These are the slides of the talk I gave on discriminating between models of inflation using space based gravitational wave detectors, at KEK in Tskuba University, Japan.
Second Order Perturbations During Inflation Beyond Slow-rollIan Huston
This is a talk I gave at the University of Sussex in June 2011. It outlines the newly released numerical code Pyflation and the results published in arXiv:1103.0912.
International Conference on Monte Carlo techniques
Closing conference of thematic cycle
Paris July 5-8th 2016
Campus les Cordeliers
Slides of Richard Everitt's presentation
International Conference on Monte Carlo techniques
Closing conference of thematic cycle
Paris July 5-8th 2016
Campus les cordeliers
Jere Koskela's slides
Detailing Coherent, Minimum Uncertainty States of Gravitons, as Semi Classical Components of Gravity Waves, and How Squeezed States Affect Upper Limits to Graviton Mass /• We present what is relevant to squeezed states of initial space time and how that affects both the composition of relic GW and also gravitons. A side issue to consider is if gravitons can be configured as semi classical “particles”, which is akin to the Pilot particles model of Quantum Mechanics as embedded in a larger non linear “deterministic” background.
First Presented Saturday, September 3, 2011 at the G999 Conference, Philadelphia, PA http://ggg999.org/
Next Presentation : Friday, September 9, SAN MARINO WORKSHOP ON ASTROPHYSICS AND COSMOLOGY
FOR MATTER AND ANTIMATTER
http://www.workshops-hadronic-mechanics.org/
San Marino, N. Italy
Discussion: The Nature of Semi-classical Nature of Gravity Reviewed; And Can We Use a Graviton Entanglement Version of the EPR System to Answer if The Graviton is Classical or Quantum in Origin?
Publisher’s note: Dr. Beckwith: I am honored that you have seen fit to acknowledge me in your presentations of today, September 3, 2011 at the G999 Conference in Philadelphia, and Friday, September 9, at the San Marino Workshop on Astrophysics and Cosmology For Matter and Antimatter, in San Marino, N. Italy.
In my view, these are rather extraordinary postulations, especially the probability of extra-universal black hole gravitational origins, along with expansion beyond Hawking of Quantum Wave theory and the “quantizing” of gravity. The latter may very well lead to a thorough reexamination of our concept of space and time, and that the latter might not be so unidirectional afterall. The potential is breathtaking, as it represents steps forward in proving the existence of subspace, the possibility of concomitant multi-universiality and 5D dimensionality of black holes.
Can time manipulation and Biefeld-Brown suggested electro-gravitics as a spacecraft propulsive methodology be far behind? I think not…
Second Order Perturbations During Inflation Beyond Slow-rollIan Huston
This is a talk I gave at the University of Sussex in June 2011. It outlines the newly released numerical code Pyflation and the results published in arXiv:1103.0912.
International Conference on Monte Carlo techniques
Closing conference of thematic cycle
Paris July 5-8th 2016
Campus les Cordeliers
Slides of Richard Everitt's presentation
International Conference on Monte Carlo techniques
Closing conference of thematic cycle
Paris July 5-8th 2016
Campus les cordeliers
Jere Koskela's slides
Detailing Coherent, Minimum Uncertainty States of Gravitons, as Semi Classical Components of Gravity Waves, and How Squeezed States Affect Upper Limits to Graviton Mass /• We present what is relevant to squeezed states of initial space time and how that affects both the composition of relic GW and also gravitons. A side issue to consider is if gravitons can be configured as semi classical “particles”, which is akin to the Pilot particles model of Quantum Mechanics as embedded in a larger non linear “deterministic” background.
First Presented Saturday, September 3, 2011 at the G999 Conference, Philadelphia, PA http://ggg999.org/
Next Presentation : Friday, September 9, SAN MARINO WORKSHOP ON ASTROPHYSICS AND COSMOLOGY
FOR MATTER AND ANTIMATTER
http://www.workshops-hadronic-mechanics.org/
San Marino, N. Italy
Discussion: The Nature of Semi-classical Nature of Gravity Reviewed; And Can We Use a Graviton Entanglement Version of the EPR System to Answer if The Graviton is Classical or Quantum in Origin?
