ST. LOUIS (AP) — Milwaukee's last NL MVP thinks he is about to get some company.Led by Christian Yelich's silky smooth game Dallas Keuchel Jersey , the Brewers are closing in on the playoffs.Yelich homered and drove in a career-high six runs, and surging Milwaukee hurt St. Louis' postseason chances with a 12-4 victory over the Cardinals on Tuesday."Just trying to help the boys out any way you can," Yelich said. "A lot of guys came through in some big situations for us. That was a lot of fun tonight."Ryan Braun went deep twice and Jesus Aguilar also connected as Milwaukee lowered its magic number for clinching a playoff spot to one. But the Brewers are thinking bigger after their fifth win in six games put them just a half-game back of the NL Central-leading Chicago Cubs."I think the division is kind of what's on our mind right now," manager Craig Counsell said, "and I think we've earned that with the last two days."Milwaukee (91-67) leads Colorado by 3½ games for the top wild card. St. Louis (87-71) dropped a half-game behind Colorado for the second NL wild card after the Rockies routed Philadelphia 10-3."We have four games to play, and we feel good about the club we have and the kind of guys we have and the character we have and the talent we have," Cardinals manager Mike Shildt said. "We'll control what we can control and we'll get after it and let the chips fall."Following Wednesday's series finale, Milwaukee closes the regular season with a weekend set at home against lowly Detroit. St. Louis also is off Thursday before beginning a three-game series against the slumping Cubs at Wrigley Field.Yadier Molina hit a three-run homer for the Cardinals, who had won six of seven before their 6-4 loss to Milwaukee on Monday night. Marcell Ozuna went 3 for 3 with a walk.St. Louis left the bases loaded in the fifth and seventh innings when Jedd Gyorko came up empty each time. Third baseman Mike Moustakas turned an inning-ending double play on Gyorko's chopper near the bag in the fifth http://www.astrosfanproshop.com/authentic-jeff-bagwell-jersey , and Gyorko flied to right for the final out of the seventh."This one's kind of on me," Gyorko said. "I've got to find a way to get the job done in those situations."Milwaukee scored 11 runs with two out, including its first six. Yelich came up with the bases loaded in the fourth and tripled into the gap in right-center, lifting the Brewers to a 6-0 lead. After Lorenzo Cain was intentionally walked with two out in the ninth, Yelich hit a three-run drive to center."It's part of the game, but at the same time, it's a challenge," Yelich said of the walk to Cain. "As a competitor, you want to answer the challenge."Yelich is batting .346 with seven homers and 28 RBIs in 22 games this month. He has 33 homers Lance McCullers Jersey , 104 RBIs, 110 runs and 181 hits on the year, all career highs.Braun, who won the MVP award in 2011, said he thinks Yelich is about to join the club."I can't foresee a scenario in which he doesn't win it," Braun said. "He's been so incredible for us in every facet of the game."Taylor Williams (1-3), the second of six Milwaukee relievers after Gio Gonzalez pitched four innings, got three outs for the win. The Brewers used nine pitchers in their series-opening victory.Cardinals rookie Austin Gomber (6-2) was tagged for five runs and seven hits in 3 2/3 innings.LONG GONECardinals first baseman Matt Carpenter and Shildt were ejected in the seventh inning. Carpenter argued with home plate umpire Will Little after he was called out on strikes, and then was thrown out as he walked toward the dugout."I understand the importance of staying in these games http://www.astrosfanproshop.com/authentic-mike-scott-jersey ," Carpenter said. "So when I get thrown out of the game as I'm walking away, that's something I have a problem with."TRAINER'S ROOMBrewers: All-Star reliever Jeremy Jeffress was available after he missed the series opener because of neck spasms.Cardinals: 2B Kolten Wong returned to the starting lineup for the first time since he left Friday's victory over San Francisco with cramps in both hamstrings. He struck out swinging as a pinch hitter in the ninth inning of Monday's loss. "Obviously at this point of the season no one's 100 percent," he said. "For me as long as I know I can run and I can play I'm going to be out there."UP NEXTBrewers right-hander Jhoulys Chacin (14-8, 3.61 ERA) and Cardinals right-hander John Gant (7-6, 3.53 ERA) pitch Wednesday night in the series finale. Chacin is 0-3 with a 4.34 ERA in four September starts. Gant lasted just 2 2/3 innings in his last outing, yielding one run and four hits Friday against the Giants. DENVER (AP) — Dodgers closer Kenley Jansen went to the hospital with an irregular heartbeat before a game Thursday against Colorado and will be sent back to Los Angeles for more tests.Manager Dave Roberts said after the Dodgers' 8-5 win over the Rockies that Jansen was at the hotel earlier in the day and didn't feel right. Jansen called the trainer and was taken to the hospital.Jansen has dealt with an irregular heartbeat in the past and will visit his cardiologist in Los Angeles. Roberts said the right-hander had his medication with him on the trip."We're holding out hope it's not too serious," Roberts said. "In talking to the trainer right now, his mood is fine."Roberts didn't know if the team would make a roster move in Jansen's absence. Los Angeles plays three more games at Coors Field before returning home.The Dodgers used Scott Alexander to close the ninth Thursday. Jansen is tied for the NL lead with 32 saves this season."Anytime you're talking about the heart and irregular heartbeat, you've got to be very careful with it Alex Bregman Jersey ," Roberts said. "You're talking about altitude (in Denver). That's why we want to be proactive and get him back" to Los Angeles.Jansen spent a month on the disabled list during the 2011 season because of blood thinning medication he was taking to treat an irregular heartbeat. In 2012, he was out again after suffering an irregular heartbeat during an Aug. 27 appearance in Colorado. He underwent a procedure on his heart on Oct. 23, 2012, to return it to normal rhythm."Once he sees the doctors, the cardiologists — we'll know more about his timetable then," Roberts said.


Sanction and Disbursement are the two main steps involved for availing a home loan from ICICI. Sanction: Based on various parameters, ICICI will evaluate the loan application. Post evaluation it would release the sanction letter mentioning the amount of loan sanctioned and other terms and conditions of sanction.
Disbursement: The amount sanctioned is disbursed in one or more installments subject to legal & technical verification of the property and other terms and conditions mentioned in the sanction letter.
What are the main steps in the disbursement process?
Once you have identified the property that you want to buy, you may get in touch with the officer mentioned in the sanction letter to under the disbursement process.
Based on the nature of the transaction (purchase from builder / resale etc.), the officer will inform you of the property documents such as Sale deed, No Objection Certificate, Own Contribution Receipt, etc. which would be required for the disbursement of the loan.
In addition to the property documents, you will be required to execute transaction documents including the Credit Facility Application Form and other documents required for disbursement of the facility. These documents will be provided to you by ICICI.
Also, any other special condition mentioned in your sanction letter before any disbursement can be made needs to be fulfilled.
Note : We recommend you retain photocopies of all documents being submitted by you for your future reference. In case you require your documents during the loan tenure the retrieval period is 6 days at a nominal charge.
You are required to repay the loan by way of Equated Monthly Installments ("EMIs") compromising of principal and interest. EMIs commence from the month following the month in which the loan has been disbursed to you in full, i.e. the last tranche of the loan has been disbursed to you.
The EMI is payable on 1st or 10th of every month depending on the date selected by you.
For example, if the loan is disbursed in full to you in April 2008, the EMI will start from May 2008.
It is necessary for you to ensure timely EMI payment. The repayment history is monitored by Credit Information Bureau Of India Limited (CIBIL) as a regulatory requirement. Irregular repayment history would impact your future loan requirements.


text:
This Ain't Heaven Yet
It was During the time of floods in Joplin Missouri. Our neighbors, a dad and three young boys were struggling. Mom had just passed away from cancer. They didn't understand a Loving God who would take there wife and mother away from them. We thank Our Father honey got to pray with her when she was in Hospice care as she asked Jesus into her heart! All this had me thinking about all the people who struggle with believing in a loving God while there is so much suffering in the world. That night when listening to a bible study on the radio a Pastor was speaking on the same subject when he said This Ain't Heaven Yet! I said that's a great song!!
The first verse of the song is from my own experience.
The second verse is My wife and daughter! A single, never married mother since she was a teenager, with a small little girl who two years later would become my Wife and Daughter, boarded a Greyhound bus in Chicago bound for California. They left everything behind but their suitcase to escape the beatings of an abusive boyfriend who often threatened to kill and has spent most of our marriage in prison. Honey and I married in Jan. 1991 and I have been blessed to be Kellee's aka lil'mama Father since she was eight, Thank You Father! I Love You Honey & Kellee and this verse always makes me cry!
The third verse is Our neighbors verse as well as all who have said goodbye to loved ones in this life!
The song ends with the promise we have!! Our Hope and Salvation, JESUS CHRIST!! That this is not the end! In this life we will suffer but for those who put there trust in Him, He will always be there for us, OUR GOOD SHEPARD!! Someday we who put our trust in Him have the promise of eternity with Him!!! In Heaven with no more sorrow, no more pain and he will wipe all the tears from eyes!!
Thank You Father!!! and We Pray We See You All There!!! We Love You!! As Honey Always Says Our Father Loves You Best!!!!
I got stones all in my passway just like the old blues man said it
and I've watered my couch with my tears just like in Psalms 6 where I read it
and I've had so much regret, ah but this ain't heaven yet
Well the bus just pulled in to the station mother and child just climbed on board
They've just got one suitcase between them and a prayer up to the Lord
and they've got so much to forget, ah but this ain't heaven yet
Well there gatherin' in the graveyard there gonna lay her body down
and as the family's weepin' as the tears fall to the ground
I'm thinkin' as they wept, ah but this ain't heaven yet
Well in heaven there'll be no more sorrow
and in heaven there'll be no more death
and he'll wipe away all the tears
all the tears that we've wept, ah but this ain't heaven yet

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: Avoid nesting promises with Firebase My aim is send verification email after signUp automatically.
I'm making this be Firebase function.
My code is at below.
firebase.auth().signInWithEmailAndPassword(email, password)
        .then(userCredential => {
            console.log(userCredential);

            Firebase.auth().sendSignInLinkToEmail(email, actionCodeSettings)
            .then(() => {
                window.localStorage.setItem('emailForSignIn', email);
                res.status(200).send(userCredential);
                res.status(200).send(userCredential);
                return;
            })
            .catch(function(error) {
                // Some error occurred, you can inspect the code: error.code
            });
            return;
        })
        .catch(error => {
            console.log(error)
            res.status(400).send(error);
})

But this handling is warned by lint.
If my aproach was invalid, please correct me.
And below is the screen of my code.

A: You can do something like that:
firebase.auth().signInWithEmailAndPassword(email, password)
  .then(userCredential => {
    console.log(userCredential);
    return Firebase.auth().sendSignInLinkToEmail(email, actionCodeSettings)
  })
  .then(res => {
    window.localStorage.setItem('emailForSignIn', email);
    res.status(200).send(userCredential);
    res.status(200).send(userCredential);
    return;
  })
  .catch(error => {
    console.log(error)
    res.status(400).send(error);
  })

redpajama_set_name: RedPajamaStackExchange


text:
\section{Introduction}
\label{intro}

The detection of primordial gravitational waves in the Cosmic Microwave
Background (CMB) offers an exciting possibility for probing the physics around
the GUT scale. In particular, the inflationary
paradigm \cite{Sta80,Kazanas,newinf} predicts the generation of
nearly scale-invariant primordial tensor and scalar perturbations
with the tensor-to-scalar ratio $r$ less than the order
of 0.1 \cite{GWper,oldper}.
Since the spectral index $n_s$ of scalar perturbations was measured
by the Planck satellite in
high precision ($n_s=0.9603\pm 0.0073$ at 68 \%\, CL \cite{Planck}),
the precise bounds on $r$ from the ongoing and upcoming
CMB B-mode polarization experiments \cite{QUIET,BICEP,Polar,Lite}
are the next important step for approaching the origin of inflation.

Many of the single-field inflationary models proposed so far belong to
a class of Horndeski theories \cite{Horndeski}-- the most general
Lorentz-invariant scalar-tensor theories with second-order equations of motion.
In fact, the leading-order power spectra of tensor and scalar perturbations
were derived for inflationary models in the framework of
Horndeski theories \cite{KYY,GaoSteer}.
These results were employed to place observational constraints on
individual models (such as slow-roll inflation \cite{newinf},
k-inflation \cite{kinf}, Starobinsky inflation \cite{Sta80},
Higgs inflation \cite{Higgs,Germani}) from the
WMAP and Planck data \cite{Reza,Kuro}.

There exist more general modified gravitational theories beyond
the Horndeski domain.
Choosing the so-called unitary gauge in which the perturbation of
a scalar field $\phi$ on the flat Friedmann-Lema\^{i}tre-Robertson-Walker
(FLRW) background vanishes, the Horndeski Lagrangian can be expressed
in terms of geometric scalar variables appearing in the 3+1
Arnowitt-Deser-Misner (ADM) decomposition of space-time \cite{Piazza}.

In Horndeski theories the coefficients in front of such ADM scalars have
two particular relations, but it is possible to consider extended theories with
arbitrary coefficients: Gleyzes-Langlois-Piazza-Vernizzi (GLPV)
theories \cite{Gleyzes}.
In general space-time the equations of motion in GLPV theories should
contain derivatives higher than second order, but the Hamiltonian analysis
based on linear perturbations on the flat FLRW background
shows that GLPV theories have one scalar
propagating degree of freedom without ghost-like Ostrogradski
instabilities \cite{Gleyzes,Lin,GlHa,GaoHa}.
This second-order property also holds for the odd-type perturbations
on the spherically symmetric background \cite{KaGe}.

