Casimir effect meets the cosmological constant - Ulf Leonhardt, Weizmann Institute of Science, 5 May 2022
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Casimir effect meets the cosmological constant
Ulf Leonh rdt, Weizm nn Institute of Science, 5 M y 2022
a
a
a
The discovery came as a complete surprise even to the Nobel Laureates themselves. What they saw would be
*
Dark energy − cosmological constant
like throwing a ball up in the air, and instead of having it come back down, watching as it disappears more
and more rapidly into the sky, as if gravity could not manage to reverse the ball’s trajectory. Something simi-
lar seemed to be happening across the entire Universe.
dark energy
gravitation
galaxy
supernova
acceleration acceleration
big bang decreases increases
universe
percent
100
80
matter
60
40 dark matter
20 dark energy
0
l Foundation.
14 billion 5 billion present
years ago years ago
Figure 1. The world is growing. The expansion of the Universe began with the Big Bang 14 billion years ago, but slowed down during the
l
.............................................................. Reduced water potential due to
The limits to tree height the turgor of living plant cells that
leaf expansion11. To determine if th
declines with height, we estimat
George W. Koch1, Stephen C. Sillett2, Gregory M. Jennings2
potentials from pressure–volume m
& Stephen D. Davis3
declined linearly with height, h, as
1
Department of Biological Sciences and the Merriam-Powell Center for þð1:30 ^ 0:07Þ, n ¼ 4 trees, rang
Environmental Research, Northern Arizona University, Flagstaff, Arizona 86011, 0.48 MPa at 110 m. At night when
USA turgor gradient was less steep,
2
Department of Biological Sciences, Humboldt State University, Arcata, þð1:39 ^ 0:19Þ, and turgor was
California 95521, USA midday.
3
Natural Science Division, Pepperdine University, Malibu, California Given the role of turgor in lea
90263-4321, USA height may underlie the distinct ver
.............................................................................................................................................................................
redwoods (Fig. 2). Leaf shape varied
Trees grow tall where resources are abundant, stresses are minor,
lower crown to small and scale-like
and competition for light places a premium on height growth1,2.
variation in terms of the leaf mass:
The height to which trees can grow and the biophysical determi-
increased exponentially over a fou
nants of maximum height are poorly understood. Some models
LMA ¼ (37.1 ^ 12.3)exp(0.0260
predict heights of up to 120 m in the absence of mechanical
0.0001 # P # 0.003, n ¼ 5 trees).
damage3,4, but there are historical accounts of taller trees5.
the highest published value for te
Current hypotheses of height limitation focus on increasing
variation in LMA has been attr
water transport constraints in taller trees and the resulting
canopies13,14. In our study trees, we
reductions in leaf photosynthesis 6 . We studied redwoods
(DSF), an index of direct solar ra
(Sequoia sempervirens), including the tallest known tree on
photographs, decreased by 14% o
Earth (112.7 m), in wet temperate forests of northern California.
decrease in height. Relative to w
Our regression analyses of height gradients in leaf functional
light on LMA was small, howeve
characteristics estimate a maximum tree height of 122–130 m
explained variation of within-crow
barring mechanical damage, similar to the tallest recorded trees
analysis including DSF (P ¼ 0.002
of the past. As trees grow taller, increasing leaf water stress due to
(P , 0.0001) as independent variab
gravity and path length resistance may ultimately limit leaf
samples from five redwoods over 1
expansion and photosynthesis for further height growth, even
vations (Fig. 3) also support the hy
with ample soil moisture.
more important than light environ
According to the cohesion-tension theory, water transport in
plants occurs along a gradient of negative pressure (tension) in the ture in redwood: (1) leaves of a 2-m
dead, tube-like cells of the xylem, with transpiration, water adhesion in soil near the top of a 95-m-t
to cell walls, and surface tension providing the forces necessary to
l
.............................................................. Reduced water potential due to
The limits to tree height the turgor of living plant cells that
leaf expansion11. To determine if th
declines with height, we estimat
George W. Koch1, Stephen C. Sillett2, Gregory M. Jennings2
potentials from pressure–volume m
& Stephen D. Davis3
declined linearly with height, h, as
1
Department of Biological Sciences and the Merriam-Powell Center for þð1:30 ^ 0:07Þ, n ¼ 4 trees, rang
Environmental Research, Northern Arizona University, Flagstaff, Arizona 86011, 0.48 MPa at 110 m. At night when
USA turgor gradient was less steep,
2
Department of Biological Sciences, Humboldt State University, Arcata, þð1:39 ^ 0:19Þ, and turgor was
California 95521, USA midday.
3
Natural Science Division, Pepperdine University, Malibu, California Given the role of turgor in lea
90263-4321, USA height may underlie the distinct ver
.............................................................................................................................................................................
redwoods (Fig. 2). Leaf shape varied
Trees grow tall where resources are abundant, stresses are minor,
lower crown to small and scale-like
and competition for light places a premium on height growth1,2.
variation in terms of the leaf mass:
The height to which trees can grow and the biophysical determi-
increased exponentially over a fou
nants of maximum height are poorly understood. Some models
LMA ¼ (37.1 ^ 12.3)exp(0.0260
predict heights of up to 120 m in the absence of mechanical
0.0001 # P # 0.003, n ¼ 5 trees).
damage3,4, but there are historical accounts of taller trees5.
the highest published value for te
Current hypotheses of height limitation focus on increasing
variation in LMA has been attr
water transport constraints in taller trees and the resulting
canopies13,14. In our study trees, we
reductions in leaf photosynthesis 6 . We studied redwoods
(DSF), an index of direct solar ra
(Sequoia sempervirens), including the tallest known tree on
photographs, decreased by 14% o
Earth (112.7 m), in wet temperate forests of northern California.
decrease in height. Relative to w
Our regression analyses of height gradients in leaf functional
light on LMA was small, howeve
characteristics estimate a maximum tree height of 122–130 m
explained variation of within-crow
barring mechanical damage, similar to the tallest recorded trees
analysis including DSF (P ¼ 0.002
of the past. As trees grow taller, increasing leaf water stress due to
(P , 0.0001) as independent variab
gravity and path length resistance may ultimately limit leaf
samples from five redwoods over 1
expansion and photosynthesis for further height growth, even
vations (Fig. 3) also support the hy
with ample soil moisture.
more important than light environ
According to the cohesion-tension theory, water transport in
plants occurs along a gradient of negative pressure (tension) in the ture in redwood: (1) leaves of a 2-m
dead, tube-like cells of the xylem, with transpiration, water adhesion in soil near the top of a 95-m-t
to cell walls, and surface tension providing the forces necessary to
Casimir force in nature: gecko feet
The discovery came as a complete surprise even to the Nobel Laureates themselves. What they saw would be
*
Dark energy − cosmological constant
like throwing a ball up in the air, and instead of having it come back down, watching as it disappears more
and more rapidly into the sky, as if gravity could not manage to reverse the ball’s trajectory. Something simi-
lar seemed to be happening across the entire Universe.
