text:
\section{Introduction} \label{sec.Intro}
The search for primordial gravitational waves (PGWs) is one of the main goals of modern cosmology as they can both provide a substantial evidence for primordial inflation and shed light on its physical nature \cite{Kamionkowski:2015yta,Baumann:2014cja,Caldwell:2018giq,Franciolini:2018ebs}. In fact long-wavelength gravitational waves are predicted by primordial cosmic inflation \cite{Guth:1980zm,Starobinsky:1980te,Linde:1981mu,Vilenkin:1983xq} and, at least in the simplest models, the scale at which inflation occurs is itself related to the amount of PGWs \cite{Sahni:1990tx, Lyth:2009zz,Mukhanov:2005sc,Dodelson:2003ft,Weinberg:2008zzc,Martin:2013tda,Kamionkowski:2015yta,Mirbabayi:2014jqa,Ozsoy:2014sba}. The missing evidence of the B-modes in the Cosmic Microwave Background (CMB) polarization originated from the inflationary tensor modes and, in general, a combined analysis of the Planck and BICEP2/Keck array (BK15) data \cite{Ade:2018gkx}, allow us to set only an upper bound on the amplitude of PGWs on the CMB scales $r<0.07$ at 95\% C.L. at the pivot scale $k_* = 0.05\,\rm{Mpc^{-1}}$ \cite{Akrami:2018odb}. Nevertheless, in the upcoming decade, a new generation of CMB experiments such as BICEP3 \cite{BICEP3}, CLASS \cite{CLASS} , SPT-3G \cite{SPT-3G}, Advanced ACTPol \cite{ACTPol}, LBIRD \cite{LBIRD} and CMB-S4 \cite{CMB-S4}  are expected to bring the sensitivity to the tensor amplitude down to $r \sim 0.01 - 0.001$ possibly leading to its first detection.

In deriving such bounds the slow roll consistency relation $n_{\rm t}=-r/8$ is usually assumed, basically leading to an almost scale independent slightly red tilted primordial tensor spectrum $\mathcal P_{\rm t} (k)$. However this relation can be violated in many non standard models of inflation and when it is relaxed the Planck data only weakly constrain the tensor tilt to $-0.55<n_{\rm T}<2.54$ at 95\% C.L. \cite{Akrami:2018odb}. Nevertheless, always in \cite{Akrami:2018odb}, it was shown that a significant improvement in the upper bound on the tensor tilt can be obtained considering the LIGO/VIRGO data on the stochastic background of gravitational waves $\Omega_{\rm GW}$, \textit{i.e.} the analogous of CMB for gravitational waves \cite{Caprini_2018}. Indeed, while a direct detection of the stochastic background has not been provided yet, in the frequency range $f\in\left(20\,\rm{-}\,85.8\right)$ Hz, which corresponds to the wavenumber range  $k \in \left(1.3\,\rm{-}\,5.5\right) \times 10^{16} \,\rm{Mpc}^{-1}$, the first and second observing runs of the LIGO/VIRGO collaboration set an upper bound on the stochastic background
 \begin{equation}
 \Omega_{\rm{GW}} (k_{\rm LV}) \leq 1.7 \times 10^{-7}
 \label{LVlimit}
 \end{equation}
at 95 \% C.L.  \cite{Akrami:2018odb,TheLIGOScientific:2016dpb}.
The Planck Collaboration, including the LIGO/VIRGO limit \eqref{LVlimit} as a half-gaussian prior on the tensor tilt, under the assumption of scale independence, derived the improved upper bound  $n_T < 0.53$ at 95\% C.L. \cite{Akrami:2018odb}. Notice that these constraints are obtained marginalizing over the probability distribution of $r$, that is typically sampled assuming a flat prior $r\in[r_{\min}\,,\,r_{\rm max}]$ with $r_{\rm min}\lesssim 10^{-3}\simeq 0$. This makes the constraint on the tensor tilt subject to misunderstanding as if there is no detection of $r$ no reliable constraint can be derived. Moreover, another important assumption beyond this analysis, is to consider the tensor tilt as scale independent extending the well known power law relation
\begin{equation}
\mathcal P_{\rm t}(k) = r\, A_{\rm s}\, \left( \frac{k}{k_{*}}\right)^{n_{\rm t}}
\label{no_runnings}
\end{equation}
(with $A_S = 2.1\times 10^{-9}$ the scalar amplitude at the pivot scale $k_*=0.05\rm{Mpc}^{-1}$) from the CMB scales ($k\sim 0.05 \, \rm{Mpc}^{-1})$ all the way up to the small scales probed by the gravitational interferometers ($k\sim 10^{16} \, \rm{Mpc}^{-1}$).  This is clearly an approximation as Eq. \eqref{no_runnings} is just a first order expansion: depending on the model of inflation, $n_{\rm t}$ can also acquire a slight scale dependence and non-linearities may break the power-law relation. It is therefore timely to investigate the impact of higher-order corrections on the constraints one can derive exploiting small-scale data on Gravitational Waves. We show that due to the huge difference in the scales probed by CMB and GW data, the higher-order terms in the spectrum, the so-called tensor runnings  \cite{Giare:2019snj,Kuroyanagi:2011iw}, albeit small on the CMB scales, may give non-negligible contributions on the ultrahigh $k$ probed by gravitational detectors, drastically changing the final predictions.

The paper is organized as follows: in Sec. \ref{sec.ScaleDependence} we discuss the constraints on the inflationary parameters from gravitational wave experiments allowing the possibility to have a scale dependent tilt. We show that the results strongly depend on the assumption of scale-independence.
In Sec. \ref{sec.Examples} we study the scale dependence in two different physical models of inflation. In particular in Sec. \ref{sec.Starobinsky} we consider the Starobinsky model showing that it predicts a negligible scale dependence as expected in the standard slow roll paradigm. In Sec. \ref{sec.ParticleProduction} instead we study a more elaborated scenario of inflation that employs a pseudo scalar axion naturally coupled to gauge fields. We show that in this model the tensor tilt can acquire a non negligible scale dependence leading to appreciable corrections on small scales.
In Sec. \ref{sec.Conclusion} we present our conclusion.

\section{Scale dependence} \label{sec.ScaleDependence}
Along with B-modes polarization, primordial tensor fluctuations may have imprinted also the stochastic background of gravitational waves, the analogous of CMB for gravitational waves \cite{Caprini_2018}. The energy density of the universe due to PGWs at the present time and at a given scale $k=2\pi\,f$ is given by \cite{Akrami:2018odb,Bartolo:2016ami,Cabass:2015jwe,Stewart:2007fu,Graef:2018fzu}
 \begin{equation}
 \Omega_{\mathrm{GW}}(k) \doteq \frac{1}{\rho_{c}} \frac{\mathrm{d} \rho_{\mathrm{GW}}}{\mathrm{d} \log k}\simeq \frac{\mathcal P_{\mathrm{t}} (k)} {24 z_{\mathrm{eq}}}
 \label{Omega_GW}
 \end{equation}
where $z_{\rm{eq}}\simeq 3400$ is the redshift at the matter-radiation equivalence \cite{Akrami:2018odb}. Using Eq. \eqref{no_runnings}, it is easy to see that, under the assumption of scale-independent tilt, a constraint on the amplitude of the stochastic background $\Omega_{\rm GW}(k)$ can be translated into an upper bound on the tensor tilt
\begin{equation}
 n_{\rm t} <\frac{\ln\left(\frac{24\,z_{\rm eq}\,\Omega_{\rm GW}(k)}{r\,A_{\rm S}}\right)}{\ln\left(\frac{k}{k_*}\right)} \lesssim 0.39 + 0.025\times \log(1/r),
 \label{blue_tilt}
 \end{equation}
where in the last inequality we considered the LIGO/VIRGO limit \eqref{LVlimit}.
Note that the constraint \eqref{blue_tilt} is derived without any assumption on the tensor amplitude\footnote{This is a different approach with respect to those performed in \cite{Akrami:2018odb} where the upper bound $n_{\rm t}<0.53$ at 95\% C.L. was derived marginalizing over the distribution of $r$. Anyway we see that for $r\sim 10^{-2} - 10^{-3}$ we basically recover the same result, see also Fig. \ref{fig:figure1}.}: if future measurements will reveal evidence for $r \ne 0$, its detection will immediately place a well defined upper bound on the tensor tilt that however, because of its logarithmic dependence, is not drastically sensitive to the precise value of the tensor amplitude, as one can also appreciate in Fig. \ref{fig:figure1}. The physical reason beyond this weak dependence on the tensor amplitude is actually that a large positive tilt will strongly amplify the GWs production on the ultrahigh $k$ probed by gravitational detectors, easily compensating a small (but of course not vanishing) $r$ on the CMB scales. On the other hand, it is also true that constraints on $n_{\rm t}$ cannot be derived for a vanishing tensor amplitude, and in fact we may see that taking the limit $r\to 0$, the right side of Eq. \eqref{blue_tilt} logarithmically diverges, as expected.

As we said, the constraint \eqref{blue_tilt} as well as the upper bound  $n_{\rm t}<0.53$ at 95\% C.L. \cite{Akrami:2018odb} is derived assuming a constant tilt over a range of about eighteen order of magnitude, namely $k\in[0.05\,,\,1.3\times 10^{16}]\,\rm{Mpc^{-1}}$. In order to include a possible scale dependence, we generalize the power law parametrization to the following expansions:
 \begin{equation}
 \mathcal P_{\rm t}(k) =  r\, A_{\rm s}\,\left(\frac{k}{k_*}\right)^{n_{\rm t}(k_*) + \sum_{n=1}^{\infty} \frac{\alpha_n^{\rm t}(k_*)} {(n+1)!} \left[\log\left(\frac{k}{k_*}\right)\right]^n}
 \label{Tensor}
 \end{equation}
 where
 \begin{equation}
 \alpha_n^{\rm t}(k_*)\doteq \left( \frac{d}{d\log k}\right)^n\,n_{\rm t}(k)\bigg \rvert _{k=k_*}
 \end{equation}
 is the $n$-order tensor running of the tensor tilt\footnote{In what follows we will usually avoid to specify that the spectral tilt and the runnings are computed on the pivot scale $k_*$ and, to simplify the notation, we will only write $n_{\rm t}$ and $\alpha_n^{\rm t}$.}. When the runnings are included, the upper bound \eqref{blue_tilt} is modified to

 \begin{equation}
 n_{\rm t}\lesssim \frac{\ln\left(\frac{24\,z_{\rm eq}\,\Omega_{\rm GW}(k)}{r\,A_{\rm S}}\right)}{\log\left(\frac{k}{k_*}\right)}  -  \sum_{n=1}^{\infty} \frac{\alpha_n^{\rm t}(k_*)} {(n+1)!} \left[\log\left(\frac{k}{k_*}\right)\right]^n.
 \label{runnings}
 \end{equation}
 Clearly, in order to exactly compute the sum expansion and to check its convergence we need to estimate all the runnings $\{\alpha_{n}^{\rm{t}}\}$ and this is possible only fixing a specific model of inflation.  Nevertheless we can appreciate how the generic $n$-order running must at least satisfy the condition $|\alpha_n^{\rm t}/n_{\rm t}|\ll (n+1)! / \log^n(k/k_*)$
to give a negligible contribution at the generic scale $k$. This should become a non trivial requirement, above all on the ultra-high $k$ as those probed by gravitational interferometers.
 \begin{figure}[h]
 	\centering
 	\includegraphics[width=0.75\linewidth]{figure1.pdf}
 	\caption{Constraints on the tensor tilt from the LIGO/VIRGO limit on the stochastic background \eqref{LVlimit}. The yellow dots represent the upper bounds on $n_{\rm t}$ for different values of $r$ when scale-dependence is ignored. When a scale dependence $d\log n_{\rm t} / d\log k\ne0$ is considered the yellow dots move on the red lines at constant $r$.}
 	\label{fig:figure1}
 \end{figure}