The full action of GLPV theories cannot be
generally mapped to that of Horndeski theories \cite{GlHa,Tsu14}
under the so-called disformal transformation \cite{Beken,Garcia},
so the two theories are not equivalent to each other.
It is possible, however, to derive the two non-Horndeski pieces in the GLPV
Lagrangian separately from a subset of the Horndeski Lagrangian
under the disformal transformation.

Another example outside the Horndeski domain is
Ho\v{r}ava-Lifshitz gravity \cite{Horava}, in which an anisotropic scaling
in time and space plays a role for the realization of
a power-counting renormalizable theory.
In this case there are spatial derivatives up to 6-th order, with which
Lorentz invariance is explicitly broken.
The building blocks of Ho\v{r}ava-Lifshitz gravity are the three-dimensional
ADM geometric scalars invariant under a foliation-preserving diffeomorphism.

The effective field theory (EFT) of cosmological perturbations
is a powerful framework to deal with low-energy degrees of freedom
in a systematic and unified way \cite{Cremi,Cheung,Weinberg}.
This approach is not only useful to parametrize
higher-order correlation functions of curvature perturbations generated during
inflation \cite{Smith,Cremi2,Planck} but also to perform a systematic study
for the physics of a late-time
cosmic acceleration induced by the modification of gravity \cite{Quin}-\cite{Kase14}.
In fact, recent studies \cite{Bloom,Piazza,Gleyzes,Gergely,Gao,RT14} showed
that the EFT approach can encompass a wide range of theories including
Horndeski/GLPV theories and Ho\v{r}ava-Lifshitz gravity.

The EFT approach of Ref.~\cite{Piazza} is based upon
a general Lagrangian $L$ in unitary gauge that depends on the
lapse $N$ and several ADM geometric scalars constructed from
the extrinsic curvature $K_{\mu \nu}$ and the three-dimensional
intrinsic curvature ${\cal R}_{\mu \nu}$. The action expanded up to
second order in scalar metric
perturbations shows that the theory has one scalar degree of freedom
with spatial derivatives higher than second order in general, while
the time derivatives are of second order.
Both Horndeski and GLPV theories satisfy conditions for the absence
of such higher-order spatial derivatives \cite{Piazza,Gergely,Tsuji14}.

The original projectable version of Ho\v{r}ava-Lifshitz gravity, in which
the lapse $N$ depends on the time $t$ alone, is plagued by
the strong coupling problem \cite{ins1,ins2}.
In the non-projectable version where $N$ depends on both time
and space, the acceleration vector $a_i=\nabla_i \ln N$ does not vanish
and hence several scalar quantities like $\eta\,a_i a^i$ can be present
in the Lagrangian ($\eta$ is a constant) \cite{Sergey}.
In this case there are some parameter spaces of $\eta$ in which
the strong coupling problem in the original theory can be alleviated.
This strong coupling still remains an open issue
without realizing a truly renormalizable and UV complete
theory \cite{Soti,Sergey2}.

The non-projectable version of Ho\v{r}ava-Lifshitz gravity
can be incorporated in the EFT approach of Ref.~\cite{Piazza}
by taking into account additional geometric
scalar quantities (associated with spatial derivatives up to 6-th order)
to the Lagrangian \cite{Gao,RT14}.
In Ref.~\cite{Kase14} the second-order action for scalar perturbations
was derived for the generic EFT Lagrangian
encompassing Horndeski/GLPV theories and Ho\v{r}ava-Lifshitz gravity.
This result can be useful for the computation of the primordial
scalar power spectrum generated during inflation and for discussing
conditions under which the ghosts and instabilities are absent
(see Ref.~\cite{Tsuji14} for a review).

In this paper we employ such a general EFT approach
for the study of tensor perturbations on the flat FLRW
background. Our analysis is more generic
than those of Refs.~\cite{Tsuji14,Gao,Cre} in that higher-order
spatial derivatives appearing in Ho\v{r}ava-Lifshitz gravity
are explicitly taken into account for the computation of
the second-order action of tensor perturbations.
Unlike Ref.~\cite{Wands}, we do not consider the terms
associated with the broken spatial diffeomorphism.
We provide a general formula for the inflationary power spectrum
of tensor modes by taking into account slow-roll
corrections to the leading-order spectrum.

This paper is organized as follows.
In Sec.~\ref{theorysec} we present the action of our EFT approach
and briefly review how several modified gravitational theories
are incorporated in our general framework.
In Sec.~\ref{persec} we derive the second-order action and
the equation of motion for tensor perturbations.
In Sec.~\ref{gwspesec} we obtain the spectrum of gravitational
waves generated during inflation and in Sec.~\ref{appsec}
we apply the results to concrete modified gravitational theories.
Sec.~\ref{conclude} is devoted to conclusions.

\section{The general EFT action of modified gravity}
\label{theorysec}

The EFT of cosmological perturbations is based upon
the 3+1 decomposition of space-time described by
the line element \cite{ADM}
\begin{equation}
ds^{2}=g_{\mu \nu }dx^{\mu }dx^{\nu}
=-N^{2}dt^{2}+h_{ij}(dx^{i}+N^{i}dt)(dx^{j}+N^{j}dt)\,,
\label{ADMmetric}
\end{equation}
where $N$ is the lapse function, $N^i$ is the shift vector, and
$h_{ij}$ is the three-dimensional spatial metric.
Introducing a unit vector $n_{\mu}=(-N,0,0,0)$ orthogonal to the constant
$t$ hypersurfaces $\Sigma_t$, the induced metric $h_{\mu \nu}$
on $\Sigma_t$ can be expressed of the form $h_{\mu \nu}=g_{\mu \nu}
+n_{\mu}n_{\nu}$.

The extrinsic curvature is defined by
$K_{\mu \nu}=h^{\lambda}_{\mu} n_{\nu;\lambda}
=n_{\nu;\mu}+n_{\mu} a_{\nu}$,
where a semicolon represents a covariant derivative and
$a_{\nu} \equiv n^{\lambda}n_{\nu;\lambda}$
is the acceleration vector.
The scalar quantities that can be constructed from the
extrinsic curvature are the trace of $K_{\mu \nu}$ and
the square of $K_{\mu \nu}$, i.e.,
\begin{equation}
K \equiv {K^{\mu}}_{\mu}\,,\qquad
{\cal S} \equiv K_{\mu \nu} K^{\mu \nu}\,.
\end{equation}

The internal geometry of $\Sigma_t$ is characterized
by the three-dimensional Ricci tensor
${\cal R}_{\mu \nu}={}^{(3)}R_{\mu \nu}$,
which is dubbed the intrinsic curvature.
{}From ${\cal R}_{\mu \nu}$ we can construct
the following scalar quantities:
\begin{equation}
{\cal R} \equiv
{{\cal R}^{\mu}}_{\mu}\,,\qquad
{\cal Z} \equiv {\cal R}_{\mu \nu}
\mathcal{R}^{\mu \nu}\,, \qquad
{\cal U} \equiv \mathcal{R}_{\mu \nu}K^{\mu \nu}\,.
\end{equation}
Since it is possible to express the Riemann tensor
$R_{\mu \nu \lambda \sigma}$ in terms of the Ricci tensor
and scalar in three dimensions, we do not need to consider
scalar combinations associated with
$R_{\mu \nu \lambda \sigma}$.

In Ho\v{r}ava-Lifshitz gravity there are other scalar quantities
that generate spatial derivatives up to 6-th order:
\begin{equation}
{\cal Z}_1 \equiv \nabla_i {\cal R} \nabla^i {\cal R}\,,\qquad
{\cal Z}_2 \equiv \nabla_i {\cal R}_{jk} \nabla^i {\cal R}^{jk}\,.
\label{Zdef}
\end{equation}
We can also take into account the terms like
${\cal R}^{j}_{i} {\cal R}^{k}_{j}  {\cal R}^{i}_{k}$
and ${\cal R} {\cal R}^{j}_{i} {\cal R}^{i}_{j}$, but they are
irrelevant to the dynamics of linear scalar perturbations
on the flat FLRW background.
Hence we do not incorporate those terms in the
following analysis.

In the original Ho\v{r}ava-Lifshitz gravity \cite{Horava} the space-time
foliation is preserved by the space-independent reparametrization
$t \to t'(t)$, so the lapse $N$ is assumed to be a function of time $t$
alone (which is called the projectability condition).
This can be extended to a non-projectable version in which
the lapse depends on both the spatial coordinate $x^i$ ($i=1,2,3$)
and $t$ \cite{Sergey}. Since the acceleration $a_i=\nabla_i \ln N$
does not vanish in this case, we can also consider
the following scalar combinations:
\begin{eqnarray}
& &
\alpha_1 \equiv a_i a^i\,,\qquad
\alpha_2 \equiv a_i \Delta a^i\,,\qquad
\alpha_3 \equiv {\cal R} \nabla_i a^i\,,\nonumber \\
& &
\alpha_4 \equiv a_i \Delta^2 a^i\,,\qquad
\alpha_5 \equiv \Delta {\cal R} \nabla_i a^i\,,
\label{aldef}
\end{eqnarray}
where $\Delta \equiv \nabla_i \nabla^i$.

The action of general modified gravitational theories that depends on
the above mentioned scalar quantities is given by
\begin{equation}
S=\int d^4 x \sqrt{-g}\,L \left( N, K, {\cal S}, {\cal R}, {\cal Z}, {\cal U},
{\cal Z}_1, {\cal Z}_2, \alpha_1, \cdots, \alpha_5; t \right),
\label{action}
\end{equation}
where $g$ is a determinant of the metric $g_{\mu \nu}$ and
$L$ is a Lagrangian. The action (\ref{action}) encompasses
Horndeski/GLPV theories and Ho\v{r}ava-Lifshitz gravity.
In the following we will present explicit forms of the
Lagrangians in these theories.

First of all, Horndeski theories are described by the Lagrangian
\begin{eqnarray}
L &=& G_2(\phi,X)+G_{3}(\phi,X)\square\phi \nonumber \\
&& +G_{4}(\phi,X)\, R-2G_{4,X}(\phi,X)\left[ (\square \phi)^{2}
-\phi^{;\mu \nu }\phi _{;\mu \nu} \right] \nonumber \\
& &
+G_{5}(\phi,X)G_{\mu \nu }\phi ^{;\mu \nu}
+\frac{1}{3}G_{5,X}(\phi,X)
[ (\square \phi )^{3} \nonumber \\
& &-3(\square \phi )\,\phi _{;\mu \nu }\phi ^{;\mu
\nu }+2\phi _{;\mu \nu }\phi ^{;\mu \sigma }{\phi ^{;\nu}}_{;\sigma}]\,,
\label{Lho}
\end{eqnarray}
where $\square \phi \equiv (g^{\mu \nu} \phi_{;\nu})_{;\mu}$, and
$G_{j}$ ($j=2,\cdots,5$) are functions in terms of a scalar
field $\phi$ and its kinetic energy
$X=g^{\mu \nu}\partial_{\mu} \phi \partial_{\nu} \phi$,
$R$ and $G_{\mu\nu}$ are the Ricci scalar and
the Einstein tensor in four dimensions, respectively.
Here and in the following, a lower index of $L$
denotes the partial derivatives with respect to the scalar quantities
represented in the index, e.g., $G_{j,X} \equiv\partial G_{j}/\partial X$.
In unitary gauge we have $\phi=\phi(t)$ and $X=-\dot{\phi}(t)^2/N^2$,
where a dot represents a derivative with respect to $t$.
Hence the dependence of $\phi$ and $X$ in the action (\ref{Lho})
is interpreted as that of the lapse $N$ and the time $t$.
In fact, we can express the Lagrangian (\ref{Lho})
of the form \cite{Piazza,Gergely,Gleyzes}
\begin{eqnarray}
L &=&
A_2(N,t)+A_3(N,t)K+A_4(N,t) (K^2-{\cal S}) \nonumber \\
& & +B_4(N,t){\cal R} +A_5(N,t) K_3 \nonumber \\
& & +B_5(N,t) \left( {\cal U}-K {\cal R}/2 \right)\,,
\label{LH}
\end{eqnarray}
where $K_3=K^3-3KK_{\mu \nu}K^{\mu \nu}
+2K_{\mu \nu}K^{\mu \lambda}{K^{\nu}}_{\lambda}$.
Horndeski theories have the following correspondence
\begin{eqnarray}
& & A_2=G_2-XF_{3,\phi}\,,\label{A2}\\
& & A_3=2(-X)^{3/2}F_{3,X}-2\sqrt{-X}G_{4,\phi}\,,\label{A3}\\
& & A_4=-G_4+2XG_{4,X}+XG_{5,\phi}/2\,,\label{A4} \\
& & B_4=G_4+X(G_{5,\phi}-F_{5,\phi})/2\,,\label{B4} \\
& & A_5=-(-X)^{3/2}G_{5,X}/3\,,\\
& & B_5=-\sqrt{-X}F_{5}\,,\label{B5}
\label{AB}
\end{eqnarray}
where $F_3$ and $F_5$ are auxiliary functions satisfying
$G_3=F_3+2XF_{3,X}$ and $G_{5,X}=F_5/(2X)+F_{5,X}$.
{}From Eqs.~(\ref{A4})-(\ref{B5}) the following
two relations hold
\begin{equation}
A_4=2XB_{4,X}-B_4\,,\qquad
A_5=-XB_{5,X}/3\,,
\label{ABcon}
\end{equation}
under which the number of 6 independent functions reduces to 4.

