Assessing the Effects of the Uncertainty in Reheating Energy Scale on Primordial Spectrum and CMB

8 Pages·0.21 MB·English

by  Bo Feng

Assessing the Effects of the Uncertainty in Reheating Energy Scale on Primordial
Spectrum and CMB
Bo Feng,1 Xiaobo Gong,1,2 Xiulian Wang1
1 Institute of High Energy Physics, Chinese Academy of Science, P.O. Box 918-4, Beijing 100039, P.R. China
2 Department of Physics, Peking University, Beijing 100871, P.R. China
The details of reheating energy scale ρ is largely uncertain today, independent of inflation
reh
models. This would induce uncertainty in predicting primordial spectrum. Such uncertainty could
be very large, especially for spectra with large running nS. We find that for some inflation models
with a large dlnnS(k)/dlnk, ρreh could behighly restricted bycurrent CMB observations.
3
0
0 The cosmic microwave background radiation(CMBR) anisotropy for the l-th multipole Cl by definition is related
2
to the angular correlation function[1]:

n
a ∞
1
J h∆(~n )∆(~n )i≡ (2l+1)C P[cos(~n ·~n )], (1)
1 2 4π l l 1 2
7 Xl=0

1 and for scalar temperature modes
v
1 2π 4π dk
1 ClS ≡ l(l+1)C˜l = (2l+1)Z k Tl2(k)PS(k), (2)
1
1
where T (k) is the transfer function and P (k) is the primordial spectrum from inflation. The scalar spectrum index
0 l S
3
is defined as n (k)≡1+dlnP (k)/dlnk.
0 s S
/ Usuallyonecanspecifythekmodesduringinflationbasingonthee-foldsnumberN(k)beforetheendofinflation[2]:
h
p
- k 1016GeV V(k)1/4 1 V1/4
o N(k)=62−ln −ln +ln − ln end , (3)
r a0H0 V(k)1/4 V1/4 3 ρ1/4
t end reh
s
a where V(k), V denote the inflaton potential at k = aH and at the end of inflation respectively, ρ is the
end reh
:
v
energydensitywhenreheatingends,resumingastandardbig bangevolution. Commonlyonetakesa =1,wherethe
i 0
X
′ ′
subscript 0 denotesthepresentvalue. Foragivenmodelofinflation(whichendsnaturally),P (a/a )orP (N)can
S end S
r
a
be exactly formulated, however N(k) cannot be specified accurately in the lack of details of reheating. Consequently
, we are unable to tell precisely when a mode k leaves out horizon , and there exists an uncertainty in predicting the
primordial spectrum and its effects on C˜ . In this paper we study in detail two effects related to the uncertainty of
l
N(k) : the normalization of inflaton’s potential and the effects on the spectrum of CMBR.
Commonlyoneassumesn =constantwithoutusingspecificinflationmodelswhenfittingcosmologicalparameters
s
to observations. An analysis by Wang et al. [3] gave that current observations indicated n =0.91+0.15. However,for
s −0.07
a general inflation model, it does not give exactly n =constant. Sometimes significant scale-dependence of n (k) is
s s
possible and allowedby current data[4, 5, 6]. It is often important to relate primordialspectra from specific inflation
models with CMBR observations for constraining model parameters, making distinctive predictions and excluding
some certain inflation models.
When applying k to P , a typicalmethod is to specify a mode aroundthe COBE pivot scale k ≈7.0a H [7].
S COBE 0 0
1/4 1/4
The three undecided energy scales V(k ), V and ρ give a typical range of N(k ) = 40 ∼ 60[2], it is
COBE end reh COBE
2
even possible that N(k ) = 0 ∼ 60 [4] if there exists thermal inflation[8]. The fit to CMB data can give an
COBE
estimation of V(k )[7] for specific inflation models[4]. However the reheating energy scale cannot be restricted
COBE
stringently by observations today. Phenomenologically ρ1/4 can be in the range of 1MeV ∼ 1016GeV, which based
reh
on Eq.3, gives rise to an uncertainty of 15 in N .
COBE
If the slow rolling(SR) parameter ǫ is small enough, the effect of changing N(k ) is exactly like a horizontal
COBE
translationintheP (k)−kpicture. Whentheprimordialspectrumisscaleinvariant,thetranslationchangesnothing.
S
As long as P (k) is tilted, choosing a H or a H as k would globally change the value of P (k). When n (k)
S 1 1 2 2 COBE S s