2011
dark energy
gravitation
galaxy
supernova
acceleration acceleration
big bang decreases increases
universe
percent
100
80
matter
60
40 dark matter
20 dark energy
0
l Foundation.
14 billion 5 billion present
years ago years ago
Figure 1. The world is growing. The expansion of the Universe began with the Big Bang 14 billion years ago, but slowed down during the
The Cosmological Principle: homogeneity and isotropy
The Cosmological Principle: homogeneity and isotropy
Take an arbitrary point in homogeneous space
Pick another point and consider their distance a
aAcceleration on particle at distance a: Gauss’ law
aAcceleration on particle at distance a: Gauss’ law
aRelativistic acceleration on particle at distance a
aEinstein’s cosmological constant Λ - “dark energy”
Einstein’s cosmological constant Λ - “dark energy”
Einstein’s eld equations with cosmological term
fiEinstein’s cosmological constant Λ - “dark energy”
Einstein’s cosmological constant Λ - “dark energy”
Einstein’s cosmological constant Λ - “dark energy”
Einstein’s cosmological constant Λ - “dark energy”
Einstein’s cosmological constant Λ - “dark energy”
Acceleration!Einstein’s cosmological constant Λ - “dark energy”
70% of all ε at present timeThe discovery came as a complete surprise even to the Nobel Laureates themselves. What they saw would be
*
Dark energy − cosmological constant
like throwing a ball up in the air, and instead of having it come back down, watching as it disappears more
and more rapidly into the sky, as if gravity could not manage to reverse the ball’s trajectory. Something simi-
lar seemed to be happening across the entire Universe.
dark energy
gravitation
galaxy
supernova
acceleration acceleration
big bang decreases increases
universe
percent
100
80
matter
60
40 dark matter
20 dark energy
0
l Foundation.
14 billion 5 billion present
years ago years ago
Figure 1. The world is growing. The expansion of the Universe began with the Big Bang 14 billion years ago, but slowed down during theThe gravity of vacuum fluctuations
SOVIET PHYSICS USPEKHI VOLUME 11, NUMBER 3 NOVEMBER-DECEMBER 1968
530.12:531.51
THE COSMOLOGICAL CONSTANT AND THE THEORY OF ELEMENTARY PARTICLES
Ya. B. ZEL'DOVICH
Institute of Applied Mathematics, USSR Academy of Sciences
Usp. Fiz. Nauk 95, 209-230 (May, 1968)
I NTEREST in gravitation theory with a cosmological of matter and of the radius of the world at the present
constant was revived in 1967. Three papers were time.
published, by Petrosian, Salpeter, and Szekeres in the At first glance such an explanation is on the whole
USA[1J and by Shklovskii [2] and Kardashev [3] in the unlikely. It must be borne in mind, however, that other
USSR, in which universe evolution models in such a attempts at explaining the predominant absorption with
theory (the A models) are considered. The stimulus z = 1.95 are at present no less far-fetched and arti-
for the revival of the theory was provided by new ob- ficial. In a paper at the 13th Congress of the Inter-
servational data on remote quasistellar sources nation Astronomical Union in Prague (August 1967),
(quasars and quasags, QSR and QSG in the English - Burbidge spoke of z = 1.95 as an argument in favor of
language literature)* It turned out, first of all, that the local theory of quasars. According to the local
for these objects the connection between the brightness theory, the distance from us to the quasars is less than
and the red shift does not fit the simple models without 100 Mpsec, and the red shift of the emission and ab-
a cosmological constant (and without assumptions con- sorption lines is of gravitational origin and is connected
cerning the evolution of the quasars!). In addition, as with the work function of the quanta from the gravita-
noted by the Burbidges , in ten quasars whose spectra tional field of the quasar [ 5 j *. However, no concrete
have revealed absorption lines the red shift of these model which yields precisely z = 1.95, or at least an
lines z = (A - Ao)/Ao lies in the narrow range 1.94 equal value of z for the quasars with different masses
during different stages of evolution, was proposed byThe gravity of vacuum fluctuations
SOVIET PHYSICS USPEKHI VOLUME 11, NUMBER 3 NOVEMBER-DECEMBER 1968
530.12:531.51
THE COSMOLOGICAL CONSTANT AND THE THEORY OF ELEMENTARY PARTICLES
Ya. B. ZEL'DOVICH
Institute of Applied Mathematics, USSR Academy of Sciences
Usp. Fiz. Nauk 95, 209-230 (May, 1968)
I NTEREST in gravitation theory with a cosmological of matter and of the radius of the world at the present
constant was revived in 1967. Three papers were time.
published, by Petrosian, Salpeter, and Szekeres in the At first glance such an explanation is on the whole
USA[1J and by Shklovskii [2] and Kardashev [3] in the unlikely. It must be borne in mind, however, that other
USSR, in which universe evolution models in such a attempts at explaining the predominant absorption with
theory (the A models) are considered. The stimulus z = 1.95 are at present no less far-fetched and arti-
for the revival of the theory was provided by new ob- ficial. In a paper at the 13th Congress of the Inter-
servational data on remote quasistellar sources nation Astronomical Union in Prague (August 1967),
(quasars and quasags, QSR and QSG in the English - Burbidge spoke of z = 1.95 as an argument in favor of
language literature)* It turned out, first of all, that the local theory of quasars. According to the local
for these objects the connection between the brightness theory, the distance from us to the quasars is less than
and the red shift does not fit the simple models without 100 Mpsec, and the red shift of the emission and ab-
a cosmological constant (and without assumptions con- sorption lines is of gravitational origin and is connected
cerning the evolution of the quasars!). In addition, as with the work function of the quanta from the gravita-
noted by the Burbidges , in ten quasars whose spectra tional field of the quasar [ 5 j *. However, no concrete
have revealed absorption lines the red shift of these model which yields precisely z = 1.95, or at least an
lines z = (A - Ao)/Ao lies in the narrow range 1.94 equal value of z for the quasars with different masses
during different stages of evolution, was proposed byWeconstant
The cosmological have aproblem
problem
Steven Weinberg
Theory Group, Department of Physics, University of Texas, Austin, Texas 7871Z
Astronomical observations indicate that the cosmological constant is many orders of magnitude smaller
than estimated in modern theories of elementary particles. After a brief review of the history of this prob-
lem, five different approaches to its solution are described.