GLPV \cite{Gleyzes} generalized Horndeski theories in such a way
that the coefficients $A_4$, $B_4$, $A_5$, and $B_5$ are not
necessarily related to each other.
The general action (\ref{action}) can incorporate
both Horndeski and GLPV theories described by the
Lagrangian (\ref{LH}).

The action (\ref{action}) also covers Ho\v{r}ava-Lifshitz gravity
given by the Lagrangian
\begin{eqnarray}
\hspace{-0.5cm}
L &=&
\frac{M_{\rm pl}^2}{2} ( {\cal S}-\lambda K^2
+{\cal R}+\eta_1 \alpha_1) \nonumber \\
\hspace{-0.5cm}
& &
-\frac12 \left( g_2 {\cal R}^2+g_3 {\cal Z}+
\eta_2 \alpha_2+\eta_3 \alpha_3\right) \nonumber \\
\hspace{-0.5cm}
& &
-\frac{1}{2M_{\rm pl}^2} \left( g_4 {\cal Z}_1+g_5 {\cal Z}_2+
\eta_4 \alpha_4+\eta_5 \alpha_5 \right)\,,
\label{lagHo}
\end{eqnarray}
where $M_{\rm pl}=2.435 \times 10^{18}$~GeV is the reduced
Planck mass, and $\lambda$, $\eta_1,\cdots,\eta_5$,
$g_2,\cdots,g_5$ are constants.
The original Ho\v{r}ava-Lifshitz gravity \cite{Horava} corresponds
to the case $\eta_1=\cdots=\eta_5=0$,
whereas its healthy extension \cite{Sergey}
involves the dependence of acceleration.

\section{The second-order action for tensor perturbations}
\label{persec}

\subsection{Cosmological perturbations}

The perturbed line element involving the
four scalar perturbations $\delta N$,
$\psi$, $\zeta$, $E$, and tensor perturbations $\gamma_{ij}$ can
be written of the form
\begin{eqnarray}
ds^{2} &=&
-(1+2\delta N) dt^{2}+2\partial_i \psi dx^{i}dt \nonumber \\
& &+a^{2}(t)[ (1+2\zeta)\hat{h}_{ij}+2\partial_{i} \partial_{j}E]
dx^{i}dx^{j}\,,
\label{permet}
\end{eqnarray}
where $a(t)$ is the scale factor, and
\begin{equation}
\hat{h}_{ij}=\delta_{ij}+\gamma _{ij}
+\frac{1}{2} \delta^{mk} \gamma_{im} \gamma_{kj}\,,
\label{gra}
\end{equation}
with ${\rm det}\,\hat{h}=1$.  The tensor perturbation $\gamma_{ij}$ is
traceless and divergence-free, i.e., $\gamma _{ii}=\partial _{i}\gamma
_{ij}=0$.  The last term on the r.h.s.\
of Eq.~(\ref{gra}) was introduced for the simplification of
calculations, but it does not affect the second-order action of
tensor modes \cite{Maldacena}.

Under the infinitesimal coordinate transformation $t \to t+\delta t$ and
$x^i \to x^i+\delta^{ij}\partial_{j}\delta x$, the metric perturbation $E$
transforms as $E \to E-\delta x$.
Throughout the paper we choose the gauge $E=0$
to fix the spatial threading $\delta x$.

The field perturbation $\delta \phi$ transforms as
$\delta \phi \to \delta \phi-\dot{\phi}\,\delta t$ under the gauge
transformation. In Horndeski and GLPV theories the
unitary gauge $\delta \phi=0$ is chosen to fix the time slicing $\delta t$.
In the projectable version of
Ho\v{r}ava-Lifshitz gravity \cite{Horava}, the lapse $N$ is a function
of $t$ alone and hence $\delta N=0$.  In the non-projectable
Ho\v{r}ava-Lifshitz gravity \cite{Sergey}, there is no such
restriction for the gauge choice.
The different gauge choices associated with the temporal
coordinate transformation do not affect the second-order action of
tensor perturbations presented later.

On the flat FLRW background described by the line element
$ds^2=-dt^2+a^2(t) \delta_{ij}dx^idx^j$, the extrinsic curvature and
the intrinsic curvature are given, respectively, by
$\bar{K}_{ij}=H\bar{h}_{ij}$ and $\bar{{\cal R}}_{ij}=0$,
where a bar represents the background values and
$H=\dot{a}/a$ is the Hubble parameter.
Then, the scalar quantities appearing in the
Lagrangian $L$ of Eq.~(\ref{action}) are $\bar{N}=1$, $\bar{K}=3H$,
$\bar{{\cal S}}=3H^{2}$, $\bar{{\cal R}}=\bar{{\cal Z}}=\bar{{\cal U}}=0$,
$\bar{{\cal Z}}_1=\bar{{\cal Z}}_2=0$, and
$\bar{\alpha}_1=\bar{\alpha}_2=\cdots=\bar{\alpha}_5=0$.

Expanding the action (\ref{action}) up to second order in scalar
perturbations for the spatial gauge choice $E=0$, we can obtain the
equations of motion for the background and linear scalar perturbations
without fixing the temporal gauge.  Varying the first-order perturbed
action with respect to $\delta N$ and $\delta \sqrt{h}$, respectively,
the background equations are given by \cite{Piazza,Gergely,Kase14}
\begin{eqnarray}
&&\bar{L}+L_{,N}-3H{\cal F}=0\,,
\label{back1} \\
&&\bar{L}-\dot{{\cal F}}-3H{\cal F}=0\,,
\label{back2}
\end{eqnarray}
where
\begin{equation}
{\cal F} \equiv L_{,K}+2HL_{,{\cal S}}\,.
\end{equation}
The linear scalar perturbation equations derived by varying the
second-order action in terms of $\delta N$, $\psi$, and $\zeta$ are
presented in Ref.~\cite{Kase14}.

\subsection{The second-order tensor action}

Let us derive the second-order action of Eq.~(\ref{action}) for
tensor perturbations.
Regarding the extrinsic curvature, tensor modes
satisfy the relations $K=3H$ and $\delta
K^{i}_{j}=\delta^{ik}\dot{\gamma}_{kj}/2$.
Up to first order, the
three-dimensional Ricci tensor reads
\begin{equation}
{\cal R}_{ij}=-\frac12 a^2 \Delta \gamma_{ij}\,.
\end{equation}
The three-dimensional Ricci scalar from tensor perturbations is
a second-order quantity, which is given by
\begin{equation}
{\cal R}=\frac{1}{4} \delta^{ik} \delta^{jl} \gamma_{ij}
\Delta \gamma_{kl}\,.
\end{equation}
Then the quantity ${\cal Z}_1$ is fourth-order in perturbations.

On using the above relations, the second-order action
for tensor modes reduces to
$S_h^{(2)}=\int d^4 x\,a^3L_h^{(2)}$, where
$L_h^{(2)}=L_{,{\cal S}} \delta K^{i}_{j}
\delta K^{j}_{i}+{\cal E} {\cal R}
+L_{,{\cal Z}} {\cal R}^{i}_{j} {\cal R}^{j}_{i}
+L_{,{\cal Z}_2} {\cal Z}_2$ with
\begin{equation}
{\cal E} \equiv L_{,\cal R}+\frac12 \dot{L_{,\cal U}}
+\frac32 H L_{,\cal U}\,.
\end{equation}
More explicitly, it is given by
\begin{equation}
S_{h}^{(2)}=
\int d^4 x\,\frac{a^3}{4} \delta^{ik} \delta^{jl}
\left( L_{,{\cal S}} \dot{\gamma}_{ij} \dot{\gamma}_{kl}
+\gamma_{ij} {\cal O}_t \gamma_{kl} \right)\,,
\label{Sh2}
\end{equation}
where
\begin{equation}
{\cal O}_t \equiv {\cal E} \Delta+L_{,{\cal Z}}\Delta^2
-L_{,{{\cal Z}_2}}\Delta^3\,.
\end{equation}
Note that there are no contributions to $S_{h}^{(2)}$ from
the scalars (\ref{aldef}).
The condition for avoiding the tensor ghost
corresponds to $L_{,\cal S}>0$.

Varying the action (\ref{Sh2}) with respect to $\gamma_{ij}$,
we obtain the equation of motion
\begin{eqnarray}
& &
\ddot{\gamma}_{ij}+\left( 3H+\frac{\dot{L_{,\cal S}}}{L_{,\cal S}}
\right)\dot{\gamma}_{ij}-c_t^2 \Delta \gamma_{ij} \nonumber \\
& &
-\frac{L_{,{\cal Z}}}{L_{,\cal S}} \Delta^2 \gamma_{ij}
+\frac{L_{,{\cal Z}_2}}{L_{,\cal S}} \Delta^3 \gamma_{ij}=0\,,
\label{teneq}
\end{eqnarray}
where
\begin{equation}
c_t^2 \equiv \frac{{\cal E}}{L_{,\cal S}}\,.
\end{equation}
In the absence of spatial derivatives higher than second order,
$c_t$ exactly corresponds to the propagation speed of gravitational
waves. In order to avoid the small-scale Laplacian
instability in this case, we require that $c_t^2>0$.

General Relativity corresponds to $G_4=M_{\rm pl}^2/2$
and $G_5=0$ in the Horndeski Lagrangian (\ref{Lho}), i.e.,
$-A_4=B_4=M_{\rm pl}^2/2$ and $A_5=B_5=0$
in Eq.~(\ref{LH}).
In this case we have $L_{,{\cal S}}=M_{\rm pl}^2/2$,
${\cal E}=M_{\rm pl}^2/2$, $c_t^2=1$, $L_{,{\cal Z}}=0$, and
$L_{,{\cal Z}_2} =0$, so
Eq.~(\ref{teneq}) reduces to
$\ddot{\gamma}_{ij}+3H\dot{\gamma}_{ij}
-\Delta \gamma_{ij}=0$.

\section{The inflationary tensor modes}
\label{gwspesec}

In this section we derive the power spectrum of tensor perturbations
generated during inflation.

\subsection{The power spectrum in Fourier space}

We expand the tensor perturbation
$\gamma_{ij} ({\bm x}, \tau)$ into the Fourier series as
$\gamma_{ij} ({\bm x}, \tau)=\int \frac{d^3k}{(2\pi)^{3/2}}
e^{i {\bm k} \cdot {\bm x}}\hat{\gamma}_{ij}({\bm k}, \tau)$, where
\begin{equation}
\hat{\gamma}_{ij}({\bm k}, \tau)= \sum_{\lambda=+,\times}
\hat{h}_{\lambda} ({\bm k}, \tau) e_{ij}^{(\lambda)} ({\bm k})\,.
\label{gamex1}
\end{equation}
Here, ${\bm k}$ is a comoving wavenumber,
$\tau \equiv \int a^{-1} dt$ is the conformal time,
and $e_{ij}^{(\lambda)}$ ($\lambda=+, \times$) are
symmetric polarization tensors.
The polarization tensors are transverse ($k_je_{ij}^{(\lambda)}=0$)
and traceless ($e_{ii}^{(\lambda)}=0$) with the normalization satisfying
$e^{(\lambda)}_{ij} (\bm{k})  e^{*(\lambda')}_{ij} (\bm{k})
= \delta_{\lambda \lambda'}$.
We write the Fourier mode $\hat{h}_{\lambda} ({\bm k}, \tau)$
of the form
\begin{equation}
\hat{h}_{\lambda} ({\bm k}, \tau)=
h_{\lambda} (k, \tau) a_{\lambda} ({\bm k})+
h_{\lambda}^* (k, \tau) a_{\lambda}^{\dagger} (-{\bm k})\,,
\label{gamex2}
\end{equation}
where the annihilation and creation operators
$a_{\lambda} ({\bm k})$ and $a_{\lambda}^{\dagger}
({\bm k}')$ obey the commutation relation
$[a_{\lambda} ({\bm k}),a_{\lambda'}^{\dagger}
({\bm k}')]=\delta_{\lambda \lambda'}\delta^{(3)}
({\bm k}-{\bm k}')$.

