Infrared divergence of pure Einstein gravity contributions to cosmological density power spectrum

J. Hwang, H. Noh, D. Jeong

1Phys. Rev. Lett. 103, 121301, arXiv:0902.4285

Linear order: Lifshitz (1946)/Bonnor(1957)

Second order: Peebles (1980)/Noh-Hwang(2004)

Third order: Hwang-Noh (2005)

Physical Review D, 72, 044012 (2005). Pure General Relativistic corrections

(comoving-synchronous gauge)

2

Curvature perturbation in the comoving gauge

~10-5

(K=0, comoving gauge)

General relativistic contributions to second-order power spectrum:

132211

312

22

12

321

)Re(2||||||

PPP

P

Pure General Relativistic contribution!

EinsteinNewton

EinsteinNewton

PPP ,13,1313

,3,33

Relativistic/Newtonian4

Density power spectrum to second-order:

Physical Review D, 77,123533 (2008)

Pure General Relativistic corrections P13,Einstein

Newtonian

P22 +

P13,Newton

K=0 =Λ:

5

P13,Einstein

P22

P11

P13,Newton

7Phys. Rev. Lett. 103, 121301, arXiv:0902.4285

Dashed line: negative

P13,Einstein

P22

P11

P13,Newton

PTotal

P22 + P

13,Newton

9Phys. Rev. Lett. 103, 121301, arXiv:0902.4285

P13,Einstein

Large-scale limit (k→0, r→∞):

P13,Newton

Leading orderNewton

Next-to-leading orderNewton

10

In the large scale we discover an infrared divergence in the next-to-leading-order power spectrum due to pure Einstein gravity contribution appearing in the third-order perturbation.

Despite cancellations of the leading-order and next-to-leading order terms in P

13,Newton, no such cancellations occur in

P13,Einstein

.

11

P13,Einstein

P13,Newton

Leading order Newton

Next-to-leading orderNewton

12

P13,Einstein

Small-scale limit (k→∞, r→0):

P13,Newton

Leading order, Einstein

Next-to-leading order, Einstein

Leading order, Newton

13

P22

In the small-scale the Einstein's gravity contribution is still negligibly small.

Due to cancellation of the leading-order and next-to-leading order terms in P

13,Einstein, despite a cancellation between P

22 in

P13,Newton

.

14

P13,Einstein

P13,Newton

Leading order Einstein

Next-to-leading orderEinstein

15

16