Ghana has adopted re-afforestation as one of the main activities for the Nationally Determined Contributions (NDCs) to reduce carbon emissions.
To this end, a number of young people are being engaged by government to grow trees in most parts of the country, Mr Peter Abum Sarkodie, Executive Director of the Environmental Protection Agency, (EPA) stated in Accra, on Wednesday.
The NDCs, which were at the heart of the 2015 Paris Agreement; represents efforts by each country to reduce national emissions and adapt to the impacts of climate change.
It also set out the actions that countries must plan and undertake to achieve the agreement's objectives, focused on limiting the rise in average global temperatures to well below two degree Celsius, and ideally, to 1.5 degree Celsius.
Mr Sarkodie stated at a media encounter on the outcomes of the 23rd Session of the United Nations Climate Change Conference (COP 23) held in November 2017, in Bonn, Germany.
The EPA Executive Director noted that, negotiations on ways and means of achieving ambitious emission reduction targets of Parties towards a temperature goal of two degree centigrade was the core objective of the Bonn conference.
Professor Kwabena Frimpong-Boateng, Minister of Environment, Science, Technology and Innovation as well as Climate Change Experts, Negotiators, Media, and Government Officials constituted Ghana's delegation.
Mr Sarkodie, who was also at the COP23, noted that, the Bonn conference, among other things, also sought to implement the call for climate change action that was adopted by the parties during the COP22 in Marrakesh Morocco.
On the outcomes of COP23, he said the 192 parties to the Agreement approved the taking into consideration the special circumstance of all developing countries in the implementation of the Paris Agreement.
The Parties also agreed to implement the NDCs and enhance south-south cooperation purposely to share experiences and best practices and called for the strengthening of the Warsaw International mechanism on loss and damage.
The African Group of Negotiators expressed concern about the lack of pre-2020 action, noting that, only eight per cent had ratified the Doha amendment.
Mr Sarkodie said there was the need for Ghana to increase high-level negotiations to attract climate change funds that would help finance adaption projects.
He urged the media to play a critical role in helping to educate the public on the need to reduce global emissions and save the environment from the effect of climate change.
Mr Kyekyeku Yaw Oppong Boadi, National Focal Person on Climate Change said Ghana's team, which was part of the African Group of Negotiators, negotiated on issues of adaption, mitigation, agriculture, response measures, finance, and on loss and damage.
He said the country was also working at linking its NDCs to the Sustainable Development Goals as a holistic approach to climate change adaptation.
Mr Antwi-Boasiako Amoah, a Principal Programme Officer at EPA in charge of Climate Vulnerability and Adaptation said, as part of implementation of the NDCs, the agency was engaging the assemblies through the MMDAs in capacity building programmes on climate change issues to enable them to help in the implementation of the NDCs at the grass-root level.
Dr Raymond Kasei, Lecturer at Regional Institute of Population Studies (RIPS), University of Ghana, said Government's 'One Village One Dam programme,' and the 'Planting for Food and Jobs,' should be adopted as critical NDCs.


text:
Home > Culture > Arts & Design
New book battles tiger mom syndrome with laughs
"Sh-tty Mom"
Authors: Laurie Kilmartin, Karen Moline, Alicia Ybarbo, Mary Ann Zoellner
Publisher: Abrams Image
One mom lured her daughter to bed - at age 8 - with a trail of chocolate chips up her townhouse stairs. Another found herself on the floor of a plane screaming "Save the Jelly Bellys!'' They spilled as she fed them to her preschooler to fend off in-flight earache meltdowns.
A third mom who avoids sweets and high fructose corn syrup like they were poison has her toddler begging for his "medicine candy'' when it's time to take the dreaded Tylenol.
Don't go diving for your dog-eared parenting guides or Googling Dr. Sears on the above, but if you're looking for a little validation of your crappiest mom moments as not such a huge deal after all, hunt down "Sh-tty Mom,'' the unofficial companion to last year's best-seller "Go the (Bleep) to Sleep.''
The latest in irreverent parenting books is part parody and part painful truth. It's written by four very busy, often tired working moms looking to offer some overdue laughs as counterpoint to today's parenting-to-perfection mania.
"Even the helicopter moms and tiger moms have their shitty mom moments. Maybe they missed their daughter's fifth ballet practice that week or were five minutes late to her piano lesson,'' said co-writer Mary Ann Zoellner. "The whole idea of the book is to just give yourself a break.''
Zoellner and Alicia Ybarbo, both segment producers for the "Today'' show, cooked up the idea with Karen Moline one night over margaritas about three years ago. One among them was avoiding her kids' bedtime madness back home.
The three were paired by an agent with Laurie Kilmartin, a joke master for "Conan'' and single mom to a 5-year-old in Burbank, California.
"The target audience is women who are tired of books coming out that tell you you're kind of doing everything wrong and if you only did this, this and this you would be doing a better job and your kids would turn out better,'' Kilmartin said. "I think tiger mom put everyone over the deep end, and so it was sort of a pushback.''
Written in guide format, the book includes easily consumed chapters and tips like these:
1) Send your daughter sick to day care. Not bubonic plague sick. Just sniffle sick. And maybe a little fever, something you wouldn't know about because you avoid the thermometer to preserve deniability.
2) Pretend you don't know your son in the playground after a sand-throwing incident, leaving the discipline to the victim's mom because she seems really good at it.
3) Play dead while your daughter scream-whispers "Are you awake?'' at 5 a.m., at least until you've trained her to head for daddy's side.
One chapter is titled "Stop Looking for a Great Babysitter and Settle for One Who Shows Up On Time.'' Another, "How to Deal with Moms Who Exercise,'' suggests - if you're in survival mode - limiting friendships to other "bare-minimum moms,'' avoiding those who throw their abs in your face.
There's "This Tradition Must Die: Handwritten Thank-you Notes'' and "Unspeakable Evil: Private Birthday Party - with a Bouncy Castle - at a Public Park.'' Has your son stormed the castle uninvited? "Keep a low profile until your kid gets busted. It's best not to outwardly condone grifty behavior.''
The book out in mid-September just hit The New York Times best-seller list at 12th in its category and the five co-writers are fielding supportive e-mail, including one from Alicia Hunter in Palm Beach Gardens, Florida.
She has a 9-year-old and a 6-year-old and immediately recognized herself in print. Among her favorite book bits:
"The chapter on hating the zoo/playing with my kids in general when they ask me to use my imagination,'' Hunter said in an interview, "Let's play school! Let's play house! Let's play grocery store! Unless it's a board game WE actually like, please do not ask us to play a game that involves using our imagination. We used that up by the time we were 13.''
Count Hunter, a beauty and weight-loss coach, among moms who never felt a kinship with those who read each piece of paper that comes home from school, have dry erase boards listing deadlines for homework and fulfill the teacher's wish list of supplies five minutes after said supplies are requested.
The two daughters of Jelly Belly mom Robyn Roth-Moise in Manhattan are grown now, but she bought several copies of the book for friends.
"I can remember having to go to the store and my daughter had chicken pox and there was no one to watch her,'' she said. "So I did what every mom does, wrapped her up and prayed no one noticed her. Did I feel guilty? Yes. Did I have another option? No.''
Corporate trainer Liz Wolfe in New York has an 11-year-old boy and an 8-year-old girl, the latter being the follower of chocolate chips - not, for the record, directly to bed but to the bathroom sink first to brush her teeth.
More in Arts & Design
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Coneflower and butterfly painted with DecoArt Stylin' Fabric Paints.
Housetop, night sky with full moon and stars.
Patchwork chicken nesting on watermelon slice.
Wooden Christmas spoons embellished with wooden spools and shapes.
Wooden Christmas spoons embellished with wooden spools and shapes... nutcracker, snowman, gingerbread.
German shepherd dog with blue chew toy in pastel / color pencil.
German shepherd dog in pastel pencil.
Patriotic Americana red white and blue rooster with watermelon slice and berries.
Sheep, flowers, and quilted patchwork border.
3 dimensional nativity manger scene on glass.
Bluebirds wearing earmuffs and wrapping presents.
Blocks painted with pumpkin goblin and skeleton ghoul.
3 Ornaments - Santa, sleigh, and snowman snowflake.
Bear scene with wood cutout overlay.
Moose scene with wood cutout overlay.
Eagle scene with wood cutout overlay.
Wolf scene with wood cutout overlay.
Simple snowman painted on recycled pallet board.


text:
Q: underscore.js _.where() equivalent in D3 or vanilla for some reason im geting exceptiones using underscore.js in Microstrategy's Visualization Builder. So im searching for an equivalent function to _.where. How can i rewrite the function _.where() in plain javascript or D3.js. Does anyone has an idea?
Maybe its to obvious that i can't find it :)
Thanks for help!

redpajama_set_name: RedPajamaStackExchange


text:
Strictly Dishonorable is a romantic comedy play written by Preston Sturges and first produced on Broadway in 1929.  It was adapted for the screen twice, first in 1931, then again in 1951. The play was Sturges' second Broadway production, and the first of his plays to be made into a film. The Attic Theater Company revived the show at The Flea Theater in the summer of 2014.

Broadway production
Sturges wrote the play shortly after being fired as the assistant stage manager for a road production of play called Frankie and Johnny. At liberty in Chicago, he started by writing two lines of dialogue: "What are your intentions?" "Strictly dishonorable."  He wrote the play quickly, in about six days of work, and called it originally Come, Come, Isabelle, giving the ingenue his grandmother's maiden name, Isabelle Perry.