On the quasi de-Sitter background the conformal time
is given by $\tau \simeq -1/(aH)$, so that the asymptotic
past and future correspond to $\tau \to -\infty$ and
$\tau \to 0$, respectively. The tensor power spectrum
${\cal P}_h (k)$ is defined by the vacuum expectation value
of $\hat{\gamma}_{ij}$ in the $\tau \to 0$ limit, as
$\langle 0| \hat{\gamma}_{ij} ({\bm k}_1, 0)
 \hat{\gamma}_{ij} ({\bm k}_2, 0)  |0
\rangle=(2\pi^2/k_1^3)\delta^{(3)}
({\bm k}_1+{\bm k}_2) {\cal P}_h (k_1)$.
On using Eqs.~(\ref{gamex1}) and (\ref{gamex2}),
it follows that
\begin{equation}
{\cal P}_h (k)=\frac{k^3}{2\pi^2}
\left( |h_+(k,0)|^2+|h_\times (k,0)|^2 \right)\,.
\label{powerspe}
\end{equation}

\subsection{Equation of motion for a canonical field}

A canonically normalized field $v_{\lambda} (k, \tau)$ is defined by
\begin{equation}
v_{\lambda} (k, \tau)\equiv z h_{\lambda}(k, \tau)\,,\qquad
z \equiv a \sqrt{L_{,\cal S}/2}\,.
\label{vz}
\end{equation}
Then the kinetic term in the action (\ref{Sh2}) can be
expressed as $S_K=\int d\tau d^3x
\sum_{\lambda=+,\times}|v_{\lambda}'|^2/2$, where
a prime represents a derivative with respect to $\tau$.
{}From Eq.~(\ref{teneq}) each Fourier component
$v_{\lambda} (k,\tau)$ obeys the equation of motion
\begin{equation}
v_{\lambda}''+\left[ \mathcal{K}(k,\tau) -\frac{z''}{z}
\right]v_{\lambda}=0\,,
\label{veq}
\end{equation}
where the function $\mathcal{K}(k,\tau)$ is defined as
\begin{equation}
\mathcal{K}(k,\tau) \equiv
c_t^2 k^2 \left( 1+c_1 \frac{k^2}{a^2M_{\rm pl}^2}
+c_2 \frac{k^4}{a^4M_{\rm pl}^4} \right)\,,
\label{keff}
\end{equation}
and
\begin{equation}
c_1 \equiv -\frac{L_{,\cal Z}M_{\rm pl}^2}{{\cal E}}\,,\qquad
c_2 \equiv -\frac{L_{,{\cal Z}_2}M_{\rm pl}^4}{{\cal E}}\,.
\end{equation}

In the context of low-energy effective field theories, we will discuss
the case where $\mathcal{K}(k,\tau)\simeq c_t^2k^2$,
such that the linear form of the dispersion relation, $\omega=c_t
k$, is not modified by the nonlinear terms in Eq.~(\ref{keff})
well below the cut-off of the theories.
Otherwise, we would need to know the
UV-completion of the theories, or our treatment would break down. In
fact, the non-linear terms are suppressed for the physical wavenumber
$k_{\mathrm{phys}}=k/a$ much below the cut-off value
$k_{\mathrm{phys}}^{\mathrm{max}}\equiv M_{\rm pl}/|c_1|^{1/2}$ or
$M_{\rm pl}/|c_2|^{1/4}$.  In Ho\v{r}ava-Lifshitz gravity
\cite{Horava} and in the trans-Planckian physics studied in
Refs.~\cite{Martin,Niemeyer,Kowalski,Tanaka,Star},
$k_{\mathrm{phys}}^{\mathrm{max}}$ is close to $M_{\rm pl}$, i.e.,
$|c_1| \sim |c_2| \sim O(1)$.  In the EFT approach to inflation
advocated by Weinberg \cite{Weinberg}, the cut-off scale is
slightly smaller than $M_{\rm pl}$, say $\sqrt{\epsilon}M_{\rm pl}$,
where $\epsilon=-\dot{H}/H^2$ is the slow-roll parameter typically of
the order of $0.01$.

Since we have the application to Ho\v{r}ava-Lifshitz gravity and
the trans-Planckian physics in mind,
we shall focus on the situation in which the cut-off
scale $k_{\mathrm{phys}}^{\mathrm{max}}$ is much larger than
the Hubble parameter $H$ during inflation.
In this case the Hubble radius crossing occurs in the
linear regime of the dispersion relation (i.e., $\mathcal{K} \simeq c_t^2 k^2$),
so that the second and third terms in the parenthesis of
Eq.~(\ref{keff}) are regarded as small corrections to the first term.
In other words, the parameter defined by
\begin{equation}
\sigma \equiv \frac{c_1 H_k^2}{M_{\rm pl}^2}
\label{sigma}
\end{equation}
is much smaller than 1, where $H_k$ is the Hubble
parameter at $c_tk=aH$.
Under this condition the EFT approach to inflation
can be justified.

According to the previous discussion, we will solve Eq.\ (\ref{veq})
iteratively, and write its solution in the form
\begin{equation}
v_{\lambda}=v_{\lambda}^{(0)}+v_{\lambda}^{(1)}\,,
\end{equation}
where the leading-order perturbation $v_{\lambda}^{(0)}$
obeys the equation of motion
\begin{equation}
v_{\lambda}^{(0)''}+\left( c_t^2 k^2 -\frac{z''}{z} \right)
v_{\lambda}^{(0)}=0\,.
\label{v0eq}
\end{equation}
The field $v_{\lambda}^{(1)}$ induced by the nonlinear
corrections to Eq.~(\ref{keff}) satisfies
\begin{eqnarray}
& &
v_{\lambda}^{(1)''}+\left( c_t^2 k^2 -\frac{z''}{z} \right)
v_{\lambda}^{(1)} \nonumber \\
& &
=-c_t^2 \frac{k^4}{a^2M_{\rm pl}^2}
\left( c_1 + c_2 \frac{k^2}{a^2M_{\rm pl}^2}\right)
v_{\lambda}^{(0)}.
\label{v1eq}
\end{eqnarray}
In order to solve Eq.~(\ref{v0eq}), we take into account the slow-roll
inflationary corrections to the leading-order solution on the de
Sitter background \cite{Stewart}.  We then substitute the
leading-order solution into Eq.~(\ref{v1eq}) to obtain an iterative
solution of $v_{\lambda}^{(1)}$.

\subsection{Solutions to the tensor equations of motion}

In the following we consider the situation in which
the parameters defined by
\begin{equation}
\epsilon \equiv -\frac{\dot{H}}{H^2}\,,\qquad
\epsilon_{{\cal S}} \equiv \frac{\dot{L_{,\cal S}}}{HL_{,{\cal S}}}\,,
\qquad
s \equiv \frac{\dot{c_t}}{Hc_t}
\end{equation}
are much smaller than unity during inflation.
The smallness of $\epsilon$ comes from the quasi de Sitter
background. Dividing Eq.~(3.4) by $2H^2L_{,\cal S}$,
the term $\epsilon_{\cal S}$ appears in addition to $\epsilon$.
Hence $\epsilon_{\cal S}$ is at most the same order
as $\epsilon$.

The tensor propagation speed
square in Horndeski theories can be estimated as
$c_t^2=1+O(\epsilon)$, so the parameter $s$ is of the
order of $\epsilon^2$ (see Sec.~\ref{Hoper}). In GLPV theories, $c_t^2$
generally differs from 1.
As we will see in Sec.~\ref{GLPVper}, it is possible to obtain
the Einstein frame with $c_t^2$ equivalent to 1
under the so-called disformal transformation.
Provided that the cosmological background in the Einstein frame
is quasi de Sitter, we will show that the
variation of $c_t^2$ in the original frame is small, i.e.,
$|s| \ll 1$.

The quantity $z''/z$, up to next to leading-order corrections,
can be estimated as
\begin{equation}
\frac{z''}{z}=2(aH)^2 \left( 1-\frac12 \epsilon
+\frac34 \epsilon_{\cal S} \right)\,.
\end{equation}
Introducing a dimensionless variable
\begin{equation}
y \equiv \frac{c_tk}{aH}\,,
\end{equation}
its time derivative obeys $y'=-aHy(1-\epsilon-s)$.
Then, Eq.~(\ref{v0eq}) can be expressed as
\begin{eqnarray}
& &
(1-2\epsilon-2s) y^2\,\frac{d^2 v_{\lambda}^{(0)}}{dy^2}
-sy \frac{dv_{\lambda}^{(0)}}{dy} \nonumber \\
& &+\left( y^2-2+\epsilon-\frac32 \epsilon_{\cal S} \right)
v_{\lambda}^{(0)}=0\,.
\label{vlameq}
\end{eqnarray}
Here and in the following, we drop contributions of the slow-roll
corrections of the order of $\epsilon^2$.  In other words, we deal
with the first-order slow-roll parameters as constants.

The solution to Eq.~(\ref{vlameq}), after neglecting
non-linear terms in the slow-roll parameters, is given by
\begin{eqnarray}
v_{\lambda}^{(0)} (y)
&=& y^{(1+s)/2} \{ \alpha_k
H_{\nu}^{(1)}[(1+\epsilon+s)y] \nonumber \\
& &~~~~~~~~~~~
+\beta_k H_{\nu}^{(2)}[(1+\epsilon+s)y]\}\,,
\label{vso}
\end{eqnarray}
where $\alpha_k$ and $\beta_k$ are integration constants,
$H_{\nu}^{(1)}(x)$ and $H_{\nu}^{(2)}(x)$ are Hankel functions of the
first and second kinds respectively, and
\begin{equation}
\nu=\frac32+\epsilon+\frac12 \epsilon_{\cal S}
+\frac32 s\,.
\end{equation}
The Bunch-Davies vacuum corresponds to the choice $\beta_k=0$. On
using the property $H_{\nu}^{(1)} (x \gg 1) \simeq -\sqrt{2/(\pi
  x)}\,e^{i[x+(3-2\nu)\pi/4]}$, the solution in the asymptotic past
reads
\begin{equation}
v_{\lambda}^{(0)}(y \gg 1) \simeq -\alpha_k \sqrt{\frac{2}{\pi}}
\frac{y^{s/2}}{\sqrt{1+\epsilon+s}}
e^{i [(1+\epsilon+s)y+(3-2\nu)\pi/4 ]}\,.
\end{equation}
 The coefficient $\alpha_k$ is determined by the Wronskian condition
$v_{\lambda}^{(0)}v_{\lambda}^{(0)*'}
-v_{\lambda}^{(0)*}v_{\lambda}^{(0)'}=i$,
such that (up to second order in the slow-roll parameters)
\begin{equation}
\alpha_k =-\frac14 \sqrt{\frac{\pi}{c_{tk}k}}\,(2+\epsilon+s)\,,
\label{alde}
\end{equation}
where $c_{tk}$ is the value of $c_t$ at $c_tk=aH$ (i.e., at $y=1$).
For the derivation of Eq.~(\ref{alde}) we used the property that any
time-dependent function $f(\tau)$ on the quasi de Sitter background
can be expanded around $y=1$ (denoted by the subscript $k$), as
$f(\tau)=f(\tau_k)-(\dot{f}/H_k) \ln (\tau/\tau_k)$ \cite{Chen}.  For
$\mu$ much smaller than 1 the quantity $y$ is also expanded as
$y^{\mu} \simeq 1+\mu \ln (\tau/\tau_k)$, so the variation of $c_t$,
$H$, and $L_{,\cal S}$ can be quantified as
\begin{equation}
c_t=c_{tk} y^{-s}\,,\quad
H=H_k y^{\epsilon}\,,\quad
L_{,\cal S}=L_{,{\cal S}k}y^{-\epsilon_{\cal S}}\,.
\label{ctre}
\end{equation}

Substituting Eq.~(\ref{alde}) and $\beta_k=0$ into Eq.~(\ref{vso}), we
obtain
\begin{eqnarray}
v_{\lambda}^{(0)} (y)
&=& -\frac{\sqrt{\pi}}{2} \frac{aH}{(c_tk)^{3/2}}
\left( 1+\frac12 \epsilon+\frac12 s \right) \nonumber \\
&& \times y^{3/2} H_{\nu}^{(1)}[(1+\epsilon+s)y]\,.
\label{vlamso}
\end{eqnarray}
Using the property $H_{\nu}^{(1)}(x \to 0)=-(i/\pi)\Gamma(\nu)
(x/2)^{-\nu}$ and the relations (\ref{ctre}), the solution
$h_{\lambda}^{(0)}(y)=v_{\lambda}^{(0)}(y)/z$ long after the Hubble
radius crossing ($y \to 0$) reduces to
\begin{equation}
h_{\lambda}^{(0)} (0)=i \frac{H_k}{\sqrt{2\pi L_{,{\cal S}k}}}
\frac{2^\nu \Gamma(\nu)}{(c_{tk}k)^{3/2}}
(1-\epsilon-s)\,.
\label{hlambda}
\end{equation}
Expanding the function $2^{\nu}\Gamma (\nu)$ around $\nu=3/2$, it
follows that
\begin{eqnarray}
\hspace{-1.3cm}
& &h_{\lambda}^{(0)} (0)=
i \frac{H_k}{\sqrt{L_{,{\cal S}k}}}
\frac{1}{(c_{tk}k)^{3/2}} \biggl[ 1+(1-\gamma-\ln 2) \epsilon
\nonumber \\
\hspace{-1.3cm}
& &~~~~~~~~~~
+\frac12 (2-\gamma-\ln 2)\epsilon_{\cal S}
+\frac12 (4-3\gamma-3\ln 2)s \biggr],
\label{hlambda2}
\end{eqnarray}
where $\gamma=0.5772...$ is the Euler-Mascheroni constant.