CONTENTS II. EARLY HISTORY
I. Introduction
After completing his formulation of
1
II. Early History 1
in 1915—1916, Einstein (1917) attempted
III. The Problem 2 theory to the whole universe. His guid
IV. Supersymmetry, Supergravity, Superstrings 3 that the universe is static: "The most im
V. Anthropic Considerations 6 we draw from experience is that the rel
A. Mass density 8 the stars are very small as compared w
8. Ages 8 "
light. No such static solution of his
C. Number counts 8
VI. Adjustment Mechanisms
could be found (any more than for Ne
9
VII. Changing Gravity 11 tion), so he modified them by adding a
VIII. Quantum Cosmology 14 ing a free parameter A. , the cosmological
IX. Outlook
Acknowledgments
20
21 R„—'g —, R —g„= —8e GT„
A,
References 21
Now, for A, &0, there was a static solutiWeconstant
The cosmological have aproblem
problem
Steven Weinberg
Theory Group, Department of Physics, University of Texas, Austin, Texas 7871Z
Astronomical observations indicate that the cosmological constant is many orders of magnitude smaller
than estimated in modern theories of elementary particles. After a brief review of the history of this prob-
lem, five different approaches to its solution are described.
CONTENTS II. EARLY HISTORY
I. Introduction
After completing his formulation of
1
II. Early History 1
in 1915—1916, Einstein (1917) attempted
III. The Problem 2 theory to the whole universe. His guid
IV. Supersymmetry, Supergravity, Superstrings 3 that the universe is static: "The most im
V. Anthropic Considerations 6 we draw from experience is that the rel
A. Mass density 8 the stars are very small as compared w
8. Ages 8 "
light. No such static solution of his
C. Number counts 8
VI. Adjustment Mechanisms
could be found (any more than for Ne
9
VII. Changing Gravity 11 tion), so he modified them by adding a
VIII. Quantum Cosmology 14 ing a free parameter A. , the cosmological
IX. Outlook
Acknowledgments
20
21 R„—'g —, R —g„= —8e GT„
A,
References 21
Now, for A, &0, there was a static solutiWeconstant
The cosmological have aproblem
problem
Steven Weinberg
Theory Group, Department of Physics, University of Texas, Austin, Texas 7871Z
Astronomical observations indicate that the cosmological constant is many orders of magnitude smaller
than estimated in modern theories of elementary particles. After a brief review of the history of this prob-
lem, five different approaches to its solution are described.
CONTENTS II. EARLY HISTORY
I. Introduction
After completing his formulation of
1
II. Early History 1
in 1915—1916, Einstein (1917) attempted
III. The Problem 2 theory to the whole universe. His guid
IV. Supersymmetry, Supergravity, Superstrings 3 that the universe is static: "The most im
V. Anthropic Considerations 6 we draw from experience is that the rel
A. Mass density 8 the stars are very small as compared w
8. Ages 8 "
light. No such static solution of his
C. Number counts 8
VI. Adjustment Mechanisms
could be found (any more than for Ne
9
VII. Changing Gravity 11 tion), so he modified them by adding a
VIII. Quantum Cosmology 14 ing a free parameter A. , the cosmological
IX. Outlook
Acknowledgments
20
21 R„—'g —, R —g„= —8e GT„
A,
References 21
Now, for A, &0, there was a static solutiWeconstant
The cosmological have aproblem
problem
Steven Weinberg
Theory Group, Department of Physics, University of Texas, Austin, Texas 7871Z
Astronomical observations indicate that the cosmological constant is many orders of magnitude smaller
than estimated in modern theories of elementary particles. After a brief review of the history of this prob-
lem, five different approaches to its solution are described.
CONTENTS II. EARLY HISTORY
I. Introduction
After completing his formulation of
1
II. Early History 1
in 1915—1916, Einstein (1917) attempted
III. The Problem 2 theory to the whole universe. His guid
IV. Supersymmetry, Supergravity, Superstrings 3 that the universe is static: "The most im
V. Anthropic Considerations 6 we draw from experience is that the rel
A. Mass density 8 the stars are very small as compared w
8. Ages 8 "
light. No such static solution of his
C. Number counts 8
VI. Adjustment Mechanisms
could be found (any more than for Ne
9
VII. Changing Gravity 11 tion), so he modified them by adding a
VIII. Quantum Cosmology 14 ing a free parameter A. , the cosmological
IX. Outlook
Acknowledgments
20
21 R„—'g —, R —g„= —8e GT„
A,
References 21
Now, for A, &0, there was a static solutiWeconstant
The cosmological have aproblem
problem
Steven Weinberg
Theory Group, Department of Physics, University of Texas, Austin, Texas 7871Z
Astronomical observations indicate that the cosmological constant is many orders of magnitude smaller
than estimated in modern theories of elementary particles. After a brief review of the history of this prob-
lem, five different approaches to its solution are described.
CONTENTS II. EARLY HISTORY
I. Introduction
After completing his formulation of
1
II. Early History 1
in 1915—1916, Einstein (1917) attempted
III. The Problem 2 theory to the whole universe. His guid
IV. Supersymmetry, Supergravity, Superstrings 3 that the universe is static: "The most im
V. Anthropic Considerations 6 we draw from experience is that the rel
A. Mass density 8 the stars are very small as compared w
8. Ages 8 "
light. No such static solution of his
C. Number counts 8
VI. Adjustment Mechanisms
could be found (any more than for Ne
9
VII. Changing Gravity 11 tion), so he modified them by adding a
VIII. Quantum Cosmology 14 ing a free parameter A. , the cosmological
IX. Outlook
Acknowledgments
20
21 R„—'g —, R —g„= —8e GT„
A,
References 21
Now, for A, &0, there was a static solutiA general argument? : general covariance of the vacuum
The vacuum
The vacuum state | 0⟩
Unruh-Fulling-Davies effect: view of an accelerated observer
FOCUS | REVIEW ARTICLES
PUBLISHED ONLINE: 31 MARCH 2011"|"DOI: 10.1038/NPHOTON.2011.39
The Casimir effect in microstructured geometries
Alejandro W. Rodriguez1,2, Federico Capasso1* and Steven G. Johnson2
In 1948, Hendrik Casimir predicted that a generalized version of van der Waals forces would arise between two metal plates due
to quantum fluctuations of the electromagnetic field. These forces become significant in micromechanical systems at submi-
crometre scales, such as in the adhesion between movable parts. The Casimir force, through a close connection to classical pho-
tonics, can depend strongly on the shapes and compositions of the objects, stimulating a decades-long search for geometries
in which the force behaves very differently from the monotonic attractive force first predicted by Casimir. Recent theoretical
and experimental developments have led to a new understanding of the force in complex microstructured geometries, includ-
ing through recent theoretical predictions of Casimir repulsion between vacuum-separated metals, the stable suspension of
objects and unusual non-additive and temperature effects, as well as experimental observations of repulsion in fluids, non-
additive forces in nanotrench surfaces and the influence of new material choices.