Sturges submitted the play to producer Brock Pemberton, whom he had worked for as a stage manager, and Pemberton accepted it.  Rehearsals took three weeks and there was an additional week out of town, during which Sturges made numerous changes at Pemberton's insistence.  The opening night audience was not responsive and Sturges, waiting at the back of the house, was concerned that the comedy was not going over.  He left before the curtain came down and went out to drink, deliberately not reading the reviews.  It was not until he called Pemberton in the morning to ask for two tickets for a friend that he found out that the play was a smash hit, garnering rave reviews.

The play had opened on September 18, 1929 – although Sturges recalled it as opening on the 19th – at the Avon Theatre and ran until January 1931, logging 557 performances. It was directed by Pemberton and Antoinette Perry, after whom the Tony Awards are named.

The success of Strictly Dishonorable, which earned him over $300,000, changed Sturges' life immediately: The aura of sudden celebrity bestowed on me by Strictly Dishonorable attracted photographers, reporters, gossip columnists, professional panhandlers, producers, job offers, and a written demand from my biological father, Mr. Biden, for immediate repayment of the sums he has dispensed on my behalf when I was about a year old.  Offers came from film world, so Sturges picked up some fast money by writing film scripts. Shortly after, a play from his trunk, Recapture, went into production and opened to receive "the most violently destructive notices I had seen in years."

Broadway cast
The opening night cast of Strictly Dishonorable was:

 John Altieri as Giovanni
 Carl Anthony as Judge Dempsey
 Tullio Carminati as Count Di Ruvo
 Louis Jean Heydt as Henry Greene
 Muriel Kirkland as Isabelle Parry
 Edward J. McNamara as Patrolman Mulligan
 William Ricciardi as Tomaso Antiovi
 Marius Rogati as Mario

Cast notes:
 William Ricciardi was the only actor from the Broadway cast to play his role in the 1931 film adaptation.

See also
Preston Sturges
Strictly Dishonorable (1931 film)
Strictly Dishonorable (1951 film)

References

External links

Plays by Preston Sturges
1929 plays
Comedy plays
American plays adapted into films
Plays set in New York City

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I was offered a Key for VRobot in Sept. 2017 well now they have added the PlayStation VR version which I would like to request and I am told by the Dev. I would be accepted but I Can Not Re-Submit a Request for the new platform because I previously Rejected Their PC Key offer. Please help!!!
We are working on a system update that will let existing owners request a key on a different platform. Watch this space.
Thanks, this would be great!


The taxicabs of New York City, with their distinctive yellow paint, are a widely recognized icon of the city. There are more than 13,000 taxis operating in the city, not including over 40,000 other for-hire vehicles. Taxicabs are operated by private companies and licensed by the Taxi and Limousine Commission (TLC), a New York City government agency. "Medallion taxis," the familiar yellow cabs, are the only vehicles in the city permitted to pick up passengers in response to a street hail.
The first taxicab company in New York was the New York Taxicab Company, which in 1907 imported 600 gasoline-powered cars from France. The cars were painted red and green. Within a decade several more companies opened business and taxicabs began to proliferate. The fare was 50 cents a mile, a rate only affordable to the relatively wealthy. Previous taxis, including the one that killed Henry Bliss in 1899, were electric.
By the 1920s industrialists recognized the potential of the taxicab market. Automobile manufacturers like General Motors and the Ford Motor Company began operating fleets. The most successful manufacturer, however, was the Checkered Cab Manufacturing Company. Founded by Morris Markin, Checker Cabs produced the large yellow and black taxis that became one of the most recognizable symbols of mid-20th century urban life. For many years Checker cabs were the most popular taxis in New York City.
By the 1930s the taxicab industry in the city was large and rife with corruption. Cabbies, many of whom at the time were Irish, Italian, or Jewish immigrants, were the frequent victims of unfair labor practices while passengers were often victims of price gouging. In 1934 more than 2,000 cabbies went on strike and occupied Times Square. In response, Mayor Fiorello H. La Guardia signed the Haas Act of 1937, which introduced official taxi licenses and the medallion system that remains in place today. In the 1960s New York City experienced many of the problems of social unrest that engulfed other American cities. Crime rates increased along with racial tensions, and cabbies often illegally avoided neighborhoods of racial minorities. As a result, a quickly growing industry of private livery services emerged. Unofficial drivers were barred from picking up people on the street, but they readily found business in under-served neighborhoods. In 1967, New York City ordered all "medallion taxis" be painted yellow to help cut down on unofficial drivers and make official taxicabs more readily recognizable.
The yellow taxi had been popularized by John D. Hertz, who started the Yellow Cab Company in 1915 and which operated in a number of cities including New York. Hertz painted his cabs yellow after he read a study identifying yellow as the most visible color from long distances. In the 1970s and 1980s both the unofficial livery services and the medallion taxicab companies began finding more and more of their drivers in the growing populations of Black, Latino, and Middle Eastern immigrants to the city as the previous generation of cabbies retired and moved out of the city. Crime in New York City had become severe at this point, and cabbies were often the victims of robberies and street crime. Bulletproof partitions between the rear passenger seat and the driver became common.
By the mid-1980s and into the 1990s the demographic changes among cabbies began to accelerate as new waves of immigrants arrived in New York. Today, according to the 2000 U.S. Census, of the 42,000 cabbies in New York 82% are foreign born: 23% are from the Caribbean (the Dominican Republic and Haiti), and 20% from South Asia (India, Pakistan, and Bangladesh).
The working conditions of cabbies have changed as crime in New York has plummeted, while the cost of medallions has increased and fewer cabbies own their taxicabs than in previous times.


I built a heavily upgrade 9 1/2 that is my bike for the last 10 years. At the moment i have new suspension with ohlins back and andreani cartridges front, mass exhaust new master cylinders brembo rcs, new brembo serie oro front discs , slippery clutch and some more visual changes. I would like to know where you found the radial fork legs and if you have any recommendation for ECU mapping. Please help me built my dream bike.
Best regards from Greece for your perfect project!!!


text:
\subsection{The constant curvature solution}

Let us  recall the results of ref.~\cite{GMV}.
Including  first order corrections
in $\alpha '$, a possible form of the effective action, in the string
frame,  is~\cite{azione,FT}
\begin{equation}\label{act0}
S=-\frac{1}{2\lambda_s^{d-1}}\int d^{d+1}x\sqrt{-g}\, e^{-\phi}
\left[ R+(\nabla \phi )^2-\frac{k\alpha '}{4}R_{\mu\nu\rho\sigma}^2
\right]\, ,
\end{equation}
where $\lambda_s$ is the string length, $\lambda_s\sim\sqrt{\alpha '}$,
$k=1,1/2$ for the bosonic
and heterotic string, respectively, and we have neglected
the antisymmetric tensor field. For type II strings $k=0$, and the
first correction starts at order $R_{\mu\nu\rho\sigma}^4~$\cite{GW}.
Performing the field redefinition $g_{\mu\nu}\rightarrow g_{\mu\nu}+
(k\alpha ')\delta g_{\mu\nu}$, $\phi\rightarrow\phi +(k\alpha ')\delta\phi$ with
\begin{equation}\label{redef1}
\delta g_{\mu\nu}= R_{\mu\nu}
-\partial_{\mu}\phi\partial_{\nu}\phi+g_{\mu\nu} (\nabla\phi )^2,
\hspace{4mm}
\delta\phi =\frac{1}{4}\left[ R+(2 d-3)(\nabla\phi )^2\right]
\end{equation}
and truncating at first order in $\alpha '$ gives the
action
\begin{equation}\label{act}
S=-\frac{1}{2\lambda_s^{d-1}}\int d^{d+1}x\sqrt{-g}\, e^{-\phi}
\left[ R+(\nabla \phi )^2-\frac{k\alpha '}{4}\left( R_{GB}^2-
(\nabla\phi )^4\right)
\right]\, ,
\end{equation}
where $R_{GB}^2=R_{\mu\nu\rho\sigma}^2-4R_{\mu\nu}^2+R^2$
is the Gauss-Bonnet term. This form is
particularly convenient because higher order derivatives in the
$\alpha '$ correction cancel after integrations by part~\cite{Zwi}.
We will discuss in some detail field redefinitions in sect.~2.2.

We now restrict ourselves
to a homogeneous and isotropic {\em FRW} background,
$ds^2=N^2(t)dt^2-a^2(t)d{\bf x}^2$ and homogeneous dilaton $\phi
=\phi (t)$. Varying the action with respect to $a,\phi$, we get two
dynamical equations of motion. Introducing $H=\dot{a}/a$
and specializing to $d=3$ for notational simplicity, they read (in
units $k\alpha '=1$):
\begin{equation}\label{din1}
-6 \dot{H} (1+H^2)+2\ddot{\phi} \left( 1+\frac{3}{2}\dot{\phi}^2\right) -12
H^2+6H\dot{\phi}-\dot{\phi}^2-\frac{3}{4}\dot{\phi}^4-6H^4 +3 H\dot{\phi}^3=0
\end{equation}
\begin{equation}\label{din2}
-12\dot{H} (1-H\dot{\phi} )+6 \ddot{\phi} (1+H^2)-18 H^2+12 H\dot{\phi}
-3\dot{\phi}^2+\frac{3}{4}\dot{\phi}^4 +12H^3\dot{\phi}-6H^2\dot{\phi}^2=0\, .
\end{equation}
The variation with respect to the lapse
function $N$ gives, instead, a constraint on the initial values:
\begin{equation}\label{vin}
6H^2-6H\dot{\phi}+\dot{\phi}^2-6H^3\dot{\phi}+\frac{3}{4}\dot{\phi}^4=0\, .
\end{equation}
The constraint is conserved by the dynamical equations of motion, and
is therefore satisfied at any time if it is satisfied at the initial
time. We can now try to solve eqs.~(\ref{din1}-\ref{vin}) with the
ansatz $H=$ const $ =y$, $\dot{\phi} =$ const $=x$. The ansatz reduces the
three differential
equations to algebraic equations in $x,y$. At first sight we
have three independent equations for two unknown variables.
However, reparametrization invariance gives one relation between these
equations such that if the constraint equation and, say, the
equation obtained with a variation with respect to $a$
(or, in the general anisotropic case, the
$d$ equations obtained with a variation  with respect to $a_i$,
$i=1,\ldots\, , d$) are
satisfied, then the equation obtained with a variation with respect to
$\phi$ is automatically satisfied.