The next step is to derive the solution to Eq.~(\ref{v1eq}) by using
the leading-order solution of Eq.~(\ref{vlamso}) on the de Sitter
background (obtained by setting $\epsilon=\epsilon_{\cal S}=s=0$
and $a=-1/(H\tau)$ with $H={\rm constant}$), i.e.,
$v_{\lambda}^{(0)}(\tau)=-i(1+ic_tk \tau)e^{-ic_tk \tau} /[\sqrt{2}
\tau (c_tk)^{3/2}]$. The speed of propagation for this mode, for
large $k$'s, coincides, by construction, with $c_t$, such that this choice
is consistent with the assumption that the corrections do not modify
the standard propagation of tensor modes.
Integrating Eq.~(\ref{v1eq}) after substitution of the leading-order solution of
$v_{\lambda}^{(0)}$, the resulting particular solution is given by
\begin{eqnarray}
\hspace{-0.7cm}
& &v_{\lambda}^{(1)} (\tau)=
\frac{e^{-ic_tk \tau}}{120\sqrt{2} c_t^4 (c_tk)^{3/2} \tau}
\frac{H^2}{M_{\rm pl}^2}  \nonumber \\
\hspace{-0.7cm}
& &~~~~~~~~~~\times
\biggl[ 5 \biggl( 5c_1c_t^2-7c_2 \frac{H^2}{M_{\rm pl}^2}
\biggr)(3i-3c_tk \tau-2c_t^3k^3 \tau^3 )  \nonumber \\
\hspace{-0.7cm}
& &~~~~~~~~~~~~
-10i \biggl( 2c_1c_t^2-7c_2 \frac{H^2}{M_{\rm pl}^2}
\biggr) c_t^4 k^4 \tau^4 \nonumber \\
\hspace{-0.7cm}
& &~~~~~~~~~~~~
-6c_2 \frac{H^2}{M_{\rm pl}^2} (7+2ic_tk \tau)
c_t^5k^5\tau^5 \biggr]\,.
\label{vfirst}
\end{eqnarray}
The correction $v_{\lambda}^{(1)} (\tau)$ has an oscillatory
part $e^{-ic_tk \tau}$, which by construction, follows the
oscillations of the dominant contribution, $v_{\lambda}^{(0)}
(\tau)$. Long after the Hubble radius crossing ($\tau \to 0$), the
perturbation $h_{\lambda}^{(1)}(\tau)=v_{\lambda}^{(1)}(\tau)/z$
approaches
\begin{equation} h_{\lambda}^{(1)}(0)=-i \frac{H}{8\sqrt{L_{,\cal
      S}}\,(c_tk)^{3/2}} \left( 5c_1 \frac{H^2}{c_t^2M_{\rm pl}^2}
  -7c_2 \frac{H^4}{c_t^4M_{\rm pl}^4} \right).
\label{hlam1}
\end{equation}
Since we are not interested in the next-order solution to
Eq.~(\ref{hlam1}), we can replace $H$, $c_t$, and
$L_{,\cal S}$ for $H_k$, $c_{tk}$, and $L_{,{\cal S}k}$,
respectively.

\subsection{The spectrum of inflationary tensor modes}

The tensor power spectrum is known by substituting
$h_{\lambda}(0)=h_{\lambda}^{(0)}(0)+h_{\lambda}^{(1)}(0)$ into
Eq.~(\ref{powerspe}), as
\begin{eqnarray}
\hspace{-0.5cm}
& &
{\cal P}_{h}(k)=\frac{H_k^2}{\pi^2 L_{,{\cal S}k}c_{tk}^3}
\biggl[ 1-2(C+1)\epsilon-C \epsilon_{\cal S}
-(3C+2)s
\nonumber \\
\hspace{-0.5cm}
& &~~~~~~~~~~~~~~~~~~~~~~~~~~
-\frac54 \frac{\sigma}{c_{tk}^2}
+\frac{7c_2}{4c_1^2}  \frac{\sigma^2}{c_{tk}^4}\biggr]\,,
\label{Phf}
\end{eqnarray}
where $C=\gamma-2+\ln 2=-0.729637...$ and $\sigma$
is defined by Eq.~(\ref{sigma}).
The leading-order power spectrum is given by
${\cal P}_h^{\rm lead}(k)=H_k^2/(\pi^2 L_{,{\cal S}k}c_{tk}^3)$.

The last two terms in the square bracket of Eq.~(\ref{Phf}), which
correspond to the corrections induced by spatial derivatives
higher than second order, are suppressed by the factor
$\sigma \approx H_k^2/(k_{\mathrm{phys}}^{\mathrm{max}})^2$.
Provided that $\sigma/c_{tk}^2 \ll \epsilon$,
these terms are smaller than the slow-roll corrections.

We introduce the tensor spectral index $n_t$, as
\begin{equation}
n_t \equiv \frac{d \ln {\cal P}_{h}(k)}{d\ln k}
\biggr|_{c_{t}k=aH}\,.
\end{equation}
On using the property $d\ln k/dt|_{c_tk=aH}=H(1-\epsilon-s)$
and defining the following slow-roll parameters
\begin{equation}
\eta \equiv \frac{\dot{\epsilon}}{H\epsilon}\,,\qquad
\eta_{\cal S} \equiv \frac{\dot{\epsilon}_{\cal S}}
{H \epsilon_{\cal S}}\,,\qquad
\delta_s \equiv \frac{\dot{s}}{Hs}\,,
\end{equation}
it follows that
\begin{eqnarray}
n_t &=&
-2\epsilon-\epsilon_{\cal S}-3s-2\epsilon^2
-5\epsilon s-\epsilon \epsilon_{\cal S}-\epsilon_{\cal S}s
-3s^2 \nonumber \\
&&-2(C+1) \epsilon \eta-C\epsilon_{\cal S}\eta_{\cal S}
-(3C+2)s \delta_s \nonumber \\
& & +\frac{5}{2c_{tk}^2} \sigma (\epsilon+s)
-\frac{7c_2}{4c_1^2}  \frac{1}{c_{tk}^4}
\sigma^2 (\epsilon+s)\,,
\label{ntf}
\end{eqnarray}
which should be evaluated at $c_tk=aH$.
The leading-order spectral index is given by
$n_t^{\rm lead}=-2\epsilon-\epsilon_{\cal S}-3s$.

\section{Application to concrete theories}
\label{appsec}

We estimate the inflationary tensor power spectrum
and its spectral index in concrete modified gravitational
theories by using the general results derived
in Sec.~\ref{gwspesec}.

\subsection{Theories with higher-order spatial derivatives}

Let us consider the theories described by the Lagrangian
\begin{eqnarray}
L &=&
\frac{M_{\rm pl}^2}{2} ( {\cal S}-\lambda K^2
+{\cal R})+A_2(N,t)+A_3(N,t)K \nonumber \\
& &
+\frac{M_{\rm pl}^2}{2} \eta_1 \alpha_1
- \frac12 \left( g_2 {\cal R}^2+g_3 {\cal Z}+
\eta_2 \alpha_2+\eta_3 \alpha_3\right) \nonumber \\
& &
-\frac{1}{2M_{\rm pl}^{2}}\left( g_4 {\cal Z}_1+g_5 {\cal Z}_2+
\eta_4 \alpha_4+\eta_5 \alpha_5 \right)\,.
\label{lagHo2}
\end{eqnarray}
For $A_2=A_3=0$ this corresponds to the
Lagrangian (\ref{lagHo}) of Ho\v{r}ava-Lifshitz gravity,
including both the projectable ($\alpha_i=0$) and
non-projectable ($\alpha_i \neq 0$) versions.

We take into account the terms $A_2(N,t)$ and
$A_3(N,t)K$ in Eq.~(\ref{lagHo2}) to realize inflation
by a scalar degree of freedom.
In fact, the Lagrangian $L=(M_{\rm pl}^2/2)R+
G_2(\phi,X)+G_3(\phi,X) \square \phi$
of the kinetic braiding theories \cite{Vikman} reduces to
Eq.~(\ref{lagHo2}) with $\lambda=1$,
$A_2=G_2-XF_{3,\phi}$, $A_3=2(-X)^{3/2}F_{3,X}$,
$\eta_1=\cdots=\eta_5=0$ and $g_2=\cdots=g_5=0$ in unitary gauge,
where we used the fact that the four-dimensional Ricci scalar
is expressed as $R={\cal S}-K^2+{\cal R}$
up to a boundary term. The field $\phi$ is responsible
for the cosmic acceleration as it happens for k-inflation ($G_3=0$)
and potential-driven slow-roll inflation ($G_3=0$ and $G_2=-X/2-V(\phi)$).

Since $L_{,\cal S}={\cal E}=M_{\rm pl}^2/2$, $c_1=g_3$, and $c_2=g_5$,
Eqs.~(\ref{Phf}) and (\ref{ntf}) read
\begin{eqnarray}
{\cal P}_{h}(k) &=& \frac{2H_k^2}{\pi^2 M_{\rm pl}^2}
\left[ 1-2(C+1) \epsilon
-\frac{5}{4}\sigma+\frac{7g_5}{4g_3^2}\sigma^2 \right]\,,
\label{Ph1}
\nonumber \\ \\
n_t &=& -2\epsilon-2\epsilon^2
-2(C+1) \epsilon \eta
+\frac{5}{2} \sigma \epsilon
-\frac{7g_5}{4g_3^2} \sigma^2 \epsilon\,,
\label{nt1}
\nonumber \\
\end{eqnarray}
where $\sigma=g_3H_k^2/M_{\rm pl}^2$.
If $g_3=g_5=0$, then the last two terms in Eqs.~(\ref{Ph1})
and (\ref{nt1}) vanish.
In this case, the above tensor power spectrum reduces to
the one in standard slow-roll inflation \cite{Stewart}.

The contributions from the terms $A_2(N,t)$ and $A_3(N,t) K$
do not directly appear in Eqs.~(\ref{Ph1})-(\ref{nt1}), but they affect
the tensor power spectrum indirectly through the background
equations of motion (\ref{back1})-(\ref{back2}).

Since the leading-order spectrum is ${\cal P}_h^{\rm lead} (k)
=2H_k^2/(\pi^2 M_{\rm pl}^2)$, the energy scale of inflation
is directly known from the measurement of primordial
gravitational waves. More concretely, we have
$H_k/M_{\rm pl} \simeq \pi \sqrt{r{\cal P}_{s}(k)/2}$,
where ${\cal P}_s(k) \simeq 2.2 \times 10^{-9}$ is the
observed scalar power spectrum \cite{Planck}
and $r={\cal P}_h^{\rm lead}(k)/{\cal P}_s(k)$
is the tensor-to-scalar ratio. On using the observational bound
$r \lesssim 0.2$ \cite{Planck}, we have that
$H_k/M_{\rm pl} \lesssim 4 \times 10^{-5}$.
Hence, for $|g_3|, |g_5| \lesssim 1$, the corrections induced by
spatial derivatives higher than second order are suppressed
compared to the slow-roll corrections (typically
of the order of 0.01).

Provided that $H$ decreases during inflation, the tensor spectrum
is red-tilted ($n_t \simeq -2\epsilon<0$).
{}From the background Eqs.~(\ref{back1}) and (\ref{back2}),
we obtain $M_{\rm pl}^2 (3\lambda-1) \dot{H}=A_{2,N}+3HA_{3,N}$.
If $\lambda>1/3$, then the condition $\dot{H}<0$ translates to
$A_{2,N}+3HA_{3,N}<0$. In unitary gauge the field kinetic
energy is given by $X=-N^{-2}\dot{\phi}^2$, so the Hubble parameter
decreases for $A_{2,X}+3HA_{3,X}<0$.

\subsection{Horndeski theories}
\label{Hoper}

In unitary gauge the Lagrangian (\ref{Lho}) of Horndeski theories
is equivalent to Eq.~(\ref{LH}) with the relations (\ref{A2})-(\ref{B5}).
On using the fact that the term $K_3$ is given by
$K_3=3H ( 2H^2-2KH+K^2-{\cal S})$ up to quadratic order
in the perturbations on the flat FLRW background, we have
$L_{,\cal S}=G_4 (1+\epsilon_1)$ and
${\cal E}=G_4 (1+\epsilon_2)$, where
\begin{eqnarray}
\hspace{-0.5cm}
\epsilon_1 &\equiv& -\frac{2XG_{4,X}}{G_4}
-\frac{X G_{5,\phi}}{2G_4}
+\frac{H(-X)^{3/2} G_{5,X}}{G_4}\,,\\
\hspace{-0.5cm}
\epsilon_2 &\equiv& \frac{X G_{5,\phi}}{2G_4}
-\frac{X G_{5,X} \ddot{\phi}}{G_4}\,.
\end{eqnarray}
The terms $\epsilon_1$ and $\epsilon_2$, which
involve $X$, work as the slow-roll corrections to
the leading-order contribution $G_4$.
In fact, all these terms appear on the r.h.s. of the
background equation for $\epsilon$
(Eq.~(9) of Ref.~\cite{DeTsu}), so
they are the same order as $\epsilon$.
The tensor propagation speed square is
given by $c_t^2 \simeq 1-\epsilon_1+\epsilon_2+O(\epsilon^2)$,
and hence $s=\epsilon_2 \eta_2/2-\epsilon_1 \eta_1/2+O(\epsilon^3)$,
where $\eta_j \equiv \dot{\epsilon}_j/(H \epsilon_j)$ with $j=1,2$.
In the following we set $G_4=(M_{\rm pl}^2/2)F(\phi,X)$,
where $F(\phi,X)$ is a dimensionless function with
respect to $\phi$ and $X$. Then the slow-roll parameter
$\epsilon_{\cal S}$ can be expressed as
$\epsilon_{\cal S}=\epsilon_{F}+\epsilon_1 \eta_1
+O(\epsilon^3)$, where $\epsilon_{F} \equiv \dot{F}/(HF)$.

The tensor power spectrum and its spectral index,
up to next to leading-order terms, read
\begin{eqnarray}
{\cal P}_h (k)
&=& \frac{2H_k^2}{\pi^2 M_{\rm pl}^2 F}
\left[ 1-2(C+1)\epsilon-C\epsilon_{F}
+\frac{\epsilon_1}2-\frac{3\epsilon_2}{2} \right],
\label{Ph2} \nonumber \\ \\
n_t
&=& -2\epsilon-\epsilon_{F}-2\epsilon^2-\epsilon \epsilon_{F}
+\frac12 \epsilon_1 \eta_1-\frac32 \epsilon_2 \eta_2
\nonumber \\
& &-2(C+1) \epsilon \eta-C \epsilon_{F} \eta_{F}\,,
\end{eqnarray}
where $\eta_{F} \equiv \dot{\epsilon}_{F}/(H \epsilon_{F})$.

