. Article. SCIENCE CHINA Physics Mechanics & Astronomy November 205 Vol. 58 No. : 040 doi: 07/s4-05-572- The (p q) inflation model HUANG QingGuo * State Key Laboratory of Theoretical Physics Institute of Theoretical Physics Chinese Academy of Sciences Beijing 0090 China Received August 25 205; accepted August 28 205 In this paper we propose a new inflation model named (p q) inflation model in which the inflaton potential contains both positive and negative powers of inflaton field in the polynomial form. We derive the accurate predictions of the canonical single-field slow-roll inflation model. Using these formula we show that our inflation model can easily generate a large amplitude of tensor perturbation and a negative running of spectral index with large absolute value. inflation tensor-to-scalar ratio spectral index running of spectral index PACS number(s): 98.80.-k 98.80.Cq 04.0.-w Citation: Huang Q G. The (p q) inflation model. Sci China-Phys Mech Astron 205 58: 040 doi: 07/s4-05-572- Introduction Inflation was proposed to solve the several puzzles e.g. horizon problem flatness problem and so on in the hot big bang model. The simplest version is the so-called canonical single-field slow-roll inflation model 2 in which the inflaton field slowly rolls down its potential. On the other hand the quantum fluctuations during inflation can provide tiny primordial density perturbations which seed the anisotropies in the cosmic microwave background (CMB) and the largescale structure of our universe. Since the Hubble parameter during inflation is roughly a constant both the spectra of scalar and tensor perturbations are nearly scale-invariant. Usually the small scale dependence of scalar power spectrum is measured by the spectral index and then the amplitude of scalar power spectrum can be parametrized by ( ) ns k P s = A s () k p where k p is the pivot scale and A s is the amplitude at k = k p. The size of tensor perturbation is characterized by the tensorto-scalar ratio r which is defined by r P t /P s wherep t is *Corresponding author (email: huangqg@itp.ac.cn) the amplitude of tensor power spectrum. Combining with WMAP Polarization data 4 the Planck data 5 released in the early of 20 imply = 0.90 ± 07 at the scale k = 5 Mpc. Recently BICEP2 7 found an excess of B-mode power over the base lensed- ΛCDM expectation in the range l 0 50. Cross correlating BICEP2 against 00 GHz maps from the BICEP experiment the microwave mission by the polarized dust is disfavored at.7σ. The observed B-mode power spectrum is well fitted by a lensed-λcdm+tensor model with r = 0.20 +7 5. (2) Since the contaminant on the B-mode spectrum from the polarized dust gives a similar behavior in the multipoles around l 80 8 9 it is still difficult to distinguish the signal of primordial gravitational waves from the dust. If the signal from BICEP2 is confirmed to be originated from the primordial gravitational waves by the upcoming data sets it must be a breakthrough for the basic science. Because the primordial gravitational waves can also make a contribution to Cl TT on the low multipoles it suppresses the contribution to Cl TT from scalar perturbation and hence > for the scalar power spectrum is preferred on the large scales 0. In order to reconcile the differentvaluesof spectral indexat differentscales c Science China Press and Springer-Verlag Berlin Heidelberg 205 phys.scichina.com link.springer.com
Huang Q G Sci China-Phys Mech Astron November (205) Vol. 58 No. 040-2 the spectral index should be scale dependent: d (k) =.0447 +295 297 () = 25 ± 09 (4) at k = 02 Mpc in ref. where d / is the running of spectral index. From the above results we can estimate the spectral index at k = 5 Mpc namely (k = 5 Mpc ).0447 25 ln(5/02) = 0.9 which is consistent with the result from Planck at k = 5 Mpc 5. See more discussion about the constraint on the running of spectral index in ref. 72 and other investigations about BICEP2 in ref.. Even though now theexcessinbicep2isconfirmed to be mainly originated from the polarized thermal dust 4 an order of O(0 2 ) running of spectral index is still allowed by Planck data released in 205 5 and a visible tensor-to-scalar ratio in the near future may also imply a non-zero running of spectral index. Usually the canonical single-field slow-roll inflation model predicts O(( ) 2 ) 0 which is much smaller than that in eq. (4). In the literatures there two possible ways to achieve a negative running of spectral index with large absolute value: one is the space-time noncommutative inflation 7 9 the other is inflation with modulations 20 22. In this letter we will propose a new inflation model called (p q) inflation which can also easily achieve a large negative running of spectral index. This letter will be organized as follows. The introduction and qualitative discussion on (p q) inflation model will be given iect. 2. The numerical predictions of (2) inflationmodelshowupinsect.. Discussion is included iect. 4. The more accurate formula for the runnings of spectral indexes of both scalar and tensor perturbations are given in the appendix. 