac{16}{3}\left(\frac{m_{P}}{M}\right)^{2}\,.$ (26)

Thus, the value $M\approx 1.88\,m_{P}$ results in $w\simeq\frac{1}{2}$, which
means that there is a stiff period of the Universe history when the GW modes
re-entering the horizon correspond to a peak with $\Omega_{\rm GW}(f)\propto
f^{2/5}$ [cf. Eq. (19)], which is not as sharp as the one due to kination and
can be extended to observable frequencies (see Fig. 1).

A multiple of mechanisms can be responsible for reheating at the appropriate
time. In Ref. [11], Ricci reheating is assumed as an example [16], where the
Universe is reheated by the decay of a spectator field non-minimally coupled
to gravity. It is shown that the appropriate $T_{\rm reh}$ is obtained with
non-minimal coupling $\xi\simeq 30$.

##  6 Hyperkination

There is another example of creating enhanced primordial GWs by inflation,
this time by truncating a peak in the GW spectrum generated by a stiff period.
I have investigated this with collaborators in Ref. [17]. It can be done as
follows.

In Palatini modified gravity we consider

${\cal L}=\frac{1}{2}m_{P}^{2}R+\frac{1}{2}\alpha
R^{2}+\frac{1}{2}\xi\varphi^{2}R-\frac{1}{2}\partial_{\mu}\varphi\,\partial^{\mu}\varphi-V(\varphi)\,,$
(27)

where $\alpha$ and $\xi$ are non-perturbative coefficients. Switching to the
Einstein frame we obtain

${\cal
L}=\frac{1}{2}m_{P}^{2}R-\frac{1}{2}\partial_{\mu}\phi\,\partial^{\mu}\phi+\frac{1}{4}\alpha\frac{h^{2}+4\alpha
V}{h^{2}m_{P}^{4}}(\partial_{\mu}\phi\,\partial^{\mu}\phi)^{2}-\frac{Vm_{P}^{4}}{h^{2}+4\alpha
V}\,,$ (28)

where $h(\varphi)=m_{P}^{2}+\xi\varphi^{2}$ and we employed the field
redefinition

$\frac{{\rm d}\phi}{{\rm d}\varphi}=\sqrt{\frac{hm_{P}^{2}}{h^{2}+4\alpha
V}}\,.$ (29)

In the above there is a strange quartic kinetic term. Such a term can be
considered in general k-inflation models (no need for Palatini modified
gravity) [18].

The EoM is

$\displaystyle\left[1+3\alpha\left(1+\frac{4\alpha
V}{h^{2}}\right)\frac{\dot{\phi}^{2}}{m_{P}^{4}}\right]\ddot{\phi}+3\left[1+\alpha\left(1+\frac{4\alpha
V}{h^{2}}\right)\frac{\dot{\phi}^{2}}{m_{P}^{4}}\right]H\dot{\phi}$
$\displaystyle+\;3\alpha^{2}\frac{\dot{\phi}^{4}}{m_{P}^{4}}\frac{\rm d}{{\rm
d}\phi}\left(\frac{V}{h^{2}}\right)+\frac{\rm d}{{\rm
d}\phi}\frac{Vm_{P}^{4}}{h^{2}+4\alpha V}$ $\displaystyle=0\,.$ (30)

Then, from the energy-momentum tensor we can obtain the energy density and
pressure of the field, which read

$\displaystyle\rho_{\phi}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[1+\frac{3}{2}\alpha\left(1+\frac{4\alpha
V}{h^{2}}\right)\frac{\dot{\phi}^{2}}{m_{P}^{4}}\right]\dot{\phi}^{2}+\frac{Vm_{P}^{4}}{h^{2}+4\alpha
V}\,,$ (31) $\displaystyle p_{\phi}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[1+\frac{1}{2}\alpha\left(1+\frac{4\alpha
V}{h^{2}}\right)\frac{\dot{\phi}^{2}}{m_{P}^{4}}\right]\dot{\phi}^{2}-\frac{Vm_{P}^{4}}{h^{2}+4\alpha
V}\,.$ (32)

After exiting the inflationary plateau, the inflaton field $\phi$ becomes
dominated by its kinetic energy density, i.e. it becomes oblivious to the
potential $V$. Then the above reduce to

$\left(1+3\alpha\frac{\dot{\phi}^{2}}{m_{P}^{4}}\right)\ddot{\phi}+3\left(1+\alpha\frac{\dot{\phi}^{2}}{m_{P}^{4}}\right)H\dot{\phi}=0$
(33)

and

$\rho_{\phi}=\frac{1}{2}\left(1+\frac{3}{2}\alpha\frac{\dot{\phi}^{2}}{m_{P}^{4}}\right)\dot{\phi}^{2}\quad{\rm
and}\quad
p_{\phi}=\frac{1}{2}\left(1+\frac{1}{2}\alpha\frac{\dot{\phi}^{2}}{m_{P}^{4}}\right)\dot{\phi}^{2}\,.$
(34)