Compared to Eq.~(\ref{Ph1}),  the leading-order power
spectrum ${\cal P}_h^{\rm lead} (k)=2H_k^2/(\pi^2 M_{\rm pl}^2 F)$
of Eq.~(\ref{Ph2}) is divided by the term $F$.
This term is associated with the conformal factor $\Omega^2$
under the transformation
$\hat{g}_{\mu \nu}=\Omega^2 (\phi, X)g_{\mu \nu}$.
In the following we study the case in which the conformal factor
depends on $\phi$ alone, i.e., on $t$ in unitary gauge.
This assumption is justified provided that the $X$ dependence
in $\Omega^2$ works only as slow-roll corrections to the
leading-order $\phi$-dependent term.
Under the conformal transformation
$\hat{g}_{\mu \nu}=\Omega^2(t)g_{\mu \nu}$,
the coefficients $A_4$ and $B_4$ in Eq.~(\ref{LH})
transform, respectively, as \cite{GlHa}
\begin{eqnarray}
\hat{A}_4 &=&\Omega^{-2} A_4
\left( 1-\frac{3\dot{\Omega}}{N\Omega}\frac{A_5}{A_4} \right)\,,
\label{A4tra} \\
\hat{B}_4 &=&\Omega^{-2} B_4
\left( 1+\frac{\dot{\Omega}}{2N\Omega}\frac{B_5}{B_4} \right)\,,
\label{B4tra}
\end{eqnarray}
where $A_4=-G_4[1+O(\epsilon)]$ and
$B_4=G_4[1+O(\epsilon)]$ from Eqs.~(\ref{A4}) and (\ref{B4}).
Since the second terms in the parentheses of Eqs.~(\ref{A4tra})
and (\ref{B4tra}) can be regarded as slow-roll corrections,
we have $\hat{A}_4=-\Omega^{-2}G_4 [1+O(\epsilon)]$
and $\hat{B}_4=\Omega^{-2}G_4 [1+O(\epsilon)]$.
Choosing the conformal factor $\Omega^2=
2G_4/M_{\rm pl}^2=F$, it follows that
$\hat{A}_4=-(M_{\rm pl}^2/2)[1+O(\epsilon)]$ and
$\hat{B}_4=(M_{\rm pl}^2/2)[1+O(\epsilon)]$.

Under the conformal transformation
$\hat{g}_{\mu \nu}=\Omega^2(t)g_{\mu \nu}$,
the structure of the Lagrangian (\ref{LH}) is preserved
with the modified leading-order coefficients
$\hat{A}_2=\Omega^{-4} A_2$,
$\hat{A}_3=\Omega^{-3}A_3$,
$\hat{A}_5=\Omega^{-1}A_5$, and
$\hat{B}_5=\Omega^{-1}B_5$ in the presence
of slow-roll corrections (involving the derivative
$\dot{\Omega}/(N\Omega)$) \cite{GlHa}.
This means that, for the choice $\Omega^2=F$,
the leading-order tensor spectrum in the transformed (Einstein)
frame can be derived by setting $L_{,\cal S}=M_{\rm pl}^2/2$
and $c_t=1$ in Eq.~(\ref{Phf}), i.e., $\hat{{\cal P}}_h^{\rm lead} (k)=
2\hat{H}_k^2/(\pi^2 M_{\rm pl}^2)$.
Since the Hubble parameters in two frames are related to
each other as $\hat{H}=[H+\dot{F}/(2NF)]/\sqrt{F}$,
the spectrum $\hat{{\cal P}}_h^{\rm lead} (k)$
is equivalent to
${\cal P}_h^{\rm lead} (k)=2H_k^2/(\pi^2 M_{\rm pl}^2 F)$
at leading order in slow-roll.
Provided that the null energy condition is not violated in the
Einstein frame the Hubble parameter $\hat{H}$ decreases,
in which case the tensor power spectrum is red-tilted.

The above properties can be notably seen in the Higgs inflationary scenario
with the scalar-field potential $V(\phi)=(\lambda/4)(\phi^2-v^2)^2$
and the function $F=1+\zeta \phi^2/M_{\rm pl}^2$,
where $\zeta$ is a non-minimal coupling \cite{Higgs}
(see also Refs.~\cite{Unruh}).
In order to realize the self-coupling $\lambda$ of the order
of 0.1, the non-minimal coupling is constrained to be
$\zeta=O(10^4)$ from the CMB normalization.
For $\zeta \gg 1$ the quantity $F$ is related to the number of
e-foldings $N_e$ from the end of inflation,
as $F \simeq 4N_e/3$ \cite{Komatsu}, which
is much larger than 1 on scales relevant to the CMB anisotropies.
The action in the Einstein frame is characterized by a canonically
normalized field with the potential $\hat{V}=V(\phi)/F^2$ \cite{Maeda},
in which case the tensor spectrum
${\cal P}_h^{\rm lead} (k)=
2\hat{H}_k^2/(\pi^2 M_{\rm pl}^2)$ is red-tilted due to
the decrease of $\hat{H}$.

\subsection{GLPV theories}
\label{GLPVper}

Let us proceed to the GLPV theories in unitary gauge, i.e.,
the Lagrangian (\ref{LH}). In this case
the functions $L_{,\cal S}$ and ${\cal E}$
are given by $L_{,\cal S}=-A_4 (1+\epsilon_1)$ and
${\cal E}=B_4(1+\epsilon_2)$ respectively, where
\begin{equation}
\epsilon_1=\frac{3HA_5}{A_4}\,,\qquad
\epsilon_2=\frac{\dot{B}_5}{2B_4}\,.
\end{equation}
Provided that $\epsilon_1$ and $\epsilon_2$ are
regarded as slow-roll corrections to the leading-order
terms of $L_{,\cal S}$ and ${\cal E}$, we have
$c_t^2=-(B_4/A_4)(1-\epsilon_1+\epsilon_2$).
The difference from Horndeski theories is that
$A_4$ and $B_4$ are not related with each other,
so $c_t^2$ generally differs from 1.
Then the leading-order tensor spectrum is given by
\begin{equation}
{\cal P}_h^{\rm lead}(k)=\frac{H_k^2}
{\pi^2 |A_4|\,c_{tk,{\rm lead}}^3} \,,
\label{tenGLPV}
\end{equation}
where $c_{tk,{\rm lead}}^2=-B_4/A_4$.

We perform the disformal transformation given by
$\tilde{g}_{\mu \nu}=g_{\mu \nu}+\Gamma(\phi,X)
\partial_{\mu}\phi \partial_{\nu} \phi$, where
$\Gamma(\phi,X)$ is a function in terms of
$\phi$ and $X$ \cite{Beken,Garcia}.
In Ref.~\cite{GlHa} it was shown that the structure of the
GLPV action is preserved under this
transformation\footnote{In the presence of an additional
matter there is a mixing between the sound
speeds of the scalar field $\phi$ and matter in GLPV theories
even for the metric frame minimally coupled to
matter \cite{Gergely,Gleyzes}.
The disformal transformation gives rise to a kinetic-type coupling
of the scalar field with matter in the transformed
frame \cite{Garcia,GlHa}, which
helps us to understand the origin of such a non-trivial mixing.
Here we do not take into account an additional matter, as
we are interested in the application to single-field inflation.}.
The coefficients $A_4$ and $B_4$ in the Lagrangian
(\ref{LH}) are transformed as
\begin{equation}
\tilde{A}_4=\sqrt{1+\Gamma X}\,A_4\,,\qquad
\tilde{B}_4=\frac{B_4}{\sqrt{1+\Gamma X}}\,.
\end{equation}
In the new frame the tensor propagation speed square is
given by $\tilde{c}_{t,{\rm lead}}^2=-\tilde{B}_4/\tilde{A}_4
=c_{t,{\rm lead}}^2/(1+\Gamma X)$.
If we choose the function
\begin{equation}
\Gamma=-\frac{1-c_{t,{\rm lead}}^2}{X}\,,
\label{Gamcho}
\end{equation}
then it follows that $\tilde{c}_{t,{\rm lead}}^2=1$.
In this case, the coefficients in Eq.~(\ref{LH}) are
transformed as $\tilde{A}_2=A_2/c_{t,{\rm lead}}$,
$\tilde{A}_3=A_3$, $\tilde{A}_4=c_{t,{\rm lead}}A_4$,
$\tilde{B}_4=B_4/c_{t,{\rm lead}}$,
$\tilde{A}_5=c_{t,{\rm lead}}^2A_5$,
and $\tilde{B}_5=B_5$.
Since $\tilde{c}_{t,{\rm lead}}^2=1$ in the new frame,
the leading-order spectrum becomes
${\cal P}^{\rm lead}_h(k)=\tilde{H}_k^2/(\pi^2 |\tilde{A}_4|)$.
If we make the conformal transformation $\hat{g}_{\mu \nu}=
\Omega^2(t) \tilde{g}_{\mu \nu}$ further with
$\Omega^2=2|\tilde{A}_4|/M_{\rm pl}^2$,
the resulting leading-order spectrum
reduces to ${\cal P}_h^{\rm lead} (k)=
2\hat{H}_k^2/(\pi^2 M_{\rm pl}^2)$.

Under the disformal transformation
$\tilde{g}_{\mu \nu}=g_{\mu \nu}+\Gamma(\phi,X)
\partial_{\mu}\phi \partial_{\nu} \phi$, the lapse
function $N$ is generally transformed to
$\tilde{N}=N\sqrt{1+\Gamma X}$ \cite{GlHa,Tsu14}.
Setting $N=1$ for the background, the choice of
$\Gamma$ in Eq.~(\ref{Gamcho}) can be interpreted as
$\tilde{N}=c_{t,{\rm lead}}$.
The Hubble parameters in the Einstein and original frames
are related with each other as $\tilde{H}=H/\tilde{N}=H/c_{t,{\rm lead}}$.
This leads to the relation $\tilde{\epsilon}=\epsilon+s$,
where $\tilde{\epsilon}=-\dot{\tilde{H}}/(\tilde{N}\tilde{H}^2)$
and $s=\dot{c}_{t,{\rm lead}}/(Hc_{t,{\rm lead}})$.
Provided that the cosmological background in the Einstein frame is
quasi de Sitter, we have that $\tilde{\epsilon} \ll 1$
and hence $|s| \ll 1$. Thus the assumption $|s| \ll 1$ used
to derive the tensor power spectrum (\ref{Phf}) is justified.

The above discussion shows that the combination of the
disformal and conformal transformations,
$\hat{g}_{\mu \nu}=\Omega^2(\phi)g_{\mu \nu}
+\Gamma(\phi,X)\partial_{\mu}\phi \partial_{\nu} \phi$,
can lead to a metric frame in which the leading-order
tensor power spectrum is of the standard form that depends
on the Hubble parameter $\hat{H}_k$ alone.
This conclusion is consistent with the recent results of
Ref.~\cite{Cre} in which the authors took the EFT approach
without having the direct connection to particular modified
gravitational theories.

\section{Conclusions}
\label{conclude}

We have studied tensor perturbations on the flat FLRW background
for the general action (\ref{action}) that encompasses most of the
modified gravitational theories proposed
in the literature--including Horndeski theories, GLPV theories, and
Ho\v{r}ava-Lifshitz gravity.
The equation of motion (\ref{teneq}), which follows from
the second-order action (\ref{Sh2}), involves the spatial derivatives
higher than second order for the theories where the Lagrangian
$L$ depends on ${\cal Z}$ or ${\cal Z}_2$.

We derived the inflationary power spectrum of tensor modes
under the condition that the cut-off scale $k_{\mathrm{phys}}^{\mathrm{max}}$ associated
with the non-linear terms of Eq.~(\ref{keff}) is much larger than
the Hubble parameter $H_k$ at $c_tk=aH$ during inflation.
On using the small parameter $\sigma$ of the order of
$H_k^2/(k_{\mathrm{phys}}^{\mathrm{max}})^2$,
the solution to Eq.~(\ref{veq}) is obtained iteratively on the de Sitter background.
Taking into account the slow-roll corrections to the
leading-order solution as well, the resulting tensor power spectrum
is given by Eq.~(\ref{Phf}) with the spectral index (\ref{ntf}).

The corrections from the higher-order spatial derivatives
to the leading-order power spectrum are suppressed by
the factor $\sigma/c_{tk}^2$. This conclusion is consistent
with the effect of modified trans-Planckian dispersion relations
on the inflationary power spectrum \cite{Niemeyer,Kowalski,Tanaka,Star}.
For $k_{\mathrm{phys}}^{\mathrm{max}}$ close to $M_{\rm pl}$ and
for $c_{tk}$ not very much smaller than 1,
the corrections induced by the spatial derivatives
higher than second order are smaller than
the slow-roll corrections arising from the deviation from
the de Sitter background.

We applied our general formula of the inflationary tensor
power spectrum to a number of concrete modified
gravitational theories. For the Lagrangian (\ref{lagHo2}), which
encompasses kinetic braiding models and Ho\v{r}ava-Lifshitz gravity,
the leading-order spectrum is directly related to $H_k$,
as ${\cal P}^{\rm lead}_h(k)=2H_k^2/(\pi^2M_{\rm pl}^2)$.