In the generic anisotropic case with scale factors $a_i$ $(i=1,\ldots
\, ,d)$,  the ansatz therefore reduces the system of $(d+2)$ differential
equations to $(d+1)$ algebraic equations  in $(d+1)$
variables $H_i=y_i,\dot{\phi} =x$, and this is true at all orders in $\alpha '$.

These algebraic equations are nothing but the requirement that there is a
zero in the beta functionals of the underlying sigma model, when the
background is specialized to the form of our ansatz,
and we will therefore write them as $\beta_i({\bf g})=0$, where
$i=1,\ldots ,(d+1)$ and ${\bf g}=(x,y_1,\ldots ,y_d)$.

It is easy to generalize this result to the case of compact
dimensions. Let us first consider  the case in which the compact space
is spatially curved, e.g. consider four-dimensional  space $M^4$ times
$S^2$, with metric
\begin{equation}
ds^2=N^2(t)dt^2-\sum_{i=1}^3a_i^2(t)dx_i^2-b^2(t)\,
(d\theta^2+\sin^2\theta d\phi^2)\, .
\end{equation}
(The extension to more general cases will be obvious.)
Writing down the equations of motion we see that now $b$
enters not only in the combination $\dot{b}/b$, but also
through terms $\sim1/b^2$, which
are due to the curvature of the
sphere. Therefore the ansatz $\dot{a_i}/a_i=$ constant, $\dot{\phi} =$ constant
can only be consistent if $b=$ const, or $\dot{b}/b=0$.
Again for the ansatz $\dot{a_i}/a_i =y_i,
\dot{\phi}=x, b={\rm const}$
the relation between equations of motion derived from
reparametrization invariance eliminates one equation and we have a
number of algebraic equations equal to the number of
variables $y_i,x,b$.

The same happens if,  instead, we compactify on a torus. In this case the
explicit dependence on the scale factor, which forces it to be
constant, comes from the winding modes, whose energy grows
with the scale factor.

In conclusion, the existence of a solution with  $H_i,\dot{\phi}$ constant
(and $b$ constant, for compact dimensions)
depends on whether the  algebraic equations discussed above have real
solutions. For the action~(\ref{act}) this is indeed the case, and the
solution turns out to be an attractor of the lowest order pre-big-bang
solution, which is therefore regularized~\cite{GMV}. However, the
inclusion of higher orders in $\alpha '$ or field redefinitions of the
type used in eq.~(\ref{redef1}) produce different algebraic equations,
which may or may not have real solutions. These issues, which were
already noted in ref.~\cite{GMV}, will be further discussed in the next
section.

\subsection{Scheme dependence of the results}
In principle, we would like to know the beta functions
$\beta_i({\bf g})$ exactly, while what
we have is a perturbative expansion  in powers of $\alpha '$.
Somewhat
optimistically, one might still hope to find a zero which, in
units $k\alpha '=1$, is located at $g_i\ll 1$, thus justifying a
perturbative treatment. Unfortunately, an even more fundamental
obstacle stands in the way. The problem is that, if we work at finite
order in $\alpha '$, the coefficients of the algebraic equation, or of
the expansion of the functions $\beta_i$, are subject to
ambiguities.
A straightforward way to understand this point
is to observe that we can perform
fields redefinitions that mix different orders in $\alpha '$.
The most general form of such redefinitions, at
order $\alpha '$, is~\cite{MT}
$g_{\mu\nu}\rightarrow g_{\mu\nu}+(k\alpha ')
\delta g_{\mu\nu}$, $\phi\rightarrow\phi +(k\alpha ')\delta\phi$, with
\begin{equation}\label{redef2}
\delta g_{\mu\nu}= a_1 R_{\mu\nu}
+a_2\partial_{\mu}\phi\partial_{\nu}\phi+a_3g_{\mu\nu} (\nabla\phi )^2
+a_4g_{\mu\nu}R+a_5g_{\mu\nu}\Box\phi\, ,
\end{equation}
\begin{equation}
\delta\phi = b_1R+b_2(\nabla\phi )^2+b_3\Box\phi\, .
\end{equation}
It is not necessary to include a term $\nabla_{\mu}\partial_{\nu}\phi$ in
eq.~(\ref{redef2}) because it can be reabsorbed by a general
coordinate transformation~\cite{MT}. After this redefinition, the new
action, truncated at order $\alpha '$, is (in units $k\alpha '=1$)
$$
S=-\int d^{d+1}x\sqrt{-g}\, e^{-\phi}
\left[ R+(\nabla \phi )^2-\frac{1}{4}R_{\mu\nu\rho\sigma}^2
+c_1R_{\mu\nu}R^{\mu\nu}+c_2R^2+c_3(\nabla\phi )^4+\right.
$$
\begin{equation}\label{act2}
\left. +c_4R^{\mu\nu}
\partial_{\mu}\phi\partial_{\nu}\phi+c_5R(\nabla\phi )^2+c_6R\Box\phi
+c_7\Box\phi(\nabla\phi )^2+c_8(\Box\phi )^2
\right]\, ,
\end{equation}
with the coefficients $c_i$ functions of $a_i,b_i$. (We have
eliminated terms that can be reduced to the above terms after
integrations by part or use of Bianchi identity). Despite the fact
that we have eight coefficients $c_i$ and eight parameters $a_1,\ldots
\, ,a_5, b_1,b_2,b_3$, we cannot fix the $c_i$ at arbitrary values since,
from the explicit expression of the $c_i$ as functions of $a_1,\ldots
\, ,b_3$, one finds that they satisfy a relation
$c_2+c_3+c_7+c_8=c_5+c_6$. Within this constraint, however, the $c_i$
can be chosen at will with the appropriate field redefinitions. The
existence of a relation between the $c_i$
means that there is a one-parameter family of field redefinitions
which, at order $\alpha '$, leaves the action invariant. It is readily
found to be
\begin{equation}
\delta g_{\mu\nu}=\zeta g_{\mu\nu}\left( R-(\nabla\phi )^2+2\Box\phi
\right)\, ,
\end{equation}
\begin{equation}
\delta\phi =\zeta\left( \frac{d-1}{2}R-\frac{d+1}{2}(\nabla\phi )^2
+d\Box\phi \right)\, ,
\end{equation}
with $\zeta$ a real parameter.
We now ask whether the existence of a zero of the functions
$\beta_i$, at a given order in $\alpha '$, is affected by the field
redefinitions, and the answer is positive. For instance, we can fix $c_1=1,
c_2=-1/4,c_3=1/4$, and vary $c_4$ (which does not enter the relation
between the $c_i$)  setting all others $c_i$ to zero.
For $c_4=0$ we have the action used in~\cite{GMV},
and there is a zero of the beta functions. In $d=3$, for the isotropic
case, it is located at $x\simeq 1.404,y\simeq 0.616$.
Increasing $c_4$ this zero disappears
(escaping at infinity in the $(x,y)$ plane) at a critical value
$c_4\simeq 0.05$.

Therefore, while we expect that, at all orders in $\alpha '$, the
existence of a zero in the beta functions is independent of  field
redefinitions, this is not true
for the truncation at any finite order in $\alpha '$.

There are other useful ways to understand the
existence of ambiguities in the
perturbative expansion of the
functions $\beta_i$, and these different points of view
are believed to be equivalent.
First, from the point of view of the underlying sigma
model, they are due to the dependence of the perturbative coefficients
of the beta functionals on the
renormalization scheme. This dependence
 starts at two loops, i.e. from the terms
$\sim R^2$ in the action, eq.~(\ref{act2}).

The effective low energy action, on the other hand, is constructed in
such a way as to reproduce the string theory $S$-matrix elements.
From this point of view, the ambiguities in the coefficients come
from the fact that some coefficients cancel in the computation of
on-shell amplitudes~\cite{DR,MT,FOTW}. Suppose for instance that we want to
fix the coefficients of the operators
$R_{\mu\nu\rho\sigma}^2,R_{\mu\nu}^2$ and $R^2$ in the effective
action. We would then expand these operators
 around the flat metric, $g_{\mu\nu}=
\eta_{\mu\nu}+h_{\mu\nu}$ and compute their contribution to the
3-point and 4-point amplitudes. However, the $\sim h^3$ part of
$\sqrt{-g}R_{\mu\nu}R^{\mu\nu}$ and of $\sqrt{-g}R^2$ vanish;
therefore, the coefficients $c_1,c_2$ in
the action~(\ref{act2}) cannot be determined from the
knowledge of the  string amplitude with three on-shell gravitons.
Both $\sqrt{-g}R_{\mu\nu}R^{\mu\nu}$ and  $\sqrt{-g}R^2$
are non-vanishing when expanded at order  $h^4$. However, the string
amplitude includes one-particle reducible graphs with graviton
exchange;  to reproduce it we must therefore sum
the contact and the exchange terms derived from the effective
action. In the sum, the coefficients
$c_1,c_2$ cancel (see~\cite{FOTW} for a detailed computation), and
so they cannot be determined from the comparison with string
amplitudes. The coefficient of $R_{\mu\nu\rho\sigma}^2$ is instead fixed by this
procedure, at the value $-1/4$.