Publisher’s note: Dr. Beckwith: I am honored that you have seen fit to acknowledge me in your presentations of today, September 3, 2011 at the G999 Conference in Philadelphia, and Friday, September 9, at the San Marino Workshop on Astrophysics and Cosmology For Matter and Antimatter, in San Marino, N. Italy.
In my view, these are rather extraordinary postulations, especially the probability of extra-universal black hole gravitational origins, along with expansion beyond Hawking of Quantum Wave theory and the “quantizing” of gravity. The latter may very well lead to a thorough reexamination of our concept of space and time, and that the latter might not be so unidirectional afterall. The potential is breathtaking, as it represents steps forward in proving the existence of subspace, the possibility of concomitant multi-universiality and 5D dimensionality of black holes.
Can time manipulation and Biefeld-Brown suggested electro-gravitics as a spacecraft propulsive methodology be far behind? I think not…
Reflection and Transmission of Thermo-Viscoelastic Plane Waves at Liquid-Soli...IDES Editor
The present paper is aimed at to study the reflection and transmission characteristics of plane waves at liquid-solid interface. The liquid is chosen to be inviscid and the solid
half-space is homogeneous isotropic, thermally conducting viscoelastic. Both classical (coupled) and non-classical (generalized) theories of linear thermo-viscoelasticity have been employed to investigate the characteristics of reflected and transmitted waves. Reflection and transmission coefficients are obtained for quasi-longitudinal ( qP ) wave. The numerical computations of reflection and transmission coefficients are carried out for water-copper structure with the help of Gauss-elimination by using MATLAB software and the results have been presented graphically.
WAVELET-PACKET-BASED ADAPTIVE ALGORITHM FOR SPARSE IMPULSE RESPONSE IDENTIFI...bermudez_jcm
Presented at IEEE ICASSP-2007:
This paper proposes a wavelet-packet-based (WPB) algorithm for efficient identification of sparse impulse responses with arbitrary frequency spectra. The discrete wavelet packet transform (DWPT) is adaptively tailored to the energy distribution of the unknown system\'s response spectrum. The new algorithm leads to a reduced number of active coefficients and to a reduced computational complexity, when compared to competing wavelet-based algorithms. Simulation results illustrate the applicability of the proposed algorithm.
The standard Galerkin formulation of the acoustic wave propagation, governed by the Helmholtz partial differential equation (PDE), is indefinite for large wavenumbers. However, the Helmholtz PDE is in general not indefinite. The lack of coercivity (indefiniteness) is one of the major difficulties for approximation and simulation of heterogeneous media wave propagation models, including application to stochastic wave propagation Quasi Monte Carlo (QMC) analysis. We will present a new class of sign-definite continuous and discrete preconditioned FEM Helmholtz wave propagation models.
Neuron-computer interface in Dynamic-Clamp experimentsSSA KPI
AACIMP 2010 Summer School lecture by Anton Chizhov. "Physics, Chemistry and Living Systems" stream. "Neuron-Computer Interface in Dynamic-Clamp Experiments. Models of Neuronal Populations and Visual Cortex" course. Part 1.
More info at http://summerschool.ssa.org.ua
This a short presentation for a 15 minutes talk at Bayesian Inference for Stochastic Processes 7, on the SMC^2 algorithm.
http://arxiv.org/abs/1101.1528
Reflection and Transmission of Thermo-Viscoelastic Plane Waves at Liquid-Soli...IDES Editor
The present paper is aimed at to study the reflection and transmission characteristics of plane waves at liquid-solid interface. The liquid is chosen to be inviscid and the solid
half-space is homogeneous isotropic, thermally conducting viscoelastic. Both classical (coupled) and non-classical (generalized) theories of linear thermo-viscoelasticity have been employed to investigate the characteristics of reflected and transmitted waves. Reflection and transmission coefficients are obtained for quasi-longitudinal ( qP ) wave. The numerical computations of reflection and transmission coefficients are carried out for water-copper structure with the help of Gauss-elimination by using MATLAB software and the results have been presented graphically.
WAVELET-PACKET-BASED ADAPTIVE ALGORITHM FOR SPARSE IMPULSE RESPONSE IDENTIFI...bermudez_jcm
Presented at IEEE ICASSP-2007:
This paper proposes a wavelet-packet-based (WPB) algorithm for efficient identification of sparse impulse responses with arbitrary frequency spectra. The discrete wavelet packet transform (DWPT) is adaptively tailored to the energy distribution of the unknown system\'s response spectrum. The new algorithm leads to a reduced number of active coefficients and to a reduced computational complexity, when compared to competing wavelet-based algorithms. Simulation results illustrate the applicability of the proposed algorithm.