To study how constraints on $n_{\rm t}$ derived under the assumption of scale independence are modified in presence of a slight scale dependence, we derive the upper bound on the tensor tilt by the LIGO/VIRGO limit \eqref{LVlimit} for different value of the tensor to scalar ratio, varying $d\log n_{\rm t} / d\log k$ - that represents the rate of change of the tensor tilt with respect to the scale - in a range $d\log n_{\rm t} / d\log k\in \left[-0.04\,,\,0.04\right]$. We find that, because of the huge difference between the scales probed by CMB and GW data, also a small departure from scale independence ($\lesssim 4\%$) can significantly change the final results, see also Fig. \ref{fig:figure1}.  In particular, a small negative (positive) running\footnote{We recall that $d\log n_{\rm t}/d\log k\doteq \alpha_1^{\rm t} / n_{\rm t}$.}, suppressing (amplifying) the amplitude of PGWs on small scales, can remarkably worsen (improve) the upper bound derived under the assumption $d\log n_{\rm t} / d\log k =0$ (yellow dots in Fig. \ref{fig:figure1}). This is clearly translated into a strong degeneracy between scale-dependence and the tensor amplitude (see Fig. \ref{fig:figure1}) that can be broken only by an independent measurement of $r$ from future CMB experiments.
We conclude that the small-scale constraints on $n_{\rm t} $ may be very sensitive to the assumption of scale independence: non-negligible contributions can arise from non-linear corrections and they cannot be always ignored when constraints on the inflationary parameters are derived exploiting GW data.
\section{Examples} \label{sec.Examples}
To further validate our discussion, in this section we study two physical models of inflation. We first analyze the Starobinsky model that, being pure slow roll, by definition predicts an almost scale independent slightly red tilted spectrum. It represents an example of models where higher order corrections are typically negligible also on small scales. However this cannot be true in more elaborated scenarios: as a counterexample we study a model of particle production where non linear corrections lead to a non negligible scale dependence.
\subsection{Starobinsky Inflation}\label{sec.Starobinsky}
In the simplest framework of single field slow roll inflation one can compute the primordial spectra both for scalar and tensor perturbations to obtain \cite{Riotto:2018pcx,Martin:2013tda,Guth:1985ya,Mukhanov:1990me,Starobinsky:1979ty,Mukhanov:2013tua,Bartolo:2001rt}:
\begin{equation}
\mathcal{P} _ {\rm s } = \left(\frac{1}{8\pi^2 M_{p}^2}\right)\left(\frac{H^2}{\epsilon_1}\right)
\label{scalar_spectrum}
\end{equation}
\begin{equation}
\mathcal{P} _ {\rm t } = \left(\frac{2}{\pi^2 M_{p}^2}\right)H^2
\label{tensor_spectrum}
\end{equation}
where $M_{\rm p}$ is the reduced Planck mass ($=2.435\times10^{18}$ GeV), $H$ is the Hubble parameter, $\epsilon_1\doteq-\dot H /H^2$ is the first of the slow roll parameters $\{\epsilon_1\,...,\epsilon_n\}$ defined as $\epsilon_{n>1}\doteq d\log\epsilon_{n-1} /d\log k$. All these relations are to be considered calculated at the horizon crossing.
We also introduce the usual slow roll relations for the tensor to scalar ratio $r$, the tensor tilt $n_{\rm t}$ and the scalar tilt $n_{\rm s}$ \cite{Martin:2013tda,Giare:2019snj}:
\begin{equation}
n_{\rm t}\doteq \frac{d\log \mathcal P_{\rm t}}{d\log k}=-2\epsilon_1 =- \frac{r}{8}
\label{nt}
\end{equation}

\begin{equation}
n_{\rm s} -1 \doteq \frac{d\log\mathcal P_{\rm s}}{d\log k}=-2\epsilon_1-\epsilon_2.
\label{ns}
\end{equation}
In this subsection we want to estimate the impact of the scale dependence choosing a specific slow roll model of inflation, namely the Starobinsky model \cite{Starobinsky:1980te,Renzi:2019ewp} that predicts the following well known relations:
\begin{equation}
n_{\rm s}-1\simeq -\frac{2}{N} , \quad r\simeq\frac{12}{N^2}
\end{equation}
where $N$ is the e-fold number of inflation that we can fix since we measure $n_{\rm s}\simeq0.96$ with good precision \cite{Akrami:2018odb}.  The tensor tilt and its runnings read
\begin{equation}
n_{\rm t}\simeq -\frac{3}{2}\left(\frac{1}{N}\right)^2 ; \quad  \alpha_n^t\simeq -\frac{3}{2} (n+1)! \left(\frac{1}{N}\right)^{n+2}
\end{equation}
The sum expansion that quantifies the scale dependence of the tensor tilt can be easily computed to be
\begin{equation}
\sum_{n=1}^{\infty} \frac{\alpha_n^{\rm t}(k_*)} {(n+1)!} \left[\log\left(\frac{k}{k_*}\right)\right]^n\simeq  n_{\rm t} \left(\frac{\frac{1}{N}\log(\frac{k}{k_*})}{1-\frac{1}{N}\log(\frac{k}{k_*})}\right)
\end{equation}

\begin{figure}[h]
	\centering
	\includegraphics[width=0.65\linewidth]{figure2.pdf}
	\caption{$\Omega_{\rm{GW}}(k)$ in the Starobinsky model both including (solid lines) and neglecting (dashed lines) the runnings. The scale dependence is negligible.}
	\label{fig:figure2}
\end{figure}
In Fig. \ref{fig:figure2} we plot $\Omega_{\rm{GW}}(k)$ from the CMB scales all the way up to the GW scales both including (black solid line) and neglecting (gray dashed line) the runnings. As one can see the runnings lead to negligible corrections also on small scales. This is not surprising since by definitions the slow roll paradigm predicts an almost scale independent slightly red tilt. We actually study this model to provide an example of negligible scale dependence and to show how the situation can be drastically different in more elaborated scenarios as those discussed in the next subsection.

\subsection{Particle Production} \label{sec.ParticleProduction}
In this subsection we want to provide a counterexample studying a different physical model of inflation that employs a pseudo scalar axion naturally coupled to gauge fields. In this model a mechanism of particle production takes place during the rolling inflation and this can be translated into a blue spectrum of gravitational waves. We will show that the tensor tilt can acquire a non trivial scale dependence as well.
We start giving a brief description of the model, more details can be found in \cite{Anber:2009ua,Pajer:2013fsa,Mukohyama:2014gba,Namba:2015gja,Ferreira:2015omg,Peloso:2016gqs}. We consider a simple theory of a Pseudo Nambo Goldstone Boson inflation. In this model the inflaton field $\phi$ and the axion $\psi$ are minimally coupled to gravity and the axion is also coupled with a $U(1)$ gauge field in a way consistent with symmetries\footnote{Note that the axion is not the inflaton itself but another distinct field.}. The action of the theory is
\begin{equation}
S=\int d^{4} x \sqrt{-g}\left[\frac{M_{p}^{2}}{2} R-\frac{1}{2}(\partial \phi)^{2}-V(\phi)-\frac{1}{2}(\partial \psi)^{2}-U(\psi)-\frac{1}{4} F^{\mu\nu}F_{\mu\nu}-\frac{\psi}{4 f} F_{\mu\nu} \tilde{F}^{\mu\nu}\right]
\end{equation}
$F_{\mu\nu}$ and $\tilde{F}^{\mu\nu}\doteq\frac{1}{2} \epsilon^{\mu \nu \alpha \beta} F_{\alpha \beta}$ are the field-strength tensor of the gauge field and its dual, respectively; $f$ is the axion decay constant while $V(\phi)$ and $U(\psi)$ are the inflation and axion potential. We also assume a flat FRW metric and that both the inflaton and the axion take a homogeneous vacuum expectation value (vev) while the gauge field carries no vev. Under this assumption the equations of motion for the inflaton and the axion are
\begin{equation}
\ddot{\bar{\phi}}+3 H \dot{\bar{\phi}}+V^{\prime}(\bar{\phi})=0
\end{equation}
\begin{equation}
\ddot{\bar{\psi}}+3 H \dot{\bar{\psi}}+U^{\prime}(\bar{\psi})=0
\end{equation}
where the prime denotes the derivatives with respect to the argument and the over-dots denote the derivatives with respect to time. We also assume that  the contribution of the axion on the background evolution is negligible compared to that of the inflaton i.e $\left| U \right|\ll V$ and $\dot{\bar{\psi}}^{2} \ll \dot{\bar{\phi}}^{2}$. We introduce the parameter
\begin{equation}
\xi \equiv \frac{\dot{\bar{\psi}}}{2 H f}
\end{equation}
that will play a crucial role in our future discussion. We assume $\xi$ to be nearly but not exactly scale independent:
\begin{equation}
\xi_1\doteq\frac{d\log\xi}{d\log k}=\frac{\dot \xi}{\xi\,H}\ll 1.
\label{xi1}
\end{equation}
We instead  assume $\xi_1$ to be constant  i.e. $d\log\xi_1/d\log k\approx0$. In our future discussion we restrict our attention to the case $\xi>1$ that allows a blue tensor tilt. We are not going to discuss in details  the  peculiarities of this model  such as the gauge quanta production \cite{Anber:2009ua} that are reviewed also in \cite{Mukohyama:2014gba,Peloso:2016gqs} and the references within, but for our aim it is sufficient to observe that, in order to avoid a significant back-reaction of the produced gauge quanta to the background dynamics, we have to require that
\begin{equation}
\frac{\mathrm{e}^{\pi \xi}}{\xi^{5 / 2}} \ll \frac{13.5}{\sqrt{\epsilon_1 \mathcal{P}_0}} \frac{f}{M_{p}}
\label{decay const}
\end{equation}
where $\mathcal P_0= \left(\frac{1}{8\pi^2 M_{p}^2}\right)\left(\frac{H^2}{\epsilon_1}\right)$ is the primordial scalar spectrum without source (i.e. as predicted by the slow roll inflation). The scalar and tensor spectra for this model are \cite{Barnaby:2012xt, Mukohyama:2014gba,Sorbo:2011rz}:
\begin{equation}
\mathcal P_{\rm s} \simeq \mathcal{P}_0\left(1+c_{\rm s}\,\epsilon_1^{2} \,\mathcal{P}_0 \,\frac{\mathrm{e}^{4 \pi \xi}}{\xi^{6}}\right)\bigg \rvert _{k=k_*}
\label{Ps}
\end{equation}
\begin{equation}
r\simeq\frac{16\, \epsilon_1 \,\left(1+c_{\rm t} \,\epsilon_1 \, \mathcal{P}_0 \frac{\mathrm{e}^{4 \pi \xi}}{\xi^{6}}\right)}{\left(1+c_{\rm s}\,\epsilon_1^{2} \,\mathcal{P}_0 \,\frac{\mathrm{e}^{4 \pi \xi}}{\xi^{6}}\right)} \bigg \rvert _{k=k_*}
\label{Pt}
\end{equation}
where $c_{\rm s}= 2.5 \cdot 10^{-6} $ and $c_{\rm t}=3.4 \cdot 10^{-5}$ are constants.
We compute the spectral tilts from the relation \eqref{Ps} and \eqref{Pt} taking the logarithm derivatives:
\begin{align}
n_{\rm s}-1\doteq \frac{d\log\mathcal P_{\rm s}}{d \log k}\bigg \rvert _{k=k_*}&=\frac{d\log\mathcal P_0}{d\log k} + \frac{c_{\rm s} } {1+c_{\rm s}\,\epsilon_1^2 \,\frac{\mathrm{e}^{4 \pi \xi}}{\xi^{6}}}\,\frac{d}{d\log k}\left(\epsilon_1^2 \mathcal P_0 \frac{\mathrm{e}^{4 \pi \xi}}{\xi^{6}} \right)\\
&=-2(1+f_{\rm s} )\epsilon_1-(1-f_{\rm s})\epsilon_2+f_{\rm s} (4\pi\xi-6)\xi_1\\
&\simeq -2\epsilon_1-\epsilon_2
\label{ns_blue}
\end{align}
and
\begin{align}
n_{\rm t}\doteq \frac{d\log\mathcal P_{\rm t}}{d \log k}\bigg \rvert _{k=k_*}&=\frac{d\log\epsilon_1}{d\log k} + \frac{d\log\mathcal P_0}{d\log k} + \frac{c_{\rm t} } {1+c_{\rm t}\,\epsilon_1 \,\frac{\mathrm{e}^{4 \pi \xi}}{\xi^{6}}}\,\frac{d}{d\log k}\left(\epsilon_1 \mathcal P_0 \frac{\mathrm{e}^{4 \pi \xi}}{\xi^{6}} \right)\\
&=-2(1+f_{\rm t})\epsilon_1+f_{\rm t}(4\pi\xi-6)\xi_1
\label{nt_blue}
\end{align}
where  the functions
\begin{equation}
f_{\rm s}\doteq \frac{c_{\rm s}\,\mathcal P_0\,\epsilon_1^2\,  \frac{\mathrm{e}^{4 \pi \xi}}{\xi^{6}} } {1+c_{\rm s}\,\mathcal P_0\,\epsilon_1^2\,  \frac{\mathrm{e}^{4 \pi \xi}}{\xi^{6}} }\ll 1
\label{fs}
\end{equation}
and
\begin{equation}
f_{\rm t}\doteq\frac{c_{\rm t}\,\mathcal P_0\,\epsilon_1\,  \frac{\mathrm{e}^{4 \pi \xi}}{\xi^{6}} }{1+c_{\rm t}\,\mathcal P_0\,\epsilon_1\,  \frac{\mathrm{e}^{4 \pi \xi}}{\xi^{6}} }
\label{ft}
\end{equation}
weigh the corrections to the slow roll predictions for the scalar and tensor parameters respectively.
In what follows we fix $\mathcal P_{s}$ and $n_s$ to the observed values $\mathcal{P}_{\rm s}\simeq 2.1\times 10^{-9}$ and $n_{\rm s}\simeq 0.96$ \cite{Akrami:2018odb}. We also fix the tensor to scalar ratio $r$ to reference value $r\simeq 10^{-2}$ and $\xi_1\simeq 5 \times 10^{-3}\ll1$. Note that our results are marginally sensitive to the value of $r$ and $\xi_1$ and that we are not interested into a parameter analysis for this specific model: our task is simply to show that also in physical models of inflation scale dependence can be non-negligible.