F
luctuation-induced electromagnetic forces between neutral produce an induced dipole moment p2 ~ d–3 (ref. 4). Assuming
bodies become more and more important as micromechani- positive polarizabilities, the direction of the dipole fields means
cal and microfluidic devices enter submicrometre scales. These that these two dipoles are oriented so as to attract each other,
forces are known by several different names, depending on the with an interaction energy that scales as d–6. This leads to the van
regime in which they operate, including van der Waals, Casimir– der Waals ‘dispersion’ force, and similar considerations apply to
Polder and, more generally, Casimir forces (of which van der Waals particles with permanent dipole moments that can rotate freely.
forces are special cases)1–4. Casimir forces arise from electromag- The key to more general considerations of Casimir physics is to
netic waves created by quantum and thermal fluctuations5–21. The understand that this d–6 picture of van der Waals forces makes two
dramatic progress made in the theoretical understanding and crucial approximations that are not always valid: it employs the
measurement of Casmir forces over the past ten years may soon quasi-static approximation to ignore wave effects, and also ignores
allow them to be exploited in novel microelectromechanical sys- multiple scattering if there are more than two particles.
tems (MEMS) and microfluidic devices8,22–24. The quasi-static approximation assumes that the dipole moment
Experimentally, Casimir forces have been measured with ever p1 polarizes the second particle instantaneously, which is valid if d is
greater precision25–34 in microstructured geometries that increas- much smaller than the typical wavelength of the fluctuating fields.
ingly deviate from the original parallel-plate configuration26. They However, the finite wave propagation speed of light must be taken
have even been measured in fluids that allow the sign of the force into account when d is much larger than the typical wavelength, as
to change35. Theoretically, the calculation of Casimir forces was tra- shown in Fig. 1b, and it turns out that the resulting Casimir–Polder
ditionally limited to planar or near-planar geometries, but recent interaction energy asymptotically scales as d–7 for large d (ref. 47).FOCUS | REVIEW ARTICLES
PUBLISHED ONLINE: 31 MARCH 2011"|"DOI: 10.1038/NPHOTON.2011.39
The Casimir effect in microstructured geometries
Alejandro W. Rodriguez1,2, Federico Capasso1* and Steven G. Johnson2
In 1948, Hendrik Casimir predicted that a generalized version of van der Waals forces would arise between two metal plates due
to quantum fluctuations of the electromagnetic field. These forces become significant in micromechanical systems at submi-
crometre scales, such as in the adhesion between movable parts. The Casimir force, through a close connection to classical pho-
tonics, can depend strongly on the shapes and compositions of the objects, stimulating a decades-long search for geometries
in which the force behaves very differently from the monotonic attractive force first predicted by Casimir. Recent theoretical
and experimental developments have led to a new understanding of the force in complex microstructured geometries, includ-
ing through recent theoretical predictions of Casimir repulsion between vacuum-separated metals, the stable suspension of
objects and unusual non-additive and temperature effects, as well as experimental observations of repulsion in fluids, non-
additive forces in nanotrench surfaces and the influence of new material choices.
F
luctuation-induced electromagnetic forces between neutral produce an induced dipole moment p2 ~ d–3 (ref. 4). Assuming
bodies become more and more important as micromechani- positive polarizabilities, the direction of the dipole fields means
cal and microfluidic devices enter submicrometre scales. These that these two dipoles are oriented so as to attract each other,
forces are known by several different names, depending on the with an interaction energy that scales as d–6. This leads to the van
regime in which they operate, including van der Waals, Casimir– der Waals ‘dispersion’ force, and similar considerations apply to
Polder and, more generally, Casimir forces (of which van der Waals particles with permanent dipole moments that can rotate freely.
forces are special cases)1–4. Casimir forces arise from electromag- The key to more general considerations of Casimir physics is to
netic waves created by quantum and thermal fluctuations5–21. The understand that this d–6 picture of van der Waals forces makes two
dramatic progress made in the theoretical
VOLUME 78, NUMBER 1
understanding and crucial approximations that
PHYSICAL REVIEW LETTERS
are not always valid: it employs the
6 JANUARY 1997
measurement of Casmir forces over the past ten years may soon quasi-static approximation to ignore wave effects, and also ignores
allow them to be exploited in novel microelectromechanical sys- multiple scattering if there are more than two particles.
tems (MEMS) and microfluidic devices8,22–24. The quasi-static approximation assumes that the dipole moment
Demonstration of the Casimir Force in the 0.6 to 6 mm Range
Experimentally, Casimir forces have been measured with ever p1 polarizes the second particle instantaneously, which is valid if d is
greater precision25–34 in microstructured geometries that increas- much smaller than the typical wavelength of the fluctuating fields.
S. K. Lamoreaux*
ingly deviate from the original parallel-plate configuration26. They However, the finite wave propagation speed of light must be taken
Physics Department, University of Washington, Box 35160, Seattle, Washington 98195-1560
have even been measured in fluids that allow the sign28ofAugust
(Received 1996) into account when d is much larger than the typical wavelength, as
the force
to change35. Theoretically, the calculation of Casimir forces was tra- shown in Fig. 1b, and it turns out that the resulting Casimir–Polder
The vacuum stress between closely spaced conducting surfaces, due to the modification of the zero-
ditionally limited to planar or near-planar geometries, but recent interaction energy asymptotically
point fluctuations of the electromagnetic field, has been conclusively demonstrated. The measurement
scales as d –7
for large d (ref. 47).Precision measurements and manipulations of Casimir forces
Fig. 1. Stable Casimir equilibrium enabled by a low–refractive index
coating layer. (A) By coating a thin layer of Teflon on a gold substrate, a stable
Casimir equilibrium is formed so that a gold nanoplate can be trapped at an
R ES E A RC H equilibrium position in ethanol. (B) Casimir interaction energy between the gold
PHYSICS forces can be repulsive at short separation dis-
Zhao et al., Science 364, 984–987
tances (2019)
and attractive 7 June 2019
at long distances.
