Gravity-Driven Cosmology
N. C. Tsamis (Crete)
R. P. Woodard (Florida)
CRETE-09-11, CRETE-09-13
1
Particle Production
• pair production from the vacuum: 0 → 2E
⇒
energy non-conservation
• uncertainty relation: ∆E ∆t ≥ h̄
h̄ violation is not detectable
⇒
∀ ∆t ≤ 2E
• extension to cosmological spacetimes:
E(t, k) =
q
m2 c4 + h̄2 c2 |k|2 a−2(t)
Z t+∆t
t
dt′ 2E(t′ , k) ≤ h̄
⊲ virtual particle lifetime increases as the mass decreases
⊲ virtual particle lifetime increases if a(t) grows
2
• the de Sitter spacetime case: a(t) = eH0 t
⊲ massless lifetime bound:
−H0 ∆t
h
2 c |kphys| × 1 − e
i
≤ H0
⊲ thus, massless virtual particles with:
k
| ≤ H0
a(t)
⇒
real particle production
c |kphys| = c |
may never recombine
• what is the rate for such massless particle production?
key distinguishing principle is conformal invariance
3
• conformal transformations with parameter Ω(x):
gµν → Ω2 gµν
,
Aµ → Aµ
,
3
−2
ψb → Ω
ψb
,
φ → Ω−1 φ
• conformal co-ordinates: dη = a−1(t) dt
⇒ conformal metric tensor: gµν (η, x) = a2(η) ηµν
⇒ a conformally invariant lagrangian is the same as in
flat space when expressed in terms of conformally
rescaled fields with Ω = a−1
⇒ production rates from such massless theories are highly
suppressed for growing a(t):
dη dn
Γ
dn
=
=
dt
dt dη
a
Γ ≡
number of virtual particles emerging per unit conformal time η
4
η
_
H −1
Short wavelength (λphys < H −1 ) graviton pairs (violet) recombine.
Long wavelength (λphys > H −1) graviton pairs (red) cannot recombine.
• graviton uniqueness: massless, non-conformally invariant
also massless & minimally coupled scalar, exotics
5
• how many such gravitons are produced?
⊲ number of infrared gravitons of a particular mode:
NIR(t, k) ∼
!2
H
t
0
H0 e
2k
≫ 1
⊲ number of hubble volumes at time t :
NH (t) = e3H0 t ≫ 1
⊲ number of infrared gravitons of any kind per hubble volume
is about one
• what is their effect on a local observer?
⊲ infrared graviton production ⇒ source imprint
⇒ secular effect ⇒ gravitational response
• which observables capture the effect?
expansion rate, acceleration observable
6
A Newtonian Model
• classical background: de Sitter spacetime
⊲ causal horizon at H0−1
⊲ manifold T 3 × ℜ
• free graviton kinematics: infrared
⊲ degrees of freedom are polarization and wave number k
⊲ infrared modes have H0 ≤ k ≤ H0 eH0t
• free graviton dynamics: massless minimally coupled scalar
⊲ go to co-ordinates where background becomes conformally
flat
⊲ time evolution of mode k is that of a harmonic oscillator
of frequency k and time dependent mass H0 a2(η)
⊲ assume conformal flat space vacuum and do QM on this
system
7
• physical energy of infrared mode k at time t:
H02 H0 t
1 −H0 t
ke
+
e
Ek =
2
4k
• newtonian energy density and potential of mode k at t:
k e−H0 t
Ek − 1
H04 −2H0 t
2
=
e
ρk =
V3(t)
4k
⇒
π GH05
ϕk = −
k3
• total infrared newtonian energy density and potential at t:
Z H exp[H t]
0
0
H04
1
2
ρIR =
dk k ρk =
3
2
8π 2
π H0 H0
Z H exp[H t]
0
0
GH02
1
2
dk k ϕk = −
ϕIR =
H0 t
3
2
π
π H0 H0
8
• total infrared newtonian interaction energy density at t:
GH06
H0 t
ρnewton = ρIR × ϕIR = −
3
8π
⊲ ρnewton is negative and ever-increasing
⊲ ρnewton starts much smaller than ρIR:
ρnewton
∼ GH02 (H0 t)
,
GH02 ≤ 10−12
ρIR
but eventually dominates
⊲ agreement with quantum gravitational perturbative result
9
Quantum Gravity
• effective lagrangian and parameters
√
1
LGR =
