Instability of de Sitter space and corpuscular nature of gravity

I   S  

C N  G

Wolfgang Mück U  N “F II” INFN, S  N TFI 2017 — Parma

    (  )

G, H 

any geodesic observer in dS feels an isotropic heat bath of particles, T =_2πH,

horizon is thermodynamically stable

G, P 

spontaneous nucleation of black holes,r∼ 1/H, S2_{× S}2 _{instanton, nucleation rate ∼ exp(−π/ΛG)}

A, I, T 

graviton propagator in dS is infrared divergent, quantum corrections change dS to Minkowski

. . . P, R, S -

accelerated expansion of the universe, ∼ 68% dark energy,

    (  )

G, H 

any geodesic observer in dS feels an isotropic heat bath of particles, T =_2πH,

horizon is thermodynamically stable G, P 

spontaneous nucleation of black holes,r∼ 1/H, S2_{× S}2 _{instanton, nucleation rate ∼ exp(−π/ΛG)}

A, I, T 

graviton propagator in dS is infrared divergent, quantum corrections change dS to Minkowski

. . . P, R, S -

accelerated expansion of the universe, ∼ 68% dark energy,

    (  )

G, H 

any geodesic observer in dS feels an isotropic heat bath of particles, T =_2πH,

horizon is thermodynamically stable G, P 

spontaneous nucleation of black holes,r∼ 1/H, S2_{× S}2 _{instanton, nucleation rate ∼ exp(−π/ΛG)}

A, I, T 

graviton propagator in dS is infrared divergent, quantum corrections change dS to Minkowski

. . .

P, R, S -

accelerated expansion of the universe, ∼ 68% dark energy,

    (  )

G, H 

any geodesic observer in dS feels an isotropic heat bath of particles, T =_2πH,

horizon is thermodynamically stable G, P 

spontaneous nucleation of black holes,r∼ 1/H, S2_{× S}2 _{instanton, nucleation rate ∼ exp(−π/ΛG)}

A, I, T 

graviton propagator in dS is infrared divergent, quantum corrections change dS to Minkowski

. . . P, R, S -

accelerated expansion of the universe, ∼ 68% dark energy,

    (  )

P 

physical dS vacuum should be time–asymmetric, cosmological constant evaporates

G, S 

IR divergences in dS lead to large loop corrections, IR regime requires non–perturbative treatment (similarity with information paradox in BHs) P 

IR divergences in dS lead to catastrophic particle production and breaking of dS symmetry

D G 

constant Λ > 0 is incompatible with corpuscular picture of dS (“quantum N–portrait”),

decay of coherent graviton state by condensate depletion

    (  )

P 

physical dS vacuum should be time–asymmetric, cosmological constant evaporates

G, S 

IR divergences in dS lead to large loop corrections, IR regime requires non–perturbative treatment (similarity with information paradox in BHs) P 

IR divergences in dS lead to catastrophic particle production and breaking of dS symmetry

D G 

constant Λ > 0 is incompatible with corpuscular picture of dS (“quantum N–portrait”),

decay of coherent graviton state by condensate depletion

    (  )

P 

physical dS vacuum should be time–asymmetric, cosmological constant evaporates

G, S 

IR divergences in dS lead to large loop corrections, IR regime requires non–perturbative treatment (similarity with information paradox in BHs) P 

IR divergences in dS lead to catastrophic particle production and breaking of dS symmetry

D G 

constant Λ > 0 is incompatible with corpuscular picture of dS (“quantum N–portrait”),

decay of coherent graviton state by condensate depletion

    (  )

R 

IR divergence in graviton propagator is removed by

spontaneous deformation of dS background

F, S, S 

cosmological expansion in dS produces soft gravitons, which change the vacuum,

dS Page timetdS∼ M2pH−3 is the relevant time scale D, G, Z 

dS is a classical approximation (coherent state) of some quantum evolution,

quantum break time tdS∼ Mp2H−3 M 

renormalized stress energy tensor for conformally coupled scalar in dS implies

˙

    (  )