In Horndeski theories, where the tensor propagation speed is 1
at leading-order in slow-roll, ${\cal P}^{\rm lead}_h(k)$
involves a dimensionless factor $F=2G_4/M_{\rm pl}^2$
in the denominator. Under the conformal transformation
$\hat{g}_{\mu \nu}=Fg_{\mu \nu}$, the spectrum
in the Einstein frame simply reduces to
${\cal P}^{\rm lead}_h(k)=2\hat{H}_k^2/(\pi^2M_{\rm pl}^2)$.

In GLPV theories the leading-order tensor spectrum (\ref{tenGLPV})
involves the terms $A_4$ and $c_{tk,{\rm lead}}^2=-B_4/A_4$.
We showed that, under the disformal transformation
$\hat{g}_{\mu \nu}=\Omega^2(\phi)g_{\mu \nu}
+\Gamma(\phi,X)\partial_{\mu}\phi \partial_{\nu} \phi$,
it is possible to find a frame in which
$\hat{c}_{tk,{\rm lead}}^2=1$ and $\hat{A}_4=-M_{\rm pl}^2/2$
up to slow-roll corrections.
Thus the prediction of inflationary tensor modes is robust
in that there exists the metric frame in which
the leading-order spectrum is simply proportional to
$\hat{H}_k^2$ in a vast class of modified gravitational theories.

\section*{Acknowledgements}

This work is supported by the Grant-in-Aid for Scientific
Research from JSPS (No.~24540286)
and by the cooperation programs of Tokyo University of Science
and CSIC.

redpajama_set_name: RedPajamaArXiv


text:
Shelly Peiken
Posted in MusicWorld on December 31, 1999 by Hanna Bolte
A Grammy-nominated songwriter, Shelly Peiken has proven to be one of the top female songwriters in the pop and rock world. Having written or co-written songs with or for Meredith Brooks, The diVinyls, Christina Aguilera, Brandy, *NSYNC and The Pretenders, Peiken is one of the new players in the pop songwriting world.
Describing herself as always being a songwriter, Peiken's first big break came in the mid-'80s when Taylor Dayne recorded "Carry Your Heart." Though it wasn't a single, it did appear on Dayne's multi-platinum album and kick-started Peiken's career. Since then, Peiken has gone on to cross musical genres, writing or co-writing for artists including Celine Dion, Curtis Stigers, Abra Moore, Kelly Price and Reba McEntire, as well as international recording artists Viktor Lazlo (East West France), Ivan Matais (Arista U.K.), Louise Hoffsten (BMG Sweden) and Elisabeth (BMG Denmark).
Her most public songwriting breakthrough came with the song "Bitch," which she co-wrote with Meredith Brooks. "Bitch" became a gold single, received a Grammy nomination for Best Rock Song, was a #2 single on Billboard's Hot 100 chart, a #1 video on MTV and VH-1, and catapulted Brooks to pop/rock stardom.
Since then, Peiken has co-written with Brooks on her latest CD, as well as writing "Under My Tree" and "I Guess It's Christmas Time" for *NSYNC; the Grammy nominated "Almost Doesn't Count" for Brandy; the single "Human" for the Pretenders; the current #1 Billboard single "What A Girl Wants" for Christina Aguilera; and "Turn The Page" for Aaliyah for the Music of The Heart soundtrack. Coming up, she has tracks being used by Mark Wills, Aguilera, Brandy, Innocence (RCA), M2M (Atlantic), Devon (Sony), The Young Americans (Qwest), Colapsis (Cherry/Universal), Nobody's Angel (Universal), Vocal Point (Dreamworks) and Josh (RCA).
SOURCEMusicWorld TAGS Pop Rock International Musicworld Hitmaker *NSYNC Aaliyah Christina Aguilera Artists Brandy Meredith Brooks Shelly Peiken
Connect with BMI & Professional Songwriters


text:
COVID-19 and the contribution of health and care staff from overseas
The pandemic has again highlighted the crucial role that EU and international staff play in UK health and social care services. Our colleagues who come to the UK from across the world must be and feel valued.
Migration Advisory Committee: response to the call for evidence from the RCP
The RCP has outlined the existing workforce challenges faced by the NHS, its reliance on recruitment from overseas and our concerns for the UK to attract physicians to the NHS in the future.
Acute care toolkit 13: Acute care for adolescents and young adults
This toolkit will help acute trusts to develop and initiate models of developmentally appropriate healthcare.
Sedation in under 19s: using sedation for diagnostic and therapeutic procedures - NICE guideline
Sedation in under 19s: using sedation for diagnostic and therapeutic procedures offers evidence-based advice on the care and treatment of children and young people under the age of 19 having sedation for therapeutic or diagnostic procedures. This guideline was previously called Sedat
Bedwetting in under 19s - NICE guideline
This guideline covers assessing and treating bedwetting in people aged under 19. It aims to reduce bedwetting and the distress this causes by explaining what to ask in an assessment, what advice to provide, and which treatments are effective.


text:
Loungo Matlhaku (* 24. März 1995 in Khakhea) ist eine botswanische Leichtathletin, die sich auf den Sprint spezialisiert hat.

Sportliche Laufbahn
Erste internationale Erfahrungen sammelte Loungo Matlhaku bei den Juniorenafrikameisterschaften 2011 in Gaborone, bei denen sie in 12,01 s die Silbermedaille im 100-Meter-Lauf gewann und in 25,48 s Achte über 200 Meter wurde. Zudem gewann sie mit der botswanischen 4-mal-100-Meter-Staffel in 48,46 s die Bronzemedaille. Anschließend nahm sie an den Jugendweltmeisterschaften in Villeneuve-d'Ascq nahe Lille teil, bei denen sie über 100 Meter in der ersten Runde ausschied, während sie über 200 Meter bis in das Halbfinale gelangte, in dem sie mit 24,71 s ausschied. Daraufhin wurde sie bei den Commonwealth Youth Games in Douglas Vierte und Fünfte. Zwei Jahre später gewann sie bei den Juniorenafrikameisterschaften in Bambous in 24,62 s die Bronzemedaille im 200-Meter-Lauf. 2014 schied sie bei den Juniorenweltmeisterschaften in Eugene in beiden Disziplinen im Vorlauf aus. Bei den Afrikameisterschaften in Marrakesch schied sie in 24,70 s im Halbfinale über 200 Meter aus und gewann mit der botswanischen 4-mal-400-Meter-Staffel in 3:40,28 min die Bronzemedaille hinter Nigeria und Kenia. Bei den IAAF World Relays 2015 auf den Bahamas belegte sie mit 3:35,76 min Rang fünf im B-Finale.

Bei den Afrikameisterschaften 2016 in Durban gelangte sie in 11,67 s auf den achten Platz über 100 Meter und wurde in 24,18 s Siebte über 200 Meter. Zudem wurde sie mit der 4-mal-100-Meter-Staffel in 46,10 s Fünfte. Bei den IAAF World Relays 2017 auf den Bahamas kam sie mit der gemischten Staffel nicht ins Ziel. 2018 nahm sie erstmals an den Commonwealth Games im australischen Gold Coast teil und schied mit 23,98 s im Halbfinale aus und gewann mit der Staffel in 3:26,86 min die Bronzemedaille hinter Jamaika und Nigeria. Im August schied sie bei den Afrikameisterschaften in Asaba mit 12,08 s im Vorlauf aus und wurde in 3:42,16 min Vierte mit der Staffel. Im Jahr darauf erreichte sie bei der Sommer-Universiade in Neapel das Halbfinale über 100 Meter, in dem sie mit 11,63 s ausschied. Im August belegte sie bei den Afrikaspielen in Rabat in 23,84 s Rang sieben über 200 Meter. Bei den World Athletics Relays 2021 im polnischen Chorzów verpasste sie mit 3:34,99 min den Finaleinzug in der 4-mal-400-Meter-Staffel. 2022 erreichte sie bei den Afrikameisterschaften in Port Louis das Halbfinale über 100 Meter und schied dort mit 11,57 s aus und belegte mit der 4-mal-100-Meter-Staffel in 45,93 s den fünften Platz und schied über 200 Meter mit 24,19 s in der Vorrunde aus.

In den Jahren 2014 und 2021 wurde Matlhaku Botswanische Meisterin im 100-Meter-Lauf sowie 2017 über 200 Meter.

Bestleistungen
 100 Meter: 11,55 s (−0,5 m/s), 1. März 2018 in Johannesburg
 200 Meter: 23,25 s (+0,1 m/s), 22. Mai 2016 in Gaborone

Weblinks

100-Meter-Läufer (Botswana)
200-Meter-Läufer (Botswana)
4-mal-100-Meter-Staffel-Läufer (Botswana)
4-mal-400-Meter-Staffel-Läufer (Botswana)
Teilnehmer an den Commonwealth Games (Botswana)
Teilnehmer an den Afrikaspielen (Botswana)
Universiadeteilnehmer (Botswana)
Teilnehmer der Sommer-Universiade 2019
Botswanischer Meister (Leichtathletik)
Botswaner
Geboren 1995
Frau

redpajama_set_name: RedPajamaWikipedia


NEW YORK (CBSNewYork) — Kale, coconuts, and avocados could be bad for you.
Experts say there's something metallic lurking in those foods, in our water, and elsewhere that could be making us feel miserable.
Anxiety, trouble focusing, exhaustion, and an all around mental funk are familiar symptoms for a lot of people.
"People will come in with very non-specific symptoms; they feel moody, have depression, difficulty falling asleep," Dr. Svetlana Kogan explained.
After some investigation Dr. Kogan — an internist — said she found that copper is increasingly the culprit.
"Copper is an essential element," she explained.
Like iron, our bodies need copper for normal growth and health, Dr. Kogan said.
Certain foods like shellfish, leafy greens, cashews, and chickpeas are naturally high in copper.
"These are wonderful things to have as part of our diet, but if somebody is consuming excess they could certainly be at risk," she said.
"If you had six cups of coffee and you're feeling jittery, that's the feeling to describe copper toxicity," she explained.
An increase in vegetable based diets may also be contributing to copper overload, Kogan said.
Environmental exposure to things like copper water pipes may also be a factor.
"We always recommend to have your water tested," said Rob Zadotti.
Zadotti, co-owner of Gold Medal Service, said if the PH of your water is not within a safe range it will start to deteriorate the lining of your copper pipes.
While you might not be able to taste the copper in your water, there are other telltale signs it's there.
"Bluish stains or greenish stains on their tubs," Mike Jackson explained.
Jackson is a technical field supervisor at Gold Medal, he recommended a water filtration system.
As for treating copper toxicity, Kogan prescribed an infrared sauna.
"Patient is literally sweating out the toxins," she said.
More importantly — as she outlines in her book 'Diet Slave No More' — Kogan said it's important to eat foods that are balanced.
"Full of rainbow colored vegetables and fruits, and avoid focusing on one group," she said.
Prolonged use of estrogen medications such as birth control pills may also contribute to an increased level of copper in the body.
Several studies have shows that daily exposure to copper can lead to Alzheimer's Disease and other former of dementia.


text:
Book Review: Russia's Last Gasp
Russia's Last Gasp: The Eastern Front 1916–17, Prit Buttar, Osprey Publishing, Oxford, U.K., 2016, $30
Observers often complain that World War I histories concentrate on the Western Front to the exclusion of fighting elsewhere. Such big-name scholars as John Keegan, Norman Stone and Dominic Lieven have answered the call, but their efforts pale in comparison to those of British amateur historian Prit Buttar. Russia's Last Gasp is the third of his exhaustively researched and lucidly written four-volume series on the Eastern Front, which spans nearly 2,000 pages.
The two fronts differed not in the nature of combat (trench warfare was also standard in the East) but in infrastructure, described as primitive at best in the East. Buttar does not quarrel with the description of the Russian soldier in 1914 as poorly trained, pitifully undersupplied and badly led. By 1916, however, industry was making reasonable quantities of weapons; supply had improved from disastrous to barely adequate; and many senior generals had achieved basic competence. The German army was better, but the Russians were no longer pushovers.
Buttar focuses much of his narrative on the fairly well known Brusilov Offensive, launched in June 1916 to relieve pressure on France and then defend Verdun. An imaginative strategist, Aleksey Brusilov rejected the traditional prolonged artillery barrage and general advance against defenders who knew what was coming. Concealing preparations, he began with a short, targeted barrage and sent his best units against the weak points of an Austro-Hungarian army far less formidable than that of Germany.
The first weeks recorded impressive advances and a haul of prisoners. This surprised Russia's high command no less than the enemy. Both sides knew what they had to do, but as usual the Germans worked faster, sending forces and hectoring Austria to transfer units back from Italy. Beset by repeated delays, the wider Russian offensive fell back on traditional tactics and was the usual bloody failure. Brusilov received some reinforcements but his targeted offensive petered out in the fall. Perhaps its sole accomplishment was to persuade Romania to enter the war on Russia's side, but German forces quickly crushed the former.
Buffs of the Eastern Front will devour Buttar's nuts-and-bolts account of the fighting. While observing the academic tradition of listing every commander and unit name in an operation, the author breaks it up with vivid anecdotes and astute observations.
—Mike Oppenheim


Standard prostate-specific antigen (PSA) tests for prostate cancer are far from ideal, sometimes resulting in unnecessary biopsies, and even failing to detect some cancers altogether. With the goal of developing a more capable alternative, a team of researchers has turned to a machine it calls the Odoreader, which is designed to analyze urine samples to provide a non-invasive prostate cancer test.
The new research is a collaboration between the University of Liverpool and the University of the West of England in Bristol. It focuses on a device called the Odoreader, which was used back in 2013 to detect bladder cancer by analyzing the odors in urine, with a 100 percent success rate.
The researchers believe that the device could significantly improve prostate cancer diagnostics, by offering a non-invasive test that's potentially much more accurate than existing methods. The machine, which is described as being " like an electronic nose," uses a gas chromatography sensor combined with specially developed algorithms to detect whether a patient has cancer from a single urine sample.
The gas from the urine sample travels along a 30-m (98-ft) long column inside the device, with the different compounds present travelling through it at different rates, breaking up the sample into a more easily readable format.
The researchers tested the machine on urine samples from 155 men. The Odoreader analysis diagnosed 58 of the men as having prostate cancer, 24 as having bladder cancer and 73 as having haematuria or a poor stream without cancer. This amounted to an accuracy of 90 percent for prostate cancer and over 95 percent accuracy for bladder cancer. Those numbers are significantly higher than other diagnosis techniques such as PSA testing, which is 65 to 75 percent accurate.
"If the test succeeds at full medical trial it will revolutionize diagnostics," said the North Bristol NHS Trust's Raj Prasad. "An accurate urine test would mean that many men who currently undergo prostate biopsy may not need to do so."
If a device that "sniffs" out prostate cancer sounds familiar, that's because we've seen studies take a similar approach before. In 2014, Finnish researchers used a machine called the ChemPro 100 to analyze gases directly above urine samples. Those results weren't quite as promising, with the machine accurately diagnosing between 67 and 78 percent of cases.
In light of the high accuracy of the Odoreader in the new study, the researchers are now looking to fund a full-scale clinical trial. The findings of the work are published online in the Journal of Breath Research.