(−2Λ + R ) −g + (counterterms)
16πG
⊲ classical gravity without matter only “knows” about Λ
⊲ the strength of quantum effects is set by G
⊲ the corresponding mass scales are:
1
Λ 4
,
M ≡
G
8πG
⊲ the dimensionless coupling constant is: ǫ ≡ GΛ
⊲ perturbation theory is valid iff :
1
2
MPl
≡
ǫ ≡ GΛ < 1
⇔
M
MPl
!4
< 1
10
• background geometry:
ds2 = −dt2 + a2(t) dx · dx = −dt2 + exp [ 2 b(t) ] dx · dx
= a2(η) (−dη 2 + dx · dx)
• quantum corrections:
h state | γµν (t, x) dxµdxν | state i = gµν (t, x) dxµ dxν
γµν (t, x)
is the metric operator
• quantum-induced stress tensor:
⊲ imprint of geometrically significant differences between
classical and quantum backgrounds
⊲ defined from the deficit by which the quantum background
gµν fails to obey the classical equations of motion:
1
8πG Tµν ≡ Rµν − gµν R + gµν Λ
2
11
• quantum-induced energy density and pressure:
T00(t) = −ρ(t) g00
,
i
1 h 2
ḃ (t) − Λ
ρ(t) =
8πG
T0i(t) = 0
,
,
Tij (t) = p(t) gij
i
1 h
p(t) =
−2b̈(t) − ρ(t)
8πG
• quantum-induced expansion rate:
a′(η)
ȧ(t)
H(t) ≡
= ḃ(t) = 2
=
a(t)
a (η)
s
8πG
Λ
+
ρ(t)
3
3
12
• perturbative results for the de Sitter background
adS (t) = ebdS (t) = eH0 t = adS (η) = −
1
H0 η
,
H02 ≡
1
Λ>0
3
−1
i ≤ 1 H −1
• spacetime manifold is T 3 × ℜ: − 1
H
<
x
2 0
2 0
(one causal volume)
• initial value problem:
bdS (0) = 0
,
ḃdS (0) = H0
| statei = | Bunch Davies vacuum at t = 0 i ≡ |0i
• fluctuating field ψµν (x):
γµν ≡ a2 (ηµν + κ ψµν )
,
κ2 ≡ 16πG
13
• results for large observation times (infrared limit) :
ρdS (t) = −ǫH04 [ # (H0 t) + O(1) ] + O(ǫ2)
pdS (t) = ǫH04 [ # (H0 t) + O(1) ] + O(ǫ2)
HdS (t) = H0
n
1 − ǫ2
h
#′ (H0 t) + O(1)
i
o
3
+ O(ǫ )
⊲ the rate of expansion decreases by an amount which
becomes non-perturbatively large at late times
⊲ the perturbation theory breakdown occurs when the
effective coupling constant becomes of order one :
ǫ2 H0 t1 ∼ 1
⇒
MPl 8
N1 ≡ H0 t1 ∼
≫ 60
M
14
• why does the effect start at two loops?
⊲ screening represents the gravitational attraction between
virtual infrared gravitons ripped from the vacuum
⊲ the graviton production process is a one-loop effect
⊲ the gravitational response to this production cannot occur
until the next loop order
• infrared logarithms (ln a)
⊲ factors of (H0 t) = ln a that appear in various perturbative
results
15
• infrared logarithms rule :
in an interaction involving N undifferentiated gravitons, along
with any number of other fields, each new factor of the coupling constant squared can produce at most N additional
infrared logarithms
⊲ e.g., quantum gravity in de Sitter background:
√
basic interaction :
G h ∂h ∂h
each additional power of G brings at most one extra infrared
logarithm ⇒
for instance, the general form of the H(t) corrections is:
n
H(t) = H0 1 −
∞
X
ℓ=2
ǫ
ℓ
ℓ−1
X
k=0
k
cℓk (H0 t)
o
ℓ is the loop order
cℓk are pure numbers of O(1)
16
• non-perturbative method
⊲ summation of leading infrared logarithms series:
n
H(t)|leading log = H0 1 − ǫ
∞
X
ℓ−1
cℓ, ℓ−1 (ǫH0 t)
ℓ=2
o
⊲ Starobinskiı̆ developed a stochastic technique which
sums the leading infrared logarithms of scalar λφ4
• non-perturbative method extensions
⊲ scalar models with bounded below potentials
(small, constant increase of the vacuum energy)
⊲ yukawa theory
(unbounded decrease of the vacuum energy)
⊲ scalar quantum electrodynamics
(small, constant decrease of the vacuum energy)
⊲ quantum gravity?