R 

IR divergence in graviton propagator is removed by

spontaneous deformation of dS background

F, S, S 

cosmological expansion in dS produces soft gravitons, which change the vacuum,

dS Page timetdS∼ M2pH−3 is the relevant time scale

D, G, Z 

dS is a classical approximation (coherent state) of some quantum evolution,

quantum break time tdS∼ Mp2H−3 M 

renormalized stress energy tensor for conformally coupled scalar in dS implies

˙

    (  )

R 

IR divergence in graviton propagator is removed by

spontaneous deformation of dS background

F, S, S 

cosmological expansion in dS produces soft gravitons, which change the vacuum,

dS Page timetdS∼ M2pH−3 is the relevant time scale D, G, Z 

dS is a classical approximation (coherent state) of some quantum evolution,

quantum break time tdS∼ M2pH−3

M 

renormalized stress energy tensor for conformally coupled scalar in dS implies

˙

    (  )

R 

IR divergence in graviton propagator is removed by

spontaneous deformation of dS background

F, S, S 

cosmological expansion in dS produces soft gravitons, which change the vacuum,

dS Page timetdS∼ M2pH−3 is the relevant time scale D, G, Z 

dS is a classical approximation (coherent state) of some quantum evolution,

quantum break time tdS∼ M2pH−3 M 

renormalized stress energy tensor for conformally coupled scalar in dS implies

˙

instability of dS

classicalization corpuscular picture soft gravitons asymptotic symmetries gravitational memory spontaneous symmetry breaking horizon thermodynamics (BH) evaporation

O

 _{peculiar things we know}

Unruh effect, BH evaporation

 _{de Sitter instability}

review of two approaches

 _{corpuscular picture of gravity}

QFT   –

peculiarities

notions of particles and the vaccum are observer dependent Unruh effect

cosmological particle production presence of horizons

entanglement, mixed states, open quantum systems thermodynamics, Hawking radiation

information paradox, firewall argument

QFT   –

peculiarities

notions of particles and the vaccum are observer dependent Unruh effect

cosmological particle production presence of horizons

entanglement, mixed states, open quantum systems thermodynamics, Hawking radiation

information paradox, firewall argument It’s not Quantum Gravity

QFT   –

“failure” of QFT at horizons

BHs: classical background = coarse grained picture breaks down at the horizon

fuzzballs (Mathur) quantumN–portrait (Dvali, Gomez) BH = BEC of gravitons N∼ (M/Mp)2 λ∼ rs cosmological (apparent) horizons ?

QFT   –

“failure” of QFT at horizons

BHs: classical background = coarse grained picture breaks down at the horizon

fuzzballs (Mathur) quantumN–portrait (Dvali, Gomez) BH = BEC of gravitons N∼ (M/Mp)2 λ∼ rs cosmological (apparent) horizons ?

QFT   –

“failure” of QFT at horizons

BHs: classical background = coarse grained picture breaks down at the horizon

fuzzballs (Mathur) quantumN–portrait (Dvali, Gomez) BH = BEC of gravitons N∼ (M/Mp)2 λ∼ rs cosmological (apparent) horizons ?

QFT   –

“failure” of QFT at horizons

BHs: classical background = coarse grained picture breaks down at the horizon

fuzzballs (Mathur) quantumN–portrait (Dvali, Gomez) BH = BEC of gravitons N∼ (M/Mp)2 λ∼ rs cosmological (apparent) horizons ?

U 

[B, D][W]

massless scalar in 2d Minkowski space

ds2_{= −dU dV} U = T − X V = T + X

positive frequency modes (ω > 0)

uM_in,ω∼ e−iωU uM_out,ω∼ e−iωV

V₌ 0 U=0 V U φM= X ω>0 

uMin,ωain,ω+uMin,ωa†in,ω+uMout,ωaout,ω+uMout,ωa†out,ω 

U 

R

massless scalar in a Rindler wedge (R)

U = −1 ae −au_{<0 V =} 1 ae av_>0 ds2_{= −e}a(v−u)_{du dv} positive frequency modes (σ > 0)

uR_in,σ∼ e−iσu uR_out,σ∼ e−iσv

φR=X

σ>0

uR_in,σbR_in,σ+uR_out,σbR_out,σ+c.c.