\subsection{Relation with scale factor duality}
Scheme dependence should not change physical results if we had the exact
expression for the beta functions. It enters into play when we
truncate at finite order in $\alpha '$. When  scheme dependence
appears,  some scheme will be `better' than others, in the sense
that the results obtained at finite order in this scheme
will be closer to the exact result.  Since scale factor duality plays
an important role in motivating the cosmological model that we are
discussing, one might hope that a scheme that respects scale factor
duality at a given order in $\alpha '$ will be better, in the above
sense.

Scale factor duality has been generalized to $O(\alpha ')$
in refs.~\cite{Mei,MK}.
The analysis of ref.~\cite{Mei} is very general, and
refers to the full $O(d,d)$ symmetry.  The result of
ref.~\cite{Mei} is that there is one, and only one action invariant
under scale factor duality at order $\alpha '$, and it is given by the
action~(\ref{act2}) with $c_1=1,c_2=-1/4,
c_3=1/4,c_4=1, c_5=-1/2,c_6=0,c_7=-1/2,c_8=0$. At
the same time, the duality transformation must acquire $\alpha '$
corrections.
If we restrict to {\em FRW}
metrics and vanishing antisymmetric tensor field, then
$a_i\rightarrow 1/a_i$ (or $\log a_i\rightarrow-\log a_i$) must be generalized as
\begin{equation}\label{dua}
\log a_i\rightarrow -(\log a_i)-k\alpha '
\left(\frac{\dot{a}_i}{a_i}\right)^2\, .
\end{equation}
The transformation of the dilaton is fixed from the condition that
$\phi -\sum_i\log a_i$ is invariant.
In $d=3$, for isotropic metric, the
 equations  $\beta_i(x,y)=0$ for this dual action have  two real
solutions\footnote{Real solutions for this action
exist for any $d$. This can be proved numerically up to large values of $d$,
where the results matches with a large $d$ expansion.}
 at $(x\simeq 1.526,y\simeq 1.913)$ and at
$(x\simeq 6.201,y\simeq 1.931)$ (plus the solutions $(-x,-y)$, which
are always associated with the solution $(x,y)$).
The existence of a pair of solutions (plus the sign-reversed pair)
is related to the scale factor duality of the action. However,
for the action that we are considering,  scale factor duality is valid
only at order $\alpha '$, and not exactly, since the transformation
(\ref{dua}), when applied to the $O(\alpha ')$ terms of the action, generates
$O(\alpha '^2)$ terms that are truncated. Thus the two solutions
are not exactly dual to each other.

Integrating the equations of motion numerically, one finds~\cite{GMV}
 that these solutions do not act as a late time attractor of the
lowest order solution, which still diverges at some finite
value of time.

To further explore the relation with duality
we have tried  a different generalization
of duality at $O(\alpha ')$. In fact, the action found in ~\cite{Mei}
and the form of the duality transformation~(\ref{dua}) are uniquely
fixed only if we consider the general case of non-isotropic metrics. If
however we restrict to isotropic metrics, a transformation
\begin{equation}\label{dua2}
\log a\rightarrow -(\log a)-\lambda k\alpha '
\left(\frac{\dot{a}}{a}\right)^2\, ,
\end{equation}
with arbitrary $\lambda$, leaves the action (\ref{act2})
invariant at order  $\alpha '$ if  the coefficients $c_i$ are
$c_1=1,c_2=-1/4,c_3=1/4+(\lambda -1)/18,
c_4=1+(\lambda -1)/9, c_5=-c_4/2,c_6=0,c_7=-2c_3,c_8=0$.
(The relation between the $c_i$ is still satisfied.)
For $\lambda =1$ we recover the action
and the transformation  of ref.~\cite{Mei}. However,
for every transformation~(\ref{dua2}) with given $\lambda$ we now
have one action that is
invariant.\footnote{Basically, we have a much less rigid structure
because many expressions which are independent in the anisotropic case,
such as $\sum_iH_i^2$ and $(\sum_i H_i)^2$, collapse to the same expression in
the isotropic case, and we can arrange cancellations between them.}
In a sense, this enlarged family of transformations is
less interesting than the transformation found in~\cite{Mei}, because
the latter is the only one that can be generalized to the anisotropic
case and to the inclusion of the antisymmetric tensor field. However,
this family of transformations also includes the case $\lambda=0$,
i.e. the transformation $a\rightarrow 1/a$,
without $\alpha '$ corrections, which is appealing for its
simplicity. Furthermore, since this transformation does not generate
higher orders in $\alpha '$, it is an exact invariance of the action
(\ref{act2}), with the appropriate choice of $c_i$, rather than an
invariance at $O(\alpha ')$.

We have therefore studied the equations of motion for the action
with $\lambda =0$ duality. We have found that
the functions $\beta_i(x,y)$ have a pair of zeros (plus the sign
reversed pair), and these zeros
are related by exact duality invariance, as it should. But, again,
these solutions
do not act as late time attractors of the lowest order  solution,
which instead runs into a singularity. The type of singularity is however
different from the one encountered for the case $\lambda =1$.
It is interesting to understand in some detail
what happens. The dynamical equations
of motion can be written in matrix form
\begin{equation}
A \pmatrix{  \ddot{\phi}\cr  \dot{H}\cr}
= \pmatrix{  f_1\cr  f_2\cr}
\end{equation}
where $A$ is a $2\times 2$ matrix that depends on $H,\dot{\phi}$, and $f_1,f_2$
are functions of $H,\dot{\phi}$. At a finite value of time we find that $\det
A=0$, and therefore the equations become singular.
As a simpler example of a similar situation, consider the differential
equation $(1-h)\dot{h}=h$ for some function $h(t)$. For small $h$ the
solution grows like $e^t$, until the 'determinant' $(1-h)$ vanishes,
so the situation is similar to our case. Here however we can integrate
the equation exactly, and find $t(h)=\log h -h$. Above a critical
value  $t=-1$, this relation cannot be inverted while,
 below this value, there are
two branches $h_+(t)$ and $h_-(t)$ that merge at $t=-1$.

This suggests that what happens in our case is that the doubling of
solutions due to duality transformations with $\lambda =0$ produces
pairs of solutions of
the dynamical equations of motion, which
merge at some critical value
of time, and thereafter move into the complex plane. (These solutions
are not dual to each other. One corresponds to a small initial value of $H$
and $\dot{H}>0$, and the other to $H$ large, and $\dot{H}<0$,
while duality changes $H\rightarrow -H$). If this conjecture is correct,
then it appears that duality works against the regularization
mechanism. In any case, neither a scheme that respects
duality with $\lambda =1$ nor a scheme that respects duality with $\lambda
=0$ provides a realization of the regularization mechanism,
at $O(\alpha ')$.

\renewcommand{\theequation}{3.\arabic{equation}}
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\section{The physics behind  the regularization of the singularity}
The conclusion of the previous sections is that perturbative
$\alpha '$ corrections can in principle regularize the singularity;
however, to determine whether this actually happens we
should know the functions $\beta_i$ at all orders in $\alpha '$,
because results at finite order are scheme dependent. The attempt to
fix the scheme using scale factor duality as a guiding principle does
not give encouraging results. We  therefore ask whether there
are general physical principles that suggest that the singularity is
indeed regularized. Generally speaking, we expect that
string theory eliminates all unwanted singularities. However, is the
physical mechanism that eliminates the singularity related to
perturbative $\alpha '$ corrections? Or should we look for a different
type of corrections? In this section we   discuss this
question.

\subsection{Effective action and massive string modes}
As it is well known, the low energy action of string theory can be
obtained either by computing the beta functions of the sigma model or
from the requirement of reproducing the $S$-matrix elements. The two
methods are equivalent, as was proved in~\cite{BNY}. Still,
the physical pictures underlying the two computations are quite
different. In particular, in the first method the massive string modes
seem to play no role at all.

In fact,  what one does in order
to include massive modes at the sigma model
level  is the following. The sigma model
action is written including the interaction with the background fields
representing the massive modes~\cite{FT,Tse},
\begin{eqnarray}
S&=\frac{1}{\alpha '}
\int d^2\sigma\sqrt{-h}&\left[ g_{\mu\nu}(X)h^{ij}\partial_iX^{\mu}\partial_jX^{\nu}
+B_{\mu\nu}(X)\epsilon^{ij}\partial_iX^{\mu}\partial_jX^{\nu}+
\phi (X) ^{(2)}R+\right.\nonumber\\
& &\left. \hspace*{3mm}+
\frac{1}{\alpha '}F_{\mu\nu\rho\sigma}(X)h^{ij}h^{kl}
\partial_iX^{\mu}\partial_jX^{\nu}\partial_kX^{\rho}\partial_lX^{\sigma}
+\ldots\right]\, ,
\end{eqnarray}
where together with the metric, dilaton and antisymmetric field, we
have displayed an example of a field at the next mass level.
Once we include massive fields at the first excited level, the whole
tower of massive fields must be included, since they are associated
with non-renormalizable interactions from the point of view of the
2-dimensional theory, and all possible terms are in principle
generated by the sigma model loop (i.e. $\alpha '$) expansion.

However, when we compute higher loops in the beta
functionals of the  massless modes,
the  massive modes play no role: the operators
associated with massive modes are non-renormalizable and do not mix
with the massless sector. More generally, to renormalize background
fields at a given mass level, we need only fields at the same or lower
mass level~\cite{BFLP}.