We use the Eqs. \eqref{Ps} and \eqref{Pt} in order to explicit $\epsilon_1$ and $\mathcal P_0$ as functions of $\xi$. This means that when $\xi$ changes, $\epsilon_1(\xi)$ and $\mathcal P_0(\xi)$ change in such a way that $\mathcal P_s$ and $r$  remain constant. Moreover because of Eq. \eqref{nt_blue} also $n_{\rm t}$ is only a function of $\xi$. Being $n_{\rm s}$ fixed by observations, we can also use the relation \eqref{ns_blue} in order to find $\epsilon_2$ as a function of $\xi$ so that when $\xi$ changes, $\epsilon_2(\xi)$ changes leaving $n_{\rm s}$ fixed to its observed value. So in this model all the inflationary parameters\footnote{The inflationary parameters are to be considered evaluated to the pivot scale $k=k_{*}=0.05\rm{ Mpc}^{-1}$ which means that also the parameter $\xi$ in the equations above is computed on the CMB scales $\xi=\xi(k=k_*)$.} become known functions of $\xi$. We plot them in Fig. \ref{fig:figure3} letting $\xi$ vary in the range $\xi\in[1\,,\,7]$.

\begin{figure}[h]
	\centering
	\includegraphics[width=0.9\linewidth]{figure3.pdf}
	\caption{The parameters of the model as functions of $\xi$.}
	\label{fig:figure3}
\end{figure}

First of all we want to stress that we have carefully checked that the scalar spectrum \eqref{Ps} remains essentially equal to $\mathcal P_0$ (that is what predicted by the single field slow roll inflation). As a matter of fact,  if we decompose the scalar spectrum $\mathcal P_{\rm s}=\mathcal P_0 + \mathcal P_{\rm{s,\,sourced}}$ we find that  the sourced term induces corrections that are extremely small compared to the vacuum contribution $\mathcal P_{\rm{s,\,sourced}}\sim 10^{-4}\,\mathcal P_0$ for all the values of $\xi$. In other words, the corrections to the scalar spectrum are completely negligible ($f_{\rm s}\approx 0$), and the scalar parameters are essentially equal to that obtained in the simplest slow roll models. This can be understood by noting that the scalar corrections are suppressed by a factor $\epsilon_1^2\,\mathcal P_0$ and that $\epsilon_1$ exponentially decreases with $\xi$ in order to keep $r$ fixed, see also Fig.  \ref{fig:figure3}.  The fact that the scalar spectrum is essentially indistinguishable from the single field slow roll models is crucial since in this way all the tight constraints on the scalar perturbations (e.g. their high level of gaussianity) are respected as well \cite{Barnaby:2012xt,Shiraishi:2019yux}.
On the other hand the corrections to the tensor spectrum can be dominant for an appreciable range of the parameter space, allowing also a blue tensor tilt, see Fig. \ref{fig:figure3}. The sourced tensor modes could also leave a sizable non-gaussianity of nearly equilateral shape on the CMB anisotropies and polarization. The amount of non-gaussianity is controlled by the parameter $f_{\rm{NL}}$ estimated as \cite{Mukohyama:2014gba,Cook:2013xea}:
\begin{equation}
f_{\rm{NL}}\simeq 1.1\times 10^{-14}\left(\epsilon_1\,\frac{e^{2\pi\xi} }{\xi^3}\right)^3
\end{equation}
and its shape given in Fig. \ref{fig:figure3}, as well.
We estimate the scale dependence of the tensor tilt performing a second order computation and deriving the expression for the tensor running $\alpha_1^{\rm t}\doteq dn_{\rm t}/d\log k$ and the running of the running $\alpha_2^{\rm t}\doteq d\alpha_1^{\rm t} / d\log k$:
\begin{equation}
\alpha_1^{\rm t}\doteq\frac{dn_{\rm t}}{d\log k}\bigg \rvert _{k=k_*}=-2(1+f_{\rm t})\epsilon_1\epsilon_2 -2 f^{\prime}_{\rm t}\epsilon_1 +f^{\prime}_{\rm t}(4\pi\xi-6)\xi_1 +4\pi f_{\rm t} \xi\xi_1^2
\label{alpha_1}
\end{equation}
\begin{equation}
\alpha_2^{\rm t}\doteq\frac{d\alpha_1^{\rm t}}{d\log k}\bigg \rvert _{k=k_*}=-2(1+f_{\rm t})\left(\epsilon_1\epsilon_2^2+\epsilon_1\epsilon_2\epsilon_3\right)-4 f^{\prime}_{\rm t}\epsilon_1\epsilon_2-2 f^{\prime \prime}_{\rm t}\epsilon_1 +f^{\prime \prime}_{\rm t} (4\pi\xi-6)\xi_1 +8\pi f^{\prime}_{\rm t}\xi\xi_1^2 +4\pi f_{\rm t} \xi \xi_1^3
\label{alpha_2}
\end{equation}
where we have defined:
\begin{equation}
f^{\prime}_{\rm t}\doteq\frac{d f_{\rm t}}{d\log k}\bigg \rvert _{k=k_*}=\left[ \frac{-2\epsilon_1 + (4\pi\xi - 6)\xi_1}{1+c_{\rm t}\mathcal P_{0} \epsilon_1 \frac{\mathrm{e}^{4 \pi \xi}}{\xi^{6}}  } \right] f_{\rm t}
\end{equation}
and
\begin{equation}
\begin{aligned}
f^{\prime \prime}_{\rm t}\doteq \frac{df^{\prime}_{\rm t}}{d\log k}\bigg \rvert _{k=k_*} =&\left[ \frac{-2\epsilon_1 + (4\pi\xi - 6)\xi_1} {1+c_{\rm t}\mathcal P_{0} \epsilon_1 \frac{\mathrm{e}^{4 \pi \xi}}{\xi^{6}}  } \right]^2 f_{\rm t}+\\
&+\left[ \frac{(1+c_{\rm t} \mathcal P_0 \epsilon_1 \frac{\mathrm{e}^{4 \pi \xi}}{\xi^{6}} ) (-2\epsilon_1\epsilon_2+4\pi\xi\xi_1^2) - c_{\rm t} \mathcal P_0 \epsilon_1 \frac{\mathrm{e}^{4 \pi \xi}}{\xi^{6}}\left[-2\epsilon_1+(4\pi\xi-6)\xi_1\right]^2} {\left(1+c_{\rm t}\mathcal P_{0} \epsilon_1 \frac{\mathrm{e}^{4 \pi \xi}}{\xi^{6}}\right)^2  } \right]f_{\rm t}
\end{aligned}
\end{equation}

In this model the tensor tilt can acquire a non trivial scale dependence. In fact, depending on the parameters, $d\log n_{\rm t}/d\log k \simeq 0.1$, see Fig. \ref{fig:figure3}. As we discussed in Sec. \ref{sec.ScaleDependence}, this can lead to non negligible corrections on small scales.

As explained before all these quantities are known functions\footnote{ Note that we parametrized the slow parameter $\epsilon_3$ appearing in \eqref{alpha_2} as $\epsilon_3=\gamma\,\epsilon_2$. Letting $\gamma$ vary in a range $\gamma\in\left[-1,1\right]$ no significant changes in $\alpha_2^{\rm t}$ are observed. We therefore fixed $\epsilon_3\simeq 0$.} of $\xi$ or equivalently $n_{\rm t}$. However, since for large values $\xi\gtrsim 5$ the backreaction becomes typically non negligible as well as the primordial non gaussianity, we decide to restrict our attention to a safer region of the parameter space. We therefore fix  $n_{\rm t}\simeq0.1$ (or equivalently $\xi\simeq 3.5$) in such a way that both backreaction and non gaussianity are still under control, see Fig. \ref{fig:figure3}.  In this way the running  $\alpha_1^{\rm t}\simeq 0.01$ and the running of running $\alpha_2^{\rm t}\simeq 3\times 10^{-6}$ are fixed as well. We let evolve $\Omega_{\rm GW}(k)$ from the CMB scales all the way up to the ultra high $k$ probed by the ground based interferometers both including and neglecting $ \alpha_1^{\rm t}$ and $\alpha_2^{\rm t}$, see Fig. \ref{fig:figure4}.  The importance of scale dependence in this model is evident as $\Omega_{\mathrm{GW}}$ differs by many orders of magnitude when the non-linear corrections are considered, possibly becoming visible to future gravitational wave experiments such as LISA \cite{Audley:2017drz} and Einstein Telescope \cite{Punturo:2010zz}.

While we have shown that the impact of the second order running $\alpha_2^{\rm t}$ is completely negligible, see also Fig.\ref{fig:figure4},  one may ask if the higher order terms $\alpha^{\rm t}_{n>2}$ can instead give an appreciable contribution possibly changing the shape of $\Omega_{\rm GW} (k)$. For our aim it is sufficient to note that being the tensor tilt only a function of $\xi$, the derivative with respect to the scale can be written as $d/d\log k=\left(d\xi/d\log k\right) \,d/d\xi=\left(\xi\xi_1\right) d/d\xi$ and that the overall factor $\xi \xi_1\sim 10^{-2}$ will further suppress the higher order derivatives. We therefore expect such terms to be smaller and smaller at least in this range of the parameters space. We leave the detailed analysis of the sum expansion convergence suitable for future works.

\begin{figure}[ht!]
	\centering
	\includegraphics[width=0.65\linewidth]{figure4.pdf}
	\caption{$\Omega_{\rm{GW}}(k)$ in the particle production model both including and neglecting the first two runnings. The scale dependence is not negligible.}
	\label{fig:figure4}
\end{figure}

\section{Conclusion} \label{sec.Conclusion}
Along with the  B-modes polarization, a satiable amount of primordial gravitational waves may have imprinted also the stochastic background of gravitational waves.
Being the small scale experiments on gravitational waves such as LIGO and VIRGO (and in future LISA or ET) sensitive to the stochastic background, they can be used together with the CMB data to put constraints on the inflationary parameters. In particular from the LIGO/VIRGO bound on $\Omega_{\rm GW}$, Eq.\eqref{LVlimit},  under the assumption of a scale independent tilt, an upper bound $n_{\rm t}<0.53$ at 95\% C.L. is derived combining the CMB and GW data \cite{Akrami:2018odb}.
In this paper we focused on the robustness of assuming the tensor tilt as scale independent over a range of about eighteen order of magnitude $k\in\left[0.05 \,,\,\sim10^{16}\right]\,	\rm{Mpc}^{-1}$.  We have shown that when the assumption of scale independence is relaxed, the constraints may become very sensitive to the higher order corrections in the power law expansion and that even a tiny scale dependence can significantly change the above mentioned constraints. Due to the huge distance between the scales probed by the CMB and the GW interferometers, also a relatively small departure ($\sim 4\%$) from scale independence can significantly affect the shape of primordial spectrum possibly leading to non-negligible corrections. We concluded that scale dependence cannot always be ignored when constraints on the inflationary parameters are derived  from small scales data. We have also provided two examples in completely different physical models of inflation. We have first considered  the Starobisnky model showing that the scale dependence is negligible as one expects in the simplest slow roll paradigm. The Starobinsky model (and in general the standard slow roll models) provides an example where the non-linear corrections are typically negligible even on the ultra high $k$ probed by the ground based interferometers. However this is not always true. As a counterexample, we have analyzed a physical model of blue inflation which employs a pseudo scalar axion naturally coupled to gauge fields showing that in this model the tensor tilt can acquire a non trivial scale dependence leading to non negligible corrections on small scales.