U 

L

massless scalar in a Rindler wedge (L)

U = 1 ae av_{>0 V = −}1 ae au _<0 ds2_{= −e}a(v−u)_{du dv} positive frequency modes (σ > 0)

uL_in,σ∼ e−iσu uL_out,σ∼ e−iσv

φL=X

σ>0

uL_in,σbL_in,σ+uL_out,σbL_out,σ+c.c.

U 

Minkowski Hilbert space in terms of Rindler modes

uRout,σ∼      e−iσ a ln(aV) V > 0 0 V < 0 uL_in,σ∼      0 V > 0 eiσa ln(−aV) V < 0 uR out,σ , uLin,σ R–out, L–in sector

L

R

φM= X

σ>0

U 

Fourier components with respect to V

uR_out,σ(±ω)∼ Z_∞

0

e±iωVe−iσa ln(aV)dV

uR_out,σ(ω)∼ i eπσ2a Z_∞ 0 dy e−ωye−iσa ln(ay) uR_out,σ(−ω)∼ −i e−πσ2a Z_∞ 0 dy e−ωye−iσa ln(ay) V V repeat for uL

in,σ,uRout,σ,uLin,σ

positive frequency combinations

uR_out,σ+e−πσa _uL

in,σ uLin,σ+e −πσ

a _uR

U 

Fourier components with respect to V

uR_out,σ(±ω)∼ Z_∞

0

e±iωVe−iσa ln(aV)dV

uR_out,σ(ω)∼ i eπσ2a Z_∞ 0 dy e−ωye−iσa ln(ay) uR_out,σ(−ω)∼ −i e−πσ2a Z_∞ 0 dy e−ωye−iσa ln(ay) V V repeat for uL

in,σ,uRout,σ,uLin,σ

positive frequency combinations

uR_out,σ+e−πσa _uL

in,σ uLin,σ+e −πσ

a _uR

U 

Bogoliubov transformation d1σ=  1 − e−2πσa −1/2 bRout,σ−e− πσ a _bL in,σ†
d2σ=  1 − e−2πσa −1/2 bL_in,σ−e−πσa bR out,σ†

|Mi is entangled state in |Ri × |Li Fock space

d1

σ|Mi = d2σ|Mi = 0 ⇒ |Mout,σi = X n pn(σ)|nRout,σi ×|nLin,σi pn(σ) = 1 − e− σ T
−1/2e− nσ 2T T = a 2π thermal density matrix

ρ_R=trL|Mout,σihMout,σ| = X

n

|pn(σ)|2 |nRout,σihnRout,σ|

Unruh temperature

U 

R L

R: ds2 _{= −e}av _{dv dU}

|Mi thermal for R–out modes vacuum for R–in modes

R

L R: ds2 = −e−au du dV

|Mi thermal for R–in modes vacuum for R–out modes

R L

R: ds2 _{= −e}a(v−u)_{dv du}

|Mi thermal for R–out modes thermal for R–in modes

U 

R L

R: ds2 _{= −e}av _{dv dU}

|Mi thermal for R–out modes vacuum for R–in modes

R

L R: ds2 = −e−au du dV

|Mi thermal for R–in modes vacuum for R–out modes

R L

R: ds2 _{= −e}a(v−u)_{dv du}

|Mi thermal for R–out modes thermal for R–in modes

E CFT   

[P]

generic metric with horizon ds2= −f (r) dt2+ f 0_(r)2 4a2_{f (r)}dr 2_+g ab(r, x) dxadxb f (rh) =0 horizon

interacting scalar field ∇2_{φ −V}0_{(φ) =0} −∂2 t + ∂2ξ
φ = 0 2d massless scalar ξ = _2a1 lnf (r) f (r) > 0 _{r→ r} h CFT know–how: ds2_=C(x+_,x−_)dx+_dx− EM tensor hT±±i = − 1 12πC 1/2_∂2 ±C−1/2 hT+−i = 1 24π∂+∂−lnC