The standard Galerkin formulation of the acoustic wave propagation, governed by the Helmholtz partial differential equation (PDE), is indefinite for large wavenumbers. However, the Helmholtz PDE is in general not indefinite. The lack of coercivity (indefiniteness) is one of the major difficulties for approximation and simulation of heterogeneous media wave propagation models, including application to stochastic wave propagation Quasi Monte Carlo (QMC) analysis. We will present a new class of sign-definite continuous and discrete preconditioned FEM Helmholtz wave propagation models.
Neuron-computer interface in Dynamic-Clamp experimentsSSA KPI
AACIMP 2010 Summer School lecture by Anton Chizhov. "Physics, Chemistry and Living Systems" stream. "Neuron-Computer Interface in Dynamic-Clamp Experiments. Models of Neuronal Populations and Visual Cortex" course. Part 1.
More info at http://summerschool.ssa.org.ua
This a short presentation for a 15 minutes talk at Bayesian Inference for Stochastic Processes 7, on the SMC^2 algorithm.
http://arxiv.org/abs/1101.1528
Atidan is a Microsoft Gold Certified Partner established since 2005 with offices in Mumbai, Pune, Philadelphia and NewYork. We have several fortune 1000 companies as our clients in North America. Our focus is SharePoint Collaboration, BI and Mobile applications.
These are the slides of the talk I presented at Gifu High School in Japan as part of the Science Dialogue initiative of the Japanese Society for the Promotion of Science.
The talk is aimed at High School students whose first language is not English.
The talk contains videos which are not activated in this slideshow.
New Presentation From Astrophysicist Dr. Andrew Beckwith: "Detailing Coherent...GLOBAL HEAVYLIFT HOLDINGS
Detailing Coherent, Minimum Uncertainty States of Gravitons, as Semi Classical Components of Gravity Waves, and How Squeezed States Affect Upper Limits to Graviton Mass /• We present what is relevant to squeezed states of initial space time and how that affects both the composition of relic GW and also gravitons. A side issue to consider is if gravitons can be configured as semi classical “particles”, which is akin to the Pilot particles model of Quantum Mechanics as embedded in a larger non linear “deterministic” background.
First Presented Saturday, September 3, 2011 at the G999 Conference, Philadelphia, PA http://ggg999.org/
Next Presentation : Friday, September 9, SAN MARINO WORKSHOP ON ASTROPHYSICS AND COSMOLOGY
FOR MATTER AND ANTIMATTER
http://www.workshops-hadronic-mechanics.org/
San Marino, N. Italy
Discussion: The Nature of Semi-classical Nature of Gravity Reviewed; And Can We Use a Graviton Entanglement Version of the EPR System to Answer if The Graviton is Classical or Quantum in Origin?
Publisher’s note: Dr. Beckwith: I am honored that you have seen fit to acknowledge me in your presentations of today, September 3, 2011 at the G999 Conference in Philadelphia, and Friday, September 9, at the San Marino Workshop on Astrophysics and Cosmology For Matter and Antimatter, in San Marino, N. Italy.
In my view, these are rather extraordinary postulations, especially the probability of extra-universal black hole gravitational origins, along with expansion beyond Hawking of Quantum Wave theory and the “quantizing” of gravity. The latter may very well lead to a thorough reexamination of our concept of space and time, and that the latter might not be so unidirectional afterall. The potential is breathtaking, as it represents steps forward in proving the existence of subspace, the possibility of concomitant multi-universiality and 5D dimensionality of black holes.
Can time manipulation and Biefeld-Brown suggested electro-gravitics as a spacecraft propulsive methodology be far behind? I think not…
Detailing Coherent, Minimum Uncertainty States of Gravitons, as Semi Classical Components of Gravity Waves, and How Squeezed States Affect Upper Limits to Graviton Mass /• We present what is relevant to squeezed states of initial space time and how that affects both the composition of relic GW and also gravitons. A side issue to consider is if gravitons can be configured as semi classical “particles”, which is akin to the Pilot particles model of Quantum Mechanics as embedded in a larger non linear “deterministic” background.