Stable Casimir equilibria were predicted in
Stable Casimir equilibria theory by arranging one of the interacting ob-
jects enclosed by another (16, 17) so that the
and quantum trapping surrounding repulsive Casimir forces could
shroud the object at the center. This special
topological requirement limits possible applica-
Rongkuo Zhao1*, Lin Li1*, Sui Yang1*, Wei Bao1*, Yang Xia1, Paul Ashby2, tions and also makes experimental verification
Yuan Wang1, Xiang Zhang1,3† extremely difficult. Because Casimir forces at large
separations are mainly contributed by low electro-
The Casimir interaction between two parallel metal plates in close proximity is usually thought magnetic frequencies and at small separations
of as an attractive interaction. By coating one object with a low–refractive index thin film, we by high frequencies, a stable Casimir equilibriumOptical analogue of expanding, spatially flat universe
n=aLifshitz theory
Renormalization: bare vacuum energy is infinite
S | FOCUS NATURE PHOTONICS DOI: 10.1038/NPHOTON.2011.39
b Casimir–Polder (waves/retardation) c Casimir effect (macroscopic bodies)
p1
Wave effects
p2
Induced
dipole
p1
p2 p2
Multiple scattering
p3
Waals, Casimir–Polder and Casimir forces, whose origins lie in the quantum fluctuations of dipoles.
ting electromagnetic dipole field, which in turn induces a fluctuating dipole p2 on a nearby particle, leading to van
When the particle spacing is large, retardation/wave effects modify the interaction, leading to Casimir–Polder
ract, the non-additive field interactions lead to a breakdown of the pairwise force laws. c, In situations consisting
ween the many fluctuating dipoles present within the bodies leads to Casimir forces.
the contribution of the elec- U ~ −p (E − E0), which is finite even for a point dipole (whereas
oint energy (U) of the parallel E and E0 themselves diverge at the source point). This energy mustS | FOCUS NATURE PHOTONICS DOI: 10.1038/NPHOTON.2011.39
Renormalization: compare vacuum energy at finite distance
b Casimir–Polder (waves/retardation) c Casimir effect (macroscopic bodies)
p1
Wave effects
p2
Induced
dipole
p1
p2 p2
Multiple scattering
p3
with
Waals, Casimir–Polder and Casimir forces, whose origins lie in the quantum fluctuations of dipoles.
ting electromagnetic dipole field, which in turn induces a fluctuating dipole p2 on a nearby particle, leading to van
When the particle spacing is large, retardation/wave effects modify the interaction, leading to Casimir–Polder
ract, the non-additive field interactions lead to a breakdown of the pairwise force laws. c, In situations consisting
ween the many fluctuating dipoles present within the bodies leads to Casimir forces.
the contribution of the elec- U ~ −p (E − E0), which is finite even for a point dipole (whereas
oint energy (U) of the parallel E and E0 themselves diverge at the source point). This energy must
d an attractive force between be integrated over fluctuations at all frequencies, multiplied by an
netic modes that have nodes appropriate frequency distribution such as a Bose–Einstein dis-
vity, the mode frequencies (ω) tribution, which includes the effects of thermal fluctuations at
he plates, giving rise to a pres- non-zero temperatures. The key fact is that computing Casimir
interactions reduces to solving classical scattering problems, and
this fact carries over to more general problems involving interac-
–27 2 tions between macroscopic bodies — such bodies consist of manyS | FOCUS NATURE PHOTONICS DOI: 10.1038/NPHOTON.2011.39
Renormalization: compare vacuum energy at finite distance
b Casimir–Polder (waves/retardation) c Casimir effect (macroscopic bodies)
p1
Wave effects
p2
Induced
dipole
p1
p2 p2
Multiple scattering
REVIEW ARTICLES | FOCUS
TURE PHOTONICS DOI: 10.1038/NPHOTON.2011.39
p3
NATURE PHOTONICS DOI: 10.1038/NPHOTON.2011.39
c Casimir effect (macroscopic bodies)
a van der Waals (quasistatic fields) b Casimir–Polder (waves/retardation) c Casimir effect (macroscopic bodies)
Waals, Casimir–Polder and Casimir forces, whose origins lie in the quantum fluctuations
p1
with
of dipoles.
ting electromagnetic dipole field, which in turn induces a fluctuating dipole p2 on a nearby particle, leading to van
When the particle spacing is large, retardation/wave effects modify Wavethe
effects
interaction, leading to Casimir–Polder
p2 Fluctuating
ract, the non-additive field interactions lead to a breakdown of the pairwise force laws. c, In situationspconsisting
dipole 2
ween the many fluctuating dipoles present within the bodies leads to Casimir forces.
Induced
dipole
p1
the
p2 contribution of the elec-
p1 U ~ −p (E − Ep0), which is finite even for a point dipole (whereas
p2
oint energy (U) of the parallel E and E0 themselves diverge at the source point). This energy must
2
edscattering
an attractive force between be integrated over fluctuations at all frequencies, multiplied by an
Multiple scattering
netic modes that have nodes appropriate frequency distribution such as a Bose–Einstein dis-
vity, the mode frequencies (ω) tribution, which includes the effects of thermal p3
fluctuations at
he plates, giving rise to a pres- non-zero temperatures. The key fact is that computing Casimir
interactions reduces to solving classical scattering problems, and
origins lie in the quantum fluctuations of dipoles.
this van
Figure 1 | Relationship between factder
carries
Waals,over to more general
Casimir–Polder problems
and Casimir forces,involving interac-
whose origins lie in the quantum fluctuations of dipoles.
duces a fluctuating dipole p2 on a nearby particle, leading to van
–27 a, A tionsabetween
2 fluctuating dipole p induces macroscopic
fluctuating bodies
electromagnetic —field,
dipole suchwhich
bodies consist
in turn inducesof many
a fluctuating dipole p on a nearby particle, leading to vanEinstein’s field equations: all energy is supposed to curve space-time
All energy should gravitateNo gravity from the bare vacuum.