E CFT   

[P]

generic metric with horizon ds2= −f (r) dt2+ f 0_(r)2 4a2_{f (r)}dr 2_+g ab(r, x) dxadxb f (rh) =0 horizon

interacting scalar field ∇2_{φ −V}0_{(φ) =0} −∂2 t + ∂2ξ
φ = 0 2d massless scalar ξ = _2a1 lnf (r) f (r) > 0 _{r→ r} h CFT know–how: ds2_=C(x+_,x−_)dx+_dx− EM tensor hT±±i = − 1 12πC 1/2_∂2 ±C−1/2 hT+−i = 1 24π∂+∂−lnC

BH: T 

M

hTµνi = 0

Hartle–Hawking vacuum, time–symmetric

R L

hTUvi = hTUUi = 0, hTvvi = −

a2

48π

Unruh vacuum, time–asymmetric, evaporation

R L

hTuvi = 0, hTvvi = hTuui = −

a2

48π

BH : P    

Schwarzschild BH emits Hawking quanta with thermal spectrum

T = (8πM)−1 A = 16πM2∼ S ( h =G = 1) Planck’s formula dL dω = #A 8π2 ω3 eω/T₋₁ L = #π 2 120AT 4 L = −dM dt = # 15 · 211_πM −2 Page time tPage=5 · 211π#−1M3 number flux dΓ dω = 1 ω dL dω Γ = dN dt = #ζ(3) 4π2 AT 3₌ #ζ(3) 128π4M −1

number of emitted quanta

N = ZtPage 0 Γdt = 120ζ(3) π3 M 2_{∼ S}

BH : P    

Schwarzschild BH emits Hawking quanta with thermal spectrum

T = (8πM)−1 A = 16πM2∼ S ( h =G = 1) Planck’s formula dL dω = #A 8π2 ω3 eω/T₋₁ L = #π 2 120AT 4 L = −dM dt = # 15 · 211_πM −2 Page time tPage=5 · 211π#−1M3 number flux dΓ dω = 1 ω dL dω Γ = dN dt = #ζ(3) 4π2 AT 3₌ #ζ(3) 128π4M −1

number of emitted quanta

N = ZtPage 0 Γdt = 120ζ(3) π3 M 2 _{∼ S}

 S P 

dS in comoving coordinates ds2 = −(1 −H2r2)dt2+ (1 −H2r2)−1dr2+r2dΩ2 ideal fluid: −p = ε = 3H 2 8πG apparent horizon rh=H −1 Mh=Vhε Vh= 4 3πr 3 h Sh= Ah 4 hG Ah=4πr 2 h dMh=ThdSh−p dVh Th= − h 2πH horizon thermodynamics −Th dSh dt = #π2 120 h3AhT 4 h ⇒ r3 h(t) = rh3(0) + #L2 P 160πt horizon evaporation dS Page time tdS∼ L−2P H −3

 S P 

dS in comoving coordinates ds2 = −(1 −H2r2)dt2+ (1 −H2r2)−1dr2+r2dΩ2 ideal fluid: −p = ε = 3H 2 8πG apparent horizon rh=H −1 Mh=Vhε Vh= 4 3πr 3 h Sh= Ah 4 hG Ah=4πr 2 h dMh=ThdSh−p dVh Th= − h 2πH horizon thermodynamics −Th dSh dt = #π2 120 h3AhT 4 h ⇒ r3 h(t) = rh3(0) + #L2 P 160πt horizon evaporation dS Page time tdS∼ L−2P H −3