However, since in string theory we have an infinite tower of massive
modes, while the effective low energy action is written in terms of
massless fields only, it is clear that we have somehow integrated over
the massive modes. This physical interpretation is confirmed by
the computation of the effective action through the comparison with
$S$-matrix elements.
In this case the effective action, at all orders in
$\alpha '$ but without $e^{\phi}$ corrections, is obtained by
requiring that it reproduces the
string amplitudes at genus zero, with an arbitrary number of
insertions of vertex operators for the massless fields. In the field
theory language, we expect that the $\alpha '$
corrections to the  amplitudes can be represented as a sum of
tree level exchange graphs with  massive string modes
in the intermediate state. This is
indeed the case, as  can be read from  the computation of Gross and
Witten~\cite{GW} of  graviton scattering in type II superstring
theory. In this case the tree amplitude is~\cite{GS,GW}
\begin{equation}\label{ampl}
A=\frac{G_N^2}{128}\left[
\frac{\Gamma (-s/8)\Gamma (-t/8)\Gamma (-u/8)}
{\Gamma (1+\frac{1}{8}s)\Gamma (1+\frac{1}{8}t)\Gamma (1+\frac{1}{8}u)}
\right] K(\epsilon^{(i)},k^{(j)})\tilde{K}(\epsilon^{(i)},k^{(j)})\, ,
\end{equation}
where $G_N$ is the gravitational constant, $s,t,u$ the Mandelstam
variables and $K,\tilde{K}$ are kinematical factors.

Expanding the term in brackets for small momenta,
one gets a leading term  $-2^9/(stu)$, which, when inserted back
in eq.~(\ref{ampl}), gives the tree level
scattering amplitude derived from the lowest order effective action;
and  a correction term $-2\zeta (3)$, which gives the leading
correction to the effective action; from the kinematical factors
$K,\tilde{K}$ one finds that it corresponds to a term in the effective
action quartic in the Riemann tensor~\cite{GW} (remember that for type II
superstrings the coefficient $k$ in eq.~(\ref{act0}) is zero,
i.e. there is no $\sim R_{\mu\nu\rho\sigma}^2$ correction).

Writing the Riemann zeta function as $\zeta
(3)=\sum_{n=1}^{\infty}n^{-3}$, we see that the correction can be
interpreted as a sum over an infinite tower of intermediate states,
and that states at level $n$ give a contribution $\sim 1/n^3$.

This means that the $\alpha '$ corrections are  dominated by
the first few massive string states. For instance, summing over the
first 10 excited states, we get $\sum_{n=1}^{10}1/n^3\simeq
1.1975\ldots$, to be compared with the value of the Riemann zeta
function $\zeta (3)\simeq 1.2020\ldots $.

Having obtained a physical picture of the mechanism that  is
responsible for the perturbative
$\alpha '$ corrections to the effective action
(integration over the first few massive modes), we proceed in the next
section to discuss the physical mechanism that is responsible for the
regularization of the singularity.

\subsection{Small distances vs. large energy singularities}
It can be useful to distinguish between small distance singularities
and large energy singularities. In field theory small distances means
large energies, and the distinction is therefore  meaningless. In string
theory, however, the behavior of amplitudes in high energy scattering
suggests the existence of a generalized uncertainty principle of the
form~\cite{GUP}
\begin{equation}
\Delta x\ge \frac{1}{\Delta p}+{\rm const.}G_N\Delta p\, .
\end{equation}
(Such an uncertainty principle is also suggested by  general
properties of quantum black holes~\cite{mm}.)
Therefore in string theory
the connection between high energies and small distances
is not completely trivial and it is useful to consider the two cases
separately.

The most basic
reason why we expect strings to regulate singularities at
small distances is that the very notion of invariant point-like event
is not meaningful in string theory. While in quantum
field theory an invariant event can be defined through the splitting
of a particle in two, no such concept exist for strings, since the
point in space-time where a string splits in two depends on the
Lorentz frame used~\cite{GSW}.

At large energies, the regulating mechanism is
instead the exponential growth
of the density of states. The asymptotic density of states has the
form
\begin{equation}\label{dens}
d(M)\sim\left(\frac{M}{M_0}\right)^{-b}e^{M/M_0}\, ,
\end{equation}
where $M_0=c/\sqrt{\alpha '}$ and $b,c$ are numerical constants, which
depend on the string theory under consideration. An example of how the
density of states prevents  quantities with
dimension of energy from growing  arbitrarily large    is given by
the fact that we cannot raise the temperature of a system beyond
the Hagedorn temperature, $T_{\rm HAG}=M_0$,
because otherwise the canonical partition
function diverges.

Another interesting example is given by the existence of a limiting
value of the electric field for an open bosonic string.
In the presence of an external electric field $E$,  the
rate for charged-string pair production
is in fact~\cite{BP}
\begin{equation}\label{rate}
w\sim\sum_Sq_S\sum_{k=1}^{\infty}
(-1)^k\left(\frac{|\epsilon |}{k}\right)^{(d+1)/2}
\exp\left\{ -\frac{\pi k}{|\epsilon |}(M_S^2+\epsilon^2)\right\}\, .
\end{equation}
The sum goes over all physical string states $S$, with mass $M_S$, and
$q_S$ is a factor that depends on the charge of the state $S$;
in the weak field limit, $\epsilon\simeq eE+o(E^3)$, where $e$ is the
total electric charge of the string,
and one recovers the Schwinger result. As long
as $\epsilon$ is finite, the factor $\exp (-\pi kM_S^2/|\epsilon|)$
ensures the convergence of the rate, even if the sum over the states
$S$, for large masses, is an integral with the exponentially growing
density of states~(\ref{dens}). However, at a critical value of the
electric field, $\epsilon$ goes to infinity and  the rate therefore
diverges.

The cosmological singularity is a large energy
singularity: the Hubble constant, and therefore the curvature or the
energy density, diverges on the lowest order solutions of the equations
of motion. We therefore expect that the regulating mechanism has to do
with the existence of an infinite set of massive states; in fact, we
expect something  similar to what happens in eq.~(\ref{rate}),
since the existence of a maximum electric field and of a maximum
gravitational field do not seem to be two fundamentally different
problems.

So, the first conclusion is that we should not necessarily expect that
perturbative $\alpha '$ corrections regulate the singularity. As we
discussed in sect.~3.1, these corrections can be accounted for with an
accuracy better than one per cent summing over the first ten mass
levels. The regularization of the singularity, instead, must be
crucially related to the existence of an infinite tower of massive
states. Of course, this does not mean that the mechanism proposed
in~\cite{GMV} cannot regulate the singularity. It simply means that we
do not have any  {\em a priori} argument that allows us to
confidently state that the mechanism will  operate.

The second point suggested by the above considerations is to look for
massive modes production in a gravitational field. This will be
discussed in the next section.

\renewcommand{\theequation}{4.\arabic{equation}}
\setcounter{equation}{0}
\section {Production of highly excited modes}
A time-varying gravitational field produces particles.
The general mechanism is the existence of a non-trivial Bogoliubov
transformation between the in and the out vacuum.
For instance, in De~Sitter space with Hubble constant $H$, a detector
moving on a geodetic sees a thermal bath of particles
at a temperature
\begin{equation}\label{TH}
T(H)=\frac{H}{2\pi}\, .
\end{equation}
This suggests that in De~Sitter space
a massive string mode with mass $M$ is created
with a probability  proportional  to $\exp (-M/T(H))$.
One might wonder whether a truly stringy computation of massive modes
creation could give a different answer from the field theory
result~(\ref{TH}). However, Lawrence and Martinec~\cite{LM}
(see also~\cite{Mar}) used string
field theory to
compute massive modes production in a FRW space with
\begin{equation}\label{Ceta}
ds^2=C(\eta )\left( d\eta^2-d{\vec{x}}^2\right)\, ,\hspace{10mm}
C(\eta )=A+B\tanh (\rho\eta )\, ,
\end{equation}
and they obtained the same result that holds
in this model in the field theory limit, discussed e.g. in
ref.~\cite{BD}.

The above result suggests that
the rate of energy density production due to massive modes creation
during the expansion is approximately
\begin{eqnarray}
\dot{\rho}&\simeq &\frac{1}{\lambda_s^d}
\int_{M_0}^{\infty}dM\, M d(M) e^{-2\pi M/H}\sim\nonumber\\
\label{drho}&\sim & \frac{M_0^2}{\lambda_s^d}
\int_1^{\infty} dx\, x^{1-b}e^{-\gamma x}\, ,
\end{eqnarray}
where
\begin{equation}\label{gam}
\gamma =\frac{2\pi M_0}{H}-1=\frac{2\pi c}{H\sqrt{\alpha '}}-1
\end{equation}
and $d(M)$ is the density of states, given asymptotically by
eq.~(\ref{dens}).
This formula, which has been suggested in~\cite{LM} as valid for
all the mass levels,
is also qualitatively confirmed by the explicit computation of the
Bogoliubov coefficients for the graviton production, i.e. at mass level
zero~\cite{GV,BGGV}. In this case, in fact, one finds that $|\beta_k|^2$
is a number of order 1 for energies of the order of the string mass
 (red-shifthing the frequency at the present epoch, this means
that $|\beta_k|^2=O(1)$ if the physical frequency, as seen today, is
of order 10 GHz), and has an exponential cutoff for larger
frequencies. Actually, the spectrum also shows some deviations from an
exact black body spectrum (it goes like $\omega^3$ for low frequencies
but as $\omega^{3-2\mu}$, where $\mu$ depends on details of the string
phase, for intermediate frequencies.) These deviations, however, are
not very relevant to the present discussion.\footnote{In an expanding
Universe one should also add to $\dot{\rho}$ a term $-aH\rho$
representing the dilution of string density due to the
expansion~\cite{Tur}, and $a=d-1$ for long strings.  However,
this term is not very important for the qualitative discussion
presented below, as we checked also numerically.}

Equation~(\ref{drho}) clearly has a regularizing effect on the growth of
$H$. When $H$ exceeds a critical value $H_c$, given by
$H_c/(2\pi )=M_0$, the
production rate diverges. This result can also be interpreted by noting
that $H/(2\pi )$ is the Hawking temperature in  De~Sitter
space, and therefore the condition on $H_c$ is the requirement that the
Hawking temperature does not exceed the Hagedorn temperature,
$T_{\rm  HAG}=M_0$.