2 The (p q)inflation model From the dynamics of inflaton field during inflation one can easily find that the evolution of inflaton field is related to the tensor-to-scalar ratio by Δφ N rdn (5) 8 0 where N is the number of e-folds before the end of inflation. In this letter we work in the unit of 8πG =. The above relation is called the Lyth bound 2. If the signal from BICEP2 stands after further cosmological observations and turns out to be primordial the Lyth bound implies that a large field inflation model should be called for. It is well-known that the inflation model with potential V(φ) φ n is a typical large field inflation model where n can be positive 24 or negative 25. Generically the potential can take the form V(φ) = + c φ + c 0 + c φ + c 2 φ 2 +. () When the inflaton field is not so large the terms with negative powers are dominant and the slow-roll parameter ɛ decreases with time and hence the inflation cannot naturally exist if these terms are always dominant. However the terms with positive powers become important when the inflaton field φ goes to the region far away from φ = 0. One may expect that the inflationendedandouruniversewasre-heatedaroundthe local minimum of the potential where c 0 is set by the condition that the potential equals zero at its local minimum. In this letter we only consider a special form of the potential in eq. () namely V(φ) = λ s φp+q φ q + (φ φ) p + V c (7) where both p and q are positive s is a positive dimensionless parameter and V c is fixed by the condition that the minimum of potential is equal to zero. Here we also suppose (φ φ) p = (φ φ ) p.theinflation model driven by the potential in eq. (7) is called (p q) inflation model. For s the local minimum of the potential is located at ( ) q φ = φ m φ + p s p φ (8) and the potential in eq. (7) can be roughly written by ( φ p+q ) V(φ) λ s φ q φ p + (φ φ) p (9) up to the order of O(s +/(p ) ). The term with negative power is roughly the same as that with positive power when φ = φ c s /q φ (0) and the contribution to dv(φ)/dφ from the term with negative power is comparable to that from the term with positive power when ( ) q φ = φ T p s q+ φ. () Therefore roughly speaking the term (φ φ) p is dominant when φ φ T and the number of e-folds from φ = φ T to the end of inflation is roughly given by N T (φ 2 φ 2 T )/(2p). If N T is much larger than the number of e-folds corresponding to the CMB scales e.g. N CMB 0 or equivalently φ 2pN CMB the relevant predictions of (p q) inflation model are the same as those of the chaotic inflation with potential V(φ) φ p. On the other hand the term of sφ p+q /φ q becomes dominant when φ < φ c and the spectral index is 2 q 2(N N T ) + which is larger than one if 0 < q < 2. So the (p q)inflation model can possibly achieve a scalar power spectrum which is blue tilted on the very large scales and red tilted on the small scales.
Huang Q G Sci China-Phys Mech Astron November (205) Vol. 58 No. 040- For simplicity we introduce a new variable which is related to φ by and then the potential in eq. (9) reads where φ/φ (2) V = s( q ) + ( ) p () V V(φ)/λφ p. (4) Now the slow-roll parameters can be written by ( V ) 2 ɛ = (5) 2φ 2 V η =. () φ 2 V The number of e-folds before the end of inflation is given by N φ 2 V N V d = φ 2 s( q ) + ( ) p d. (7) N sq q + p( ) p In the limit of s 0 the (p q)inflation model reduces to the chaotic inflation model with potential V(φ) φ p as well. In general we cannot get the analytical formula for the above integration. For an instance the numerical predictions of (2) inflation model will be discussed in the next section. Numerical predictions of (2) inflation model In this section let s consider a simple example namely (2) inflation model in which the potential takes the form ( φ ) V(φ) = λ s φ φ2 + (φ φ) 2 (8) and then V V s( ) + ( ) 2 (9) where s. The shape of inflaton potential in (2) inflation model is illustrated in Figure. Now the slow-roll parameters are V ( ) 2.0.5.0.0.5 2.0 Figure The potential of inflaton field in the (2) inflation model. The grey dashed curve corresponds to the case with s = 0. ɛ η ξ σ 2φ 2 φ 2 φ 4 φ ( s 2 ) 2 + 2( ) (20) s( ) + ( ) 2 2 + 2s s( ) + ( ) 2 (2) s 4 (s 2 + 2( )) s( ) + ( ) 2 2 (22) 24s 5 s 2 + 2( ) 2 s( ) + ( ) 2 (2) where the value of field is related to the number of e-folds beforethe end of inflation by N φ 2 s( ) + ( ) 2 N s 2 + 2( ) d. (24) The amplitude of the scalar power spectrum is given by P s = λφ4 2π 2 (s( ) + ( ) 2 ) (s 2 + 2( )) 2. (25) In this model there are three parameters namely λ s and φ. But there are four observables i.e. P s r and.themore accurate formula for these four observables are given in the appendix. For example considering that the tensor-to-scalar ratio and the spectral index are r = 0.22 and =.0447 respectively at the pivot scale k p = 02 Mpc assuming to correspond to N = 0 we get φ 7.7 ands = 08 and then it predicts = 22 which is nicely consistent with the constraint from data in eq. (4). In order to see how the spectral index runs with the perturbatiocales or equivalently the number of e-folds before the end of inflation we plot the spectral index as a function of N in Figure 2. From this figure we clearly see that the spectral index in our model can run from > to < in a few number of e-folds. Using the normalization P sobs 2.2 0 9 we find λ 2.54 0. More generally we take the parameters φ and s as two free parameters and figure out the predictions of (2) inflation model. Considering φ 5 25 and s 0 4 0.2.0.08.0.04.02.00 0.98 50 52 54 5 58 0 2 N Figure 2 The spectral index varies with the number of e-folds N before the end of inflation where φ 7.7 ands = 08.