S    S

[R]

physical graviton modes propagate like a minimally coupled massless scalar field

 − m2
φ = 0 ds2 = 1 H2_τ2 −dτ 2_{+ dx}2
τ∈ (−∞, 0) mode expansion φ(x) =X k h uk(x)ak+uk(x)a†_k i

positive frequency modes uk(τ, x)∼ (−τ)3/2e−ik·xHν(1)(−kτ)

with ν2 ₌ 9 4−m

2_H−2

S    S

[R]

physical graviton modes propagate like a minimally coupled massless scalar field

 − m2
φ = 0 ds2 = 1 H2_τ2 −dτ 2_{+ dx}2
τ∈ (−∞, 0) mode expansion φ(x) =X k h uk(x)ak+uk(x)a†_k i

positive frequency modes uk(τ, x)∼ (−τ)3/2e−ik·xHν(1)(−kτ)

with ν2 ₌ 9 4−m

2_H−2

S    S

scalar propagator in dS is IR divergent form = 0

GSK(x, y) =   iF(x, y) GR_{(x, y)} GA_{(x, y) 0}    − m2
_{F(x, y) = 0}  − m2
_GR,A_{(x, y) = δ(x, y)} F(x, y) = 1 2[hφ(x)φ(y)i + hφ(y)φ(x)i] Schwinger–Keldysh formalism (CTP) k→ 0 : F(k; τ1, τ2)∼  H2 2k3 m = 0 H2 2k3(k2τ1τ2) m2 3H2 m 6= 0 F(x, y) is ill–defined for m = 0 !

S    S

scalar propagator in dS is IR divergent form = 0

GSK(x, y) =   iF(x, y) GR_{(x, y)} GA_{(x, y) 0}    − m2
_{F(x, y) = 0}  − m2
_GR,A_{(x, y) = δ(x, y)} F(x, y) = 1 2[hφ(x)φ(y)i + hφ(y)φ(x)i] Schwinger–Keldysh formalism (CTP) k→ 0 : F(k; τ1, τ2)∼  H2 2k3 m = 0 H2 2k3(k2τ1τ2) m2 3H2 m 6= 0 F(x, y) is ill–defined for m = 0 !

S    S

scalar propagator in dS is IR divergent form = 0

GSK(x, y) =   iF(x, y) GR_{(x, y)} GA_{(x, y) 0}    − m2
_{F(x, y) = 0}  − m2
_GR,A_{(x, y) = δ(x, y)} F(x, y) = 1 2[hφ(x)φ(y)i + hφ(y)φ(x)i] Schwinger–Keldysh formalism (CTP) k→ 0 : F(k; τ1, τ2)∼  H2 2k3 m = 0 H2 2k3(k2τ1τ2) m2 3H2 m 6= 0 F(x, y) is ill–defined for m = 0 !

S    S

deformed dS ds2= 1 H2_τ2− dτ 2_{+f (τ) dx}2 f (τ) ≈ 1 spontaneous deformation for example, f (τ) = τ τ0 ε ε 1 (ddS)φ =0 ⇒ (_(dS)−m2)φ =0 m2∼ εH2

S    S

deformed dS ds2= 1 H2_τ2− dτ 2_{+f (τ) dx}2 f (τ) ≈ 1 spontaneous deformation for example, f (τ) = τ τ0 ε ε 1 (ddS)φ =0 ⇒ (_(dS)−m2)φ =0 m2∼ εH2

S    S

Consider pure trace components 8h − 8τh0−4τ2h00= 8f 0_−4τf00 H2√_κτ + 2√κH2h2_ijτ2 =! 0 classical source tadpole hhij(x)hij(x)i∼ F(x, x) ∼ 1 H2_τ4_ε

tadpole cancellation fixes

ε2+2κH

2

15π2 =0

S    S

Consider pure trace components 8h − 8τh0−4τ2h00= 8f

0_−4τf00 H2√_κτ + 2

√

κH2h2_ijτ2 =! 0

classical source tadpole

hhij(x)hij(x)i∼ F(x, x) ∼ 1

H2_τ4_ε

tadpole cancellation fixes

ε2+2κH

2

15π2 =0

S    S

Consider pure trace components 8h − 8τh0−4τ2h00= 8f

0_−4τf00 H2√_κτ + 2

√

κH2h2_ijτ2 =! 0

classical source tadpole

hhij(x)hij(x)i∼ F(x, x) ∼ 1

H2_τ4_ε

tadpole cancellation fixes

ε2+2κH

2

15π2 =0

Q     S

[D, G, Z]