(unknown yet, very complicated)
• another approach: phenomenological construction
17
Gravitational versus Scalar Inflation
• initial conditions
⊲ gravitational case:
Λ>0
,
matter | < Λ
|Tµν
(i.e. matter stress-energy does not “overwhelm” Λ)
⇒ both are robust conditions
⊲ scalar case: inflaton field must be homogeneous over
more than one hubble region and without kinetic energy
⇒ quite unlikely
• potential issues
⊲ scalar case: inflaton potential needs to be flat
⇒ constraints on couplings
gravity case: Λ is a constant
⊲ scalar case: inflaton potential needs Vmin ∼ 0
⇒ cosmological constant problem
gravity case: screening of Λ
18
• potential issues (contd.)
⊲ scalar case: inflaton potential is arbitrary
⇒ not predictive
gravity case: gravitation exists
• reheating issues
⊲ scalar case: inflaton transfers energy via coupling to matter
⇒ different couplings to different matter fields
⇒ generation of Veff from quantum corrections
⊲ scalar Veff example:
– start with V (φ) = λ φ4 , |λ| < 10−14
– couple φ to matter with, say, strength g
⇒ generation of Veff = ± g 2 φ4 ln(φ/µ)
( + for bosons, − for fermions; instability for the latter for large φ )
h
i
2
λ ± g ln(φ/µ) φ4
⇒ Vtot =
⇒ fine-tuning to make bracketed term very small
19
• reheating issues (contd.)
⊲ gravity case: gravitation couples universally but is weak
⇒ normally, reheating is generated non-gravitationally
→ exception: if - besides the zero-mode - all the modes
participate in the reheating, gravity can become strong
and capable of naturally potent reheating
→ danger: gravity may reheat the universe before it stopped
inflating
20
A Simple Model for Early Cosmology
• the phenomenological goal:
what is the most cosmologically significant part of the
effective field equations?
1
Gµν ≡ Rµν − gµν R = −Λ gµν + 8πG Tµν [g]
2
• the perfect fluid form:
Tµν [g] = (ρ + p) uµ uν + p gµν
is determined completely by providing:
(i) the gravitationally induced energy density ρ[g](x)
(ii) the gravitationally induced energy pressure p[g](x)
(iii) the 4-velocity field uµ[g](x):
g µν uµuν = −1
(timelike and normalized)
21
• why go to the effective field equations directly?
⊲ very hard to identify “in-in” effective action terms
⊲ avoid changes in the effective newton constant G
• the perfect fluid conservation:
D µ Tµν = 0
there are 4 conservation equations and 5 independent
quantities in Tµν (ρ, p, normalized uµ)
⇒ must only specify one quantity; we choose the pressure
22
• requirements on the pressure:
⊲ should not alter the initial value problem
⊲ should be non-local,
– effect is inherently non-local
– any local modification simply renormalizes Λ
⊲ should be simple,
– a simple non-local operator is the inverse of:
√
1
µν
≡ √
−g ∂ν )
∂µ ( g
−g
(defined with retarded boundary conditions)
– a simple scalar it can act on is the curvature scalar R:
X ≡
1
R
⊲ should reproduce our perturbative de Sitter result
23