is scale independent, different choice of k will change P (k) to cP (k), where c is a constant, acting also like a
COBE S S
vertical shift of P (k) in the P (k)−k picture. In Fig.1 we show the effect of choosing different N . Assuming
S S COBE
the Hubble constant H remains the same on the shown scale, although P (N) is exactly known, N =50 or 60
S COBE
will give different value for P (k). Assuming constant n and no gravitational waves, Bunn and White’s fitting to
S s
COBE observation gives[9]:
|δ (k )|=1.94×10−5×Ω−0.785−0.05lnΩ0exp[−0.95(n −1)−0.169(n −1)2], (4)
H COBE 0 s s
with an uncertainty less than 10% , where
2g(Ω )
δ (k)= 0 P1/2(k) (5)
H 5 Ω S
0
and
5 1 209Ω Ω2
g(Ω )= Ω ( + 0 − 0 +Ω4/7)−1, (6)
0 2 0 70 140 140 0
with less than 5% uncertainty for g(Ω )/Ω . P (k ) has been stringently restrictedfor given n and Ω . On the
0 0 S COBE s 0
other hand, taking single-field inflation model V =V f(φ/µ) as a example, one has
0
V3
P (k)∝ ∝V . (7)
S V2 0
φ
For different N , V has to be chosen differently to match the COBE normalization. For n = 0.84, ∆N = 15
COBE 0 s
the uncertainty on the amplitude of V is about e15(1−0.84) ≈11 .
0
InsomesensetheuncertaintyofV matterslittle,sincecurrentinflationtheoriescannotphysicallygiveexactvalue
0
of V . V can even be used as a free parameter to be normalized by CMB and LSS observations. For n =constant,
0 0 s
the effects caused by the uncertainty of N will be fully cancelled by normalization. However, in the case of
COBE
dn /dlnk 6=0,evenifV is best used,two normalizedprimordialspectra P (k) with differentchoosingof N(k )
s 0 S COBE
still cannot fully overlap.
Now let us study the effects on CMB. For simplicity we have fixed the cosmological parameters as h = .64,Ω =
Λ
0.66,Ω h2 =0.020 and Ω =0[3] in the following studies.
b k
Firstly whenthe running of n is very small,the uncertainty of ρ cannotbring forth observableeffects in CMB.
S reh
In Fig.2 we give an example
P (k)=A(k/k )0.91−1−0.001ln(k/k⋆), (8)
S ⋆
3
where the constant A is to be normalized. C denotes C when setting N = N . For assessing the difference,
0l l COBE ⋆
we have used the cosmic variance values, where [10, 11, 12]
2
∆C˜ = C˜. (9)
l r2l+1 l
′
The two specified lines stand for choosing N = N +15 and N −15 respectively. They are both within the
COBE ⋆ ⋆
cosmic variance of C0l for l <∼ 2000 and are hard to distinguish from C0l. We’ve also checked the effect of choosing
different n in Eq.8, and find the result remains the same for 0.84 < n < 1.06. In this sense, for many inflation
⋆ ⋆
models in which dn /dlnk are very small, such as chaotic and natural inflation, the effect caused by the uncertainty
s
of reheating temperature is negligible, provided V can be normalized.
0
Secondly we rebuild a primordial spectrum with a constant running dn /dlnk =−0.022:
S
P (k)=A(k/k )1.014−1−0.011ln(k/k⋆). (10)
S ⋆
n runs from 1.06 to 0.84 on the scale relevant to CMB and LSS observations(k=3×10−4 ∼6 hMpc−1) [13] when
S
choosing k =k . P (k) with constant index n =0.91+0.15 is allowed by current observations[3], phenomenologi-
⋆ COBE S S −0.07
cally such a running should also satisfy current observations. In Fig.3, C stands for C when choosing k =k ,
0l l COBE ⋆
and the two dashed lines are its cosmic variance limits. One can see C is hardly distinguishable from the constant
0l
index spectrum case in which n =0.97. As the figure also shows, one e-fold of uncertainty in N can not make
S COBE
difference out of cosmic variance if A can be best normalized. And it is obvious that if N varies largerthan 10,
COBE
i.e. when n is globally larger than 1.06 or less than 0.84, this spectrum will be excluded by current observations.
S
Significant scale dependent primordial spectrum can explain the tentatively observed feature at k ∼ 0.05Mpc−1
[14, 15], and in some degree solve the small scale problems of the CDM model on dark halo densities and dwarf
satellites[16], but it also can be easily excluded by current observations if N(k) changes. The location of bump on