linearize gravity (with Λ) around Minkowski αβ_µνhαβ= −2Ληµν

metric = approximation of dS

for tcl Λ−1/2

gravitons with Fierz–Pauli mass αβ_µνhαβ+m2(hµν− ηµνh) = 0

graviton dynamics valid fortcl m−1 They agree up to an additive constant! m =√Λ

Q     S

[D, G, Z]

linearize gravity (with Λ) around Minkowski αβ_µνhαβ= −2Ληµν

metric = approximation of dS

for tcl Λ−1/2

gravitons with Fierz–Pauli mass αβµνhαβ+m2(hµν− ηµνh) = 0

graviton dynamics valid fortcl m−1

Q     S

[D, G, Z]

linearize gravity (with Λ) around Minkowski αβ_µνhαβ= −2Ληµν

metric = approximation of dS

for tcl Λ−1/2

gravitons with Fierz–Pauli mass αβµνhαβ+m2(hµν− ηµνh) = 0

graviton dynamics valid fortcl m−1 They agree up to an additive constant! m =√Λ

Q     S

Quantum resolution of dS trace component of Fierz–Pauli graviton Φcl= Λ √ 4πm2 cos(mt) ( h =G = 1) hN|Φ|Ni = Φcl Φ =X k 1 √ 2Vωk  ake−ikx+a†ke ikx

coherent state |Ni = e

√ N(a0−a†0)|0i N = VΛ 2 8πm3 with m = √ Λ, V∼ Λ−3/2 N∼ 1 hGΛ

Q     S

decay of the coherent state

N _{N − 1}

p = (m/2, p) p0= (m/2, −p)

decay rate Γ ∼ Λ1/2

quantum break time

C   G

[D, G] object massM radiusR N = (M/MP)2 ω =2 h/R gravitational field

= graviton coherent state

gravitational energy Eg=Nω E_g < M BH R = 2GM Eg = M “BE condensation” condensate depletion

C   G

[D, G] object massM radiusR N = (M/MP)2 ω =2 h/R gravitational field

= graviton coherent state

gravitational energy Eg=Nω E_g < M BH R = 2GM Eg = M “BE condensation” condensate depletion

C   G

static, spherically symmetric metric Misner–Sharp mass function

ds2= −f (r) dt2+f (r)−1dr2+r2dΩ2 f (r) = 1 −2G m(r) r

horizon f (rh) =0 ⇒ rh=2Gm(rh)

N = [m(rh)/MP]2 gravitons with meanenergy ω = 2 h/rh Quantum N–portrait

For dS, turn the argument around: Dark Energy density

m =4π 3 R 3_{ε, Nω =} m2 M2 P 2 h R =m ⇒ ε = 3 8πGR2

C   G

static, spherically symmetric metric Misner–Sharp mass function

ds2= −f (r) dt2+f (r)−1dr2+r2dΩ2 f (r) = 1 −2G m(r) r

horizon f (rh) =0 ⇒ rh=2Gm(rh)

N = [m(rh)/MP]2 gravitons with meanenergy ω = 2 h/rh Quantum N–portrait

For dS, turn the argument around: Dark Energy density

m =4π 3 R 3 ε, Nω = m 2 M2 P 2 h R =m ⇒ ε = 3 8πGR2

C   G

2 masses, m1, m2, of radiiR1,R2, distance r12  R1,R2

# of gravitons N = (m1+m2) 2 M2 P = m 2 1 M2 P + m 2 2 M2 P + 2m1m2 M2 P gravitational energy Eg = m2₁ M2 P 2 h R1 + m 2 2 M2 P 2 h R2 + 2m1m2 M2 P h r12

total mass Newtonian gravitational potential

m1+m2= m1− m2 1 M2 P 2 h R1 −Gm1m2 r12 ! | {z } E1 + m2− m2 2 M2 P 2 h R2 −Gm1m2 r12 ! | {z } E2 +Eg