First Presented Saturday, September 3, 2011 at the G999 Conference, Philadelphia, PA http://ggg999.org/
Next Presentation : Friday, September 9, SAN MARINO WORKSHOP ON ASTROPHYSICS AND COSMOLOGY
FOR MATTER AND ANTIMATTER
http://www.workshops-hadronic-mechanics.org/
San Marino, N. Italy
Discussion: The Nature of Semi-classical Nature of Gravity Reviewed; And Can We Use a Graviton Entanglement Version of the EPR System to Answer if The Graviton is Classical or Quantum in Origin?
Calculating Non-adiabatic Pressure Perturbations during Multi-field InflationIan Huston
This talk was given at the March 2012 UK Cosmology meeting at the University of Sussex.
It describes work done in collaboration with Adam Christopherson published in Physical Review D and available of the arXiv at http://arxiv.org/abs/1111.6919 .
Specific aspects of the evolution of antimatter globular clusters domainsOrchidea Maria Lecian
Specific aspects of the evolution of antimatter globular clusters
domains
Authors: Maxim Yu. Khlopov , Orchidea maria Lecian
Speaker: Orchidea Maria Lecian
Talk delivered at The Fourth Zeldovich virtual meeting, September 7-11, 2020
11 Sep 2020
Abstract: In the Affleck-Dine-Linde scenario of baryosynthesis, there exists a
possibility of creation of sufficiently large regions with antibaryon
excess . Such regions can evolve in antimatter globular clusters.
They appear only in the result of domain evolution and only at the
stage of galaxy formation. Their number is demonstrated to
increase as a function of time, being determined by the mechanism
of antibaryon excess generation and on the properties of the
inflaton filed and related ones; the number of clusters depends also
on the Hubble parameter at the inflationary stage, as well as on
the field initial conditions in the Einstein field Equations. Possible
evolution of the domains of the antimatter globular clusters
provides observational constraints on the mechanisms of inflation,
baryosynthesis and evolution of antibaryon domains in
baryon-asymmetrical Universe.
http://www.icranet.org/index.php?option=com_content&task=view&id=1254
http://www.icranet.org/zeldovich4
Talk given at Kobayashi-Maskawa Institute, Nagoya University, Japan.Peter Coles
Cosmic Anomalies
Observational measurements of the temperature variation of the Cosmic Microwave Background across the celestial sphere made by the Wilkinson Microwave Anisotropy Probe (WMAP) and, more recently Planck, have played a major part in establishing the standard "concordance" cosmological model. However, extensive statistical analysis of these data have also revealed some tantalising anomalies whose interpretation within the standard framework is by no means clear. In this talk, I'll discuss the significance of the evidence for some aspects of this anomalous behaviour, offer some possible theoretical models, and suggest how future measurements may provide firmer conclusions.
Similar to Further discriminatory signature of inflation (20)
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
Home assignment II on Spectroscopy 2024 Answers.pdf
Further discriminatory signature of inflation
1. A Further
Discriminatory
Signature of Inflation
Laila Alabidi
Yukawa Institute of Theoretical Physics
talk presented at KEK
19th of March 2012
2. Based on
various papers with D.H. Lyth (Lancaster U), James
Lidsey and Ian Huston (Queen Mary, University of
London).
and most recently
01203.???? (very soon we hope!)
with
Kazunori Kohri (KEK), Misao Sasaki (YITP),Yuuiti
Sendouda (Hirosaki U.)
4. Some parameter
definitions
Scale factor
distance distance
dt = a(t)d0
at time t at initial time
a
˙ derivative w.r.t.
Hubble parameter H= a
time
Conformal time dτ 2 = a2 dt2
a H
Conformal Hubble parameter H= a = a
derivative
w.r.t. conformal time
5. 0s
98
Inflation: brief review
:1
e
Pr
“Cosmological Principle”-- the universe
has to be both homogeneous and
isotropic, simply because it seemed to
be!
Original matter/radiation
perturbations which sourced the
evolution of structure were put in by
hand.
6. Inflation: a brief
80s
19
review
t:
s
Po
Paradigm: the near exponential expansion of the
universe at t=10-37s
Effect: the comoving Hubble Horizon decreases.
d 1
dt aH 0
Result: explanation of:
Universal homogeneity, causality and
isotropy.