Thus, when the quadratic kinetic term dominates one can effectively set
$\alpha=0$ and we have regular kination with $w=1$. However, when the quartic
kinetic term dominates we can effectively consider only the $\alpha$-depended
terms. Then, $w=p_{\phi}/\rho_{\phi}=\frac{1}{3}$. We call this period
hyperkination. Eq. (19) suggests that the corresponding part of the GW
spectrum is flat as in the radiation era. This means that the peak generated
by kination has been truncated, which implies that the kinetic regime can last
longer without disturbing BBN. As such, the GW signal can be amply boosted at
observable frequencies as shown in Fig. 1.

Figure 1: Plot of $\Omega_{\rm GW}(f)$ superimposed with the observational
expectations of LISA and Advanced LIGO (taken by Ref. [19]). The frequency
range extends up to the scale of inflation, corresponding to the largest
possible frequency, of GW which re-enter the horizon right at the end of
inflation (assumed at the energy scale of grand unification). The uplift in
the spectrum at low frequencies corresponds to the matter era of the hot Big
Bang. The case when inflation is directly followed by the radiation era of the
hot Big Bang (prompt reheating) is depicted by the low horizontal dashed thick
line. As shown, the predicted spectrum is well beyond the observational
capabilities of LISA and Advanced LIGO. The case when an early era of kination
follows right after inflation corresponds to the purple line. As shown, there
is a sharp peak in the spectrum ($\Omega_{\rm GW}(f)\propto f$), which however
cannot be larger than the upper horizontal dashed line, which is the BBN
constraint $\Omega_{\rm GW}<10^{-6}$. Thus, the kination peak cannot extend to
low frequencies and is not near the expected observations of LISA and Advanced
LIGO. The case of a stiff period with $w\approx\frac{1}{2}$ following right
after inflation is depicted with the red line. The peak of GW spectrum is
milder ($\Omega_{\rm GW}(f)\propto f^{2/5}$), which means that it can be
extended to lower frequencies without violating the BBN bound. As shown, the
spectrum will be detectable by both LISA and Advanced LIGO. Finally, the case
when a period of hyperkination (with a flat spectrum) and then regular
kination follows the end of inflation is shown with a green line, which well
overlaps with the expected observations of LISA and Advanced LIGO. In all
cases, after reheating, the usual radiation era of the hot Big Bang begins and
the GW spectra become flat as the frequency is lowered.

##  7 Conclusions

Cosmic Inflation resolves the fine-tunings of the hot Big Bang and provides
seeds for structure formation. Inflation is spectacularly verified by CMB
observations. Another generic prediction of inflation is a superhorizon
spectrum of primordial gravitational waves (GWs) generated through particle
production. The form of the resulting GW spectrum depends on the post-
inflation history. However, when GW modes re-enter the horizon during
radiation domination they form a flat spectrum, too faint to be observable at
present.

A stiff period in the Universe history enhances primordial GWs forming a peak
in their spectrum. Non-oscillatory inflation is followed by such a period,
dominated by the inflaton’s kinetic energy density, called kination, but the
frequencies of the peak are too high. The GW peak can be extended to
observable frequencies if the stiff period is milder than that of kination,
with $w\approx 1/2$. A model realisation of this possibility considers two
flat directions which intersect at an ESP and give rise to the hybrid
mechanism with Planckian waterfall VEV, which is also a kinetic pole of the
waterfall field, as in $\alpha$-attractors.

Another possibility to obtain a boost in primordial GWs down to observable
frequencies is by considering higher order kinetic terms, as with k-inflation.
This is possible to realise in Palatini modified gravity. Considering
$R+R^{2}$ gravity and a non-minimally coupled scalar field, results in
additional quartic kinetic terms. When the quartic kinetic terms dominate,
this gives rise to to hyperkination. Hyperkination is followed by regular
kination, when the kinetic terms become canonical. The resulting truncated GW
peak can be extended to observable frequencies without disturbing BBN.

Forthcoming observations of Advanced LIGO, LISA, DesiGO and BBO may well
detect the primordial GWs generated by inflation. Detection of primordial GWs
will not only confirm a prediction of inflation but offer tantalising evidence
of the quantum nature of gravity, because the Bunch-Davis vacuum of virtual
gravitons is assumed as an initial condition for the generation of GWs during
inflation by particle production.

### Acknowledgements

This work was funded in part by STFC with the consolidated grant:
ST/X000621/1.

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