No evidence for the bare vacuum.
Neither does it do mechanical
work nor gravitate.Casimir cosmology: turn from space
S | FOCUS NATURE PHOTONICS DOI: 10.1038/NPHOTON.2011.39
b Casimir–Polder (waves/retardation) c Casimir effect (macroscopic bodies)
p1
Wave effects
p2
Induced
dipole
p1
p2 p2
Multiple scattering
p3
Waals, Casimir–Polder and Casimir forces, whose origins lie in the quantum fluctuations of dipoles.
ting electromagnetic dipole field, which in turn induces a fluctuating dipole p2 on a nearby particle, leading to van
When the particle spacing is large, retardation/wave effects modify the interaction, leading to Casimir–Polder
ract, the non-additive field interactions lead to a breakdown of the pairwise force laws. c, In situations consisting
ween the many fluctuating dipoles present within the bodies leads to Casimir forces.
the contribution of the elec- U ~ −p (E − E0), which is finite even for a point dipole (whereas
oint energy (U) of the parallel E and E0 themselves diverge at the source point). This energy mustCasimir cosmology: turn from space to time
NATURE PHOTONICS DOI: 10.1038/NPHOTON.2011.39
ardation) c Casimir effect (macroscopic bodies)
p2
p2
Multiple scattering
p3
es, whose origins lie in the quantum fluctuations of dipoles.
in turn induces a fluctuating dipole p2 on a nearby particle, leading to van
ardation/wave effects modify the interaction, leading to Casimir–Polder
lead to a breakdown of the pairwise force laws. c, In situations consisting
ent within the bodies leads to Casimir forces.
~ −p (E − E0), which is finite even for a point dipole (whereas
and E0 themselves diverge at the source point). This energy must
e integrated over fluctuations at all frequencies, multiplied by an
ppropriate frequency distribution such as a Bose–Einstein dis-
ibution, which includes the effects of thermal fluctuations at
on-zero temperatures. The key fact is that computing Casimir
teractions reduces to solving classical scattering problems, and
is fact carries over to more general problems involving interac-
ons between macroscopic bodies — such bodies consist of many
uch dipoles, and correspondingly one must solve many scattering
roblems for many current sources or incident waves. This has
ree consequences, which are discussed in more detail below.
rst, it is evident that standard computational techniques from
assical electromagnetism can be used to solve for the Green’s
nction and hence the Casimir energy, although many classical
roblems must be solved to yield a single U. Second, the non-
dditivity is clear because classical scattering involves solvingCasimir cosmology: continuous n(t)
Surprises in Casimir physics with three students and a postdoc William Simpson Itay Griniasty Yael Avni Efi Shahmoon PhD student PhD student Masters postdoc 2010-14 2013-19 2015-17 2014
Two years of work in solitude
Casimir force in cosmology?
Some say in ice…”*
*Robert Frost, Fire and Ice
What’s the difference?
What is the fate of the Universe? Probably it will end in ice if we are to believe this year’s Nobel Laureates.
They have carefully studied several dozen exploding stars, called supernovae, in faraway galaxies and
have concluded that the expansion of the Universe is speeding up.
The discovery came as a complete surprise even to the Nobel Laureates themselves. What they saw would be
like throwing a ball up in the air, and instead of having it come back down, watching as it disappears more
and more rapidly into the sky, as if gravity could not manage to reverse the ball’s trajectory. Something simi-
lar seemed to be happening across the entire Universe.
dark energy
gravitation
galaxy
supernova
acceleration acceleration
big bang decreases increases
universe
percent
100
80
matter
60
40 dark matter
20 dark energy
0
on.
14 billion 5 billion presentEquivalence principle: space-time is the same for everything
Equivalence principle: space-time is the same for everything Horizon Planck
The result(s) UL, Ann. Phys. (NY) 411, 167972 (2019) D. Berechya & UL, MNRAS 507, 3473 (2021) UL 2022
Renormalised vacuum energy depends on changes in n
NATURE PHOTONICS DOI: 10.1038/NPHOTON.2011.39
ardation) c Casimir effect (macroscopic bodies)
p2
p2
Multiple scattering
p3
es, whose origins lie in the quantum fluctuations of dipoles.
in turn induces a fluctuating dipole p2 on a nearby particle, leading to van
ardation/wave effects modify the interaction, leading to Casimir–Polder
lead to a breakdown of the pairwise force laws. c, In situations consisting
ent within the bodies leads to Casimir forces.
~ −p (E − E0), which is finite even for a point dipole (whereas
and E0 themselves diverge at the source point). This energy must
e integrated over fluctuations at all frequencies, multiplied by an
ppropriate frequency distribution such as a Bose–Einstein dis-
ibution, which includes the effects of thermal fluctuations at
on-zero temperatures. The key fact is that computing Casimir
teractions reduces to solving classical scattering problems, and
is fact carries over to more general problems involving interac-
ons between macroscopic bodies — such bodies consist of many
uch dipoles, and correspondingly one must solve many scattering
roblems for many current sources or incident waves. This has
ree consequences, which are discussed in more detail below.
rst, it is evident that standard computational techniques from
assical electromagnetism can be used to solve for the Green’s
nction and hence the Casimir energy, although many classicalThe anomaly
PH YSICAI, RK VIE% 0 Vol UME 17, 5 UMBER 6 15 MARCH 1978
Trace anomaly of a conformally invariant quantum field in curved spacetime
Robert M. %aid
Enrico Fermi Institute, University of Chicago, Chicago, Illinois 60637
{Received 19 August 1977)
%e analyze a point-separation prescription for renormalizing the stress-energy operator T„„ofa quantum
field in curved spacetime, based on the assumption that the expectation value G(x, x') =
(+(x)iIi(x ') + P(x ')~Ii(x))
has the form of a Hadamard elementary solution. An error is pointed out in the work of Adler, Lieberman,
and Ng: The "locally determined" piece G~(x, x') and "boundary-condition-dependent" piece 6(x, x')
of G(x, x') do not separately satisfy the wave equation in x', as required in their proof of the
conservation of the boundary-condition-dependent contribution to T„„.This error affects the point-separation
renormalization prescription given in my previous paper describing an axiomatic approach to stress-energy
renormalization. It is now seen that this prescription yields a stress-energy tensor whose divergence is not
zero but is the gradient of a local curvature term, However, this deficiency can be corrected by subtracting
off this local curvature term times the metric tensor; as a direct consequence the trace of T„, becomes
nonvanishing. Given this result it is shown that any prescription for renormalizing T,
which is consistent
with conservation (axiom 3), causality (axiom 4), and agreement with the formal expression for the matrix
element between orthogonal states (axiom 1) must yield precisely this trace, modulo the trace of a conserved
local curvature term. Hence, for consistency with the first four axioms and dimensional considerations, we
find that the trace of the stress tensor of the conformally invariant scalar field must be T"„
= ~ ~C ~R — R. This confirmsEnergy conservation: a thermodynamic argument
Energy conservation: a thermodynamic argument
Energy conservation: a thermodynamic argument
FriedmanThe anomaly: a thermodynamic argument
ATURE PHOTONICS DOI: 10.1038/NPHOTON.2011.39
n) c Casimir effect (macroscopic bodies)
p2
p2
ultiple scattering
ose origins lie in the quantum fluctuations of dipoles.