Origin of structure: allows for a quantum to
classical transition of vacuum fluctuations.
10. Overview
Inflation: the parameters.
Inflation models I.
Observation: the bounds.
Results I: from CMB and related data sets.
Going beyond the CMB constraints: the
induced gravitational waves.
Inflation models II.
Results II: prospects from gravitational wave
experiments.
11. Inflation: the
requirements
Inflation requires an accelerated rate of change of expansion
a0
¨
recall the acceleration equation? energy density
a
¨ 4πG
=− (ρ + 3P )
a 3 pressure
negative ρ is nonsensical, but negative pressure is not
a scalar field!
potential scalar field
1 ˙2 1 ˙2
ρ = φ + V (φ) P = φ − V (φ)
2 2
Just considering canonical kinetic terms for now!
12. Inflation: slow roll
¨ ˙ dV (φ)
φ + 3H φ + =0
dφ
and we can define
2
}
1 V
=
2 V
V
η= 1
V
V V
ξ=
V2
a ‘ is a derivative with respect
to the “inflaton” field φ
13. Number of e-folds
Time re-parametrisation
A measure of how long inflation lasts
V
ae
N = ln ai = Hdt V,φ dφ
Start time is the time that scales of cosmological interest leave
the horizon
15. background
φ = φ0 + δφ perturbation
quantity
imprint themselves on the background (via the Einstein
equations) generating what we call the curvature perturbation
ζ(φ)
the characteristics and future evolution of which is
determined by the model of inflation
17. Spectrum and Spectral
scale on whichIndex the ‘spectrum’
perturbation is defined
2π 2 (3)
ζ(k)ζ(k ) = k3 δ (k − k )Pζ (k)
1 V
Pζ = 24π2
The ‘s pectral index’ defines the scale dependance of the
spectrum P
d ln ζ
ns − 1 = d ln k
ns = 1 + 2η − 6
The scale dependance of which is called the ‘running’
dns 2 2
ns = d ln k ns = 16η − 24 − 2ξ
18. Gravitational waves
Traceless, transverse part of the perturbed spatial metric
with a spectrum
H 2
Pgrav = 8 2π
The ratio to the scalar spectrum is defined as
Pgrav
Pζ ≡r
In terms of slow roll parameters
r = 16
20. Primordial Black Holes
The spectrum on small scales has not been ‘well’ measured
If spectrum very large (0.03) then PBHs will form.
Have an ‘upper’ bound due to astrophysical constraints
Pζend 10 −2
21. PBH pre-requisites
For an enhanced spectrum towards the end of
inflation:
Pζe → 10−2
Then a decreasing slow roll parameter is required:
→0
and a running of the spectral index:
n 0
s
22. Classification of single
field models
“Small” field vs. “Large” field models
∆φ = φend − φ∗
“Observable” gravitational waves generated up to 4-efolds
after horizon exit
∆φ ∼ 0.5Mpl r ∼ 0.1
Gravitational waves on the order of 0.1 will be detectable soon!
24. Tree level potential
p p0, inflaton rolls away from the origin
φ
V = V0 1−
µ
Taylor expand about the vacuum, then
assume one of the p’s dominates.
Logarithmic potential
2
gsφ
V = V0 1+ ln
2π Q p0 logarithmic and exponentialtowards the
p0, logarithmic, towards the origin
p0, inflaton rolls inflaton rolls , inflaton
Dvali, Shafi Schaefer (1994) origin
rolls towards the origin
Exponential potential
V = V0 1 − e−qφ/Mpl
Dimopoulos, Lazarides, Lyth Ruiz de Austri (2003) Stewart (1995); Lazarides Panagiotakopoulos (1995)
2 p−1
1 − ns = *not so if μmpl
N p−2
25. Hilltop-type inflation models
p q
V = V0 (1 + ηp φ − ηq φ )
Kohri et al (2007)
The Running Mass Model End of
2
µ0
+ A0 2 Inflation
V = V0 1 − φ Scales
2
leave
A0
+ 2
φ 2
horizon
2(1 + α ln φ)
Stewart (1996)
These models have bet ween 3 and 4 independent parameters
so they can be ‘fit’ to data
these are the only 2 models which can lead to PBH
Kohri et al (2007) and Drees et al (2011)
formation.