P may be shifted to different k by the uncertainty of N , 15 e-fold of uncertainty can make the bump out of
S COBE
the range of CMB and LSS observation scale, being fully unobservable, or to any other mode where k6=0.05Mpc−1,
makingthe primordialspectrumdisagreewiththe LSSandCMBobservations. Inthe BSI[17]modelwhichhasbeen
used to solve the CDM puzzle on small scales, one requires a red-tilted P (k) which is near scale invariant at scale
S
k = 3×10−4 ∼ 6 hMpc−1 and damps severely at larger k, however the location of such feature is still sensitive to
ρ .
reh
Now let us study a specific inflation model considered in ref. [5]:
φ Nφ
V(φ)=V (1+cos +δcos ). (11)
0
f f
For the calculations here we’ve set N =300, δ =5×10−5 and f =0.4M . Slow rolling is well satisfied for N >50,
Pl
using the SR parameter ǫ:
M2 V
ǫ= Pl( φ)2, (12)
16π V
where V ≡∂V/∂φ. One can easily get
φ
8
P (k)= V(k)/ǫ(k). (13)
S 3M4
Pl
4
With the background values of the cosmologicalparameters mentioned above, COBE normalization predicts
V(k)14 ≈6.4ǫ41 exp[−0.95(ns−1)/2−0.169(ns−1)2/2]×1016GeV (14)
around k . For 0.84 < n < 1.06 the exponential factor is negligible. The normalization to full current CMB
COBE S
and LSS data differs little with the COBE normalization. In Fig.4 the Hubble constant H remains almost constant
for N > 40 and it changes no more than one percent on the full shown scale, rendering dlnk ≈ dN. V(N) is also
almost constant for N >40 and V1/4(N >40)/V1/4 ≈2.0, disregarding the exact values of V . In the current CMB
end 0
and LSS observation scale, the minimum value of k is about k =H , the maximum possible value N(k ) is
0 0 0
1016GeV
N(k )≈62−ln( )+ln2≈60.0 (15)
0 6.4×1014GeV
where ǫ ∼ 10−8. The corresponding spectrum index is n (k ) ≈ 0.96. As one can see from Fig.4, the primordial
S 0
spectrum with a certain range where N(k ) > 60.0 should also fit current CMB data well, but the details of this
0
model has excluded the possibility. In the figure, N = 52.6 corresponds to n = 0.84. It is also obvious that for
S
n (k ) ≤ 0.84, the model is excluded by current observations, and n (k ) > 0.84 gives ρ1/4 > 105GeV. To avoid
S 0 S 0 reh
too many gravitinos, one requires ρ1/4 <1010GeV[18], i.e. N(k )< 56.5 , n (k )≤ 0.91. Phenomenologically when
reh 0 S 0
thermal inflation follows, N(k ) can be much smaller, however, such spectra cannot satisfy current CMB data, as
0
shown in Fig.5. Only a small range between the two dotted lines may be acceptable. In this range the spectrum has
a large running |dn /dlnk|∼0.01, such an n cannot be excluded by current CMB data, as can be seen from Fig.6
S S
that it performs exactly like the constantn =0.9 case. Meanwhile such a tilted spectrum exhibits the possibility of
S
solving the CDM puzzle on small scales.
In summary, the uncertainties of ρ has induced some difficulty in formulating P (k) from inflation model. This
reh S
led to additional degeneracy in the inflation parameter V which otherwise could be highly restricted for specific
0
inflation models[19]. For models in which dn /dlnk is small enough, the uncertainty is negligible. However, for
s
models where the spectrum is significant scale-dependent, the difference induced by reheating can be very large as
showninthispaper. Furthermore,forsomespecificinflationmodelswhichhavelargerunningn ,currentobservations
S
provide a constraint on the reheating temperature. Before concluding we should point that some aspects relevant
to the uncertainty have been partly mentioned in Refs[4, 7, 20, 21], but our paper is different from those mentioned
above since we mainly focus on the uncertainty of reheating and on restriction of ρ .
reh
We thank Dr. Mingzhe Li and Prof. Xinmin Zhang for discussions. We also acknowledge the using of CMBfast
code[22]. ThisworkissupportedinpartbyNationalNaturalScienceFoundationofChinaandbyMinistryofScience
and Technology of China.
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6