n induces a fluctuating dipole p2 on a nearby particle, leading to van
on/wave effects modify the interaction, leading to Casimir–Polder
o a breakdown of the pairwise force laws. c, In situations consisting
ithin the bodies leads to Casimir forces.Casimir cosmology: astrophysics, analogues & more
Itai Efrat Nikolay Ebel
Dror Berechya
Masters Masters
Masters 2020
2019-21 2020-
PhD student
2021-PHYSICAL REVIEW B 104, 235432 (2021)
Editors’ Suggestion
Van der Waals anomaly: Analog of dark energy with ultracold atoms
Itai Y. Efrat and Ulf Leonhardt
Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 761001, Israel
(Received 16 September 2021; revised 5 December 2021; accepted 7 December 2021; published 27 December 2021)
In inhomogeneous dielectric media the divergence of the electromagnetic stress is related to the gradients of
ε and µ, which is a consequence of Maxwell’s equations. Investigating spherically symmetric media we show
that this seemingly universal relationship is violated for electromagnetic vacuum forces such as the generalized
van der Waals and Casimir forces. The stress needs to acquire an additional anomalous pressure. The anomaly
is a result of renormalization, the need to subtract infinities in the stress for getting a finite, physical force. The
anomalous pressure appears in the stress in media like dark energy appears in the energy-momentum tensor in
general relativity. We propose and analyze an experiment to probe the van der Waals anomaly with ultracold
atoms. The experiment may not only test an unusual phenomenon of quantum forces but also an analog of dark
energy, shedding light where nothing is known empirically.
DOI: 10.1103/PhysRevB.104.235432
I. INTRODUCTION vanish, and the stress would get lost in renormalization. SoComparison with astronomical data
MNRAS 507, 3473–3485 (2021) https://doi.org/10.1093/mnras/stab2345
Advance Access publication 2021 August 6
Lifshitz cosmology: quantum vacuum and Hubble tension
‹
Dror Berechya and Ulf Leonhardt‹
Weizmann Institute of Science, Rehovot 7610001, Israel
Downloaded from https://academic.oup.com/mnras/a
Accepted 2021 August 9. Received 2021 June 15; in original form 2020 November 24
ABSTRACT
Dark energy is one of the greatest scientific mysteries of today. The idea that dark energy originates from quantum vacuum
fluctuations has circulated since the late ’60s, but theoretical estimations of vacuum energy have disagreed with the measured
value by many orders of magnitude, until recently. Lifshitz theory applied to cosmology has produced the correct order of
magnitude for dark energy. Furthermore, the theory is based on well-established and experimentally well-tested grounds in
atomic, molecular and optical physics. In this paper, we confront Lifshitz cosmology with astronomical data. We find that the
dark–energy dynamics predicted by the theory is able to resolve the Hubble tension, the discrepancy between the observed
and predicted Hubble constant within the standard cosmological model. The theory is consistent with supernovae data, Baryon
Acoustic Oscillations and the Cosmic Microwave Background. Our findings indicate that Lifshitz cosmology is a serious
candidate for explaining dark energy.Comparison with Pantheon supernova data
42
40
38
36
RMSD CDM = 0.22
34
RMSDLT = 0.15
0.0 0.1 0.2 0.3 0.4
z
Hubble diagramCasimir cosmology: continuous n(t), varying Λ
parameter w is model-dependent and data that may perhaps reveal systematic CM
w = –1 corresponds to a cosmological errors or shortcomings of the standard the
constant) along the way. The model six-parameter model? met
calibrated on early-Universe observations Several talks meeting
addressed thereport
first question. nex
predicts the present-day value of several In addition to the well-known small the
COSMOLOGYcosmological parameters, some of which difference between the inference of H0 from that
can be empirically measured locally low- and high-angular-resolution Planck of o
Tensions between the early and late Universe
(for z < 1) with little or no model and Wilkinson Microwave Anisotropy Probe para
dependence. In particular, the model data, all of the early-Universe
A Kavli Institute for Theoretical Physics workshop in July 2019 directed attention to the Hubble constant
data seem to som
calibrated
discrepancy. withshowed
New results data from
that it the
doesPlanck
not appear to depend be
mission consistently
on the predicting
use of any one a low
method, team value of H0.
or source. or n
Proposed predicts
solutionsthe Hubble
focused constant,
on the today’s era.
pre-recombination The Atacama Cosmology Telescope (ACT) won
expansion rate, to a remarkable and the South Pole Telescope (SPT) are in earl
N
early a century of cosmological 1% precision, 67.4 ± 0.5 km s–1 Mpc–1. agreement with Planck, and any CMB data
research has led us to a standard Is this whole story right? used to calibrate the sound horizon and
NATURE
model ASTRONOMY
of cosmology, the Λ cold| VOL 3 | OCTOBER 2019
The simplest test| 891–895 | www.nature.com/natureastronomy
of this paradigm, from subsequently the BAOs leads to a low H0
dark matter model, or ΛCDM, with its end to end, is to compare the absolute scale of ~67–68.5 km s–1 Mpc–1, even without
six free parameters and several ansatzes. provided through the application of early- Planck. A completely independent and
This model is dominated by dark Universe physics (for example, the physical statistically consistent value of H0 can be
components (energy and matter) with still size of the sound horizon used to interpret obtained by using light-element abundances
uncertain physics. With some simplifying the CMB and BAOs) to the absolute scale to calibrate the sound horizon, BAOs and
assumptions about the uncertain bits, the measured by the Hubble constant in the other lower-redshift probes.