\section*{Acknowledgments}
WG and AM thank Antonio Riotto for the useful collaboration and for his precious suggestions that contributed to the realization of this article. WG thanks Marco Peloso and Fabrizio Renzi for their comments and valid clarifications. AM and WG are supported by "Theoretical Astroparticle Physics" (TAsP), iniziativa specifica INFN.

\section{Introduction} \label{sec.Intro}
The search for primordial gravitational waves (PGWs) is one of the main goals of modern cosmology as they can both provide a substantial evidence for primordial inflation and shed light on its physical nature \cite{Kamionkowski:2015yta,Baumann:2014cja,Caldwell:2018giq,Franciolini:2018ebs}. In fact long-wavelength gravitational waves are predicted by primordial cosmic inflation \cite{Guth:1980zm,Starobinsky:1980te,Linde:1981mu,Vilenkin:1983xq} and, at least in the simplest models, the scale at which inflation occurs is itself related to the amount of PGWs \cite{Sahni:1990tx, Lyth:2009zz,Mukhanov:2005sc,Dodelson:2003ft,Weinberg:2008zzc,Martin:2013tda,Kamionkowski:2015yta,Mirbabayi:2014jqa,Ozsoy:2014sba}. The missing evidence of the B-modes in the Cosmic Microwave Background (CMB) polarization originated from the inflationary tensor modes and, in general, a combined analysis of the Planck and BICEP2/Keck array (BK15) data \cite{Ade:2018gkx}, allow us to set only an upper bound on the amplitude of PGWs on the CMB scales $r<0.07$ at 95\% C.L. at the pivot scale $k_* = 0.05\,\rm{Mpc^{-1}}$ \cite{Akrami:2018odb}. Nevertheless, in the upcoming decade, a new generation of CMB experiments such as BICEP3 \cite{BICEP3}, CLASS \cite{CLASS} , SPT-3G \cite{SPT-3G}, Advanced ACTPol \cite{ACTPol}, LBIRD \cite{LBIRD} and CMB-S4 \cite{CMB-S4}  are expected to bring the sensitivity to the tensor amplitude down to $r \sim 0.01 - 0.001$ possibly leading to its first detection.

In deriving such bounds the slow roll consistency relation $n_{\rm t}=-r/8$ is usually assumed, basically leading to an almost scale independent slightly red tilted primordial tensor spectrum $\mathcal P_{\rm t} (k)$. However this relation can be violated in many non standard models of inflation and when it is relaxed the Planck data only weakly constrain the tensor tilt to $-0.55<n_{\rm T}<2.54$ at 95\% C.L. \cite{Akrami:2018odb}. Nevertheless, always in \cite{Akrami:2018odb}, it was shown that a significant improvement in the upper bound on the tensor tilt can be obtained considering the LIGO/VIRGO data on the stochastic background of gravitational waves $\Omega_{\rm GW}$, \textit{i.e.} the analogous of CMB for gravitational waves \cite{Caprini_2018}. Indeed, while a direct detection of the stochastic background has not been provided yet, in the frequency range $f\in\left(20\,\rm{-}\,85.8\right)$ Hz, which corresponds to the wavenumber range  $k \in \left(1.3\,\rm{-}\,5.5\right) \times 10^{16} \,\rm{Mpc}^{-1}$, the first and second observing runs of the LIGO/VIRGO collaboration set an upper bound on the stochastic background
 \begin{equation}
 \Omega_{\rm{GW}} (k_{\rm LV}) \leq 1.7 \times 10^{-7}
 \label{LVlimit}
 \end{equation}
at 95 \% C.L.  \cite{Akrami:2018odb,TheLIGOScientific:2016dpb}.
The Planck Collaboration, including the LIGO/VIRGO limit \eqref{LVlimit} as a half-gaussian prior on the tensor tilt, under the assumption of scale independence, derived the improved upper bound  $n_T < 0.53$ at 95\% C.L. \cite{Akrami:2018odb}. Notice that these constraints are obtained marginalizing over the probability distribution of $r$, that is typically sampled assuming a flat prior $r\in[r_{\min}\,,\,r_{\rm max}]$ with $r_{\rm min}\lesssim 10^{-3}\simeq 0$. This makes the constraint on the tensor tilt subject to misunderstanding as if there is no detection of $r$ no reliable constraint can be derived. Moreover, another important assumption beyond this analysis, is to consider the tensor tilt as scale independent extending the well known power law relation
\begin{equation}
\mathcal P_{\rm t}(k) = r\, A_{\rm s}\, \left( \frac{k}{k_{*}}\right)^{n_{\rm t}}
\label{no_runnings}
\end{equation}
(with $A_S = 2.1\times 10^{-9}$ the scalar amplitude at the pivot scale $k_*=0.05\rm{Mpc}^{-1}$) from the CMB scales ($k\sim 0.05 \, \rm{Mpc}^{-1})$ all the way up to the small scales probed by the gravitational interferometers ($k\sim 10^{16} \, \rm{Mpc}^{-1}$).  This is clearly an approximation as Eq. \eqref{no_runnings} is just a first order expansion: depending on the model of inflation, $n_{\rm t}$ can also acquire a slight scale dependence and non-linearities may break the power-law relation. It is therefore timely to investigate the impact of higher-order corrections on the constraints one can derive exploiting small-scale data on Gravitational Waves. We show that due to the huge difference in the scales probed by CMB and GW data, the higher-order terms in the spectrum, the so-called tensor runnings  \cite{Giare:2019snj,Kuroyanagi:2011iw}, albeit small on the CMB scales, may give non-negligible contributions on the ultrahigh $k$ probed by gravitational detectors, drastically changing the final predictions.

The paper is organized as follows: in Sec. \ref{sec.ScaleDependence} we discuss the constraints on the inflationary parameters from gravitational wave experiments allowing the possibility to have a scale dependent tilt. We show that the results strongly depend on the assumption of scale-independence.
In Sec. \ref{sec.Examples} we study the scale dependence in two different physical models of inflation. In particular in Sec. \ref{sec.Starobinsky} we consider the Starobinsky model showing that it predicts a negligible scale dependence as expected in the standard slow roll paradigm. In Sec. \ref{sec.ParticleProduction} instead we study a more elaborated scenario of inflation that employs a pseudo scalar axion naturally coupled to gauge fields. We show that in this model the tensor tilt can acquire a non negligible scale dependence leading to appreciable corrections on small scales.
In Sec. \ref{sec.Conclusion} we present our conclusion.

\section{Scale dependence} \label{sec.ScaleDependence}
Along with B-modes polarization, primordial tensor fluctuations may have imprinted also the stochastic background of gravitational waves, the analogous of CMB for gravitational waves \cite{Caprini_2018}. The energy density of the universe due to PGWs at the present time and at a given scale $k=2\pi\,f$ is given by \cite{Akrami:2018odb,Bartolo:2016ami,Cabass:2015jwe,Stewart:2007fu,Graef:2018fzu}
 \begin{equation}
 \Omega_{\mathrm{GW}}(k) \doteq \frac{1}{\rho_{c}} \frac{\mathrm{d} \rho_{\mathrm{GW}}}{\mathrm{d} \log k}\simeq \frac{\mathcal P_{\mathrm{t}} (k)} {24 z_{\mathrm{eq}}}
 \label{Omega_GW}
 \end{equation}
where $z_{\rm{eq}}\simeq 3400$ is the redshift at the matter-radiation equivalence \cite{Akrami:2018odb}. Using Eq. \eqref{no_runnings}, it is easy to see that, under the assumption of scale-independent tilt, a constraint on the amplitude of the stochastic background $\Omega_{\rm GW}(k)$ can be translated into an upper bound on the tensor tilt
\begin{equation}
 n_{\rm t} <\frac{\ln\left(\frac{24\,z_{\rm eq}\,\Omega_{\rm GW}(k)}{r\,A_{\rm S}}\right)}{\ln\left(\frac{k}{k_*}\right)} \lesssim 0.39 + 0.025\times \log(1/r),
 \label{blue_tilt}
 \end{equation}
where in the last inequality we considered the LIGO/VIRGO limit \eqref{LVlimit}.
Note that the constraint \eqref{blue_tilt} is derived without any assumption on the tensor amplitude\footnote{This is a different approach with respect to those performed in \cite{Akrami:2018odb} where the upper bound $n_{\rm t}<0.53$ at 95\% C.L. was derived marginalizing over the distribution of $r$. Anyway we see that for $r\sim 10^{-2} - 10^{-3}$ we basically recover the same result, see also Fig. \ref{fig:figure1}.}: if future measurements will reveal evidence for $r \ne 0$, its detection will immediately place a well defined upper bound on the tensor tilt that however, because of its logarithmic dependence, is not drastically sensitive to the precise value of the tensor amplitude, as one can also appreciate in Fig. \ref{fig:figure1}. The physical reason beyond this weak dependence on the tensor amplitude is actually that a large positive tilt will strongly amplify the GWs production on the ultrahigh $k$ probed by gravitational detectors, easily compensating a small (but of course not vanishing) $r$ on the CMB scales. On the other hand, it is also true that constraints on $n_{\rm t}$ cannot be derived for a vanishing tensor amplitude, and in fact we may see that taking the limit $r\to 0$, the right side of Eq. \eqref{blue_tilt} logarithmically diverges, as expected.

As we said, the constraint \eqref{blue_tilt} as well as the upper bound  $n_{\rm t}<0.53$ at 95\% C.L. \cite{Akrami:2018odb} is derived assuming a constant tilt over a range of about eighteen order of magnitude, namely $k\in[0.05\,,\,1.3\times 10^{16}]\,\rm{Mpc^{-1}}$. In order to include a possible scale dependence, we generalize the power law parametrization to the following expansions:
 \begin{equation}
 \mathcal P_{\rm t}(k) =  r\, A_{\rm s}\,\left(\frac{k}{k_*}\right)^{n_{\rm t}(k_*) + \sum_{n=1}^{\infty} \frac{\alpha_n^{\rm t}(k_*)} {(n+1)!} \left[\log\left(\frac{k}{k_*}\right)\right]^n}
 \label{Tensor}
 \end{equation}
 where
 \begin{equation}
 \alpha_n^{\rm t}(k_*)\doteq \left( \frac{d}{d\log k}\right)^n\,n_{\rm t}(k)\bigg \rvert _{k=k_*}
 \end{equation}
 is the $n$-order tensor running of the tensor tilt\footnote{In what follows we will usually avoid to specify that the spectral tilt and the runnings are computed on the pivot scale $k_*$ and, to simplify the notation, we will only write $n_{\rm t}$ and $\alpha_n^{\rm t}$.}. When the runnings are included, the upper bound \eqref{blue_tilt} is modified to

 \begin{equation}
 n_{\rm t}\lesssim \frac{\ln\left(\frac{24\,z_{\rm eq}\,\Omega_{\rm GW}(k)}{r\,A_{\rm S}}\right)}{\log\left(\frac{k}{k_*}\right)}  -  \sum_{n=1}^{\infty} \frac{\alpha_n^{\rm t}(k_*)} {(n+1)!} \left[\log\left(\frac{k}{k_*}\right)\right]^n.
 \label{runnings}
 \end{equation}
 Clearly, in order to exactly compute the sum expansion and to check its convergence we need to estimate all the runnings $\{\alpha_{n}^{\rm{t}}\}$ and this is possible only fixing a specific model of inflation.  Nevertheless we can appreciate how the generic $n$-order running must at least satisfy the condition $|\alpha_n^{\rm t}/n_{\rm t}|\ll (n+1)! / \log^n(k/k_*)$
to give a negligible contribution at the generic scale $k$. This should become a non trivial requirement, above all on the ultra-high $k$ as those probed by gravitational interferometers.
 \begin{figure}[h]
 	\centering
 	\includegraphics[width=0.75\linewidth]{figure1.pdf}
 	\caption{Constraints on the tensor tilt from the LIGO/VIRGO limit on the stochastic background \eqref{LVlimit}. The yellow dots represent the upper bounds on $n_{\rm t}$ for different values of $r$ when scale-dependence is ignored. When a scale dependence $d\log n_{\rm t} / d\log k\ne0$ is considered the yellow dots move on the red lines at constant $r$.}
 	\label{fig:figure1}
 \end{figure}