• relevant spacetimes and parameters:
ds2 = gµν (t) dxµdxν = −dt2 + a2(t) dx · dx
d
ȧ(t)
=
ln a(t)
a(t)
dt
H(t) ≡
q(t) ≡ −
T00 = ρ
,
T0i = 0
R = 12H 2 + 6Ḣ
X =
1
,
Ḣ(t)
a(t) ä(t)
=
−1
−
ȧ2(t)
H 2(t)
,
1
Tij = gij p
uµ = −δµ0
;
t′
1
= −
dt′ 3 ′
dt′′ a3(t′′ )
a (t ) 0
0
Z t
Z
Z t′
h
i
1
′′ 3 ′′
2 ′′
2 ′′
′
R = −
dt a (t ) 12H (t ) + 6Ḣ (t )
dt 3 ′
a (t ) 0
0
Z t
24
• the field and conservation equations:
3H 2 = Λ + 8πG ρ
−2Ḣ − 3H 2 = −Λ + 8πG p
ρ̇ = −3H (ρ + p)
⇒
t
1
ρ(t) = −p(t) + 3
dt′ a3(t′ ) ṗ(t′ )
a (t) 0
Z
• the de Sitter correspondence limit:
HdS (t) = H0 > 0
RdS = 12H 2
,
,
1
qdS (t) = −1
|dS = −
Z t
0
,
dt′ e−3H0
adS (t) = eH0 t
t′
Z t′
0
′′
dt′′ e3H0 t
i
4h
−3H
t
0
XdS = −4H0 t +
1−e
≃ −4 ln[ adS (t) ] + O(1)
3
25
• the de Sitter physical ansatz for the source:
p[gdS ](t) = Λ2 f [−ǫ XdS ](t)
,
ǫ ≡ GΛ
reproduces the monotonically unbounded perturbative result:
(up to a positive coefficient)
H 2 ≃ H02 { 1 − 32πǫ2 ln(adS ) + O(ǫ3) }
provided f is some monotonically unbounded function:
f [−ǫ XdS ] = −ǫ XdS + O(ǫ2)
⊲ XdS is monotonically increasing and negative
⊲ f, p are monotonically increasing and positive
26
• the general physical ansatz for the source
for a general geometry, we take the gravitationally induced
pressure to equal:
p[g](x) = Λ2 f [−ǫ X](x)
,
X ≡
1
R
f [−ǫ X] = −ǫ X + O(ǫ2)
⊲ given p we determine ρ and uµ by conservation
(up to their initial value data)
• the linear and exponential models
two simple choices for f are:
(i) the linear model :
(ii) the exponential model :
f [−ǫ X] ≡ −ǫ X
f [−ǫ X] ≡ e−ǫ X − 1
27
• numerical evolution
⊲ the relevant equation is:
2Ḣ + 3H 2 = 3H02 { 1 − 8πǫ f [−ǫ X] }
,
X ≡
1
R
⊲ its discretization involves constants:
step size in Hubble units ⇒
δ ≡ H0 ∆t
ǫ ≡ GΛ = 3GH02
coupling constant ⇒
⊲ and the basic variables with their initial value data:
a(t) → a(i ∆t) = ebi
(ρ + p)(t) → [ρ + p](i ∆t)
,
,
b0 = 0
[ρ + p]0 = 0
[ρ + p]i+1 = e−3bi { [ρ + p]i − ǫ ∆Xi f ′[−ǫ Xi ] }
⊲ all quantities of interest and their initial values
can be determined from the above
28
• numerical results
⊲ discretized evolution equation:
3 2
∆ bi = [ δ − (∆bi)2 ] − 12πδ 2ǫ f [−ǫ Xi ]
2
2
⊲ results are for the exponential model:
f (x) = ex − 1
f −1(x) = ln(1 + x)
⇒
,
f ′ (x) = ex
for the following choice of input parameters and step range:
δ =
1
1000
,
ǫ =
1
200
;
i ∈ [0, 350000]
⊲ numerical integration used MATHEMATICA
29
• the source behaviour: growth → oscillations
X
50 000
100 000
150 000
200 000
250 000
300 000
t
350 000
-100
-200
-300
-400
X
-437.5
-438.0
-438.5
200 000
250 000
300 000
t
350 000
30
• the curvature behaviour: decrease → oscillations
R
R
0.000012
6. ´ 10-8
0.00001
4. ´ 10-8
8. ´ 10-6
2. ´ 10-8
6. ´ 10-6
4. ´ 10-6
200 000
2. ´ 10-6
250 000
300 000
t
350 000
-2. ´ 10-8
50 000
t
100 000 150 000 200 000 250 000 300 000 350 000
-4. ´ 10-8
⊲ oscillations are centered around R = 0
⊲ oscillations have an envelope falling like t−1