35. Future data may alleviate some of the
degeneracy, e.g. PLANCK or CMBPOL. But
not fully.
Further signatures of inflation models.
Be more philosophical and ask what
makes a model ‘natural’.
36.
37. The Data Fantasy
•Space based detectors of gravitational waves, DECIGO and
LISA.
• DECIGO and LISA cover a frequency range of approx. 10-3 to
101.5 Hz.
•This corresponds to scales which leave the horizon at the
end of inflation.
•Funding has not been approved yet but ...
•Proposals and white papers for these projects have
appeared on the arXiv!
39. Anatomy of Induced Gravitational
Waves
These are induced by scalar perturbations entering the horizon
after inflation
(2)
ds2 = 2 (1)
a (τ ) − 1 + 2Φ + 2Φ (2)
dτ + 2Vi dτ dxi + · · ·
2
(1) (2) 1
+ 1 − 2Φ − 2Φ δij + hij dxi dxj
2
contains 1st
and 2nd order
terms
Einstein Eqns
40. Get the equation of motion:
2
hij + 2Hhij − ∇ hij = −4Sij source term
just 2nd order
Sij = 4Φ∂i ∂j Φ + 2∂i Φ∂j Φ + · · ·
4
+ ∂i (Φ + HΦ)∂j (Φ + HΦ)
3H2 (1 + w)
Where the Φ is a first order quantity given by
6(1 + w) 1 2
Φk + Φk + wk Φk = 0
1 + 3w τ
Note: the source term depends only on the (1) scale and
(2) epoch of re-entry (w, the equation of state).
41. Fourier transform
calculate spectra
the equation
2
2π
hk hk = 3 δ(k + k )Ph (k, τ )
k
Spectrum of induced gravitational waves
scalar spectra
integral over all k-space
from inflation
Ph (k, τ ) ∼ ˜
dk ˜ ˜
dµPζ (k)Pζ (|k − k|)I1 (τ )I2 (τ )
cosine of angle integrals over
between modes conformal time
independent of inflation
model
The time integrals are model independent, and require
only Φ and thus the equation of state.
Ananda et al (2007), Baumann et al (2007), Acquaviva et al (2003), Saito et al (2009)
42. ˜
k
Time integrals
v= y = 1 + v 2 − 2vµ
k
Matches results found on Maple
x = kτ and previous Authors results
asymptote to
zero!
44. Scales
V model picture
leave
End of
horizon
Inflation
ε decreases and the
scalar spectrum increases
field
45. Selection Criteria:
Hilltop-Model
V = V0 (1 + ηp φp − ηq φq )
Pre-set couplings to {0,1}.
Reject values that violate WMAP.
Reject values which require super-
planckian evolution.
Unique values then selected which
maximise the spectrum after N e-folds.
46. Selection criteria:
Running Mass
µ +A 2
A 0 0
0 2
V = V0 1 − φ + 2
φ2
2 2(1 + α ln φ)
Less intense criteria: pick values which
satisfy WMAP.
Evolve until just before the spectrum
hits the PBH bound.
Terminate inflation and evaluate N.
51. Present Day Spectrum
The Energy density (per logarithmic
interval) is:
1 dρGW energy density
of GW
ΩGW (k, τ ) =
ρc d log f critical
energy density
This is related to the primordial spectrum as:
2 transfer function
(2) a(τ )k 2
ΩGW = t (k, τ )Ph (k)
aeq keq
For scales kkeq this gives us the relation:
(2) 1
ΩGW = Ph (k)
1 + ze Baumann et al (2007)
52. The Running Mass Model
PBH Bound ns’=0.002
ns’=0.005
0.007ns’0.012
Models are
terminated at
N=64
DM N=43
29N39
53. For comparison ...
DE
BBO
CIG O
The spectra of primordial gravitational waves look
very different!
this image is taken from Kuroyanagi et al (2010)
54. Hilltop models
lines=fractional q, stars and circles = integral q
N=
55
60
N=
55. Conclusions
Hilltop model: generates potentially obser vable induced
GWS
reasonable e-folds.
also p=2,q=3 within the reach of BBO/DECIGO.
Running Mass:
within the reach of BBO/DECIGO for N50.
DM production measurable by LISA.