To study how constraints on $n_{\rm t}$ derived under the assumption of scale independence are modified in presence of a slight scale dependence, we derive the upper bound on the tensor tilt by the LIGO/VIRGO limit \eqref{LVlimit} for different value of the tensor to scalar ratio, varying $d\log n_{\rm t} / d\log k$ - that represents the rate of change of the tensor tilt with respect to the scale - in a range $d\log n_{\rm t} / d\log k\in \left[-0.04\,,\,0.04\right]$. We find that, because of the huge difference between the scales probed by CMB and GW data, also a small departure from scale independence ($\lesssim 4\%$) can significantly change the final results, see also Fig. \ref{fig:figure1}.  In particular, a small negative (positive) running\footnote{We recall that $d\log n_{\rm t}/d\log k\doteq \alpha_1^{\rm t} / n_{\rm t}$.}, suppressing (amplifying) the amplitude of PGWs on small scales, can remarkably worsen (improve) the upper bound derived under the assumption $d\log n_{\rm t} / d\log k =0$ (yellow dots in Fig. \ref{fig:figure1}). This is clearly translated into a strong degeneracy between scale-dependence and the tensor amplitude (see Fig. \ref{fig:figure1}) that can be broken only by an independent measurement of $r$ from future CMB experiments.
We conclude that the small-scale constraints on $n_{\rm t} $ may be very sensitive to the assumption of scale independence: non-negligible contributions can arise from non-linear corrections and they cannot be always ignored when constraints on the inflationary parameters are derived exploiting GW data.
\section{Examples} \label{sec.Examples}
To further validate our discussion, in this section we study two physical models of inflation. We first analyze the Starobinsky model that, being pure slow roll, by definition predicts an almost scale independent slightly red tilted spectrum. It represents an example of models where higher order corrections are typically negligible also on small scales. However this cannot be true in more elaborated scenarios: as a counterexample we study a model of particle production where non linear corrections lead to a non negligible scale dependence.
\subsection{Starobinsky Inflation}\label{sec.Starobinsky}
In the simplest framework of single field slow roll inflation one can compute the primordial spectra both for scalar and tensor perturbations to obtain \cite{Riotto:2018pcx,Martin:2013tda,Guth:1985ya,Mukhanov:1990me,Starobinsky:1979ty,Mukhanov:2013tua,Bartolo:2001rt}:
\begin{equation}
\mathcal{P} _ {\rm s } = \left(\frac{1}{8\pi^2 M_{p}^2}\right)\left(\frac{H^2}{\epsilon_1}\right)
\label{scalar_spectrum}
\end{equation}
\begin{equation}
\mathcal{P} _ {\rm t } = \left(\frac{2}{\pi^2 M_{p}^2}\right)H^2
\label{tensor_spectrum}
\end{equation}
where $M_{\rm p}$ is the reduced Planck mass ($=2.435\times10^{18}$ GeV), $H$ is the Hubble parameter, $\epsilon_1\doteq-\dot H /H^2$ is the first of the slow roll parameters $\{\epsilon_1\,...,\epsilon_n\}$ defined as $\epsilon_{n>1}\doteq d\log\epsilon_{n-1} /d\log k$. All these relations are to be considered calculated at the horizon crossing.
We also introduce the usual slow roll relations for the tensor to scalar ratio $r$, the tensor tilt $n_{\rm t}$ and the scalar tilt $n_{\rm s}$ \cite{Martin:2013tda,Giare:2019snj}:
\begin{equation}
n_{\rm t}\doteq \frac{d\log \mathcal P_{\rm t}}{d\log k}=-2\epsilon_1 =- \frac{r}{8}
\label{nt}
\end{equation}

\begin{equation}
n_{\rm s} -1 \doteq \frac{d\log\mathcal P_{\rm s}}{d\log k}=-2\epsilon_1-\epsilon_2.
\label{ns}
\end{equation}
In this subsection we want to estimate the impact of the scale dependence choosing a specific slow roll model of inflation, namely the Starobinsky model \cite{Starobinsky:1980te,Renzi:2019ewp} that predicts the following well known relations:
\begin{equation}
n_{\rm s}-1\simeq -\frac{2}{N} , \quad r\simeq\frac{12}{N^2}
\end{equation}
where $N$ is the e-fold number of inflation that we can fix since we measure $n_{\rm s}\simeq0.96$ with good precision \cite{Akrami:2018odb}.  The tensor tilt and its runnings read
\begin{equation}
n_{\rm t}\simeq -\frac{3}{2}\left(\frac{1}{N}\right)^2 ; \quad  \alpha_n^t\simeq -\frac{3}{2} (n+1)! \left(\frac{1}{N}\right)^{n+2}
\end{equation}
The sum expansion that quantifies the scale dependence of the tensor tilt can be easily computed to be
\begin{equation}
\sum_{n=1}^{\infty} \frac{\alpha_n^{\rm t}(k_*)} {(n+1)!} \left[\log\left(\frac{k}{k_*}\right)\right]^n\simeq  n_{\rm t} \left(\frac{\frac{1}{N}\log(\frac{k}{k_*})}{1-\frac{1}{N}\log(\frac{k}{k_*})}\right)
\end{equation}

\begin{figure}[h]
	\centering
	\includegraphics[width=0.65\linewidth]{figure2.pdf}
	\caption{$\Omega_{\rm{GW}}(k)$ in the Starobinsky model both including (solid lines) and neglecting (dashed lines) the runnings. The scale dependence is negligible.}
	\label{fig:figure2}
\end{figure}
In Fig. \ref{fig:figure2} we plot $\Omega_{\rm{GW}}(k)$ from the CMB scales all the way up to the GW scales both including (black solid line) and neglecting (gray dashed line) the runnings. As one can see the runnings lead to negligible corrections also on small scales. This is not surprising since by definitions the slow roll paradigm predicts an almost scale independent slightly red tilt. We actually study this model to provide an example of negligible scale dependence and to show how the situation can be drastically different in more elaborated scenarios as those discussed in the next subsection.

\subsection{Particle Production} \label{sec.ParticleProduction}
In this subsection we want to provide a counterexample studying a different physical model of inflation that employs a pseudo scalar axion naturally coupled to gauge fields. In this model a mechanism of particle production takes place during the rolling inflation and this can be translated into a blue spectrum of gravitational waves. We will show that the tensor tilt can acquire a non trivial scale dependence as well.
We start giving a brief description of the model, more details can be found in \cite{Anber:2009ua,Pajer:2013fsa,Mukohyama:2014gba,Namba:2015gja,Ferreira:2015omg,Peloso:2016gqs}. We consider a simple theory of a Pseudo Nambo Goldstone Boson inflation. In this model the inflaton field $\phi$ and the axion $\psi$ are minimally coupled to gravity and the axion is also coupled with a $U(1)$ gauge field in a way consistent with symmetries\footnote{Note that the axion is not the inflaton itself but another distinct field.}. The action of the theory is
\begin{equation}
S=\int d^{4} x \sqrt{-g}\left[\frac{M_{p}^{2}}{2} R-\frac{1}{2}(\partial \phi)^{2}-V(\phi)-\frac{1}{2}(\partial \psi)^{2}-U(\psi)-\frac{1}{4} F^{\mu\nu}F_{\mu\nu}-\frac{\psi}{4 f} F_{\mu\nu} \tilde{F}^{\mu\nu}\right]
\end{equation}
$F_{\mu\nu}$ and $\tilde{F}^{\mu\nu}\doteq\frac{1}{2} \epsilon^{\mu \nu \alpha \beta} F_{\alpha \beta}$ are the field-strength tensor of the gauge field and its dual, respectively; $f$ is the axion decay constant while $V(\phi)$ and $U(\psi)$ are the inflation and axion potential. We also assume a flat FRW metric and that both the inflaton and the axion take a homogeneous vacuum expectation value (vev) while the gauge field carries no vev. Under this assumption the equations of motion for the inflaton and the axion are
\begin{equation}
\ddot{\bar{\phi}}+3 H \dot{\bar{\phi}}+V^{\prime}(\bar{\phi})=0
\end{equation}
\begin{equation}
\ddot{\bar{\psi}}+3 H \dot{\bar{\psi}}+U^{\prime}(\bar{\psi})=0
\end{equation}
where the prime denotes the derivatives with respect to the argument and the over-dots denote the derivatives with respect to time. We also assume that  the contribution of the axion on the background evolution is negligible compared to that of the inflaton i.e $\left| U \right|\ll V$ and $\dot{\bar{\psi}}^{2} \ll \dot{\bar{\phi}}^{2}$. We introduce the parameter
\begin{equation}
\xi \equiv \frac{\dot{\bar{\psi}}}{2 H f}
\end{equation}
that will play a crucial role in our future discussion. We assume $\xi$ to be nearly but not exactly scale independent:
\begin{equation}
\xi_1\doteq\frac{d\log\xi}{d\log k}=\frac{\dot \xi}{\xi\,H}\ll 1.
\label{xi1}
\end{equation}
We instead  assume $\xi_1$ to be constant  i.e. $d\log\xi_1/d\log k\approx0$. In our future discussion we restrict our attention to the case $\xi>1$ that allows a blue tensor tilt. We are not going to discuss in details  the  peculiarities of this model  such as the gauge quanta production \cite{Anber:2009ua} that are reviewed also in \cite{Mukohyama:2014gba,Peloso:2016gqs} and the references within, but for our aim it is sufficient to observe that, in order to avoid a significant back-reaction of the produced gauge quanta to the background dynamics, we have to require that
\begin{equation}
\frac{\mathrm{e}^{\pi \xi}}{\xi^{5 / 2}} \ll \frac{13.5}{\sqrt{\epsilon_1 \mathcal{P}_0}} \frac{f}{M_{p}}
\label{decay const}
\end{equation}
where $\mathcal P_0= \left(\frac{1}{8\pi^2 M_{p}^2}\right)\left(\frac{H^2}{\epsilon_1}\right)$ is the primordial scalar spectrum without source (i.e. as predicted by the slow roll inflation). The scalar and tensor spectra for this model are \cite{Barnaby:2012xt, Mukohyama:2014gba,Sorbo:2011rz}:
\begin{equation}
\mathcal P_{\rm s} \simeq \mathcal{P}_0\left(1+c_{\rm s}\,\epsilon_1^{2} \,\mathcal{P}_0 \,\frac{\mathrm{e}^{4 \pi \xi}}{\xi^{6}}\right)\bigg \rvert _{k=k_*}
\label{Ps}
\end{equation}
\begin{equation}
r\simeq\frac{16\, \epsilon_1 \,\left(1+c_{\rm t} \,\epsilon_1 \, \mathcal{P}_0 \frac{\mathrm{e}^{4 \pi \xi}}{\xi^{6}}\right)}{\left(1+c_{\rm s}\,\epsilon_1^{2} \,\mathcal{P}_0 \,\frac{\mathrm{e}^{4 \pi \xi}}{\xi^{6}}\right)} \bigg \rvert _{k=k_*}
\label{Pt}
\end{equation}
where $c_{\rm s}= 2.5 \cdot 10^{-6} $ and $c_{\rm t}=3.4 \cdot 10^{-5}$ are constants.
We compute the spectral tilts from the relation \eqref{Ps} and \eqref{Pt} taking the logarithm derivatives:
\begin{align}
n_{\rm s}-1\doteq \frac{d\log\mathcal P_{\rm s}}{d \log k}\bigg \rvert _{k=k_*}&=\frac{d\log\mathcal P_0}{d\log k} + \frac{c_{\rm s} } {1+c_{\rm s}\,\epsilon_1^2 \,\frac{\mathrm{e}^{4 \pi \xi}}{\xi^{6}}}\,\frac{d}{d\log k}\left(\epsilon_1^2 \mathcal P_0 \frac{\mathrm{e}^{4 \pi \xi}}{\xi^{6}} \right)\\
&=-2(1+f_{\rm s} )\epsilon_1-(1-f_{\rm s})\epsilon_2+f_{\rm s} (4\pi\xi-6)\xi_1\\
&\simeq -2\epsilon_1-\epsilon_2
\label{ns_blue}
\end{align}
and
\begin{align}
n_{\rm t}\doteq \frac{d\log\mathcal P_{\rm t}}{d \log k}\bigg \rvert _{k=k_*}&=\frac{d\log\epsilon_1}{d\log k} + \frac{d\log\mathcal P_0}{d\log k} + \frac{c_{\rm t} } {1+c_{\rm t}\,\epsilon_1 \,\frac{\mathrm{e}^{4 \pi \xi}}{\xi^{6}}}\,\frac{d}{d\log k}\left(\epsilon_1 \mathcal P_0 \frac{\mathrm{e}^{4 \pi \xi}}{\xi^{6}} \right)\\
&=-2(1+f_{\rm t})\epsilon_1+f_{\rm t}(4\pi\xi-6)\xi_1
\label{nt_blue}
\end{align}
where  the functions
\begin{equation}
f_{\rm s}\doteq \frac{c_{\rm s}\,\mathcal P_0\,\epsilon_1^2\,  \frac{\mathrm{e}^{4 \pi \xi}}{\xi^{6}} } {1+c_{\rm s}\,\mathcal P_0\,\epsilon_1^2\,  \frac{\mathrm{e}^{4 \pi \xi}}{\xi^{6}} }\ll 1
\label{fs}
\end{equation}
and
\begin{equation}
f_{\rm t}\doteq\frac{c_{\rm t}\,\mathcal P_0\,\epsilon_1\,  \frac{\mathrm{e}^{4 \pi \xi}}{\xi^{6}} }{1+c_{\rm t}\,\mathcal P_0\,\epsilon_1\,  \frac{\mathrm{e}^{4 \pi \xi}}{\xi^{6}} }
\label{ft}
\end{equation}
weigh the corrections to the slow roll predictions for the scalar and tensor parameters respectively.
In what follows we fix $\mathcal P_{s}$ and $n_s$ to the observed values $\mathcal{P}_{\rm s}\simeq 2.1\times 10^{-9}$ and $n_{\rm s}\simeq 0.96$ \cite{Akrami:2018odb}. We also fix the tensor to scalar ratio $r$ to reference value $r\simeq 10^{-2}$ and $\xi_1\simeq 5 \times 10^{-3}\ll1$. Note that our results are marginally sensitive to the value of $r$ and $\xi_1$ and that we are not interested into a parameter analysis for this specific model: our task is simply to show that also in physical models of inflation scale dependence can be non-negligible.