31
• the expansion rate behaviour: decrease → oscillations
H
H
0.0010
0.00006
0.0008
0.00005
0.00004
0.0006
0.00003
0.0004
0.00002
0.0002
0.00001
50 000
t
100 000 150 000 200 000 250 000 300 000 350 000
200 000
250 000
300 000
t
350 000
⊲ there is net expansion
⊲ there are short periods of H < 0 (novel feature)
32
• the Ḣ(t) behaviour: increase → oscillations
è
è
H
H
5. ´ 10-9
5. ´ 10-9
t
50 000 100 000 150 000 200 000 250 000 300 000 350 000
200 000
-5. ´ 10-9
-5. ´ 10-9
-1. ´ 10-8
-1. ´ 10-8
250 000
300 000
t
350 000
⊲ during the oscillations era we see that:
R(t) and 6Ḣ(t) are almost equal ⇒ R = 12H 2 + 6Ḣ ≃ 6Ḣ
33
• the end of inflation: q(t) goes to positive values
q
300
200
100
200 000
250 000
300 000
t
350 000
-100
-200
-300
q
1.0
0.5
110 000
120 000
130 000
140 000
150 000
160 000
t
170 000
-0.5
-1.0
34
• the scale factor:
30
25
20
15
10
5
200 000
250 000
300 000
350 000
⊲ the evolution of the scale factor ratio [a(t)/a(150000)]
during the oscillation regime versus a linear interpolation
35
• analytical results
• criticality:
⊲ f is monotonically unbounded, therefore
∃ Xcr :
1 − 8πǫ f [−ǫ Xcr ] = 0
⇒
Xcr = −
1 −1 1
f (
)
ǫ
8πǫ
⊲ inflationary evolution dominates roughly until criticality
⊲ close to criticality the induced pressure p is small ⇒
can expand f around Xcr and perturbatively solve for the
subsequent evolution:
2Ḣ + 3H 2 = 3H02 { 1 − 8πǫ f [−ǫ Xcr − ǫ(X − Xcr )] }
≃ 24πǫ2 H02 (X − Xcr ) f ′[−ǫ Xcr ]
⊲ resulting evolution equation is that of a damped oscillator:
2
R̈ + 2H Ṙ + (ω − Ḣ ) R ≃ 0
,
ω ≡ ǫH0
q
′
72π fcr
36
• asymptotic solution:
lim a(t) ≃
t≫1
lim H(t) ≃
t≫1
lim Ḣ(t) ≃
t≫1
lim R(t) ≃
t≫1
K1
1
K3 + K2 t −
sin(ωt + ϕ) + O( )
2
6ω
t
1
K1 cos(ωt + ϕ)
1
−
+ O( 2 )
t
6K2 ω
t
t
K1 sin(ωt + ϕ)
1
+ O( 2 )
6K2
t
t
1
K1 sin(ωt + ϕ)
+ O( 2 )
K2
t
t
• analytical predictions:
⊲ as inflation exit approaches, oscillations become significant
1
⊲ inflation ended before criticality was reached: qcr = + 2
⊲ at criticality the system is underdamped
⊲ during this evolution: H 2 < |Ḣ| < ω 2
37
• consistency with numerical analysis: criticality
⊲ numerical side:
Criticality
0.030
0.025
0.020
0.015
0.010
0.005
155 000
160 000
165 000
170 000
175 000
t
180 000
-0.005
Determining the critical point 1 − 8πǫf [−ǫXi ] = 0 for the exponential model
at the critical point (step i = 160942):
X[160942] = −438.50
,
q[160942] = 0.50
⊲ analytical side: complete agreement
1
1
Xcr = − ln (1 +
) ∼ − 438.50
ǫ
8πǫ
,
1
qcr = +
2
38
• consistency with numerical analysis: frequency
⊲ numerical side:
R
-8
6. ´ 10
4. ´ 10-8
2. ´ 10-8
200 000
250 000
300 000
t
350 000
-2. ´ 10-8
-4. ´ 10-8
Six oscillations have occured between steps i = 174291 and i = 342478
342478 − 174291
2π
T =
∆t =
6
ω
⇒
2.24 × 10−4
ω =
∆t
⊲ analytical side: complete agreement
ǫδ q
ǫδ h
1 i 21
2.25 × 10−4
′
ω =
72π fcr =
72π (1 +
∼
)
∆t
∆t
8πǫ
∆t
39
• scalar perturbations