The effect that we are discussing is non-perturbative in $\alpha
'$. In fact, for $\gamma >0$ the integral in eq.~(\ref{drho}) is an
incomplete gamma function, and from its known asymptotic expansion
we find that, in the limit $\alpha '\rightarrow 0$,
\begin{equation}\label{limit}
\dot{\rho}\sim\left(\frac{1}{\alpha '}\right)^{\frac{d}{2} +1}
\exp\left\{ -
\frac{2\pi c}{H\sqrt{\alpha '}}\right\}\, ,
\end{equation}
and it is therefore non-analytic in $\alpha '$ at $\alpha '= 0$.

To understand in more detail the evolution of the cosmological model,
we  tentatively  use the expression~(\ref{drho}) for the energy
density produced, as a backreaction term in the cosmological
equations.
The equations of motion, without perturbative $\alpha '$ corrections,
and including  matter sources with $T_{\mu}^{\nu}=(\rho
(t),-\delta_i^j p(t))$, are~\cite{GV,TV} (in the isotropic case, in $d$
spatial dimensions)
\begin{equation}\label{vin2}
\dot{\bar{\phi}}^2-d H^2=2\lambda_s^{d-1}e^{\bar{\phi}}\bar{\rho}
\end{equation}
\begin{equation}
\label{dyn21}2\ddot{\bar{\phi}}-\dot{\bar{\phi}}^2-d H^2=0
\end{equation}
\begin{equation}
\label{dyn22}\dot{H}-H\dot{\bar{\phi}} =\lambda_s^{d-1}e^{\bar{\phi}}\bar{p}
\end{equation}
where $\bar{\phi} =\phi -d\log a, e^{\bar{\phi}}\bar{\rho}=e^{\phi}\rho,
e^{\bar{\phi}}\bar{p}=e^{\phi}p$. We use  eq.~(\ref{drho})
for the time derivative of
$\rho$; the pressure $p$ is determined by the
conservation of the energy momentum tensor, which gives
\begin{equation}\label{cons}
\dot{\rho}+d H (\rho +p)=0\, .
\end{equation}
This is equivalent to requiring that the
constraint equation~(\ref{vin2}) is conserved by the dynamical
equations of motion~(\ref{dyn21}) and (\ref{dyn22}).
Since we know $\dot{\rho}$ rather than $\rho ,p$, these are in principle
integro-differential equations. However,
we can combine
eq.~(\ref{dyn22}) and eq.~(\ref{vin2}) so that only the combination
$\rho +p$ appears, which is expressed through $\dot{\rho}$ using
eq.~(\ref{cons}), and we are left with ordinary differential equations.
Writing everything in terms of $\phi$ rather than
$\bar{\phi}$, the dynamical equations of motion can then be written as
\begin{equation}\label{dyn3}
2\dot{H}+\dot{\phi}^2-2(d+1)H\dot{\phi} +d(d+1)H^2=-\frac{2}{d}\lambda_s^{d-1}e^{\phi}
\frac{\dot{\rho}(H)}{H}\, ,
\end{equation}
\begin{equation}
2\ddot{\phi}-2d\dot{H}-\dot{\phi}^2+2dH\dot{\phi} -d(d+1)H^2=0\, .
\end{equation}
The constraint equation
\begin{equation}
\dot{\phi}^2+d(d-1)H^2-2 d H\dot{\phi} =2\lambda_s^{d-1}e^{\phi}\rho
\end{equation}
is still an integro-differential equation. However, we impose the
constraint at the initial time $t_0\ll -1$ (in units $\lambda_s=1$),
when the right-hand side is
negligible. The constraint is then automatically preserved by the
evolution (we use its conservation as a check of the numerical
integration).  We have then
integrated  these equations numerically. The results are qualitatively
the same for any $d$ and any value of $b,c$ in
eqs.~(\ref{drho}) and (\ref{gam}) that
we have tried.  The plots
 that we present refer to $d=3$ and to the
values $b=10, c=10$ (this value of $c$ is responsible for the  values
$\sim O(10)$ of the $H$-scale in Fig.~1). Similar results holds for
$c=[ (2+\sqrt{2}) \pi ]^{-1}\simeq 0.09$, which is the value
for the heterotic string.

Figure~1 shows the behavior of the Hubble parameter versus cosmic time
$t$. At large negative values of $t$ we recover of course the lowest
order solution, since the production of massive string states is
exponentially small; $H(t)$ then reaches a maximum value and then very
abruptly bounces off and goes to zero at a finite value of $t$. At
this point the  numerical
integration stops.\footnote{The integration stops because the integral
in eq.~(\ref{drho}) is only defined for $\gamma >0$. For negative
values of $H$ we could replace $H$ is eq.~(\ref{gam}) with
$|H|$. This is motivated by the fact that, for instance,
in the model defined in
eq.~(\ref{Ceta}) the production rate is the same independently of
whether we consider a model with a scale factor increasing in time or
decreasing in time with the same rate. However, as we will now
discuss, a more crucial modification of our equations is needed.}
The behavior of $\dot{\phi}$ is shown in fig.~2,
while fig.~3 shows $\dot{\rho} (t)$.

These plots show that some important aspects of the regularization
mechanism are still missing from our equations. In fact, even if $H$
does not diverge, $\dot{H}$ becomes very large
when $H$ goes to zero, and the Ricci scalar ${\cal R}=-6(\dot{H}+
2H^2)$ in this regime is dominated by $\dot{H}$ and is large.
The missing ingredient can be traced back to eq.~(\ref{TH}). This form
of the temperature holds for  De~Sitter space, where $H$ is constant
and $\dot{H}=0$. It is inappropriate at a stage of the cosmological
evolution when $H$ is small compared to
the critical value but $\dot{H}$ is
large. Indeed, at this stage our equations predict no massive mode
creation, while a large $\dot{H}$, as any strong time-varying
gravitational field, will produce particles, much the same as a large
$H$. This is quite clear physically, but it is anyway
interesting to check this point in a toy model. One can consider for
instance a model with a scale factor such that $H=A/\cosh
(\omega t)$, with $A,\omega >0$.
If $A\ll 1$ but $A\omega\gg 1$ the Hubble constant is
always small during the evolution while $\dot{H}$ reaches a maximum
value $\dot{H}_{\rm max}=
A\omega/\sqrt{2}\gg 1$. A straightforward computation gives a
production rate of massive particles $\sim\exp (-M/T)$ with
$T\simeq \dot{H}_{\rm max}$.

So, at least at the level of phenomenological description of the
string phase, one should generalize eq.~(\ref{TH}) to
\begin{equation}\label{TH2}
T=T(H,\dot{H},\ddot{H},\ldots,\dot{\phi},\ddot{\phi},\ldots )\, ,
\end{equation}
where all higher order derivatives of $H$  appear. Note that we
also included a dependence on $\dot{\phi}$ because, changing frame,
say from the string to the Einstein frame, the derivative of the dilaton
field contributes to the gravitational field, and therefore in a
generic frame even $\dot{\phi}$ must contribute to massive modes
production. For physical reasons, we expect that $T$ will be a sum
of positive definite contributions. For instance, a possible plausible
form could be
\begin{equation}\label{TH3}
T=\frac{1}{2\pi}\left( H^2+c_1\dot{H}^2+c_2\dot{\phi}^2+\ldots
\right)^{1/2}\, ,
\end{equation}
with $c_1,c_2>0$.
Now it becomes technically impossible to solve the differential
equations, since already including a term $\dot{H}$ into $T$, the
highest derivative of $H$ in the differential equations appears as
argument of the incomplete gamma functions, which makes the equations
intractable, even numerically. However, the qualitative effects of
eq.~(\ref{dyn3}), combined with an expression of the form of
eq.~(\ref{TH2},\ref{TH3})
for the temperature,  can be understood as follows.

When any of the derivative of $H$  or $\phi$ exceeds a critical
value of order one in units $\lambda_s=1$,
the right-hand side of eq.~(\ref{dyn3})
diverges, while the left-hand side is finite. So there can be no
solution where $H,\dot{H},\dot{\phi},\ldots$ are larger than their
critical value. Similarly to what happens to $H$ in fig.~1, when
$\dot{H},\dot{\phi},$ etc. approach their respective
critical values, the effect
of the term $\dot{\rho}$ in eq.~(\ref{dyn3}) is to repel them away.
Furthermore, $H$ cannot go to zero. Indeed, unless all
higher derivatives $\dot{H},\dot{\phi},\ldots$ vanish (as it happens
at $t\rightarrow -\infty$ if the evolution starts from the string perturbative
vacuum), then $\dot{\rho}(H,\dot{H},\dot{\phi},\ldots )$
does not vanish for $H\rightarrow 0$, contrarily
to what happens when we use the expression $T(H)=H/2\pi$, see
eq.~(\ref{limit}). Therefore the right-hand side of eq.~(\ref{dyn3})
diverges if $H\rightarrow 0$ while the left-hand side is finite, because
$\dot{H},\dot{\phi}$ are bounded. Therefore, there can be no solution
with $H$ approaching zero.
So, we expect a solution in which the scale factor in the string frame
is always expanding, $H>0$, possibly in a very complicated way, but
still without divergences in the curvature or in any other
quantity.\footnote{We stress that this is not a proof that there will
be a non-singular solution. In principle it may happen that at some
value of $t$ two different solutions `collide' and move into the
complex plane, as we have seen in sect.~2.3 for the equation with
perturbative $\alpha '$ corrections.}

The results obtained trying to incorporate  massive mode
production as a backreaction on the metric and dilaton field show
that massive modes may have a very important effect on the
evolution of the cosmological model.
However, the very complexity of the equations that describe the
backreaction of massive modes on the massless modes $\phi, g_{\mu\nu}$
indicates that a treatement as backreaction is not
appropriate. If we enter
a full stringy regime, with an infinite tower of excited
massive fields,  it is not surprising that a
description of the cosmological model in terms of classical evolution
of  massless fields runs into difficulties.

