Soft limits in multi-field inflation

Transcription

1 Soft limits in multi-field inflation David J. Mulryne Queen Mary University of London based on arxiv: and forthcoming work with Zac Kenton

2 Soft limits in multi-field inflation David J. Mulryne Queen Mary University of London based on arxiv: and forthcoming work with Zac Kenton

3 Intro to soft limits Observations constrain correlations Soft limits occur when we consider a correlation with a large hierarchy of scales Often simple to calculate theoretically, and can lead to beautiful relations between correlations Important for comparison with observation, as range of scales probed increases, and for specific observations (halo bias)

4 Intro to soft limits Simplest example (Maldacena 03; Creminelli, Zaldarriaga 04; Cheung et al. 08) Power spectrum Bispectrum

5 Intro to soft limits Simplest example (Maldacena 03; Creminelli, Zaldarriaga 04; Cheung et al. 08) Power spectrum Bispectrum

6 Intro to soft limits Another example (e.g. Suyama and Yamaguchi 08; Smith et al. 11; Aassassi et al. 12; Yamaguchi 12) Trispectrum h ~k1 ~k2 ~k3 ~k4 i =(2 ) 3 ( ~ k 1 + ~ k 2 + ~ k 3 + ~ k 4 )T (k 1,k 2,k 3,k 4 )

7 Intro to soft limits Another example (e.g. Suyama and Yamaguchi 08; Smith et al. 11; Aassassi et al. 12; Yamaguchi 12) Trispectrum h ~k1 ~k2 ~k3 ~k4 i =(2 ) 3 ( ~ k 1 + ~ k 2 + ~ k 3 + ~ k 4 )T (k 1,k 2,k 3,k 4 ) f nl lim k 1!0 B (k 1,k 2,k 3 ) P (k 1 )P (k 2 ), nl lim ~ k 1 + ~ k 2!0 T (k 1,k 2,k 3,k 2 ) P (k 1 )P (k 3 )P ( ~ k 1 + ~ k 2 ) nl ( 6 5 f nl ) 2

8 Intro to soft limits Another example (e.g. Suyama and Yamaguchi 08; Smith et al. 11; Aassassi et al. 12; Yamaguchi 12) Trispectrum h ~k1 ~k2 ~k3 ~k4 i =(2 ) 3 ( ~ k 1 + ~ k 2 + ~ k 3 + ~ k 4 )T (k 1,k 2,k 3,k 4 ) f nl lim k 1!0 B (k 1,k 2,k 3 ) P (k 1 )P (k 2 ), nl lim ~ k 1 + ~ k 2!0 T (k 1,k 2,k 3,k 2 ) P (k 1 )P (k 3 )P ( ~ k 1 + ~ k 2 ) nl ( 6 5 f nl ) 2 Extensions to higher point functions and multiple soft limits, other fields (e.g. Mirbabayi and Zaldarriaga et al. 14; Joyce et al. 14)

9 Calculating observables with multiple fields Want to track correlations of the fluctuations, δφ etc, ultimately want curvature perturbation, ζ, power-spectrum, bispectrum, trispectrum etc. Tools required: In-In (e.g. Maldacena 2003) and δn formalism (e.g. Lyth and Rodriguez 2005) h k l i = 2 2 k 3 P (k + l) f nl h i h i 2 n s = dlnp dlnk h i h i 3! NL, g NL = dn s dlnk f nl = 5 6 N,i N,ij N,j (N,k N,k ) 2

10 Calculating observables with multiple fields Want to track correlations of the fluctuations, δφ etc, ultimately want curvature perturbation, ζ, power-spectrum, bispectrum, trispectrum etc. Tools required: In-In (e.g. Maldacena 2003) and δn formalism (e.g. Lyth and Rodriguez 2005) h k l i = 2 2 n s = dlnp dlnk = dn s dlnk k 3 P (k + l) (e.g. Wands et al., 00) f nl h i h i 2 h i h i 3 f nl = 5 6! NL, g NL N,i N,ij N,j (N,k N,k ) 2

11 Calculating observables with multiple fields In-In calculations can give us correlations at horizon crossing for k1~k2~k3 f nl h i h i 2 n s = dlnp dlnk h i h i 3! NL, g NL = dn s dlnk f nl = 5 6 N,i N,ij N,j (N,k N,k ) 2