N = 50
COBE
0.1 N = 60
0.1 COBE

>-----

0.01 S
S P
P
0.01

1E-3

1E-3
70 60 50 40 -10 -5 0 5 10
N ln[k/k ]
COBE
FIG.1: Left: theprimordialspectrumPS asafunctionofthee-fold numberN;Right: PS asafunctionofthewavenumberk.
The figureillustrates theeffect on PS(k) when choosing different NCOBE.

1.5
N' =N - 15
1.4 N'COB E = N C O B E +15
COBE COBE
1.3

1.2
1.1
$(cid:19)L1.0
$(cid:18)L
0.9
0.8
0.7

0.6
0.5
10 100 1000
L
FIG.2: TheratioofdifferentCMBangularpowerspectraClrelativetoC0l. C0lstandsfornS(kCOBE)=0.91anddnS/dlnk=
−0.002, theregion between thetwo dashed lines are allowed by its cosmic variance.
7

1.5
1.4 NCOB E = N * - 1
N =N+1
1.3 nsC=OB0E.97 *

1.2
1.1
$(cid:18)$L(cid:19)L01..90
0.8
0.7

0.6
0.5
10 100 1000
L
FIG. 3: C0l stands for nS(kCOBE)=1.014 and dnS/dlnk=−0.022.

0.1 1E-8
0.01 e 1E-9 P ( N )
S
1E-3 1E-10
1E-4
1E-11
1E-5
1E-6 1E -12
1E-7 1E-13
1E-8 1E-14

1E-9 1E-15
70 60 50 40 30 20 10 70 60 50 40 30 20 10

0.00
2.5
2.0 n S -0.02 dnS/d l n k
1.5 -0.04
1.0 -0.06
0.5 -0 .08
0.0 -0.10
-0.5 -0.12

-1.0 -0.14
70 60 50 40 30 20 10 65 60 55 50 45 40
N
N
FIG. 4: The SR parameter ǫ , spectrum PS, scalar spectra index nS and its running dnS/dlnk with N = 300, δ = 5×10−5
and f =0.4MPl for V(φ)=V0(1+cosφf +δcosNfφ), the region within the two dashed lines is roughly allowed by the model
and current observations for k0 =H0.
8

6000

5000

4000
(cid:3)
p$(cid:18)(cid:21)L3000
(cid:14)(cid:20)(cid:12)
2000
L(cid:11)L

1000

0
10 100 1000

L
FIG.5: CMBangularpowerspectraforchoosingN(kCOBE)differently,assumingtheexistenceofthermalinflationsometime
after theinflation in Eq.11. From left top to bottom, the lines stand for N(kCOBE)=37 , 27 and 32 respectively.

1.5
1.4 n s = 0 . 9
1.3

1.2
1.1
1.0
$(cid:19)L
$(cid:18)L0.9
0.8

0.7

0.6
0.5
10 100 1000
L(cid:3)
FIG. 6: C0l standsfor N(k0)≈60 for thePS(N) shown in Fig.4 .