We use the Eqs. \eqref{Ps} and \eqref{Pt} in order to explicit $\epsilon_1$ and $\mathcal P_0$ as functions of $\xi$. This means that when $\xi$ changes, $\epsilon_1(\xi)$ and $\mathcal P_0(\xi)$ change in such a way that $\mathcal P_s$ and $r$  remain constant. Moreover because of Eq. \eqref{nt_blue} also $n_{\rm t}$ is only a function of $\xi$. Being $n_{\rm s}$ fixed by observations, we can also use the relation \eqref{ns_blue} in order to find $\epsilon_2$ as a function of $\xi$ so that when $\xi$ changes, $\epsilon_2(\xi)$ changes leaving $n_{\rm s}$ fixed to its observed value. So in this model all the inflationary parameters\footnote{The inflationary parameters are to be considered evaluated to the pivot scale $k=k_{*}=0.05\rm{ Mpc}^{-1}$ which means that also the parameter $\xi$ in the equations above is computed on the CMB scales $\xi=\xi(k=k_*)$.} become known functions of $\xi$. We plot them in Fig. \ref{fig:figure3} letting $\xi$ vary in the range $\xi\in[1\,,\,7]$.

\begin{figure}[h]
	\centering
	\includegraphics[width=0.9\linewidth]{figure3.pdf}
	\caption{The parameters of the model as functions of $\xi$.}
	\label{fig:figure3}
\end{figure}

First of all we want to stress that we have carefully checked that the scalar spectrum \eqref{Ps} remains essentially equal to $\mathcal P_0$ (that is what predicted by the single field slow roll inflation). As a matter of fact,  if we decompose the scalar spectrum $\mathcal P_{\rm s}=\mathcal P_0 + \mathcal P_{\rm{s,\,sourced}}$ we find that  the sourced term induces corrections that are extremely small compared to the vacuum contribution $\mathcal P_{\rm{s,\,sourced}}\sim 10^{-4}\,\mathcal P_0$ for all the values of $\xi$. In other words, the corrections to the scalar spectrum are completely negligible ($f_{\rm s}\approx 0$), and the scalar parameters are essentially equal to that obtained in the simplest slow roll models. This can be understood by noting that the scalar corrections are suppressed by a factor $\epsilon_1^2\,\mathcal P_0$ and that $\epsilon_1$ exponentially decreases with $\xi$ in order to keep $r$ fixed, see also Fig.  \ref{fig:figure3}.  The fact that the scalar spectrum is essentially indistinguishable from the single field slow roll models is crucial since in this way all the tight constraints on the scalar perturbations (e.g. their high level of gaussianity) are respected as well \cite{Barnaby:2012xt,Shiraishi:2019yux}.
On the other hand the corrections to the tensor spectrum can be dominant for an appreciable range of the parameter space, allowing also a blue tensor tilt, see Fig. \ref{fig:figure3}. The sourced tensor modes could also leave a sizable non-gaussianity of nearly equilateral shape on the CMB anisotropies and polarization. The amount of non-gaussianity is controlled by the parameter $f_{\rm{NL}}$ estimated as \cite{Mukohyama:2014gba,Cook:2013xea}:
\begin{equation}
f_{\rm{NL}}\simeq 1.1\times 10^{-14}\left(\epsilon_1\,\frac{e^{2\pi\xi} }{\xi^3}\right)^3
\end{equation}
and its shape given in Fig. \ref{fig:figure3}, as well.
We estimate the scale dependence of the tensor tilt performing a second order computation and deriving the expression for the tensor running $\alpha_1^{\rm t}\doteq dn_{\rm t}/d\log k$ and the running of the running $\alpha_2^{\rm t}\doteq d\alpha_1^{\rm t} / d\log k$:
\begin{equation}
\alpha_1^{\rm t}\doteq\frac{dn_{\rm t}}{d\log k}\bigg \rvert _{k=k_*}=-2(1+f_{\rm t})\epsilon_1\epsilon_2 -2 f^{\prime}_{\rm t}\epsilon_1 +f^{\prime}_{\rm t}(4\pi\xi-6)\xi_1 +4\pi f_{\rm t} \xi\xi_1^2
\label{alpha_1}
\end{equation}
\begin{equation}
\alpha_2^{\rm t}\doteq\frac{d\alpha_1^{\rm t}}{d\log k}\bigg \rvert _{k=k_*}=-2(1+f_{\rm t})\left(\epsilon_1\epsilon_2^2+\epsilon_1\epsilon_2\epsilon_3\right)-4 f^{\prime}_{\rm t}\epsilon_1\epsilon_2-2 f^{\prime \prime}_{\rm t}\epsilon_1 +f^{\prime \prime}_{\rm t} (4\pi\xi-6)\xi_1 +8\pi f^{\prime}_{\rm t}\xi\xi_1^2 +4\pi f_{\rm t} \xi \xi_1^3
\label{alpha_2}
\end{equation}
where we have defined:
\begin{equation}
f^{\prime}_{\rm t}\doteq\frac{d f_{\rm t}}{d\log k}\bigg \rvert _{k=k_*}=\left[ \frac{-2\epsilon_1 + (4\pi\xi - 6)\xi_1}{1+c_{\rm t}\mathcal P_{0} \epsilon_1 \frac{\mathrm{e}^{4 \pi \xi}}{\xi^{6}}  } \right] f_{\rm t}
\end{equation}
and
\begin{equation}
\begin{aligned}
f^{\prime \prime}_{\rm t}\doteq \frac{df^{\prime}_{\rm t}}{d\log k}\bigg \rvert _{k=k_*} =&\left[ \frac{-2\epsilon_1 + (4\pi\xi - 6)\xi_1} {1+c_{\rm t}\mathcal P_{0} \epsilon_1 \frac{\mathrm{e}^{4 \pi \xi}}{\xi^{6}}  } \right]^2 f_{\rm t}+\\
&+\left[ \frac{(1+c_{\rm t} \mathcal P_0 \epsilon_1 \frac{\mathrm{e}^{4 \pi \xi}}{\xi^{6}} ) (-2\epsilon_1\epsilon_2+4\pi\xi\xi_1^2) - c_{\rm t} \mathcal P_0 \epsilon_1 \frac{\mathrm{e}^{4 \pi \xi}}{\xi^{6}}\left[-2\epsilon_1+(4\pi\xi-6)\xi_1\right]^2} {\left(1+c_{\rm t}\mathcal P_{0} \epsilon_1 \frac{\mathrm{e}^{4 \pi \xi}}{\xi^{6}}\right)^2  } \right]f_{\rm t}
\end{aligned}
\end{equation}

In this model the tensor tilt can acquire a non trivial scale dependence. In fact, depending on the parameters, $d\log n_{\rm t}/d\log k \simeq 0.1$, see Fig. \ref{fig:figure3}. As we discussed in Sec. \ref{sec.ScaleDependence}, this can lead to non negligible corrections on small scales.

As explained before all these quantities are known functions\footnote{ Note that we parametrized the slow parameter $\epsilon_3$ appearing in \eqref{alpha_2} as $\epsilon_3=\gamma\,\epsilon_2$. Letting $\gamma$ vary in a range $\gamma\in\left[-1,1\right]$ no significant changes in $\alpha_2^{\rm t}$ are observed. We therefore fixed $\epsilon_3\simeq 0$.} of $\xi$ or equivalently $n_{\rm t}$. However, since for large values $\xi\gtrsim 5$ the backreaction becomes typically non negligible as well as the primordial non gaussianity, we decide to restrict our attention to a safer region of the parameter space. We therefore fix  $n_{\rm t}\simeq0.1$ (or equivalently $\xi\simeq 3.5$) in such a way that both backreaction and non gaussianity are still under control, see Fig. \ref{fig:figure3}.  In this way the running  $\alpha_1^{\rm t}\simeq 0.01$ and the running of running $\alpha_2^{\rm t}\simeq 3\times 10^{-6}$ are fixed as well. We let evolve $\Omega_{\rm GW}(k)$ from the CMB scales all the way up to the ultra high $k$ probed by the ground based interferometers both including and neglecting $ \alpha_1^{\rm t}$ and $\alpha_2^{\rm t}$, see Fig. \ref{fig:figure4}.  The importance of scale dependence in this model is evident as $\Omega_{\mathrm{GW}}$ differs by many orders of magnitude when the non-linear corrections are considered, possibly becoming visible to future gravitational wave experiments such as LISA \cite{Audley:2017drz} and Einstein Telescope \cite{Punturo:2010zz}.

While we have shown that the impact of the second order running $\alpha_2^{\rm t}$ is completely negligible, see also Fig.\ref{fig:figure4},  one may ask if the higher order terms $\alpha^{\rm t}_{n>2}$ can instead give an appreciable contribution possibly changing the shape of $\Omega_{\rm GW} (k)$. For our aim it is sufficient to note that being the tensor tilt only a function of $\xi$, the derivative with respect to the scale can be written as $d/d\log k=\left(d\xi/d\log k\right) \,d/d\xi=\left(\xi\xi_1\right) d/d\xi$ and that the overall factor $\xi \xi_1\sim 10^{-2}$ will further suppress the higher order derivatives. We therefore expect such terms to be smaller and smaller at least in this range of the parameters space. We leave the detailed analysis of the sum expansion convergence suitable for future works.

\begin{figure}[ht!]
	\centering
	\includegraphics[width=0.65\linewidth]{figure4.pdf}
	\caption{$\Omega_{\rm{GW}}(k)$ in the particle production model both including and neglecting the first two runnings. The scale dependence is not negligible.}
	\label{fig:figure4}
\end{figure}

\section{Conclusion} \label{sec.Conclusion}
Along with the  B-modes polarization, a satiable amount of primordial gravitational waves may have imprinted also the stochastic background of gravitational waves.
Being the small scale experiments on gravitational waves such as LIGO and VIRGO (and in future LISA or ET) sensitive to the stochastic background, they can be used together with the CMB data to put constraints on the inflationary parameters. In particular from the LIGO/VIRGO bound on $\Omega_{\rm GW}$, Eq.\eqref{LVlimit},  under the assumption of a scale independent tilt, an upper bound $n_{\rm t}<0.53$ at 95\% C.L. is derived combining the CMB and GW data \cite{Akrami:2018odb}.
In this paper we focused on the robustness of assuming the tensor tilt as scale independent over a range of about eighteen order of magnitude $k\in\left[0.05 \,,\,\sim10^{16}\right]\,	\rm{Mpc}^{-1}$.  We have shown that when the assumption of scale independence is relaxed, the constraints may become very sensitive to the higher order corrections in the power law expansion and that even a tiny scale dependence can significantly change the above mentioned constraints. Due to the huge distance between the scales probed by the CMB and the GW interferometers, also a relatively small departure ($\sim 4\%$) from scale independence can significantly affect the shape of primordial spectrum possibly leading to non-negligible corrections. We concluded that scale dependence cannot always be ignored when constraints on the inflationary parameters are derived  from small scales data. We have also provided two examples in completely different physical models of inflation. We have first considered  the Starobisnky model showing that the scale dependence is negligible as one expects in the simplest slow roll paradigm. The Starobinsky model (and in general the standard slow roll models) provides an example where the non-linear corrections are typically negligible even on the ultra high $k$ probed by the ground based interferometers. However this is not always true. As a counterexample, we have analyzed a physical model of blue inflation which employs a pseudo scalar axion naturally coupled to gauge fields showing that in this model the tensor tilt can acquire a non trivial scale dependence leading to non negligible corrections on small scales.