12 Calculating observables with multiple fields In-In calculations can give us correlations at horizon crossing for k1~k2~k3 f nl h i h i 2 n s = dlnp dlnk h i h i 3! NL, g NL = dn s dlnk f nl = 5 6 N,i N,ij N,j (N,k N,k ) 2

13 Calculating observables with multiple fields In-In calculations can give us correlations at horizon crossing for k1~k2~k3 f nl h i h i 2 n s = dlnp dlnk h i h i 3! NL, g NL = dn s dlnk f nl = 5 6 N,i N,ij N,j (N,k N,k ) 2

14 Calculating observables with multiple fields In-In calculations can give us correlations at horizon crossing for k1~k2~k3 f nl h i h i 2 n s = dlnp dlnk = dn s dlnk (3) ij (k 1) ij H 2 2k 3 1 h i, (3) ijk 0 h i 3 f nl = 5 6! NL, g NL N,i N,ij N,j (N,k N,k ) 2

15 Calculating observables with multiple fields In-In calculations can give us correlations at horizon crossing for k1~k2~k3 f nl h i h i 2 n s = dlnp dlnk = dn s dlnk (3) ij (k 1) ij H 2 2k 3 1 h i, (3) ijk 0 h i 3 f nl = 5 6! NL, g NL N,i N,ij N,j (N,k N,k ) 2 (Seery and Lidsey 05; Lyth 05)

16 Calculating soft observables with multiple fields Analytic calculations difficult for correlations at times after horizon crossing (c.f. results of Byrnes et al. 09; Dias et al. 12)

17 Calculating soft observables with multiple fields Analytic calculations difficult for correlations at times after horizon crossing (c.f. results of Byrnes et al. 09; Dias et al. 12)

18 Calculating soft observables with multiple fields Analytic calculations difficult for correlations at times after horizon crossing (c.f. results of Byrnes et al. 09; Dias et al. 12)

19 Calculating soft observables with multiple fields Analytic calculations difficult for correlations at times after horizon crossing (c.f. results of Byrnes et al. 09; Dias et al. 12)

20 Calculating soft observables with multiple fields Analytic calculations difficult for correlations at times after horizon crossing (c.f. results of Byrnes et al. 09; Dias et al. 12) (3) ij (k 1) il jl H 2 2k 3 1, (3) ijk?

21 Calculating soft observables with multiple fields Analytic calculations difficult for correlations at times after horizon crossing (c.f. results of Byrnes et al. 09; Dias et al. 12) (3) ij (k 1) il jl H 2 2k 3 1, (3) ijk? (Kenton and DJM, 15)

22 Calculating soft observables with multiple fields Analytic calculations difficult for correlations at times after horizon crossing (c.f. results of Byrnes et al. 09; Dias et al. 12) (3) ij (k 1) il jl H 2 2k 3 1, (3) ijk? (Seery and Lidsey 05; Lyth 05)

23 Calculating soft observables with multiple fields Analytic calculations difficult for correlations at times after horizon crossing (c.f. results of Byrnes et al. 09; Dias et al. 12) (3) ij (k 1) il jl H 2 2k 3 1, (3) ijk? (Kenton and DJM, 15)

24 Calculating soft observables with multiple fields Final step is to calculate squeezed

25 Calculating soft observables with multiple fields Final step is to calculate squeezed

26 Calculating soft observables with multiple fields Final step is to calculate squeezed (Kenton and DJM, 15)

27 Calculating soft observables with multiple fields Final step is to calculate squeezed (Kenton and DJM, 15) Note similarity to Maldacena relation for curvature perturbation (c.f. previous results: Allen et al. 05; Li and Wang '08)

28 Example: interacting curvaton Bispectrum in squeezed limit (relevant for e.g. Halo Bias)

29 Example: interacting curvaton Spectral index of the Halo Bias

30 Current work extending to trispectum Many more soft limits

31 Current work extending to trispectum Many more soft limits

32 Current work extending to trispectum Many more soft limits

33 Current work extending to trispectum Many more soft limits

34 Current work extending to trispectum Many more soft limits

35 Current work extending to trispectum Many more soft limits Important to be able to calculate to compare with observation, and provides new insight to Suyama-Yamaguchi relation

36 Conclusion Soft limits lead to interesting consistency relations We also simply need to be able to calculate correlations away from near equilateral configurations (to compare models with observations) We have presented an explicit discussion for the multiple field bispectrum in arxiv: In ongoing work we are extending to the trispectrum