If you're selling a product that has to ship to the customer, you have a perfect opportunity to suggestive sell. Contact the customer just prior to the shipment going out to let them know it's on its way and then suggest another item that they may benefit from.
This is such a great idea for any company using customer service reps or others who have the opportunity to interact with customers on a regular basis. These people may not feel comfortable cold-calling or selling complex items, but that shouldn't stop them from selling "add-on" items. It is one of those very simple ideas that you can use to increase sales by 2%. More importantly, it's probably one of the easiest and best ways to increase your numbers because it doesn't require any additional selling expenses.


text:
Quiz Me!
Professor Severus Snape's Sorcerer-Tastic, Muggalicious Mid-Summer Movie Quiz is upon us. Now go take it and post answers before Snape turns you into a 3-headed goat or resets your blog's template changes.
1) Second-favorite Stanley Kubrick film.
At this moment, The Killing is leaning in to the wire to be my fave, so I guess that makes Clockwork Orange the runner-up. Man, the field is in tight behind them though...
2) Most significant/important/interesting trend in movies over the past decade, for good or evil.
Hard to deny the onset of DV and its use by major filmmakers, but I find the most interesting trend is the desire to make films (even comedies) longer than 2 hours - ie. typically way longer than necessary. A huge generalization of course, but I've been surprised at how the studios are allowing these longer films. The previous line of thinking was to keep them shorter to squeeze in additional showings, but I guess they've already given that up since each film is preceded by 15 minutes of commercials and 5 trailers. Perhaps their studies have shown that there's a benefit to the longer film? More snack trips due to the frequent bathroom breaks?
3) Bronco Billy (Clint Eastwood) or Buffalo Bill Cody (Paul Newman)?
Haven't seen either, but Altman and Newman just can't possibly be bad...Can they?
4) Best Film of 1949.
The Third Man. Yep, boring answer. Don't care. Love the film. Gun Crazy is pretty dang wicked too.
5) Joseph Tura (Jack Benny) or Oscar Jaffe (John Barrymore)?
Both great characters from two great comedies both starring the amazing Carole Lombard. Benny beats Barrymore for me though.
6) Has the hand-held shaky-cam directorial style become a visual cliché?
Only when the director doesn't know how to use it.
7) What was the first foreign-language film you ever saw?
Some strange slapstick French comedy when I was about 8. I'd love to know what it was, it's just that I can't remember a single thing about it. Likely I would hate the thing now. Granted, French was my second language growing up and I was actually going to a French elementary school at that time, so perhaps that doesn't count...
8) Charlie Chan (Warner Oland) or Mr. Moto (Peter Lorre)?
Neither - simply because I haven't seen a single one of their movies. Still, Peter Lorre would be hard to beat.
9) Favorite World War II drama (1950-1970).
The Dirty Dozen is way too much fun to be a straight drama, so I'll go with Frakenheimer's The Train.
10) Favorite animal movie star.
The dogs who portrayed Quill, the seeing eye dog (as puppy and as the older working dog). Not a great film, but far better than I ever expected it to be.
11) Who or whatever is to blame, name an irresponsible moment in cinema.
The ending of Julio Medem's Chaotic Ana when the heroine...
[gross spoiler ahead]
...defecates on the face of a U.S. Senator.
[/gross spoiler ahead]
Now that's subtle political commentary...
12) Best Film of 1969.
Z or Double Suicide...Z or Double Suicide...Z or Double Suicide...Um, I'll need some more time on this one.
13) Name the last movie you saw theatrically, and also on DVD or Blu-ray.
Unfortunately, the last theatrical screening I saw was Ice Age 3: Dawn Of The Dinosaurs - not terrible, but just not overly satisfying. On DVD it was the Danish film How To Get Rid Of The Others - pretty much the same reaction for that. Though Louise Mieritz was quite excellent in it.
14) Second-favorite Robert Altman film.
Nashville. Short Cuts is still at number one. Again, there's a lot of competition.
15) What is your favorite independent outlet for reading about movies, either online or in print?
Blogs, blogs, blogs and more blogs. I like blogs.
16) Who wins? Angela Mao or Meiko Kaji? (Thanks, Peter!)
Does it matter who the competition is? Kaji can wipe you out with a stare and a sneer.
17) Mona Lisa Vito (Marisa Tomei) or Olive Neal (Jennifer Tilly)?
I'm quite fond of them both, but Mona Lisa is memorable in every scene she has.
18) Favorite movie that features a carnival setting or sequence.
Carnival Of Souls is the first thing that jumped to mind so I'm sticking with it.
19) Best use of high-definition video on the big screen to date.
Michael Mann was the first name that came to mind and Collateral sure is purty (a good film too), but two others overtook it after some reflection: Helvetica (they made the Helvetica font look good...) and Russian Ark (a single continuous, glorious-looking 90 minute shot).
20) Favorite movie that is equal parts genre film and a deconstruction or consideration of that same genre.
Again, let's go with my first thought: Behind The Mask: The Rise Of Leslie Vernon. It's a mockumentary and a spoof of horror films that becomes its own horror film towards the end.
Monty Python's Life Of Brian makes me laugh like just about no other movie, except perhaps that scene in the same year's The In-Laws where Peter Falk is describing a great chicken sandwich. In the end though, Breaking Away is probably my favourite of that year. I can't help but think that so many of the "indie comedies" of this past decade are simply trying too hard to be this film - and they can't touch it.
22) Most realistic and/or sincere depiction of small-town life in the movies.
A Gentle Breeze In The Village is a beautiful sweet tale of a young girl transitioning from her one room schoolhouse into adulthood. I don't know how realistic this depiction of a small Japanese mountain village is, but it sure is sincere. Directed by Nobuhiro Yamashita (who also did Linda Linda Linda), I desperately hope this gets wider North American release.
23) Best horror movie creature (non-giant division).
Damn, I love those Ju-On ghosts.
24) Second-favorite Francis Ford Coppola film.
Well, if you count The Godfather I and II as a single film, then Apocalypse Now would be my second favourite. If you don't count them as one, then I cannot fulfill the requirements of this question.
25) Name a one-off movie that could have produced a franchise you would have wanted to see.
The Incredibles. How great are those characters?
26) Favorite sequence from a Brian De Palma film.
The car crash in Blow Out is pretty great, but I'd be lying if I said it was anything else but the "Odessa Steps" sequence in The Untouchables.
27) Favorite moment in three-strip Technicolor.
The ballet sequence in The Red Shoes. Wowee.
28) Favorite Alan Smithee film. (Thanks, Peter!)
I like the fact that Michael Mann credits the "altered for TV" versions of his films (at least both Heat and The Insider anyway) to Alan Smithee.
29) Crash Davis (Kevin Costner) or Morris Buttermaker (Walter Matthau)?
I like Crash Davis. I like Costner as Crash Davis. It's still not even close though.
30) Best post-Crimes and Misdemeanors Woody Allen film.
I was surprised by how much I really liked Vicky Christina Barcelona - whenever the narrator wasn't talking that is. As a matter of fact, simply due to the narrator, Husbands And Wives takes the top slot.
Helluva year. Mr. Death - The Rise And Fall Of Fred A. Leuchter Jr. is possibly my favourite documentary ever and Being John Malkovich is one of the most original films I've ever seen, but it was the year Magnolia came out and I love every one of its 188 minutes.
32) Favorite movie tag line.
I'm absolutely at a loss for anything except for "In space, no one can hear you scream".
33) Favorite B-movie western.
That Budd Boetticher set was great wasn't it? Any of those would do fine, so I'll pick Decision At Sundown.
34) Overall, the author best served by movie adaptations of her or his work.
I can't quite think of anyone else but Jane Austen right now...Pride And Prejudice, Emma and Sense And Sensibility all have very good versions (some with more than one).
35) Susan Vance (Katharine Hepburn) or Irene Bullock (Carole Lombard)?
Easiest question of the whole set - Carole Lombard. OK, Irene Bullock, but I can't quite separate the two of them. Lombard makes a whiny, spoiled, rich girl endearing and attractive. I still don't quite get Hepburn, but to be fair I'm coming off having recently seen Desk Set so that doesn't help at all.
36) Favorite musical cameo in a non-musical movie.
Jeff Beck and The Yardbirds in Antonioni's Blow-Up.
37) Bruno (the character, if you haven't seen the movie, or the film, if you have): subversive satire or purveyor of stereotyping?
I'm kinda tired of thinking about it to be honest. I've soured on the whole angle of putting people in awkward positions so that we can feel superior to them. It just lends itself to the wider issue of always looking for failure in others. Having said that, it can also be very funny at times (though I haven't seen Bruno). Therefore I end up feeling kinda hypocritical and that's why I'm tired of the question.
38) Five film folks, living or deceased, you would love to meet. (Thanks, Rick!)
The first 5 names that sprung to mind: Barbara Stanwyck, Jacques Tati, Seijun Suzuki, Jodie Foster, Martin Scorsese.
What, you think I'm going to miss an opportunity to post a screenshot of Stanwyck?
Labels: Blogging Fun
Ohhh I love reading these, but never write them (mostly because I could only answer 1/2 the questions!). Love the screen caps. One day I knock Carnival of Souls off my list!
James Yates said...
Bob, fantastic list! And "The Third Man" is NOT a boring answer; it's a brilliant film. I'd have to think about it, but "On the Town" would definitely be in the Top Five.
And you know my baseball obsessions...Kevin Costner as Crash Davis is one of my favorite casting/characters of all time. Everyone always cites his "I believe in..." monologue from "Bull Durham," but my favorite scene depicts him telling Nuke, with pure honesty, "You couldn't hit water if you fell out of a f**king boat."
Yeah, "Carnival Of Souls"! Do it! Do it!
I'm a solid fan of Bull Durham too Jamie - there's some great moments. Unfortunately, I haven't liked any of Shelton's follow ups (that I've seen anyway). Neither Tin Cup nor White Men Can't Jump had characters that were likeable.
He does have something in development called "Our Lady Of The Ballpark" though...
Sequences of a Titular Nature #1
Terror-ible, but fun...
"The Space Amoeba" - Spirit of Ed Wood Blog-a-thon
Toronto After Dark 2009 - First Wave of Announcements
New and Improved G.I. Joe Trailer
An Independence Day Blow Out


The Cobra Electronics 5200 Action Camera is designed with high definition quality for people who love adventure and makes for the perfect camping gift. There are many accessories included with this product and are as follows; waterproof housing, a spare housing backdoor, a waterproof case clip, 3 straight joint mounts, basic buckle, j-hook buckle, tripod adapter, tripod mount, flat adhesive mount, curved adhesive mount, USB cable, Velcro strap, zip ties, wire tether, a cleaning cloth and extra adhesive stickers.
Records HD video at 1080p/30fps (interpolated), 720p/30fps, 480p/60fps.
Up to 12-Megapixel photo quality (or 10MP / 8MP / 5MP / 3MP).
Auto-Looping Function: records over itself for continuous video recording.
1.5 inch LCD display screen provides ease of viewing and use.
Waterproof down to 98 feet (30 meters) for those accident prone personalities.
120°A + HD wide-angle lens.
Adjustable ISO, exposure and white balance.
Great for your next RV camping trip.
Dimensions: 1.17" x 1.61" x 2.33".
Included Accessories: Waterproof Case, Camera Holder, Camera Holder Clip, Straight Joint Mount x3, Basic Buckle Mount, Tripod Mount, Tripod Mount Adapter, J-Hook Buckle, Flat Adhesive Mount, Curved Adhesive Mount, USB Cable, Velcro Strap, Wire Tether, 3M Stickers, User Manual, Lithium-ion Battery, Spare Waterproof Backdoor, Zip Ties.