• tensor perturbations
• long-range force
• reheating
40
An Improved Model for Cosmology
• post-inflationary evolution
must consider constant ε spacetimes:
Ḣ
ε ≡ − 2
H
⇒
εdS = 0 ,
εrad = 2 ,
a(t) = ain [ 1 + ε Hin(t −
3
εmat =
2
1
tin) ] ε
Hin
H(t) =
1 + ε Hin(t − tin)
R(t) = 6 (2 − ε) H 2(t)
R(t) = +36ε (1 − ε)(2 − ε) H 4(t)
R00(t) = −3 (1 − ε) H 2(t)
41
• simple model inadequacy
⊲ oscillations after the end of inflation are not a problem
⊲ average expansion a(t) ∼ t after the end of inflation
⇒
no reheating
⇒
a problem
⊲ assume, as usual, that energy flows from the gravitational
to the matter sector
• transitions
⊲ inflation → radiation, say at t = tr
⊲ radiation → matter, say at t = tm
42
• the total pressure
⊲ recall simple induced pressure ansatz:
2
p[g](x) = Λ f [−ǫ X](x)
,
X ≡
1
R
f [−ǫ X] = −ǫ X + O(ǫ2)
⊲ the total pressure equals:
Λ
+ p[g](x)
8πG
o
Λ n
1 − 8πGΛ f [ − GΛ (Xcr + ∆X) ]
= −
8πG
Λ
′
≃ − × (GΛ)2 fcr
∆X
G
ptot = −
⊲ source change after transition to matter domination:
3
4
∆X(t) ≡ X(t) − Xcr = − ln[ 1 + Hm(t − tm) ] + O(1)
3
2
43
• the sign problem
⊲ since f is monotonically increasing and unbounded
⇒
ptot > 0
when
X(t) < Xcr ≪ 0
observation implies the reverse
⊲ need f to be monotonically increasing and unbounded
to cancel an arbitrary bare Λ
• the magnitude problem
⊲ total pressure magnitude produced is unacceptably large:
ptot
≃
pnow
GΛ H0 2 ′
′ × ∆X
fcr ∆X ≃ 1086 × fcr
Hnow
3 H2
⊲ have used: pnow ≃ − 8πG
now
H0 ∼ 1013GeV
,
Hnow ∼ 10−33eV
′ ≥1
⊲ for all models analyzed: fcr
44
• decreasing the magnitude
⊲ in our induced pressure ansatz Λ = 3H02
H0 can be 55 orders larger than Hnow
⊲ replace a Λ with a dynamical scalar S without disturbing
the relaxation mechanism:
p[g](x) = Λ2 f [−GΛ X](x) :
−GΛ X = −GΛ
1
R
−→
−G
1
(R × S ) = −
GΛ
S
R×
Λ
45
• changing the sign
⊲ the curvature scalar R is positive during both inflation
and matter domination
⊲ simple quantities behaviour:
R
R
R00
INFLATION
RADIATION
MATTER
+12H 2
0
+ 3H 2
0
0
−3H 2
3H 2
27 4
−
H
2
3 2
+ H
2
⊲ simplest choice that does the job seems to be R00
46
• improved ansatz
⊲ the scalar S must be evaluated far back in the past
⇒
use integral curves χµ [g](x) of a timelike
4-velocity field V µ(x)
⇒
...
⊲ we shall consider p[g](x) = Λ2 f [−GΛ Y ](x) where:
1 1 n
Y [g](x) ≡
R(x) ×
Λ
o
µ
∗
ν
∗
∗
− V (χ(τ , x)) V (χ(τ , x)) Rµν (χ(τ , x))
⊲ for F RW spacetimes and for τ ∗[g](x) associated with,
9 of the time from xµ back to the IVS:
e.g. 10
1 1 1
R(t) × R00( 10 t) ≡ Xcr + ∆Y
YF RW [g](t) = −
Λ
( V µ → δ µ0 )
47
• late time acceleration
⊲ compute total pressure in the improved ansatz:
′
′
2
ptot ≃ − GΛ3 fcr
∆Y ≃ − 200 GΛ2 fcr
Hm
1
′ =
⊲ for the exponential model: fcr
8πGΛ
⊲ the pressure ratio is:
t ≫ tm
⇒
2
H
200
ptot
m
′ ×
8π(GΛ)2 × fcr
≃
pnow
3
Hnow
200
′
≃
8π(GΛ)2 × fcr
× 1010
3
2
≃ × 1012 × GΛ
3
⊲ physical values of GΛ = M 4 MP−4
l easily achieve equality
48
• prospects, etc
49