C   D M

[C, C, G, W.M., T]

“baryonic” mass µ inside a universe filled with dark energy

rh=2G(md+ µ) N = m2 d M2 P + 2µmd M2 P + µ 2 M2 P dark energy md = m2 d M2 P 2 h rh+ δ + 2µmd M2 P h λ horizon shift

new length scale 2µG = rµ λ  rh δ rh  1 −rµ λ  = rµ λ − rµ rh 1 equation, 2 unknowns

C   D M

[C, C, G, W.M., T]

“baryonic” mass µ inside a universe filled with dark energy

rh=2G(md+ µ) N = m2 d M2 P + 2µmd M2 P + µ 2 M2 P dark energy md = m2 d M2 P 2 h rh+ δ + 2µmd M2 P h λ horizon shift

new length scale 2µG = rµ λ  rh δ rh  1 −rµ λ  = rµ λ − rµ rh 1 equation, 2 unknowns

C   D M

[C, C, G, W.M., T]

generic solution must satisfyrµ λ  rh δ rh = rµ λ + (1 − γ) r2 µ λ2 + γ r2 µ λ2 − rµ rh +O rµ3/λ3
γ: O(1) constant δ≈ q γ−1_r µrh λ≈ √ γrµrh

dS horizon shift⇒ total apparent matter in dS universe

rh≈ H−1−GM = H−1− δ ⇒ M ≈ s

2µ γGH

C   D M

[C, C, G, W.M., T]

generic solution must satisfyrµ λ  rh δ rh = rµ λ + (1 − γ) r2 µ λ2 + γ r2 µ λ2 − rµ rh +O rµ3/λ3
γ: O(1) constant δ≈ q γ−1_r µrh λ≈ √ γrµrh

dS horizon shift⇒ total apparent matter in dS universe

rh≈ H−1−GM = H−1− δ ⇒ M ≈ s

2µ γGH

C   D M

(one) evidence of DM: galaxy rotation curves

baryonic Tully–Fisher relation

v_f4(r) ≈ a0Gµ(r) a0≈ 1 6H MOND g = ν(gN/a0)gN      ν(x  1)∼ x−1/2 ν(x  1)∼ 1 Verlinde 2016

elastic reaction of space–time to presence of matter

C   D M

(one) evidence of DM: galaxy rotation curves

baryonic Tully–Fisher relation

v_f4(r) ≈ a0Gµ(r) a0≈ 1 6H MOND g = ν(gN/a0)gN      ν(x  1)∼ x−1/2 ν(x  1)∼ 1 Verlinde 2016

elastic reaction of space–time to presence of matter

C   D M

(one) evidence of DM: galaxy rotation curves

baryonic Tully–Fisher relation

v_f4(r) ≈ a0Gµ(r) a0≈ 1 6H MOND g = ν(gN/a0)gN      ν(x  1)∼ x−1/2 ν(x  1)∼ 1 Verlinde 2016

elastic reaction of space–time to presence of matter

C   D M

(one) evidence of DM: galaxy rotation curves

baryonic Tully–Fisher relation

v_f4(r) ≈ a0Gµ(r) a0≈ 1 6H MOND g = ν(gN/a0)gN      ν(x  1)∼ x−1/2 ν(x  1)∼ 1 Verlinde 2016

elastic reaction of space–time to presence of matter

C   D M

static, spherically symmetric anisotropic fluid space–time

pk∼ ε ∼ √ HGµ 4πG 1 r2 µ: baryonic mass match BTF kinematics ε = 3H 2 8πG+ pHGµ/6 4πG α r2 pk= − 3H2 8πG+ pHGµ/6 4πG 1 − α r2 dS asymptotics horizon shift Hrh≈ 1 − pHGµ/6 1 + α