\section*{Acknowledgments}
WG and AM thank Antonio Riotto for the useful collaboration and for his precious suggestions that contributed to the realization of this article. WG thanks Marco Peloso and Fabrizio Renzi for their comments and valid clarifications. AM and WG are supported by "Theoretical Astroparticle Physics" (TAsP), iniziativa specifica INFN.

redpajama_set_name: RedPajamaArXiv


I got home two hours late last night. My "adopted son" had brought a FRIEND along for supper, and I'd been running late in the morning. He'd asked me if I would make a turkey for him if he bought it.
When he said he wanted it for Wednesday, I told him I was still in school.
So, he forgot to buy it and drop it off on Monday, and I thought he'd changed his mind.
When I got home extremely late on Tuesday night, I had a note in the door... he'd brought one and gotten it into the fridge (no mean trick!) IT WAS STILL FROZEN SOLID.
My preset has never worked, so he came over and started it at 1, figuring to eat at 5 (VERY SMALL FEMALE BIRD). Fortunately, he came in at four to check on her, and she was done already. Generally, I use a very low oven (250) to do a turkey so it isn't so dry, but this time, I'd put it at 325. She literally fell off the bone.
I just couldn't get done at school, so was home two hours late... and he'd brought a FRIEND... Oops!
AARGG! He was at a table in the library, when his one time rival brought his wife in to see him and a group of select students holding "class".
Do the credits then roll? It has a climaxy feeling to it.


New Brunswick IT Services provides IT support to various departments within the New Brunswick Chancellor's office.
We provide a wide range of services including email services, database management and development, web hosting, content management, collaboration services, file and print sharing, security and compliance, application development and deployment, technology consulting, IT procurement, identity and directory management, project management, workstation management, network management, security services and repair/maintenance services of various IT equipment.
We currently operate with full time staff as well as a number of student employees. Our team supports the IT needs of our customers in over 60 units located in 70 buildings across all 5 campuses. We are also home to the ITS Software Development Office. This office consists of full time staff developers supported by student developers.
Our main office is located on Livingston Campus, in the RUNet Field Offices. The address is 63 Road 1, Edison, NJ 08817.
Our mission is to provide a single point of contact for all of your IT needs and eliminate the complexity and confusion that navigating IT at Rutgers can sometimes be. We also strive to provide excellent customer service and innovative IT solutions.


While most hospitals have policies in place to prevent health care-associated infections, clinicians often fail to follow evidence-based guidelines established to prevent these infections, according to research from Columbia University School of Nursing published in the February issue of the American Journal of Infection Control. The study, the most comprehensive review of infection control efforts at U.S. hospitals in more than three decades, found lax compliance even in intensive care units, where patients are more likely to be treated with devices linked to preventable infections—such as central lines, urinary catheters, and ventilators.
A team led by Patricia Stone, PhD, MPH, RN, FAAN, Centennial Professor of Health Policy at Columbia Nursing, investigated compliance with evidence-based policies to prevent infection in 1,653 ICUs at 975 hospitals nationwide. The study focused on three of the most common preventable infections: central line-associated bloodstream infections, ventilator-associated pneumonia, and catheter-associated urinary tract infections. Despite decades of research establishing best practices for prevention of these infections, approximately one in 10 hospitals lack checklists to prevent bloodstream infections, and one in four lack checklists to help avoid pneumonia in ventilator patients. Even worse, these checklists are followed only about half of the time, the study found.
Health care-associated infections kill an estimated 100,000 Americans a year and create approximately $33 billion in excess medical costs. The U.S. Centers for Disease Control and Prevention first linked infection rates to prevention programs in the 1970s. Research since then has shown that checklists and other targeted infection-control practices can make a significant dent in infection rates—but only if compliance rates among clinicians are high.
Compliance could be improved through two approaches most hospitals aren't using: electronic monitoring systems and staff certified in infection control. The study found that only about one-third of the ICUs have an electronic surveillance system to track compliance with infection-prevention policies at the clinician level. Previous research has found that electronic monitoring systems that provide report cards on compliance are effective at getting clinicians to follow the rules and lowering infection rates. At the same time, more than one-third of hospitals also failed to employ a full-time clinician certified in infection prevention to supervise compliance, the study found.
"Every hospital should see this research as a call to action. It's just unconscionable that we're not doing every single thing we can, every day, for every patient, to avoid preventable infections," Stone says.
For central line-associated bloodstream infections, or CLABSI, the study found that more than 90 percent of ICUs had checklists for sterile insertion but the policies were followed only about half of the time. Catheters, also known as central lines, deliver life-saving medicines and nutrition. Without proper insertion, utilization, and maintenance, catheters can also transmit deadly infections to the bloodstream. Simple infection-prevention measures include handwashing before handling the catheter and immediately changing the dressing around the central line if it gets wet or dirty.
Compliance rates were no better for preventing ventilator-associated pneumonia, or VAP, the study found. Overall, three out of four ICUs had checklists for protecting against infections linked to ventilators, machines that force air into the lungs of patients unable to breathe on their own. The ICUs followed their own checklists just half the time. Keeping patients elevated in bed, with the head higher than the feet, is one simple precaution that can help prevent pneumonia.
Patients with urinary catheters fared even worse, the study found. Guidelines to prevent catheter-associated urinary tract infections, or CAUTI, are newer, and there are not yet universally accepted checklists to follow at the bedside. About one-third of hospitals had no prevention polices in place to prevent these infections. Even at hospitals that did establish guidelines, the measures were followed less than 30 percent of the time.
Watch a video of Patricia Stone discussing the findings.
This article originally appeared on the Columbia University School of Nursing website.


text:
Council steps in to help secure new Rosemount Resource Centre
The Rosemount Resource Centre, Derry. DER4814MC007
Brendan McDaid
Derry City & Strabane District Council is expected to offer to make up a shortfall in funding for a planned new community facility for Rosemount.
SDLP councillor for the area, Shauna Cusack,has congratulated council on its decision to provide the money needed in order to progress the major rebuild works at Rosemount Resource Centre.
However, the 'Journal' understands that the decision emanating from a council committee meeting last week, would have to be ratified by the full council towards the end of this month.
Colr. Cusack said: "Letters of Offer have been forwarded by the Department of Communities offering over £900,000 in order to deliver a new state of the art centre for those living in and the around Rosemount area.
"However there was still a considerable shortfall which needed to be found in order to progress the project.
"Last week in council I asked officers to ensure, by any means possible, that this money be found, otherwise the Department funding could be lost and this would have a domino effect on capital projects elsewhere in the city and district.
"I'm delighted that this money has been secured and approved which will see this exciting project get underway.
"I appreciate this was not an easy ask given the centre is not a council-owned facility, but the services it will provide to the local community fit precisely into our strategic plans and ambitions."
Derry & Strabane District Council last month pledged to help look at ways of addressing the funding shortfall for the new Rosemount Resource Centre building.
The project involves demolishing the old Rosemount Resource Centre and building a new modern community facility.
At the earlier meeting, Colr. Cusack maintained that without council making up the shortfall by the end of March, the project could not progress.
Chief Executive of the council, John Kelpie, said at the time that council had received correspondence from the Resource Centre on February 22 and officer teams were meeting with DfC "to understand the scale of the issue; to understand how that gap could be bridged or whether there would be other solutions to it and certainly bringing the knowledge we can to the table."
The Department said that back in October the new building will house current services and facilities as well as a new men's health shed, a fishing club, reading club and other 'Healthy Mind' programmes.
Bomb alert in Derry over
Planning for the Rosemount Resource Centre project was approved back in May last year.
Cate Conway shares tearful video of hug Stephen Clements gave her at end of their final radio programme together

redpajama_set_name: RedPajamaCommonCrawl


Show me your war space! Seriously, I'm always interested in how other people organize their gaming space. I'll show you mine, now you show me yours!
One 27" monitor, two 22" monitors. MWO on the big screen, TS3 and the forums on the right, maps and my 123rd Hind Paw paperwork on the left. The smaller monitor is a 10" Android tablet that controls my gaming rig. It also doubles as an input for Elite: Dangerous and DCS World.
Yes, I'm one of those heathens who plays MWO with a joystick.
The mug is from the Bradly ADATS program in the early '90s, and I found it about a year ago meandering through eBay. Coolest change-holder that ever.
I've been debating rotating one of my monitors. So many textbooks are digital now and it would make reading them so much easier.
I can tell you that having a monitor in Portrait mode is amazing for webpages, spreadsheets, databases, coding... it's REALLY nice. But getting your backgrounds to behave is a little more problematic.
I usually run minimalist wallpapers, so rotating them should be a piece of cake.
i cannot deal with simple wallpapers. here are a few of my favorites. (god please let the spoiler code work,otherwise, imageflood inbound.
I am so old school.
i got one monitor, one ashtray, a handfull of beer bottles and thats about it.
oh, and i am too lazy to stand up and take apicture to proof it.
Wow Aiden, I find 1 27" monitor to actually be too big for me. So I couldn't do 3.
Wow Aiden, I find 1 27" monitor to actually be too big for me.
I don't play on all 3 at once, that's for sure.
Srs bsns side of things. Hard to study when MWO is right there so I have a separate desk for studying and what not. School's on break, so comics time!


We welcome you to celebrate the High Holy Days with Kol Ami. Each year, we celebrate the cycle of the Jewish holidays. Our services are filled with prayers, music, and are a time to reflect on our lives. When we unite as a congregation we create a sacred space in time as a family to share our strength and console one another.
Oneg (a joyful social gathering) often follow holiday services and are a special way to celebrate the holidays. At these events, not only do the children enjoy the activities but the adults are delighted to participate in the festivities.
Apples and honey will be available after the morning service. Tashlich ceremony to follow.
*Please note that the afternoon series of discussion groups is intended for adult participants.
1What is Tashlich? The ceremony of Tashlich is a centuries-old ritual for Jews to symbolically cast their sins away. Traditionally held outdoors near a stream or river, we take crumbs of bread, think about the aspects of our lives from the past year that we are not happy with, and cast the bread crumbs into the flowing water. As the water takes the crumbs away, we allow ourselves to let go of our sins. We hope you will join us for this brief but uplifting ceremony.
Located in the UUCF following the Ne'ilah service.
Please indicate when purchasing tickets if you plan to join us. A Sign-Up Genius link will be emailed to you after you RSVP to sign up for food and/or desserts. Bagels, cream cheese, drinks, and paper goods will be provided.