Finally we stress that, even if our results allow to obtain some
understanding of the physical mechanism that regularizes the
singularity of the lowest order solution, still this mechanism
cannot be the full
story, since it does not provide a solution that, besides being possibly
non-singular, is also smoothly connected with the present radiation
dominated phase. This is the so-called graceful exit
problem~\cite{gra1,gra2,BM,BM2}. It can be posed as follows. From
eq.~(\ref{vin2}) one sees that there are two solutions for
$\dot{\bar{\phi}}$, given by
\begin{equation}\label{branch}
\dot{\bar{\phi}}=\pm\left(
dH^2+2\lambda_s^{d-1}e^{\phi}\rho\right)^{1/2}\, .
\end{equation}
These two solutions are usually referred to as the $(+)$ and $(-)$
branches, depending on the choice of sign in front of the square
root. The pre-big-bang solution starts on the $(+)$ branch; this
ensures that the solution start from a regime of low curvature and
couplings, and  goes
through a phase of inflation driven by the kinetic energy of the
dilaton field. The final state, however, should correspond to a
radiation dominated FRW model with constant dilaton, and therefore
$\dot{\bar{\phi}}=\dot{\phi}-dH<0$, and we are on the $(-)$
branch. Therefore at least one branch change must take
place. Eq.~(\ref{branch}) shows that a necessary condition for this to
happen is that, at some point in the evolution, the matter density
$\rho$, in the string frame, be negative. Further global conditions
are necessary to ensure that no further branch change back to the
$(+)$ branch takes place, and have been discussed in
refs.~\cite{BM,BM2}. However, we see that, if we include massive
modes as the only source of energy density, $\rho$ is necessarily
positive and no branch change can occur. Other source of corrections,
like $\alpha '$ correction, provide an effective contribution to
$\rho$ which is negative and therefore allows branch
changes. Still, to satisfy the global conditions for a succesful exit
completion, $e^{\phi}$ corrections must also be included. An
example in which the combined effect of $\alpha '$ corrections and
$e^{\phi}$ corrections provides a successful exit has been discussed
in~\cite{BM2}. So, it is clear that the effect that we have studied in
this paper is only an ingredient
of the mechanism that finally can produce a
well-motivated and non-singular cosmological model, that eventually
matches standard cosmology.

\renewcommand{\theequation}{5.\arabic{equation}}
\setcounter{equation}{0}
\section {Conclusions}
The lowest-order cosmological solutions obtained from the string
effective action are singular, and therefore corrections to the lowest
order action must play  a crucial role.
In string theory there are two kind of corrections, `stringy
corrections', parametrized by $\alpha '$, and loop corrections,
parametrized by  $e^{\phi}$. In this paper we have drawn attention to
the fact that $\alpha '$ corrections come into two types, perturbative
and non-perturbative, and that they have a distinct physical
origin. Perturbative corrections are related to the integration over
the first few massive modes while non-perturbative corrections comes
from the exponentially growing density of massive states.  As we
discussed in sect.~3, this
physical interpretation makes clear that non-perturbative $\alpha '$
corrections are expected to have a regularizing effect.

The explicit verification of the regularization mechanism meets
difficulties both for pertubative and non-perturbative
$\alpha '$ corrections. In the former case, a truncation at any finite
order in perturbation theory gives scheme-dependent results, so that
an all-order computation is in principle required;
since an all-order perturbative computation is anyway faced with the
problem of the convergence and resummation
of the asymptotic series, this is already a
hint of the fact that the full answer requires the inclusion
of non-perturbative effects.

In the case of non-perturbative corrections related to massive modes
production, the technical difficulty that we have
to face is the inclusion of the effect of all higher order derivative
of $H$ and $\phi$ in the computation of the production rate of massive
modes. However, independently of the technical difficulties,
the mechanism has a clear physical interpretation, which
suggests that it  regularizes the lowest order solution.

We conclude by pointing out briefly a possible phenomenological
consequence of the  scenario in which in the string phase we have
highly excited string modes.
The massive string states produced will eventually decay.
The decay of the excited string levels has been studied  in
ref.~\cite{WTM}. The result is that, for large level number $N$,
the  width is dominated  by
decays in which one of the products is massless, i.e. by transitions
of the form $N\rightarrow N-1$ with emission of a graviton or another massless
particle.  From the mass formula $M_N^2\sim N$,
the level spacing is $\Delta M_N\sim 1/M_N$. If a string
at an excited level $N$ decays at level  $N-1$
with emission of a graviton of frequency $\omega=\Delta M_N$, we get
transitions at the frequencies $f_N=f_0/\sqrt{N}$. The
most naive estimate of the frequency $f_0$, as seen today,
is obtained by red-shifting
the string mass from a string time to the present epoch, which
gives  $f_0=\omega_{\rm today}/(2\pi )$
 of the order of $10-10^2$ GHz, which is also the typical cutoff
value of the gravitons produced by quantum vacuum
fluctuations~\cite{GV,BGGV,AB}.
However, this radiation is  produced
within the string phase, rather than at the transition between the
string phase and the ${\em FRW}$ phase,
and a further red-shift must be included, to
take into account the expansion from the time of production to the
onset of a standard {\em FRW} cosmology. We
cannot compute it, in the absence of a knowledge of the mechanism that
terminates the string phase. There is, however, the interesting
possibility that, because of this further redshift and/or the factor
$1/\sqrt{N}$ in $f_N$,
 this gravitational radiation might be shifted  in
the frequency band accessible to the LIGO/Virgo gravitational
wave detectors, which are expected to operate in the 6~Hz--1~kHz region.

Concerning the intensity of this gravitational
radiation, it can be estimated as follows.
If there is not a subsequenty inflationary phase below the string
phase, also the photons that we observe today in the CMBR have  been
produced during the large curvature phase of the model, by
massive modes decay. Since at this scale the rates for photon and
graviton production are comparable, this suggests that the overall
intensity of the relic gravitational wave background produced by the
decay of massive string modes has an energy density comparable to the
energy density of the 2.7~K radiation.

\vskip 2 cm

\section*{Acknowledgements}
I am grateful to Alessandra Buonanno, Stefano Foffa,
Krzysztof Meissner, Riccardo Sturani,
Carlo Ungarelli and Gabriele Veneziano for very useful discussions. I
am especially grateful to  Jorge Russo for invaluable discussions.
I also thank the referee for stimulating comments.