version of the model calibrated on physics local, late-time Universe. Because the As far as the second question is
prior to recombination (63% dark matter, Universe has only one true scale, and concerned, some curiosities among high-
15% photons, 10% neutrinos and 12% in light of the uncertain physics of the redshift probes at the level of ~2σ were
atoms) is used to predict the physical size of dark sector, comparing the two calibrated identified. The most compelling ones
density fluctuations in the plasma — that is, at opposite ends of the Universe’s history appear to be the departure from unity of
how far sound, or any perturbation in the is natural and potentially insightful. the nuisance parameter Alens, which is used
photon–baryon fluid, could have travelled (To determine whether a measurement to match CMB anisotropies (temperature
from the beginning of the Universe to is truly derived from the ‘early’ or ‘late’ fluctuations around z ≈ 1,000) and
recombination — the sound horizon and its Universe it is necessary to trace back its CMB-lensing data (from the deflection of
overtones, as well as the primordial baryon chain of calibration — a useful check is to CMB photons by gravitational masses,
density. By comparing the fluctuation determine whether or not it depends on, for such as clumps of dark matter). Ifare
arXiv:2112.04510
ΛIndirect determination of Hubble constant from
Cosmic Microwave Background
ΛE. Di Valentino, MNRAS 502, 2065 (2021)
Correlations in the Cosmic Microwave Background
6000
5000
4000
DT T [µK2 ]
3000
2000
1000
0
600 60
300 30
DT T
0 0
-300 -30
-600 -60
2 10 30 500 1000 1500 2000 2500Correlations in the Cosmic Microwave Background
6000
5000
4000
DT T [µK2 ]
3000
2000
1000 d = 46.2 Gly
0 t = 13.8 Gy
600 60
300 30
DT T
0 0
-300 -30
-600 -60
2 10 30 500 1000 1500 2000 2500Correlations in the Cosmic Microwave Background
6000
5000
4000
DT T [µK2 ]
3000
2000
1000
0
600 60
300 30
DT T
0 0
-300 -30
-600 -60
2 10 30 500 1000 1500 2000 2500Observed Hubble constant is consistent with Lifshitz theory
80
75
H0
70
65 Lifshitz theory
0.00 0.01 0.02 0.03 0.04 0.05
Direct measurements of H0 usinge various metods:
Cepheids - SN Ia TRGB - SN Ia Tully Fisher
SBF SN II Time-delay Lensing2
10 theories
Classical and Quantum Gravity
Class. Quantum Grav. 38 (2021) 153001 (110pp) https://doi.org/10.1088/1361-6382/ac086d
Topical Review
In the realm of the Hubble tension—a
∗
review of solutions
Eleonora Di Valentino1,∗∗ , Olga Mena2 , Supriya Pan3 , Luca
4 5 6
Visinelli , Weiqiang Yang , Alessandro Melchiorri ,
7 8,9 8,10,11
David F Mota , Adam G Riess and Joseph Silk
1
Institute for Particle Physics Phenomenology, Department of Physics, Durham
University, Durham DH1 3LE, United Kingdom
2
IFIC, Universidad de Valencia-CSIC, 46071, Valencia, Spain
3
Department of Mathematics, Presidency University, 86/1 College Street, Kolkata2
10 theories, new fields, new GR, new cosmology
Classical and Quantum Gravity
Class. Quantum Grav. 38 (2021) 153001 (110pp) https://doi.org/10.1088/1361-6382/ac086d
Topical Review
In the realm of the Hubble tension—a
∗
review of solutions
Eleonora Di Valentino1,∗∗ , Olga Mena2 , Supriya Pan3 , Luca
4 5 6
Visinelli , Weiqiang Yang , Alessandro Melchiorri ,
7 8,9 8,10,11
David F Mota , Adam G Riess and Joseph Silk
1
Institute for Particle Physics Phenomenology, Department of Physics, Durham
University, Durham DH1 3LE, United Kingdom
2
IFIC, Universidad de Valencia-CSIC, 46071, Valencia, Spain
3
Department of Mathematics, Presidency University, 86/1 College Street, KolkataNo new fields.
No new fields, but new ideas in fields as old as QED
No new fields, but new ideas in fields as old as QED
Why only QED?No new fields, but new ideas in fields as old as QED
Why only QED?
Experimental tests?No new fields, but new ideas in fields as old as QED
Why only QED?
Experimental tests?
Astronomical tensions?No new fields, but new ideas in fields as old as QED
Why only QED?
Experimental tests?
Astronomical tensions?
Renormalization?No new fields, but new ideas in fields as old as QED
Why only QED?
Experimental tests?
Astronomical tensions?
Renormalization?
…Some say in ice…”*
*Robert Frost, Fire and Ice
What is the fate of the Universe? Probably it will end in ice if we are to believe this year’s Nobel Laureates.
They have carefully studied several dozen exploding stars, called supernovae, in faraway galaxies and
have concluded that the expansion of the Universe is speeding up.
The discovery came as a complete surprise even to the Nobel Laureates themselves. What they saw would be
like throwing a ball up in the air, and instead of having it come back down, watching as it disappears more
and more rapidly into the sky, as if gravity could not manage to reverse the ball’s trajectory. Something simi-
lar seemed to be happening across the entire Universe.
dark energy
gravitation
galaxy
supernova
acceleration acceleration
big bang decreases increases
universe
percent
100
80
matter
60
40 dark matter
20 dark energy
0
on.
14 billion 5 billion presentSome say in ice…”*
*Robert Frost, Fire and Ice
Let there be light!
What is the fate of the Universe? Probably it will end in ice if we are to believe this year’s Nobel Laureates.
They have carefully studied several dozen exploding stars, called supernovae, in faraway galaxies and
have concluded that the expansion of the Universe is speeding up.
The discovery came as a complete surprise even to the Nobel Laureates themselves. What they saw would be
like throwing a ball up in the air, and instead of having it come back down, watching as it disappears more
and more rapidly into the sky, as if gravity could not manage to reverse the ball’s trajectory. Something simi-
lar seemed to be happening across the entire Universe.
dark energy
gravitation
galaxy
supernova
acceleration acceleration
big bang decreases increases
universe
percent
100
80
matter
60
40 dark matter
20 dark energy
0
on.
14 billion 5 billion presentYou can also read