Huang Q G Sci China-Phys Mech Astron November (205) Vol. 58 No. 040-4 r 0.4 0. 0.2 0. 0.95.00.05.0.5 0 2 4 5 0.95.00.05.0.5 can be fixed by the three observables P s and r. Therefore the running of spectral index is predicable in the (2) inflation model and we find that the predictedrunningof spectral index can fit the data quite well. Or equivalently our model can explain four observables by tuning only three parameters. It is quit non-trivial. In this article we propose a new inflation model phenomenologically. The next step is to understand this new inflation model from the viewpoint of fundamental physics. Usually it is hard to understand the term with negative power from the viewpoint of field theory. In order to understand such kind of term we believe that a deeper insight on the field theory is needed. However the inverse power term typically exists itring theory. For example the potential of inflaton field in brane inflation 2 takes the form V(φ) = V 0 ( μ 4 /φ 4 ). The term of /φ 4 can be easily understood from the viewpoint of gravity in the bulk. Unfortunately there is a minus sign for the inverse power term which is originated from the fact that the gravity between brane and anti-brane is attractive. How to understand a positive term with negative power is still an open question. We expect that it may shed a light on the new fundamental physics. Figure r 0.4 0. 0.2 0. 5 4 2 0 (Color online) The predictions of (2) inflation model. we plot the numerical predictions of (2) inflation model in Figure. FromFigure we see that the (2) inflation model can easily generate a negative running of spectral index with large absolute value. 4 Discussion The chaotic inflation model in which the potential takes the form V(φ) φ p with p > 0 predicts a red tilted scalar power spectrum with negligible running of spectral index. In this letter we construct a simple phenomenological inflation model which can easily achieve both a large tensor-to-scalar ratio and a negative running of spectral index with large absolute value. In the literatures these two points can be also realized in the inflation model with modulation 20 22. However there are too many free parameters to be tuned such as the amplitude the period and the phase of modulation. In the (p q) inflation model for example the (2) inflation model there are only three parameters namely λ s and φ which This work was supported by the project of Knowledge Innovation Program of Chinese Academy of Sciences and the National Natural Science Foundation of China (Grant Nos. 082504 22545 and 502). Guth A H. The inflationary universe: A possible solution to the horizon and flatness problems. Phys Rev D 98 2: 47 5 2 Linde A D. A new inflationary universe scenario: A possible solution of the horizon flatness homogeneity isotropy and primordial monopole problems. Phys Lett B 982 08: 89 9 Albrecht A Steinhardt P J. Cosmology for grand unified theories with radiatively induced symmetry breaking. Phys Rev Lett 982 48: 220 22 4 Hinshaw G Larson D Komatsu E et al. Nine-year wilkinson microwave anisotropy probe (WMAP) observations: Cosmological parameter results. Astrophys J Suppl 20 208: 9 5 Ade P A R Aghanim N Armitage-Caplan C et al. Planck 20 results. XVI. Cosmological parameters. arxiv:07 astro-ph.co Ade P A R Aghanim N Armitage-Caplan C et al. Planck 20 results. XXII. Constraints on inflation. arxiv:082 astro-ph.co 7 Ade P A R Aikin R W Barkats D et al. Detection of B-mode polarization at degree angular scales by BICEP2. Phys Rev Lett 204 2: 240 8 Mortonson M J Seljak U. A joint analysis of Planck and BICEP2 B modes including dust polarization uncertainty. arxiv:405.5857 astroph.co 9 Flauger R Hill J C Spergel D N. Toward an understanding of foreground emission in the BICEP2 Region. arxiv:405.75 astroph.co 0 Cheng C Huang Q G. Probing the primordial Universe from the lowmultipole CMB data. arxiv:405.049 astro-ph.co Cheng C Huang Q G Zhao W. Constraints on the extensions to the base ΛCDM model from BICEP2 Planck and WMAP. Sci China-Phys Mech Astron 204 57: 40 45 2 Hu B Hu J W Guo Z K et al. Reconstruction of the primordial power
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