\newpage

redpajama_set_name: RedPajamaArXiv


text:
Arts Roundup
The "Newport Jazz Festival: Now 60" tour comes to town boasting a great lineup of jazz stars.
By Isabel Dunkerley	- Mar 24th, 2014 12:51 pm
In 1954, a music event was started which at the time was called "First Annual American Jazz Festival," and was held in Newport, Rhode Island. Socialite Elaine Lorillard and her husband Louis Lorillard had hired jazz impresario George Wein to organize what became the first outdoor jazz festival ever held in the U.S. It grew into a celebrated annual event, the "Newport Jazz Festival," which was broadcast on Voice Of America radio and had some of its performances recorded and issued by record labels. The festival is held at Fort Adams State Park, often during the same month as its sister festival, the Newport Folk Festival.
Newport Jazz Festival's talented ensemble will be performing at the Sharon Lynne Wilson Center for the Arts on March 29.
Wein, now 88, still produces the jazz festival and continues to hear live jazz in the clubs of NYC on a regular basis and has focused the programming of the festival towards the young composers and musicians who perform nightly in those venues. This year, in celebration of Newport Jazz Festival's 60th Anniversary, a handpicked group of musicians are doing on a 17-city tour across the U.S., with more dates across North America and Europe being added this summer. Clarinetist and saxophonist Anat Cohen leads the all-star, multigenerational ensemble featuring vocalist/pianist Karrin Allyson, bassist Larry Grenadier, pianist Peter Martin, guitarist Mark Whitfield, trumpeter Randy Brecker, and drummer Clarence Penn. Collectively, these musicians have recorded hundreds of albums with a vast range of styles. This concert should offer an entertaining update on the state of an original American art form, jazz.
Sharon Lynne Wilson Center for the Arts: Newport Jazz Festival: Now 60! Saturday, March 29 at 8 p.m. Tickets are $35, available online or call (262) 781-9520.
Milwaukee Symphony Orchestra: Beethoven: "Eroica" Symphony March 28-30 at The Pabst Theater. Tickets range from $21.25-$86.25, available online or call (414) 291-7605.
Florentine Opera: Julius Caesar March 28 & 30 at the Marcus Center for the Performing Arts. Tickets are $39.15-$135.15, available online or call 414-291-5700 ext. 224.
Milwaukee Youth Symphony Orchestra: Davidson Chamber Ensemble Recitals: March 30 at 1 p.m., 3 p.m., 5 p.m. and 7 p.m. at the Milwaukee Youth Arts Center. Tickets are $12, $10 for students and seniors.
UWM's Peck School of the Arts presents Fine Arts Quartet Sunday, March 30 at 3 p.m. at Helen Bader Concert Hall. Tickets are $10, $5 for students, available online or call (414) 229-4308.
UWM's Peck School of the Arts: MKE Unplugged Mike Mangione and the Union with opening act Macyn Taylor, Thursday March 27 in room 250 at the Helene Zelazo Center for the Performing Arts. Opening act at 7:30 p.m., headliner at 8 p.m. Admission is free.
UWM's Peck School of the Arts: Unruly Music at UWM Music Recital Hall March 27-29. Tickets are $10, $5 for students, free for majors. For tickets call the Peck School of the Arts box office at (414) 229-4308. https://www4.uwm.edu/psoa/music/unrulymusic/
Italian Community Center: Cure for Cabin Fever Series presents Bob Hirschi & Groove Therapy Thursday, March 27 at 6:30 p.m. Free admission and parking for guests.
Mitchell Park Conservatory (The Domes) "Music Under Glass" series presents The Noyz Boyz and Gyrlz, 6:30 to 9 p.m. on Thursday, March 27. Tickets are $6.50, $5 for seniors/students.
Turner Hall Ballroom: James Vincent McMorrow with special guest Aidan Knight Friday, March 28. Doors open at 7 p.m. and tickets are $15, available online or call (414) 286-3663.
Off The Wall Theatre: Romeo & Juliet March 27 – April 6. Tickets are $25, available online or call (414) 484-8874.
Milwaukee Repertory Theater: Rep Lab at Stiemke Studio March 28-31. Tickets are $10, available online or call (414) 224-9490.
First Stage: CRASH, March 28-April 13. Tickets range from $12-$32, available online or call (414) 267-2961.
Schauer Center presents Cinderella & Other Tales by New York Theatre Ballet Friday March 28 at 7:30 p.m. Tickets range from $23-$27 for adults, $10 for students, available online or call (262) 670-0560, ext. 3.
Milwaukee Art Museum: Art in Bloom: A Tribute to Art and Flowers, March 27-30. Admission is $5 for members, $17 for adults, $14 for seniors and students with ID, free for kids 12 and younger.
Museum of Wisconsin Art: Handmade for Hard Times exhibition, March 28-June 15. One time member fees for admission start at $12 and are good for one full year.
Skylight Music Theatre presents Taste of the Ward 2014 Thursday, March 27 from 5:30 to 7:30 p.m. Tickets are $45, $30 for young professionals ages 21-45, available online.
South Milwaukee Performing Arts Center: An Evening with John McGivern on Saturday, March 29 at 7:30 p.m. Tickets range from $27-$37, available online or call (414) 766-5049. All proceeds benefit St. Matthew School & Parish.
Charles Allis Art Museum: Prospect Avenue Mansions Lecture Thursday, March 27 at 6:30 p.m. Tickets are $15 and can be purchased in advance online. Remaining tickets will be available the night of the lecture.
Skylight Music Theatre: Hydrogen Jukebox, through March 30
Waukesha Civic Theatre: Curtains, through March 30
Lynden Sculpture Garden: The Mingei Tradition in the Midwest: Warren MacKenzie and Beyond, through March 30
Sunset Playhouse: Joseph and the Amazing Technicolor Dreamcoat, through April 6
Tory Folliard Gallery:"The Figure in Clay" and "Concerning Nature," through April 19
Museum of Wisconsin Art: Beth Lipman: Precarious Possessions, through April 13
Portrait Society: Ceramic exhibition, through April 12
Fireside Dinner Theatre: Mary Poppins, through April 20
Walker's Point Center for the Arts: Featured Member Exhibition: Paul Matzner, through April 5
Wisconsin Lutheran College Schlueter Art Gallery: Pfister Artist in Residence Exhibition, through April 11
Milwaukee Repertory Theater: Ain't Misbehavin', through May 18
Charles Allis Art Museum: FORWARD: A Survey of Wisconsin Art Now, through June 29
Categories: Art, Arts & Entertainment, Classical, Dance, Dining, Guides, Milwaukee This Week, Museums & Galleries, Music, Performance Venues, Theater


text:
The Cornell Daily Sun (https://cornellsun.com/2018/03/30/editorial-its-time-to-reform-student-assembly-elections/)
EDITORIAL | It's Time to Reform Student Assembly Elections
By The Cornell Daily Sun | March 30, 2018
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Forty-eight hours after polls closed in this year's Student Assembly elections, the student body is no closer to knowing just who will represent them for the next twelve months. And now, The Sun has learned that results may not be public until as late as after Spring Break.
The reason for this extended protraction is a challenge to the campaign of presidential candidate Varun Devatha '19. Devatha was disqualified from the election late Wednesday for using a Cornell University logo in campaign materials, a violation of Article I, Section B, Rule 5 of the Student Assembly Election Rules, but has since appealed his disqualification.
The letter of the rules is clear: the use of the logo is prohibited. It is, in fact, so important to the elections committee that it is stated twice. If Devatha did indeed use the logo as is alleged, he is in violation. But whatever the outcome of his final appeal, the entire episode raises serious questions about the S.A. elections process.
Under the current rules, there is no method of reprimand other than an effective death sentence for a campaign. This all-or-nothing approach can produce an incredibly undemocratic result, by which the will of the voters can be overturned by an unelected committee operating in a confidential setting. Last semester, a candidate for freshman representative received enough votes to be seated, but was disqualified for reasons the elections committee refused to disclose.
Although any member of the Cornell community may request a review of settled challenges, photographs, copies and transcriptions are prohibited, and the review must be held in-person at Day Hall, rendering any "accessibility" virtually moot. It is similarly egregious that the final reports on challenges — the rationale for disqualifications — are kept totally confidential. Decisions of such import should be made in the open, not behind closed doors.
While we do not know (and may never know) whether Devatha or his opponent, Dale Barbaria '19, received the most votes, the possibility that a misused logo could invalidate the choice of thousands of students is deeply disturbing.
Ex post facto disqualifications are undemocratic and should be eliminated. To avoid this situation in the future, the Student Assembly should consider instituting a campaign blackout period before polls open, as many countries across the world do. This period of enforced silence — perhaps 48 hours — in which candidates and their supporters could not actively campaign, would allow the elections committee to review any challenges and publicly announce any disqualifications before voting commences, thus preventing students from wasting their votes.
The fate of Devatha's campaign is in the hands of the elections committee now. Regardless of the outcome of the appeal, we have seen how clearly flawed the current system is, and whomever sits on the assembly in the coming year should take action to correct those flaws. While not a cure-all, a campaign blackout period would be a good place to start.
Everything You Need to Know About Voting in the 2018 Midterm Elections
By Amanda H. Cronin October 21, 2018
November 6 is fast approaching, but you probably still have lots of questions on the when's and where's and who's and how's. The Sun has put together some information to help you answer those questions.
EDITORIAL | Inform the Public, Release the Report
By The Cornell Daily Sun April 12, 2018
The Elections Committee's refusal to comply with the ruling of the judicial codes counselor in the Varun Devatha '19 disqualification case is direly unfortunate and demonstrates an embarrassing lack of regard for reason on the part of our student government. Furthermore, it represents an impractical and dangerous seizure of power by a small, unelected council, and the public statement from the members of the committee does little to inspire any confidence in that body's decision to uphold Devatha's disqualification for violating election rules regulating promotional materials.


text:
MLB Rumors: Talks Have Stopped Between Orioles and Mark Trumbo
Posted on December 18, 2016 by Bryan Zarpentine
Image via foxsports.com
Just when it looked like the Baltimore Orioles were ready to turn their complete focus to re-signing free agent Mark Trumbo, negotiations appear to have reached an impasse. Earlier this week, it was believed that Trumbo and the Orioles had agreed on the framework of a four-year deal to keep last season's home run champion in Baltimore. However, a report that surfaced Saturday night states that the offer has been taken off the table, as talks between the two sides have stalled.
Trumbo is arguably the second best free agent left on the market behind Edwin Encarnacion, who is also having trouble finding a home. After leading the majors with 47 home runs last season following a trade to Baltimore from the Mariners last offseason, it's been surprising to see Trumbo receive so little interest on the free agent market this winter. However, Trumbo's numbers in 2014 and 2015 were nowhere close to what he did last season, so some teams may view him as a one-season wonder.
While both Trumbo and the Orioles seemed willing to settle on a four-year deal, there was reportedly a large gap in money between the two sides. Reports indicate that Baltimore's ceiling on a four-year deal was around $55 million, which is far below what Trumbo and his representatives were expecting to receive. With the sides so far apart on money, the Orioles pulled the offer with there being little traction in a deal getting done any time soon.
Of course, the Orioles have used this negotiating tactic before, most notably last season with Chris Davis. The team intentionally backed off from Davis as a way to pull him back to the bargaining table, and the two sides ultimately agreed to a seven-year deal. However, Trumbo has spent just one season in Baltimore, so he's not as attached to the organization as a player like Davis, which means this move may not be as effective.
Trumbo has also received interest from several other teams this offseason. The Indians, Mariners, Rangers, and Rockies are the teams with a known interest in Trumbo. With the Orioles backing off, Trumbo's representatives may look to accelerate talks with one of the other teams. However, the Rockies have already spent big on Ian Desmond, the Rangers would prefer not to sign a big-money free agent this winter, and the Indians and Mariners are a couple of teams with limited payroll flexibility to make Trumbo an offer far beyond what the Orioles had on the table.
Nevertheless, with several potential suitors, Trumbo may not need the Orioles as much as they need him. Baltimore is poised to lose both Trumbo and Matt Wieters from a team that relies on hitting the long ball. The Orioles have added Welington Castillo behind the plate, making Wieters unlikely to return, and so it would only make sense for Baltimore to use its remaining resources to secure the return of Trumbo. However, things have not played out that way, and after the team's four-year offer was pulled, Trumbo's return to the Orioles is in serious jeopardy.