Community activists, educators, and concerned parents met at AAMA high school to protest the name change of Mexican-American studies to "Ethnic Studies: An Overview of Americans of Mexican Descent".
The name change, proposed by Beaumont Republican Board of Education member David Bradley has caused friction between segments of the Mexican-American community and the Texas Estate Board of Education. Latino activists in six Texas cities, have been organizing a movement to keep the name of the Mexican-American studies course.
"I don't subscribe to hyphenated Americanism… I find hyphenated Americanism to be divisive," said Bradley.
Tony Diaz, founder of the Librotraficante movement expressed his discontent in the name change by calling it "illogical". The Librotraficante movement, promotes Mexican-American literature on the educational system.
"It simply makes no sense. It is like calling TEX-MEX food, Texan food with Mexican ingridients," Diaz said.
The Texas State Board of Education will have full board meeting on June 12, 2018 at the Texas State Capitol, Room 1- 104.
Community activists from San Antonio, Forth Worth, Houston, El Paso, Austin and San Juan Texas will caravan to the meeting to protest the name changed by Bradley.
Parents have also expressed their frustration towards Bradley.
Maria Delgado, a parent who attended the press conference believes the name change gives a bad message to the community.
"I find it offensive how Michael Bradley changed the name without consulting Mexican-Americans", said Maria Delgado.


Center. The victim, who had just attended case conferences in court, died at the scene.
that this is not the type of man that can be replaced.
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At Stephen Bilkis and Associates we have our New York Criminal Lawyers ready to help in an Injury Law case, to assist you in getting what's yours.


text:
MIA podcast
Peter Breggin and Michael Cornwall – Stop the Psychiatric Abuse of Children
Miranda Spencer - December 14, 2019
An interview with Drs. Peter Breggin and Michael Cornwall who discuss their new initiative, Stop the Psychiatric Abuse of Children (SPAC!). SPAC! was formed in response to the introduction of the Monarch eTNS, an electrical stimulation device worn on a child's forehead at night that was fast-tracked by FDA with little testing.
Bridging Critical and Conceptual Psychiatry: An Interview with Awais Aftab
Justin Karter - July 3, 2020
MIA's Justin Karter interviews psychiatrist Awais Aftab about how "conceptual competence" uses philosophy to transform psychiatry.
Addressing the Roots of Racial Trauma: An Interview with Psychologist Lillian Comas-Díaz
Hannah Emerson - August 14, 2019
MIA's Hannah Emerson interviews Comas-Díaz on the need for culturally competent care in a medicalized and individualistic society.
The Global 'Mental Health' Movement – Cause For Concern
Justin Karter - October 10, 2018
On October 10th, 2018, World Mental Health Day, The Lancet Commission on Global Mental Health and Sustainable Development published a report outlining a proposal to "scale up" mental health care globally.
Moving Global Mental Health "Outside Our Heads"
On MIA Radio we interview Dr. Derek Summerfield, honorary senior lecturer at the Institute of Psychiatry in London, former Research Associate at the Refugee Studies Centre at the University of Oxford and consultant at Oxfam.
World Benzodiazepine Awareness Day 2020
James Moore - July 11, 2020
This week on MIA Radio, we present the second part of our podcast to join in the events for World Benzodiazepine Awareness Day 2020...
Angela Peacock – Medicating Normal
James Moore - July 5, 2020
An interview with Angela Peacock who talks of her experiences of being prescribed benzodiazepines, her journey off multiple medications, her continuing work in veterans advocacy and her thoughts about the film Medicating Normal which will be screened on World Benzodiazepine Awareness Day, July 11.
Jeffrey Michael Friedman: Trauma and Forced Psychiatric Treatment
Emily Cutler - April 14, 2018
An interview with Jeffrey Michael Friedman, clinical social worker and activist in the psychiatric survivors movement who provides trauma-informed therapy to victims of abuse and violence, including those who have survived abuses within the mental health system.
John Read: What the Science and Evidence Tell Us About Electroshock
James Moore - September 9, 2017
Professor John Read talks about his research interests and in particular, the science and evidence base for Electroconvulsive Therapy (Electroshock).
How Culture Influences Voice Hearing: An Interview with Stanford Anthropologist Tanya Luhrmann
Ayurdhi Dhar, PhD - November 13, 2020
Ayurdhi Dhar interviews Tanya Luhrmann about cultural differences in voice-hearing, diagnosis and damaged identities, and conflicts in psychiatry.
Stuart Shipko – SSRI Withdrawal: Shooting the Odds
James Moore - September 12, 2020
We interview Dr. Stuart Shipko, a psychiatrist and author who has a particular interest in the side effects and withdrawal effects of SSRI antidepressants and the need for informed consent when prescribing.
Leading Psychology in Existential Times: An Interview with Kirk Schneider
Justin Karter - September 22, 2020
MIA's Justin Karter interviews humanistic-existential psychologist Kirk Schneider about how psychology can play a role in confronting the political, social, and climate crises facing humankind.
The Management of Captive Populations with Psychiatric Drugs: An Interview with Anthony Ryan Hatch
Leah Harris, - January 15, 2020
Dr. Anthony Ryan Hatch is a sociologist and associate professor of Science in Society, African American studies and Environmental Studies at Wesleyan University, who...
Professor John Read: The Royal College of Psychiatrists and Antidepressant Withdrawal
James Moore - March 9, 2018
Patients, academics and psychiatrists formally complain that the president of the UK Royal College of Psychiatrists has misled the public over antidepressant safety. Professor John Read talks to us about recent events.
How to Know What We Don't Know: An Interview with Psychologist and Novelist Jussi...
Gavin Crowell-Williamson - July 29, 2020
MIA's Gavin Crowell-Williamson interviews the neuropsychologist and novelist Jussi Valtonen about how novels can lead us to see the limits of our understanding.
Global Mental Health – The Hypocrisy of Mental Health in The Age of Austerity
Zenobia Morrill - October 24, 2018
Dr. China Mills shares her reactions to recent events focused on Global Mental Health, elaborating on deeper issues with the framing of mental health as a "burden" and the underlying implications of coloniality, technology, and medicalization.
Wendy Dolin – Making Akathisia a Household Word
James Moore - January 25, 2020
An interview with Wendy Dolin who talks about the work of MISSD, the Medication-Induced Suicide Prevention and Education Foundation in Memory of Stewart Dolin, a non-profit founded to raise awareness of the tragic consequences of drug-induced akathisia.
Healing Youth with Nature and Connection: An Interview with Peter Mayfield
James Moore - December 22, 2019
An interview with Peter Mayfield, founder and Executive Director of the Gateway Mountain Center. Peter talks of his journey from mountaineering to his role as an educator and mentor, and how enabling children and adolescents to connect with nature has such a profound effect on their health and wellbeing.
Anatomy of a Psychiatrist
James Moore - November 10, 2018
An interview with Dr. Sandy Steingard, Medical Director at Howard Center and Clinical Associate Professor of Psychiatry at the College of Medicine of the University of Vermont. Dr. Steingard serves as Board Chair of the Foundation for Excellence in Mental Health Care and is editor of the book 'Critical Psychiatry, Controversies and Clinical Implications' due in 2019.
Dr. George Atwood: Shattered Worlds, the Experience of Personal Annihilation
Zenobia Morrill - December 2, 2017
Dr. George Atwood has devoted a substantial part of his life to the study and treatment of what he refers to as 'so-called psychosis' and has authored or coauthored several books, including The Abyss of Madness published in 2011 and more than one hundred articles.
Poverty, Pathology and Pills: An Interview with Dr. Felicity Thomas and Dr. Richard Byng
Tim Beck, PhD - June 12, 2019
MIA's Tim Beck interviews Dr. Felicity Thomas and Dr. Richard Byng about their report, Poverty, Pathology, and Pills, which situates increasing rates of mental health diagnosis and psychiatric prescriptions within socioeconomic and policy trends across the UK.
Julia Rucklidge: Nutrition, Mental Health and TED
An interview with Dr Julia Rucklidge, professor of clinical psychology at the University of Canterbury in New Zealand and Director of the Mental Health and Nutrition Research Group.
Psychological Support for Psychiatric Drug Withdrawal
James Moore - December 4, 2019
We discuss the release of guidance which has been specifically written to support UK psychological therapists and their clients in having discussions about taking and withdrawing from psychiatric drugs. The guidance is a collaboration between counsellors, therapists, psychologists, peer support specialists and psychiatrists.
Dr. Russell Razzaque: Breaking Down is Waking Up
James Moore - April 9, 2018
An interview with Dr. Russell Razzaque, consultant psychiatrist and associate medical director in east London who, together with colleagues, is leading a pioneering multi-centre Open Dialogue pilot in the UK National Health Service.
Climate Change, Mental Health and Collective Action: An Interview with Jennifer Freeman
Akansha Vaswani - October 4, 2019
In an interview with MIA's Akansha Vaswani, narrative therapist Jennifer Freeman calls for a shift away from individualistic approaches to 'eco-anxiety' and toward responses that connect us all to a counter-tsunami of action for the planet.


Dj Khaled merchendise now available in one place. Are you a fan of the king of snapchat? Do you follow DJ kHALLED on instagram? Facebook? Do you get your major keys from his twitter? Do you have the we the best music collection?
Like me you would like to have more items endorsed by Dj khaled? We have selected some of the top most requested items by the fan luv around the world.
One of the major key is always to stay fresh.
To begin our recommended products list, we thought the best way to start is with the headphones. DJ Khaled is known for his music and what best way to listen to his records than with his headphones collections to get the intended sound.
Throughout DJ khaled career he made deals with two headphones companies B&O and Beats. Both are providing amazing sound quality.
The B&O headphones collection come in 4 amazing color both for men and women. See the collection right here.
The Beats by Dre unity on the other hand aim at promoting unity around the world.
DJ Khaled being a positive person and a promoter of positivity wanted through these headphones to push further the unity agenda on the young world.
In 2017 Dj Khaled made another deal with Beatbydre to release a new limited headphone edition in connection to the release of his 10th studio album grateful. The headphones are a collaboration between the We the Best team and the Apple team.
#WETHEBEST #GRATEFUL #BEATS @wethebestmusic @beatsbydre 🎧🔑. #GRATEFUL #JUNE23RD THIS FRIDAY!
After the selection of DJ Khaled headphones, it is only understandable that we look at some of the DJ Khaled items to make you look fresh and cool.
They don't want you to stay fresh and cool with the DJ Khaled merchendise collection, so we made sure you stay fresh, cool and different with our selection of DJ Khaled hoodies and Tshirts.
To end our selection, it was unconceivable to leave you bear foot knowing that Dj Khaled is one of the top shoe connoisseur in the game. We didn't pick the mountain top collection (coming soon) in this selection but we made sure your feet's stay fly and expressive. We have selected some of the most expressive slide in the DJ Khaled merchandise collection for you here.
Now that you are fly from head to toe with our top selections, you might want to check the smooth coco butter for a fresh skin and the repellant of "they". Don't miss to update your "we the best music" music catalogue/playlist collection for your up to date We the best sound.
MORE #1s! MORE #🔑s FOR YOUR HEAD TOP!! It's summer time !! THEY🚷 out there heavy!! Get your THEY🚷 BLOCK NOW AT #WETHEBESTSTORE.COM … This Friday !!! BE GREAT!! BE GRATEFUL!!!! #GRATEFUL #JUNE23!!
I also know among the fan luv world we have some of the Ciroc lovers. Well we have not forget you. You can get that wonderful Ciroc mango here. As for my champagne drinkers grab that smooth Belaire gold over here. For our flowers lovers make sure to check our post on the Dj Khaled garden flowers.
One last thing – be fly, be fresh and drink responsibly.


This is a program designed to reduce the fear of falling and increase activity levels among older adults. This class includes eight two-hour sessions for a small group led by a trained facilitator. Participants must attend at least 6 of the classes to graduate. Classes begin Wednesday, January 2nd and will be held every Wednesday and Friday in January from 1:00 pm to 3:00 pm. Classes are free but are limited to 15 people and you must sign up to attend. Classes are free but are limited to 15 people and you must sign up to attend. To register for the class please call 520-459-8210.
If you are unable to make this session but would like to attend the next scheduled class please call to be put on the waiting list.


"Caroline is a MASTER at her craft. She facilitates, she makes us think, she makes us resolve problems and she makes us more efficient. Together she transforms our working environment in a better culture of performance.
"Playing the role of Meeting Facilitator, Caroline constantly kept in mind the result we were expecting.
I had the privilege to run over 15 strategic executive meetings with her and, all of them were conducted with professionalism, integrity and a huge sense of achievement.
"We worked with Caroline to deliver a creative and impactful conclusion to our 2011 AGM.
Caroline developed a Visual Metaphor exercise which saw teams of employees channel their inner artist to bring our corporate vision to life on canvass.
The result was an enduring expression of the passion our people have for achieving our strategic goals as an organization and ultimately making a significant impact on the lives of our patients.
Employees left feeling inspired about the year ahead and the possibilities we would be creating together!